tag:blogger.com,1999:blog-53033074821589225652018-11-01T21:50:52.895-07:00Math Mama Writes...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger560125tag:blogger.com,1999:blog-5303307482158922565.post-31119513984073600172018-09-08T16:17:00.002-07:002018-10-04T19:46:13.164-07:00Geometric Construction of the Regular Pentagon<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-oYXNhVqh9us/W4ywV5RDpEI/AAAAAAAAEnc/FMxwu0EqB-k8HFhzUjIQkZE0Fy27evQXgCLcBGAs/s1600/1st%2Bpentagon.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="721" data-original-width="985" height="292" src="https://2.bp.blogspot.com/-oYXNhVqh9us/W4ywV5RDpEI/AAAAAAAAEnc/FMxwu0EqB-k8HFhzUjIQkZE0Fy27evQXgCLcBGAs/s400/1st%2Bpentagon.png" width="400" /></a></div><br />I never particularly enjoyed geometric constructions when I was younger. It may be because I had a tendency to press down too hard, and mess up the width my compass was set to. Like so many things, technology has a fix for that.<br /><br />You can do geometric constructions on a number of sites, and in <a href="https://www.geogebra.org/" target="_blank">geogebra</a>. My favorite site, which <a href="https://mathmamawrites.blogspot.com/2013/07/euclidean-construction-science-vs-magic.html" target="_blank">I discovered about 5 years ago</a>, is <a href="http://sciencevsmagic.net/">sciencevsmagic.net</a>. I having been telling students about it for years, and decided this summer to play with it again, to see how much I still like it. (A lot, it turns out.)<br /><br />There are 40 challenges, in sets of 4. This time around I got every shape but the pentagon. I asked for hints (from math friends on facebook) which I then avoided reading because I wanted to do it on my own. I got nowhere that way, and finally followed instructions for how to construct it. Every other shape I have constructed makes sense on its own, but the construction of a regular pentagon must be proved with algebra. Ok, I don't feel so bad about 'cheating'.<br /><br />I kept constructing it over and over, as I tried to understand it better. The first time (which you see above) I used <a href="http://www.opentutorial.com/How-to-Construct-a-Pentagon/construction" target="_blank">these instructions</a> (which I found by following links from wikipedia) and constructed it at sciencevsmagic, achieving my final shape challenge. Then I did it again using geogebra. I needed to figure out how it worked, and was at a loss. <a href="http://www.cut-the-knot.org/pythagoras/pentagon.shtml" target="_blank">Cut-the-knot</a>, a site I've gone to many times with my math questions, had a different construction - which I followed on both sciencevsmagic and geogebra - and an explanation (by Scott Brodie) that I kept working through as I wrote this.<br /><br />[Sadly, Alexander Bogomolny, the creator of the cut-the-knot site, has died. I hope the math blogging community can find a way to maintain his amazing site. We miss you, Alexander.]<br /><br />I will explain this as I understand it. My explanation will mostly just restate what I learned from these sites. But it may help some people, since some of the reasoning steps in Scott's explanation were hard for me to follow.<br /><br />The constructions are worth doing first, so you can get a feel for what's happening as you do it. But when you're done, the question is still open - is this <i>really</i> a regular pentagon? The following explanation proceeds in two major steps. First it looks at relationships in the regular pentagon and the pentagram (which is a regular pentagon with a five-sided star inside it). Then it looks at the relationships built by the construction method, and we finally see that the side length given in the two ways is the same. Bingo!<br /><br /><br /><br /><br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-leVPM5Oo39Y/W49YYua6XFI/AAAAAAAAEoM/iAakg5reoMYIvOFTrUdzP7a_KvtsFc5kACLcBGAs/s1600/pentagon%2Bv2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="696" data-original-width="709" height="314" src="https://3.bp.blogspot.com/-leVPM5Oo39Y/W49YYua6XFI/AAAAAAAAEoM/iAakg5reoMYIvOFTrUdzP7a_KvtsFc5kACLcBGAs/s320/pentagon%2Bv2.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><b>The Construction</b><br />If you're ready to follow this, I trust that you already know how to construct the perpendicular bisector of a line segment. I will leave out the construction details for each of those.<br /><ul><li>Starting with a circle with center O, construct a horizontal diameter, AB.</li><li>Now construct its perpendicular bisector, CD.</li><li>Construct the perpendicular bisector of AO, intersecting it at E. </li><li>Construct the circle with center at E, through C, labeling its intersection with AB as F.</li><li>Construct the circle with center at C, through F, labeling its intersection with the original circle (above B) as G.</li><li>Go around the original circle, constructing this same size circle from each new point. (Center at G, through C, gives H at the other intersection with the original circle. Etc.)</li></ul>The question remains: Does this really construct a regular pentagon, or is it maybe just pretty close?<br /><br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Blx1DUz_W-Q/W49ZLf3UsPI/AAAAAAAAEoU/wSKrp1CNzqocUpUTfxEqgN2ZX1oXaeMHwCLcBGAs/s1600/pentagram%2Bv2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="700" data-original-width="692" height="400" src="https://4.bp.blogspot.com/-Blx1DUz_W-Q/W49ZLf3UsPI/AAAAAAAAEoU/wSKrp1CNzqocUpUTfxEqgN2ZX1oXaeMHwCLcBGAs/s400/pentagram%2Bv2.png" width="395" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><br /><b>Relationships in a regular pentagram</b><br /><ul><li>The angles in a regular pentagon are each 108<span class="ILfuVd yZ8quc">°, so each external angle, like </span>∠HMN, is 72<span class="ILfuVd yZ8quc">°. That makes each angle at a point, like </span><span class="ILfuVd yZ8quc">∠NHM, 36</span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°. Since </span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠HMG is 108</span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°, </span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠MHG and </span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠MGH must each be 36</span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°. </span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"> Since angles </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠GLH and </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">∠LGH are each 72</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">°</span></span>, </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Δ</span></span></span></span></span></span></span></span></span></span></span></span></span>HGL has two equal sides, HL and HG (which also equal CM, HI, etc).</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">ΔCHI is similar to </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Δ</span></span></span></span></span></span></span></span></span></span></span></span></span>IHM. So corresponding sides are in proportion. We get CH/HI=HI/HM, giving us HI<sup>2</sup> = CH*HM = CH(CH-CM) = CH(CH-HI).</span></span></span></span></span></span></span></span></span></span></span></span></span></li><li><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">Let CH = x*HI. Then the equation above gives us </span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">HI<sup>2</sup> =x*HI(x*HI-HI), which gives us a (perhaps) familiar equation: 1 = x</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>-x, whose solution (in this case, where x is clearly greater than 1) is x = (1+</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2 aka φ. (At this point, I get excited. φ, aka the golden ratio, shows up in such different contexts!)</li><li>So CH = φ*HI, and (similar triangles) HI = φ*HM and HM = φ*MN.</li><li>We now have relationships between all the sides of the pentagon, star, and line segments on the star, but these are not yet connected to the radius of the circle.</li><li>Now consider ΔODH. Two sides are radii, and the angle at O is 36°. Another triangle similar to all the others we've found. So OD = φ*DH.</li><li>Also note that ∠DHC (inscribed angle on the diameter) is a right angle.</li><li>Let's switch to simple variables names now. Let r=OD, s=CG, t=DH, and d=CI=CH.</li><li>Then d= φ*s, r= φ*t, and t<sup>2</sup>+d<sup>2</sup> = (2r)<sup>2</sup>, giving (r/φ)<sup>2</sup> +(φ*s)<sup>2</sup> = 4r<sup>2</sup>.</li><li>This gives us s<sup>2</sup> = 1/φ<sup>2</sup>*(4r<sup>2</sup> - (r/φ)<sup>2</sup>) = (4/φ<sup>2</sup> - 1/φ<sup>4</sup>)*r<sup>2</sup>. </li></ul><br /><br /><br /><b>Analyzing our construction</b><br /><ul><li>We will find the length of s = CG = CF.</li><li>OE=1/2*r. And EC = <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√(OE</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>+OC</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>) = </code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√(5/4)*r = </code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2*r = EF. OF = (<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2- 1/2)*r.</li><li>So <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">s<sup>2</sup> = </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>CG<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = CF<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = OC<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> + OF<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><sup>2</sup></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> = (1+(<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>/2- 1/2)<code><sup>2</sup>)*r<sup>2</sup>.</code><code></code></li><li>We have two very messy expressions for s<sup>2</sup>, one from the regular pentagon/pentagram and one from our construction. Are they equal? It may help to write out positive and negative powers of <span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc">φ </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>to help simplify the first expression. Yes! They are both equal to (5-<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2*r<sup>2</sup>, making the side length of the regular pentagon inscribed in a circle of radius 1 equal the square root of (5-<span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><span class="ILfuVd yZ8quc"><code class="selectable">√5</code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></code></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>)/2.</li><li>And that proves that our construction created a regular pentagon. (Whew!) </li></ul><br /><br />I've completed 32 of the 40 challenges at <a href="https://sciencevsmagic.net/geo/" target="_blank">sciencevsmagic</a>. I still haven't figured out how to make some of the shapes within the original circle, and still haven't found the least moves for some of the shapes. I'm so glad there's still something to work toward the next time I revisit this.<br /><br /><br /><br />[Blogger doesn't deal well with superscripts and square roots. Once again, I am noticing that I ought to learn some LaTex. Sigh.] <br /><ul></ul><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-29660198845747253032018-07-01T18:20:00.000-07:002018-07-01T18:20:23.674-07:00Math Teachers at Play #118<br /><a href="https://2.bp.blogspot.com/-teS_QhiW1AI/Wzl15QPAiEI/AAAAAAAAEWw/8Iq6MsiunOQyBUY6MtTgB1orslYj3CtegCLcBGAs/s1600/118%2Bv4.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="225" data-original-width="225" height="320" src="https://2.bp.blogspot.com/-teS_QhiW1AI/Wzl15QPAiEI/AAAAAAAAEWw/8Iq6MsiunOQyBUY6MtTgB1orslYj3CtegCLcBGAs/s320/118%2Bv4.jpg" width="320" /></a>In two more months, we'll hit MTAP #120, which <i>should</i> be the ten-year anniversary mark. Is it?<br /><br /><br /><br />The number 118 ...<br />... factors to 59*2.<br />... is 1110110 in base two, 11101 in base three, 1312 in base four, 433 in base five, and 226 in base 7.<br />Not particularly exciting...<br /><br /><br /><br />Aha! Can you make 118 using four 4's? (I did.) I wonder if you can make it using five 5s, or ...? <br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><br /><b>Summertime Learning</b><br /><a href="https://www.mathmammoth.com/lessons/summer_math.php" target="_blank">Summer Math Resources</a> from Math Mammoth's Maria Miller. <br /><br />Denise Gaskins offers us one of her FAQs, on <a href="https://denisegaskins.com/2018/06/15/faq-forgetting-what-they-learned/" target="_blank">forgetting what they learned</a>. My favorite of all her suggestions? Play games! <br /><br />On <a href="https://emergentmath.com/2018/06/25/routines-the-driving-beat-of-your-class/" target="_blank">Routines</a> and <a href="https://emergentmath.com/2018/06/28/lessons-the-stuff-were-used-to-but-better/" target="_blank">Lessons</a> <br />I like Geoff's perspective: "What’s the shortest amount of time you could possibly do the talking? Go with that. And maybe subtract a few more minutes." After 30 years of teaching, I am still learning, and I still get excited when I see something that might make a difference for my students. This looks helpful, and perfect for my summer meditations on teaching.<br /><br /><br /><br /><b>Young Math </b><br />Can you imagine doing a college math lesson with 2nd graders, and having it work out well? Manan Shah did it! <a href="http://mathmisery.com/wp/2018/06/14/set-theory-for-second-grade/" target="_blank">His lesson was on set theory.</a><br /><a href="https://www.mathmammoth.com/lessons/summer_math.php" target="_blank"><br /></a>Ahh... A whole blog on magical math books. And she wrote <a href="https://mathbookmagic.com/2018/06/18/uncovering-the-complexities-of-counting-with-a-magical-counting-book/" target="_blank">about Christopher Danielson's new book, How Many?</a> If you like this, she also has <a href="https://mathbookmagic.com/blog-book-list/" target="_blank">a list of every book she has posted about</a>. Thank you, Kelly Darke!<br /><br />Remember playing the card game War? If you're a parent, those memories may not be so fond. Kids love it and parents can get sooo bored. Kent wrote <a href="https://www.gamesforyoungminds.com/blog/war-card-game" target="_blank">a great post about his interactions with his son around Addition, Subtraction, and Multiplication War</a>. (If you want a good summary of even more mathy variations on war, <a href="https://denisegaskins.com/2006/12/29/the-game-that-is-worth-1000-worksheets/" target="_blank">check out Denise Gaskins' classic post.</a>)<br /><br /><a href="https://medium.com/@jamestanton/the-global-math-project-uplifting-mathematics-for-all-9253d383e5e6" target="_blank">The first ever Global Math Week</a> went well, with its exploding dots exploding around the world. Thank you, James Tanton, for getting people all over the globe into math.<br /><br />How do you talk about numbers with young children? There are so many ways! Here's another:<a href="http://realteachingmeansreallearning.blogspot.com/2018/06/dice-chats.html" target="_blank"> Counting with Dice</a>, from Dave Martin. <br /><br /><br /><br /><br /><a href="https://3.bp.blogspot.com/-cZfAlzi1CUs/Wzl15qHyafI/AAAAAAAAEXI/lObvsKEgyEUFA1zUBDilQt9NOkawImIeQCEwYBhgL/s1600/118%2Bv3.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="768" height="180" src="https://3.bp.blogspot.com/-cZfAlzi1CUs/Wzl15qHyafI/AAAAAAAAEXI/lObvsKEgyEUFA1zUBDilQt9NOkawImIeQCEwYBhgL/s320/118%2Bv3.jpg" width="320" /></a><b>Calculus</b><br />Which is bigger, e<sup><span class="st">π</span></sup> or <span class="st">π<sup>e</sup>?</span> Sure, you can check it on a calculator. Or <a href="http://blog.mrwaddell.net/archives/1835" target="_blank">you can use areas to see why it's true</a>. Lovely! (from Glenn Waddell)<br /><br />The Fundamental Theorem of calculus says that you can figure out areas by using anti-derivatives. I do a project to help students understand it. Sam Shah does even more. <a href="https://samjshah.com/2018/06/21/part-iii-the-final-turn/" target="_blank">This is Part III of his lessons for this topic.</a> I recommend clicking on the links to Parts I and II first.<br /><span style="background-color: #ead1dc;"><br /></span><br /><br /><b><br /></b><b><br /></b><b>A Few More Goodies...</b><br />Dan McKinnon shares his notes from his <a href="http://www.mathrecreation.com/2018/06/origami-workshop-again.html" target="_blank">Origami Workshop</a>. For those who enjoy doing origami, can you find something new here? <br /><br />Logic and Math go together so well. Check out this <a href="https://mathsvenns.com/nearest-integer-one-decimal-place-two-decimal-places/" target="_blank">blog full of Venn Diagrams to fill in</a>.<br /><br />In his post <a href="https://pballew.blogspot.com/2018/06/from-surds-to-ab-surds.html" target="_blank">From Surds to Ab-surds</a>, Pat Bellew looks at an interesting relationship, which looks like bad simplifying but is still correct. He suggests it as a challenge in an algebra course to produce more correct 'bad simplifying' equations. Hmm, I want to think about how to do that.<br /><br /><br /><br /><br /><b>Until Next Month...</b><br />Our sister blog carnival, <a href="http://aperiodical.com/2018/06/carnival-of-mathematics-158/" target="_blank">The Carnival of Mathematics</a>, often includes posts that are above my head. This month, our big sister is quite approachable. Enjoy!<br /><br />If you have suggestions for next month's MTAP, share via the form at the <a href="https://denisegaskins.com/mtap/" target="_blank">Carnival home page</a>. Sharing in the carnival, or hosting, is a great way to increase connections in the #mtbos/#iteachmath community.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-31276521474713485332018-01-07T22:00:00.001-08:002018-01-07T22:00:40.043-08:00Logic Puzzle - What Does Your Friend See?<div><div><div><div><div>I love logic puzzles, and was drawn by <a href="http://nautil.us//blog/the-logic-puzzle-you-can-only-solve-with-your-brightest-friend" target="_blank">the title saying this was a hard one</a>. Usually Nautilus is well-written, but their version of this puzzle isn't as good (in my opinion) as <a href="https://mindyourdecisions.com/blog/2016/03/20/the-seemingly-impossible-escape-sunday-puzzle/" target="_blank">the original, blogged about by Presh Talwalkar</a>.</div><div> </div><div>The Nautilus version of the puzzle says to imagine your brightest friend. I imagined a friend I know likes logic puzzles (Sharon), but since I wanted a different initial than mine for notation purposes, I imagined another smart friend who <i>might</i> like logic puzzles (Linda). And I began scribbling away with S's (for Sue) and L's.<br /></div>The answer I got is different than the answer the author got because we made different assumptions. Mine were based on his wording, his were based on Presh's wording.</div><div><br /></div><b>Puzzle #1 </b>(with Sue's interpretation):</div>You’ve been caught snooping around a spooky graveyard with your best friend. The caretaker, a bored old man fond of riddles (and not so fond of trespassers), imprisons each of you in a different room inside the storage shed, and, taking your phones, says, “Only your mind can set you free.” <br /><br />To you, he gestures toward a barred window. Through it, you can see 12 statues. Out of your friend’s window, which overlooks the opposite side of the graveyard, she can see eight. Neither of you know the other’s count.The caretaker tells you each, individually, that together you can see either 18 or 20 statues. Unfortunately, there’s no way to tell your friend how many you can spot. <br /><br />The only way for you both to escape is for one of you to give the total number of visible statues. Get it wrong, and neither of you ever leave. The caretaker asks you each once a day [Sue assumed neither person knew who was asked first], and you can choose to answer or to pass. If you both pass on a given day, the question—are there 18 or 20?—is posed to each of you again the next day, and the next, and so on, until you get it right or wrong.<br /></div><b>Puzzle #2</b> <a href="https://mindyourdecisions.com/blog/2016/03/20/the-seemingly-impossible-escape-sunday-puzzle/" target="_blank">is at Presh's blog</a>. (No way to copy-paste that one.) It's now Alice and Bob, and they know that Alice gets asked first, so they'd both be released before Bob is asked if she has it right.<br /></div>I think I have found a solution to (my version of) Puzzle #1. I would love to hear other folks reasoning before posting anything. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-57196971611274875702017-12-31T21:33:00.003-08:002017-12-31T22:27:21.455-08:00Arithmetic, by Paul Lockhart<div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;"><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-hXXBQJafQOw/WknIG70g7lI/AAAAAAAAD78/Rta2KpJ6PWAB3zn2mUdmjKeV2DyerhPcwCLcBGAs/s1600/arithmetic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="192" data-original-width="128" src="https://4.bp.blogspot.com/-hXXBQJafQOw/WknIG70g7lI/AAAAAAAAD78/Rta2KpJ6PWAB3zn2mUdmjKeV2DyerhPcwCLcBGAs/s1600/arithmetic.jpg" /></a></div><br /></div><div style="margin-bottom: 0; margin-top: 0;"><br /><br />This book is part history and part philosophy of arithmetic. He also includes a few exercises in italics (one every few pages). </div><div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;">I love his easygoing tone (arithmetic is not mathematics, it's an art or craft, something like knitting, I enjoy it, and I hope you will), his low-key sense of humor, and his perspective.</div><div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;">I loved his book <i>Measurement</i>, and look forward to someday working through all the challenging problems in it. This one is much easier, and yet it's not boring for me.</div><div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;">It would be a lovely book to read to kids and think through together.</div><div style="margin-bottom: 0; margin-top: 0;"><br /></div><div style="margin-bottom: 0; margin-top: 0;">(It came out this year in hardcover. $22.95. I love the cover, though I know I shouldn't judge a book by that...)</div><div style="margin-bottom: 0; margin-top: 0;"><br /><br /><br /><br />[Edited to add: <br /><div style="margin-bottom: 0; margin-top: 0;">I wrote this review before finishing the book, because I was so excited. I've read a few more pages now. Sadly, Lockhart is sometimes sexist. On page 45 he mentions the (Japanese) emperor's concubines, and says "Now, <i>this</i> is why people do arithmetic!" (To please the concubines.) No. It's not. And I thought better of you, Paul Lockhart. </div><div style="margin-bottom: 0; margin-top: 0;"><br /></div>I still think I'm going to love 99% of this book...]</div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-89776316923837675272017-11-04T12:27:00.000-07:002017-11-04T12:28:14.288-07:00Re-reading Archimedes CodexI am re-reading <a href="https://www.bookfinder.com/search/?ac=sl&st=sl&ref=bf_s2_a1_t1_1&qi=29KqANHAecX,oVkeOt4wCqLVadg_1497963026_1:6:5&bq=author%3Dreviel%2520netz%26title%3Darchimedes%2520codex%2520how%2520a%2520medieval%2520prayer%2520book%2520is%2520revealing%2520the%2520true%2520genius%2520of%2520antiquity%2527s%2520greatest%2520scientist" target="_blank"><i>The Archimedes Codex.</i></a> It is fabulous. A detective story of history, science, and math. (I wrote <a href="http://mathmamawrites.blogspot.com/2015/06/book-review-archmedes-codex.html" target="_blank">my first review of it two years ago</a>.)<br /><br />Both authors have a sweet nerdy guy sense of humor, gently self-deprecating, piercing when it should be.<br /><blockquote class="tr_bq">"What are readers today [of science] afraid of? They are afraid of equations. With good reason: they were force-fed such equations for several, terrible years of their childhood and adolescence." </blockquote><br /><div class="text_exposed_show">I actually think he (Netz) is wrong here. I love math, and I love the beauty of some equations, but equations are still intimidating when you don't yet see what they're saying.<br /><br />Here's another math-related quote, pointing in a different direction: <br /><blockquote class="tr_bq">"Archimedes wrote out this problem in verse! A poet-mathematician! - the thought seems to us absurd, but it was natural for Archimedes, whose entire science was built on a sense of play and beauty, on hidden meanings."</blockquote>Not surprisingly, I disagree with him here too. <a href="http://mathmamawrites.blogspot.com/2010/02/math-poetry-wiki.html" target="_blank">Some of us</a> like <a href="http://mathmamawrites.blogspot.com/2009/03/math-poems.html" target="_blank">mixing poetry and math</a>. And the problem written up this way was a silly one. Archimedes was kind of like Martin Gardner with a few of his problems, and it makes sense he'd be playful in his presentation of it. Kind of like <a href="http://mathmamawrites.blogspot.com/2009/12/holiday-logic-puzzles.html" target="_blank">this one by Mike Shenk</a> (sorry for the xmas reference this early).<br /><br /></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-60303062256520126332017-08-14T21:38:00.000-07:002017-08-14T21:47:18.356-07:00First Day, Again<br />First day of classes. I was not as excited as usual. But I had my prep done, and once I got in the classroom, I loved it like always.<br /><br />25 people in beginning algebra. I tell them how math is not about memorizing but about making sense. I get them talking in groups about the first time math didn’t make sense and got past them.<br /><br />I ask them to estimate the percentage of the population that’s uncomfortable with math. First one says 100%. I almost laugh, but manage not to. Percentages all over, 32% up to another at 95%. I tell them I don’t know either but I guess 70 to 80%. So that means most elementary teachers are nervous when they teach fractions, and then they pass it on. I think I see a few nods.<br /><br />I ask how to make meaning of 3 -5. Someone says you go past 0. I say “You’re talking about a number line, right?” And I draw it. I say I like that, but how can we give some real world meaning to this problem. Someone else says “Debt”, but I hear “Death” at first. I shake my head at my bad hearing. (I hope it doesn’t interfere with my teaching.) And I flesh it out. “Yeah, you’ve got $3 in your pocket. And you want to buy a $5 something. What does the 3 -5 tell you?” A student says, “How much you need to borrow.” I add in the temperature model, which I tell them might have worked better for my students back in Michigan who had experienced 3 degrees, and going down 5 degrees from there.<br /><br />Then I give them 31 – 52. They discuss in their groups. It’s easier than I meant it to be.<br /><br />We discuss the syllabus in between other things. I give them a sheet full of magic squares that use negative numbers. Some use fractions. I’ll need to check in tomorrow to see how they did with those.<br /><br />I think the class went well. If they really feel good about it, they’ll end up thinking I’m their best teacher ever. That only happens about a third of the time with this course. I can hope…<br /><br />Then I had to run to my next class. Statistics. I had them average the ages of 5 other students, and type their average onto my computer. I averaged their averages. Which is not the same as just averaging everyone. But it often comes close. Then I put a number line up, and we each put a dot at our age. That’s a dot plot. I showed how it’s skewed right. And talked about how median is a better way to show the center than mean (which is the average they had done).<br /><br />I had an hour to eat lunch and chill.<br /><br />Then I taught calculus. I love that class! Every time. Draw y = x squared. Draw a tangent line at x=3. Estimate the slope of the tangent line. What makes this different from algebra is that we need the idea of infinity here. Ahh… Happy me. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-60599483224795629622017-06-30T22:12:00.000-07:002017-07-01T15:43:31.988-07:00Math Teachers at Play #109 (a blog carnival)<span style="font-size: small;"><b>Who is 109?</b> 109 is a twin prime, twinned with 107. <span style="font-size: x-small;">(from numbergossip.com)</span></span><br /><br /><br /><span style="font-size: small;"><br /></span><br /><a href="https://2.bp.blogspot.com/-dP-LUEneJY4/WVKXWU4NMWI/AAAAAAAADck/7eib_zjkWHUSgC5jJz1_bA-xeJVmxS4IACLcBGAs/s1600/pipe%2Borgan.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="1234" data-original-width="1600" height="246" src="https://2.bp.blogspot.com/-dP-LUEneJY4/WVKXWU4NMWI/AAAAAAAADck/7eib_zjkWHUSgC5jJz1_bA-xeJVmxS4IACLcBGAs/s320/pipe%2Borgan.jpg" width="320" /></a><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/949.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="949" /></a> If 109 is written in Roman notation (CIX), then it becomes reflectable along the line it is written on. </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/949.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="949" /></a> The pipe organ at the Cathedral of Notre Dame in Paris has 109 stops. </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/950.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="950" /></a> When chilled below minus 109°F, CO<sub>2</sub> becomes a solid, called dry ice. </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/951.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="951" /></a> 109 equals the square root of 11881 or 118 - 8 - 1. </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/15936.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="15936" /></a> The only three-digit prime formed by concatenation of consecutive numbers. [<a href="https://primes.utm.edu/curios/ByOne.php?submitter=Silva">Silva</a>] </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/18156.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="18156" /></a> 109 = 1*2+3*4+5*6+7*8+9. [<a href="https://primes.utm.edu/curios/ByOne.php?submitter=Silva">Silva</a>] </span><br /><span style="font-size: small;"><a href="https://primes.utm.edu/curios/cpage/12541.html"><img alt="+" src="https://primes.utm.edu/gifs/check.gif" title="12541" /></a> The Sun is just over 109 times the diameter of the Earth. [<a href="https://primes.utm.edu/curios/ByOne.php?submitter=Friedman">Friedman</a>] </span><br /><span style="font-size: x-small;"> (from https://primes.utm.edu/curios/page.php/109.html)</span><br /><span style="font-size: small;"><br /></span><span style="font-size: small;"> </span><br /><br /><span style="font-size: small;"><b>A Puzzle</b>: Can you make 109 from four 4's? (I don't promise that it's possible...)</span><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-I91iL-YpHOM/WVKYIv7WLlI/AAAAAAAADcs/09NTs37-iHk1eHOlnkyE9oozhllTLBaCgCLcBGAs/s1600/gggg%2Bgranny.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="624" data-original-width="1024" height="195" src="https://4.bp.blogspot.com/-I91iL-YpHOM/WVKYIv7WLlI/AAAAAAAADcs/09NTs37-iHk1eHOlnkyE9oozhllTLBaCgCLcBGAs/s320/gggg%2Bgranny.jpg" width="320" /></a></div><br /><br /><br /><br /><span style="font-size: small;"><a href="https://twitter.com/tweetsauce/status/661169211310518272?lang=en" target="_blank">At age 109, Augusta Bunge became</a> the youngest living great great great great grandmother. Is that mom to the fifth power? </span><br /><br /><br /><br /><br /><br /><br /><br /><i><b>Math Teachers at Play</b></i> is a monthly blog carnival, hosted at a different blog each month. <span style="font-size: small;">I was hoping to give you 109 math links this month, but life intervened (parenting...) long before I got through my storehouse of cool stuff. There are plenty of goodies here, but not as many as I'd hoped.</span><br /><br /><span style="font-size: small;"> </span><br /><div class="MsoListParagraph" style="margin-left: 0.25in; text-indent: -0.25in;"><span style="font-size: small;"><span style="font-family: "times";"></span></span></div><span style="font-size: small;"> </span><br /><div class="MsoListParagraphCxSpFirst" style="margin-left: 0.25in; text-indent: -0.25in;"><br /></div><span style="font-size: small;"> </span><br /><br /><b> Books</b><br />There has been an explosion of super cool mathy books since I last hosted MTAP. Here are some I know about. I am embarrassed to admit that I haven't read most of them, and so I can't guarantee how cool they are. Let me know in the comments.<br /><ul><li><a href="http://learnersinbloom.blogspot.com/2016/07/upcoming-book-club-and-math-circle-math.html" target="_blank">Math and Magic in Wonderland</a>, by Lilac Mohr (This is volume 1, there is now a volume 2. I've been meaning to buy this for the past year, and finally did, just now.) </li><li><a href="http://naturalmath.com/avoid-hard-work/" target="_blank">Avoid Hard Work</a>, by Maria Droujkova, James Tanton, and Yelena McManaman</li><li><a href="https://beastacademy.com/" target="_blank">Beast Academy</a> (Fabulous curriculum, don't believe the grade levels (3 to 5), adults can find challenges in these)</li><li><a href="http://bentobooks.com/mathgirls/" target="_blank">Math Girls 3</a>, by Hiroshi Yuki (I love the first two. Turns out the 3rd one is out.)</li><li><a href="https://new.myubam.com/p/5003/this-is-not-a-math-book" target="_blank">This is Not a Math Book</a>, by Anna Weltman</li><li><a href="https://tabletopacademy.net/playful-math-books/" target="_blank">Any of Denise Gaskins books</a></li></ul><br /><br /><b>Animations</b><br /><ul><li><a href="http://www.businessinsider.com/7-gifs-trigonometry-sine-cosine-2013-5" target="_blank">GIFs for Trig concepts </a></li><li><a href="http://www.decisionsciencenews.com/2017/06/19/counterintuitive-problem-everyone-room-keeps-giving-dollars-random-others-youll-never-guess-happens-next/" target="_blank">A simulation of money moving randomly</a> </li><li><a href="https://undergroundmathematics.org/divisibility-and-induction/picture-this" target="_blank">Explore Euclid's Algorithm for finding GCF,</a> using a diagram instead of numbers. </li><li>I do not yet understand <a href="http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/" target="_blank">this animation of 'involutes'</a>, but I want to read it and think about it again. </li></ul><br /><b>Puzzles & Games </b><br /><ul><li><a href="https://solveme.edc.org/Mobiles.html" target="_blank">Solving equations as represented by mobiles (online)</a></li><li><a href="https://mathhombre.blogspot.com/2017/01/pentiremeter.html" target="_blank">Reduce the perimeter game</a> (John Golden) </li><li><a href="https://mathlesstraveled.com/2016/06/17/the-route-puzzle/" target="_blank">Find the number: Primes, Triangle Numbers, more</a></li><li><a href="http://www.numbrcise.com/numbrcise-puzzles/" target="_blank">Equation Crossnumber puzzles</a></li><li><a href="http://blog.tanyakhovanova.com/2016/05/a-travel-puzzle/" target="_blank">Rate, Time, and Distance Puzzle</a> </li><li><a href="http://musingmathematically.blogspot.com/2016/05/min-max-imize-classroom-game-for-basic.html" target="_blank">Highest / Lowest Result Game</a></li><li><a href="https://sonatamathematique.wordpress.com/2016/04/22/favorite-problems-1-pirates/" target="_blank">Pirate Puzzle</a> </li></ul><br /><br /><b>Early Math</b><br /><ul><li>Skip counting and lots more: If you like <a href="http://gregtangmath.com/index" target="_blank">Greg Tang</a>'s books, he's provided a page full of lovely resources for you to play with.</li><li><a href="http://tjzager.com/2017/06/17/representations-and-manipulatives-and-tools-and-things/" target="_blank">A good problem to help resistant adults take a fresh look at multi-digit multiplication</a></li><li><a href="http://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">Why Number Sense is the most important math skill</a></li><li><a href="http://blog.mathedpage.org/2016/05/fractions.html" target="_blank">Fractions</a> (Henri Picciotto) </li><li><a href="https://kristinlabeau.com/number-sense-what-is-it-and-why-is-it-so-important/" target="_blank">Number Sense</a> (Kristin LaBeau) </li><li><a href="https://denisegaskins.com/2017/06/20/10-ways-to-play-math-with-play-doh/" target="_blank">Play Math with Play Doh</a> (Lucy Ravitch)</li></ul><br /><br /><b>Geometry</b><br /><ul><li><a href="http://mymathclub.blogspot.com/2016/10/12-triangles-and-their-link-to.html" target="_blank">Triangles and Pythagorean Triples in Math Club</a> </li><li><a href="https://sonatamathematique.wordpress.com/2016/05/01/favorite-problems-2-triangles/" target="_blank">Solution process for a hard problem involving three triangles</a></li><li><a href="http://blogs.ams.org/blogonmathblogs/2016/04/20/fold-your-way-to-glory/" target="_blank">What shapes can you get with one cut on a folded paper? </a></li><li><a href="https://samjshah.com/2016/04/20/a-new-insight-on-the-famous-painted-block-problem/" target="_blank">Insight: How deep can you go? Painted Block Problem</a> (Sam Shah)</li><li><a href="http://matematicas11235813.luismiglesias.es/2017/06/18/18617-teselado-mosaicos-matematicas-arte-geometria-worldtessellationday/" target="_blank">Some lovely tesselations</a> (Luis Iglesias) </li></ul><br /><br /><b>Probability & Statistics</b><br /><ul><li><a href="https://matheducators.stackexchange.com/questions/1325/impressive-common-misleading-interpretations-in-statistics-to-make-students-awar?newsletter=1&nlcode=478309%7c101e" target="_blank">Misleading Interpretations (Stack Exchange) </a></li><li><a href="https://matharguments180.blogspot.com/2016/06/514-probability-game.html" target="_blank">Fun Probability Question</a></li><li><a href="https://onegoodthingteach.wordpress.com/2016/05/06/an-interesting-math-club-problem/" target="_blank">Math Club Probability Problem</a> </li><li><a href="http://blogs.cofc.edu/owensks/2016/04/22/modeling-fun-with-paper-fish/" target="_blank">Estimating Population Size with Capture / Recapture</a> </li></ul><br /><br /><b>Writing in Math Class </b><br /><ul><li><a href="http://busynessgirl.com/elaborations-for-creative-thinking-in-stem/" target="_blank">Problem Elaborations: One way to get students writing in math class (Maria Andersen)</a></li><li><a href="http://mrhonner.com/archives/16315" target="_blank">Math Haiku</a></li><li><a href="https://denisegaskins.com/2017/06/12/math-humor-and-copywork/" target="_blank">Math Copywork and Humor </a>(Denise Gaskins) </li></ul><br /><br /><b>Math and ...</b><br /><ul><li><a href="https://www.nytimes.com/2017/06/10/opinion/sunday/cuny-ending-the-curse-of-remedial-math.html" target="_blank">Ending the Curse of Remedial Math (NYT)</a> </li><li><a href="https://blog.supplysideliberal.com/post/31224211784/the-logarithmic-harmony-of-percent-changes-and" target="_blank">How logarithms explain economics</a> </li><li><a href="https://rootsoftheequation.wordpress.com/2016/12/21/vietnamese-age-and-the-sat" target="_blank">Is it hard for you to understand how math tests can be culturally biased? Here's an example.</a> (James Cleveland) </li><li><a href="http://www.johndcook.com/blog/2016/05/03/family-tree-numbering/" target="_blank">Family Tree Numbering</a> (John D. Cook) </li></ul><br /><b>On Teaching </b><br /><ul><li><a href="http://hyperbolicguitars.blogspot.com/2012/07/zen-and-fermats-last-theorem.html" target="_blank">'Showing' doesn't work. (Mike Thayer)</a></li><li><a href="https://www.nature.com/articles/nchem.2788.epdf?author_access_token=GkqcEB_h_vNGj_nd4juBdNRgN0jAjWel9jnR3ZoTv0PKX0DnafBppEkrPagykiwxk3FuydjDvDwk4TDN-Jwo1WHUwM_avG3XaTIIJ8o3x7OahEvhhwtSSKs30N4vYdOU" target="_blank">Playing with slide rules improves estimating skills.</a></li><li><a href="http://blog.mathedpage.org/2016/06/forward-design.html" target="_blank">Forward</a> and <a href="http://blog.mathedpage.org/2016/06/the-perils-of-backward-design.html" target="_blank">Backward Design</a> (Henri Picciotto) </li><li><a href="https://blog.peerinstruction.net/2016/06/07/increasing-student-engagement-by-design/" target="_blank">Designing a Course with Peer Instruction and Project-Based Learning </a></li><li><a href="https://samjshah.com/2016/05/29/pitching-college-math-courses/" target="_blank">Getting students excited about higher math</a> (Sam Shah)</li><li><a href="https://rootsoftheequation.wordpress.com/2016/05/04/shear-madness/" target="_blank">Improvisational Theater and Improvisational Teaching</a></li><li>One of the principles of improv is "Yes, and...". <a href="https://deltascape.blogspot.com/2016/05/how-does-mathematician-see-world.html" target="_blank">Here's a way to use that</a> to get students playing with math. </li><li><a href="http://fawnnguyen.com/always-sometimes-never/" target="_blank">Encouraging discussion with Always, Sometimes, Never</a></li></ul><br /><b><br /></b><b>Random Stuff</b><br /><ul><li><a href="https://opencalculus.wordpress.com/2016/04/29/last-day-at-aim-meet-some-of-my-new-friends" target="_blank">Links to people doing good work in Open Resources</a></li><li><a href="http://spot.pcc.edu/math/clm/section-velocity.html" target="_blank">Calculus Lab Manual </a></li><li><a href="http://betterexplained.com/articles/understanding-quakes-fast-inverse-square-root/" target="_blank">Using Newton's Method for a game</a></li><li>Fermat's Last Theorem was proved by Andrew Wiles in 1994. Though Fermat thought he had proved it, we are pretty sure he didn't. James Propp writes <a href="https://mathenchant.wordpress.com/2016/05/16/fermats-last-theorem-the-curious-incident-of-the-boasting-frenchman" target="_blank">a lovely summary of what Fermat probably knew</a>. </li></ul><br />I have more but it's bedtime and June is ending. Would you like to see your favorite blog post in next month’s playful math blog carnival? <a href="https://denisegaskins.com/mtap/mtap-submission-form/" target="_blank">Submissions are always open!<br /></a><br /><br /><br /><br /><span style="font-size: x-small;">(Note: Edited on 7/1 to add a few forgotten links, and fix a few broken links.)</span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com7tag:blogger.com,1999:blog-5303307482158922565.post-52881048698576622012017-06-06T09:37:00.000-07:002017-06-06T09:37:05.175-07:00What I learned at CAP's Community of PracticeCAP is <a href="http://accelerationproject.org/" target="_blank">California Acceleration Project</a>. Check out their <a href="http://accelerationproject.org/Publications" target="_blank">publications</a>. The first time I attended one of their conferences, I struggled with the word acceleration. It does not mean getting through the material faster. It means getting to the good stuff faster - shortening the pathway of required prerequisite courses students must take before taking a college level course. Their work is mainly with math and English, the two subjects that generally hold students back.<br /><br />In math, the college level course for someone not interested in STEM is statistics. Students take a placement test, and the large majority (86% at my college) are placed in remedial courses, anywhere from 1 to 4 levels below the statistics course. Imagine a student starting 3 levels below, at pre-algebra, which is where over half of our students are put by the placement test. If we had phenomenal success rates, with 90% passing each course, and phenomenal persistence rates, with 90% going on to the next course, we'd still only get 43% of these students finishing statistics (.9^8 = .43). What happens to the other 57%? Usually they give up on college, for at least a while.<br /><br />Because housing is pretty segregated in the U.S., and that makes k12 education pretty segregated, with people of color getting less resources dedicated to their schools, this becomes a civil rights issue. CAP is dedicated to: changing the way we place students (many who do badly on the placement test can still pass a college statistics course), developing models for co-requisite courses that students can take with statistics to improve their success rates, and developing radically shortened and improved remedial pathways (creating a pre-statistics course that prepares students with just enough algebra and lots of data analysis).<br /><br />I have been attending their workshops whenever I can for the past few years. This past weekend I went with two other math faculty and 5 English faculty. Even though I've seen much of the information before, I still got a lot out of it. (Maybe I'm a slow learner!)<br /><br />Here's something I put together yesterday at the request of our dean for equity, which summarizes some of the important points I learned...<br /><br /><br /><div dir="ltr" id="docs-internal-guid-c9ed0335-7e43-0c7c-03e5-e9d94866dc11" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Planning a High-Impact Course</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">More important than any one course are these 3 principles:</span><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Create separate pathways for STEM and non-STEM.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Place students as high in the sequence as possible.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Shorten the sequence as much as possible.</span></div></li></ul><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">CAP’s 5 design principles</span></div><ol style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Backward design</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Low-stakes, collaborative practice</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Relevant, thinking-oriented curriculum</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Just-in-time remediation</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Intentional support for affective needs</span></div></li></ol><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">[For more detail, see: </span><a href="http://accelerationproject.org/Publications/ctl/ArticleView/mid/654/articleId/12/Toward-a-Vision-of-Accelerated-Curriculum-and-Pedagogy-High-Challenge-High-Support-Classrooms-for-Underprepared-Students" style="text-decoration: none;"><span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: underline; vertical-align: baseline;">http://accelerationproject.org/Publications/ctl/ArticleView/mid/654/articleId/12/Toward-a-Vision-of-Accelerated-Curriculum-and-Pedagogy-High-Challenge-High-Support-Classrooms-for-Underprepared-Students</span></a><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> ]</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">High Performing Math Classrooms (Internationally)</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">James Stigler on high performing math countries. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">All have these things in common:</span></div><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Productive struggle</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Explicit connections</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Deliberate practice, increasing variation and complexity over time</span></div></li></ul><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">[For an NPR piece on Stigler’s work, see: </span><a href="http://www.npr.org/sections/health-shots/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning" style="text-decoration: none;"><span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: underline; vertical-align: baseline;">http://www.npr.org/sections/health-shots/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning</span></a><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> ]</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Lesson Planning (CAP)</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Given a topic you want students to learn through groupwork,</span></div><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Identify the prerequisite skills needed,</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Decide whether these will be addressed through productive struggle (ie not addressed overtly), targeted group activity, or just-in-time mini-lecture, and how you’ll do that,</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Plan main activity,</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Plan closure (vital for making explicit connections)</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Note: Over-scaffolding brings down the thinking level required.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">(Sue has a form from CAP. If this link works, it’s to all the CAP materials: </span><a href="https://app.box.com/s/965xg12luwsgjgmeq86px8oonsr9yolm" style="text-decoration: none;"><span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: underline; vertical-align: baseline;">https://app.box.com/s/965xg12luwsgjgmeq86px8oonsr9yolm</span></a><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> )</span></div></li></ul><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Thinking Levels</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">The thinking levels mentioned above come from a study by Quasar. Here’s the relevant info:</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 12pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">“This research yielded two major findings: (1) mathematical tasks with high-level cognitive demands were the most difficult to implement well, frequently being transformed into less-demanding tasks during instruction; and (2) student learning gains were greatest in classrooms in which instructional tasks consistently encouraged high-level student thinking and reasoning and least in classrooms in which instructional tasks were consistently procedural in nature.” (Stein p. 4)</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">QUASAR Task Analysis Guide (adjusted slightly to address statistical thinking as well as mathematical thinking)</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Lower-Level Cognitive Demand</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Memorization Tasks </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Involve either reproducing previously learned fact, rules, formula, or definitions; </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure; </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Not ambiguous; clear and direct instructions to reproduce previous material; </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">No connection to the concepts or meaning that underlie the fact, rules, formula, or definitions. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Procedures Without Connections Tasks </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Algorithmic; direct instructions to use a procedure or the use of the procedure is evident based on prior instruction, experience, or placement of the task. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require limited cognitive demand for successful completion. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">There is little ambiguity about what needs to be done and how to do it. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">No connections to concepts or meaning that underlie the procedure being used. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Focused on correct answers rather than developing mathematical or statistical understanding. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require no explanations, but may require students to “show work”.</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Higher-Level Cognitive Demand</span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Procedures With Connections Tasks</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Focus students’ attention on the use of procedures or concepts for the purpose of developing deeper levels of understanding of mathematical or statistical concepts and ideas. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Usually are represented in multiple ways (e.g. graphs, tables, numerical summaries, verbal descriptions). </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Making connections among multiple representations helps to develop meaning. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require some degree of cognitive effort. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Although general procedures may be followed, they cannot be followed mindlessly. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline;">Doing–Mathematics or Doing–Statistics Tasks</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;"> </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require complex and non-algorithmic thinking (i.e. there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked-out example). </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require students to explore and understand the nature of mathematical or statistical concepts, processes, or relationships. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Demand self-monitoring or self-regulation of one’s own cognitive processes. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require students to access and make appropriate use of relevant knowledge and experiences </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline;">Require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required. </span></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><br /></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-33812288731554717822017-03-04T15:20:00.001-08:002017-03-04T15:20:31.818-08:00At the Julia Robinson Math Festival Today<a href="http://jrmf.org/index.php" target="_blank">Julia Robinson Math Festival</a>s invite kids to play with math puzzles that start easy and offer harder questions as you go along. Today's festival was at Bentley School in Lafayette. (Some festivals are open to the public, and are much bigger.)<br /><br />I was working the Pilgrim's Puzzle table. We had this puzzle to work on.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-SvcnOr6eJw4/WLtJaOJoBzI/AAAAAAAAC60/g0CceMG04kwKajMbE3oHmG7CODnP-Ze-wCLcB/s1600/pilgrim%2527s%2Bpuzzle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-SvcnOr6eJw4/WLtJaOJoBzI/AAAAAAAAC60/g0CceMG04kwKajMbE3oHmG7CODnP-Ze-wCLcB/s1600/pilgrim%2527s%2Bpuzzle.png" /></a></div>It was really fun watching kids and parents get engaged with it. Some paths give you fractions, and then taking away 2 can give you something like 1/8 - 2, which can be pretty confusing for a 3rd grader.<br /><br />The first time I tried to help a kid with a problem like that I was not able to find an image that made this sensible. When B was stuck with a problem like this, I came up with anti-matter apples. It worked! We imagined 1/8th of an apple, and imagined two anti-matter apples. We cut the 2nd one into 8 pieces, took one of those pieces and exploded it with the regular 1/8th slice to make a poof and then nothing. So we had one anti-matter apple and ... 7/8ths of another, which we wrote as -1 7/8. Done.<br /><br />I will be teaching beginning algebra in the fall. I don't think I've ever found an image for negative fractions that worked as well as I think this one will. I'm excited.<br /><br />Here's B and me.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-C1bzhBTsp7I/WLtLja-XC7I/AAAAAAAAC7A/tLxU3DehExI3uzlhWLHsqIQ6fLKa8PyPgCLcB/s1600/me%2Bat%2Bjrmf.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://3.bp.blogspot.com/-C1bzhBTsp7I/WLtLja-XC7I/AAAAAAAAC7A/tLxU3DehExI3uzlhWLHsqIQ6fLKa8PyPgCLcB/s320/me%2Bat%2Bjrmf.jpg" width="320" /></a></div><br /><br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-25955902614088930042016-12-20T20:19:00.003-08:002016-12-20T20:19:35.323-08:00The Cat in Numberland is Back in PrintOne of my favorite mathy kids' books is back in print. In <i><a href="https://shop.cricketmedia.com/books/The-Cat-In-Numberland.html" target="_blank">The Cat in Numberland</a> </i>we visit Hotel Infinity, with its infinite rooms, all full and yet they always seem to be able to make room for new guests.<br /><br />Good for ages 5 to adult. (It's $19.95 plus over $9 shipping. I think they should charge less for this slim volume, but this book is so wonderful, it's worth it.)<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-X3olPyClPbk/WFoCfGDlT6I/AAAAAAAACiQ/PbQ335yQM3goug9aVeNpxXOeLQmdl0rNwCLcB/s1600/cat%2Bin%2Bnumberland.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-X3olPyClPbk/WFoCfGDlT6I/AAAAAAAACiQ/PbQ335yQM3goug9aVeNpxXOeLQmdl0rNwCLcB/s1600/cat%2Bin%2Bnumberland.png" /></a></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-38126262874230944632016-10-16T18:50:00.000-07:002016-10-16T18:50:14.372-07:00Fun Question from Michelle at Prairie Creek Community School"Recently in math, we were working on the Deka Tree, a tree that has 10 trunks with 10 branches with 10 twigs with 10 leaves. One trunk, one branch, one twig, and one leaf is cut off...how many leaves are left?"<br /><br />I found this question in <a href="http://prairiecreek.typepad.com/herons/2016/10/rhythms-of-the-forest.html" target="_blank">Michelle's post about doing Forest School</a>. And she talked more about it in <a href="http://prairiecreek.typepad.com/herons/2016/10/the-deca-tree.html" target="_blank">this post</a>.<br /><br /><br />I have over fifty tabs open with interesting goodies. I hope to find time soon to sort them out and share...Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-37533249355588601992016-08-21T08:20:00.000-07:002016-08-21T13:49:56.177-07:00Calculus: From Secant Lines to the TangentOur semester began on Monday. I'm teaching Calculus I (as always, because it's my favorite class), Statistics, and Algebra for Statistics. All three classes were a joy to teach this week, even though I was a bit underprepared because of the chaos in my personal life.<br /><br />On Thursday I was working on wrapping up the exercise from <a href="http://scholarworks.gvsu.edu/books/10/" target="_blank">Active Calculus</a> that the students had been working on since Tuesday. I showed the velocity curve we'd been exploring on Desmos, and limited the domain to the appropriate times, 0 to 3 seconds (which I learned how to do with the face project I described in <a href="http://mathmamawrites.blogspot.com/2016/08/calculus-reviewing-functions-with-desmos.html" target="_blank">my previous blog post</a>). I had a little trouble remembering how to make a secant line attached to one stable point and one moving point, but I got it. (And helped the students get it. This took some hard thinking for many of them.)<br /><br />Then I had a wonderful surprise. When I pulled the moving point over the stable point, the line disappeared and Desmos said "x= undefined or undefined" (not sure where their stutter came from...). I gasped. I hadn't expected that, and it was a perfect way to start talking about this problem calculus has of needing two points to figure slope, but needing to use just one point from the function in order to have a tangent. I got to talk about Newton and Bishop Berkeley and fluxions and infinitely close. It was great fun for me. On Monday I'll find out how much the students got out of it.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://www.desmos.com/calculator/g7giaub7bi" style="margin-left: 1em; margin-right: 1em;" title="View with the Desmos Graphing Calculator"> <img height="400" src="https://s3.amazonaws.com/calc_thumbs/production/g7giaub7bi.png" style="border-radius: 5px; border: 1px solid rgb(204, 204, 204);" width="400" /></a></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-65368167884209990892016-08-21T07:40:00.003-07:002016-08-21T07:40:47.375-07:00Calculus: Reviewing Functions with Desmos<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-yDLcmF2ZmkA/V7m1NP7y2ZI/AAAAAAAACHc/ILpTu5AiilE0MLaIVAkB1kCSh057E07LACLcB/s1600/me%2Bon%2Bdesmos.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="198" src="https://2.bp.blogspot.com/-yDLcmF2ZmkA/V7m1NP7y2ZI/AAAAAAAACHc/ILpTu5AiilE0MLaIVAkB1kCSh057E07LACLcB/s200/me%2Bon%2Bdesmos.png" width="200" /></a></div>I like to dive into the calculus ideas in my calc I course, so I do not start with review. (I use a just-in-time approach, reviewing what we need when we need it.)<br /><br />But I know that the students' understanding of functions is weak and needs to be brought to mind. So I was excited about having them outline their own faces in Desmos as a homework assignment, which I learned about at Twitter Math Camp from Deb Boden (@debboden).<br /><br /><br /><br /><br /><br />Here are my instructions:<br /><div class="description user_content teacher-version enhanced"><div style="text-align: center;"><span style="background-color: #fce5cd;"><strong>Desmos Graph of Yourself</strong></span></div><span style="background-color: #fce5cd;"></span><ol><li><span style="background-color: #fce5cd;">Set up an account on <a class="external" href="https://www.desmos.com/" rel="noreferrer" target="_blank" title=""><span><span>desmos.com</span><span class="screenreader-only"></span></span><span class="ui-icon ui-icon-extlink ui-icon-inline" title="Links to an external site."></span></a>. (It’s free.)</span></li><li><span style="background-color: #fce5cd;">Upload a selfie into desmos. (Click the + in the upper left corner of the desmos calculator screen to add your image. Photos with you facing front are easiest to use.)</span></li><li><span style="background-color: #fce5cd;">Use various functions to outline features of your face. (At least: lines, arcs of circles and ellipses, parabolas, and trig functions. Try including exponential and log functions, hyperbolas, and cubics.)</span></li><li><span style="background-color: #fce5cd;">When you’re done, you can hide your photo to display the icon you’ve created. You can also hide the axes by clicking on the wrench in the upper right corner.</span></li><li><span style="background-color: #fce5cd;">Add a link to your completed desmos work on our class google doc: [Link removed, for student privacy. I didn't need google, actually. They could have submitted directly to Canvas. But we may need the google doc for our viewable collection.] (My icon is linked there. You can check it out to figure out how to do this.)</span></li><li><span style="background-color: #fce5cd;">We’ll share these in class and see how many classmates everyone can recognize.</span></li></ol><span style="background-color: #fce5cd;"></span><span style="background-color: #fce5cd;">Every time I got stuck, I googled my question. For example, "Desmos function restrictions" helped me make short pieces of the curves I used. If you are still stuck, start a discussion item here.</span><br /><span style="background-color: #fce5cd;"></span><br /><span style="background-color: #fce5cd;"></span><br /><span style="background-color: #fce5cd;"><b>Rubric</b></span><br /><span style="background-color: #fce5cd;">50% Required function types (lines, arcs of circles and ellipses, parabolas, and trig functions) 10% each (Extras can bring this score up to 60%)</span><br /><span style="background-color: #fce5cd;"></span><span style="background-color: #fce5cd;">15% Good Match with Photo</span><br /><span style="background-color: #fce5cd;"></span><span style="background-color: #fce5cd;">15% Visually Engaging</span><br /><span style="background-color: #fce5cd;"></span><span style="background-color: #fce5cd;">20% for using Desmos and Canvas (our "learning management system")</span><br /><br />(I didn't post the rubric until after they turned in their work, but I will next time.)<br /><br />31 out of 44 students turned it in. I am loving Canvas, which our college just started using. (We used d2l before and I hated it. Yes, I have strong feelings about things.) I took hours grading this, but once I get good at it, I think I could do this in about an hour. Canvas made it easy. And now I know that a third of my class is having trouble. So I know I need to intervene somehow. Good information to have.<br /><br />Here are some of my favorites...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-bD3DHthgiJc/V7m81HjI_oI/AAAAAAAACIA/ehn4RJCKFloanqEOHQEY47ThRPoPE5IjgCLcB/s1600/student1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://2.bp.blogspot.com/-bD3DHthgiJc/V7m81HjI_oI/AAAAAAAACIA/ehn4RJCKFloanqEOHQEY47ThRPoPE5IjgCLcB/s200/student1.png" width="167" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-y1oGWLd32bY/V7m81IenG7I/AAAAAAAACH8/M1uIC9uAqE0BqL6itH559Qo5OEe8bD-zgCLcB/s1600/student2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://3.bp.blogspot.com/-y1oGWLd32bY/V7m81IenG7I/AAAAAAAACH8/M1uIC9uAqE0BqL6itH559Qo5OEe8bD-zgCLcB/s200/student2.png" width="162" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-4CQTc3EWl3k/V7m81Lc4neI/AAAAAAAACH4/97Q6RuffVBgIAlLDEP6Ap8xTm-2VRf2xwCLcB/s1600/student3.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://2.bp.blogspot.com/-4CQTc3EWl3k/V7m81Lc4neI/AAAAAAAACH4/97Q6RuffVBgIAlLDEP6Ap8xTm-2VRf2xwCLcB/s1600/student3.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-DJFE4jQ3cVc/V7m81fFmJcI/AAAAAAAACIE/BI_sTjWHlZIEWYYnRCzmV90HbXjSp_2bgCLcB/s1600/student4.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-DJFE4jQ3cVc/V7m81fFmJcI/AAAAAAAACIE/BI_sTjWHlZIEWYYnRCzmV90HbXjSp_2bgCLcB/s1600/student4.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Yy8c2oD-Z0c/V7m81uVHJ5I/AAAAAAAACII/GC5eE8EQ-RI5H-_FnEBwo0ImFQhRWMsuACLcB/s1600/student5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://3.bp.blogspot.com/-Yy8c2oD-Z0c/V7m81uVHJ5I/AAAAAAAACII/GC5eE8EQ-RI5H-_FnEBwo0ImFQhRWMsuACLcB/s200/student5.png" width="129" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-6Gl4q3_Nczw/V7m81nZcE4I/AAAAAAAACIM/ThlBXIVzl08iDbhrIua8aSc_QCtwUUb6wCLcB/s1600/student6.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://4.bp.blogspot.com/-6Gl4q3_Nczw/V7m81nZcE4I/AAAAAAAACIM/ThlBXIVzl08iDbhrIua8aSc_QCtwUUb6wCLcB/s200/student6.png" width="189" /></a></div><br /></div>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-44133389427650636802016-07-22T20:05:00.000-07:002016-07-22T20:05:35.402-07:00My Favorites: Becoming Invisible & Math Relax On the last day of Twitter Math Camp 2016, I got to do a ten-minute presentation about two of my favorite teaching ideas. This is a quick way for me to share the links.<br /><br /><a href="https://drive.google.com/file/d/0B4Lou9CsLnQxaTdjODd6cFkxMjQ/view?usp=sharing" target="_blank">Becoming Invisible </a> is a great collection of things you can say when you're trying to hand the floor over to the students.<br /><a href="http://mathmamawrites.blogspot.com/2009/09/math-relax-guided-visualization-for.html" target="_blank"><br /></a><a href="http://mathmamawrites.blogspot.com/2009/09/math-relax-guided-visualization-for.html" target="_blank">Math Relax</a> is my audio track to help students get over the anxiety some of them feel during math tests. After my talk, people mentioned some lovely anxiety-busters, including giving the student a nice stone to play with. I might just make a basket of magic calming stones...<br /><br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com1tag:blogger.com,1999:blog-5303307482158922565.post-27326833840775531592016-04-13T18:25:00.001-07:002016-04-16T23:01:14.619-07:00Kahoot I heard about <a href="https://getkahoot.com/" target="_blank">Kahoot</a> from my colleague, who heard about it from his wife who teaches third grade. It's a game site with lots of content already available. I looked up logarithms yesterday, <a href="https://create.kahoot.it/?_ga=1.136995064.990966096.1460488149&deviceId=f761c75b-eb18-471e-b1e8-e0cb73ea629a#quiz/b188f973-7be4-4a29-868b-9a34074722ec" target="_blank">found a kahoot* I liked</a>, and played it with my pre-calculus class.<br /><br />[To find a kahoot you like, choose Public Kahoots in the black bar at the top, search on a term like logarithms, click on"<span class="filter-teachers pull-right">Only show Kahoots made by teachers?", and search the list. I've been looking for the ones with high counts on the favourites list, but I might find better criteria later. Once you find one you like, favorite it right away. There doesn't seem to be an easy mechanism to get back to it later.]</span><br /><br /><span class="filter-teachers pull-right">We are about 2/3 rds of the way through the semester. The energy is a bit low about now. This game livened things up and kept us focused on mathematical ideas. The students loved it. </span><br /><br />This evening, I made <a href="https://create.kahoot.it/?_ga=1.244598221.990966096.1460488149&deviceId=f761c75b-eb18-471e-b1e8-e0cb73ea629a#quiz/6048bced-6996-49e4-8d4d-e833d34c7dfe" target="_blank">a pretty simple kahoot</a> to go along with my <a href="http://mathmamawrites.blogspot.com/2010/04/murder-mystery-project-for-logarithms.html" target="_blank">murder mystery</a>, which we're starting in precalc right now. I'll use this kahoot next week, when we're farther along in the murder mystery.<br /><br /><br /><br />______<br />*A kahoot is a gamified quiz. Each question is set up with multiple answers. Students use a pin shown on the screen to sign in using their phones. They get points for right answers based on how quickly they answer.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com3tag:blogger.com,1999:blog-5303307482158922565.post-69803894222965197442016-03-07T15:14:00.001-08:002016-03-09T16:34:00.376-08:00A Huge Bunch of Lovely LinksI have so many tabs with cool math posts, I don't know if I can possibly get them all into this collection. (I never seem to have enough time to finish, and then more goodies accumulate.)<br /><br /><b><br /></b><b>Math & Teaching Ideas I might use</b><br /><ul><li>I want to show this to my pre-calc class as part of their intro to trig: <a href="https://www.facebook.com/worldofskipper/videos/241010499356393/" target="_blank">sailboat leans to get under bridge</a></li><li>Wild Maths has a lovely collection of questions with photos: <a href="http://wild.maths.org/tags/move-it-prove-it" target="_blank">Move it to prove it </a></li><li><a href="http://samjshah.com/2016/01/17/snug-angles/" target="_blank">Snug angles</a>, from Sam Shah. He's doing it with a geometry class. I'm wondering if it might be good at the beginning of the trig unit. </li><li>As I prepared for Math Jam, our 3-day pre-semester math boost, I found lots of cool ideas here: <a href="http://map.mathshell.org/lessons.php" target="_blank">Math Assessment Project</a> (assessment doesn't sound promising, but these activities have lots of open-ended questions)</li><li>I just learned that if you start with the harmonic series, which diverges, and <a href="https://collegemathteaching.wordpress.com/2016/01/14/trimming-a-divergent-series-into-a-convergent-one/" target="_blank">take out all the terms with 9s in the denominator</a>, <a href="http://blogs.ams.org/blogonmathblogs/2015/12/21/mind-blowing-math-reminiscence" target="_blank">you'll get a converging series</a>. Too weird. I don't understand it yet, but I sure want to. (for Calc II)</li><li><a href="https://makingmathpeople.wordpress.com/2016/01/17/2016-blogging-initiative-week-2" target="_blank">Geometric construction on sciencevsmagic.net.</a> (I knew about this site, but reading this post made me decide to use it in Math Jam to get them playing around.)</li><li><a href="http://www.whatdowedoallday.com/2016/01/mobius-strip-hearts.html" target="_blank">Mobius Hearts.</a> Too fun not to do. (I didn't use it, though. Too overwhelmed this past month to do anything new...)</li><li>Sam Shah made<a href="http://explore-math.weebly.com/" target="_blank"> </a><a href="http://explore-math.weebly.com/" target="_blank">this fabulous website</a>, Explore Math, that pulls together gobs of cool math resources from the web. He has his students pick one (or was it a few?) to play around with and report on. I believe I'm going to do this in pre-calc.</li><li>What is proof? <a href="http://blog.amathknauft.com/2016/01/martin-asked-on-twitter-whether.html" target="_blank">Here's a good conversation</a> about proving the Pythagorean Theorem with visuals. Includes the best video I've ever seen, on my favorite proof.</li><li><a href="http://www.appetite-for-instruction.com/my-favorite-trig-tale/" target="_blank">Trig Fairy Tales</a> (having students write them) </li><li><a href="https://plus.maths.org/content/ping-pong-balls-and-super-powers" target="_blank">Infinity is so weird!</a> (infinite ping pong balls in, infinite ping pong balls out, how many left in?)</li><li><a href="https://plus.maths.org/content/population-growth" target="_blank">Infinite sums and China's demographics</a> </li><li><a href="http://tube.geogebra.org/material/simple/id/134244" target="_blank">Algebra Aerobics Stick Figure in Geogebra</a></li></ul><b></b><br /><b></b><br /><b>Problem Solving</b><br /><ul><li><a href="https://plus.maths.org/content/dropping-eggs-solution" target="_blank">Finding out how far you can drop an egg without breaking it</a></li><li><a href="http://musingmathematically.blogspot.com/2016/02/candies-pennies-and-inequalities.html" target="_blank">Systems of equations, using a problem with no solution</a></li><li><a href="https://mikesmathpage.wordpress.com/2016/01/18/my-favorite-watching-problem-solving-ideas-develop/" target="_blank">On problem solving, with videos</a>. I might give the absolute value problem in precalc, as a challenge.</li><li><a href="http://musingmathematically.blogspot.com/2016/02/my-favourite-surface-area-question.html" target="_blank">Doubling surface area, a good question </a></li><li><a href="http://considerlearning.com/2016/02/08/5-minute-problems-to-five-year-problems/" target="_blank">What's the longest time you've ever spent solving a problem?</a></li><li><a href="http://aperiodical.com/2016/02/open-season-pancake-flipping/" target="_blank">Flipping pancakes</a> </li></ul><br /><b><br /></b><b> </b><br /><b>Using Desmos</b><br /><ul><li><a href="http://mrhonner.com/archives/15951" target="_blank">An introduction to desmos</a> </li><li><a href="http://blog.amathknauft.com/2016/01/designing-and-assessing-desmos-calculus.html" target="_blank">Linearization in Calculus</a>, an amazingly detailed lesson using desmos, with commentary about how students did with it</li><li>I do a unit in trig called Days Of Our Lives, using minutes of daylight on each day of the year as data, and getting students to construct an equation for it. This <a href="https://student.desmos.com/activitybuilder/student/56ad0a34dd023fde0ba35760" target="_blank">Moon Illumination project</a> someone made on desmos using the activity builder looks like something I could imitate. (Where did they get their data? Who made this?)</li><li><a href="http://musingmathematically.blogspot.com/2016/01/desmos-art-project.html" target="_blank">Desmos art project </a></li></ul><br /><br /><b>On Teaching</b><br /><ul><li><a href="https://problemproblems.wordpress.com/2015/12/27/the-problems-of-writing/" target="_blank">Michael Pershan, on writing about teaching</a><b> </b></li><li><a href="http://blog.peerinstruction.net/2016/01/08/how-to-help-people-remember-what-they-learn/" target="_blank">How to help people remember what they learn (using retrieval practice)</a></li><li>How do you respond to wrong answers? <a href="http://profteacher.com/2016/01/16/explanatory-power-of-the-hierarchy-of-student-needs/" target="_blank">This post helps me think about that.</a></li><li>A good summary of <a href="https://www.brainpickings.org/2014/01/29/carol-dweck-mindset/" target="_blank">Dweck's Mindset research</a> </li><li><a href="https://researchinpractice.wordpress.com/2016/01/28/lessons-from-bowen-and-darryl" target="_blank">Ben Blum-Smith on the strategies used at PCMI</a>. "when students are talking to the room it is always students that Bowen and Darryl have preselected to present a specific idea they have already thought about. They <i>never</i> ask for hands, and they never cold-call. <i>This means they already know more or less what the students are going to say." </i>And then <a href="http://cheesemonkeysf.blogspot.com/2016/02/lessons-from-lessons-from-bowen-and.html" target="_blank">Elizabeth responded</a>. I loved her katamari.<i><br /></i></li><li><a href="http://prairiecreek.typepad.com/herons/2016/02/lets-talk-about-it.html" target="_blank">Using sentence starters for math conversations</a> with 4th and 5th grade students </li><li><a href="http://www.fractiontalks.com/p/how-to.html" target="_blank">Fraction talks </a></li><li><a href="http://learn-always.com/2016/01/20/my-favourite-getting-students-talking-to-each-other-about-math-mtbos/" target="_blank">Getting students talking to each other</a> </li><li><a href="http://education.lms.ac.uk/2016/02/ronnie-brown-answer-to-a-students/" target="_blank">Getting students not to fear confusion</a> </li><li><a href="http://www.cbc.ca/news/health/physical-activity-class-lessons-1.3460346" target="_blank">Physical activity during lessons improves learning</a> (research with elementary students, but I imagine it would help my college students too. Yikes! I don't like this perspective: "the researchers found no differences on reading scores. They think activity works better for subjects with a lot of memorization and repetition." Math should not have lots of memorization!)</li><li><a href="https://www.washingtonpost.com/local/education/teachers-are-using-theater-and-dance-to-teach-math--and-its-working/2016/02/22/61f8dc0c-d68b-11e5-b195-2e29a4e13425_story.html" target="_blank">More movement and math</a>...</li><li>If I were a high school teacher, I'd seriously consider this. <a href="http://blog.amathknauft.com/2016/02/notes-and-homework-structures-calc-bc.html" target="_blank">Metacognition and homework</a></li><li><a href="http://www.tandfonline.com/doi/full/10.1080/10511970.2015.1027837" target="_blank">On Metacognition</a> (download pdf, interesting part for me is sections 3 and 4) </li></ul><a href="http://blog.peerinstruction.net/2016/01/08/how-to-help-people-remember-what-they-learn/" target="_blank"><br /></a><b><br /></b><b>Science</b><br /><ul><li><a href="http://ncase.me/emoji-prototype/?remote=-K71iWhftjtOIfx6b2Fo" target="_blank">Simulation, mathematically modelling</a> how chemistry and growth work together</li></ul><br /><b>Statistics</b><br /><ul><li><a href="http://drhagen.com/blog/the-missing-11th-of-the-month/" target="_blank">The missing 11th of the month</a></li><li><a href="http://markkreie.blogspot.com/2016/01/my-favorite-linear-regression-movies.html" target="_blank">Linear Regression and Movies</a></li><li><a href="http://www.johndcook.com/blog/2016/02/20/the-empty-middle-no-one-is-average/" target="_blank">Why no one is average</a> </li></ul><br /><br /><b>Estimation & Elementary</b><br /><ul><li><a href="http://gfletchy.com/the-apple/" target="_blank">How many blocks will equal an apple?</a> (3-act lessons, with video) </li><li><a href="https://aerecord.wordpress.com/2016/01/18/my-favorite-activity-number-talks/" target="_blank">Number Talks</a></li><li>Pre-algebra: <a href="http://www.mathedpage.org/manipulatives/slides/lg-2d-arithmetic/lab-gear-2d-arithmetic/assets/player/KeynoteDHTMLPlayer.html#15" target="_blank">Working with signed numbers</a></li></ul><br /><br /><b>Math for Parents</b><br /><ul><li><a href="http://education.lms.ac.uk/2015/08/parents-math-anxiety-can-undermine-childrens-math-achievement/" target="_blank">Parents’ Math Anxiety Can Undermine Children’s Math Achievement</a></li><li>Fractions may be elementary (previous topic), but the idea of fractions is also the first math concept that messes a lot of people up. Here's <a href="http://gdaymath.com/courses/fractions-are-hard/" target="_blank">James Tanton's new collection on fractions</a>. </li><li><a href="http://education.lms.ac.uk/2016/02/when-did-you-stop/" target="_blank">When did you stop</a> playing around with mathy ideas?</li><li> A video about <a href="http://gfletchy.com/2016/03/04/the-progression-of-addition-and-subtraction/" target="_blank">what kids learn in the early grades about addition and subtraction</a> (Please let me know what you think!)</li><li><a href="http://ww2.kqed.org/mindshift/2013/10/01/finding-the-beauty-in-math/" target="_blank">Finding the Beauty in Math</a> </li></ul><br /><b>Social Justice</b><br /><ul><li><a href="https://sonatamathematique.wordpress.com/2016/02/21/an-evolution-of-my-reaction/#comment-723" target="_blank">On responding to people's surprise that I'm a math teacher</a></li><li><a href="http://considerlearning.com/2016/01/27/an-actual-response-to-chief-justice-roberts/" target="_blank">How affirmative action makes for a better physics education</a></li><li>"<a href="http://hiphopchess.blogspot.com/2016/02/why-dont-black-kids-like-math-and.html" target="_blank">Why Black kids don't like math...</a>" </li><li><a href="http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/270/169" target="_blank">The master's tools...</a> (Dr. Danny Martin's talk at NCTM conference)</li></ul><br /><br /><br /><b>Playing with Math</b><br /><ul><li>As usual, <a href="http://gameaboutsquares.com/" target="_blank">this game (called this game is about squares)</a> is more about logic than about math. What I'm finding interesting is how impossible it seems, and then when I (and others) go away and come back, it can suddenly seem so easy.</li><li> Tracy Zager wrote a great post on <a href="https://tjzager.wordpress.com/2016/01/05/my-criteria-for-fact-based-apps" target="_blank">evaluating math fact apps</a>. Lots of good ones are mentioned in the comments. [My comment: I would really love to be able to find this app online so I can recommend it. I have this game on my phone. It seems to be called 1 Whole. There are rectangular shapes that fill with liquid. You push one toward another and they go together if the sum is less than or equal to one. You watch the liquid rise. If it’s 1, it goes away and you get points. You keep going until the screen is full of things that won’t combine (sum > 1). There is no time pressure, the conceptual basis seems strong to me, and mistakes aren’t allowed. No penalties, no bad sounds, it just won’t work. I think it’s pretty good. I wish I could find it online. Cna anyone help me?]</li><li>Kids like doing the simple math involved in thinking about the Collatz Conjecture. [Start with any number (whole, >1). If odd, triple it and add 1. If even, cut in half. Repeat. Does this always end up at 1? Conjecture is 'yes'.] Mathematicians don't know the answer, but they like to explore the question in sophisticated ways. Here's a <a href="http://gottwurfelt.com/2016/01/10/logarithmic-approximations-for-collatz/" target="_blank">post on what sorts of functions come close</a> to modeling the number of steps it takes to get to 1 from each number.</li><li>This game would have made it into my book, I think. <a href="https://mindfull.wordpress.com/2016/01/10/cross-over-a-game-for-practicing-addition-and-subtraction/" target="_blank">Cross Over</a> looks like it has enough strategy to entertain us jaded adults, and it's for addition and subtraction practice. Coolo.</li><li>Not math. Go. <a href="http://aperiodical.com/2016/01/learning-to-play-go/" target="_blank">Learning to play go</a>. </li><li>New game for iphone (really, it's logic not math), <a href="http://blog.tanyakhovanova.com/2016/02/ringiana/" target="_blank">Ringiana </a></li><li>I love <a href="https://mikesmathpage.wordpress.com/2016/02/04/i-think-you-can-share-the-surreal-numbers-with-kids/comment-page-1/#comment-1992" target="_blank">surreal numbers</a>. I need to come back and read this more carefully when I have more time to play with it. </li><li>A silly little game. Totally violates Tracy's criteria (nothing timed). But mathy folk may like it. <a href="http://isthisprime.com/game/" target="_blank">How many primes can you identify in a minute</a> (with no mistakes)? (Use y and n for y and no.)</li></ul><br /><br /><b>Books</b><br /><ul><li>Here's a great list of <a href="http://aperiodical.com/2016/01/books-a-14-year-old-whos-good-at-maths-might-enjoy/" target="_blank">fun math books</a>, compiled with a 14-year-old in mind, but almost all good for adult mathophiles too. I think <a href="https://mikesmathpage.wordpress.com/2016/01/18/some-book-suggestions-for-a-14-year-old-who-loves-math/" target="_blank">this list</a> came from the same question and has a different set of books.</li><li>My publisher is having a <a href="http://naturalmath.com/goods/" target="_blank">sale</a>. All 5 books published by <a href="http://naturalmath.com/goods/" target="_blank">Natural Math</a> for $50 total. What a great way to expand your playful math collection. </li></ul><ul></ul>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com4tag:blogger.com,1999:blog-5303307482158922565.post-6139186327488816372016-01-16T22:14:00.000-08:002016-01-16T22:14:23.621-08:00My Favorite Course (to teach): Calculus<b><span style="color: #cc0000;"><span style="font-size: large;"><span style="color: #3d85c6;">Why is calculus my favorite?</span> Let me count the ways ...</span></span></b><br /><ol><li>It tells a story.</li><li>It has cool historical connections,</li><li>... and great connections to science.</li><li>It's a good time to help students start to see what proof means.</li><li>I keep learning more.</li></ol><br /><br /><b>Calculus Tells a Story...</b><br />...if we let it. And the conventional textbooks don't. So I used two different creative commons texts (Boelkins and Hoffman), some of my own materials, and a few things from some of my favorite bloggers, and I made a coursepack to use for the first three weeks. I gave a talk about it at the Joint Mathematics Meeting a week ago. As part of my preparation for that, I made a new blog page. Click 'calculus' above, and you'll see all of my materials, including the slides from my talk, links to the creative commons texts I used, and lots more.<br /><br />What stories does calculus tell? It takes one of the central concepts from algebra, that of slope, and twists it so it will work for curves. To do that, we need to consider two points that are "infinitely close together," whatever that means. So we have to delve into the weirdness of "infinitely close." Once we get good at all that, we can find out where things reach their maximum and minimum values, and use that to graph all sorts of curves. We also use that to optimize, to get the most volume with the least surface area (when building boxes), for instance. And then we play with finding areas of strange shapes, and how that's connected to slopes. <br /><br /><b><br /></b><b><br /></b><b>Calculus has cool historical connections, and great connections to science.</b><br />Archimedes figured out all sorts of things that are really a part of calculus (call it proto-calculus), and used the 'method of exhaustion' which is a foundation for what we now do with limits. Newton and Leibniz are credited with inventing calculus, even though lots of what we do in Calculus I had already been figured out. The main thing they discovered was what we call the Fundamental Theorem of Calculus, which says that areas and rates of change are inverse functions. It makes sense that two different people invented calculus because it was needed at the time for the science questions that were being considered: lenses and light, paths of planets, gravity, angle to shoot a cannon, volume of the Earth. And then it took 150 years to get that limit thing just right, and another 150 years (in 1960 Abraham Robinson invented non-standard analysis) to prove that Newton's original conception (of fluxions) wasn't so far off.<br /><a href="http://2.bp.blogspot.com/-AYDhD3yloxg/Vpsp8ICsNzI/AAAAAAAABpA/rubX2d-kln0/s1600/220px-CircleArea.svg.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="http://2.bp.blogspot.com/-AYDhD3yloxg/Vpsp8ICsNzI/AAAAAAAABpA/rubX2d-kln0/s200/220px-CircleArea.svg.png" width="184" /></a><br /><br /><br /><b>It's a good time to help students start to see what proof means.</b><br />Did you realize that the two 'formulas' we all know for circles are very different sorts of creatures? The first, C=2*pi*r, is really just a restatement of a definition. pi is <i>defined</i> to be C(ircumference) over D(iameter), so it takes 2 or 3 algebraic steps to get to C=2*pi*r. But the other, A = pi*r<sup>2</sup>, should be proved. The simplest almost-proof comes from cutting the circle up and rearranging it.<br /><br /><br /><br /><b>I keep learning more.</b><br />I learned two cool things while preparing for that talk: <a href="http://mathmamawrites.blogspot.com/2016/01/newton-and-notion-of-limit-he-knew-more.html" target="_blank">Newton had a clearer conception of limits than we usually think</a>, and <a href="http://mathmamawrites.blogspot.com/2015/12/the-roots-of-calculus-archimedes.html" target="_blank">Archimedes' calculation of an approximation for pi</a> was easier to follow than I would have imagined, and really simple and beautiful (in our modern notation).<br /><br />And to make this post a fun one for all you MTBOS folks, here's the worksheet I designed to share with my calculus class (.<a href="https://drive.google.com/file/d/0B4Lou9CsLnQxQTR1X0RzU1VCZWc/view?usp=sharing" target="_blank">doc</a> and .<a href="https://drive.google.com/file/d/0B4Lou9CsLnQxWV9UMVVza0Nldlk/view?usp=sharing" target="_blank">pdf</a>), leading them through Archimedes' first few steps as he worked toward the 96-gon to approximate pi. Go ahead, try it and put your answer for the 96-gon in the comments. (I couldn't find it anywhere else online!)<br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-i10vclQgGmw/VpsvZavb-hI/AAAAAAAABpU/sbwLX3foai0/s1600/pi%2Bworksheet%2Bimage.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="457" src="http://2.bp.blogspot.com/-i10vclQgGmw/VpsvZavb-hI/AAAAAAAABpU/sbwLX3foai0/s640/pi%2Bworksheet%2Bimage.png" width="640" /></a></div><br /><br /><br /><br />_____<br /><span style="font-size: x-small;">*(There's a better way to show word docs, right? Someone tell me. I should know that after all these years of blogging!)</span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-11228907619096759132016-01-02T09:07:00.000-08:002016-01-02T09:07:00.814-08:00Newton and the Notion of Limit (he knew more than I thought he did)Preparing to give a math talk has been very educational for me. I posted about ten days ago about finally figuring out <a href="http://mathmamawrites.blogspot.com/2015/12/the-roots-of-calculus-archimedes.html" target="_blank">how Archimedes calculated pi with his 96-gon</a>.<br /><br />Now I just found out that <a href="http://www.sciencedirect.com/science/article/pii/S0315086000923012" target="_blank">Newton wrote more about limits than we're usually led to believe</a>. In 1687, Newton wrote:<br /><br /><blockquote class="tr_bq"><span style="font-size: small;"><span style="font-family: "Times";">"Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...." </span></span></blockquote><br />This quote comes from Bruce Porciau's paper, <a href="http://www.sciencedirect.com/science/article/pii/S0315086000923012" target="_blank">Newton and the Notion of Limit</a>, in Historia Mathematica. He gives much more evidence that Newton understood the limit concept pretty well.<br /><br />I guess I can still say that it took the best minds in all the world 150 years to come up with a precise definition of limit. But Bishop Berkeley's complaint ...<br /><blockquote class="tr_bq">"And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?"</blockquote>... now seems to me more the product of a small mind and less the careful quest for precision of a mathematician. Now I lean more toward thinking Newton (and Leibniz?) got it, but it took 150 years for a mathematician to create a precise definition that would convince all the other mathematicians. Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-37513673826811674512016-01-02T08:01:00.001-08:002016-01-02T08:01:12.059-08:00Joint Mathematics Meetings in Seattle this coming weekI leave on Wednesday for the <a href="http://jointmathematicsmeetings.org/jmm" target="_blank">Joint Mathematics Meetings</a> in Seattle. I'm giving a talk there on using creative commons textbooks in calculus. Friday, 1:20pm, room 620. I'd like to meet online friends there!Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-77443934460629170002015-12-27T17:51:00.002-08:002015-12-27T17:51:58.773-08:00Does your kid hate math? Try a new angle.Long before I became a parent, in my teaching (of community college students), a number of them told me how bad they were at math even though their mom or dad taught it. I figured the parents pushed too much or something. (Blame the parents much, do we?) I ‘knew’ I wouldn’t do that.<br /><br />Well, I don’t think I pushed. But my son hates math, and is consequently way behind his peers. (He unschooled for years and there was no ‘behind’. But he chose to go to a regular middle school this year, where the other kids have mostly had the standard schooling.) So when two people I respect got into a meaty conversation about this, my antennae popped up. They’ve allowed me to share this conversation, which occurred in a closed group on Facebook called <a href="https://www.facebook.com/groups/1001mathcircles/" target="_blank">1001 Math Circles</a>. (Ask to join if you’d like - group description: A place to share and discuss your #mathcircles plus learn more about the Natural Math principles! Run by Shelley Nash and Maria Droujkova of NaturalMath.com.)<br /><br /><br /><br /><br /><i>Lhianna</i>: Hi. I'm a homeschool mom of daughters 7 and 13. I absolutely love math and creative problem-solving and my oldest daughter hates it. My failure to transfer my love of math to her drove me to find better ways of teaching and sharing the beauty and excitement that I see. I found out about Math Circles and have done <a href="http://themathcircle.org/" target="_blank">the summer training camp with Bob and Ellen Kaplan</a> for several years now. I run Math Circles around Philadelphia as time and opportunity allow. I love getting inspired by all the great ideas of a wonderful math community like this one. Thanks for letting me join!<br /><br /><br /><i>Maria</i>: Lhianna, welcome! The Kaplans’ community is wonderful. Maybe we can have a live chat sometime about your circles? When someone hates math, there is usually what I call a grief story. Even with homeschooling, our children can get enough grief "second-hand" from us, or from the society... When I ask people who hate math what happened to them, they usually do know, and tell their stories. Do you know what happened to your 13-year-old? And what does your 7 year-old like to do? It's such interesting age for girls!<br /><br /><br /><i>Lhianna</i>: My 7 year-old loves logic problems. (The island of knights and knaves kind. I have a special fondness for all of Raymond Smullyan's books!) She likes unit origami (especially the sonobe units). And she seems fascinated by anything to do with parity. Also building with geometric shapes of all kinds.<br /><br />I think my 13 year-old has a deep fear of getting things wrong in any subject and in general in life. In other subjects she finds ways around it. But it is especially devastating for mathematical exploration. You really have to try many different avenues and be able to look at your failures and analyze them to arrive at a solution in math. Math is about exploring what is unknown to you and she can't stand that. She prefers the familiar.<br /><br />It has been an interesting journey for me. I started thinking how lucky she is to get an exploratory background in math. I then realized my own shortcomings that, while I loved to explore math, I hadn't been able to communicate that idea to my child. Which led me on a wonderful journey of discovering Math Circles and many more amazing people and sources full of creative ideas about learning math.<br /><br />But as my daughter continued to hate it (and trying to do math with other people too, not just me), I also learned that math is not for everyone like I originally thought. It's ok now that she doesn't like math! That is a homeschooling journey to learn and accept this. (When she does do some math she is perfectly able to learn and understand the concepts. She just has zero interest and will not voluntarily spend any time on math study).<br /><br />I am currently dragging her through "The Art of Problem Solving" book series so she can have enough math to go on to higher education. (And it's a pretty decent series for a textbook!) I am very much an amateur. I am constantly learning and open to new ideas. Any suggestions would be greatly helpful.<br /><br /><br /><i>Maria</i>: Lhianna, thank you for sharing. Yes, I am with you - love of math for its own sake isn't for everyone (just like any other area); but I do feel that everyone can feel good doing some math-rich activities in their own ways. I see a pattern in your interaction with math and with your 13 year-old. Do most of your math activities center on problem-solving?<br /><br />In contrast, have you ever tried math activities that don't involve problems, solutions, answers, or unknowns? There are activities where you: (1) only work with what you know, and (2) don't seek any answers or solutions. When I say that now, can you picture 4-5 examples of activities that I am talking about?<br /><br /><br /><i>Lhianna</i>: Not off the top of my head. What kinds of activities are you thinking about?<br /><br /><br /><i>Maria</i>: Logic is so lovely! Smullyan's books made a difference for many people. <a href="http://naturalmath.com/camplogic/" target="_blank"><i><b>Camp Logic</b></i></a>, which we published this year, is one of our most popular books, too. I just sent three big boxes of it to groups. Next year, "Bright, Brave, Open Minds" will be out, by Julia Brodsky - there are very lovely logic activities in there, too. Here are a few things to try from that book:<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-uPUmmRWhwfE/VoCVCbiH4eI/AAAAAAAABn0/4cg41Z8ccU0/s1600/brodsky%2Bi%2Bam%2Blying.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://3.bp.blogspot.com/-uPUmmRWhwfE/VoCVCbiH4eI/AAAAAAAABn0/4cg41Z8ccU0/s640/brodsky%2Bi%2Bam%2Blying.jpg" width="414" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-5WNZSdiRlfE/VoCVDhUDAUI/AAAAAAAABn8/hCgHa2ovlb0/s1600/brodsky%2Bdinosaurs.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="497" src="http://1.bp.blogspot.com/-5WNZSdiRlfE/VoCVDhUDAUI/AAAAAAAABn8/hCgHa2ovlb0/s640/brodsky%2Bdinosaurs.jpg" width="640" /></a></div><br /><br /><br /><i>Lhianna</i>: I see my 13 year-old use math in other activities (she really likes to cook and make up her own recipes which involves experimentation and therefore doubling and tripling many measurements as well as analyzing the ratios of one ingredient to another). Is this what you are talking about? Or math games? She likes to play SET.<br /><br /><br /><i>Maria</i>: Lhianna, so the goal is to find math-rich activities that: (1) are not problem-solving, and (2) center on what you already know, and yet (3) are open and can be made uniquely yours. Let’s see if we can find a fresh angle on what your daughter can try…<br /><ul><li>Storytelling. You tell what you know; you make the story interesting, fun, pretty, and may invent details, but you know your story (and math therein). Vi Hart videos are like that. Or storybooks like The Cat in Numberland.</li><li>Illustrations. Take something you know. Illustrate it with a picture, comic, video, toys, interpretive dance smile emoticon Basically, represent it by some medium you like. A lot of math comics are illustrations of math jokes, for example.</li><li>Programming. Take a formula or pattern you know and use, and make your computer (spreadsheet, solver, etc.) do it for you.</li><li>Scavenger hunt. Find some math idea you know (e.g. ratio) in what you like (e.g. Star Wars, your favorite park, or your room). Or find a lot of math ideas in one book, movie, room... Make a curated collection. There are a lot of those online. Have you tried that sort of approach? How did it go?</li></ul>SET is a very good game too. To use this as an example of doing what you like and know... We do this activity where we make our own set of SET cards from scratch, using our own shapes and themes. On the one hand, it's something you know. On the other, the amount of delicious a-ha moments you have along the way is just incredible!<br /><br /><br /><i>Lhianna</i>: Great idea! Thanks. And thanks for the advice. I will start looking for activities and examples that follow along the lines of familiar but open. I appreciate the new perspective.<br /><br /><br /><i>Maria</i>: I would love to hear what else you find, because you have such a thoughtful approach to the whole thing! Moving the focus to, "Love SET, like Vi Hart videos, like Tangram puzzles..." (from, "hate math").<br /><br /><br /><br />Do you have a kid who hates math? Do any of these ideas sound like something you might want to try out with them? <br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-73946664094019545982015-12-24T15:58:00.001-08:002015-12-24T15:58:43.484-08:00Question for my ReadersLately, when I'm trying to write a post, I often get shifted over to some sort of ad. Does that happen to any of you reading my posts? If it does, I may move my blog over to Wordpress.Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com8tag:blogger.com,1999:blog-5303307482158922565.post-55768467239837351942015-12-23T21:31:00.000-08:002015-12-28T19:41:53.779-08:00The Roots of Calculus - ArchimedesArchimedes did a lot that nowadays looks like calculus...<br /><br />He determined the value of pi very precisely, by starting with a hexagon inscribed in a circle, then a 12-sided polygon, then he kept doubling the number of sides until he got to a 96-gon. A procedure like this is called the 'method of exhaustion', and it looks a lot like what we do nowadays with limits.<br /><br />I am embarrassed to admit that I couldn't figure out how he did it. (I think I was focusing on area, and that might be harder.) I just found <a href="https://www.youtube.com/watch?v=_rJdkhlWZVQ" target="_blank">a great video by David Chandler</a> (whose youtube channel is Math Without Borders).<br /><br />Here's a summary:<br />Start with a hexagon inscribed in a circle of radius 1 (giving diameter of 2). The perimeter of the hexagon will be 6. This gives a lower bound on pi, which is the ratio of circumference to diameter. We know the circumferences is bigger than this perimeter of 6, so pi is bigger than 6/2 = 3.<br /><br />If you cut one of the triangles that made the hexagon into two, you get a radius that crosses a side of the hexagon at right angles. You can use the Pythagorean Theorem (twice) to find the new side length. Repeat 3 times and you're at the 96-gon. Archimedes had none of our technology, and little or none of our algebraic symbolism, so the calculations were much harder for him. We can do all this on a spreadsheet, and up comes pi (if you have a column for the perimeter over the diameter). So satisfying!<br /><br />If this doesn't make sense, watch <a href="https://www.youtube.com/watch?v=_rJdkhlWZVQ" target="_blank">this lovely video</a>. Thank you, David!<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/_rJdkhlWZVQ/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/_rJdkhlWZVQ?feature=player_embedded" width="320"></iframe></div><br /><br />Archimedes did a lot more than find a value for pi! What's your favorite bit of calculus that started out with Archimedes?Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com9tag:blogger.com,1999:blog-5303307482158922565.post-12688593991900354262015-12-21T09:08:00.000-08:002015-12-21T18:18:34.784-08:00Fun Mathy BooksIs it too late to suggest good holiday gifts<i><b>? </b></i>Here are some books I think you might like.<br /><br /><br /><br /><i><b><a href="http://www.amazon.com/This-Not-Maths-Book-Activity/dp/1782402055" target="_blank">This is Not a Math Book</a></b></i>, by Anna Weltman<br /><br /><i><b><a href="http://www.abebooks.com/servlet/BookDetailsPL?bi=17662548102" target="_blank">Patterns of the Universe: A Coloring Adventure in Math and Beauty</a></b></i>, by Alex Bellos <br /><br /><a href="http://www.abebooks.com/servlet/BookDetailsPL?bi=17481782380" target="_blank"><b><i>Mathematical Mindsets</i></b></a>, by Jo Boaler<br /><i><br /></i><a href="https://www.bookbyte.com/textbooks/intentional-talk-how-to-structure-and/9781571109767-1571109765" target="_blank"><i><b>Intentional Talk: How to Structure and Lead Productive Mathematical Discussions</b></i></a>, by Elham Kazemi <i><br /></i><br /><br />Dan MacKinnon wrote a lovely review of a book I hadn't heard of before, at his blog, Math Recreation. Here's the beginning of it...<br /><blockquote class="tr_bq">In <i><b><a href="http://www.bookfinder.com/search/?ac=sl&st=sl&ref=bf_s2_a1_t1_1&qi=.Mzh9tzyG8iHT9ljpY0FCP6W5As_1450617514_1:33:543&bq=author%3Divan%2520moscovich%26title%3Dpuzzle%2520universe" target="_blank">The Puzzle Universe: A History of Mathematics</a></b></i>* <b>in 315 Puzzles </b>(TPU), <a href="http://yoz.com/wired/2.09/features/moscovich.html">Ivan Moscovich</a> stretches the concept of puzzles to encompass almost anything that combines curiosity and playfulness (<i>playthinks</i> is his preferred term for this more general category of puzzling items). No surprise - these playful curiosities are inherently mathematical. In an informal and accessible way, Moscovich details the development of these puzzles, revealing their surprising family resemblances and the deep mathematics behind their playful exterior. [<a href="http://www.mathrecreation.com/2015/12/a-universe-of-puzzles.html" target="_blank">read the rest at Dan's blog...</a>] </blockquote> <br />And of course, there's my book, <b><a href="http://naturalmath.com/playingwithmath/" target="_blank"><i>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</i></a></b><i>, </i>along with <a href="http://naturalmath.com/goods/" target="_blank">all the other cool books at Natural Math</a>.<br /><br /><br /><br /><br /><br /><span style="font-size: x-small;">_________</span><br /><span style="font-size: x-small;">*This link goes to bookfinder.com, which will point to other sites. It's the best way I know of to find the least expensive copy available. (My other links point to the sites that were cheapest at bookfinder on the day I wrote this.)</span>Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com2tag:blogger.com,1999:blog-5303307482158922565.post-46992092513026887362015-12-21T08:53:00.001-08:002015-12-21T09:03:56.799-08:00Lots of LinksFor months I've been saving cool things in tabs in my browser. I think I was up to over 80 tabs when I started cleaning up yesterday. Here are the goodies... <br /><br /><ul><li><a href="http://wild.maths.org/solve-very-old-problem" target="_blank">Trisect the Angle</a> (using origami axioms)</li><li><a href="https://rootsoftheequation.wordpress.com/2015/12/14/growth-vs-fixed-mindset-on-npr/" target="_blank">Growth Mindset</a> (a conversation with Neil deGrasse Tyson)</li><li><a href="http://denisegaskins.com/2015/12/15/understanding-math-area-of-a-rectangle/#comment-143037" target="_blank">Denise Gaskins on Understanding Math</a> (She references an article by Richard Skemp that differentiates between 'instrumental' and 'relational' understanding.)</li><li><a href="https://plus.maths.org/content/what-are-sigma-levels" target="_blank">What are sigma levels?</a> (statistics)</li><li><a href="http://www.npr.org/sections/thetwo-way/2015/08/14/432015615/with-discovery-3-scientists-chip-away-at-an-unsolvable-math-problem?utm_source=facebook.com&utm_medium=social&utm_campaign=npr&utm_term=nprnews&utm_content=20150814" target="_blank">A new pentagon for tiling the plane</a></li><li><a href="https://christopherdanielson.wordpress.com/2015/12/16/project-pentagon/" target="_blank">Project Pentagon</a> (Christopher Danielson is playing around, and thinking about the math.)</li><li><a href="http://www.intmath.com/integration/6b-fundamental-theorem-calculus-interactive.php" target="_blank">Fundamental Theorem of Calculus with proofs and an applet</a></li><li><a href="http://www.geogebra.org/student/m59882" target="_blank">What is a radian?</a> (geogebra applet)</li><li><a href="http://mathybeagle.com/2015/12/02/making-groups-work/" target="_blank">Teaching students to work well in groups</a></li><li><a href="http://mathwithbaddrawings.com/2015/10/28/the-differentiation-a-survivors-tale/" target="_blank">The Differentiation: A Survivor's Tale</a> </li><li><a href="http://www.intmath.com/blog/mathematics/wallis-pi-and-quantum-theory-10494" target="_blank">John Wallis, Pi, and Quantum Theory</a> (I need to read this again, and the next one)</li><li><a href="https://plus.maths.org/content/ramanujan" target="_blank">Ramanujan and Fermat's Last Theorem</a></li><li><a href="http://blogs.scientificamerican.com/roots-of-unity/teaching-the-controversy-is-5-3-five-3s-or-three-5s/" target="_blank">Is 5x3 Five Threes or Three Fives?</a> (Scientific American)</li><li>Steven Strogatz, in Scientific American, on <a href="http://www.newyorker.com/tech/elements/einsteins-first-proof-pythagorean-theorem" target="_blank">Einstein's First Proof</a> (my favorite proof of the Pythagorean Theorem, based on symmetry)</li><li><a href="https://plus.maths.org/content/secret-club-diverse-triangles-0" target="_blank">Using theater exercises to teach math</a> (Malke would like this!)</li><li><a href="http://drawingonmath.blogspot.com/2015/11/sine-and-cosine-waves-with-activity.html" target="_blank">Trig graphs on Desmos</a> (using their new <a href="https://teacher.desmos.com/activitybuilder/" target="_blank">activity builder</a>)</li><li><a href="http://www.theatlantic.com/education/archive/2015/11/math-showing-work/414924/" target="_blank">If you can't explain it, does that mean you don't understand it?</a></li><li><a href="http://untilnextstop.blogspot.com/2015/11/what-it-means-to-slow-down-problem.html" target="_blank">What it means to slow down a (calculus) problem</a></li><li><a href="http://youcubed.org/">youcubed.org</a> is Jo Boaler's new site (her new book is <i>Mathematical Mindsets</i>, which I hope to review soon)</li><li><a href="https://plus.maths.org/content/plus-advent-calendar-door-4-konigsberg-movie" target="_blank">A video on the Konigsberg Bridges Problem</a></li><li><a href="http://prairiecreek.typepad.com/herons/2015/10/turtle-triangles.html" target="_blank">Turtle Triangles</a> (on programming using turtle)</li><li><a href="http://blogush.edublogs.org/2015/11/01/it-takes-courage-to-play-in-a-world-that-does-not-play/" target="_blank">It takes courage to play in a world that does not play</a></li><li><a href="http://samjshah.com/2015/11/05/playing-with-blocks-three-dimensional-visual-sequences/" target="_blank">High school students playing with blocks</a> (3D visual sequences)</li><li><a href="http://www.epsilon-delta.org/2015/10/related-rates-related-to-you.html" target="_blank">Making related rates relevant by using students' names</a></li><li><a href="http://musingmathematically.blogspot.com/2015/10/wodb-polynomial-functions.html" target="_blank">Which one doesn't belong? (with polynomial functions)</a></li><li><a href="http://homeschoolerpost.com/16105/130218/a/the-emotional-connection-to-math" target="_blank">Pam Sorooshian on emotions and math</a></li><li><a href="http://blog.plover.com/aliens/dd/intro.html" target="_blank">A message to the aliens</a> </li><li><a href="http://mathpages.com/rr/s8-01/8-01.htm" target="_blank">Kepler, Napier, and the Third law</a> (I'm trying to learn more of the history of calculus, to help me teach calculus more effectively. This article is good.)</li><li><a href="https://bookzoompa.wordpress.com/2015/05/03/the-animated-equation-book/" target="_blank">Function flip books </a>(I thought I was done, but twitter is dangerously good!)</li></ul><br /><br /><b>Games, Puzzles, & Problems</b><br /><ul><li><a href="http://gamedesign.jp/flash/chatnoir/chatnoir.html" target="_blank">Chat Noir</a> (Can you corral the cat? I did it once. Can't do it again.)</li><li><a href="http://wild.maths.org/drips" target="_blank">Drips</a> (a nim game)</li><li><a href="http://mrhonner.com/archives/15551" target="_blank">How many sides of a pentagon can you see?</a></li><li><a href="http://nautil.us/issue/30/identity/how-to-solve-the-hardest-logic-puzzle-ever" target="_blank">A very hard truth and lies logic puzzle</a></li><li><a href="http://matharguments180.blogspot.com/2015/10/497-factor-grids.html" target="_blank">Factor Grid</a> (I wonder if I could make up my own versions of this) </li><li><a href="http://cemc.uwaterloo.ca/resources/potw.php" target="_blank">Some good problems of the week</a> (this site changes each week)</li><li><a href="http://mrhonner.com/archives/7717" target="_blank">A simple trig challenge</a> (I need to save this for my precalc class)</li><li><a href="http://matharguments180.blogspot.com/2015/12/512-factory-ratios-3.html" target="_blank">Factory Ratios</a><a href="http://www.ohiorc.org/for/math/stella/setintro/problem.aspx?id=415#" target="_blank">Speed of sound</a></li><li><a href="http://samjshah.com/2015/09/03/blermions-cyclic-quadrilaterals-and-cross-chords/#comment-98592" target="_blank">Blermions (an approach to some geometry questions)</a></li><li><a href="http://dailydesmos.com/2015/11/04/parabola-of-lines-1-advanced/" target="_blank">Can you make this graph?</a></li></ul><br /><br />I have to admit that I skip the Intermediate Value Theorem when I teach Calc I (please tell me if you think I'm short-changing my students), but here are two great posts about it. <a href="http://blogs.scientificamerican.com/roots-of-unity/math-on-the-run/" target="_blank">If you ran a race at an average pace of 3:07 per kilometer, did you run any single kilometer in exactly 3:07?</a> (from Scientific American) and <a href="https://christopherdanielson.wordpress.com/2015/12/08/a-new-calculus-activity-builder-activity/" target="_blank">an activity using Desmos</a> (from Christopher Danielson).<br /><br /> <br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0tag:blogger.com,1999:blog-5303307482158922565.post-89944915388211293142015-10-25T12:37:00.001-07:002015-10-25T15:11:12.666-07:00Math Teachers at Play, #91<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-QWruwGBIqGg/Vi0SUnWqbeI/AAAAAAAABm4/mdwWFZG3TJ8/s1600/91.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="http://1.bp.blogspot.com/-QWruwGBIqGg/Vi0SUnWqbeI/AAAAAAAABm4/mdwWFZG3TJ8/s200/91.jpg" width="200" /></a></div><span style="font-size: x-large;">Number 91</span> feels like we're closing in on 100. <a href="https://plus.maths.org/content/maths-minute-power-powers" target="_blank">The last time I hosted MT@P</a>, we were at #71 and I managed to include 71 posts. I wasn't quite that ambitious this time. (Old math posts don't go stale. You might enjoy browsing through a bunch of <a href="http://denisegaskins.com/mtap/" target="_blank">the old Math Teachers at Play blog carnivals</a>. And don't forget our partner carnival: the <a href="http://aperiodical.com/category/columns/carnival-of-mathematics/" target="_blank">Carnival of Mathematics</a>.) <br /><br />If there are 14 people in a group, and each shakes hands with each other, there will be 91 handshakes. (Can you see why?)<br /><br />91 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13<br />(which makes it triangular)<br /><br />and<br /><br />91 = 7 * 13<br />(the middle and last numbers in the sum above)<br /><br />Will this always happen for triangular numbers?<br /><br /><br /><br /><br /><h3>Games & Puzzles</h3><ul><a href="http://4.bp.blogspot.com/-2d1H38gvef8/Vi0fgrMfN8I/AAAAAAAABnQ/q6qR6eKFUAk/s1600/number-tile-puzzles-primary.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://4.bp.blogspot.com/-2d1H38gvef8/Vi0fgrMfN8I/AAAAAAAABnQ/q6qR6eKFUAk/s1600/number-tile-puzzles-primary.png" /></a><li><b>Shannon Duncan</b>, a 6th grade math & science teacher, shares <a href="http://blog.mindresearch.org/blog/game-based-learning-tips-from-math-educator" target="_blank">4 Reasons to Promote Math Success through Games</a> at the <i>MIND Research Institute</i> blog, illustrating her ideas with some of the games she has her students playing. I especially like the first point - making a mind-body connection.</li><li><b>John Golden</b> (@mathhombre) shares <a href="http://mathhombre.blogspot.com/2015/10/angle-of-coincidence.html" target="_blank">Angle of Coincidence</a> at his blog, <i>Math Hombre</i>, about an angle identification game he's developing. Ask your students to playtest it and give him feedback! John also wrote about the start of the semester, and included a game called <a href="http://mathhombre.blogspot.com/2015/09/a-sorted-beginning.html" target="_blank">In or Out?</a> that looks fun.</li><li><b>Jeff Trevaskis</b> shares a <a href="https://webmaths.wordpress.com/2015/10/18/multiplication-tic-tac-toe-in-3-acts/" target="_blank">Multiplication Tic-Tac-Toe Game</a> at his blog, <i>webmath<b>.</b></i><b> </b></li><li><b>Carole Fullerton</b> shares <a href="https://mindfull.wordpress.com/2015/10/17/number-tile-puzzles-primary-and-intermediate/" target="_blank">Number Tile Puzzles</a> at her blog, <i>Mathematical Thinking</i>.<b> </b></li><li><b>Gray Antonick</b> interviewed <a href="http://wordplay.blogs.nytimes.com/2015/06/01/salomon/" target="_blank">Paul Salomon in the New York Times Numberplay column</a>, about his Imbalance Puzzles, one of many puzzles and games featured in <i><b>Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers</b></i> (my book, published in April!).</li></ul><h3> </h3><h3>Arithmetic</h3><ul><li><b>Denise Gaskins</b> (@letsplaymath) shares an old favorite, <a href="http://denisegaskins.com/2008/09/22/things-to-do-hundred-chart/" target="_blank">30+ Things To Do with a Hundred Chart</a>, at her blog, <i>Let's Play Math</i>.</li><li><b>Brian Bushart</b> (@bstockus) shares <a href="https://bstockus.wordpress.com/2015/01/" target="_blank">Fraction Number Sense</a> at his blog, <i>Teaching To the Beat of a Different Drummer</i>. </li><li><b>Lior Pachter</b> shares <a href="https://liorpachter.wordpress.com/2015/09/20/unsolved-problems-with-the-common-core/" target="_blank">Unsolved math Problems and the Common Core</a> at his blog, <i>Bits of DNA</i>. (Lior writes about computational biology. I found this post thanks to Andrew Knauft, at <a href="http://blog.amathknauft.com/2015/10/share-from-repository-weekly_18.html" target="_blank"><i>LimSoup</i></a>.)</li></ul><h3><b> </b></h3><h3><b>Geometry </b></h3><ul><li><b>Stephen Cavadino </b>(@srcav) shares <a href="https://cavmaths.wordpress.com/2015/10/21/parallelograms/" target="_blank">Parallelograms</a> at his blog, <i>cavmaths</i>, on a student's creative way to find the area of a parallelogram.</li><li><b>Ioana I Pantiru</b> (@LThMathematics) shares <a href="https://lifethroughamathematicianseyes.wordpress.com/2015/10/17/playing-with-paper-folding/" target="_blank">Playing with Paper Folding</a> at her blog, <i>Life Through a Mathematician's Eyes</i>, showing the steps of an origami construction. In her post, <a href="https://lifethroughamathematicianseyes.wordpress.com/2015/10/15/maths-class-everywhere-project/" target="_blank">Maths Class Everywhere</a>, she asks readers to take her survey of math classes around the world. </li><li><b>Curmudgeon</b> shares <a href="http://matharguments180.blogspot.com/2015/10/498-circles-on-lattice.html" target="_blank">Circles on a Lattice</a>, at their blog, <i>Math Arguments 180</i>. I wonder if this would make a good problem for a math circle... </li><li><b>Greg Blonder</b>, a professor of manufacturing and product design, shares <a href="https://plus.maths.org/content/trisecting-angle-ruler" target="_blank">Trisecting the Angle With a Straightedge</a>, at <i>Plus Maths</i>.</li><li>There have been lots of posts in the past few months about classifications of pentagons (<a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile" target="_blank">here's one</a>), because a new (15th) type of pentagon that will tile the plane was recently found. Here's <a href="http://mathtourist.blogspot.com/2010/06/tiling-with-pentagons.html" target="_blank">a good background post</a>, from before the discovery, from the <i>Mathematical Tourist</i>.</li></ul><h3> </h3><br /><h3>It's All Connected</h3><ul><li><b>Miss D</b> shares <a href="http://www.missdtheteacher.blogspot.co.nz/2015/10/age-of-ultron.html" target="_blank">The Age of Ultron</a> at her blog, <i>Miss D the Teacher</i>, about teaching a unit on artificial intelligence in a way that gets at the deep ideas and really gets students thinking, partly through connecting math, science, and art. </li><li><b>Henri Picciotto</b> (@hpicciotto) posts about <a href="http://blog.mathedpage.org/2015/10/more-on-programming-in-education.html" target="_blank">Computer Programming and Math Education</a>. </li><li>What is the distance to Mars? It changes depending where the two planets are in their orbits. <b>John D. Cook</b> <a href="http://www.johndcook.com/blog/2015/10/24/distance-to-mars/" target="_blank">explains the math</a>. </li><li><b>Michelle</b> shares <a href="http://prairiecreek.typepad.com/herons/2015/10/making-time-for-the-serindipitous.html" target="_blank">Making Time for the Serendipitous</a> at <i>The Rookery</i>.</li> </ul><ul> </ul><h3>Ideas for Learning ...</h3><ul><li><b>Kate Snow</b> (@katesmathhelp) shares <a href="http://kateshomeschoolmath.com/how-to-teach-your-kids-to-read-math-and-be-more-independent-too/" target="_blank">How to Teach Your Kids to Read Math</a> at her blog, <i>Kate's Homeschool Math Help</i>. I'm still trying to teach my college students how to read math, with some of the same tips. </li><li><b>Manan</b> (@shalock) shares <a href="http://mathmisery.com/wp/2015/08/31/becoming-mathematically-fluent/" target="_blank">Becoming Mathematically Fluent</a> at his blog, <i>Math Misery.</i></li><li><b>Shecky</b> (@sheckyr) shares <a href="http://math-frolic.blogspot.com/2015/10/true-deep-beauty-comes-only-with.html" target="_blank">True Deep Beauty ...</a> at his blog, <i>Math-Frolic</i>, about the how our understanding of math deepens.</li><li><b>Chris Rime</b> is making <a href="https://partiallyderivative.wordpress.com/2015/09/30/october-2015-problem-calendars/" target="_blank">monthly math calendars</a> (Algebra I, II, and Geometry), available as doc or pdf at his blog, <i>Partially Derivative</i>.</li></ul><br /><h3>... And Teaching</h3><ul><li><b>Tom Bennison</b> (@DrBennison) shares <a href="http://blog.ifem.co.uk/how-to-enjoy-your-nqt-year/" target="_blank">How to enjoy your NQT Year</a> at his blog, <i>Mathematics and Coding</i>. [I had to look up NQT. It means newly qualified teacher, and in England and Wales, you are "inducted" in your NQT year, (generally) your first year of paid teaching.] I like his suggestion to make time for doing some math(s) yourself. </li></ul><br /><br /><br /><h3>Announcements</h3>I'm going to the <a href="http://jointmathematicsmeetings.org/jmm" target="_blank">Joint Mathematics Meetings</a> in January in Seattle. I'd love to connect with other bloggers who are going. There's a <b>math poetry reading</b> plus art exhibit on Thursday evening at 5:30. You can get all the details from <a href="http://poetrywithmathematics.blogspot.com/2015/10/jmm-seattle-1-7-16-poetrymathart.html" target="_blank">JoAnne Growney's Intersections blog</a>.<br /><br />Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.com0