tag:blogger.com,1999:blog-28538309911063091172024-03-13T06:41:30.221-07:00Johnny and Mary Do MathsSecondary MathematicsChris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.comBlogger56125tag:blogger.com,1999:blog-2853830991106309117.post-16921726301082779472017-09-22T06:26:00.001-07:002017-09-22T06:26:48.688-07:00A Multiplication Shortcut
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<div style="margin: 0px 0px 13px;">
<span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;">Suppose I want to multiply 8 and 256.<span style="margin: 0px;"> </span>I note that 8 <span style="margin: 0px;">× </span>256 = 2<sup>3</sup> <span style="margin: 0px;">× </span>2<sup>8</sup>
= 2<sup>11</sup> = 2048.<span style="margin: 0px;"> </span>I have turned
multiplication into addition. A voice in my head says, “Hey, that could be a
useful trick.” <span style="margin: 0px;"> </span>Another voice says “What
if you want to multiply 7 and 652? Stupid!”<span style="margin: 0px;">
</span>It is true that 7 = b<sup>x</sup> or <span style="margin: 0px;"> </span>652 = b<sup>y </sup>have no whole number solutions
(except the obvious and useless b = b<sup>1</sup>.)<span style="margin: 0px;"> </span>But we do not have to restrict ourselves to
whole numbers.</span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;"><span style="margin: 0px;">Consider the equations √</span>10 <span style="margin: 0px;">×</span> <span style="margin: 0px;">√</span>10 = 10 and 10<sup>0.5</sup> <span style="margin: 0px;">×</span> 10<sup>0.5</sup>
= 10<sup>1</sup> = 10.<span style="margin: 0px;"> </span>In these
equations <span style="margin: 0px;">√</span>10
and 10<sup>0.5</sup> have exactly the same job.<span style="margin: 0px;">
</span>We can consider them equivalent.<span style="margin: 0px;">
</span>Hence the exponent ½ (or 0.5) represents a square root.<span style="margin: 0px;"> </span>Similarly <span style="margin: 0px;"> </span>the exponent <span style="margin: 0px;">⅓</span> represents a cube root. In
general, the exponent n/m represents the m<sup>th</sup> root of a number to the
power n. For example, 10<sup>5/6</sup> is the sixth root of 100 000.<span style="margin: 0px;"> </span>Note that 100 000 = 10<sup>5</sup>. </span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;">Lets look again at the product of 7 and 652. </span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;">P = 7 <span style="margin: 0px;">×</span> 652</span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;">P = 10<sup>a</sup> <span style="margin: 0px;">×</span> 10<sup>b</sup> = 10<sup>a+b</sup>.<span style="margin: 0px;"> </span></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;">I want 7 = 10<sup>a</sup> and I want 652 = 10<sup>b</sup>.
In other words I want the logarithms of 7 and of 652 in base 10.<span style="margin: 0px;"> </span>By definition of a logarithm y = b<sup>x</sup>
<span style="margin: 0px;"><span style="margin: 0px;">ó</span></span> log<sub>b</sub> y = x.<span style="margin: 0px;"> </span>Therefore 7 = 10<sup>a</sup> <span style="margin: 0px;"><span style="margin: 0px;">ó</span></span> log<sub>10</sub> 7= a<span style="margin: 0px;"> </span>and 652 = 10<sup>b</sup> <span style="margin: 0px;"><span style="margin: 0px;">ó</span></span> log<sub>10</sub> 652= b.<span style="margin: 0px;"> </span>The numbers a and b are 0.845 098 040 and <span style="margin: 0px;"> </span>2.814 247 595 respectively.<span style="margin: 0px;"> </span><span style="margin: 0px;"> </span><span style="margin: 0px;"> </span></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;">Therefore P = 10<sup> 0.845 098 040 +2.814 247 595 <span style="margin: 0px;"> </span></sup>= 10<sup>3.659345635</sup> = 4563.999 = 4564 <span style="margin: 0px;"> </span></span></div>
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<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;"><span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;">Ok, that’s impressive. But where did you get log<sub>10</sub>
7= 0.845 098 040 <span style="margin: 0px;"> </span>and log<sub>10</sub> 652
= 2.814 247 595? </span><span style="margin: 0px;"><span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif;"> </span><span style="font-family: "Calibri","sans-serif"; font-size: 11pt; line-height: 115%; margin: 0px;"><span style="font-family: "Helvetica Neue", Arial, Helvetica, sans-serif; font-size: small;">Agreed
, that’s the hard part.<span style="margin: 0px;"> </span>John Napier was
the first person to calculate a table of logarithms. He published his table in
the year 1614; after 20 years of work!</span> <span style="margin: 0px;"> </span></span></span><span style="margin: 0px;"> </span></span></div>
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<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Picture from <span style="color: blue;">https://math.stackexchange.com/questions/47927/motivation-for-napiers-logarithms</span></span></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Napier's "<span lang="EN" style="margin: 0px;">invention was quickly and widely met with acclaim. The works of Bonaventura
Cavalieri (Italy), Edmund Wingate (France), Xue Fengzuo (China), and Johannes
Kepler's <i>Chilias logarithmorum</i> (Germany) helped spread the concept.” <a href="https://en.wikipedia.org/wiki/History_of_logarithms"><span style="color: blue;">https://en.wikipedia.org/wiki/History_of_logarithms</span></a></span></span></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Napier’s original table did not use base 10.<span style="margin: 0px;"> </span>A few years later Napier worked with Henry Briggs
<span style="margin: 0px;"> </span>to produce a table of common (base 10) logarithms.
<span style="margin: 0px;"> </span>Briggs continued after Napier died and published
their table in 1624.</span></div>
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<span style="font-family: calibri;"><div style="margin: 0px 0px 13px;">
Picture from</div>
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<a href="http://www.cbi.umn.edu/hostedpublications/Tomash/Images%20web%20site/Image%20files/B%20Images/pages/Briggs.Arithmetica%20logarithmica.1624.critica%20logarithms.htm"><span style="color: blue;">http://www.cbi.umn.edu/hostedpublications/Tomash/Images%20web%20site/Image%20files/B%20Images/pages/Briggs.Arithmetica%20logarithmica.1624.critica%20logarithms.htm</span></a></div>
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<b></b><i></i><u></u><sub></sub><sup></sup><strike></strike><br /></div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-21914755958093097422017-06-27T04:28:00.000-07:002017-06-27T04:28:07.736-07:00Arnold Sunrise Problem
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<div style="line-height: 14.4pt; margin: 0px;">
<span style="font-family: calibri;">In a interview in 1995 the notable Russian
mathematician Vladimir Igorevich Arnold<span style="margin: 0px;"> </span>recalled
a problem set <span style="margin: 0px;"> </span>by his schoolteacher, I.
V. Morozkin, when he (Arnold) was 11 or 12 years old. (<span lang="EN" style="margin: 0px;"><a href="http://www.ams.org/notices/199704/arnold.pdf"><span style="color: #001ba0; margin: 0px; text-decoration: none;">An Interview with Vladimir Arnold</span></a></span>)</span></div>
<br />
<div style="line-height: 14.4pt; margin: 0px;">
<br /></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">“Two old women started at sunrise and each walked at a
constant velocity. One went from A to B and the other from B to A. They met at noon
and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9
p.m. At what time was the sunrise on this day?”</span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Arnold says “I spent a whole day thinking on this.” <span style="margin: 0px;"> </span>I presume, from Arnold’s statement, the <i style="mso-bidi-font-style: normal;">sunrise problem</i> was an extension problem<span style="margin: 0px;"> </span>set for high achievers after they completed <span style="margin: 0px;"> </span>routine exercises.<span style="margin: 0px;"> </span>That Arnold spent a day thinking about the
problem implies Mr Morozkin did not teach the class a technique to solve it. Even
as an extension exercise the <i style="mso-bidi-font-style: normal;">sunrise
problem </i>seems far beyond the curriculum for 11-12 year olds in most
countries now or then. The year was 1949. <span style="margin: 0px;"> </span></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">The <i style="mso-bidi-font-style: normal;">sunrise problem</i>
seems to require fairly advanced abstract thinking at an age when Piaget believed
that children were just making the transition from the concrete operational stage
of development to the formal operation stage. Arnold said his solution was based
on what <span style="margin: 0px;"> </span>are “now called scaling arguments”
and “came as a revelation.” <span style="margin: 0px;"> </span>Did he draw
something like this?</span></div>
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<a href="https://3.bp.blogspot.com/-9On_cDLRRak/WVJBMg7mBoI/AAAAAAAAMKQ/xIe6bOFuC5oJiIROJshTUIxky6QGWuOeQCLcBGAs/s1600/scaling.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="77" data-original-width="231" src="https://3.bp.blogspot.com/-9On_cDLRRak/WVJBMg7mBoI/AAAAAAAAMKQ/xIe6bOFuC5oJiIROJshTUIxky6QGWuOeQCLcBGAs/s1600/scaling.png" /></a></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;"><br /></span></div>
<span style="font-family: calibri;"></span>
<br />
<div style="margin: 0px 0px 13px;">
T<span style="font-family: calibri;">he women (lets us call them Ekaterina and Yelena) are
walking at constant velocity.</span></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Using Arnold's hint we can write V<sub><span style="font-size: x-small;">E </span></sub>= kV<sub><span style="font-size: x-small;">Y</span></sub>. </span></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Rearranging,
k= V<sub><span style="font-size: x-small;">E </span></sub>/V<sub><span style="font-size: x-small;">Y</span></sub> . </span></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Then from the diagram, or otherwise, <span style="margin: 0px;"> </span>t/9 = 4/t. Obviously, t = 6.</span></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Sunrise occurs 6 hours before noon, that is 06.00 or 6 am. </span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="font-family: calibri;">Actually, I have omitted one step. (AP/t)/(AP/9) = (PB/4)/(PB/t
) But this seems trivial.</span></div>
<b></b><i></i><u></u><sub></sub><sup></sup><strike></strike><span style="font-family: calibri;"></span>Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-34977710354155040152017-05-13T00:59:00.000-07:002017-05-13T00:59:01.475-07:00Using Algebra and Geometry <br />
<h2 style="margin: 13px 0px 0px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-size: medium;"><span style="color: #4f81bd;"><span style="font-family: "cambria";">Question<span style="margin: 0px;"> </span></span></span></span></span></h2>
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<div style="margin: 0px 0px 13px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">In the regular hexagon below, compare the
shaded area to the entire area.<span style="margin: 0px;"> </span></span></span></div>
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<a href="https://1.bp.blogspot.com/-keuD8lQ3hpw/WRa3eH7LXJI/AAAAAAAALyw/Hb4h6g5GUPMNPl2aKnxamR7JH3GKpEwZQCEw/s1600/hex.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="190" src="https://1.bp.blogspot.com/-keuD8lQ3hpw/WRa3eH7LXJI/AAAAAAAALyw/Hb4h6g5GUPMNPl2aKnxamR7JH3GKpEwZQCEw/s320/hex.png" width="320" /></a></div>
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<br /></div>
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</div>
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">There appear to be three tasks. </span></span><br />
<br />
<div style="margin: 0px 0px 0px 48px; text-indent: -0.25in;">
<span lang="EN-US" style="font-family: "symbol"; margin: 0px;"><span style="margin: 0px;">·<span style="font-size-adjust: none; font-stretch: normal; font: 7pt "Times New Roman"; margin: 0px;">
</span></span></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">Find the grey area.<span style="margin: 0px;"> </span></span></span></div>
<br />
<div style="margin: 0px 0px 0px 48px; text-indent: -0.25in;">
<span lang="EN-US" style="font-family: "symbol"; margin: 0px;"><span style="margin: 0px;">·<span style="font-size-adjust: none; font-stretch: normal; font: 7pt "Times New Roman"; margin: 0px;">
</span></span></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">Find the total area. </span></span></div>
<br />
<div style="margin: 0px 0px 13px 48px; text-indent: -0.25in;">
<span lang="EN-US" style="font-family: "symbol"; margin: 0px;"><span style="margin: 0px;">·<span style="font-size-adjust: none; font-stretch: normal; font: 7pt "Times New Roman"; margin: 0px;">
</span></span></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">Compare the two results.</span></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">Where to start? It is often a good idea to
make a rough sketch.<span style="margin: 0px;"> </span></span></span></div>
<br />
<h2 style="margin: 13px 0px 0px;">
<span lang="EN-US" style="margin: 0px;"><span style="color: #4f81bd; font-family: "cambria"; font-size: medium;">Central congruent triangles</span></span></h2>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">We don’t know if the hexagon has sides of 3
cm or 2 miles or ….<span style="margin: 0px;"> </span>Actually it doesn’t
matter.<span style="margin: 0px;"> </span>Each of the sides is 1s.</span></span><span lang="EN-US" style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; margin: 0px; position: relative; top: 4.5pt;"><v:shape id="_x0000_i1025" style="height: 15pt; width: 14.25pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image002.png"></v:imagedata></v:shape></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";"><span style="margin: 0px;"> </span><span style="margin: 0px;"> </span>Partition
the hexagon into 6 identical equilateral triangles (One has been shaded
pink).<span style="margin: 0px;"> </span>The apothem (or “radius”) is the
height of the pink triangle, h_p</span></span><span lang="EN-US" style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; margin: 0px; position: relative; top: 7pt;"><v:shape id="_x0000_i1025" style="height: 16.5pt; width: 12pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"></v:imagedata></v:shape></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">. </span></span></div>
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<a href="https://3.bp.blogspot.com/-DrGOpdYfp-0/WRa3eDeNIOI/AAAAAAAALyw/4OKyVruIXDE2tr0kpOFmeSeCj6E96vGmQCEw/s1600/central.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="193" src="https://3.bp.blogspot.com/-DrGOpdYfp-0/WRa3eDeNIOI/AAAAAAAALyw/4OKyVruIXDE2tr0kpOFmeSeCj6E96vGmQCEw/s320/central.png" width="320" /></a></div>
<div style="margin: 0px 0px 13px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";"><br /></span></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="margin: 0px;"><v:shape id="Picture_x0020_4" o:spid="_x0000_i1028" style="height: 199.5pt; mso-wrap-style: square; visibility: visible; width: 330pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image004.png">
</v:imagedata></v:shape></span><span lang="EN-US" style="margin: 0px;"></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">The area of the pink triangle can be found
using A= 1/2
(side)(side)<span style="margin: 0px;"> </span>sin<span style="font-family: "cambria math" , "serif"; margin: 0px;">
</span>(included angle). Your </span></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">calculation should show the area of a
pink triangle is √3/4</span></span><span lang="EN-US" style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; margin: 0px; position: relative; top: 10pt;"><v:shape id="_x0000_i1025" style="height: 29.25pt; width: 12pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"></v:imagedata></v:shape></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">. Therefore the area of
the entire hexagon is<span style="margin: 0px;"> </span>6√3/4<span lang="EN-US" style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; margin: 0px; position: relative; top: 10pt;"><v:shape id="_x0000_i1025" style="height: 29.25pt; width: 12pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"></v:imagedata></v:shape></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";"></span></span> </span></span><span lang="EN-US" style="font-size: 14pt; line-height: 115%; margin: 0px;">.</span></div>
<br />
<h2 style="margin: 13px 0px 0px;">
<span lang="EN-US" style="margin: 0px;"><span style="color: #4f81bd; font-family: "cambria"; font-size: medium;">The grey area </span></span></h2>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">Now consider the grey triangle. <span style="margin: 0px;"> </span>The sum of the interior angles is
180(n - 2)°=720°.<span style="margin: 0px;"> </span></span></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">Therefore one interior angle is 120°. <span lang="EN-US" style="margin: 0px;">The area of the grey triangle is 1/2 (side)(side)<span style="margin: 0px;"> </span>sin<span style="font-family: "cambria math" , "serif"; margin: 0px;"> </span>(included angle) = 1/2 (1)(1)<span style="margin: 0px;"> </span>sin<span style="font-family: "cambria math" , "serif"; margin: 0px;"> </span>(120°) = √3/4<span lang="EN-US" style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; margin: 0px; position: relative; top: 10pt;"><v:shape id="_x0000_i1025" style="height: 29.25pt; width: 12pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"></v:imagedata></v:shape></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">.</span></span> </span></span></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";"><span lang="EN-US" style="margin: 0px;"></span></span></span></div>
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";"><span style="margin: 0px;"> </span><span style="margin: 0px;"> </span></span></span><span lang="EN-US" style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; margin: 0px; position: relative; top: 8.5pt;"><v:shape id="_x0000_i1025" style="height: 24.75pt; width: 130.5pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image011.png"></v:imagedata></v:shape></span><span lang="EN-US" style="margin: 0px;"></span><br />
<span lang="EN-US" style="margin: 0px;"><span style="color: #4f81bd; font-family: "cambria"; font-size: medium;"><b>Solution</b></span></span><br />
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">The ratio of<span style="margin: 0px;"> </span>kX to
X<span style="margin: 0px;"> </span>is √3/4<span style="margin: 0px;"> </span>:<span style="margin: 0px;">
</span>6√3/4<span style="margin: 0px;"> </span>so k is 1 <span style="font-family: "cambria math" , "serif"; margin: 0px;">∶</span>
6 =<span style="margin: 0px;"> </span>1/6 .</span></span></div>
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";"></span></span><br />
<div style="margin: 0px 0px 13px;">
</div>
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">
</span></span>
<br />
<h2 style="margin: 13px 0px 0px;">
<span lang="EN-US" style="margin: 0px;"><span style="color: #4f81bd; font-family: "cambria"; font-size: medium;">Aha!</span></span><span lang="EN-US" style="margin: 0px;"></span></h2>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">Is the figure below a cube or a
hexagon?<span style="margin: 0px;"> </span>Actually we can consider the
hexagon to be the ‘projection’ <span style="margin: 0px;"> </span>(or
shadow) of a cube.<span style="margin: 0px;"> </span><span style="margin: 0px;">Immediately, we see the shaded area is one-sixth </span></span></span><span lang="EN-US" style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; margin: 0px; position: relative; top: 10pt;"><v:shape id="_x0000_i1025" style="height: 26.25pt; width: 9pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image016.png"></v:imagedata></v:shape></span><span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">of the
total area.</span></span></div>
<div style="margin: 0px 0px 13px;">
<br />
<div align="center">
<table border="1" cellpadding="0" cellspacing="0" style="border-collapse: collapse; border-image: none; border: medium; margin: 0px 0px 0px -4px;">
<tbody>
<tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="background-color: transparent; border-image: none; border: 1px solid rgb(0, 0, 0); margin: 0px; padding: 0in 5.4pt; width: 218.45pt;" valign="top" width="291">
<div style="line-height: normal; margin: 0px;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Calibri;"><a href="https://4.bp.blogspot.com/-IQ9r3UfNpzc/WRa3d1mAsCI/AAAAAAAALyw/BNvCjfhr29Ut9K8g3Lo-Fvkdsw3dovDuwCEw/s1600/cube.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="258" src="https://4.bp.blogspot.com/-IQ9r3UfNpzc/WRa3d1mAsCI/AAAAAAAALyw/BNvCjfhr29Ut9K8g3Lo-Fvkdsw3dovDuwCEw/s320/cube.png" width="320" /></a></span></div>
<br />
</td>
<td style="background-color: transparent; border-color: rgb(0, 0, 0); border-image: none; border-style: solid solid solid none; border-width: 1px 1px 1px 0px; margin: 0px; padding: 0in 5.4pt; width: 220.55pt;" valign="top" width="294">
<div style="line-height: normal; margin: 0px;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Calibri;"><a href="https://1.bp.blogspot.com/-B8ZuByewR-4/WRa3el-s_CI/AAAAAAAALyw/p6R5SE8IUjQA64vwQpDC_BUb1cbEMUlOwCEw/s1600/teselation.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="258" src="https://1.bp.blogspot.com/-B8ZuByewR-4/WRa3el-s_CI/AAAAAAAALyw/p6R5SE8IUjQA64vwQpDC_BUb1cbEMUlOwCEw/s320/teselation.png" width="320" /></a></span></div>
<br />
</td>
</tr>
<tr style="mso-yfti-irow: 1; mso-yfti-lastrow: yes;">
<td colspan="2" style="background-color: transparent; border-color: rgb(0, 0, 0); border-image: none; border-style: none solid solid; border-width: 0px 1px 1px; margin: 0px; padding: 0in 5.4pt; width: 439pt;" valign="top" width="585">
<div style="line-height: normal; margin: 0px;">
<span lang="EN-US" style="margin: 0px;"><span style="font-family: Calibri;">Input:
A cube(3-D shape) casts a shadow on a wall. Output: A regular hexagon (2-D
shape) is partitioned into 6 congruent triangles.<span style="margin: 0px;"> </span>Thanks to students S.H. and A.L. for
partitioning the hexagon into congruent triangles.</span></span></div>
</td>
</tr>
</tbody></table>
</div>
<b></b><i></i><u></u><sub></sub><sup></sup><strike></strike></div>
<b></b><i></i><u></u><sub></sub><sup></sup><strike></strike><br />
<h2 style="margin: 13px 0px 0px;">
<span lang="EN-US" style="margin: 0px;"><span style="color: #4f81bd; font-family: "cambria"; font-size: medium;">Conclusion</span></span></h2>
W<span lang="EN-US" style="margin: 0px;"><span style="font-family: "calibri";">hen you have the answer, don't stop. A hard-won
solution may prompt additional insight.<span style="margin: 0px;"> </span></span></span><br />
<b></b><i></i><u></u><sub></sub><sup></sup><strike></strike><span style="font-family: "calibri";"></span>Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-57805024486041533432017-04-08T05:59:00.003-07:002017-04-08T05:59:56.703-07:00Using Your Question Bank
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "Arial","sans-serif"; margin: 0px;"><br /></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "Arial","sans-serif"; margin: 0px;">Open
the <span style="margin: 0px;"> </span>Acrobat Catalog Index.</span></div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-FMDnkFmAyfU/WOjeSNhcjsI/AAAAAAAALgc/Qr83dMxuBkgZEMaHNAmCitIWHKb9W_YwgCLcB/s1600/search1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="108" src="https://1.bp.blogspot.com/-FMDnkFmAyfU/WOjeSNhcjsI/AAAAAAAALgc/Qr83dMxuBkgZEMaHNAmCitIWHKb9W_YwgCLcB/s640/search1.png" width="640" /></a></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Arial","sans-serif"; margin: 0px;"><br /></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "Arial","sans-serif"; margin: 0px;">A
Search window opens.<span style="margin: 0px;"> </span>Since I am looking
for sequence questions I will type ‘nth term’. </span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-01m53EbDIZw/WOjeSPfXRSI/AAAAAAAALgY/FsCPGmlktjw6m06FPTUMEtJvJvUetOYdwCEw/s1600/search2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-01m53EbDIZw/WOjeSPfXRSI/AAAAAAAALgY/FsCPGmlktjw6m06FPTUMEtJvJvUetOYdwCEw/s320/search2.png" width="225" /></a></div>
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "Arial","sans-serif"; margin: 0px;"><br /></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "Arial","sans-serif"; margin: 0px;">Click
Search.</span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-OH0BmJ2yKyw/WOjeR8rFX7I/AAAAAAAALgU/0p1ADEe_7FsyjpUBnou0zKOGgj6cabJgwCEw/s1600/search3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="328" src="https://1.bp.blogspot.com/-OH0BmJ2yKyw/WOjeR8rFX7I/AAAAAAAALgU/0p1ADEe_7FsyjpUBnou0zKOGgj6cabJgwCEw/s400/search3.png" width="400" /></a></div>
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "Arial","sans-serif"; margin: 0px;"><br /></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "Arial","sans-serif"; margin: 0px;">Open
a file by clicking on it, for example “ nth term of a different sequence is 4n –
2.”</span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://1.bp.blogspot.com/-SFEsfyN_9Dg/WOjeSkCvp8I/AAAAAAAALgk/saMe-hWWV-8vaLteFHwnDT9QtSv3912ugCEw/s1600/search4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="292" src="https://1.bp.blogspot.com/-SFEsfyN_9Dg/WOjeSkCvp8I/AAAAAAAALgk/saMe-hWWV-8vaLteFHwnDT9QtSv3912ugCEw/s640/search4.png" width="640" /></a></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Arial","sans-serif"; margin: 0px;"><br /></span></div>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "Arial","sans-serif"; margin: 0px;">By
clicking on a few more search results you can quickly put together a quiz or
test on the topic.</span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-E0-H2BDWc-U/WOjeSTRd4lI/AAAAAAAALgg/gRnePHYK2vU8FHzSJ1ri2fZmSpSNAZptQCEw/s1600/search5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="236" src="https://3.bp.blogspot.com/-E0-H2BDWc-U/WOjeSTRd4lI/AAAAAAAALgg/gRnePHYK2vU8FHzSJ1ri2fZmSpSNAZptQCEw/s640/search5.png" width="640" /></a></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: "Arial","sans-serif"; margin: 0px;"><br /></span></div>
<b></b><i></i><u></u><sub></sub><sup></sup><strike></strike>Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-29753426287317894922017-04-08T05:17:00.001-07:002017-04-08T05:23:02.120-07:00Create Your Own Question BankIf you have a copy of Adobe Acrobat Professional you can create your own question bank from past year papers. The bank you will create is 'bare-bones'. It does not link to curriculum focus, level of difficulty, answer keys or examiner reports. With all those negatives, why bother? Mainly because the question bank itself runs on the free Adobe Acrobat.<br />
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="color: #333333; font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">1) Create a folder and
copy all the PDFs you want to index to the folder.</span></div>
<span lang="EN-GB" style="color: #333333; font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">2) Open Adobe Acrobat
Professional.</span><br />
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">3) Choose Advanced > Catalog....</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-vIQRxIHFxSQ/WOiz7urDzQI/AAAAAAAALfw/rPBXKUU0KnwWLNerBtgGKveBiGUBorx4wCLcB/s1600/im1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="155" src="https://3.bp.blogspot.com/-vIQRxIHFxSQ/WOiz7urDzQI/AAAAAAAALfw/rPBXKUU0KnwWLNerBtgGKveBiGUBorx4wCLcB/s400/im1.png" width="400" /></a></div>
<br />
<div style="margin: 0px 0px 13px;">
<span style="color: #333333; font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;"><img src="file:///C:/Users/user/AppData/Local/Temp/msohtmlclip1/01/clip_image001.png" v:shapes="Picture_x0020_2" /></span></div>
<span lang="EN-GB" style="color: #333333; font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">4) Click New Index...</span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-YTLSaT0Gif4/WOi0B6Qy1uI/AAAAAAAALf0/qS4aWaD2UD4jPQ4nBtAzTT0-p84RGyNaACLcB/s1600/im2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="106" src="https://1.bp.blogspot.com/-YTLSaT0Gif4/WOi0B6Qy1uI/AAAAAAAALf0/qS4aWaD2UD4jPQ4nBtAzTT0-p84RGyNaACLcB/s400/im2.png" width="400" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">In Index Title, give your index file a name.</span><br />
<span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">In Index
Description, type a few words about the type of index.</span><br />
<span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">Then browse for folders
(Any folder nested under an included folder will also be
indexed.)</span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-8HbHWSPhHtw/WOi0FB0qzbI/AAAAAAAALf4/iuf_4QCXb2kXPHar_IkAhxabFadKWLxggCEw/s1600/im3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="507" src="https://4.bp.blogspot.com/-8HbHWSPhHtw/WOi0FB0qzbI/AAAAAAAALf4/iuf_4QCXb2kXPHar_IkAhxabFadKWLxggCEw/s640/im3.png" width="640" /></a></div>
<br />
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">Click OK.<span style="margin: 0px;"> </span>you will be prompted to build.</span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-BegE2ph86DU/WOi0FHVC4RI/AAAAAAAALgI/AxpNHvM2hnoLhu6wIXlSyxGxKWRQ4pLsACEw/s1600/im4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-BegE2ph86DU/WOi0FHVC4RI/AAAAAAAALgI/AxpNHvM2hnoLhu6wIXlSyxGxKWRQ4pLsACEw/s400/im4.png" width="348" /></a></div>
<span style="font-family: "cambria";"></span><br /></div>
<div style="margin: 0px 0px 13px;">
<span style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;"></span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<span lang="EN-GB" style="color: #333333; font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">Click <b>Build</b>, and specify the location for the index file.
Click Save. When complete you will see index build successful.</span><br />
<br />
<div style="margin: 0px 0px 13px;">
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;"><span style="margin: 0px;"><a href="https://2.bp.blogspot.com/-caqd_ziQ_ro/WOi0F1vS1oI/AAAAAAAALgI/UgNR5AY7YGk-aTnfog-PBtZDwaPkmbOBQCEw/s1600/im6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-caqd_ziQ_ro/WOi0F1vS1oI/AAAAAAAALgI/UgNR5AY7YGk-aTnfog-PBtZDwaPkmbOBQCEw/s320/im6.png" width="240" /></a></span></span></div>
<span style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;"></span></div>
<a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;">Acrobat Professional has created a file with a .pdx extension and a support folder, with file(s) with .idx extensions. The IDX files contain the index entries. These files are available to any user who wants to search the index.<span style="margin: 0px;"> </span><b style="mso-bidi-font-weight: normal;"> </b></span><br />
<b><span style="font-family: "cambria";"><br /></span></b>
<b><span style="font-family: "cambria";">Note:</span></b><br />
<span style="font-family: "cambria";"><a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a>The procedure may vary slightly in a different version of Adobe Acrobat Professional. </span><b></b><br />
<div style="margin: 0px 0px 13px;">
</div>
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;"><b style="mso-bidi-font-weight: normal;"><br /></b></span></div>
<div style="margin: 0px 0px 13px;">
<span lang="EN-GB" style="font-family: "cambria" , "serif"; font-size: 12pt; line-height: 115%; margin: 0px;"><b><br /></b></span></div>
<a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><br />
<div style="line-height: normal; margin: 10px 0px;">
<br /></div>
<a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=2853830991106309117" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><br />
<br />
<br />
<br />
<br />Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-67503704646390542262013-06-01T21:01:00.000-07:002013-06-01T21:01:06.022-07:00Making Sense of Variables<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">Some of us
have no trouble with writing expressions or equations for word problems. But we are in the minority. Earlier today I was reading from a rather old
(1980s) study on this very difficulty.
One might have hoped that in the 30 plus years since then, a pedagogical
solution to the problem would have been found, but any algebra teacher can tell
you that making sense of word problems remains traumatic for many children. <o:p></o:p></span></span></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">The
following question (taken from Translation difficulties in Learning Mathematics,
American Mathematical Monthly, v88 n4 p286-90 Apr 1981) was<u> </u>given to 47
(non-science) algebra students and 150 calculus students.<o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-bottom: 10.0pt; margin-left: 1.0cm; margin-right: 47.3pt; margin-top: 0cm; tab-stops: 404.0pt;">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">Write an equation for the statement, ‘There are six times as many
students as professors at this university.’ Use S for the number of students
and P for the number of professors. <o:p></o:p></span></span></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">I was not
surprised that more than half (57%) of the algebra students got this question
wrong. But a large minority (37%) of the calculus students also had an incorrect
answer. According to the authors, the
most common mistake was 6S = P. When
asked to explain their thinking, some students drew a picture like this:<o:p></o:p></span></span></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-HBrMlsUGqqc/UarAD46xraI/AAAAAAAADG4/ukoS5ysRkXs/s1600/v1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Arial, Helvetica, sans-serif;"><img border="0" src="http://2.bp.blogspot.com/-HBrMlsUGqqc/UarAD46xraI/AAAAAAAADG4/ukoS5ysRkXs/s1600/v1.jpg" /></span></a></div>
<br />
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">The mathematics teacher might be frustrated
that the requirement (or warning? or advice?) “Use S for the <span style="font-variant: small-caps;">number </span>of students…” was not
followed. Otherwise, the drawing should
look like this:<o:p></o:p></span></span></div>
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;"><span lang="EN-US"> </span><a href="http://2.bp.blogspot.com/-rcYV2_AeJhw/UarAD3qGcwI/AAAAAAAADG0/8o6FyHM-8uM/s1600/v2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="36" src="http://2.bp.blogspot.com/-rcYV2_AeJhw/UarAD3qGcwI/AAAAAAAADG0/8o6FyHM-8uM/s640/v2.jpg" width="640" /></a></span></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">The number
of students = 6.<o:p></o:p></span></span></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: 11pt; line-height: 115%;">The
number of professors </span><span style="font-size: 11pt; line-height: 115%;"> </span> = 1 <o:p></o:p></span></span></div>
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;"><span lang="EN-US">Obviously,
6 </span><span style="font-size: 11pt; line-height: 115%;">≠</span>1. However,</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-w6XxAdMuXO8/UarCqICDzsI/AAAAAAAADHk/5erwXjcOjWk/s1600/v3a.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Arial, Helvetica, sans-serif;"><img border="0" src="http://1.bp.blogspot.com/-w6XxAdMuXO8/UarCqICDzsI/AAAAAAAADHk/5erwXjcOjWk/s1600/v3a.jpg" /></span></a></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: Arial, Helvetica, sans-serif;">Replace “The
number of students” with S and “The number of professors” with P to obtain:</span></div>
<div class="MsoNormal">
<a href="http://1.bp.blogspot.com/-RSWk1OL4Y8E/UarB8JroQpI/AAAAAAAADHU/bEIBgJIGaHU/s1600/v4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;"><img border="0" src="http://1.bp.blogspot.com/-RSWk1OL4Y8E/UarB8JroQpI/AAAAAAAADHU/bEIBgJIGaHU/s1600/v4.jpg" /></span></a></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">Make ‘S’ the
subject of the equation by multiplying both sides by P.<o:p></o:p></span></span></div>
<div class="MsoNormal">
<a href="http://4.bp.blogspot.com/-O0rhp-20cXs/UarB8CaUUmI/AAAAAAAADHY/ETmCvARTOKU/s1600/v5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;"><img border="0" src="http://4.bp.blogspot.com/-O0rhp-20cXs/UarB8CaUUmI/AAAAAAAADHY/ETmCvARTOKU/s1600/v5.jpg" /></span></a></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">Finally,
obtain, S = 6P. <o:p></o:p></span></span></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Arial, Helvetica, sans-serif;">Is this too
much work for something that’s obvious? Perhaps the joke is on us:<o:p></o:p></span></span></div>
<div class="MsoNormal" style="background: #F6F6F6; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 35.45pt; margin-right: 33.1pt; margin-top: 0cm;">
<span style="color: #333333; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><span style="font-family: Arial, Helvetica, sans-serif;">The professor is giving a lecture and
has made an assertion as part of his presentation. A student, not understanding
the basis for the assertion asks why it is true. The professor responds that
"It is obvious." Then the professor steps back, stares at the board
and ponders for several minutes. Then he turns and walks out of the lecture
hall. He is absent for a fairly long time
… Finally, just before the class is scheduled to end the professor
reappears, and announces "Yes, it is obvious." </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: #F6F6F6; margin-bottom: .0001pt; margin-bottom: 0cm;">
<a href="http://www.thescienceforum.com/mathematics/10833-math-joke.html">http://www.thescienceforum.com/mathematics/10833-math-joke.html</a></div>
<br />
<div class="MsoNormal" style="background: #F6F6F6; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #333333; font-family: "Verdana","sans-serif"; font-size: 9.0pt; line-height: 115%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><br />
<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-69452591094442960572013-04-20T22:43:00.000-07:002013-04-20T22:43:42.017-07:00Visualisation versus Algebra?<br />
<div class="MsoNormal">
Here is a Primary 5 (Grade 5) word problem from Yan Kow
Cheong in Singapore:</div>
<div class="MsoNormal">
There are 100 chickens and rabbits altogether. The chickens
have 80 more legs than the rabbits. How many chickens and how many rabbits are
there? (<a href="http://www.singaporemathplus.net/1/post/2013/03/the-chickens-and-rabbits-problem.html">http://www.singaporemathplus.net/1/post/2013/03/the-chickens-and-rabbits-problem.html</a>
). In ‘the West’ this sort of problem would not
be attempted until the students have been taught algebra (i.e., y = mx + c,
etc) so it would probably be a Grade 9 problem! In a comment on <i>another problem involving chickens and ducks, </i>Kow-Cheong states that
“the model heuristic is nothing but algebra in disguise. Instead of using the
variable x, we use a unit, part or line to represent some unknown quantity
..[which] … allows us to use visualization to solve higher-order word problems.</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-JR52Y9vBjXA/UXN8CBa4_DI/AAAAAAAADFo/VjkC_HlcIF4/s1600/chicken.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="http://4.bp.blogspot.com/-JR52Y9vBjXA/UXN8CBa4_DI/AAAAAAAADFo/VjkC_HlcIF4/s400/chicken.jpg" width="307" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
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<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]-->(<a href="http://www.singaporemathplus.com/2010/07/singapore-grade-four-question-without.html">http://www.singaporemathplus.com/2010/07/singapore-grade-four-question-without.html</a>
)</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
The same author asks if we can use the bar method, or the
Sakamoto method, to solve the following problem: Mr. Yan has <i>almost</i> twice as many chickens as cows. The total number of legs and
heads is 184. How many cows are there?</div>
<div class="MsoNormal">
(<a href="http://www.singaporemathplus.net/1/post/2013/03/the-chickens-and-rabbits-problem.html">http://www.singaporemathplus.net/1/post/2013/03/the-chickens-and-rabbits-problem.html</a>
). </div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
I must admit that my preferred technique for this sort of
problem is algebraic. </div>
<div class="MsoNormal">
Let the number of cows be k.</div>
<div class="MsoNormal">
Let the number of chickens be h.</div>
<div class="MsoNormal">
Then h = 2k – x, where x is a ‘slack’ variable such that 0 <
x << k.</div>
<div class="MsoNormal">
Legs + heads = 184</div>
<div class="MsoNormal">
(4k + 2h) + (k + h) = 184</div>
<div class="MsoNormal">
Substitute for h.</div>
<div class="MsoNormal">
(4k + 2(2k – x)) + (k + (2k – x)) = 184</div>
<div class="MsoNormal">
4k + 4k – 2x + k + 2k – 2x = 184</div>
<div class="MsoNormal">
11k – 3x = 184</div>
<div class="MsoNormal">
11k = 184 + 3x.</div>
<div class="MsoNormal">
Apparently, k < 20 and x is much less than k, so x might
be 1, 2, 3 or 4.</div>
<div class="MsoNormal">
Assuming x = 1</div>
<div class="MsoNormal">
11k = 184 + 3 = 187</div>
<div class="MsoNormal">
k = 187/11 = 17 </div>
<div class="MsoNormal">
Then h = 2k – x = 2(17) – 1 = 33.</div>
<div class="MsoNormal">
Note that if x = 2, 3 or 4, k cannot be an integer but we
obviously expect there to be a whole number of cows so the problem is solved. Mr Yan has 33 chickens and 17 cows. I do look forward to seeing the Sakamoto
solution.</div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-34065122737169147002013-02-27T15:10:00.001-08:002013-02-27T15:10:53.906-08:00An Introduction to the Vector Equation of a Line<br />
<div class="MsoNormal">
<span lang="EN-US">Given two
different points we can always draw a line segment between them. We can also sketch a line <i>given just one point</i> and a direction. How? The
instructions are ‘hiding in plain view’ in the equation y = mx + c. Take, for
example y = ½ x – 3. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Please <b>do not </b>make a table of values for x and
y.<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-nmHgmSbl4-g/US6RCPqUm1I/AAAAAAAADFA/DpaPa5jaOQ8/s1600/nt.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="56" src="http://4.bp.blogspot.com/-nmHgmSbl4-g/US6RCPqUm1I/AAAAAAAADFA/DpaPa5jaOQ8/s320/nt.jpg" width="320" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
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coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe"
filled="f" stroked="f">
<v:stroke joinstyle="miter"/>
<v:formulas>
<v:f eqn="if lineDrawn pixelLineWidth 0"/>
<v:f eqn="sum @0 1 0"/>
<v:f eqn="sum 0 0 @1"/>
<v:f eqn="prod @2 1 2"/>
<v:f eqn="prod @3 21600 pixelWidth"/>
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style='width:393.75pt;height:61.5pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="EN-US"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Tables are
slow and inefficient. Remember, we only
need one point! Which point? The
y-intercept. For the equation y = ½ x –
3, start at the point (0, c) = (0, -3).
Now what? From the gradient, m =
½, we find that the rise is 1 and the
run is 2. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">If the
gradient is written as a fraction (½, 23/7, -4/3, …) the numerator is the
‘rise’ and the denominator is the run.
If, however, the gradient is written as a decimal number (0.5, 1.6, -
3.9, …) the given number is the ‘rise’ and the run is always ‘1’. Positive runs
go to the right; negative runs to the left.
Positive rises go up; negative rises go down. <o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
With
preliminaries out of the way, here is the line: Start at the y-intercept,
sketch the run (running parallel to the x-axis) and then sketch the rise
(rising parallel to the y-axis). Label
the end point, P. Take a ruler and
connect the y-intercept and point P.</div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-6KN1IOHa4Vg/US6RCD5KXMI/AAAAAAAADE8/-_SE5esu6Oo/s1600/rr.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-6KN1IOHa4Vg/US6RCD5KXMI/AAAAAAAADE8/-_SE5esu6Oo/s1600/rr.jpg" /></a></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<!--[if gte vml 1]><v:shape id="Picture_x0020_2"
o:spid="_x0000_i1026" type="#_x0000_t75" style='width:212.25pt;height:180pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="EN-US"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US">You can
explore with the Geogebra file “mx_plus_c” to vary intercept c and gradient m
using the respectively labeled sliders.
Turn on the <i>trace function</i> for
point P by right clicking on the point and selecting “Trace On”. Move the c and m sliders to choose your
desired y intercept and gradient. After you have chosen these parameters, go
the pull-down menu, View and click “Refresh Views”. Finally, go to the x</span><sub><span lang="EN-US" style="font-size: 12.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">P</span></sub><span lang="EN-US"> slider and drag that button.
What happens?<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-a6ncdUOhvHE/US6RCMlZtuI/AAAAAAAADFE/SkOkmvpPGh8/s1600/tr.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="556" src="http://4.bp.blogspot.com/-a6ncdUOhvHE/US6RCMlZtuI/AAAAAAAADFE/SkOkmvpPGh8/s640/tr.jpg" width="640" /></a></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US">The rise
and run are continually refreshed, so the screen only shows the latest values,
but we see all the previous values of point P.
The locus of point P is the required line. Ta Dah! <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Conclusion:<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">For many of
you this essay merely re-stated the obvious.
However, the next installment will use very similar ideas to explain the
vector equation of a line: </span></div>
<div class="MsoNormal">
<span lang="EN-US"><b>p</b> = <b>q</b> + k<b>v .<o:p></o:p></b></span></div>
<div class="MsoNormal">
<span lang="EN-US"><b><br /></b></span></div>
<div class="MsoNormal">
<b><span lang="EN-US">Appendix:
Geogebra Construction<o:p></o:p></span></b></div>
<table border="1" cellpadding="0" class="MsoNormalTable" style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-insideh: .5pt solid windowtext; mso-border-insidev: .5pt solid windowtext; mso-cellspacing: 1.5pt; mso-yfti-tbllook: 1184; width: 618px;">
<tbody>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<b><span lang="EN-MY">No.<o:p></o:p></span></b></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" width="106">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<b><span lang="EN-MY">Name<o:p></o:p></span></b></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" width="348"></td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" width="129">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<b><span lang="EN-MY">current value<o:p></o:p></span></b></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">1<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Number, m<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY"> a
slider variable<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">m = 1.7<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">2<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Number, c<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY"> a
slider variable<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">c = -1<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">3<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Point, O<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY"> the
origin<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">O = (0, 0)<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">4<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Point, I<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">(0, c)<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">I = (0,
-1)<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">5<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Vector,
vector</span><sub><span lang="EN-MY" style="font-size: 12pt;">c</span></sub><span lang="EN-MY"><o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Vector[point,
point] = Vector[O, I]<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">vector</span><sub><span lang="EN-MY" style="font-size: 12pt;">c</span></sub><span lang="EN-MY"> = (0, -1)<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">6<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Number, x</span><sub><span lang="EN-MY" style="font-size: 12pt;">p</span></sub><span lang="EN-MY"><o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY"> a
slider variable<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">x</span><sub><span lang="EN-MY" style="font-size: 12pt;">p</span></sub><span lang="EN-MY"> = 2.8<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">7<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Vector run<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Vector[point,
point] = Vector[(0, c), (xp, c)]<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">run =
(2.8, 0)<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">8<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Vector
rise<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Vector[point,
point] =Vector[(xp, c), (xp, c + xp times m)]<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">rise = (0,
4.76)<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt;" valign="top">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">9<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 79.65pt;" valign="top" width="106">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">Point P<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 260.75pt;" valign="top" width="348">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">vectorc + run + rise<o:p></o:p></span></div>
</td>
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: .75pt .75pt .75pt .75pt; width: 96.95pt;" valign="top" width="129">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-MY">P = (2.8,
3.76)<o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: Arial, sans-serif; font-size: 13.5pt; line-height: 115%;">Created with</span><span lang="EN-MY" style="font-family: Arial, sans-serif; font-size: 13.5pt; line-height: 115%;"> </span><span lang="EN-MY" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-MY;"><a href="http://www.geogebra.org/" target="_blank"><span style="color: blue; font-family: "Arial","sans-serif"; font-size: 13.5pt; line-height: 115%; mso-bidi-font-size: 11.0pt;">GeoGebra</span></a><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span lang="EN-US"> A copy of the applet is stored on </span>http://www.geogebratube.org/</b></div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-12524752350341000502013-02-23T21:35:00.001-08:002013-02-23T21:35:39.200-08:00The Pentagram<br />
<div class="MsoNormal">
The
pentagram is an ancient symbol (<a href="http://mathworld.wolfram.com/Pentagram.html">http://mathworld.wolfram.com/Pentagram.html</a>)
, recently popularized by Dan Brown (<a href="http://www.danbrown.com/the-davinci-code/">http://www.danbrown.com/the-davinci-code/</a>).
Mathematicians have analysed its
properties since the time of Pythagoras. (<a href="http://www.math.tamu.edu/~dallen/history/pythag/pythag.html">www.math.tamu.edu/~dallen/history/pythag/pythag.html</a>)</div>
<div class="MsoNormal">
<span lang="EN-US">One such question
is what is the area of the blue star below?<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-u-JFS69T2Gs/USmlGJ03rSI/AAAAAAAADEI/wce7uZ51oy8/s1600/1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-u-JFS69T2Gs/USmlGJ03rSI/AAAAAAAADEI/wce7uZ51oy8/s1600/1.jpg" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
Johnny
decides to find the area of the large pentagon (ABCDE) and to subtract the area
of the five (congruent isosceles) triangles, each coloured light brown above.
ABF is one such triangle.<br />
<div class="MsoNormal">
<span lang="EN-US">The
exterior angle of a regular pentagon is 360°/5 = 72°. Hence each interior angle
is 180° - 72° = 108°. Angle EAB is one
such angle so <EAB = 108°. Call the
centre of the pentagon K. Segment KA bisects angle EAB. Hence Angle KAB = ½ ×108°
= 54°.<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-CSEznUT9yPM/USmlGCoS1sI/AAAAAAAADEM/bZqifO6oivA/s1600/2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-CSEznUT9yPM/USmlGCoS1sI/AAAAAAAADEM/bZqifO6oivA/s1600/2.jpg" /></a></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<!--[if gte vml 1]><v:shape id="Picture_x0020_2"
o:spid="_x0000_i1027" type="#_x0000_t75" style='width:237pt;height:231pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="EN-US"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US">The area of
the pentagon is 5 times the area of the green triangle. The height of the required triangle is 5 tan 54° = 6.88190….
Hence the area of the triangle is ½ (10)( 6.88190….) = 34.40954. The area of
the entire pentagon is 5×34.40954. = 172.047.<o:p></o:p></span></div>
<div class="MsoNormal">
</div>
<div class="MsoNormal">
<span lang="EN-US"><a href="http://1.bp.blogspot.com/-MhrtoLTSY10/USmlGD6vqrI/AAAAAAAADEQ/mNCWVjmbb20/s1600/3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-MhrtoLTSY10/USmlGD6vqrI/AAAAAAAADEQ/mNCWVjmbb20/s1600/3.jpg" /></a> Each base angle
of the highlighted isosceles triangle is ½ (180° -108°) = 36°.Hence, the height
of the highlighted triangle is 5 tan 36° = 3.632712. Consequently the area of the triangle is ½ (10)( 3.632712….)
= 18.163563. The area of all five
triangles is 5×34.40954. = 90.81781.<o:p></o:p></span></div>
<div class="MsoNormal">
</div>
<div class="MsoNormal">
</div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">The area of
the star is the area of the area of pentagon (ABCDE) minus the area of the five
triangles; 172.047 - 90.817 = 81.2 units<sup>2</sup> to 3 sig figs.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Mary takes
a different approach. She decides the
area of the star is five times the area of the dart KAFB (shaded grey below). A dart is simply an unusual kite. <o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-ynwQVsZNuHA/USmmHZKwwdI/AAAAAAAADEo/0_SVJJAphLY/s1600/4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-ynwQVsZNuHA/USmmHZKwwdI/AAAAAAAADEo/0_SVJJAphLY/s1600/4.jpg" /></a></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<!--[if gte vml 1]><v:shape id="Picture_x0020_3"
o:spid="_x0000_i1025" type="#_x0000_t75" style='width:200.25pt;height:206.25pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="EN-US"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">The area of
a kite is half the product of the diagonals = ½ (AB)(KF). Note that diagonal KF
is inside the polygon and diagonal AB is outside the polygon; nevertheless the
formula still holds. Diagonal AB is given (10 units). Hence, Mary only needs the
length of segment KF = 5 tan 54°- 5 tan 36° = 5(tan 54°- tan 36°)
= 6.88190 - 3.632712 = 3.249188.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Mary
calculates the area of the star = 5×½ (10)( 3.249188)= 25×3.249188 = 81.2 units<sup>2
</sup>to 3 sig figs.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Conclusion:
Johnny and Mary both used the five-fold symmetry of the shape, both used the
trigonometric ratio ‘tangent θ’, and both used the same angles (54° and 36°). Hence their algebra is essentially the same. They
only differed in the physical meaning they gave to the algebra. Johnny
calculated the area of two triangles (ABK and ABF) whereas Mary calculated the
area of the dart KAFB. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Can you
find another way to find the area of the star? Is it possible to find the area but
avoid using trigonometry? What about the
mysterious number phi (<a href="http://en.wikipedia.org/wiki/Golden_ratio">http://en.wikipedia.org/wiki/Golden_ratio</a>
).<o:p></o:p></span></div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-28374063723738334902013-02-13T05:42:00.000-08:002013-02-13T05:42:21.636-08:00Trigonometry Quiz<br />
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;">Give answers which are lengths correct to three significant figures. Give answers in degrees correct to one
decimal place. </span></div>
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;"> </span></div>
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;">1. In the right-angled triangle MNO, <!--[if gte msEquation 12]><m:oMath><m:acc><m:accPr><m:chr
m:val="̅"/><span style='mso-bidi-font-size:11.0pt;font-family:"Cambria Math","serif";
mso-ascii-font-family:"Cambria Math";mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-hansi-font-family:"Cambria Math";
mso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-ansi-language:
EN-MY;mso-fareast-language:EN-US;font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:accPr><m:e><i
style='mso-bidi-font-style:normal'><span lang=EN-MY style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-ansi-language:
EN-MY'><m:r>MO</m:r></span></i></m:e></m:acc></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-size: 11pt; position: relative; top: 2.5pt;"><!--[if gte vml 1]><v:shapetype
id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
<v:stroke joinstyle="miter"/>
<v:formulas>
<v:f eqn="if lineDrawn pixelLineWidth 0"/>
<v:f eqn="sum @0 1 0"/>
<v:f eqn="sum 0 0 @1"/>
<v:f eqn="prod @2 1 2"/>
<v:f eqn="prod @3 21600 pixelWidth"/>
<v:f eqn="prod @3 21600 pixelHeight"/>
<v:f eqn="sum @0 0 1"/>
<v:f eqn="prod @6 1 2"/>
<v:f eqn="prod @7 21600 pixelWidth"/>
<v:f eqn="sum @8 21600 0"/>
<v:f eqn="prod @7 21600 pixelHeight"/>
<v:f eqn="sum @10 21600 0"/>
</v:formulas>
<v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
<o:lock v:ext="edit" aspectratio="t"/>
</v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:17.25pt;
height:13.5pt'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--> MO = 4.5 metres and <span style="font-size: 15px;">angle</span> MON = 32<sup>o</sup>. Calculate the length of MN. </span></div>
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;"> </span> </div>
<div class="MsoNormal">
<!--[if gte vml 1]><v:shape
id="Picture_x0020_3" o:spid="_x0000_i1028" type="#_x0000_t75" style='width:187.5pt;
height:87pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-WKBIrBfOp30/URtRutjj2CI/AAAAAAAADCU/6hRg2gP08lQ/s1600/1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-WKBIrBfOp30/URtRutjj2CI/AAAAAAAADCU/6hRg2gP08lQ/s1600/1.jpg" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;">2. GHI is a right triangle. GH = 8
cm and HI = 7 cm. <span style="font-size: 15px;">Angle</span> H = 90°. </span><span style="font-family: Arial, Helvetica, sans-serif; text-align: justify;">Calculate the size of angle HIG.</span><span style="font-family: Arial, Helvetica, sans-serif; text-align: justify;"> </span><span style="font-family: Arial, Helvetica, sans-serif; text-align: justify;"> </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"> <o:p></o:p></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">3. UVWX is a dart. </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">a) Calculate the length of side UV. <span style="background-color: white;"><span style="color: #eeeeee;">nnnnnnnnnnnnnnnnnnnnnnnnnnnnnn </span></span> </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">b) Calculate the size of angle<!--[if gte msEquation 12]><m:oMath><i
style='mso-bidi-font-style:normal'><span style='font-family:"Cambria Math","serif"'><m:r>∠</m:r></span></i></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-size: 11pt; position: relative; top: 2.5pt;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:7.5pt;height:12.75pt'>
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o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--> UXW. </span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">4. </span><span lang="EN-MY"><span style="font-family: Arial, Helvetica, sans-serif;">Square-based pyramid PQRST is shown in the diagram below. All the edges are 3 cm in length. </span><o:p></o:p></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span lang="EN-MY">a) Calculate the
length of PR</span><!--[if gte msEquation 12]><m:oMath><m:acc><m:accPr><m:chr
m:val="̅"/><span style='mso-bidi-font-size:11.0pt;font-family:"Cambria Math","serif";
mso-ascii-font-family:"Cambria Math";mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-hansi-font-family:"Cambria Math";
mso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-ansi-language:
EN-MY;mso-fareast-language:EN-US;font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:accPr><m:e><i
style='mso-bidi-font-style:normal'><span lang=EN-MY style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-ansi-language:
EN-MY'><m:r>PR</m:r></span></i></m:e></m:acc></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-size: 11pt; position: relative; top: 2.5pt;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:14.25pt;height:13.5pt'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image015.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span lang="EN-MY">. <o:p></o:p></span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span lang="EN-MY">b) Calculate the vertical
height, MT</span><!--[if gte msEquation 12]><m:oMath><m:acc><m:accPr><m:chr m:val="̅"/><span
style='mso-bidi-font-size:11.0pt;font-family:"Cambria Math","serif";
mso-ascii-font-family:"Cambria Math";mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-hansi-font-family:"Cambria Math";
mso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-ansi-language:
EN-MY;mso-fareast-language:EN-US;font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:accPr><m:e><i
style='mso-bidi-font-style:normal'><span lang=EN-MY style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-ansi-language:
EN-MY'><m:r> </m:r><m:r>MT</m:r></span></i></m:e></m:acc></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-size: 11pt; position: relative; top: 2.5pt;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:18.75pt;height:13.5pt'>
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o:title="" chromakey="white"/>
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<span lang="EN-MY"><span style="font-family: Arial, Helvetica, sans-serif;">c) Calculate the angle
between PT</span></span><span lang="EN-MY"><span style="font-family: Arial, Helvetica, sans-serif;"> and the base PQRS. </span>
<o:p></o:p></span></div>
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<o:p></o:p></div>
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Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-25766578807962623362013-02-13T05:41:00.003-08:002013-02-13T05:41:52.427-08:00<br />
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Trigonometry Quiz: Answers</div>
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1. In the right-angled triangle MNO, <span lang="EN-MY">MO </span>= 4.5 metres and angle MON = 32 <sup>o</sup>. Calculate the length of MN. </div>
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Trigonometry usually begins with a sketch of a right angled
triangle and definitions of the hypotenuse and the opposite and adjacent ‘legs’
for a given angle. Next up, of course, are the sine, cosine and tangent
functions.</div>
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Consequently we begin by identifying the side opposite the
32<sup> o </sup>angle, the side adjacent to the 32<sup> o</sup> angle and the
hypotenuse. We write down the mnemonic ‘SOH CAH TOA’ to help
us recall that which function is which.</div>
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As we are given the hypotenuse and are asked for the
opposite side, our best choice is the sine function. (If you choose the cosine function you will
calculate the length of segment ON. You can
still find MN by following on with Pythagoras (a<sup>2</sup> + b<sup>2</sup> =
c<sup>2</sup>) but that option is more time-consuming. If you go with ‘tan θ’ however, you are at a
dead end.</div>
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sine 32<sup> o</sup> = MN/MO
= MN/4.5 </div>
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MN = 4.5* sine 32<sup> o</sup> = 2.384… </div>
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To get an answer with three significant figures, we ought to
jot down four figures. The digit on the right (alternatively, the digit for the
thousandths place) will be the ‘decider’. As the rule for ‘4’ is to round down,
we write MN = 2. 38 metres (to 3 significant figures). </div>
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If the teacher wanted to make the question a little bit
harder s/he could change the orientation of the triangle. Or the student could be given the opposite
side and required to find the hypotenuse.</div>
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2. GHI
is a right triangle; GH = 8 cm and HI = 7 cm; angle H = 90° <span style="text-align: justify;">Calculate the size
of angle HIG.</span><span style="text-align: justify;"> </span><span style="text-align: justify;"> </span><span style="text-align: justify;"> </span></div>
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<o:p></o:p></div>
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Again this question follows the
standard development of trigonometry.
After learning to add, we learn to subtract; after learning to multiply,
we learn to divide. After learning the
standard ‘trig’ functions, we learn the inverse functions: arc-sine, arc-cosine
and arc-tangent. <o:p></o:p></div>
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Your calculator and your text probably
have the notation sin<sup>– 1</sup>,
cos<sup>– 1</sup> and tan <sup>–
1</sup>. The latter notation emphasizes that these are inverse functions,
whereas the somewhat old-fashioned ‘arc’ notation emphasizes that the output is
an angle. <o:p></o:p></div>
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tan (θ) = 8/7<o:p></o:p></div>
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tan <sup>– 1</sup>(tan (θ)) = tan <sup>–
1 </sup>(8/7)<o:p></o:p></div>
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θ = 48.81°<o:p></o:p></div>
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After giving the required rounding,
angle HIG = 48.8° (to one decimal
place). <o:p></o:p></div>
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3. UVWX is a dart. </div>
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a) Calculate the length of
side UV. <span style="color: #f3f3f3;"> </span><span style="background-color: #f3f3f3;"><span style="color: #f3f3f3;">nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn </span> </span> </div>
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This question is best
solved with the cosine rule, c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> – 2ab
cos C. The cosine rule is an extension
of Pythagoras. The last term disappears
when C = 90°, since cos 90° = 0. The
cosine rule is (in my opinion) most easily proved with vectors, so for today we
will simply take it as it is. To apply
the formula we will re-label triangle UVW as follows:</div>
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c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup>
– 2ab cos C</div>
<div class="MsoNormal" style="margin-right: -6.9pt; tab-stops: 0cm 22.6pt 44.7pt; text-align: justify; text-justify: inter-ideograph;">
c<sup>2</sup> = 3.9<sup>2</sup> +
5.7<sup>2</sup> – 2*3.9*5.7 cos 124°</div>
<div class="MsoNormal" style="margin-right: -6.9pt; tab-stops: 0cm 22.6pt 44.7pt; text-align: justify; text-justify: inter-ideograph;">
c<sup>2</sup> = 3.9<sup>2</sup> +
5.7<sup>2</sup> + 24.86… (Note that cos 124° is negative so the last term
becomes positive.)</div>
<div class="MsoNormal" style="margin-right: -6.9pt; tab-stops: 0cm 22.6pt 44.7pt; text-align: justify; text-justify: inter-ideograph;">
c<sup>2</sup> = 72.5617…</div>
<div class="MsoNormal" style="margin-right: -6.9pt; tab-stops: 0cm 22.6pt 44.7pt; text-align: justify; text-justify: inter-ideograph;">
Some students, carelessly, write c <sup>2</sup>
to 3 figures. The table below shows that
their final answer will be marked wrong and a very ‘easy’ mark has been lost as
a result.</div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border-collapse: collapse; margin-left: 4.65pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed; mso-yfti-tbllook: 1184; width: 455px;">
<tbody>
<tr style="height: 17.25pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td nowrap="" style="border: solid windowtext 1.0pt; height: 17.25pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.55pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
If c <sup>2</sup> is written down as <o:p></o:p></div>
</td>
<td nowrap="" style="border-left: none; border: solid windowtext 1.0pt; height: 17.25pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.55pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
then ‘c’ is calculated as<o:p></o:p></div>
</td>
<td nowrap="" style="border-left: none; border: solid windowtext 1.0pt; height: 17.25pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 127.85pt;" valign="bottom" width="170">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
and ‘c’ written down to 3 significant figures as <o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 1;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.55pt;" valign="bottom" width="142">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
72.56217…<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.55pt;" valign="bottom" width="142">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
8.518343<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 127.85pt;" valign="bottom" width="170">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
8.52<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 2;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.55pt;" valign="bottom" width="142">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
72.561…<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.55pt;" valign="bottom" width="142">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
8.518274<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 127.85pt;" valign="bottom" width="170">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
8.52<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 3; mso-yfti-lastrow: yes;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.55pt;" valign="bottom" width="142">
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72.5…<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.55pt;" valign="bottom" width="142">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
8.514693<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 127.85pt;" valign="bottom" width="170">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
8.51<o:p></o:p></div>
</td>
</tr>
</tbody></table>
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b) Calculate the size of angle UXW. </div>
<div class="MsoNormal">
A syllabus certainly has more topics than a quiz
(or a test or an exam) can cover. The
examiner is unlikely to ‘waste’ a precious resource by asking the same question
twice. (Of course if your examiner has a
question bank from which a computer program samples randomly, all bets are
off. I once had a 25 question multiple
choice paper for which ‘oxytocin’ was the correct answer, thrice!)<o:p></o:p></div>
<div class="MsoNormal">
Yes, you could find the angle by doing another
cosine rule calculation. But the givens
are two sides and one angle so it is much easier to use the sine rule,
instead. The formula is shorter, and
there is no need to take a square root at the end. <o:p></o:p></div>
<div class="MsoNormal">
sin x/5.7 = sin 108°/8.2</div>
<div class="MsoNormal">
sin x = 0.661100…
</div>
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x = 41.4°
(to one decimal place). <o:p></o:p></div>
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Once again, it is advisable to keep
many more figures than required, and round off as your final step. The sine
rule is proved quite easily from the definition of sine and the formula for the
area of a triangle, but that is a story for another day.<o:p></o:p></div>
<div class="MsoNormal">
</div>
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4. <span lang="EN-MY">Square-based pyramid PQRST is shown in the diagram below. All the edges are 3 cm in length. <o:p></o:p></span></div>
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<span lang="EN-MY">This is a <i>so-called</i> 3-D problem, but in reality
three points (not all on a straight line) define a triangle and a planar (or
2-D) surface. The skill is in finding
the correct triangle. With 6 points (P,
Q, R, S, T and M) we can find <sub>6</sub>C<sub>3</sub> = 20 different
triangles, where (for example) PQR and PQS are different, but PQR and RQP are
the same. So guesswork will take too
long. You need to become proficient at looking
at a fairly cluttered diagram and extracting just the bits you need.<o:p></o:p></span></div>
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<span lang="EN-MY">a) Calculate the
length of PR. <o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-MY">To calculate PR we
only need the base PQRS. In fact, we
only need the isosceles right triangle PQR.
It is a right triangle because angle Q is the vertex of a square, hence
90°. It is isosceles, because PQ and QR
are the sides of a square so both are the same length, i.e., 3 cm. <o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-PipaLl1KaCU/URuV82FG9YI/AAAAAAAADDY/MLNRcY3fMVM/s1600/4a.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="107" src="http://2.bp.blogspot.com/-PipaLl1KaCU/URuV82FG9YI/AAAAAAAADDY/MLNRcY3fMVM/s320/4a.jpg" width="320" /></a></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
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<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">Notice that the sketch
on the right is very rough. There is no
attempt made to make angle Q look like a right angle , but it is labelled as a
right angle, which is enough. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">Since the triangle is
isosceles, both angle P and angle R = 45°.
In keeping with the theme (trigonometry) we can calculate PR with sine
or cosine. For example<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">sin 45° = 3/PR </span><span lang="EN-MY" style="font-family: Wingdings; mso-ansi-language: EN-MY; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;">ß</span><span lang="EN-MY"> </span><span lang="EN-MY" style="font-family: Wingdings; mso-ansi-language: EN-MY; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-MY"> PR = 3/(sin 45°) = 3/(1/</span><span lang="EN-MY" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-MY;">√</span><span lang="EN-MY">2) = 3</span><span lang="EN-MY" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-MY;">√</span><span lang="EN-MY">2 = 4.14 cm
(to 3 ‘sig figs’)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">Undoubtedly, most
students will use Pythagoras instead, with the same result.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">b) Calculate the
vertical height, MT.
</span><!--[if gte vml 1]><v:shape
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</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="EN-MY"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">As the length of PR
was 3</span><span lang="EN-MY" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-MY;">√</span><span lang="EN-MY">2, the length
of PM is 1.5*</span><span lang="EN-MY" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-MY;">√</span><span lang="EN-MY">2 or 2.12
cm. Point M is vertically under T. As the square is symmetrical, point M is
where the diagonals PR and QS intersect, i.e. M is the midpoint of PR.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">We could use the inverse
cosine function to find angle TPM and with the angle we could use any of sin,
cos or tan to solve for h. Or we could
use Pythagoras, base<sup>2</sup> + height<sup>2</sup> = hypotenuse<sup>2</sup> </span><span lang="EN-MY" style="font-family: Wingdings; mso-ansi-language: EN-MY; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-MY"> hypotenuse<sup>2 </sup>– base<sup>2</sup> = height<sup>2</sup>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">If we do part c) before
part b), we get another </span>isosceles right triangle, which from the
symmetry of the situation should not be terribly surprising. In other words, the height h is the same as
the base, <span lang="EN-MY">1.5*</span><span lang="EN-MY" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-MY;">√</span><span lang="EN-MY">2 or 2.12 cm. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY"> <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">c) Calculate the angle
between PT and the base PQRS. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY">As already indicated
in part b), we have hypotenuse and the side adjacent to the required angle, so
this is an inverse cosine problem.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-right: .2pt; tab-stops: 0cm 22.6pt right 467.8pt; text-align: justify; text-justify: inter-ideograph;">
cos (θ) = 2.12/3<o:p></o:p></div>
<div class="MsoNormal" style="margin-right: .2pt; tab-stops: 0cm 22.6pt right 467.8pt; text-align: justify; text-justify: inter-ideograph;">
cos <sup>– 1</sup>(cos (θ)) = cos <sup>–
1 </sup>(0.70666…)<o:p></o:p></div>
<div class="MsoNormal" style="margin-right: .2pt; tab-stops: 0cm 22.6pt right 467.8pt; text-align: justify; text-justify: inter-ideograph;">
θ = 45.0° (to 3 sf). Had you calculated with the exact
value, ( <span lang="EN-MY">1.5*</span><span lang="EN-MY" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-MY;">√</span><span lang="EN-MY">2)/3 = 1/</span><span lang="EN-MY" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-MY;">√</span><span lang="EN-MY">2, you would realise that indeed </span>θ is exactly 45°. <o:p></o:p></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span lang="EN-MY"> <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
What else could have gone into this trigonometry quiz? Reading graphs of the sine or cosine function,
bearings and scale drawings are the likely candidates. <o:p></o:p></div>
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Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-43562199128667140472013-02-05T05:45:00.001-08:002013-02-05T05:45:15.101-08:00A Logarithms Quiz <div dir="ltr" style="text-align: left;" trbidi="on">
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">I gave the following quiz in class recently <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">If log <i><sub>b</sub> D </i>= 16 and log <i><sub>b </sub>H
= </i>12<i>, </i>simplify:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(a) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub></i>(<i>DH</i>)
<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(b) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub> </i>(1/<i>H</i>) <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(c) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub></i><b>√</b><i>D</i>
<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(d) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>H</sub> D</i>. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">Two students took quite different approaches. Johnny’s quiz was as follows:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">(a) log <i><sub>b</sub> </i>(<i>DH </i>) = log <i><sub>b</sub>
</i>(<i>X </i>) + log <i><sub>b</sub> </i>(<i>X </i>) = 16 + 12 = 28<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(b) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub> </i>(1/<i>H</i>) =
log <i><sub>b</sub>
</i>(<i>H<sup>-1</sup></i>) = - log <i><sub>b</sub>
</i>(<i>H</i>) = -12<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(c) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub></i><b>√</b><i>D</i> = log <i><sub>b </sub>D <sup>0<b>.</b>5</sup></i><sup> </sup>= 0.5 log <i><sub>b</sub></i><b> </b><i>D=
0.5 × 16 = 8<o:p></o:p></i></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(d) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>H</sub> D</i> = log <i><sub>b</sub>
D ÷ l</i>og <i><sub>b </sub>H = 16/12 = 1<b>.</b>33<o:p></o:p></i></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">Mary’s quiz was as follows:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub> D </i>= 16 </span><span lang="EN-US" style="font-family: Wingdings; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;"> D = b <sup>16 </sup> and log <i><sub>b </sub>H = </i>12 </span><span lang="EN-US" style="font-family: Wingdings; font-size: 12.0pt; mso-ansi-language: EN-US; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;"> H = b <sup>12</sup><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(a) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub> </i>(<i>DH </i>) <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">= log <i><sub>b</sub> </i>(b <sup>16</sup> b <sup>12</sup>)
<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">= log <i><sub>b</sub> </i>(b <sup>16+12</sup>) <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">= log <i><sub>b</sub> </i>b <sup>28<o:p></o:p></sup></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;"> = 28<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(b) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">H = b <sup>12<o:p></o:p></sup></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">1/H = 1/ (b <sup>12</sup></span><span style="font-family: Arial, sans-serif; font-size: 12pt;">)</span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">= b</span><sup style="font-family: Arial, sans-serif;">-12</sup></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">So log <i><sub>b</sub> </i>(1/<i>H</i>)<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">= log <i><sub>b</sub> </i>b<sup>-12<o:p></o:p></sup></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">= 12<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">(c) </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">D = b <sup>16 </sup> <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">So </span><b><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-style: italic;">√</span></b><i><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">D = </span></i><b><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-style: italic;">√</span></b><i><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">(</span></i><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;"> b <sup>16</sup>)<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">= b <sup>8</sup><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-weight: bold;">So </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub></i><b>√</b><i>D<o:p></o:p></i></span></div>
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<i><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">= </span></i><span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">log <i><sub>b</sub></i> b <sup>8<o:p></o:p></sup></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">= 8<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">(d) not attempted<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US;">Johnny had a perfect score. (In our syllabus, decimal
answers are rounded to three significant figures so 4/3 = 1.33 is ‘correct’).
Mary made a small mistake, forgetting the minus sign on part (c), and omitting
part (d) entirely. However, it is more
difficult to say who has a better understanding of logarithms. Has Johnny just memorized the rules, while he
follows blindly? Or is he using the
rules because class quizzes place a premium on speed and efficiency? Is Mary a more careful mathematician because
she works from first principles? Or is
she using the index laws because she has not done her revision for the quiz and
does not actually know the rules of logarithms- which she accordingly needs to
derive, ‘on the spot’? </span></div>
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Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-34633248337963917062013-01-21T02:20:00.002-08:002013-01-21T02:20:44.341-08:00Reflect, then translate<div dir="ltr" style="text-align: left;" trbidi="on">
In the "a picture is worth 1000 words" category:<br />
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<a href="http://1.bp.blogspot.com/-6yXMJ5rRfR0/UP0WFhHetbI/AAAAAAAADCE/AZwjGDH64rI/s1600/re_+_tr.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="288" src="http://1.bp.blogspot.com/-6yXMJ5rRfR0/UP0WFhHetbI/AAAAAAAADCE/AZwjGDH64rI/s640/re_+_tr.jpg" width="640" /></a></div>
The image was created using Geogebra. Our class uses the text book Extended Mathematics for IGCSE by David Rayner.</div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-10247226231385989642012-12-16T18:29:00.000-08:002012-12-16T18:29:33.235-08:00 Some Observations about TIMSS 2011 <div dir="ltr" style="text-align: left;" trbidi="on">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">TIMSS (Trends in International Mathematics and Science
Study) is an assessment of the maths and science skills of fourth grade and
eighth grade students around the world. The Mathematics data in the table below
are taken from TIMSS 2011Exhibit 2.2: Performance at the International
Benchmarks of Mathematics Achievement 4<sup>th</sup> Grade. </span><a href="http://timssandpirls.bc.edu/timss2011/downloads/T11_IR_M_Chapter2.pdf" style="font-family: 'Times New Roman', serif;">http://timssandpirls.bc.edu/timss2011/downloads/T11_IR_M_Chapter2.pdf</a></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">GNI is gross national income. Thus GNI per capita is
the national income per person. The data in the the table below are from Wikipedia. <a href="http://en.wikipedia.org/wiki/List_of_countries_by_GNI_(PPP)_per_capita">http://en.wikipedia.org/wiki/List_of_countries_by_GNI_(PPP)_per_capita</a><o:p></o:p></span></div>
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Country<o:p></o:p></div>
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Maths
score<o:p></o:p></div>
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World Bank GNP per capita (PPP)<o:p></o:p></div>
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Notes<o:p></o:p></div>
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Singapore <o:p></o:p></div>
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43<o:p></o:p></div>
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59790<o:p></o:p></div>
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Korea, Rep. of <o:p></o:p></div>
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39<o:p></o:p></div>
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30290<o:p></o:p></div>
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Hong Kong SAR <o:p></o:p></div>
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37<o:p></o:p></div>
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51490<o:p></o:p></div>
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<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 4;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Chinese Taipei <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
34<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
41385<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 5;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Japan <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
30<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
35510<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 6;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Northern Ireland<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
24<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
36970<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for UK, not Northern
Ireland<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 7;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
England <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
18<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
36970<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for UK, not England<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 8;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
North Carolina, US<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
16<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
48890<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for US, not North
Carolina<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 9;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Florida, US<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
14<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
48890<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for US, not Florida<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 10;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Russian Federation <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
13<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
20050<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 11;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
United States <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
13<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
48890<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 12;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Finland <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
12<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
37990<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 13;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Australia <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
10<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
36910<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 14;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Denmark <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
10<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
42350<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 15;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Flanders, Belgium <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
10<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
39300<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for Belgium, not
Flanders<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 16;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Hungary <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
10<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
20380<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 17;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Lithuania<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
10<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
19690<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 18;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Ireland <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
9<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
33310<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 19;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Serbia<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
9<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
11640<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 20;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Portugal<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
8<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
24530<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 21;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Kazakhstan <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
7<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
11310<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 22;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Ontario, Canada<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
7<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
39,830<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for Canada, not
Ontario<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 23;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Romania <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
7<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
15140<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 24;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Quebec, Canada<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
6<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
39,830<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for Canada, not Quebec<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 25;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Azerbaijan<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
5<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
9020<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 26;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Dubai, UAE<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
5<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
48220<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for UAE, not Dubai<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 27;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Germany <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
5<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
40170<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 28;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Italy <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
5<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
32350<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 29;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Netherlands <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
5<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
43770<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 30;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Slovak Republic <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
5<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
22610<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 31;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Czech Republic <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
4<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
24190<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 32;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Malta <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
4<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
24170<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 33;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
New Zealand<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
4<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
29140<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 34;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Slovenia <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
4<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
27110<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 35;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Turkey <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
4<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
16730<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 36;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Alberta, Canada<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
3<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
39,830<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for Canada, not
Alberta<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 37;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Sweden <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
3<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
42350<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 38;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Armenia <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
6140<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 39;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Austria <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
41970<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 40;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Chile <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
16160<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 41;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Croatia <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
19330<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 42;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Georgia<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
5390<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 43;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Norway<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
58090<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 44;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Poland <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
20450<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 45;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Qatar <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
87030<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 46;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Saudi Arabia <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
24870<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 47;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
United Arab Emirates <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
2<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
48220<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 48;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Abu Dhabi, UAE<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
1<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
48220<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
GNI per capita for UAE, not Abu Dhabi<o:p></o:p></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 49;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Bahrain <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
1<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
21240<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 50;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Iran, Islamic Rep. of <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
1<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
11400<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 51;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Oman <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
1<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
25770<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 52;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Spain <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
1<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 90.0pt;" valign="bottom" width="120">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
31930<o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 203.8pt;" valign="bottom" width="272"></td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 53;">
<td nowrap="" style="border-top: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 4.0cm;" valign="bottom" width="151">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
Thailand <o:p></o:p></div>
</td>
<td nowrap="" style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 15.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.8pt;" valign="bottom" width="85">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
1<o:p></o:p></div>
</td>
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8390<o:p></o:p></div>
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Botswana (Grade 6)<o:p></o:p></div>
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0<o:p></o:p></div>
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14560<o:p></o:p></div>
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Honduras (Grade 6)<o:p></o:p></div>
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<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
0<o:p></o:p></div>
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3840<o:p></o:p></div>
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Kuwait <o:p></o:p></div>
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0<o:p></o:p></div>
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53820<o:p></o:p></div>
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Morocco <o:p></o:p></div>
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0<o:p></o:p></div>
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4910<o:p></o:p></div>
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Tunisia <o:p></o:p></div>
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0<o:p></o:p></div>
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9090<o:p></o:p></div>
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Yemen (Grade 4)<o:p></o:p></div>
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0<o:p></o:p></div>
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2180<o:p></o:p></div>
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Yemen (Grade 6)<o:p></o:p></div>
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0<o:p></o:p></div>
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2180<o:p></o:p></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"> </span><span style="font-family: 'Times New Roman', serif;">Three countries entered Grade 6 students for the
study. Somewhat confusingly Yemen provided Grade 4 and Grade 6 students. At
first glance, it seems unusual to lump Grade 6s in with Grade 4s, but on
reappraisal, </span><i style="font-family: 'Times New Roman', serif;">what is Grade 4 anyway</i><span style="font-family: 'Times New Roman', serif;">?
Is it an internationally recognized standard unit like the kilogram or the
ampere? Actually, no. Finns and Croatians start Primary School (Grade School) at
7 years old, while English and Australian children start Primary School at age
5. (Data from the World Bank: </span><a href="http://data.worldbank.org/indicator/SE.PRM.AGES" style="font-family: 'Times New Roman', serif;">http://data.worldbank.org/indicator/SE.PRM.AGES</a><span style="font-family: 'Times New Roman', serif;">)</span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Some of the data are for entire countries while others
are for, using TIMSS own terminology, ‘benchmarking participants.’ All of the benchmarking
participants (e.g. , North Carolina) are sub-units of sovereign states (eg
United States of America) , but (confusingly) not every sub-unit (eg North
Ireland) is classified as a ‘benchmarking
participant.’ (Well, it provides a possible
lesson digression on Venn diagrams.)<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The original table (TIMSS 2011Exhibit 2.2) actually
provides four different benchmarks (Advanced, High, Intermediate and Low) for
each country or ‘benchmarking participant’. The original table also gives the standard
error for each data point. Completely arbitrarily, I chose just the ‘Advanced’
benchmark and omitted the standard error.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The choice of economic data was, again, arbitrary. What
choices did I have? For one, I might have used either ‘GDP per capita’ or ‘GNP
per capita’ instead of GNI per capita. Secondly,
the data I used are not ‘raw’ data but have been ‘put into a blender- chopped,
diced and reassembled’ as purchasing power parity (PPP) data using
‘international dollars’. PPP is an expression of the belief that countries
manipulate their currency’s exchange rate relative to the US dollar. Therefore,
another option would have been to use actual GNI per capita (without PPP). For another thing, the data are not equally
up-to-date. Australia’s GNI data are given as “</span><span style="background-color: #f9f9f9; background-position: initial initial; background-repeat: initial initial; font-family: 'Times New Roman', serif;">2009 or 2010” whereas
most of the other countries GNI data is for 2011. For another, England and North Ireland have
separate entries for the TIMSS data, but are a single entity (both are part of
the United Kingdom) in the Wikipedia economic data. I could go on and on, but
my last point will be that some countries are given more weight than others in
my data. For example, both the USA and
Canada get three times the weight of Lithuania. The USA is counted as itself
and again as North Carolina and Florida. Canada, as such, does not appear in
the table but gets entries for Alberta, Ontario and Quebec.<o:p></o:p></span></div>
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<span style="background-color: #f9f9f9; background-position: initial initial; background-repeat: initial initial; font-family: 'Times New Roman', serif;">Enough with the caveats, what does the data
actually look like?<o:p></o:p></span></div>
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<a href="http://2.bp.blogspot.com/-GsjPWhCoH3c/UM6AvkumM7I/AAAAAAAADB0/TgcJURVfI0w/s1600/graph.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-GsjPWhCoH3c/UM6AvkumM7I/AAAAAAAADB0/TgcJURVfI0w/s1600/graph.jpg" /></a></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Remarkably, there does appear to be a correlation
between the Mathematics benchmark at Grade 4 and the GNI per capita. Using my Excel spreadsheet, I calculate the
regression coefficient to be r = 0.34. Statisticians would generally consider
this to be an example of ‘weak’ correlation.
They almost certainly would go on to calculate r<sup>2</sup>, which
would allow them to say that the “mathematics benchmark” <u>explains</u> 12% of
the “GNI per capita.”<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Remember, ‘explains’ does not mean ‘causes.’ Today’s Grade 4 Maths students cannot possibly
determine the size of today’s economy! Of
course the present Grade 4s could be a proxy for the Grade 4s of one or two generations
ago. The mathematics ability of Grade 4s of the 1970s could be a determiner of today's GNI. Another possibility is that the causation works in the other direction.
Rich economies might spend more on Grade 4 education and better paid teachers
might teach in such a way that the children do better on a standardized test of
Mathematics. Or there may be a third
variable that causes both of the graphed variables. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
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<span style="font-family: "Times New Roman","serif";">Unlike
the data comparing height and shoe size (yesterday’s blog), there should be plenty
to think about. How big is the leap from
the impersonal “How much maths education should country X provide to its
citizens to ensure its economic success?” to the much more personal “How much
maths education do I need to do ensure my
economic success?” Students might even
wonder if substituting “musical ability” or “prowess in sports” in Grade 4 in
place of “maths” would have produced a higher correlation coefficient. I do think it is unlikely that most students
would they take the same line as the Guardian newspaper. The Guardian appears unwilling
to takes sides but instead errs on the side of caution by writing “<span style="background: white;">The IEA's studies, much like similar intermittent
league tables compiled by the OECD, generally contain so much data as to
support a series of different, even contradictory, arguments.”<o:p></o:p></span></span></div>
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<span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: 'Times New Roman', serif;"><a href="http://www.guardian.co.uk/education/2012/dec/11/english-schools-slip-science-rankings">http://www.guardian.co.uk/education/2012/dec/11/english-schools-slip-science-rankings</a><o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif";">In the
light of the weak correlation seen in the data, does it make sense for
politicians to push for better maths and science scores and for more STEM (science,
technology, engineering, and mathematics) grads? Brian Vastag who is a science reporter at The
Washington Post doesn’t think so. In a story entitled “U.S. pushes for more
scientists, but the jobs aren’t there” he writes: “<span style="background-color: white; background-position: initial initial; background-repeat: initial initial;">There
are too many laboratory scientists for too few jobs.</span>” Taking the
pharmaceutical industry as an example he goes on to say: “Since 2000, U.S. drug
firms have slashed 300,000 jobs, according to an analysis by consulting firm
Challenger, Gray & Christmas. In the latest closure, Roche last month
announced it is shuttering its storied Nutley, N.J., campus — where Valium was
invented — and shedding another 1,000 research jobs.</span>”<span style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"> <span lang="EN-US">“(</span></span><span style="font-family: "Times New Roman","serif";"><a href="http://www.washingtonpost.com/national/health-science/us-pushes-for-more-scientists-but-the-jobs-arent-there/2012/07/07/gJQAZJpQUW_story.html"><span lang="EN-US">http://www.washingtonpost.com/national/health-science/us-pushes-for-more-scientists-but-the-jobs-arent-there/2012/07/07/gJQAZJpQUW_story.html</span></a></span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">) </span><span style="font-family: "Times New Roman","serif";"><o:p></o:p></span></div>
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Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-18452001975376796572012-12-15T06:44:00.000-08:002012-12-15T06:44:06.997-08:00Data in the Maths Text Book<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: 'Times New Roman', serif;"> In a typical maths text book,
data are ‘clean’, easy to use and rather boring. The example below comes from
Chapter 10 of STP National Curriculum Mathematics 9B which just happened, along
with a calculus text, to be within easy reach of my computer as I was typing. <o:p></o:p></span></div>
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<a href="http://3.bp.blogspot.com/-QWeLitGal0I/UMyGnwyPuwI/AAAAAAAADBM/hNONoiFu-xs/s1600/9B.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-QWeLitGal0I/UMyGnwyPuwI/AAAAAAAADBM/hNONoiFu-xs/s1600/9B.jpg" /></a></div>
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<span lang="EN-US" style="font-family: 'Times New Roman', serif;"><a href="http://www.nelsonthornes.com/shop/nt/stp-national-curriculum-mathematics-pupil-book-9b"><span style="color: blue;">http://www.nelsonthornes.com/shop/nt/stp-national-curriculum-mathematics-pupil-book-9b</span></a></span><span style="font-family: 'Times New Roman', serif;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: 'Times New Roman', serif;">Since there wasn't any data in the calculus text, the STP text
became the scapegoat. For the most part, I quite like the STP National
Curriculum Maths series and I am sure that I could have made my point equally
well with any of 10 or 15 other texts on my shelves.</span><span style="font-family: 'Times New Roman', serif;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Scatter
graphs are introduced by considering the hypothesis: “Tall people have larger
feet than short people.” The authors give the following data:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Person</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">1</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">2</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">3</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-left: none; border: solid black 1.0pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="top" width="43">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">4</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-left: none; border: solid black 1.0pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="top" width="43">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">5</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-left: none; border: solid black 1.0pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="top" width="43">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">6</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-left: none; border: solid black 1.0pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="top" width="43">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">7</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-left: none; border: solid black 1.0pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="top" width="43">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">8</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-left: none; border: solid black 1.0pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="top" width="43">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">9</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-left: none; border: solid black 1.0pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="top" width="43">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">10</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">11</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">12</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Height
(cm)</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">160<o:p></o:p></span></div>
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">166<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">174<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">158<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">167<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">161<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">170<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">171<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">166<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">163<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">164<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">168<o:p></o:p></span></div>
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<td style="border-top: none; border: solid black 1.0pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 76.3pt;" valign="top" width="102">
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Shoe
size (continental)</span><span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">36<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
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<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">38<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">40<o:p></o:p></span></div>
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<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">37<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">37<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">38<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">42<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">41<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">40<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">39<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">37<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; padding: 0cm 5.4pt 0cm 5.4pt; width: 32.15pt;" valign="bottom" width="43">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;">
<span style="font-family: "Times New Roman","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">39<o:p></o:p></span></div>
</td>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;"><br />
I used an Excel spreadsheet to obtain the following graph. (The authors’
graph is similar but does not include a regression line.)<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">The
authors go on to say that “The points do not all fit on a straight line… Taller
people tend to have larger feet but the relationship between height and shoe
size is not strong enough to justify the original statement.” Who
could disagree? My Excel spreadsheet calculates the correlation coefficient, r
= 0.73, which is indicative of strong correlation. As a correction, we simply
need to insert the phrase “tend to” into the original hypothesis to get, “Tall
people <i>tend to</i> have larger feet than short people.” A
good teacher will advise the students that correlation is not causation. When x
and y show strong correlation on a scatter graph x could cause y, y could cause
x, or both could be the effect of an underlying cause. Most students will
understand that height is not the cause of shoe size but that genetics and
nutrition underlie both.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Having
dispensed above – in reverse order–with ‘boring’ and ‘easy to
use’ I move on to ‘clean.’ Cleanliness is a complex topic, e.g see <a href="http://en.wikipedia.org/wiki/Data_cleansing"><span style="color: blue;">http://en.wikipedia.org/wiki/Data_cleansing</span></a> .<o:p></o:p></span></div>
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<span lang="EN-US"><span style="font-family: 'Times New Roman', serif;">My point
is that the text (and presumably the teacher and the class) just assume that
‘height’ and ‘shoe size’ are well defined. But is that the case? How
were the heights obtained? Are all the heights up-to-date? Was an effort made
to prevent slouching or standing on tippy-toes? Could it be that some people
were wearing their shoes while others were in bare feet? Were some heights measured
in feet and inches and afterwards converted to centimeters? What about hair
styles? If someone has a ‘bee-hive’ hairstyle how do we measure her
height?</span><span style="font-family: Times, Times New Roman, serif;"> </span></span></div>
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<a href="http://2.bp.blogspot.com/-BZN0tBnAbE8/UMyGodu_MuI/AAAAAAAADBQ/eRMuac_PDtM/s1600/Amy.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-BZN0tBnAbE8/UMyGodu_MuI/AAAAAAAADBQ/eRMuac_PDtM/s320/Amy.jpg" width="214" /></a></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Amy
Winehouse: <a href="http://en.wikipedia.org/wiki/File:WinehouseLA.jpg"><span style="color: blue;">http://en.wikipedia.org/wiki/File:WinehouseLA.jpg</span></a><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Next
consider the shoes. Currently, I have three pairs of shoes from
different manufacturers. Their sizes are 45, 45, and 47 but they all fit! It
is unlikely that everybody was wearing the same shoe. Unless it is part of a
school uniform or they were all interviewed while shopping at Bata. <o:p></o:p></span></div>
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<a href="http://4.bp.blogspot.com/-QIwlw7GWl3E/UMyGpHBnB9I/AAAAAAAADBY/g_-KwxBy0SE/s1600/Bata.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Times, Times New Roman, serif;"><img border="0" src="http://4.bp.blogspot.com/-QIwlw7GWl3E/UMyGpHBnB9I/AAAAAAAADBY/g_-KwxBy0SE/s1600/Bata.jpg" /></span></a></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Bata shoes: <a href="http://bata.com.my/webbata/faces/tiles/product.jsp?productID=1423534&catalogueID=68&categoryID=5137&parentCategoryID=5003"><span style="color: blue;">http://bata.com.my/webbata/faces/tiles/product.jsp?productID=1423534&catalogueID=68&categoryID=5137&parentCategoryID=5003</span></a><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">Of course it is possible interviewees were asked for their <i>average</i> shoe size. Is that a
statistic that you know about yourself? An interested student could find a
project here: what is the percentage of people who know the range and central
tendency (mean, median and/or mode) of their shoe size?<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-GB;">On the other hand, real data tend to be untidy, more difficult to
interpret and potentially more exciting. I hope to follow up
tomorrow by looking at some data from TIMSS (Trends in International
Mathematics and Science Study). TIMSS is an assessment of the maths and science
skills of fourth grade and eighth grade students around the world. <o:p></o:p></span></div>
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Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-5498096518905324462012-11-25T05:18:00.000-08:002012-11-25T05:18:17.813-08:00A Difficult Shear<div dir="ltr" style="text-align: left;" trbidi="on">
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<span lang="EN-US">Today's lesson is from Ch 9, Ex 15, Q14 David Rayner (<a href="http://www.amazon.com/Extended-Maths-Camb-Igcse-3e/dp/0199138745/ref=dp_ob_title_bk">http://www.amazon.com/Extended-Maths-Camb-Igcse-3e/dp/0199138745/ref=dp_ob_title_bk</a>)<o:p></o:p></span><br />
<span lang="EN-US"><br /></span></div>
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<span lang="EN-US">a) Find and
draw the image of the square O (0, 0), P (1, 1), Q (0, 2), R (-1, 1) under the </span></div>
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<span lang="EN-US"> <o:p></o:p></span></div>
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<span lang="EN-US">Square transformed to Parallelogram (Geogebra 4.0,
www.</span><span lang="EN-US"> </span><span lang="EN-US">geogebra.org)<o:p></o:p></span></div>
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<span lang="EN-US"><br /></span>
<span lang="EN-US">b) Show that the
transformation is a shear and find the equation of the invariant line.<o:p></o:p></span></div>
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<span lang="EN-US">Mary says, from
the drawing, it is evident that O’ = O and R’ = R. Hence the invariant line is
y = -x. If you need convincing, the
gradient of the line is (-1 - 0)/(1 - 0) </span><span lang="EN-US"> </span><span style="font-family: "Calibri","sans-serif"; font-size: 11pt; line-height: 115%; position: relative; top: 12pt;"><v:shape id="_x0000_i1025" style="height: 27.75pt; width: 79.5pt;" type="#_x0000_t75">
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</v:imagedata></v:shape></span><span lang="EN-US">=
-1 and its y intercept is 0. Next, look
at the determinant: det M </span><span lang="EN-US">= (4)(-2) – (3)(-3) = -8 + 9 = 1. For IGCSE purposes, we can understand the determinant
as an area scaling factor. Since the scale factor is unity, the image parallelogram
must have the same area as the original square. M is not a congruency, since a square is
transformed into to a parallelogram.
There is just one line of invariant points, y = -x. Finally, M preserves areas. M must
be a shear. </span></div>
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<span lang="EN-US"><br /></span>
<span lang="EN-US">Johnny says, but
perhaps there is another invariant line? Actually, a little bit more algebra
will prove this transformation has only one line of invariant points. For an invariant point, P’ = P by
definition. That is, after the
transformation, <u>the point has not moved</u>. That is exactly what we mean by
invariant!! Note that for any matrix
transformation, the point O (0, 0) must always be an invariant point since</span><br />
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<span lang="EN-US">This is exactly
the same as the equation that we found previously. Thus we can say safely there is <u>only one</u>
line of invariant points.<o:p></o:p></span></div>
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<span lang="EN-US">I summarise the
lesson by reminding the students that, of the transformations we know, both
reflections and shears have one invariant line. However, the matrix
representing the reflection will have det R<sub>e</sub> = -1 whereas, as we
have already seen, the shear will have det H = 1. <o:p></o:p></span></div>
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<span lang="EN-US">It works. Supposing that something exists, and then
seeing where it leads us is quite characteristic of algebraic geometry. Our motto is “Only a contradiction implies a
contradiction.” That is, if our supposition does not lead us into a
contradiction, it is valid. Equating elements: p =4, q = 3, k = -¾, s = ¼ . As
the first matrix has the form of a shear, the transformation must be a shear. <o:p></o:p></span><br />
<br />
A parting question: Do you think the author, David Rayner, just played around with different values for a, b, c and d until he found a shear matrix? </div>
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Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-61174681256224803172012-10-30T07:19:00.003-07:002012-10-30T07:19:50.846-07:00Quadratic Functions Test<div dir="ltr" style="text-align: left;" trbidi="on">
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">1. The equation
of a curve is <i>y </i>= 10 – <i>x</i><sup>2</sup> + 6<i>x</i>.<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY; mso-bidi-font-weight: bold;"> a) </span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Express <i>y </i>in the form <i>a
</i>– (<i>x </i>+ <i>b</i>)<sup> 2 </sup>, where <i>a </i>and <i>b </i>are
integers. <o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">The equation for y in the expanded form = The equation
for y in the given form<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">10 – <i>x</i><sup>2</sup> + 6<i>x</i> = <i>a </i>– (<i>x
</i>+ <i>b</i>)<sup> 2</sup><o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">10 – <i>x</i><sup>2</sup> + 6<i>x</i> = <i>a </i>– (<i>x</i><sup>2</sup><i> </i>+ 2bx
+ <i>b</i><sup>2</sup>)<sup> </sup>;
expand the squared term<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">10 – <i>x</i><sup>2</sup> + 6<i>x</i> = <i>a </i>–
<i>x</i><sup>2</sup><i> </i>– 2bx
– <i>b</i><sup>2</sup>; distribute
the –
sign<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">– <i>x</i><sup>2</sup> + 6<i>x</i> + 10 = – <i>x</i><sup>2</sup><i>
</i>– 2bx – <i>b</i><sup>2</sup> + a ; order terms by
their degree<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">And match up the coefficients:<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">– <i>x</i><sup>2</sup> = – <i>x</i><sup>2</sup><i>
</i>; 6<i>x</i> = –
2bx </span><span lang="EN-MY" style="font-family: Wingdings; mso-ansi-language: EN-MY; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"> b = -3; 10
= –
<i>b</i><sup>2</sup> + a </span><span lang="EN-MY" style="font-family: Wingdings; mso-ansi-language: EN-MY; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"> a
=10 + <i>b</i><sup>2</sup> = 19<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY; mso-bidi-font-weight: bold;">b) </span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Hence, find
the coordinates of the vertex of the curve. <o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">The vertex form is
Y = A(X – H)<sup>2</sup> + K <o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">The vertex form is also Y = K + A(X – H)<sup> 2</sup> by the Commutative
Law of Addition (e.g., 2 + 3 = 3 + 2)<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Compare with y = 19 – (<i>x </i>+ <i>(-3)</i>)<sup>
2 </sup><o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Obviously, K = 19,
and A = -1.<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Also – H = +(-3) <o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Therefore H = +3<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Hence, the vertex is (x, y) = (H, K) = (+3, +19)<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"><br /></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">2. </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">Solve the quadratic inequality: x</span><sup><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">2</span></sup><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"> –
8x + 12 > 0. Your answer must
include a sketch. </span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Let y = P(x) = 1</span><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">x</span><sup><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">2</span></sup><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"> –
8x + 12, with a = 1, i.e. a positive number. We have no requirement for b and
c.</span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"><o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">To find the roots change the form of the polynomial
from <i>expanded form</i> to <i>factored form</i>.<o:p></o:p></span></div>
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<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">y = (x – 6)(x – 2) in factored form<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Set y = 0 to find the roots. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">x – 6 = 0 </span><span lang="EN-MY" style="font-family: Wingdings; mso-ansi-language: EN-MY; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"> x
= 6<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">x – 2 = 0 </span><span lang="EN-MY" style="font-family: Wingdings; mso-ansi-language: EN-MY; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"> x
= 2. The roots are “critical numbers”
on the x-axis. Between the roots the
parabola is under the x-axis. To the left and right of the roots the parabola
lies above the x-axis. Check the original problem. We need the 2 separate pieces of the parabola
above the x-axis.<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-K-XP7adduEQ/UI_hg8HsedI/AAAAAAAADAE/ulHMFwq1fFs/s1600/disjoint.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-K-XP7adduEQ/UI_hg8HsedI/AAAAAAAADAE/ulHMFwq1fFs/s1600/disjoint.jpg" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Using our knowledge of set theory, we can write the
solution as {x: x < -2 U x > 6, x is a Real number}. The solution set is composed of 2 disjoint
pieces.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">3. Determine the set of values of <i>k </i>for which
the equation <i>x</i><sup>2</sup> + 2<i>x </i>+ <i>k </i>= 3<i>kx </i>– 1 has <b>no real roots</b>. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">As given, this equation is of the form, quadratic
function = linear function. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">A rough sketch shows that there can be 2, 1 or zero
solutions. (The sketch of a quadratic function is a parabola. The sketch of a
linear function is a line. Amazing!)<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-CUp6zv17crQ/UI_hhYTMg7I/AAAAAAAADAM/njyYlQCgznw/s1600/possibilities.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-CUp6zv17crQ/UI_hhYTMg7I/AAAAAAAADAM/njyYlQCgznw/s1600/possibilities.jpg" /></a></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: "Arial","sans-serif"; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="Picture_x0020_1"
o:spid="_x0000_i1026" type="#_x0000_t75" style='width:183.75pt;height:147.75pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">To solve, we bring all the terms to the left.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">We need (quadratic) – (line) = 0
has no real roots. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">New quadratic = 0 has no real roots<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Do not make the mistake of believing that in the
equation y = x<sup>2</sup> +2x </span><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">–</span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"> 3kx + k + 1that ‘x’ and ‘k’ are both
variables!!! Yes, ‘x’ is a variable,
because we can draw an x-axis. However,
k is <i>merely</i> a <i>parameter</i>.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">y = 1x<sup>2</sup> + (2 – 3k)x + (k +1) <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">y = ax<sup>2</sup> + bx + c.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">By direct comparison:
a = 1, b = (2 – 3k) and c = (k + 1)<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"> The phrase “no
real roots” should tell you to use the discriminant.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Discriminant = b<sup>2</sup> – 4ac <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">D = (2 – 3k) <sup>2</sup> – 4(1)(k + 1)<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">D = 4 – 12k + 9k<sup>2</sup> – 4k – 4 <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">D = 9k<sup>2</sup> – 16k in expanded form or D = k(9k
– 16) in factored form. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">That is, D is a polynomial function of k. In fact this will also be a parabolic
function.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Now, we can give D and k their own axes.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "Arial","sans-serif"; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="Picture_x0020_2"
o:spid="_x0000_i1025" type="#_x0000_t75" style='width:231pt;height:168.75pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image005.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-XJBQ8pjyol8/UI_hgOf2RFI/AAAAAAAAC_8/6kP5HnsNN4E/s1600/between.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-XJBQ8pjyol8/UI_hgOf2RFI/AAAAAAAAC_8/6kP5HnsNN4E/s1600/between.jpg" /></a></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;">Using set notation the solution is { 0 < k <
16/9}. Unlike the previous question,
all the points in this set form one continuous piece.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-MY" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-MY;"><br /></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
</div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-56154652024719851502012-10-21T03:28:00.000-07:002012-10-21T03:28:12.928-07:00Plant Interest<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<div class="MsoNormal">
<span style="font-family: 'Times New Roman', serif;">Here is a seedling. It has two leaves which collect
sunlight for photosynthesis.</span><span style="font-family: 'Times New Roman', serif;"> </span><span style="font-family: 'Times New Roman', serif;">Some of the
sugar created is used for the plant’s metabolism and some is used to create
DNA, protein, cellulose, </span><i style="font-family: 'Times New Roman', serif;">et cetera</i><span style="font-family: 'Times New Roman', serif;"> to make new cells and hence new leaves.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shapetype
id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
<v:stroke joinstyle="miter"/>
<v:formulas>
<v:f eqn="if lineDrawn pixelLineWidth 0"/>
<v:f eqn="sum @0 1 0"/>
<v:f eqn="sum 0 0 @1"/>
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<v:f eqn="prod @3 21600 pixelHeight"/>
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<v:f eqn="sum @8 21600 0"/>
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<v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
<o:lock v:ext="edit" aspectratio="t"/>
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style='width:345pt;height:182.25pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-2gejyhwhB9Q/UIPMLyMSs5I/AAAAAAAAC_I/LsSQolw4tjY/s1600/T0.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="143" src="http://1.bp.blogspot.com/-2gejyhwhB9Q/UIPMLyMSs5I/AAAAAAAAC_I/LsSQolw4tjY/s320/T0.jpg" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-family: 'Times New Roman', serif;">The plant at time ‘zero’.</span></div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="Picture_x0020_4"
o:spid="_x0000_i1028" type="#_x0000_t75" style='width:414pt;height:324pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-l3Jq0bsEKww/UIPMMiPqf0I/AAAAAAAAC_Q/-D9m3dJEdNY/s1600/T1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="241" src="http://2.bp.blogspot.com/-l3Jq0bsEKww/UIPMMiPqf0I/AAAAAAAAC_Q/-D9m3dJEdNY/s320/T1.jpg" width="320" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The plant at week 1.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><a href="http://4.bp.blogspot.com/-06NJzCZ3l4o/UIPMNGyCInI/AAAAAAAAC_Y/_XYhvXiK4Gk/s1600/T2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="268" src="http://4.bp.blogspot.com/-06NJzCZ3l4o/UIPMNGyCInI/AAAAAAAAC_Y/_XYhvXiK4Gk/s320/T2.jpg" width="320" /></a></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="Picture_x0020_6"
o:spid="_x0000_i1027" type="#_x0000_t75" style='width:330.75pt;height:285pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image005.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The plant at week 2.<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-unq-YOQ80GU/UIPMNk-dbQI/AAAAAAAAC_g/N_wEh06OlpI/s1600/T3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="273" src="http://1.bp.blogspot.com/-unq-YOQ80GU/UIPMNk-dbQI/AAAAAAAAC_g/N_wEh06OlpI/s320/T3.jpg" width="320" /></a></div>
<div class="MsoNormal">
</div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="Picture_x0020_7"
o:spid="_x0000_i1026" type="#_x0000_t75" style='width:340.5pt;height:315pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The plant at week 3.<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-RPr7lNAQzLk/UIPMOAO7aXI/AAAAAAAAC_o/g46weQ4_Qfs/s1600/T4.jpg" imageanchor="1" style="font-family: 'Times New Roman'; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="320" src="http://4.bp.blogspot.com/-RPr7lNAQzLk/UIPMOAO7aXI/AAAAAAAAC_o/g46weQ4_Qfs/s320/T4.jpg" width="315" /></a></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"></span></div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="Picture_x0020_8"
o:spid="_x0000_i1025" type="#_x0000_t75" style='width:372.75pt;height:370.5pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image009.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal" style="text-align: center;">
<br /></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The plant at week 4.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal" style="text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr>
<td style="border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Time (weeks)<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Leaf area (arbitrary units)<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">0<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">2<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">1<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">3<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">2<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">4<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">3<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">5<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">4<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 231.05pt;" valign="top" width="308">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">6<o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The plant is adding one new leaf every week. As it originally had two leaves, its growth
rate can be considered to be 50% per week.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">To create a formula we will<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpFirst" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: Symbol; mso-ansi-language: EN-US; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Use the letter P to represent the original number of
leaves. Therefore P = 2.<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: Symbol; mso-ansi-language: EN-US; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Use the letter R to represent the growth rate. Therefore R = 50% = 0.5<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: Symbol; mso-ansi-language: EN-US; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Use the letter T to represent the number of weeks. Therefore
0 </span><span lang="EN-US" style="font-size: 15px; line-height: 17px;"><span style="font-family: Calibri, sans-serif;"><=</span></span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> T </span><span lang="EN-US" style="font-size: 15px; line-height: 17px;"><span style="font-family: Calibri, sans-serif;"><=</span></span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> 4.</span><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: Symbol; mso-ansi-language: EN-US; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Use the letter I to represent the number of new leaves. <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: Symbol; mso-ansi-language: EN-US; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Use the letter A to represent the total number of
leaves. <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<br /></div>
<div class="MsoListParagraphCxSpLast">
<br /></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Our
formula for plant growth is I = PRT. Consequently, A = P + I.<o:p></o:p></span></div>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; margin-left: 18.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr>
<td style="border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 146.9pt;" valign="top" width="196">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">T<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 152.25pt;" valign="top" width="203">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I = PRT<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.95pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A = P + I<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 146.9pt;" valign="top" width="196">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">0<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 152.25pt;" valign="top" width="203">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I = 2 × 0.5 × 0 = 0<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.95pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A = 2 + 0 = 2<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 146.9pt;" valign="top" width="196">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">1<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 152.25pt;" valign="top" width="203">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I = 2 × 0.5 × 1 = 1<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.95pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A = 2 + 1 = 3<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 146.9pt;" valign="top" width="196">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">2<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 152.25pt;" valign="top" width="203">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I = 2 × 0.5 × 2 = 2<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.95pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A = 2 + 2 = 4<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 146.9pt;" valign="top" width="196">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">3<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 152.25pt;" valign="top" width="203">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I = 2 × 0.5 × 3 = 3<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.95pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A = 2 + 3 = 5<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 146.9pt;" valign="top" width="196">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">4<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 152.25pt;" valign="top" width="203">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I = 2 × 0.5 × 2 = 2<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.95pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A = 2 + 2 = 4<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 146.9pt;" valign="top" width="196">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">n<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 152.25pt;" valign="top" width="203">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I = 2 × 0.5 × n = n<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.95pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A = 2 + n<o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<br /></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">As you
can see, the formula matches perfectly with the observed growth. Some of you may be saying those formulas look
suspiciously like the simple interest formulas for the growth of money in a
bank account. Well, yes. Actually, these are valuable plants which
produce an anti-malarial drug.
Consequently, each leaf can be sold for exactly $1. 00.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">End of
story? Not really. A biologist examines the data. He notices
that in week 0 the plant produces 200 units of sugar, i.e. 100 units per
leaf. From the 200 units, the plant used 150 units for metabolism and 50
units to grow a new leaf. In succeeding
weeks the plant produced much more sugar. However every week the plant only
allocated the same 50 units to growth.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; margin-left: 18.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr>
<td style="border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 29.95pt;" valign="top" width="40">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">T<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 3.0cm;" valign="top" width="113">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Sugar produced<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 77.95pt;" valign="top" width="104">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Sugar used for metabolism<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.3pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Sugar used for growth of a new leaf<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.85pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Percentage of sugar used for growth<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 29.95pt;" valign="top" width="40">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">0<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 3.0cm;" valign="top" width="113">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">200<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 77.95pt;" valign="top" width="104">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">150<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.3pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">50<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.85pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">25%<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 29.95pt;" valign="top" width="40">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">1<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 3.0cm;" valign="top" width="113">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">300<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 77.95pt;" valign="top" width="104">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">250<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.3pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">50<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.85pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">17%<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 29.95pt;" valign="top" width="40">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">2<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 3.0cm;" valign="top" width="113">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">400<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 77.95pt;" valign="top" width="104">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">350<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.3pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">50<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.85pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">13%<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 29.95pt;" valign="top" width="40">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">3<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 3.0cm;" valign="top" width="113">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">500<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 77.95pt;" valign="top" width="104">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">450<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.3pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">50<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.85pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">10%<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 29.95pt;" valign="top" width="40">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">4<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 3.0cm;" valign="top" width="113">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">600<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 77.95pt;" valign="top" width="104">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">550<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.3pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">50<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.85pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">8%<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 29.95pt;" valign="top" width="40">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">n<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 3.0cm;" valign="top" width="113">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">200 + 100 n<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 77.95pt;" valign="top" width="104">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">150 + 100n<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 106.3pt;" valign="top" width="142">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">50<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 144.85pt;" valign="top" width="193">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The biologist notes growth is tending towards 0% as n
becomes larger. Does this not contradict
the previous statement that the growth rate R remains at a constant 50% every
week? No, because we are ‘comparing apples and oranges.’ The 50% growth rate is
in comparison to the original size of the plant. The declining growth rate
noted by the biologist is recalculated on a week-to-week basis.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The owner of the plant asks the biologist to use DNA
technology to increase the plant’s growth.
After months, or possibly years, of failed experiments, the genetically modified
plant grows as follows.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr>
<td style="border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 83.4pt;" valign="top" width="111">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Time (weeks)<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.85pt;" valign="top" width="94">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Leaf area <o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 307.85pt;" valign="top" width="410">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Growth on a week by week basis<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 83.4pt;" valign="top" width="111">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">0<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.85pt;" valign="bottom" width="94">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">2<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 307.85pt;" valign="top" width="410">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">n/a<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 83.4pt;" valign="top" width="111">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">1<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.85pt;" valign="bottom" width="94">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">3<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 307.85pt;" valign="top" width="410">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">(3-2)/2
× 100 = 50%<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 83.4pt;" valign="top" width="111">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">2<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.85pt;" valign="bottom" width="94">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">4.5<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 307.85pt;" valign="top" width="410">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">(4.5-3)/3
× 100 = 50%<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 83.4pt;" valign="top" width="111">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">3<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.85pt;" valign="bottom" width="94">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">6.75<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 307.85pt;" valign="top" width="410">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">(6.75-4.5)/4.5
× 100 = 50%<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 83.4pt;" valign="top" width="111">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">4<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.85pt;" valign="bottom" width="94">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">10.125<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 307.85pt;" valign="top" width="410">
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;">(10.125
- 6.75)/6.75 × 100 = 50%<o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<br /></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Some readers
may object to the leaf area being given as 4.5 in week 2. Does this imply ½ a leaf? A moment’s
consideration will show there is no cause for concern. The original plant’s
leaves were all 10 cm<sup>2</sup> in size. In the genetically modified plant, most
of the new leaves are smaller than the original two leaves, but each leaf still
contributes some growth. So in some sense, “Yes, ½ a leaf.”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Financially
sophisticated readers will see that the biologist has improved the plant’s
yield by causing the plant to switch from a linear growth model to an
exponential growth model. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The
sequence 2, 4, 6, 8, 10, … exhibits
linear growth. The equation of a line is y = mx + c. This equation is of <b>exactly the same form</b> as A = PI + P. We
match y with A, m with P, x with I and c also with P.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">The
sequence 2, 4, 8, 16, 32, … exhibits
exponential growth. The equation of an exponential curve is y = k×b<sup>x</sup>.
This equation is of <b>exactly the same
form</b> as A = P(1 + r)<sup>n</sup>. We match y with A, k with P, (1 + r) with
b and x with n. In conclusion, not only is it the case that “time is money” but
plant growth is also money! <o:p></o:p></span></div>
</div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-75445305351792278882012-10-21T01:05:00.000-07:002012-10-21T01:05:50.603-07:00Interest<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<div class="MsoNormal">
<span style="font-family: 'Times New Roman', serif;">Johnny, Mary, and Ifran each have $100. There are only
two banks in town. However, they both offer 5% simple interest.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Johnny puts his money in Ambank for 2 years. Here is
what he gets:<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"> I = PrT <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">= 100 × 0.05 × 2 <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">= 10<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A = P + I <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">= 100 + 10<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">=110.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">How much would does Mary get if she puts her money in CIMB
Bank for 2 years? </span><span style="font-family: 'Times New Roman', serif;">Obviously, the same amount.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Irfan decides to put his money in Ambank for the first
year; then withdrew everything from Ambank and put it in CIMB Bank for the
second year. What does he get?<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I<sub>A</sub>= P<sub>A</sub>rT <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">= 100 × 0.05 × 1 <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">=5<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A<sub>A</sub>= P<sub> A</sub> + I<sub> A</sub> <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">= 100 + 5<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">=105<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">I<sub>C</sub>= P<sub>C</sub>rT <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">= 105 × 0.05 × 1 <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">=5.25<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">A<sub>C </sub>= P<sub> C</sub> + I<sub> C</sub> <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">= 105 + 5.25<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">=110.25<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Irfan has made a bigger profit than Johnny and Mary
have. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Question: Where did the extra 25 cents come from? <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Answer: Compound Interest.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">When Johnny kept his money with Ambank for <u>2 years
at simple interest</u> he did not get any benefit from the interest earned in
the first year. That is, he was earning
interest on $100 in year 1 and he were still earning interest on the same $100
in year 2. Likewise, for Mary’s money at CIMB. But when Irfan took his money
out of Ambank, he started earning interest on $105 in year 2. In other words, in
year 2, he was earning interest on the principal and <i>interest on the first year’s interest</i>. Hence ‘compound’ interest.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Now banks don’t want people rushing around at the end
of the year withdrawing and depositing money.
Half of the town would take money from bank A and put it in Bank B. The
other half would take money out of Bank B and put it in Bank A. Hence, each
bank will decide to credit the customer’s account <u>at the end of the year</u>
and let him or her earn interest on the principal and interest on the interest.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; mso-ansi-language: EN-US;">Is that the end of the story? Not necessarily, suppose
a new bank opens up in town and offers monthly interest. Responding to that threat, another bank
offers weekly interest. Where will it stop?<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
</div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-56220028522695058442012-09-22T08:13:00.001-07:002012-09-22T08:20:55.288-07:00 f(x) – f(x–1) = 3x^2 – 5x<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<div class="MsoNormal">
<span lang="EN-US">After a
lesson on the factor theorem, the teacher assigns the following homework
problem. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"> “A cubic polynomial f(x) has a factor x and
f(x) – f(x–1) = 3x<sup>2</sup> – 5x. … Find f(x).”<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">(Question
28, page 97, New Additional Mathematics by Ho Soo thong and Khor Nyak Hiong)<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Johnny
begins as follows. As f(x) is cubic, he reasons that it will factorise as x(x –
<i>a</i>)(x –<i> b</i>).<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Consequently,
Johnny defines f(x) in Wx Maxima. (A free computer algebra system that he downloaded
from </span><a href="http://sourceforge.net/projects/wxmaxima/">http://sourceforge.net/projects/wxmaxima/</a>
.
Unfortunately, Johnny cannot use <span lang="EN-US">Wx Maxima</span><span lang="EN-US"> </span>on the exam.)<span lang="EN-US"><o:p></o:p></span><br />
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-98VKkj7owAQ/UF3U3vtg0kI/AAAAAAAAC-Q/-7NUn4PGJFI/s1600/ft1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-98VKkj7owAQ/UF3U3vtg0kI/AAAAAAAAC-Q/-7NUn4PGJFI/s1600/ft1.jpg" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
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<v:stroke joinstyle="miter"/>
<v:formulas>
<v:f eqn="if lineDrawn pixelLineWidth 0"/>
<v:f eqn="sum @0 1 0"/>
<v:f eqn="sum 0 0 @1"/>
<v:f eqn="prod @2 1 2"/>
<v:f eqn="prod @3 21600 pixelWidth"/>
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style='width:246.75pt;height:47.25pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="EN-US"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<br />
<br />
<br />
<br />
Note that
(%i1) is “input 1” and (%o1) is “output 1”.</div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">In a very
straight-forward way, f(x) – f(x–1) is:<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-bTl8tOzNh2I/UF3U9x0Q4aI/AAAAAAAAC-Y/66Uta-SabsM/s1600/ft2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-bTl8tOzNh2I/UF3U9x0Q4aI/AAAAAAAAC-Y/66Uta-SabsM/s1600/ft2.jpg" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
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height:46.5pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="EN-US"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">As the textbook
has told Johnny that expression %o2 is identical to 3x<sup>2</sup> – 5x, he
inputs<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-SDrweRRRwIM/UF3U-St5daI/AAAAAAAAC-g/3qYkvoatLIk/s1600/ft3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-SDrweRRRwIM/UF3U-St5daI/AAAAAAAAC-g/3qYkvoatLIk/s1600/ft3.jpg" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">Next he simplifies:<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="MsoNormal">
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-aObaI1AjPlI/UF3XOWjzm0I/AAAAAAAAC-4/2Bpn0mGrAZo/s1600/ft4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-aObaI1AjPlI/UF3XOWjzm0I/AAAAAAAAC-4/2Bpn0mGrAZo/s1600/ft4.jpg" /></a></div>
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
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id="Picture_x0020_4" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:381.75pt;
height:51pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\Chris\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="EN-US"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">On
comparing the coefficients of x Johnny sees that <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">–2<i>b</i> –2<i>a</i>
– 3 = –5 <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Thus, <i>a </i>=
1 – <i>b </i> Equation (1) <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">As there is
no constant term<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">(<i>a </i>+ 1)<i>b</i> + <i>a</i> + 1 = 0 Equation (2)<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Johnny substitutes
for <i>a</i>:<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">(1 – <i>b</i> + 1)<i>b </i>+ 1 – <i>b</i> + 1 = 0<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">– <i>b</i><sup>2</sup> +<i> b</i> + 2 = 0<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">(– <i>b </i>+ 2) (<i>b</i> + 1) = 0<o:p></o:p></span></div>
<div class="MsoNormal">
<i><span lang="EN-US">b</span></i><span lang="EN-US"> = – 1 or <i>b</i> =2 <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">To solve
for<i> a</i>, he substitutes back into <i>a</i> = 1 – <i>b</i>.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">If b = – 1, a = 2 or if b = 2, a = –1. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Johnny realizes
that it is immaterial which label goes with which number. He has established that, written in factored
form, f(x) = x(x – –1) (x – 2) = x(x + 1)(x – 2). <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">Alternatively,
Mary starts with f(x) – f(x–1) = 3x<sup>2</sup> – 5x and substitutes 0 for x,
everywhere.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">She gets
f(0) – f(0 – 1) = 3(0)<sup>2</sup> – 5(0) = 0<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">However she
knows that f(0) itself evaluates to zero: <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(0) = 0(x –
a) (x – b) = ‘zero’ times ‘other stuff’
= 0.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Therefore,<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">0 – f(0 –
1) = 0<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">It follows that
f(– 1) = 0<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Mary
applies the factor theorem, for any f(a) = 0 she knows (x –a) is a factor of f.
Therefore (x – – 1) </span><br />
<span lang="EN-US">= (x + 1) is a factor
of f.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Iteratively,
Mary lets x = 1<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(1) – f(1–1)
= 3(1)<sup>2</sup> – 5(1)<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(1) – f(0)
= –2 <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Once again
she substitutes f(0) =0, so<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(1) – 0 = –2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(1) = –2 ≠
0, so (x – 1) is not a factor of f.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Therefore
she continues and lets x = 2.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(2) – f(2–1)
= 3(2)<sup>2</sup> – 5(2)<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(2) – f(1)
= 3(2)<sup>2</sup> – 5(2)<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">She
substitutes for f(1).<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(2) – (–2)
= 2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(2) + 2 =
2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">f(2) = 0,
i.e., she knows (x – 2) is also a factor of f(x). Since f is a cubic function Mary
knows it has exactly three factors. She
writes f(x) = x(x + 1)(x – 2).<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US">Both
students have successfully completed their homework. Johnny has used the notion of function
identity and successfully solved a pair of simultaneous equations to find his
solution. He made some use of the
computer but could have expanded everything by hand if necessary. Mary has been
more traditional and done all her calculations with pencil and paper. Also, she
has directly applied the factor theorem itself, which is more in line with the
teacher’s expectations. Full marks to
both students.<o:p></o:p></span></div>
</div>
Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-22319070449423419532012-09-20T06:59:00.003-07:002012-09-20T06:59:39.051-07:00Test Answers<div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Answers<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Q1. 3<sup>x+2 </sup>= 45 + 4 × 3<sup>x </sup> <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Put all terms with 3<sup>x </sup> on the left, 3<sup>x+2 </sup> –
4 × 3<sup>x </sup> <sup> </sup>=
45 <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Factor out 3<sup>x
</sup> , 3<sup>x </sup> (3<sup>2
</sup> – 4 ) <sup> </sup>= 45<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">… Solve to
get x = 2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Q2. 2 log<sub>5</sub>r – 3 log <sub>125</sub>t =
2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Recall that 125 = 5<sup>3</sup>,<sup> </sup>so try to
get like terms on the left. This means
use the change of base formula.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">2 log<sub>5</sub>r – 3 (log <sub>5</sub>t / log <sub>5</sub>125)
= 2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">2 log<sub>5</sub>r – 3 (log <sub>5</sub>t / log <sub>5</sub>5<sup>3</sup>)
= 2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Next remember that log <sub>x</sub>x = 1, so log <sub>5</sub>5=
1<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">2 log<sub>5</sub>r – 3 (log <sub>5</sub>t / <s>log</s>
<s><sub>5</sub>5</s><sup>3</sup>) = 2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Next the 3s will cancel<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">2 log<sub>5</sub>r – <s>3</s> (log <sub>5</sub>t / <s>3</s>)
= 2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">Replace 2 with log <sub>5</sub> 25 and you are almost
finished<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">… r = 5</span><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">√</span><span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US;">t<o:p></o:p></span></div>
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<a href="http://2.bp.blogspot.com/-Dj5wrHpzeXo/UFshBbceYrI/AAAAAAAAC9c/8ip0x93sIkA/s1600/q4b.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="105" src="http://2.bp.blogspot.com/-Dj5wrHpzeXo/UFshBbceYrI/AAAAAAAAC9c/8ip0x93sIkA/s400/q4b.jpg" width="400" /></a></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></div>
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<a href="http://4.bp.blogspot.com/--baMmaON2Cs/UFshJvoVRsI/AAAAAAAAC9k/NO6hKLy0jj8/s1600/q5.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="323" src="http://4.bp.blogspot.com/--baMmaON2Cs/UFshJvoVRsI/AAAAAAAAC9k/NO6hKLy0jj8/s640/q5.jpg" width="640" /></a></div>
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<br /></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So x = 3 and y = </span><span lang="EN-US" style="font-size: 15px; line-height: 17px;"><span style="font-family: Calibri, sans-serif;">-</span></span><span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">5 . (Confirm by substituting into both equations).<o:p></o:p></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">6. 4<sup>x+2 </sup>×
2<sup>y</sup> = 8<o:p></o:p></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">2, 4 and 8 are related by 4 = 2<sup>2</sup> and 8 = 2<sup>3</sup>.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">4<sup>x+2 </sup>× 2<sup>y</sup> = 8 </span><span lang="EN-US" style="font-family: Wingdings; font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">(2<sup>2</sup>)
<sup>x+2 </sup>× 2<sup>y</sup> = 2<sup>3</sup> </span><span lang="EN-US" style="font-family: Wingdings; font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">
2(x+2) + y = 3 Eqn (1)<o:p></o:p></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Likewise, 27<sup>x – 2 </sup>× (1/3)<sup>y</sup> = 1 </span><span lang="EN-US" style="font-family: Wingdings; font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> (3<sup>3</sup>) <sup>x
– 2 </sup>× (3)<sup> – y</sup> = 3<sup>0</sup>
</span><span lang="EN-US" style="font-family: Wingdings; font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;">à</span><span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">
Eqn (2)<o:p></o:p></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Don’t forget, x<sup>0</sup> = 1 (unless 0<sup>0</sup>)<o:p></o:p></span></div>
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<span lang="EN-US" style="font-size: 14.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">… Solve for the simultaneous equations: x = 1, y = – 3.<o:p></o:p></span></div>
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Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-76855307645664825462012-02-18T20:17:00.000-08:002012-02-18T20:17:30.374-08:00Quadratics, a Discriminant Problem<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-EBIzf-EEYfI/T0B3OU8UetI/AAAAAAAAC7s/PbL_SxL8RsM/s1600/discriminant1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br />
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<span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">The line y + 2x = p has no intersection with the parabola y</span><span style="font-size: small;"><sup><span lang="EN-US" style="font-family: "Arial","sans-serif"; line-height: 115%;">2</span></sup></span><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"> = x + p. Prove that p < <b style="mso-bidi-font-weight: normal;">-</b>1/24.</span><br />
<span lang="EN-US" style="font-family: "Arial","sans-serif";"> </span> <br />
<div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">Whenever there is an intersection, the two y co-ordinates must be equal, i.e., y<sub>L</sub> = y<sub>P</sub>. According to the usual method, Mary lets y = p – 2x, so y<sup>2</sup> = (p – 2x)<sup>2</sup> = p<sup>2</sup> – 4px +4x<sup>2</sup>.</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">y<sup>2 </sup>= y<sup>2</sup>, so p<sup>2</sup> – 4px +4x<sup>2</sup> = x + p</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">Then, 4x<sup>2</sup> – 4px + p<sup>2</sup> – x – p = 0</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">4x<sup>2</sup> – (4p + 1) x + (p<sup>2</sup> – p) = 0</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">Finally, Mary uses the discriminant, b<sup>2 </sup>– 4ac, with a =4, b = – (4p + 1) and c = p<sup>2</sup> <span style="mso-spacerun: yes;"> </span>– <span style="mso-spacerun: yes;"> </span>p. </span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">b<sup>2 </sup>– 4ac < 0 </span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">(– (4p + 1))<sup>2</sup> – 4(4)(p<sup>2</sup> <span style="mso-spacerun: yes;"> </span>– <span style="mso-spacerun: yes;"> </span>p) < 0</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">16p<sup>2 </sup>+ 8p + 1 – 16p<sup>2</sup> + 16p <0</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">24p < -1</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">p < -1/24</span></div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">However, it must also be true that, whenever there is an intersection, the two x co-ordinates must be equal, i.e., x<sub>L</sub> = x<sub>P</sub>. Therefore Johnny writes:</span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-EBIzf-EEYfI/T0B3OU8UetI/AAAAAAAAC7s/PbL_SxL8RsM/s1600/discriminant1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-EBIzf-EEYfI/T0B3OU8UetI/AAAAAAAAC7s/PbL_SxL8RsM/s1600/discriminant1.jpg" /></a></div><div class="MsoNormal"><br />
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<div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">Johnny also has a quadratic equation, and he also uses the discriminant. However, he has somewhat ‘nicer’ values for a, b and c. a = 1, b =½ and c = -1.5p. </span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">b<sup>2 </sup>– 4ac < 0 </span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">(½)<sup>2</sup> – 4(1)(-1.5p) < 0</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">¼ + 6p <0</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">6p < -¼</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">p < -1/24</span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">If you believe that every second counts on the exam, it pays to notice that y</span><span style="font-size: small;"><sup><span lang="EN-US" style="font-family: "Arial","sans-serif"; line-height: 115%;">2</span></sup></span><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"> = x + p is not our typical parabola, as it opens rightwards (instead of upwards or downwards).</span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-uViumhQMUfI/T0B3ftK1X2I/AAAAAAAAC70/ynCf_sETbJQ/s1600/discriminant.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="410" src="http://1.bp.blogspot.com/-uViumhQMUfI/T0B3ftK1X2I/AAAAAAAAC70/ynCf_sETbJQ/s640/discriminant.jpg" width="640" /></a></div><div class="MsoNormal"><span style="font-family: "Arial","sans-serif"; mso-fareast-language: EN-GB; mso-no-proof: yes;"><br />
</span><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"></span></div><div class="MsoNormal"><span lang="EN-US" style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">This observation ought to give rise to the notion that working with x<sub>L</sub> = x<sub>P </sub>is likely to be easier than working with y<sub>L</sub> = y<sub>P</sub>.</span></div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><br />
</div>Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-16980336268123955162012-02-18T19:05:00.000-08:002012-02-18T19:05:38.768-08:00Averages<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-GB</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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<div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">What does the word average signify?<span style="mso-spacerun: yes;"> </span>Did you know Wikipedia lists at least 14 different kinds of averages? (<a href="http://en.wikipedia.org/wiki/Average">http://en.wikipedia.org/wiki/Average</a>)</span></div><div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">The most common average is the arithmetic mean, but it is not always appropriate.<span style="mso-spacerun: yes;"> </span>Basil runs 800 metres at 8 metres per second and another 800 m at 7 metres per second.<span style="mso-spacerun: yes;"> </span>What is his average speed?<span style="mso-spacerun: yes;"> </span>Many ‘good’ students blithely add 8 and 7 and then divide by 2 to get 7.5. </span><span lang="EN-US" style="font-family: Wingdings; mso-ansi-language: EN-US; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-char-type: symbol; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span lang="EN-US" style="mso-ansi-language: EN-US;"> <span style="font-variant: small-caps;">wrong!</span></span><span lang="EN-US" style="font-family: Wingdings; mso-ansi-language: EN-US; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-char-type: symbol; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span lang="EN-US" style="mso-ansi-language: EN-US;"> </span></div><div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">Let’s take an extreme case.<span style="mso-spacerun: yes;"> </span>Extreme cases are an essential part of your mathematical toolbox. Suppose Basil runs his first 800 m at 8 m/s and his second 800 m at the glacially slow speed of 0. 0000…1 m/s. <span style="mso-spacerun: yes;"> </span>(Imagine as many zeros as you like before the one). Is the average speed (8 + 0)/2 = 4 metres per second? <span style="mso-spacerun: yes;"> </span>Certainly not! </span></div><div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">Imagine another case.<span style="mso-spacerun: yes;"> </span>I am driving my car at 100 km per hour for 0.99999 …9 hours (or 59.999…9 minutes) and for a brief fleeting instant at 200 km per h.<span style="mso-spacerun: yes;"> </span>Is my average speed (100+200)/2 = 150 km/h?<span style="mso-spacerun: yes;"> </span>Certainly not!</span></div><div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">Instead of the ordinary mean (a + b)/2, what the student needs is a weighted mean (w<sub>1</sub> × a + w<sub>2</sub> × b)/( w<sub>1</sub> + w<sub>2</sub>). <span style="mso-spacerun: yes;"> </span>But, what are the appropriate weights?<span style="mso-spacerun: yes;"> </span>The obvious (and correct answer) are the times spent at each speed.<span style="mso-spacerun: yes;"> </span>Then we get (t<sub>1</sub> × s<sub>1</sub> + t<sub>2</sub> × s<sub>2</sub>)/( t<sub>1</sub> + t<sub>2</sub>) = (d<sub>1</sub> + d<sub>2</sub>)/( t<sub>1</sub> + t<sub>2</sub>) = total distance/total time, as taught by the physics teachers.<span style="mso-spacerun: yes;"> </span></span></div>Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-51027060732340892502012-02-11T21:19:00.000-08:002012-02-11T21:19:05.111-08:00Unusual products<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-GB</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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UnhideWhenUsed="false" Name="Medium Grid 1 Accent 6"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 6"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 6"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/> <w:LsdException Locked="false" Priority="19" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/> <w:LsdException Locked="false" Priority="21" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/> <w:LsdException Locked="false" Priority="31" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/> <w:LsdException Locked="false" Priority="32" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/> <w:LsdException Locked="false" Priority="33" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Book Title"/> <w:LsdException Locked="false" Priority="37" Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> </w:LatentStyles> </xml><![endif]--><!--[if gte mso 10]> <style>
/* Style Definitions */
table.MsoNormalTable
{mso-style-name:"Table Normal";
mso-tstyle-rowband-size:0;
mso-tstyle-colband-size:0;
mso-style-noshow:yes;
mso-style-priority:99;
mso-style-qformat:yes;
mso-style-parent:"";
mso-padding-alt:0cm 5.4pt 0cm 5.4pt;
mso-para-margin-top:0cm;
mso-para-margin-right:0cm;
mso-para-margin-bottom:10.0pt;
mso-para-margin-left:0cm;
line-height:115%;
mso-pagination:widow-orphan;
font-size:11.0pt;
font-family:"Calibri","sans-serif";
mso-ascii-font-family:Calibri;
mso-ascii-theme-font:minor-latin;
mso-fareast-font-family:"Times New Roman";
mso-fareast-theme-font:minor-fareast;
mso-hansi-font-family:Calibri;
mso-hansi-theme-font:minor-latin;}
</style> <![endif]--> <div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">We all know that 9 × 7 is equal to 7 × 9. However, the following pairs of products appear unusual.</span></div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">Example 1:</span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-ee0RPj0x_D8/TzdEkamXW-I/AAAAAAAAC7E/kqzhhtvp-ww/s1600/UP1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-ee0RPj0x_D8/TzdEkamXW-I/AAAAAAAAC7E/kqzhhtvp-ww/s1600/UP1.jpg" /></a></div><div class="MsoNormal"><br />
<span lang="EN-US"></span></div><div class="MsoNormal"><span lang="EN-US">Example 2:</span></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-mHA0unX6YgU/TzdKpX-gOqI/AAAAAAAAC7M/bK9VT7r0HQI/s1600/UP2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-mHA0unX6YgU/TzdKpX-gOqI/AAAAAAAAC7M/bK9VT7r0HQI/s1600/UP2.jpg" /></a></div><br />
</div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-GB</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
DefSemiHidden="true" DefQFormat="false" DefPriority="99"
LatentStyleCount="267"> <w:LsdException Locked="false" Priority="0" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Normal"/> <w:LsdException Locked="false" Priority="9" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="heading 1"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 2"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 3"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 4"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 5"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 6"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 7"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 8"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 9"/> <w:LsdException Locked="false" Priority="39" Name="toc 1"/> <w:LsdException Locked="false" Priority="39" Name="toc 2"/> <w:LsdException Locked="false" Priority="39" Name="toc 3"/> <w:LsdException Locked="false" Priority="39" Name="toc 4"/> <w:LsdException Locked="false" Priority="39" Name="toc 5"/> <w:LsdException Locked="false" Priority="39" Name="toc 6"/> <w:LsdException Locked="false" Priority="39" Name="toc 7"/> <w:LsdException Locked="false" Priority="39" Name="toc 8"/> <w:LsdException Locked="false" Priority="39" Name="toc 9"/> <w:LsdException Locked="false" Priority="35" QFormat="true" Name="caption"/> <w:LsdException Locked="false" Priority="10" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Title"/> <w:LsdException Locked="false" Priority="1" Name="Default Paragraph Font"/> <w:LsdException Locked="false" Priority="11" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Subtitle"/> <w:LsdException Locked="false" Priority="22" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Strong"/> <w:LsdException Locked="false" Priority="20" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Emphasis"/> <w:LsdException Locked="false" Priority="59" SemiHidden="false"
UnhideWhenUsed="false" Name="Table Grid"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Placeholder Text"/> <w:LsdException Locked="false" Priority="1" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="No Spacing"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 1"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 1"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 1"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 1"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 1"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Revision"/> <w:LsdException Locked="false" Priority="34" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="List Paragraph"/> <w:LsdException Locked="false" Priority="29" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Quote"/> <w:LsdException Locked="false" Priority="30" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Intense Quote"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 1"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 1"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 1"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 1"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 1"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 1"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 1"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 2"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 2"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 2"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 2"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 2"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 2"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 2"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 2"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 2"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 2"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 2"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 3"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 3"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 3"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 3"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 3"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 3"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 3"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 3"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 3"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 3"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 3"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 3"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 3"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 4"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 4"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 4"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 4"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 4"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 4"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 4"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 4"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 4"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 4"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 4"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 4"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 4"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 4"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 5"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 5"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 5"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 5"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 5"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 5"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 5"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 5"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 5"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 5"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 5"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 5"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 5"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 5"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 6"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 6"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 6"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 6"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 6"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 6"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 6"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 6"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 6"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 6"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/> <w:LsdException Locked="false" Priority="19" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/> <w:LsdException Locked="false" Priority="21" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/> <w:LsdException Locked="false" Priority="31" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/> <w:LsdException Locked="false" Priority="32" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/> <w:LsdException Locked="false" Priority="33" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Book Title"/> <w:LsdException Locked="false" Priority="37" Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> </w:LatentStyles> </xml><![endif]--><!--[if gte mso 10]> <style>
/* Style Definitions */
table.MsoNormalTable
{mso-style-name:"Table Normal";
mso-tstyle-rowband-size:0;
mso-tstyle-colband-size:0;
mso-style-noshow:yes;
mso-style-priority:99;
mso-style-qformat:yes;
mso-style-parent:"";
mso-padding-alt:0cm 5.4pt 0cm 5.4pt;
mso-para-margin-top:0cm;
mso-para-margin-right:0cm;
mso-para-margin-bottom:10.0pt;
mso-para-margin-left:0cm;
line-height:115%;
mso-pagination:widow-orphan;
font-size:11.0pt;
font-family:"Calibri","sans-serif";
mso-ascii-font-family:Calibri;
mso-ascii-theme-font:minor-latin;
mso-fareast-font-family:"Times New Roman";
mso-fareast-theme-font:minor-fareast;
mso-hansi-font-family:Calibri;
mso-hansi-theme-font:minor-latin;}
</style> <![endif]--> <br />
<div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">Usually, ‘reversed products’ are not the same:</span></div><div class="MsoNormal"><br />
</div></div><div class="MsoNormal"><span lang="EN-US">Example 3:</span></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-ySB-qPDwVgk/TzdKvS6kg-I/AAAAAAAAC7U/gzAER2DGdYg/s1600/UP3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-ySB-qPDwVgk/TzdKvS6kg-I/AAAAAAAAC7U/gzAER2DGdYg/s1600/UP3.jpg" /></a></div><br />
</div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><span lang="EN-US">What distinguishes the identical products from the non-identical products? Do the factors offer any clues?</span></div><div class="MsoNormal"><span lang="EN-US">Here is another look at Example 1:</span></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-mHA0unX6YgU/TzdKpX-gOqI/AAAAAAAAC7M/bK9VT7r0HQI/s1600/UP2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br />
</a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-MUsU5cxkeyA/TzdK1rVngZI/AAAAAAAAC7c/dQyWpSdmYmw/s1600/UP4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-MUsU5cxkeyA/TzdK1rVngZI/AAAAAAAAC7c/dQyWpSdmYmw/s1600/UP4.jpg" /></a></div><br />
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</style> <![endif]--> <div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">Note that 2 × 3 = 6 × 1, or in other words the product of the Units is the same as the product of the Tens. <span style="mso-spacerun: yes;"></span>When we do long multiplication our first ‘sub-product ‘(or should that be sub-total) is <span style="mso-spacerun: yes;"> </span>U × U, followed by <span style="mso-spacerun: yes;"> </span> U × T, then T × U and finally T × T. Therefore, when we ‘reflect’ 13 and 62 all of the ‘sub-products’ stay the same, although the middle pair U × T and T × U are in a different order.</span></div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">For Secondary Students:</span></div><div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">We can use algebra to give the general pattern.<span style="mso-spacerun: yes;"> </span>Let A, B, C and D represent any of the digits from 0 to 9.<span style="mso-spacerun: yes;"> </span>Please note that AB<span style="mso-spacerun: yes;"> </span>and CD are not products but concatenations (or strings). That is, in polynomial notation, AB would be written as 10 A + 1 B and likewise CD would be 10 C + 1 D.</span></div><div class="MsoNormal"><br />
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<div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">Extending the Pattern</span></div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><span lang="EN-US" style="mso-ansi-language: EN-US;">Now that we have some understanding of the pattern, it is natural to ask if it can be extended to three digit numbers.<span style="mso-spacerun: yes;"> </span>Have a go!</span></div></div><div class="MsoNormal"><br />
</div>Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0tag:blogger.com,1999:blog-2853830991106309117.post-23368518712047316562012-01-24T04:17:00.000-08:002012-01-24T04:17:07.016-08:00Do We Ever Use The Quadratic Formula?To the perennial cry of the high school mathematics student, the teacher writes:<br />
<br />
x=(-b±√(b^2-4ac))/2a<br />
<br />
The cry is of course, “When will we ever use this?” sometimes restated as, “We aint never ever gonna need this!”<br />
Maths is not real. It’s all imaginary. If you are a realist, you will never use the quadratic formula. But if you do use the quadratic formula, chances are you will use it in situations like this one.<br />
<br />
<i>A circle of radius 10 exists. Each of four congruent smaller circles are tangent to the outer circle and to same-sized neighbours. The smallest circle has the same centre as the large circle and is tangent to all four intermediate-sized circles. What is the radius of the smallest circle? See diagram. (Hint: you might want to use the quadratic formula.)</i><br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-wsTgt-ZKR-Q/Tx6fBI0XRgI/AAAAAAAAC6s/eMzqSOFYxQA/s1600/p1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-wsTgt-ZKR-Q/Tx6fBI0XRgI/AAAAAAAAC6s/eMzqSOFYxQA/s1600/p1.jpg" /></a></div><br />
<br />
A circle is not a parabola, so we appear to be a long way from the quadratic formula. However, the formula of both is second degree. A ‘plain vanilla’ circle is <span lang="EN-US">x</span><span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + <span lang="EN-US">y</span><span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span><span style="font-size: small;"><span lang="EN-US" style="line-height: 115%;"></span></span> = <span lang="EN-US">r</span><span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>. Likewise, the most basic parabola is y = <span lang="EN-US">x</span><span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>. The teacher writes both of these formulas on the board.<br />
<br />
Johnny and Mary discuss the problem. Mary notes the four intermediate circles create 4-fold symmetry. Johnny wants to use Cartesian coordinates. Together they realise the lines of symmetry must be the x-axis, the y-axis, the line y = x and the line y = -x. The teacher agrees and suggests that the use the symmetry to reduce the problem to the first quadrant. He also asks, “What is the distance from O to A?” They redraw the problem as folows.<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-8Ip6ySSoRoI/Tx6fNm_yg1I/AAAAAAAAC60/zwNm3eaDR6o/s1600/p2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-8Ip6ySSoRoI/Tx6fNm_yg1I/AAAAAAAAC60/zwNm3eaDR6o/s320/p2.jpg" width="318" /></a></div><br />
<br />
Johnny says the formula of the circle reminds him of Pythagoras: a<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + <span lang="EN-US">b</span><span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>= <span lang="EN-US">c</span><span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>. Mary suggests the distance formula, D = √ [(x2 – x1)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + (y2 – y1)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>] is likely to be important. Another student in the group says he prefers to write D2 = (x2 – x1)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + (y2 – y1)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>, thus avoiding the square root sign. <br />
The origin is (0, 0). Point A is on the line y =x, so A = (xA, yA) = (xA, xA). The distance from the origin to point A is 10, so D<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> = 100. Thus (x2 – x1)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + (y2 – y1)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> becomes (xA – 0<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + (y2 –0)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> = (xA – 0)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>+ (xA –0)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>. Therefore 100 = 2xA<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>. <br />
xA = √50 = 5√2.<br />
What are the coordinates of B?<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-SzIa5mxoxW0/Tx6fiMsyxEI/AAAAAAAAC68/MbhxGjZT_gw/s1600/p3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-SzIa5mxoxW0/Tx6fiMsyxEI/AAAAAAAAC68/MbhxGjZT_gw/s320/p3.jpg" width="316" /></a></div><br />
Just like point A, B is also on the line y = x. Consequently, B = (xB, yB) = (xB, xB) = (x, x) for ease of writing. The square of the distancefrom A to B is DAB<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> = (5√2 – x)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + (5√2 – x)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>. The square of the distancefrom BxB to B is D2 = (x – x)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + (0 – x)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>. (Not a misprint, remember y = x). Since AB and BxB (or ByB) are both radii, they are the same distance.<br />
Therefore, D<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> = DAB<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span><br />
(x– x)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + (0 – x)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> = (5√2 – x)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span>+ (5√2 – x)<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span><br />
0<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> + x<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> = (50 – 10√2x + x<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> ) + (50 – 10√2x + x<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> )<br />
x<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> = 100 – 20√2x + 2x<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> <br />
0 = 100 – 20√2x + x<span style="font-size: small;"><sup><span lang="EN-US" style="line-height: 115%;">2</span></sup></span> <br />
At last we are ready to apply the quadratic formula, with a = 1, b =–20√2 and c = 100. Then the required radius, OA<b><span style="font-family: Arial,Helvetica,sans-serif;">’</span></b> = 10 – 2x = <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-GB</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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mso-tstyle-colband-size:0;
mso-style-noshow:yes;
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mso-style-qformat:yes;
mso-style-parent:"";
mso-padding-alt:0cm 5.4pt 0cm 5.4pt;
mso-para-margin-top:0cm;
mso-para-margin-right:0cm;
mso-para-margin-bottom:10.0pt;
mso-para-margin-left:0cm;
line-height:115%;
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font-family:"Calibri","sans-serif";
mso-ascii-font-family:Calibri;
mso-ascii-theme-font:minor-latin;
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</style> <![endif]--><span style="font-size: small;"><span style="font-family: "Times New Roman","serif";">1.71572875</span></span>…(Google calculator)<br />
Whereas, a realist would have made a scale drawing and found OA’ = 1.7 to 2 significant figures.<br />
Who uses the quadratic formula? Dreamers and idealists. Don’t be too dismissive George Bernard Shaw wrote:<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-GB</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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<div class="MsoNormal" style="margin-bottom: 10.0pt; margin-left: 63.8pt; margin-right: 40.2pt; margin-top: 0cm;"><span lang="EN-MY" style="font-family: "Comic Sans MS"; mso-bidi-font-family: Arial;">The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man. </span></div>Of course there are a few loose ends. It is justified to reduce the problem to a single quadrant? Further, is it justified to insist that y = x is the axis of symmetry for the reduced problem?<br />
But these are thoughts for another day.Chris Pattersonhttp://www.blogger.com/profile/06280885648784664003noreply@blogger.com0