tag:blogger.com,1999:blog-55446619683269100272017-03-28T00:19:24.477-07:00Three J's LearningJoshua Greenenoreply@blogger.comBlogger293125tag:blogger.com,1999:blog-5544661968326910027.post-19735147103112607692017-03-06T06:19:00.002-08:002017-03-06T06:19:58.836-08:00Cryptarithmetic puzzles follow-upI was asked to write a bit about strategies and answers for the <a href="http://3jlearneng.blogspot.com/2017/02/cryptarithmetic-puzzles-for-grades-1-to.html" target="_blank">puzzles we gave two weeks ago</a>.<br /><br /><b>BIG + PIG = YUM</b><br />Because the digits in YUM are all distinct from BIG and PIG and there are only 7 letters in this puzzle, we should expect there to be many solutions.<br /><br />The easiest way to get a feel for the puzzle is to start trying values and see what develops. This was part of the idea of using this puzzle as the opening challenge.<br /><br />As we play with examples, the kids should notice these things that constrain our possible solutions:<br /><br /><ol><li>B, G, I, M, P, U, Y must all be distinct</li><li>We are adding two three digit numbers and the sum is a three digit number</li><li>B, P, and Y are all leading digits</li><li>The largest sum possible with two numbers 0 to 9 is 18.</li></ol><div>Some conclusions:</div><div>(a) G is not 0. If it was, then M would also be 0.</div><div>(b) B, P, and Y are all not 0. They are leading digits, the rules of our puzzles say they can't be zero.</div><div>(c) G + G is at most 18. It may contribute at most one ten to the calculation of U. That will only happen if G is 5 or larger.</div><div>(d) I + I is at most 18. Along with a potential ten from G+G, that means we have at most 19 coming from the tens. That will only happen if I is 5 or larger.</div><div>(e) B+G is at most 9. If there is an extra hundred coming from the tens digits, B+ G is at most 8.</div><div>(f) If I is 9, G must be less than 5. Can you see why?</div><div>(g) If G is less than 5, I cannot be 0</div><div><br /></div><div>After these observations, I'd suggest picking values of G, then seeing what values of I are allowed, then checking what remains for B and P. Because we aren't allowed to have duplicates, we quickly see that our choices are constrained.</div><div><br /></div><div>For example, if G is 1 or 2, then I is at least 3 and we get the following possible solutions (B and P can be interchanged):</div><div>431 + 531 = 962</div><div>341 + 641 = 982</div><div>351 + 451 = 802</div><div>371 + 571 = 942</div><div>381 + 581 = 962</div><div><br /></div><div>132 + 732 = 864</div><div>132 + 832 = 964</div><div>152 + 652 = 804</div><div>152 + 752 = 904</div><div>182 + 582 = 764</div><div>192 + 392 = 584</div><div>192 + 592 = 784</div><div><br /></div><div>There are some more advanced ideas that could come out of trying to count or list all of the solutions, so I'd encourage people to explore. Even this simple puzzle can be a lot of fun!</div><div><br /></div><div><b>CAT + HAT = BAD</b></div><div>The A in BAD is the key part of this puzzle. We can get two cases:</div><div>(a) A is 0 and T is 1, 2, 3 or 4</div><div>(b) A is 9 and T is 5, 6, 7 or 8.</div><div><br /></div><div>Again, while there are a lot of solutions (and counting them would be a fun challenge) they are easiest to build up by choosing A (either 0 or 9), then T, then seeing what flexibility is left for C and H. Here are some examples:</div><div><br /></div><div>301 + 401 = 702</div><div>301 + 501 = 802</div><div>301 + 601 = 902</div><div>302 + 502 = 804</div><div>302 + 602 = 904</div><div>103 + 403 = 506</div><div>395 + 495 = 890</div><div><br /></div><div><b>SAD + MAD + DAD = SORRY</b></div><div>This was a puzzle without a solution. In this case, it isn't too hard to see that SORRY has too many digits. The best explanation was given by one student:</div><div><ul><li>The largest three digit number is 999. </li><li>If we add three of them, we will at most get 2997. </li><li>SORRY has to be bigger than 10,000.</li><li>This isn't possible</li></ul><div><b>CURRY + RICE = LUNCH</b></div></div><div>Unfortunately, this also doesn't have a solution, but the reasoning is more subtle than the previous puzzle.</div><div><br /></div><div>Here, we can reason as follows:</div><div><ul><li>R cannot be 0 because it is the leading digit in RICE</li><li>Because the tens digit of RICE and LUNCH are both C, R must be 9 and we must have Y + E > 10.</li><li>This also means R + C + 1 = 10 + C.</li><li>That will mean the 100s digit of RICE must be the same as the 100s digit of the sum.</li><li>However, the 100s digit of RICE and LUNCH are different.</li></ul><div>Too bad, it was such a cute puzzle!</div></div><div><br /></div><div><b>ALAS + LASS + NO + MORE = CASH</b></div><div>This is the most challenging puzzle from this set.</div><div><br /></div><div>Some things we notice:</div><div><ol><li>There are ten letters (A C E H L M N O R S) and they must all be distinct.</li><li>We are adding three 4-digit numbers and a two digit number to produce another 4 digit number.</li><li>A, L, N, M and C are leading digits, so they can't be zeros.</li><li>The tens and hundreds digits of CASH (S and A) are also involved in the sums for those digits.</li></ol><div>Point 4 has a subtle implication, which I'll illustrate with the hundreds digits. Since L + O must be more than 0, but A is the hundreds digit of the sum, we must have some number of thousands carried over. Because A, L and M are all distinct and larger than 0, the smallest their sum can be is 1+2+3. Putting these two observations together, C must be at least 7.</div></div><div><br /></div><div>In this case, I find it helpful to put together a table showing possibilities that we have eliminated:</div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Ui8BxZMCIHI/WLuNrcMOB0I/AAAAAAAAB3s/jOeEFeo4EREai32kvI5LqNxKbwnRraJ7QCLcB/s1600/ALASLASSNOMORECASJ.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="204" src="https://3.bp.blogspot.com/-Ui8BxZMCIHI/WLuNrcMOB0I/AAAAAAAAB3s/jOeEFeo4EREai32kvI5LqNxKbwnRraJ7QCLcB/s640/ALASLASSNOMORECASJ.png" width="640" /></a></div><div>We can see some more restrictions from the fact that A + L + M must be less than 9. That means we have only the following possible triplets (ignoring order):</div><div style="text-align: center;">{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}</div><div><br /></div><div>One thing we notice is that 1 is in all of these triplets, so either A, L or M must be 1 and none of the other letters can be 1. Another thing we notice is that we don't yet have any way of differentiating A, L, or M, so any ordering of our triplets is possible. That would mean we have 24 cases to consider.<br /><br />Let's see how we would work through the cases, starting with A = 1, L = 2, M = 3, the first on our list. Now this, happens to be a stroke of luck, as we'll see.<br /><br />Starting from the thousands digit, we see that this would make C = 7, if there is a single carry from the hundreds. Indeed, we can see that this must be the value (in the case we are testing), as the carry from there could only come from L + O (plus any carry from the tens digit). Since L is at most 5, L + O is at most 14 and any carry from the tens digit must be less than 6.<br /><br />Now, in the hundreds digit, we have 2 + O + carry from the tens = 10, so O = 8 - carry from tens.<br />We know there must be at least one carry from the tens, so O is at most 7. Since 7 is already used by C, let's try 6. That means we need to get 2 hundreds carried over from the tens, so we need<br />A + N + R + carry from ones = 20, or N + R + carry from ones = 19. Since we have already used 6 and 7, the only way this is possible is if N and R are 8 and 9 (in either order) and we are carrying 2 from the ones.<br /><br />At this point, the case we've worked through has:<br />121S + 21SS + 86 + 369E = 71SH<br /><br />We still have to allocate digits 0, 4, and 5. and we know that S + S + 6 + E = 20 + H. Given our remaining digits, the biggest the left hand can be is if S is 5 and E is 4, making 20. The smallest the right hand can be is if H is 0. Fortunately, this makes the equality hold, so we get our final answer:<br /><br />1255 + 2155 + 86 + 3694 = 7150<br /><br />Through the process of checking this case, we learned more about how the carry from lower digits is restricted and it would be faster for us to check through remaining cases.<br />Let me know how many other solutions you find!<br /><br /></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b><complete id="goog_1412240271"></complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL +</complete></b></div><div><b>LOL + LOL + LOL + </b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ </complete></b><b>LOL + LOL + LOL <complete id="goog_1412240271">+ LOL + LOL = ROFL</complete></b></div><div><b><complete><br /></complete></b></div><div><complete>There are 71 LOLs, so this is 71 x LOL = ROFL. While this looks daunting, there are some ideas which take us a long way to the solution.</complete></div><div><complete><br /></complete></div><div><complete>First, ROFL has 4 digits. If L were 2, 71 x LOL would be more than 14,000, so L must be 1. In fact, ROFL is less than 9861, so LOL is smaller than 9871 / 71 which is 139. We can quickly check</complete></div><div><complete>101, 121, and 131 and see that 131 works.</complete></div><div><complete><br /></complete></div><div><complete>71 x 131 = 9301</complete></div><div><complete></complete></div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-55965637414730859812017-02-23T00:02:00.003-08:002017-02-23T00:02:29.212-08:00A bit of 3D(ish) geometryNote: this post started out focused on two recent geometry projects. However, the Desmos Function Carnival, which I originally just included as a miscellaneous item, is also worth your time.<br /><h1>Nets and solids</h1>The most ambitious project recently was led by P (the mom, of course). She found nets for 3-d shapes and supervised J1 and J2 as they created a nice display to take to school.<br /><br />Here are the nets: <a href="http://www.senteacher.org/worksheet/12/NetsPolyhedra.html" target="_blank">SenTeacher Polyhedral nets</a>. They have a collection of other printables, but this collection of nets seems to be the most interesting. Take a look and let me know if you see anything else worthwhile.<br /><br />Here's the completed board:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-676ULIQxdTs/WKzqVJy-8xI/AAAAAAAAB24/HoSERzvhoEwmLjaD1m-CcwalP90Y3QLBgCLcB/s1600/IMG_0935.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-676ULIQxdTs/WKzqVJy-8xI/AAAAAAAAB24/HoSERzvhoEwmLjaD1m-CcwalP90Y3QLBgCLcB/s400/IMG_0935.JPG" width="300" /></a></div><br />A profile picture to show that these are really 3-d:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-d8DUT_BrYxA/WKzqZPiB3uI/AAAAAAAAB28/eZ_u2gu_AwgvAHGawH2H0hyr4j9V9IeSQCLcB/s1600/IMG_0936.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-d8DUT_BrYxA/WKzqZPiB3uI/AAAAAAAAB28/eZ_u2gu_AwgvAHGawH2H0hyr4j9V9IeSQCLcB/s400/IMG_0936.JPG" width="300" /></a></div><br /><a href="http://gwydir.demon.co.uk/jo/solid/index.htm" target="_blank">Solid Shapes and Their Nets</a> (I think Jo Edkins is the author) has a nice discussion of nets and a little puzzle game to distinguish nets that fold into the platonic solids and which don't. Feel free to try to do this for the icosahedron or dodecahedron!<br /><br /><h1>A two dimensional challenge?</h1>J2 asked me about a triangle with two right angles. Of course, we all know that a triangle can't have two 90 degree angles, right? Well, this fits in my list of math lies from this old post: <a href="http://3jlearneng.blogspot.com/2015/01/23-isnt-prime-short-bedtime-story.html" target="_blank">23 isn't prime</a>.<br /><br />We talked briefly about triangles on a plane and agreed that two 90 degree angles wouldn't work. If we try by starting with a side and building two right angles on that side, we just get parallel lines. Ok, that's standard.<br /><br />However, what about this:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-JH3GiGySiCA/WKzqTOt8YRI/AAAAAAAAB20/S9NJ8hAfkT427RHUv6nZUlWTh8ck6u3AQCLcB/s1600/IMG_0930.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-JH3GiGySiCA/WKzqTOt8YRI/AAAAAAAAB20/S9NJ8hAfkT427RHUv6nZUlWTh8ck6u3AQCLcB/s400/IMG_0930.JPG" width="300" /></a></div><br />We discussed spherical geometry for a bit. Not shown is our first attempt on the other side of the tennis ball that did have two right angles, but the third wasn't. He wanted to see an equiangular triangle on a sphere. This led to further discussion of life on a sphere:<br /><br /><ul><li>what is the largest interior angle sum for a triangle?</li><li>what is the smallest interior angle sum?</li><li>are there any parallel lines?</li><li>are there any squares? are there even any rectangles?</li><li>is π (the ratio of the circumference to diameter of a circle) a constant?</li><li>do we have to think like this in real life because the earth is close to a sphere?</li></ul><div>Picking up the point about π, it occurs to me that this is another math lie. This point got a nice treatment recently by <a href="http://smbc-comics.com/comic/pi" target="_blank">SMBC</a>:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img height="320" src="https://www.smbc-comics.com/comics/1481553833-20161212.png" style="margin-left: auto; margin-right: auto;" width="132" /></td></tr><tr><td class="tr-caption" style="text-align: center;">I made this pic small so you will go to the <a href="http://smbc-comics.com/comic/pi" target="_blank">site and look at Zach Weinersmith's </a>other awesome work</td></tr></tbody></table><div class="separator" style="clear: both; text-align: left;"><b>Some open follow-ups:</b></div><div class="separator" style="clear: both; text-align: left;"></div><ul><li>what is an equiangular quadrilateral on the sphere?</li><li>what about hyperbolic geometry? I think no triangles with two right angles. If I recall correctly, triangles there have angle sums smaller than 180 degrees (or π radians, ha!)</li></ul><h1>Other Misc Math</h1>Now for my usual grab bag of other things we've been doing. Some of these are really great activities, so don't skimp on this section!<br /><b><br /></b><b>Desmos Function Carnival</b><br />We learned about the <a href="https://student.desmos.com/carnival/student-welcome/5489163c20d9898c7e223aea" target="_blank">Desmos Function Carnival activity</a> from a post by <a href="http://www.kenthaines.com/blog/2017/2/13/functions-are-finally-clicking" target="_blank">Kent Haines</a> which, in turn, we learned about via <a href="https://problemproblems.wordpress.com/2017/02/16/functions-rules-formulas/" target="_blank">Michael Pershan's post</a>. They both were writing about teaching functions to their students and have outlined a really nice sequencing of lessons, if you're into that kind of thing.<br /><br />In case you are, you might like to know that there is another flavor of Function Carnival available through the <a href="https://teacher.desmos.com/carnival/" target="_blank">Desmos teacher site</a> with 2 other activities. Each version is worth checking out because both of worthwhile activities that aren't in the other. Also, Kent links to <a href="http://map.mathshell.org/lessons.php?unit=8225&collection=8&redir=1" target="_blank">this nice graphing activity from the Shell Center</a> which also worked well with J1 and J2.<br /><br />For our purposes, I was more interested in the mathematical physics (plots of position vs time or velocity vs time) and the fun of creating wacky animations with impossible plots.<br /><br />If I can figure out how to post the animations, I will update this page, since the animations really enhance the experience. For now, let me give you some screen caps of some of their proposed graphs from the Cannon Man (height) activity:<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-Fj4cAyqkPUQ/WK6VRYq-AbI/AAAAAAAAB3c/4el-dWbcnoE7Q_D5B_OreVgf11DNf_F2gCLcB/s1600/cannon%2Bman%2Bnames.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="245" src="https://2.bp.blogspot.com/-Fj4cAyqkPUQ/WK6VRYq-AbI/AAAAAAAAB3c/4el-dWbcnoE7Q_D5B_OreVgf11DNf_F2gCLcB/s400/cannon%2Bman%2Bnames.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Have a free-form drawing tool? Make your name!</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-xBWoWYHFnfg/WK6VRGD-_xI/AAAAAAAAB3Y/kKqthAkb2tc2kTTQkvD7glwNWBNnJgr2QCLcB/s1600/cannon%2Bman%2Bunion%2Bjack.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://1.bp.blogspot.com/-xBWoWYHFnfg/WK6VRGD-_xI/AAAAAAAAB3Y/kKqthAkb2tc2kTTQkvD7glwNWBNnJgr2QCLcB/s400/cannon%2Bman%2Bunion%2Bjack.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Note: the vertical stripe doesn't really work with this animation, but the patriotic spirit is there!</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-TGj7OCFkmPA/WK6VQi9FbdI/AAAAAAAAB3U/b_DW7lmGrwo4oZYX7lclYFkpYnH3xPZ_gCLcB/s1600/particle%2Bdiagram.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="291" src="https://1.bp.blogspot.com/-TGj7OCFkmPA/WK6VQi9FbdI/AAAAAAAAB3U/b_DW7lmGrwo4oZYX7lclYFkpYnH3xPZ_gCLcB/s400/particle%2Bdiagram.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Our first attempt at <a href="https://en.wikipedia.org/wiki/Feynman_diagram" target="_blank">Quantum Field Theory</a></td></tr></tbody></table><br /><br />Here is J2 playing with the function carnival, but J3 (4 years old) enjoyed it just as much:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-7iWPM9_6h5w/WKzqSzmzvzI/AAAAAAAAB2w/_tUpXUM10Qofio6h08xVcjLmcJxDLMbbQCLcB/s1600/IMG_0929.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-7iWPM9_6h5w/WKzqSzmzvzI/AAAAAAAAB2w/_tUpXUM10Qofio6h08xVcjLmcJxDLMbbQCLcB/s320/IMG_0929.JPG" width="240" /></a></div><br /><b>Different representations</b><br />J3 is working on place value and playing with different representations of numbers. Here, she's got the 100 board, an abacus, and foam decimal models. Sometimes I challenge her to make a number, sometimes she challenges me.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/--1evcdriQJU/WKzqKfZqDxI/AAAAAAAAB2s/VpNhelRGF9o40Kcf6_Ywnq7yV97DD-SxwCLcB/s1600/IMG_0928.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/--1evcdriQJU/WKzqKfZqDxI/AAAAAAAAB2s/VpNhelRGF9o40Kcf6_Ywnq7yV97DD-SxwCLcB/s320/IMG_0928.JPG" width="240" /></a></div><br />Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-42503566588879727182017-02-21T19:37:00.000-08:002017-02-21T19:37:37.466-08:00RSM International Math ContestSometime the first week of February, the RSM offers an on-line math competition. For the second year, I had J1 and J2 work through the problems at grade 3 and grade 4 level. This post is about our thoughts on competition, but I end with a problem from the contest that had us debating.<br /><h1>Compete and win!</h1>To get it out of the way, let me explain my interest in the competition and why I had the kids enter. Above all, I was curious about the problems and expected they would be interesting challenges. We found the puzzles last year interesting, so I was pretty sure this year's collection would also be nice.<br />While I know that there are tons of excellent activities, I couldn't resist making use of this resource when someone thoughtful had put together a convenient collection in one place.<br /><br />Second, I am curious about levels and assessments. What types of questions does RSM think 3rd and 4th graders should be able to answer, but find challenging? How difficult would our two little ones find them? Since we are currently operating outside a standard curriculum framework, it is hard for me to judge where they are or what we should expect of them. Admittedly, there are other ways I could form this assessment, but they take more effort.<br /><br />Notice that I don't really care about how well they perform on the test. It is interesting information for me, but I don't <em>need</em> them to do well.<br /><br />P has a different view. She was an olympiad kid and sees competitions as the easiest way for our kids to distinguish themselves. You know the anxiety: if they don't win competitions, they won't get into Harvard, they'll end up on the street somewhere.<br /><br />Perhaps unsurprisingly, the kids are getting a mixed message about the importance and reason for participating in competitions. This year, J1 was particularly sensitive. He was very resistant to doing the RSM test. I spent a lot of time talking with him. Mainly, I wasn't hoping to convince him to do the test, but I wanted to explore other issues related to the expectations he feels on himself, his relationship with J2, and his mindset about his own learning.<br /><br />I am trying to communicate: <br /><br /><ol><li>effort and progress are important, starting point and base ability aren't. I know this is debatable, but I'm talking about the differences between my kids where I'm on pretty firm ground.</li><li>there are many aspects to math and mathematical ability. Calculating and answering questions quickly are facets, logical reasoning, strategic thinking, spatial reasoning, asking good questions, gathering data, exploring connections, etc, etc are all components, too.</li><li>time isn't important. One issue with the RSM test is that it is timed. This goes in the face of our repeated efforts to emphasize that the time it takes to solve something is not a key consideration. To help with this, I screen capped the questions so he could work on them after the official time had expired. Note, however, his recorded performance on the test was based on work within the contest rules. </li><li>Math is not about right answers.</li></ol><div>The next section might help illustrate that last point.</div><br /><h1>Which area do we want?</h1>As mentioned above, one of the questions from grade 4 cause us to argue among ourselves. I'm paraphrasing slightly:<br /><blockquote class="tr_bq">Two tennis ball machines stand on opposite ends of a 25 meter by 10 meter court. The yellow machine shoots yellow balls that stop on the court 2 meters to 16 meters from the yellow machine's side. The green machine shoots green balls that stop on the court 5 meters to 20 meters from the green machine's side. <b>Find the area of the court that has balls of either color on it.</b></blockquote>Here are some ideas from our discussion:<br /><br /><ul><li>Option A: we want to find the area of overlap, because that's the only place we could find balls of either color.</li><li>Option B: we want to find the combined areas, because that's where we could find tennis balls, either color.</li><li>Option 1: the area with green balls is a rectangle, the area with yellow balls is a rectangle.</li><li>Option 2: think of the ball machines as single points, shooting balls at various angles and various distances. the target area for each machine is an intersection of of a circular ring and a rectangle. This idea came from J2 when we were looking over the grade 4 questions after time had expired.</li></ul><div>To answer the question, choose either A xor B and choose either 1 xor 2. So, what is the right answer?</div><div><br /></div><div>I think the right answer is to have this discussion, to encourage multiple interpretations (with justification), to see how the answers compare, to think about what other things we could do to make the intended interpretation more clear (a diagram seems the most obvious), to recognize that this is part of the richness of math and it can't be represented by a single numerical answer on a timed test.</div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-50855163986311816432017-02-21T02:06:00.000-08:002017-02-21T02:06:29.794-08:00Cryptarithmetic Puzzles for Grades 1 to 4Inspired by a series of puzzles from <a href="http://mathmisery.com/" target="_blank">Manan Shah</a>, I decided to have the kids play with cryptarithmetic puzzles today. In addition to borrowing some of Manan's puzzles, I also used some from this puzzle page: <a href="http://www.brain-fun.com/Cryptic-Math-Puzzles/" target="_blank">Brain Fun</a>. I've included some more comments below about the Brain Fun puzzles.<br /><br />My main concern was whether the puzzles were at the right level. In particular, I was afraid that the puzzles would be too hard. In fact, I tried solving a bunch of them yesterday and actually found myself struggling. I'll ascribe some of that to being tired and sick. However, my intuition was to make some simpler puzzles of my own. In particular, I added:<br /><ul><li>puzzles that have many solutions: I figured that many solutions would make it easy to find at least one.</li><li>a puzzle that "obviously" has no solution. Now, obviously, the word "obviously" is a sneaky one in math, but I was pretty sure the kids could see the problem with this structure.</li></ul><div><h1>Grades 1 and 2</h1></div><div>For the younger kids, I started with a shape substitution puzzle. This is one our family explored almost 2 years ago: <a href="http://3jlearneng.blogspot.com/2015/05/twist-on-old-puzzle-and-our-number.html" target="_blank">Shape Substitution</a>. I don't recall the original source.</div><div><br /></div><div>Two reasons why I started with this. First, it has a lot of solutions, but there is an important insight that unlocks those solutions. Second, by using shapes, we can write possible number solutions inside them as we solve or guess-and-check the puzzle. This made it easier for the kids to see the connection that all squares have the same value, etc.</div><div><br /></div><br /><div>The second puzzle: BIG + PIG = YUM</div><div>Really just a warm-up practicing the rules and doing a little bit of checking that we haven't duplicated any numbers.</div><br /><div><br /></div><div>The third puzzle: CAT + HAT = BAD</div><div>Again, lots of solutions, but noticing leads to a good insight.</div><div><br /></div><div>Fourth puzzle: SAD + MAD + DAD = SORRY</div><div>This is a trick puzzle. The kids know that I like to tease them, so they are aware they need to look out for things like this. We discussed this in class and I suggested they give this puzzle to their parents.</div><div><br /></div><div>Fifth puzzle: CURRY + RICE = LUNCH</div><div>When I translated this to Thai, all the kids laughed. I was sneaking a little bit of English practice into the lesson and then they realized that it was worth trying to read all the puzzles, not just solve them.</div><div><br /></div><div><i style="font-weight: bold;">Sources: </i>I think I made up all of these puzzles (original authors, please correct me if I'm wrong).</div><div><h1>Grades 3 and 4</h1></div><div>The older kids already had experience with these puzzles. We did refresh their memory a bit with BIG + PIG = YUM</div><div><br /></div><div>I asked them to give me the rules and explain why those rules made sense. As with most games, I want to communicate that we're doing things for a reason, but those reasons can be challenged. If they think it makes sense to do it a particular way, we're open to their ideas.</div><div><br /></div><div>Second puzzle: SAD + MAD + DAD = SORRY</div><div>Same discussion as for the younger kids. When prompted, this was pretty easy for them to spot, but they weren't naturally attuned to think about whether a puzzle had solutions or how many. This led me to take a vote on all the puzzles at the end to see who thought the puzzles would have 0, 1 or many solutions.</div><div><br /></div><div><div>Third puzzle: ALAS + LASS + NO + MORE = CASH</div><div>A puzzle from Brain Fun. I think this is one of the easier ones on that page. Again, a bit of English practice.</div></div><div><br /></div><div><div><div>Fourth puzzle: LOL + LOL + LOL + .... + LOL = ROFL (71 LOLs)</div><div>This was from Manan. I think it is one of the easier ones in his collection, but it looks daunting. Turns out none of the kids in the class were familiar with (English) texting short-hand, so my attempt to be <i style="font-weight: bold;">cool</i> fell flat.</div></div></div><div><br /></div><div>Fifth puzzle: CURRY + RICE = LUNCH</div><div>Again, everyone was delighted when I translated this one. We're in Thailand, after all, so at least one puzzle had to be about food.</div><div><br /></div><div><h1>The key exercise</h1></div><div>The final assignment everyone (all four grades) was given was to make up a puzzle for me to solve. I was thinking it would be nice to have one in Thai, but we decided to keep it in English as further language practice.<br /><br />Manan wrote a nice post about having kids design their own puzzles. If it goes well, this is actually the activity that ties a lot of the learning messages together: they think about structure, they think about what allows multiple or single solutions, they apply their own aesthetic judgment, they use their knowledge of the operations, they are empowered with an open-ended task that cannot be "wrong."<br /><br />We'll see how it goes. At the very least, I expect a lot of work for myself when their puzzles come in!</div><div><br /></div><div><b>An extra sweetener</b></div><div>Two kids asked if we could use other operations than addition. That prompted me to put this on the table (also from Brain Fun):</div><div><br /></div><div>DOS x DOS = CUATRO</div><div><br /></div><h1>Brain Fun Problems</h1>The first time I'd seen the Brain Fun problems, I added them to a list and called them "basic" (see <a href="http://3jlearneng.blogspot.com/2016/11/puzzling-puzzlers.html" target="_blank">this page</a>.) When I actually went to solve them, though, they didn't seem so easy.<br /><br /><b>Big confession time: </b>I actually looked at some of the solutions. However, I was disturbed to see that the solutions involved extra information that wasn't included as part of the problem statement! For example, in THREE + THREE + FIVE = ELEVEN, the solution assumes that ELEVEN is divisible by 11. This seems to be the case for several of the puzzles involving written out arithmetic:<br /><br />TWO + TWENTY = TWELVE + TEN (assume 20 divides TWENTY and 12 divides TWELVE, I wasn't clear about whether any divisibility was assumed for TWO and TEN)<br /><br />I'm not sure if similar assumptions are allowed/required for any of the others.<br /><br />Maybe I shouldn't complain, since this assumption creates an additional constraint without which there could be further solutions. Perhaps part of the reason it doesn't sit well is aesthetic. In the 3 + 3 + 5 = 11 puzzle, 3 doesn't divide THREE and 5 doesn't divide FIVE.<br /><br />Lastly, there is a typo in the final puzzle of the Brain Fun page. That puzzle should be<br />TEN x TEN = FIFTY + FIFTYJoshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-55097574243410505082017-02-16T01:10:00.000-08:002017-02-16T01:10:08.736-08:00More Man Who Counted (gaps and notes)As previously mentioned, we have been reading <a href="https://www.amazon.com/Man-Who-Counted-Collection-Mathematical/dp/0393351475" target="_blank">The Man Who Counted</a>. While the story is good and there are nice math puzzles, we've found some of our best conversations have come from errors or weaknesses in the book. Here are three examples:<br /><h1>How old was Diophantus?</h1>In chapter 24, we encounter a puzzle to figure out how old Diophantus was when he died. In summary, the clues are:<br /><br /><ol><li>he was a child for 1/6 of his life</li><li>he was an adolescent for 1/12 of his life. (J1: "what's that?" J0: "a teenager")</li><li>childless marriage for 1/7 of his life</li><li>Five more years passed, then had a child</li><li>The child got to half its father's age, then died.</li><li>Diophantus lived for four more years</li></ol>Perhaps we are wrong about our interpretation of the clues, but we noticed two things:<br />(a) the answer is not a whole number of years.<br />(b) the answer given in the book doesn't fit the clues.<br /><br />For the first part, it seems a natural assumption of these types of puzzles that we are only working with whole number years. Sometimes, this is an interesting assumption to directly challenge.<br />Here, since the clues involve a second person (Diophantus's child) we felt whole numbers were a strong assumption. Also, the name <a href="https://en.wikipedia.org/wiki/Diophantine_equation" target="_blank">Diophantus</a>, you know?<br /><br />Each clue required some discussion for us to agree on the interpretation. The one that seems most open is the fifth clue. In particular, did the child live until its age was half of the age of its father at the time of birth or to the point that, contemporaneously, it was half its father's age?<br /><br />For completeness, I'd note that neither interpretation matches the book's answer. The first interpretation does allow a whole number answer, but it doesn't give whole numbers for all the listed segments of Diophantus's life.<br /><br />Just so you can check for yourself, the solution given in the book is 84 years old.<br /><br /><b>How do you fix it?</b><br />We discussed several possible fixes:<br /><br /><ul><li>accept answers that aren't whole numbers or require whole number segments for each clue. This allows us to take the alternative interpretation of the fifth clue (though that still isn't satisfying) or to accept the clues and just take a new answer. This isn't satisfactory because... Diophantus.</li><li>Change clue 4 or clue 5 to match the book's answer. This approach seemed to fix the puzzle without distorting it or changing the mathematics required to analyze it.</li><li>Change clue 1, 2, or 3. While possible, these seemed to open the possibility of changing the character of the puzzle. Also, these fractions were plausible based on our own experience of human life spans.</li></ul><div>Of course, an even more satisfying answer would be to introduce a further variable and make the puzzle into one that makes heavy(ier) use of the integer restriction.</div><h1>Clever Suitors</h1>In chapter 31, Beremiz is confronted by a nice logic puzzle. Three suitors are put to a test, each is blindfolded and has disc strapped to his back. The background of the discs: other than color, the discs are all identical, there are five to choose from, 2 black and 3 white. <br /><br />The first suitor is allowed to see the colors of the discs on the backs of his two competitors, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.<br /><br />The second suitor is allowed to see the disc on the back of the third suitor, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.<br /><br />Finally, the third suitor is required to identify the color of his own disc and explain his reasoning. He succeeds.<br /><br /><b>Weakness 1</b><br />As a logic puzzle, we enjoyed this. Our problems came from the context in the story. This challenge was set to the three suitors as a way of fairly judging between them by finding the most clever suitor. However, this process was clearly unfair. In fact, it is inherent in the solution that it was impossible for the first and second suitors to determine the color of their own discs.<br /><br />This led to a nice discussion about who really held the power in this process: the person who structured the problem by deciding what color disc should be on which suitor and what order they would be allowed to give their answers.<br /><br />Extensions:<br /><ul><li>consider all arrangements of discs. Are there any arrangements where none of the suitors can answer correctly?</li><li>What is the winning fraction for each suitor? If you were a suitor, would you prefer to answer first, second, or third?</li></ul><div><br /></div><div><b>Weakness 2</b></div><div>Our second objection was non-mathematical, but again related to the story context. The fundamental problem wasn't how to choose a suitor. The fundamental problem was how the king could remain peacefully friendly toward all the suitors' home nations through this process.</div><div><br /></div><div>For this discussion, we went back to the story of Helen of Sparta, which we'd read a long time ago in the <a href="https://www.amazon.com/DAulaires-Greek-Myths-Ingri-dAulaire/dp/0440406943" target="_blank">D'Aulaire's Book of Greek Myths</a>. Of course, that also led to discussion of the division of the golden apples, another puzzle we all felt surely could have been solved more effectively with some mathematical reasoning...</div><h1>The Last Matter of Love</h1>The last puzzle of the book is in chapter 33. It is another logic puzzle, again intended to test the merit of a suitor in marriage. The test:<br /><br /><ul><li>there are five people</li><li>two have black eyes and always tell the truth</li><li>three have blue eyes and always lie</li><li>the suitor is permitted to ask three of them, in turn, a "simple" question each.</li><li>the suitor must determine the eye color of all five people</li></ul><div>As a logic puzzle, we readers get some extra information:</div><div><ol><li>The first person is asked: "what are the color of your eyes?" The answer is unintelligible.</li><li>The second is asked: "What did the first person say?" The answer is "blue eyes."</li><li>The third is asked: "What are the eye colors of the first and second people?" The answers are "the first has black eyes and the second has blue eyes."</li></ol><div><b>Simple questions</b></div></div><div>Our first objection was the part about asking "simple" questions. Having developed our taste for these types of puzzles through the knights and knaves examples of <a href="https://en.wikipedia.org/wiki/Raymond_Smullyan" target="_blank">Raymond Smullyan</a> (RIP, we loved your work!!!), the third question really bothered us. If you're going to go that far, why not ask the third person for the color eyes of all five people?</div><div><br /></div><div>Personally, I would prefer that the puzzle require us to ask each person a single yes/no question.</div><div><br /></div><div>As an extension: can you solve the puzzle with that restriction? </div><div><br /></div><div><b>Getting lucky</b></div><div>Again, we felt that this puzzle didn't meet the requirements of the context: to prove the worthiness of the suitor. Putting aside the question of whether this is really an appropriate way to decide whether two people should be allowed to marry, the hero here got lucky.</div><div><br /></div><div>Extension: what eye color for the third person would have caused the suitor to fail?</div><div>Extension: what answer from the third person would have caused the suitor to fail?</div><div>Extension: for what arrangement of eye colors would the questions asked by the suitor guarantee success?</div><div>Extension: what was the suitors' probability of success, given those were the three questions asked?</div><div><br /></div><div><b>Waste</b></div><div>Our final objection was the simple waste in the first question. From a narrative perspective, this is justified and even seems made to serve the purposes of the suitor. However, it opens another idea:</div><div>can you solve the puzzle, regardless of eye color arrangement, with only two questions?</div><div><br /></div><div>Feel free to test this with yes/no questions only or your own suitable definition of a "simple" question.</div><div><br /></div><h1>The power of...</h1>As a final thought, let me say that I think errors and ambiguity in a text are a feature, not a bug. It is another great opportunity for us to emphasize that mathematics is about the power of reasoning, not the power of authority.Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-8070710959439062112017-02-14T00:30:00.002-08:002017-02-14T00:30:46.938-08:00Good games and badRecently, we have been playing the following games:<br /><br /><ol><li>Go (baduk, weiqi, หมากล้อม). For now, we are playing on small boards, usually 5x5 or smaller.</li><li>Hanabi</li><li>Cribbage</li><li>Qwirkle (not regulation play, a form of War invented by J3 and grandma)</li><li>Munchkin</li><li>UNO</li><li>Vanguard</li></ol><div>I've ordered these by my own preference. In fact, I would be delighted playing just the first two exclusively and am happy to play cribbage or Qwirkle when asked.</div><div><br /></div><div>For the other three, I find myself biting my tongue a bit and grudgingly agreeing to be part of the game. I'm in the mood for strategic depth and a moderate (but not large) amount of pure chance. Part of my feeling was echoed in a recent My Little Poppies post: <a href="http://my-little-poppies.com/gateway-games/" target="_blank">Gateway Games</a>.</div><div><br /></div><div>However, as in the MLP post, I recognize that my enjoyment of the game is only a part of the reason for the activity. I guess the kids' enjoyment counts, too. </div><div><br /></div><div>Beyond that, even the games with limited depth are helping to build habits and skills:</div><div><ul><li>executive control: assessing the situation, understanding what behavior is appropriate, understanding options and making choices.</li><li>general gaming etiquette: taking turns, use of the game materials</li><li>meta-gaming: helping and encouraging each other, making sure that the littler ones have fun, too</li><li>numeracy and literacy: every time a number or calculation comes up or when something needs to be read, they are reinforcing their observation that math and reading are all around them.</li><li>meta-meta gaming: game choice, consensus building, finding options that interest and are suitable for all the players, knowing when it is time to play and when it isn't.</li></ul><div>As a family, and a little team, they are also building a shared set of experiences and jargon as they absorb ideas from each of the games.</div></div><div><br /></div><div>All of these are, of course, enough reason to make the effort to be open minded and follow their gaming lead.</div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-77774018593666749962017-02-13T23:10:00.003-08:002017-02-13T23:10:59.629-08:00NRICH 5 Steps to 50A quick note about the game we played in first grade today: <a href="http://nrich.maths.org/10586" target="_blank">5 Steps to 50</a>.<br /><br />This is an <a href="http://nrich.maths.org/frontpage" target="_blank">NRICH</a> activity that I've had on my radar for a while. I even made a <a href="http://jgplay.pencilcode.net/home/jumpGameTree2" target="_blank">pencilcode program</a> to explore the activity in reverse. True to their other activities (check them out!!!) 5 steps to 50 requires very little explanation, is accessible to students with limited background, but has depth and richness.<br /><br /><b>Our lesson outline</b><br />I explained the basic activity and did an example at the board. To get my starting value, I had one student roll for the 10s digit and one for the 1s digit. Then we talked through together as we added 10s and 1s.<br /><br />I then distributed dice and had the kids try 3 rounds. As they worked, I confirmed several rules:<br /><br /><ol><li>the only operations allowed are +1, -1, +10, -10</li><li>we must use exactly five steps (I note that this is ambiguous on the NRICH description, they say "you <i>can</i> then make 5 jumps")</li><li>we are allowed to do the operations in any order</li><li>we can mix addition and subtraction operations</li></ol>After everyone had been through 3 rounds, we regrouped to summarize our findings:<br /><br /><ul><li>Which starting numbers can jump to 50?</li><li>Which starting numbers cannot jump to 50?</li></ul><div>We helped the kids resolve disagreements and then posed the following:</div><div><ul><li>What is the smallest number that can jump to 50?</li><li>What is the largest number that can jump to 50?</li></ul></div>For those to challenges, we kept the restriction that the numbers must be possible to generate from 2d6.<br /><br /><b>Basic level</b><br />To engage with the activity, some of the kids just started trying operations without much planning. This quickly reinforced the basic points about addition and place-value and commutativity of addition.<br /><br />For these kids, it was helpful to ask a couple of prompting questions:<br /><br /><ul><li>What do you notice? This is a standard that never gets old!</li><li>If this path doesn't get to 50, does that mean there is no path to 50?</li></ul><div>This second question, particularly, raises the interesting observation that it is easy to show when a number can jump to 50 (just show a path) but to show that no path is possible requires a different type of thinking.</div><div><br /></div><div><b>Getting more advanced</b></div><div>The next level of sophistication was really about noticing that the key consideration is the distance to 50. In particular, this identified a symmetry, where n could jump to 50 if 2*50 - n can jump to 50. Of course, the kids didn't phrase this relationship in this way....</div><div><br /></div><div>The next major step is thinking about a way to systematically write down the paths.</div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-14781773949297313772017-02-01T22:16:00.004-08:002017-02-01T22:16:34.556-08:00Perfect Play for My closest neighborJoe Schwartz at Exit10a wrote a <a href="http://exit10a.blogspot.com/2017/02/ball-dont-lie.html" target="_blank">fraction comparison post</a> that prompted me to write up more of my experience and thoughts on this game.<br /><br /><b>Let's find perfect play</b><br />This week, I intended to use the game one last time with the 4th graders as an extended warm-up to our class. The challenge I presented:<br /><br /><b><i>If we got super lucky and were given perfect cards for each round of the game, what are the best possible plays?</i></b><br /><br />My intention was to spend about 20 minutes on this. Depending on how quickly it went and the kids' reactions, I considered giving them a follow-up for a short homework: what are the best plays if we include all cards A (1) through K (13)?<br /><br /><b>How did it go?</b><br />In the end, the basic activity took the whole class. These comparisons were difficult for the kids, so we spent time talking about each different strategy for comparison:<br /><br /><ol><li>common denominators</li><li>common numerators</li><li>distance to 1</li><li>relationship to another benchmark number. Like 1/2 in Joe's 4/6 and 8/18 example, a benchmark is a "familiar friend" that should be relatively easy to see it is larger than one and smaller than another. In practice, 1/2 seems to be the most popular benchmark. </li></ol><br />For visualization, drawing on a number line seemed to work best.<br /><br />I did not assign the full deck challenge as homework. Instead, we gave them some more work with fractions of pies and bars.<br /><br /><b>What have I learned?</b><br />This game is really effective at distinguishing levels of understanding:<br />(0) some kids are totally at sea. They don't really understand what this a/b thing means, how a and b are related, etc. These kids struggle with the first round of the game when the target is 0, when the idea is to just want to make their fraction as small as possible.<br /><br />(1) Some kids have got a basic understanding of the meaning of the fraction and can play confidently when the target is 0 or 1. They might still be weak about equivalent fractions. Trying to play some spot-on equivalents when 1/3 and 1/2 are targets is a give-away.<br /><br />(2) familiar with some frequent friends: kids who can tell readily whether their plays are larger or smaller than the target for 1/3, 1/2, 3/4.<br /><br />(3) proficient: have at least one consistent strategy they can work through to make a comparison<br /><br />(4) fraction black-belts: using multiple strategies, already familiar with many of the most common comparisons.<br /><br /><b>What would I do differently?</b><br />Generally, I think it is valuable to spend more time and more models directed at the basic understanding of what fractions <b><i>mean</i></b>. The kids who were at or close to stage 4 have, over the years, been seeing diagrams of pies, cakes, chocolate bars, number lines and physical experience with baking measures and fractional inches on measuring tapes and rulers. Oh, and also actual pies (mostly pizza), cakes, cookies, and chocolate bars discussed using fractional language.<br /><br />More locally, for this game in a class of mixed levels, I would<br /><br /><ul><li>lean toward doing this more as a cooperative puzzle</li><li>re-order the targets for the rounds as 0, 1, 1/2, 3/4, 1/3, 2 (note: I don't have strong feelings about where 2 fits in this sequence)</li><li>I also would consider allowing equivalent fractions to the target as winning plays</li></ul>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com2tag:blogger.com,1999:blog-5544661968326910027.post-77217319334891125012017-02-01T19:45:00.000-08:002017-02-01T19:45:13.370-08:00Impassable Din Daeng (BKK intersections 3)It has been a while since I've done one of these. For your topological and civil engineering pleasure, I present Din Daeng intersection (แยกดินแดง):<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-m1bhvPTMb14/WJKpDDEj4BI/AAAAAAAAB2E/jMwn3ulIW7kUjJdWTrHv1Wcd9e7Ac9rTACLcB/s1600/dindaeng.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="226" src="https://2.bp.blogspot.com/-m1bhvPTMb14/WJKpDDEj4BI/AAAAAAAAB2E/jMwn3ulIW7kUjJdWTrHv1Wcd9e7Ac9rTACLcB/s400/dindaeng.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Notice how the expressway obscures key details?</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-k0UG93Mml_0/WJKpC-FzLWI/AAAAAAAAB2A/cN43U3f2csMXHTPVJaB43ZcCr5UhkWtFgCLcB/s1600/dindaeng2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="213" src="https://2.bp.blogspot.com/-k0UG93Mml_0/WJKpC-FzLWI/AAAAAAAAB2A/cN43U3f2csMXHTPVJaB43ZcCr5UhkWtFgCLcB/s400/dindaeng2.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Magnified view. We now see a tunnel, but where does it emerge?</td></tr></tbody></table><br />This intersection has a special place in my heart. Last month, J1 and J2 played in a squash tournament at the Thai-Japan Youth Centre. It isn't visible on either map I've included, but is just to the northeast of the intersection. Since we live on the west of the intersection, we needed to cross somehow.<br /><br />After three failed attempts (following the driving directions on google maps) to get through from west to east, I ended up parking our car in one of the small side streets and we just walked. The walk took about 30 minutes...Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-45303886108245631792017-02-01T06:30:00.001-08:002017-02-01T06:30:16.723-08:00ideas for upcoming classes<h4>warm-ups for all</h4><div>WODB: (1) shapes book (2) <a href="http://wodb.ca/">wodb.ca</a></div><div><b style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">any:</b><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"> Traffic lights/inverse tic tac toe/faces game</span></div><div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><span style="font-size: 12.8px;">good options here, mostly grades 1/2:</span><span style="font-size: 12.8px;"> </span><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://3jlearneng.blogspot.com/2014/08/adding-and-subtracting-games.html&source=gmail&ust=1486000547474000&usg=AFQjCNG9Y4HlaLS74fGPccPv3J7wc0L1oA" href="http://3jlearneng.blogspot.com/2014/08/adding-and-subtracting-games.html" style="color: #1155cc; font-size: 12.8px;" target="_blank">some games</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">dots & boxes (maybe with an arithmetic component)</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">loop-de-loops</div></div><div><br /></div><h4><b>Grades 1 and 2</b></h4><div><a href="https://denisegaskins.com/2015/04/21/math-game-thirty-one/" target="_blank">31 Game</a>: could be used by any grade<br /><a href="https://denisegaskins.com/2008/05/29/hit-me-math-game/" target="_blank">Integer addition game </a><br /><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://nrich.maths.org/10586&source=gmail&ust=1486000547474000&usg=AFQjCNEX_NgqPntRyIxELfBLg5e6sUqToQ" href="http://nrich.maths.org/10586" style="color: #1155cc; font-family: arial, sans-serif; font-size: 12.8px;" target="_blank">jump to 50</a><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">,</span><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"> </span><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://nrich.maths.org/221&source=gmail&ust=1486000547474000&usg=AFQjCNGsQnJZwEl1F-gBPcDF5ISnr3CLnA" href="http://nrich.maths.org/221" style="color: #1155cc; font-family: arial, sans-serif; font-size: 12.8px;" target="_blank">chain of changes puzzles</a></div><div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><div style="font-size: 12.8px;">close to 100 game: </div><div style="font-size: 12.8px;"><span style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px; line-height: 18.2px;">Equipment: A pack of cards with 10 and face cards (J,Q,K) removed.</span><br style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px;" /><div style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px;"><span style="line-height: 18.2px;">Procedure: </span></div></div><blockquote style="border: none; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div><div style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px;"><span style="line-height: 18.2px;">- Deal out 6 cards to each player</span></div></div><div><div style="color: #666666; font-family: "trebuchet ms", trebuchet, verdana, sans-serif; font-size: 14.52px;"><span style="line-height: 18.2px;">- Each player picks 4 cards from the 6 cards they were dealt to form a pair of 2-digit numbers. The goal is to get the sum of the two numbers as close to 100 as possible but </span><span style="line-height: 18.2px;">cannot exceed 100.</span></div></div><div><span style="line-height: 18.2px;"><br /></span></div></blockquote></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://mikesmathpage.wordpress.com/2014/09/03/fawn-nguyen-shares-a-really-neat-math-forum-problem/&source=gmail&ust=1486000547474000&usg=AFQjCNHKtW8n0GByoPJhaSC3souihiOAKw" href="https://mikesmathpage.wordpress.com/2014/09/03/fawn-nguyen-shares-a-really-neat-math-forum-problem/" style="color: #1155cc;" target="_blank">3 chips puzzle</a></div></div><div><br /></div><h4><b>Grades 3 and 4</b></h4><a href="https://denisegaskins.com/2015/05/08/math-games-with-factors-multiples-and-prime-numbers/" target="_blank">Factor finding game</a> (maybe warm-up?)<br /><a href="http://nrich.maths.org/5468" target="_blank">Factors and Multiples game</a><br /><a href="http://www.mathwire.com/games/contig.pdf" target="_blank">Contig for 3 and 4</a> (<a href="https://denisegaskins.com/2008/11/24/contig-game-master-your-math-facts/" target="_blank">explanation</a>).<br /><a href="https://denisegaskins.com/2011/03/16/game-times-tac-toe/" target="_blank">Times tic-tac-toe</a>: review for Grades 3 and 4<br /><a href="https://denisegaskins.com/2006/12/29/the-game-that-is-worth-1000-worksheets/" target="_blank">Fraction war for grade 4</a> (smallest card is numerator)<br /><a href="https://letsplaymath.files.wordpress.com/3012/07/multiplication-matching-cards.pdf" target="_blank">Multiplicaton models</a>: worth making for grades 3-4 for solidifying concepts? <a href="https://denisegaskins.com/2013/12/17/multiplication-models-card-game/" target="_blank">Associated games</a><br />d 2<br /><br /><br /><br /><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://3jlearneng.blogspot.com/2014/11/subtraction-games-math-class-2.html&source=gmail&ust=1486000547474000&usg=AFQjCNGDjFkVyLK6ht_MHoHpka2M93A5SQ" href="http://3jlearneng.blogspot.com/2014/11/subtraction-games-math-class-2.html" style="color: #1155cc;" target="_blank">card subtraction</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathsolutions.com/documents/978-1-935099-02-4_NL36_L1.pdf&source=gmail&ust=1486000547474000&usg=AFQjCNH4GGZhUx8SRRDLdJo8hrXhhaElwg" href="http://mathsolutions.com/documents/978-1-935099-02-4_NL36_L1.pdf" style="color: #1155cc;" target="_blank">factor game</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">card on head game</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://3jlearneng.blogspot.com/2014/12/math-games-class-5.html&source=gmail&ust=1486000547474000&usg=AFQjCNEW2vXfy7gBqZx3YKIgQfxCGRx4hQ" href="http://3jlearneng.blogspot.com/2014/12/math-games-class-5.html" style="color: #1155cc; font-size: 12.8px;" target="_blank">Pattern revealed</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://3jlearneng.blogspot.com/2015/01/calendar-tricks-and-break-bank-math.html&source=gmail&ust=1486000547474000&usg=AFQjCNFP6znksS3ily1H84D43kNYIFabXQ" href="http://3jlearneng.blogspot.com/2015/01/calendar-tricks-and-break-bank-math.html" style="color: #1155cc;" target="_blank">Calendar tricks</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">Pico Fermi Bagel</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">Magic triangle puzzles</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">damult dice</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://nrich.maths.org/179&source=gmail&ust=1486000547474000&usg=AFQjCNHAx1Lbj-3MbHcO1hgfLxG5Vx-IGw" href="http://nrich.maths.org/179" style="color: #1155cc;" target="_blank">4 dominoes puzzle</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://nrich.maths.org/2908&source=gmail&ust=1486000547474000&usg=AFQjCNHshPqexGG-MIivlsoVYoTdXZFJhA" href="http://nrich.maths.org/2908" style="color: #1155cc;" target="_blank">tables and chairs</a></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><div style="font-size: 12.8px;"><b><span style="font-size: medium;">(1) Dice <span class="il">game</span> perudo</span></b></div><div style="font-size: 12.8px;"><b>Equipment</b></div><div style="font-size: 12.8px;">- multi-player, 2-5</div><div style="font-size: 12.8px;">- Everyone gets the same number of 6 sided dice (full <span class="il">game</span> they get 5, I would start with 3)</div><div style="font-size: 12.8px;">- Everyone has a cup to shake and conceal their dice</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><b>Basic Play</b></div><div style="font-size: 12.8px;">- Simultaneously, players shake their cups and turn them over on the ground or a table. They peak in to look at their own dice, but keep them concealed from the other players.</div><div style="font-size: 12.8px;">- starting randomly (or from the person who lost a dice in the last round), players make bids, for example: two 3s. </div><div style="font-size: 12.8px;">This bid signifies that the player has 2 (or more dice) showing the value 3.</div><div style="font-size: 12.8px;">- the next player has two choices: </div><div style="font-size: 12.8px;"><ul><li style="margin-left: 15px;">call/doubt the previous player's bid: if they do this, all players show their dice. If there are enough to meet the bid, the caller loses a die. If not, then the bidder loses a die.</li><li style="margin-left: 15px;">raise, either the number of dice or the value or both get increased </li></ul><div><b>Advanced rules</b></div></div><div style="font-size: 12.8px;">- Ones are wild, they count as any number toward the target bid</div><div style="font-size: 12.8px;">- If someone drops to their last dice, they start the next round. On that round, only the number of dice can be increased in the bid, not the value. Ones are not wild on this round</div><div style="font-size: 12.8px;">- After someone bids, the next player has a third option, to call "exact." If the bid is exactly matched by the dice, then the bidder loses a die and the caller gets an extra one. If the actual dice show either more or less than the bid, the caller loses a die.</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><b>remainder jump</b></div><div style="font-size: 12.8px;"><span style="font-size: 12.8px;">we played this </span><span class="il" style="font-size: 12.8px;">game</span><span style="font-size: 12.8px;"> before, but we could give them blank boards and let them create. See the last page here: </span><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://ba-cdn.beastacademy.com/store/products/3C/printables/RemainderJump.pdf&source=gmail&ust=1486019989770000&usg=AFQjCNFUmyIglOcPjrSLQ_l_cwz1zSy6kw" href="http://ba-cdn.beastacademy.com/store/products/3C/printables/RemainderJump.pdf" style="color: #1155cc; font-size: 12.8px;" target="_blank">http://ba-cdn.<wbr></wbr>beastacademy.com/store/<wbr></wbr>products/3C/printables/<wbr></wbr>RemainderJump.pdf</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://www.beastacademy.com/store/products/5B/printables/GCF_LCM_Webs.pdf&source=gmail&ust=1486019989775000&usg=AFQjCNGWSNN0TEuiIoz-Es7mvfVZyCrjhg" href="https://www.beastacademy.com/store/products/5B/printables/GCF_LCM_Webs.pdf" style="color: #1155cc; font-size: 12.8px;" target="_blank">https://www.<wbr></wbr>beastacademy.com/store/<wbr></wbr>products/5B/printables/GCF_<wbr></wbr>LCM_Webs.pdf</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><span style="font-size: 12.8px;">another good </span><span class="il" style="font-size: 12.8px;">game</span><span style="font-size: 12.8px;">:</span><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://saravanderwerf.com/2015/12/13/5x5-most-amazing-just-for-fun-game/&source=gmail&ust=1486019989777000&usg=AFQjCNEu9-BperhhonKQ-2G3XkoJGHqfxA" href="https://saravanderwerf.com/2015/12/13/5x5-most-amazing-just-for-fun-game/" style="color: #1155cc;" target="_blank">https://saravanderwerf.com/<wbr></wbr>2015/12/13/5x5-most-amazing-<wbr></wbr>just-for-fun-<span class="il">game</span>/</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">We should still use the pyramid puzzle sometime:</div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/lesson/pyramid-puzzles/&source=gmail&ust=1486019989778000&usg=AFQjCNES2-o5XkMjJTB9AmidWA7OuPa-Kg" href="http://mathforlove.com/lesson/pyramid-puzzles/" style="color: #1155cc;" target="_blank">http://mathforlove.com/lesson/<wbr></wbr>pyramid-puzzles/</a></div></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><span style="font-size: 12.8px;">(1) double digit and double dollar:</span><div style="font-size: 12.8px;">We've done something like this, but I think there could be a good variation done trying to make 1000 baht, using 1, 2, 5, 10, 20, 50, and 100 baht units.</div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/lesson/double-digit-and-dollar-digit/&source=gmail&ust=1486019990897000&usg=AFQjCNFOB3X-FUmvlg5nmkRzue2tNBgCdw" href="http://mathforlove.com/lesson/double-digit-and-dollar-digit/" style="color: #1155cc;" target="_blank">http://mathforlove.com/lesson/<wbr></wbr>double-digit-and-dollar-digit/</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">(2) biggest rectangle. This could be used as a warm-up. For the older kids, they will probably have seen something like this, but I like the inclusion of perimeters that are even but not divisible by 4 and odd perimeters and the question about "smallest area" (here <a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/wp-content/uploads/2016/02/MFL_Rectangle-Worksheet-2_Q.pdf&source=gmail&ust=1486019990897000&usg=AFQjCNEcNELxKyJ7tl7ER2jvWu28J5BMHA" href="http://mathforlove.com/wp-content/uploads/2016/02/MFL_Rectangle-Worksheet-2_Q.pdf" style="color: #1155cc;" target="_blank">are 5 questions</a>).</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/lesson/1009569184/&source=gmail&ust=1486019990897000&usg=AFQjCNFmqnY1m7W--feMeJjewVqxzeM60g" href="http://mathforlove.com/lesson/1009569184/" style="color: #1155cc;" target="_blank">http://mathforlove.com/lesson/<wbr></wbr>1009569184/</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">(3) some of these <span class="il">games</span> are promising:</div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://danburf.files.wordpress.com/2014/11/tapson-games.pdf&source=gmail&ust=1486019990897000&usg=AFQjCNHo_EVEYa_QDFKti_GgvDcPyeIsCQ" href="https://danburf.files.wordpress.com/2014/11/tapson-games.pdf" style="color: #1155cc;" target="_blank">https://danburf.files.<wbr></wbr>wordpress.com/2014/11/tapson-<wbr></wbr><span class="il">games</span>.pdf</a></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://mathforlove.com/lesson/pyramid-puzzles/&source=gmail&ust=1486019990899000&usg=AFQjCNGR_HzFaVOiKV0gRMRvEVjR0H_Vcw" href="http://mathforlove.com/lesson/pyramid-puzzles/" style="color: #1155cc; font-size: 12.8px;" target="_blank">http://mathforlove.com/lesson/<wbr></wbr>pyramid-puzzles/</a></div><div style="font-size: 12.8px;"><br /></div></div></div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com4tag:blogger.com,1999:blog-5544661968326910027.post-69872954780258196522017-01-31T06:27:00.000-08:002017-01-31T06:27:12.662-08:00Quadratic Friends (The Man Who Counted)J1, J2, and I are currently reading <a href="https://www.amazon.com/Man-Who-Counted-Collection-Mathematical/dp/0393351475" target="_blank">The Man Who Counted</a>. Here are some quick thoughts:<br /><br /><b>Quadratic Friends</b><br />The book is a great entry point for mathematical discussions. In fact, it makes it questionable as bedtime reading, since I have to be careful to find a more narrative section to close the evening. Otherwise, we would just continue talking and they'd never get to sleep.<br /><br />Fortunately, the J's are willing to extend some of these conversations over to the next day, so we're not obligated to wrap up everything in one evening.<br /><br />Here is an example discussion: in one of the early chapters, the protagonist Beremiz talks about the special relationship between 13 and 16. Namely:<br /><blockquote class="tr_bq">13 * 13 = 169<br />1 + 6 + 9 = 16<br />16 * 16 = 256<br />2 + 5 + 6 = 13</blockquote><b>Finding more</b><br />We wondered: what other pairs of numbers share this property?<br /><br />Our first instinct was to gather data, so we started calculating some examples. We began with 0 and worked up, squaring, adding the digits, repeating. We found a couple of cases that flowed into the 13-16 relationship, for example 7. This gives a feeling that 7 is very fond of 13, but 13 only has eyes for 16. Not the usual way people think about numbers, I guess.<br /><br />Along the way, we made some interesting observations about this iterative process. I won't spoil the surprise, but would encourage you to explore yourself.<br /><br />I'd note that J1 did the calculations up to 30 in his head, while I was a bit lazy and wrote a pencilcode program.<br /><br /><b>An extension</b><br />This conversation branched in an interesting way. Squaring is a natural thing to do with numbers, but summing the digits is a bit artificial. It depends on a choice of base. So, a natural follow-up question:<br />what quadratic friends exist in other bases? This is an exploration for another day.Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-47611632012325494232017-01-24T02:03:00.000-08:002017-01-24T02:03:00.564-08:00Delta Pack the uncreative way (euclidea series)Returning with a very brief installment of geometric constructions. Basically, I was just brute-forcing the constructions in this pack, so most of the cases where I managed the minimum move constructions were just because those were straightforward. Reviewing the pack in preparation for this post, I did find a couple of ways to shave down some of the constructions, but I don't feel like there was a major breakthrough.<div><br /></div><div>Here's a snapshot showing (almost) where the V stars are:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-HZScYHlsNj8/WIclZLbW-4I/AAAAAAAAB1w/VJz1hvctHC00DCNFLLaD3Jc4RhX7xlWYwCLcB/s1600/delta%2Bpack.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="458" src="https://4.bp.blogspot.com/-HZScYHlsNj8/WIclZLbW-4I/AAAAAAAAB1w/VJz1hvctHC00DCNFLLaD3Jc4RhX7xlWYwCLcB/s640/delta%2Bpack.png" width="640" /></a></div><div><br /></div><div><br /></div><div>If you do some simple counting, you'll see that there is one more V-star missing. Since this somehow escaped me the first time, I'll leave its location as an exercise for you, too.</div><div><br /></div><div><b>Favorites</b></div><div>I've mentioned the idea from 4.5 before and I still like that construction, even though it is very simple.</div><div><br /></div><div>Constructing the two equilateral triangles also seemed nice. Constructing the inscribed and circumscribing circles from the triangle seems more like the "usual" direction of construction, so these reverse constructions appealed to me.</div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-7901438494263867422017-01-23T00:13:00.003-08:002017-01-23T00:29:55.974-08:00Fractions and Farey AdditionBenjamin Leis (who posts at <a href="http://mymathclub.blogspot.com/" target="_blank">Running a Math Club</a>) flagged this video in response to our recent fractions work: <a href="https://www.youtube.com/watch?v=0hlvhQZIOQw" target="_blank">Funny Fractions and Ford Circles (Numberphile)</a>. <br /><br /><b>Ex ante discussion ideas</b><br />The video gave me several ideas for possibly interesting conversations with the kids:<br />(1) Some basic geometry, particularly for J3. Circles that are tangent, nesting pictures, pictures that have fractal qualities.<br />(2) Comparing Farey addition and regular addition<br />(3) Well-defined operations on fractions. I always like to discuss whether the operations gives us the same results regardless of the equivalent form we start with? Farey addition is a good example where the choice of representation is important (indeed, Prof Banahon is careful to keep reminding us that he wants the fractions in lowest terms.)<br />(4) why do we want the fractions in simplest terms? Possibly relate this to the Cat in Numberland (showing rationals are countable).<br />(5) what happens if we try Farey addition of three fractions in a row: e.g., (1/5) @ (1/3) @ (1/2)? This is one of the few "naturally occurring" non-associative operations I know.<br />(6) Since associativity doesn't work, surely distribution of multiplication over Farey addition must not work, right? What about commutativity?<br />(7) Linking back with our comparison game, if a<b, how do a@c and b@c compare? If a@c < b@c, what can we say about a and b?<br />(8) what if we allow negative numbers? How should we define Farey addition, then?<br /><br /><b>How the conversation actually went</b><br />J2 was especially taken with the picture of the Ford circles and immediately had two requests: he wanted to draw them and he wanted me to create a pencilcode program to draw them.<br /><br />The former was a great activity with a lot of figuring and fraction practice. Here he is, hard at work:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-wVwuI5NYIio/WITq_iticpI/AAAAAAAAB0w/ll20sdH26HwDQrEgMO5msGuZN5WfFTbWgCLcB/s1600/IMG_0876.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-wVwuI5NYIio/WITq_iticpI/AAAAAAAAB0w/ll20sdH26HwDQrEgMO5msGuZN5WfFTbWgCLcB/s320/IMG_0876.JPG" width="240" /></a></div><br />Along the way, there was lots of discussion about where to position each fraction on the number line (he scaled with 20 cm as the unit distance from 0 to 1), and how big to make each circle. Tangency condition was a nice check on his work. He would see right away when something was wrong (which did happen several times:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-kyN-wdw6Cgk/WITrILJ4DwI/AAAAAAAAB00/4HwSfgG11_MNIpb88Ms5biNHaWPOl1hCQCLcB/s1600/IMG_0877.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://1.bp.blogspot.com/-kyN-wdw6Cgk/WITrILJ4DwI/AAAAAAAAB00/4HwSfgG11_MNIpb88Ms5biNHaWPOl1hCQCLcB/s320/IMG_0877.JPG" width="320" /></a></div><br /><br />We did talk through some of the ideas on my pre-planned list: is Farey addition well-defined on fractions (no! point 3), does associativity work (no! point 5), could we extend to negative numbers (yes, make the numerator negative seems to work best, point 8). Other areas are still open for future discussion.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-2dtyESv7sVw/WIW-5WKe9VI/AAAAAAAAB1g/PikoP2npAigu6io2UMqaDt1Rz4ij8epnwCLcB/s1600/IMG_0885.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://3.bp.blogspot.com/-2dtyESv7sVw/WIW-5WKe9VI/AAAAAAAAB1g/PikoP2npAigu6io2UMqaDt1Rz4ij8epnwCLcB/s320/IMG_0885.JPG" width="320" /></a></div><br /><br /><b>Pencilcode result</b><br />I wrote a quick program here: <a href="http://jgplay.pencilcode.net/edit/Math/FareyFord" target="_blank">FareyFord</a>. You'll notice that it doesn't actually generate Farey sequences. Instead, it creates generations of fractions, starting from 0 and 1 as the original parents. For each new generation, it uses Farey addition to create a new fraction between each adjacent pair in the previous generation.<br /><br />Here's a picture of the associated Ford circles:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-LNiaQKn3m8s/WIW6YqS_cfI/AAAAAAAAB1U/n3ZhoFbxPIUfAnXqYSmItYF93ewKc8C0gCLcB/s1600/Ford%2BCircles.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="342" src="https://1.bp.blogspot.com/-LNiaQKn3m8s/WIW6YqS_cfI/AAAAAAAAB1U/n3ZhoFbxPIUfAnXqYSmItYF93ewKc8C0gCLcB/s640/Ford%2BCircles.png" width="640" /></a></div><br />This method raised an interesting question: what is the largest denominator in each generation? If you don't know, it is cute and worth considering.<br /><br /><b>More Go (miscellaneous)</b><br />Note: this part is unrelated to fractions or farey sequences.<br /><b><br /></b>J3 wasn't in the mood to play more capture go with me, but I had an idea. I noticed in one of Nick Sibicky's lectures that one of his students was a young girl, roughly around the age of our three kids. I showed that part of the video to J3 and she made the connection: "this is something girls like me do."<br /><br />We went and played some silly games on very small boards: 1x1, 2x2, 3x3. In the picture below, we set out a blue-green alternating boundary around a 3x3 board. Then, I asked J3 how many different moves were available. She pointed first to the center, then I asked if there were any other spaces that were the same as the center, if we moved the board around or tipped ourselves upside down.<br /><br />No, so we made the center red. What other moves? She then chose a side square and figured out that there were three other places that were equivalent. Those became yellow. Finally, we figured out that the four corners were also identical, so that gave us the final picture:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-uKerzA8MTDY/WITrIEbVGrI/AAAAAAAAB04/_XAo16JkBUYTmRRRT2Y3xUuDxeIuH6QAQCLcB/s1600/IMG_0880.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-uKerzA8MTDY/WITrIEbVGrI/AAAAAAAAB04/_XAo16JkBUYTmRRRT2Y3xUuDxeIuH6QAQCLcB/s320/IMG_0880.JPG" width="240" /></a></div><br />Later, I was playing 9x9 with J2. Instead of go stones, we used Banangram tiles for the white stones. At the end of the game, we tried to make words with the captured tiles from the game. Here was one case where we could (sort of?) make a complete scrabble chain with all the captures:<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-0yxnMgjhVCY/WITr1i5NWLI/AAAAAAAAB1A/ofunAcwT2nsG6ZoezYD6QPZiDbOcAl8jgCLcB/s1600/IMG_0882.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-0yxnMgjhVCY/WITr1i5NWLI/AAAAAAAAB1A/ofunAcwT2nsG6ZoezYD6QPZiDbOcAl8jgCLcB/s320/IMG_0882.JPG" width="240" /></a></div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-91628910203992331892017-01-18T22:38:00.000-08:002017-01-18T22:38:12.264-08:00Closest neighbor one-on-oneIn my last post, I wrote about playing Denise Gaskins' closest neighbor fraction game with our 4th grade class. Yesterday, I spent time with J2 and used the game as a semi-cooperative puzzle.<br /><div><br /></div><div>This activity worked really well and the experience gave me some additional ideas about how to use the core ideas again with the 4th grade class.</div><div><br /></div><div><b>Puzzle or game?</b></div><div>First, there were only two of us, one a kid and another an adult, so that background naturally makes the activity very different. As the key modification for play, we played all of our hands open and helped each other find the fraction in each of our hands that was closest to the target for that round. Then, we worked together to determine which of those two "champions" was closest overall.</div><div><br /></div><div>Some of the consequences:</div><div><ul><li>the activity was not really competitive (see below)</li><li>J2 had to do a lot more fraction work.</li></ul><div>Let me explain the second point here. Because we were looking for the best play, J2 had to consider all of the combinations in his hand (20 choices). Some of those can be rejected quickly with simple analytical strategies depending on the target. Even this is good number sense thinking. Also, some combinations are close competitors and need to be analyzed more carefully.</div></div><div><br /></div><div>If we were playing with closed hands, he could choose two cards, play a fraction based on them, and I wouldn't be able to say anything about whether those were his best options or not.</div><div><br /></div><div>Second, while I write that "we worked together," as a sneaky dad, that means that I pretended to do work, while actually getting J2 to analyze my hand as well as his. Really, the only thing I offered was an alternative comparison strategy, once he had already worked through his own approach.</div><div><br /></div><div><b>An example of some strategies</b></div><div>We found that some of the comparisons that arise naturally in this game are quite tricky, even for me. For example, quickly tell me which is closer to 1/3: 1/5 or 4/9?</div><div><br /></div><div>We found that placing the fractions on a number line was a really helpful strategy for many of the comparisons. We also made very heavy use of the two strategies involving common numerators or common denominators.</div><div><br /></div><div>Finally, you can see in this example that J2 is comfortable mixing decimals and fractions, for example converting to 1/2 to 3.5/7 to aid some comparison:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-dP_4HiV5qQE/WIBWhBnBiFI/AAAAAAAAB0U/PilxbBW3NTIn782SeBEnbDbXPB_kM4U9wCLcB/s1600/IMG_0871.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-dP_4HiV5qQE/WIBWhBnBiFI/AAAAAAAAB0U/PilxbBW3NTIn782SeBEnbDbXPB_kM4U9wCLcB/s400/IMG_0871.JPG" width="300" /></a></div><div><br /></div><div><br /></div><div><b>Our grid</b></div><div>Through our play, we filled out this grid, taking turns putting in our best results and congratulating each other when our hand was the ultimate champion for that round:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-wACjF58Nw4k/WIBWhDgkpVI/AAAAAAAAB0Q/2cMoXCUZVnUtipe--QingicVzOXc48mZQCEw/s1600/IMG_0872.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-wACjF58Nw4k/WIBWhDgkpVI/AAAAAAAAB0Q/2cMoXCUZVnUtipe--QingicVzOXc48mZQCEw/s400/IMG_0872.JPG" width="300" /></a></div><div><br /></div><div><br /></div><div><b>Competition and Strategic thinking</b></div><div>I was particularly pleased by one comment J2 made about this overall game: "this is mostly luck, how well we can play depends on the cards we get." This comment came after one round where he had several duplicate cards in his hand, reducing the number of distinct values he could play. We've discussed elsewhere my goals of helping the kids think about game structure, so I always love it when they bring those ideas up themselves.</div><div><br /></div><div>Some thoughts about competition. While we played this game non-competitively, I'm not opposed to competition nor do I think that this game always needs to be played non-competitively. Ultimately, my litmus test is how to play in a way that is the most fun. If I were a more serious educator, I suppose I would also consider which way is the most educational, too.</div><div><br /></div><div>It won't always be obvious what is the best way to play each game. In this case, I got to benefit from the prior experience with the class and my close knowledge of J2. Many times, I'll tell the kids that there are several ways to play and we'll try them out together, then review the experience.</div><div><br /></div><div>Among other things, this is why I love handicap games like Go. By adjusting the starting advantages, we can create scenarios where it is very competitive and very fun, even though the players have very different levels of experience and current strength in the game. And also, there are things we can do together when we want a non-competitive activity.</div><div><br /></div><div><b>Ideas for going back to class</b></div><div>From this time with J2, here are my ideas about taking the game back to the 4th grade class are:</div><div><ol><li>Spend a lot of time on fraction comparison strategies before we play</li><li>Reduce the number of cards dealt to each player</li><li>play as teams</li><li>convert to open hands with a lot of talk about why we chose particular plays</li></ol></div><h1>An actual puzzle</h1>As a reward for reading down this far, here's an actual puzzle related to the closest neighbors fraction game: <br /><br />During the round where the target is 1/2, Jay plays 6/6 = 1. Was that her best play? How do we know?<br /><br /><div id="spoiler1" style="display: none;">Remember, we are playing our game with a single deck of playing cards and each player is dealt five cards</div><button onclick="if(document.getElementById('spoiler1') .style.display=='none') {document.getElementById('spoiler1') .style.display=''}else{document.getElementById('spoiler1') .style.display='none'}" title="Click to show/hide construction" type="button">Hint 1</button><br /><div id="spoiler2" style="display: none;">The target for the next round will be 3/4</div><button onclick="if(document.getElementById('spoiler2') .style.display=='none') {document.getElementById('spoiler2') .style.display=''}else{document.getElementById('spoiler2') .style.display='none'}" title="Click to show/hide construction" type="button">Hint 2</button> <br /><div id="spoiler3" style="display: none;">Burning 2 identical values might allow Jay to increase the diversity of her hand, especially if she happened to have 3 or 4 sixes.<br />Hey, life doesn't promise that all puzzles will have solutions that can be wrapped up in a nice neat package, does it? </div><button onclick="if(document.getElementById('spoiler3') .style.display=='none') {document.getElementById('spoiler3') .style.display=''}else{document.getElementById('spoiler3') .style.display='none'}" title="Click to show/hide construction" type="button">Hint 3</button>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-37090240877032814382017-01-17T04:47:00.002-08:002017-01-17T04:47:25.625-08:00My Closest Neighbor Fraction game Denise Gaskins recently flagged a post about a good fraction game: <a href="https://denisegaskins.com/2014/08/06/fraction-game-my-closest-neighbor/">My Closest Neighbor</a>. I tried this out in class today.<br /><br /><b>A pre-test</b><br />First, I wasn't sure whether the level of the game would be right for the kids. I was considering it for the 3rd and 4th graders, but had some alternative activities planned in case. To start, I posed the following questions:<br /><ul><li>Which is closest to one-half: 1/3 or 2/5? The third graders really struggled with this, so I left it alone and went to my plan B games. The fourth graders were all confident on this one.</li><li>Which is closest to 3/4: 5/11 or 11/12? This was a challenge for the fourth graders, but I thought it would be ok to play the game.</li></ul><div>In our discussion of the second question, we explored two strategies:</div><div><ol><li>making a common denominator</li><li>comparing with reference numbers</li></ol><div>The common denominator is a bit of a pain, since 11 is prime, though at least we have the fact that 4 is a factor of 12. One student soldiered through this approach, but it was difficult for the other kids to follow.</div></div><div><br /></div><div>For the second strategy, we made use of some observations that were more elementary for the kids:</div><div>(a) 5/11 < 5/10 = 1/2</div><div>(b) 3/4 is halfway between 1/2 and 1</div><div>(c) 11/12 < 1</div><div><br /></div><div>Combining these, it was easy for us to draw a rough number line, place 5/11, 1/2, 3/4, 11/12, 1 and see that 11/12 must be closer.<br /><br /><b>The game</b><br />We played three rounds: target 0, target 1/3 and target 1/2. I think this game was very challenging for the kids. Everyone had to work to figure out the best play from their hand and didn't always make the right (local) choice. For example, whether to choose 5/8 or 5/9 for a 1/2 target.<br /><br />Once everyone had played, the challenge was still just starting. They had to figure out who was closest. I structured the discussion by helping them figure out which plays were lower than the target and which were higher. For the ones that were lower, they could put them in order and only needed to consider the highest. Then, we worked on the ones higher than the target and got the lowest of those.<br /><br />In the course of this discussion, we added a third strategy to the ones listed above:<br /><br /><ul><li>making a common numerator</li></ul><div><b>Summary thoughts</b></div><div>Fraction comparison like this was still too difficult for the kids to make an engaging game. If I were to do it again, I would change to make it more of a puzzling exercise, removing competition and any sense of time pressure.</div><div><br /></div><div>Once the kids gain a bit more experience, though, I think this game has some nice features. It is particularly good for practicing fraction sense, and the multiple rounds allow some scope for strategic play.</div></div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-82525501268080949972017-01-17T02:48:00.002-08:002017-01-17T02:48:53.727-08:00100 board GoIn past posts, we've shown some of the make-shift materials we are using to play/learn Go without a proper set. Over the last two days, we had experiences that reinforce the value of this approach.<br /><br /><b>Exploring an earlier pattern</b><br />First, when playing with J3, she noticed that we could complete a repeating blue-green-yellow-red pattern around the boundary of a 5x5 board. In a follow-up conversation with J1 and J2, we explored this:<br /><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-JksT1o0tU5U/WHm6C_EhDhI/AAAAAAAABzs/Vb3vwacd9kcRpfkaGBuL-FnUL0E1R6MRwCLcB/s1600/IMG_0859.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-JksT1o0tU5U/WHm6C_EhDhI/AAAAAAAABzs/Vb3vwacd9kcRpfkaGBuL-FnUL0E1R6MRwCLcB/s320/IMG_0859.JPG" width="240" /></a></div><br />J1 explained why it would work, grouping the boundary tiles as 5 for each side of the playing square and 4 in the corners, so 5 x 4 + 4. This made it easy to see that the boundary would be a multiple of 4 and also made it easy to extend to any square board: n x 4 + 4.<br /><br />J2 had a new idea. He thought we were talking about the pattern continuing as an inward spiral. That gave us this design:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-5oIZJ2RMKwM/WHm6D8Lcj-I/AAAAAAAABz0/7o6HgHxb83krzZjjR5BBFBfDmYJiV0YWwCLcB/s1600/IMG_0861.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-5oIZJ2RMKwM/WHm6D8Lcj-I/AAAAAAAABz0/7o6HgHxb83krzZjjR5BBFBfDmYJiV0YWwCLcB/s320/IMG_0861.JPG" width="240" /></a></div>This also led to discussions about symmetry (the blue and yellow have reflectional symmetries that green and red lack) and further investigation on boards of different sizes. Interestingly, we found that, for some boards, none of the colors have a reflection symmetry.<br /><br /><b>Trying some tsumego</b><br />I set J1 and J2 the following challenge (J3 was watching): are the blue cubes alive or dead in each of these two clusters?<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-vc2imBlqJDY/WHm6KUUyqrI/AAAAAAAABz4/absV2aGol5gjbFZI__sC55uCYGsjZeDGwCLcB/s1600/IMG_0862.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-vc2imBlqJDY/WHm6KUUyqrI/AAAAAAAABz4/absV2aGol5gjbFZI__sC55uCYGsjZeDGwCLcB/s320/IMG_0862.JPG" width="240" /></a></div><br />Putting aside the interesting Go discussion that resulted, there are two consequences of doing this on the 100 board: extra unnecessary information and a built-in coordinate system. By unnecessary information, I'm talking about the letters on the white tiles and the numbers on the 100 board. This is information that is entirely orthogonal to solving the life-and-death puzzles. This is a simple toy version of one of the key modern challenges in problem solving: identifying which information is useful and which is a distraction.<br /><br />On the other hand, for talking about the puzzles, we could say things like "what if white plays a tile on 99?" For J3 who was watching, this offered another little example of the idea that numbers are all around.<br /><br /><b>Some capture fun</b><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-YjEQyMmD4YE/WHm6LaxpMcI/AAAAAAAABz8/VrO3NMMLNWQtwkFPGh7Cp2ZQzfysOudBACLcB/s1600/IMG_0863.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-YjEQyMmD4YE/WHm6LaxpMcI/AAAAAAAABz8/VrO3NMMLNWQtwkFPGh7Cp2ZQzfysOudBACLcB/s320/IMG_0863.JPG" width="240" /></a></div><div><br />For the last example, I set out some white tiles (some alone, some in groups) and asked J3 to capture them with blue cubes. After we did that, we counted the number of cubes we used to capture by moving them to cells in the 100 board (note that we removed one of the lone white tiles). A fun counting exercise, an opportunity to talk about groups of 10, and more familiarization with the layout of the 100 board.<br /><br /></div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-67159663941734899372017-01-12T00:44:00.001-08:002017-01-12T00:44:04.781-08:00Compass only non-collapsing compass (Euclidea Series)Since I've branched into this topic, I want to include some notes on additional references I've found and some results I've been able to get. First, an admission:<br /><br /><blockquote class="tr_bq">I'm still having trouble finding the intersection points of a circle and a line when the circle's center is on the line!</blockquote><b>Another resource</b><br />James King (<a href="https://www.math.washington.edu/~king/">UW home page</a>) has a nice <a href="https://www.math.washington.edu/~king/write/nwmc2007/Compass%20Constructions.pdf">session outline</a> working through compass only constructions. Be warned, there are some spoilers for some of the Euclidea challenges in that material. Unfortunately, I can't tell from write-up when and where this was used. Maybe <a href="http://www.math.washington.edu/~nwmi/outreach-alumni/meet-2007-11-17.html">this NWMI meeting</a>?<br /><br />In Professor King's notes, he describes the construction I'm struggling with as "important and difficult," so I will have to redouble my efforts.<br /><br /><b>Non-collapsing compass from collapsing compass</b><br />As an aside, perhaps you have noticed that everyone else refers to the two types of compasses as collapsible and non-collapsible. Doesn't that terminology strike you as wrong, too? The point isn't that one type of compass <i>is able to collapse</i>, but rather that it <i>always collapses</i> when not drawing a given circle. Also, a common form of "non-collapsible" compass is able to collapse!<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://ae01.alicdn.com/kf/HTB1Y8qQKXXXXXaFXFXXq6xXFXXXX/TrueColor-522002-Geometry-Drafting-Compass-Set-with-font-b-pencil-b-font-lead-good-painting-tool.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://ae01.alicdn.com/kf/HTB1Y8qQKXXXXXaFXFXXq6xXFXXXX/TrueColor-522002-Geometry-Drafting-Compass-Set-with-font-b-pencil-b-font-lead-good-painting-tool.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Just squeeze the legs together to collapse</td></tr></tbody></table><br /><br />Anyway, I was able to figure out the construction of a non-collapsing compass, just in time for Euclidea 13.2.<br /><br />I'll give a construction below. For a more mild hint, the key is the ability to reflect points through a line we've already "constructed" (where we have two points on the line).<br /><br />I've chosen to do this in GeoGebra so I can show a sequence, explain some of the reasoning, and grey out earlier parts of the construction that we don't need anymore. Check my work to make sure I didn't slip in any subtle straightedge moves!<br /><br /><div id="spoiler1" style="display: none;">Here, we are given points A, B, and C and want to construct the circle centered at B with radius AC.<br /><br />First step is two circles centered at A and B with radii AB, giving us points D and E. The line DE will be the one we are reflecting through. Key is that A and B are the reflected images of each other through this line.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-mkdwOfu-UdE/WHc_q1piWLI/AAAAAAAABzA/9xzAR3JT2Bcdn7Jle0V3MxpKmz4RNhUsgCLcB/s1600/compass1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="196" src="https://4.bp.blogspot.com/-mkdwOfu-UdE/WHc_q1piWLI/AAAAAAAABzA/9xzAR3JT2Bcdn7Jle0V3MxpKmz4RNhUsgCLcB/s320/compass1.png" width="320" /></a></div><br /></div><button onclick="if(document.getElementById('spoiler1') .style.display=='none') {document.getElementById('spoiler1') .style.display=''}else{document.getElementById('spoiler1') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide Step 1</button> <br /><div id="spoiler2" style="display: none;">Next, we draw the circle at A with radius AC. This intersects our circle centered at B in point F. F is a special point, distance AC from A and AB from B, and distance DF from D.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-3dSghcjIT0E/WHc_v_4h5TI/AAAAAAAABzE/SUyFUIU7MvEOY50nxwUQdRDwxarMgdjhACLcB/s1600/compass2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="193" src="https://4.bp.blogspot.com/-3dSghcjIT0E/WHc_v_4h5TI/AAAAAAAABzE/SUyFUIU7MvEOY50nxwUQdRDwxarMgdjhACLcB/s320/compass2.png" width="320" /></a></div><br /><br /></div><button onclick="if(document.getElementById('spoiler2') .style.display=='none') {document.getElementById('spoiler2') .style.display=''}else{document.getElementById('spoiler2') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide Step 2</button> <br /><div id="spoiler3" style="display: none;">That means the reflection of F through line DE is a point distance AC from B! We can find it using the fact that it will also be distance AB from A and distance DF from D:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-6SeQi6pdDHY/WHc_0t3-b6I/AAAAAAAABzI/HaxLczALtw8Yu8CzFii-Qzye3RdhgLWnwCLcB/s1600/compass3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="188" src="https://3.bp.blogspot.com/-6SeQi6pdDHY/WHc_0t3-b6I/AAAAAAAABzI/HaxLczALtw8Yu8CzFii-Qzye3RdhgLWnwCLcB/s320/compass3.png" width="320" /></a></div><br /><br />It almost looks like C and F are the same distance from D. That isn't true and is entirely coincidental. Try it out yourself and use the move tool to push C around and see that this is the case. In particular, you might notice that we could have made another choice for F (it is a point of intersection of two circles, we could have taken the other) and that other choice is on the opposite side of AB from C.<br /><br />I'll leave to you the satisfaction of completing the construction and drawing our target circle.<br /><br />Also, the Euclidea challenge is formulated very slightly differently: instead of point C, we are given the starting circle centered at A. That means the construction is one move shorter than what I've shown here. </div><button onclick="if(document.getElementById('spoiler3') .style.display=='none') {document.getElementById('spoiler3') .style.display=''}else{document.getElementById('spoiler3') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide Step 3</button>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-29648131249974228192017-01-10T21:51:00.002-08:002017-01-10T21:51:40.110-08:00Teaching math with GoRecently, I have been insinuating Go playing into my time with the 3 Js. This was initially motivated by a quote I saw on one of the Mathpickle pages (<a href="http://mathpickle.com/gamers/">Gamers under Inspired People</a>):<br /><blockquote class="tr_bq">Schools should experiment teaching go* instead of a regular math curriculum for one year to students around the age of 7. It is my prediction that the strong problem solving skills that this will engender will make superior students than any existing mathematics curriculum.</blockquote>Now, when we first decided to have kids, my objective was to help them develop into people with whom I would enjoy spending time. In particular, I wanted to be able to play games with them. With that in mind, the Mathpickle idea resonated with another idea from Richard Garfield (via <a href="http://mathhombre.blogspot.com/p/games.html">Math Hombre</a>):<br /><blockquote class="tr_bq">play each game so as to increase your chances of winning all games</blockquote><br />With these three ideas in mind, I went looking for a way to properly introduce Go to our clan.<br /><br /><h2>Curriculum outline</h2>Not surprisingly this is a question other gaming and math people have asked before. Quickly putting together the ideas I liked the most from other sources, we basically started following the curriculum shown in the Go GO Igo videos with Yoshihara Yukari (Umezawa Yukari at the time of filming):<br /><br /><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://www.youtube.com/watch?v%3DCH8GulBSj-k&source=gmail&ust=1484196153288000&usg=AFQjCNE5K4nVyOmxfPqIYP-hWW7sOh7AKA" href="https://www.youtube.com/watch?v=CH8GulBSj-k" style="color: #1155cc;" target="_blank"><span class="m_-1324303217220015475gmail-il"><span class="il">Yukari</span></span> Sensei Go Go Igo Part 1</a></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><span style="font-size: 12.8px;">(1) basics</span></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- placing stones</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- black vs white</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- capturing single stone</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(2) capture game</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- 6x6 board</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- first to capture wins</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- etiquette</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(3) illegal moves</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- playing where your stone will have no liberties</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- playing where the stone has no liberties but captures an opponent's stone(s)</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(4) expanded capture games</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- first to capture 3 stones wins</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- infinite capture</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- Ko rule</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(5) territory</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- counting territory at the end of the game</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(6) simple capture puzzles</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- one move</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- two moves</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- three moves</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://www.youtube.com/watch?v%3DiHrpEqmhbZY&source=gmail&ust=1484196153288000&usg=AFQjCNHUYt2ELJ6xhbc1xIcpVxIK96a8_Q" href="https://www.youtube.com/watch?v=iHrpEqmhbZY" style="color: #1155cc;" target="_blank"><span class="m_-1324303217220015475gmail-il"><span class="il">Yukari</span></span> Sensei Go Go Igo Part 2</a></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(7) Etiquette: </div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- Nigiri: choosing white vs black</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- komi and first player advantage (maybe useful to play some 5x5 or 7x7 games to make the first player advantage clear?)</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(8) eyes and false eyes</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(9) Scoring</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- Dame, </div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- kyu, </div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- Japanese vs Chinese scoring</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- agehama: stones considered captured </div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">How important is this?</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(10) standard patterns</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- stair-step (shichou)</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">- geta (also kosumi? 45 degree cut to capture enemy stones)</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://www.youtube.com/watch?v%3D2VkU7a-Nqf4&source=gmail&ust=1484196153288000&usg=AFQjCNHl6gkyw8PG5fKdoJPqMTujBkVESA" href="https://www.youtube.com/watch?v=2VkU7a-Nqf4" style="color: #1155cc;" target="_blank">Part 3</a></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(11) more puzzles/standard patterns</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><br /></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;"><a data-saferedirecturl="https://www.google.com/url?hl=en&q=https://www.youtube.com/watch?v%3DXe1EnIdIleI&source=gmail&ust=1484196153288000&usg=AFQjCNG5ZpovfNR4ChH9loIysPbnW03wIg" href="https://www.youtube.com/watch?v=Xe1EnIdIleI" style="color: #1155cc;" target="_blank">Part 4</a></div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(12) Tsumego</div></blockquote><blockquote style="background-color: white; border: none; color: #222222; font-family: arial, sans-serif; font-size: 12.8px; margin: 0px 0px 0px 40px; padding: 0px;"><div style="font-size: 12.8px;">(13) maxims</div></blockquote><br /><h2>Some early lessons</h2>Since we don't actually have a Go board or stones, we started with the electronic board <a href="https://www.gokgs.com/download.jsp">CGoban</a>. This works well for J1 and J2. We have also used J1's chess/checkers set as a makeshift 9x9 board (playing on the lines instead of the squares).<br /><br />For J3, we started playing the simple capture game using the blank side of our 100 board.<br />For the first lesson, we arranged things like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-QCRyRfgfh_E/WHW_3IVb3gI/AAAAAAAABys/qMX9M0jBeW8B4Pj2T9-DZ2PnWrSeJBTIQCLcB/s1600/IMG_0853.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-QCRyRfgfh_E/WHW_3IVb3gI/AAAAAAAABys/qMX9M0jBeW8B4Pj2T9-DZ2PnWrSeJBTIQCLcB/s320/IMG_0853.JPG" width="240" /></a></div><br />She played the centimeter cubes (which substitute for black stones) and I played the Bananagram tiles (substituting for white stones). I gave her a four stone advantage and we played three games with me starting in different places (center, corner, side) and saw that she could easily capture at least one of my stones without trouble.<br /><br />Some of J3's observations along the way:<br /><br /><ul><li>There are 11 blue tiles forming the boundary</li><li>There are 25 squares in our playing area</li><li>There are five squares along each edge of our playing area</li><li>The placement of the four handicap stones is symmetric in the playing area. There are several symmetries</li><li>Stones in the corner have two directions to live</li><li>Stones on the edge have three directions to live</li><li>Stones inside have four directions to live</li></ul><div><br /></div><div>For the second lesson, we made the board a little differently based on J3's preference for a blue-green-yellow-red pattern around the border:</div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-2UCMEduu81Q/WHW_2v741WI/AAAAAAAAByo/VxAPSaa-uc4l_K1T-agBQ6YEVf7QQlEXQCLcB/s1600/IMG_0854.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://4.bp.blogspot.com/-2UCMEduu81Q/WHW_2v741WI/AAAAAAAAByo/VxAPSaa-uc4l_K1T-agBQ6YEVf7QQlEXQCLcB/s320/IMG_0854.JPG" width="320" /></a></div><br />This time, J3 made some different observations:<br /><br /><ul><li>The pattern continues around the border (at no place, did we have to break the pattern). A more advanced question: will this always happen with our Blue-Green-Yellow-Red pattern around a square board?</li><li>The colors in opposite corners are the same (blue-blue and yellow-yellow)</li><li>There are more than 11 tiles on the border now.</li><li>Still 25 squares on the board and 5 squares along each side</li></ul><div>For the third session, J3 was willing to reduce her starting advantage and she wanted to place the handicap stones herself:</div><div><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-xPc3BJrZ3Fs/WHW_2uJX5MI/AAAAAAAAByk/d29O5KcbWJIgwuzMOUhKIgb1J47LDT_TACLcB/s1600/IMG_0855.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://4.bp.blogspot.com/-xPc3BJrZ3Fs/WHW_2uJX5MI/AAAAAAAAByk/d29O5KcbWJIgwuzMOUhKIgb1J47LDT_TACLcB/s320/IMG_0855.JPG" width="320" /></a></div><br />This is a losing position (remember, we are still playing where the first to capture at least one stone is the winner):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-WpXQ40fG5cw/WHW_6dbDjYI/AAAAAAAAByw/jLhwPsKT8c4wm-td4IbTP1hHtlpp5n9WQCLcB/s1600/IMG_0856.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://2.bp.blogspot.com/-WpXQ40fG5cw/WHW_6dbDjYI/AAAAAAAAByw/jLhwPsKT8c4wm-td4IbTP1hHtlpp5n9WQCLcB/s320/IMG_0856.JPG" width="320" /></a></div><br />She didn't take losing especially well, but this is a nice feature of playing these kinds of short games. The kids can make a mistake, they have to deal with failure, but it isn't very costly since each game only takes a couple minutes and the next game starts right away.<br /><br /><b>Some Go concepts we are still developing</b><br />At this stage, we are still working on the basic concepts:<br /><br /><ol><li>once placed, the stones don't move</li><li>only the main compass directions (north, east, south, west) are liberties. Diagonals don't give life.</li><li>liberties are shared for a group, not just the individual stones. For example, a stone surrounded by its own color is not dead (if the overall group still has liberties).</li><li>I need to remember to announce "Atari" when a stone or group has only one remaining liberty.</li></ol><h2>Observations</h2><div>From a Go/games perspective, I think it is helps to start playing a lot of low-cost games: fast games where the winning condition is easy to identify and immediate. This allows the kids to make mistakes, see clearly the consequences of those mistakes, and lose, then immediately try again.</div><div><br /></div><div>From a math perspective, there is a huge amount of elementary math that comes out of the simple games:</div><div><ol><li>counting</li><li>addition</li><li>patterns</li><li>some basic multiplication, particularly with the array model</li></ol><div>In addition, we had the usual experience with using physical manipulatives: something extra always comes up. For example, using the 100 board inspired J3 to show off to me that she can count to 100 now (using the board as a reference).</div></div><div><br /></div><div>I'm looking forward to future sessions.</div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-24624174143017512132017-01-10T20:11:00.000-08:002017-01-10T20:11:13.618-08:00Beginning Restricted Constructions (Euclidea Series)Note: This is a little bit out of order, but reflects the challenge on which we're currently working.<br /><br /><b>Choices of Construction Rules</b><br />The classic construction problems involve specific, restricted tools:<br /><br /><ol><li>unmarked straight-edge: this tool is only able to determine a line that contains two existing points (previously given or already constructed). It has no length markings and only one side.</li><li>collapsing compass: this tool is only able to determine a circle given the center and a point on the circle. Again, both center and an included point must already be given earlier in the construction. Like the straight-edge, the compass has no way to indicate the length of the radius. (side note: when I was in HS geometry, we didn't require the compasses to be collapsing).</li><li>A plane as a writing surface and infinitely fine/precise pens: these allow us to identify the intersection points of the constructed objects (circles and lines).</li></ol><br />I, and maybe you, for years considered these rules to be natural. However, there are several other construction rules that are at least as natural:<br /><br /><ul><li>Origami: moves based on folding paper. This is really a fascinating topic. Take a look at this <a href="https://www.youtube.com/watch?v=SL2lYcggGpc">Numberphile video </a>for a taste.</li><li>Straight-edge only constructions: for those unfortunate to have left their compass at home.</li><li>Compass-only constructions: easy to imagine how ancients made a good compass, but how would they have gotten a really straight edge anyway?!</li></ul><br />Several of the Euclidea challenges impose either straight-edge only or compass-only restrictions, which got me interested in this general topic again.<br /><br /><b>Examples in Euclidea</b><br />Unless I've missed some, the restriction challenges start officially in Theta pack:<br /><br /><ul><li>Drop a perpendicular (8.4): we are only given the straight-edge, but we are also given a circle.</li><li>Mid-point (8.5): find the midpoint of a segment with the straight-edge and a parallel line.</li><li>Segment trisection (9.7): same restrictions as 8.5</li><li>Midpoint (13.1): this time, we've only got the compass, no straight-edge</li><li>Some I've not yet unlocked: Tangent to Circle (13.5), Drop a Perpendicular (13.7), Line-Circle Intersection (13.8)</li></ul><div>In addition, Zeta pack has a some challenges that, while not marked as restrictions, are good warm-ups: Point Reflection (6.1), Line Reflection (6.2), and Translate Segment (6.6).</div><div><br /></div><div><b>Humans can forget things!</b></div><div>If you've made it to Zeta pack, you know that the collapsing compass is able to replicate the function of a non-collapsing compass, so that particular restriction doesn't change what is theoretically constructible (but it sure reduces a lot of extra steps and auxiliary objects in real constructions.)</div><div><br /></div><div>It turns out that the straight-edge is also unnecessary! This result is the <a href="https://en.wikipedia.org/wiki/Mohr%E2%80%93Mascheroni_theorem">Mohr-Mascheroni Theorem</a>. One really fascinating/disturbing fact about this result is that it was first proven, as far as we know, in 1672 by Mohr, but his proof was lost for over 250 years. Mascheroni independently discovered and proved the result about 120 years after Mohr.</div><div><br /></div><div>I think this is an important lesson that, with search-empowered internet, is easy to forget: human knowledge accumulation isn't always steady or certain.</div><div><br /></div><div><b>Compass-only program</b></div><div><a href="http://www.cut-the-knot.org/do_you_know/compass.shtml">Cut-the-Knot</a> (a generally excellent resource) has a good discussion of restricted constructions and outlines a program for proving the sufficiency of compass-only constructions.<br /><br />I have started to work through this list. Generally, I won't post solutions since Cut the Knot already has solutions. However, I was concerned about whether the compass they were using was collapsing or not. The solution for 6 (given three points, find a fourth point that makes them vertices of a parallelogram) clearly uses a non-collapsing compass. So, that leaves an open challenge: how to build a non-collapsing compass from just a collapsing compass.<br /><br />I'm currently stuck on this. There are two ways I could break through:<br />(a) find the point of intersection of two lines. If I could do this, I would use the ability to find perpendicular lines to move distances around.<br />(b) Find the point of intersection of a line and a circle with the center on the line.<br /><br />Note, this second one is very similar to Cut the Knot's third challenge, but that assumes the center of the circle is not on the line segment. Amazingly, this slight change makes the challenge much harder, at least for me.<br /><br /></div><div><br /></div><div><br /></div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-61452529819019221512017-01-10T20:02:00.002-08:002017-01-10T20:02:52.483-08:00Running, rates, roundingMy running session this morning gave me an idea for a kind of 3-act math discussion with J1 and J2. I will discuss this with them when they come back from camp and see what they think. I expect the last questions will be hard for them and I would like to see how much progress they can make working together.<br /><h1>First Act</h1>Today, I went running and recorded some information on my GPS. For five laps, I ran moderately fast. Here is the data: <br /><center><table><tbody><tr><th>Time</th><th> Rate</th><th> Distance</th></tr><tr><td>3:00</td><td> 12.7 kph </td><td> 635 m </td></tr><tr><td>3:00</td><td> 12.9 kph </td><td> 647 m </td></tr><tr><td>3:00</td><td> 12.6 kph </td><td> 633 m </td></tr><tr><td>3:00</td><td> 12.7 kph </td><td> 637 m </td></tr><tr><td>3:00</td><td> 12.8 kph </td><td> 645 m </td></tr></tbody></table></center><br />What do you notice?<br />What do you wonder?<br /><h1>Second Act</h1>My target was actually to run 12 kph for each of these three minute segments. After the first lap, I knew that I could run more slowly and still hit my target. I wondered, how much less than 635m could I run and still hit my target? <br />If I compare two laps, both rates and distances, can I figure out the distance I get for each 0.1 kph? Is there another way to calculate the difference in distances for each 0.1 kph? <br /><h1>Third act</h1>For some reason, this made me think about rounding that J1 had recently been studying. He is a bit disturbed about what to do with values that are halfway between the rounded levels, for example whether 15 should round up or down to the nearest ten. Since this investigation of running data involved calculations with measured values and rounding, I though it would be instructive to explore a couple of calculations: <br /><br /><ul><li>I have two distances, rounded to the nearest 10 cm of 20 cm and 10 cm. What is a reasonable range for the difference of those distances?</li><li>My GPS measured a time of 3 minutes (3:00, rounded to the nearest second) and speed of 12 kph (12.0 kph rounded to the nearest tenth of a kilometer per hour). What distance did I run? What is a reasonable range for that distance?</li></ul><div><br /></div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-46756811139437916532017-01-08T05:35:00.004-08:002017-01-08T05:35:52.914-08:00Some Reverses in Gamma (Euclidea series)I found several of the puzzles in Gamma pack to be cute, even though they aren't necessarily hard. In especially liked the "reverse" constructions, finding the triangle given the orthocenter (3.2) or given the circumcenter (3.3).<br /><br />Note: toward the latter half of this pack, I was getting anxious to see what Delta pack had in store, so I bashed through constructions for 3.5-3.8 without always finding the minimal moves solutions. Among those four challenges, I still have 5 missing stars.<br /><br /><b>Triangle from orthocenter (3.2)</b><br />Obtaining the E star gave me trouble on this one. I didn't originally get one, but figured I would try harder for these notes.<br /><br /><div id="spoiler1" style="display: none;"><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-aChe8obD8Fg/WHD5-8E2cXI/AAAAAAAAByQ/0pT7djdmSqQ4duxvKAjB5j3cDkwE0LgewCLcB/s1600/Ortho%2BReverse%2BE%2BSoln.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="169" src="https://1.bp.blogspot.com/-aChe8obD8Fg/WHD5-8E2cXI/AAAAAAAAByQ/0pT7djdmSqQ4duxvKAjB5j3cDkwE0LgewCLcB/s320/Ortho%2BReverse%2BE%2BSoln.png" width="320" /></a></div><br />The key idea is to use the E-optimal perpendicular construction from 2.6 and the vertex of the angle we're given as one of the center points. That allows us to pick up two perpendiculars for the cost of 5 E moves, leaving one last move to connect the two new vertices.<br /><br /></div><button onclick="if(document.getElementById('spoiler1') .style.display=='none') {document.getElementById('spoiler1') .style.display=''}else{document.getElementById('spoiler1') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide</button><br /><br /><b><br /></b><b>Triangle from intersection of perpendicular bisectors (3.3)</b><br /><b><br /></b><div id="spoiler2" style="display: none;"><br />Well, I actually already gave a spoiler above when I shortened the name of this challenge. If you've got the intersection of the perpendicular bisectors, then you have the center of the orthocircle, the circle that contains all the vertices of the triangle.<br /><br />Since we already have one vertex and rays where the other edges are....<br /><br /></div><button onclick="if(document.getElementById('spoiler2') .style.display=='none') {document.getElementById('spoiler2') .style.display=''}else{document.getElementById('spoiler2') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide</button><br /><b><br /></b><b>Three equal distances (3.4)</b><br />When going back to write up these notes, I didn't remember how this construction worked and was concerned I'd have trouble working through a tricky challenge. Fortunately, ....<br /><br /><div id="spoiler3" style="display: none;">The key insight is the relationship between points B, D, and M. Just think about which of our favorite construction tricks relates them and you are done. Finding E from D and M is straightforward, but keep your eyes opened for a nice surprise!<br /><br /></div><button onclick="if(document.getElementById('spoiler3') .style.display=='none') {document.getElementById('spoiler3') .style.display=''}else{document.getElementById('spoiler3') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide</button><br /><b><br /></b><b>Those V stars</b><br /><b><br /></b><div id="spoiler4" style="display: none;">Three equal distances (3.4), Forty-five degree angle (3.7), and lozenge (aka rhombus 3.8) all have V stars. In fact, 3.8 needs 4 versions to collect the V!<br /><br /></div><button onclick="if(document.getElementById('spoiler4') .style.display=='none') {document.getElementById('spoiler4') .style.display=''}else{document.getElementById('spoiler4') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide</button> <b><br /></b>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-42925630776742262772017-01-06T06:49:00.001-08:002017-01-06T06:49:06.311-08:00A construction mess (Euclidea Series)Part of my motivation for writing this series is to make a confession: some of my constructions are just a mess built of geometrically calculating a length that I've determined algebraically. Typically, my approach is to create a coordinate system, then set up a couple of equations that determine a key length for the construction, solve those equations, then use geometric operations to construct that value.<br /><br />Here is an example. Theoretically, it is a spoiler for a construction in mu pack (circle tangent to two circles 12.5) but I doubt anyone will really be able to see what is going on here:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-z1pIVWUdHO0/WG-s_q_trkI/AAAAAAAAByA/r6DBpLWt1IM6GsCSfPZRHRR6qk-NcfhggCLcB/s1600/Construction%2BMess.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="212" src="https://1.bp.blogspot.com/-z1pIVWUdHO0/WG-s_q_trkI/AAAAAAAAByA/r6DBpLWt1IM6GsCSfPZRHRR6qk-NcfhggCLcB/s400/Construction%2BMess.png" width="400" /></a></div><br /><br />Another example is construction of the regular pentagon. I know the golden ratio figures prominently, so one approach I often use is just to build that ratio between two lengths and then impose it on the basic construction template.<br /><br />Other than massively overrunning the move targets for E and L stars, the weakness of this approach is that it doesn't link with any geometric insight about the construction. In a sense, this is the power of analytic geometry: you can get results without having to find a new insight.<br /><br />Over the course of the series, I'll try to work on constructions that are geometrically insightful, but won't shy away from letting you know where I've had to push through with a brute force attack.Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-37241997784671358622017-01-05T00:20:00.000-08:002017-01-05T00:21:41.593-08:00Perpendiculars (Euclidea series)Beta pack is the only one for which we have received all of the stars. Even so, a couple of the puzzles are worth discussing because they illustrate some interesting ideas. In particular, we like Drop a Perpendicular (2.6) and Erect a Perpendicular (2.7).<br /><br /><b>Perpendicular bisectors</b><br />One of the ideas lurking is around the perpendicular bisector, which we saw way back in alpha pack (1.2). One of the things I always found interesting about the perpendicular bisector was that it was easier to construct an object with more conditions than either constructing a perpendicular alone or finding a midpoint alone.<br /><br />The other great thing about the perpendicular bisector is that it can be defined in an entirely different way: it is the locus of all points that are equidistant from our two starting points. In case that isn't clear, assume we start with two distinct points A and B. The perpendicular bisector of the segment AB is also the locus of all points C such that distance AC is equal to distance BC.<br /><br />Of course, the fact that this is a line means that we only have to find two such points to construct the perpendicular bisector (which is how you solved 1.2, right?)<br /><br /><b>Drop a perpendicular</b><br />For the 2 move solution using tools, I'll let you find a solution on your own. In case you need a hint: how many combinations of 2 moves are there anyway? You could just try them all and see what you find.<br /><br />Way back in my HS geometry class, I learned a construction for dropping a perpendicular that uses 4 elementary moves. For the E start, though, that's not good enough. The cool idea that breaks through here is to choose two totally arbitrary points on the line as centers of circles that we draw. For some reason, I get a kick out of the idea that arbitrary points can be helpful ("if the point we choose doesn't matter, how can it help to choose a point anyway?")<br /><br />In this case, while the points on the line we choose don't matter, the circle we draw with those points as centers need to have the right radius. Click the button below if you want to see how it is done and a bit more explanation.<br /><br /><div id="spoiler1" style="display: none;"><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-NYGzxobS40g/WG39dWO5X2I/AAAAAAAABxk/rJQGDfBiceEsBQGZeij4TBanwGUsVjQmQCLcB/s1600/drop%2Bperp.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="346" src="https://1.bp.blogspot.com/-NYGzxobS40g/WG39dWO5X2I/AAAAAAAABxk/rJQGDfBiceEsBQGZeij4TBanwGUsVjQmQCLcB/s640/drop%2Bperp.png" width="640" /></a></div><br />The key here is that the circle radii are equidistant from our target point and the new intersection point of the two circles. That means they are on the perpendicular bisector of the segment between those points. Another observation is that the new intersection of the two circles is the reflection of our target point through the line. Kind of cool that, no matter which two center points we choose for the two circles, the new intersection point will always be the same! </div><button onclick="if(document.getElementById('spoiler1') .style.display=='none') {document.getElementById('spoiler1') .style.display=''}else{document.getElementById('spoiler1') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide</button><br /><br /><b>Erect a perpendicular</b><br />Erecting a perpendicular also uses the idea of choosing an arbitrary point, but goes a step farther. This time, we choose <i>any point we want that is not on the line already!</i> Well, it also fails if we happen to choose a point that is already on the perpendicular, but that should be impossible.... In any case, just choose a point somewhere off to the side.<br /><br /><div id="spoiler2" style="display: none;"><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-2lkWpJAsv6s/WG4BZuBQrEI/AAAAAAAABxw/8_vNtnfBaxA1qo8qGNZWC-T80QGGj7UiwCLcB/s1600/erect%2Bperp.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="275" src="https://2.bp.blogspot.com/-2lkWpJAsv6s/WG4BZuBQrEI/AAAAAAAABxw/8_vNtnfBaxA1qo8qGNZWC-T80QGGj7UiwCLcB/s400/erect%2Bperp.png" width="400" /></a></div><br /></div><button onclick="if(document.getElementById('spoiler2') .style.display=='none') {document.getElementById('spoiler2') .style.display=''}else{document.getElementById('spoiler2') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide</button><br /><br /><br />This construction uses <a href="https://en.wikipedia.org/wiki/Thales'_theorem">Thales' Theorem</a>. I don't know exactly why I find this result to be so cool, since the proof isn't hard. For me, it transforms a circle into a family of right triangles. Given a length for the hypotenuse, all the right triangles with that hypotenuse are living right there on the arc of the circle with that length as the diameter.<br /><br />Incidentally, Thales' Theorem and the two defining properties of perpendicular bisectors will come up a lot in other Euclidea Constructions. <br /><br /><b>V stars</b><br />If you want to know where the V stars are, click below: <br /><br /><div id="spoiler3" style="display: none;">Angle of 30 degrees (2.3) and Double Angle (2.4) both have two solutions. </div><button onclick="if(document.getElementById('spoiler3') .style.display=='none') {document.getElementById('spoiler3') .style.display=''}else{document.getElementById('spoiler3') .style.display='none'}" title="Click to show/hide construction" type="button">Show/hide</button>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com1tag:blogger.com,1999:blog-5544661968326910027.post-2817002993493403032017-01-03T20:40:00.000-08:002017-01-03T20:41:39.331-08:00Inscribed Circle (Euclidea series)Sometime in the past two years, <a href="http://mathmamawrites.blogspot.com/">Sue VanHattum</a>, introduced us to <a href="http://euclidthegame.com/">Euclid the Game</a>. This is a nice series of classical construction puzzles (compass and straight-edge) built on top of Geogebra. We recently returned to it and saw a link to another version that we've been playing a lot recently:<br /><a href="https://www.euclidea.xyz/en/game">Euclidea</a>.<br /><br />In EtG, there is a nice discussion in the comments section. Euclidea doesn't have this feature, so I decided to write blog posts chronicling some of our struggles and, hopefully, starting a place for discussion. I'm not planning to write about every challenge or post answers to all of them, but am happy to take requests. Otherwise, I'll write about the puzzles I find challenging and/or interesting for some reason.<br /><br /><h2>Alpha Pack</h2><div>I think there are two things worth talking about in alpha pack: (1) the puzzle that has stumped us and (2) <a href="http://3jlearneng.blogspot.com/2017/01/inscribed-circle-euclidea-series.html#vstars">the location of the V stars.</a></div><div><br /></div><div><b>That darned square (or is it a diamond?)</b></div><div>The last puzzle in the alpha pack is the one that has stumped us. Our target is this inscribed square:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-sEXtUwVSj3k/WGx5e_DT-SI/AAAAAAAABxI/wQ-zD1q0RMsakYLeJhebr6x1mSc8K6NNwCLcB/s1600/target.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-sEXtUwVSj3k/WGx5e_DT-SI/AAAAAAAABxI/wQ-zD1q0RMsakYLeJhebr6x1mSc8K6NNwCLcB/s320/target.png" width="290" /></a></div><div><br /></div><div>The kicker is to construct it in 7 elementary moves!</div><div><br /></div><div><b>Our thought process</b></div><div>We need four line moves to draw the sides of the square. That means we have only 3 moves to find the other three vertices. We can get one by drawing the diameter of the circle, so we have two moves to find the other two vertices.</div><div><br /></div><div>We can easily find those two side vertices with three elementary moves, but are really stuck on the idea needed to get one less move.</div><div><br /></div><div><br /></div><h2>Some spoilers</h2><div><b>Six move square</b></div><div>Our approach to get the construction in 8 elementary moves serves easily to get the 6 move construction. Below, I've included the finished picture which should be enough to see the approach (it isn't very involved anyway).</div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-D_4vJhjtv1g/WGx7Cf5DYZI/AAAAAAAABxU/spXUU6A0LykGmNUE7c7gS1uWsez4_eUmwCLcB/s1600/Inscribed%2BSquare.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="https://2.bp.blogspot.com/-D_4vJhjtv1g/WGx7Cf5DYZI/AAAAAAAABxU/spXUU6A0LykGmNUE7c7gS1uWsez4_eUmwCLcB/s400/Inscribed%2BSquare.png" width="400" /></a></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><b><a href="https://www.blogger.com/null" name="vstars">V Stars</a></b></div><div>We found V stars in the following puzzles: equilateral triangle tutorial, 60 degree angle (1.1), and rhombus in a rectangle (1.5).<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /></div>Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0tag:blogger.com,1999:blog-5544661968326910027.post-53711369752442154162017-01-03T05:56:00.001-08:002017-01-03T20:04:29.814-08:00Age puzzlesSome quick puzzles created and discussed with J2 while at lunch today. For all, we made the simplifying assumptions that everyone has their birthday on the same day.<br /><br /><b>Basics/assumptions</b><br />Our friends were discussing their current ages: Jin 9, Jate 7, Panelia 6, Sophia 5, and Jane 4. The first two are boys, the latter three girls. All have their birthday on the same day of the year and are having their birthday today!<br /><br /><b>Add them up</b><br />The boys wondered: will it or has it ever happened that the sum of our ages is the same as the sum of the ages of the girls?<br /><br /><b>Doubles</b><br />A little twist on the previous question: when will the sum of the girls ages be twice that of the boys?<br /><br />For those that know algebra: can you make sense of the solution?<br /><br /><b>Halves</b><br />The previous puzzle was a bit strange. What if we go the other way: when will (or were) the sum of the ages of the boys twice that of the girls?<br /><br /><b>Murky waters with products</b><br />What is the current product of the boys ages? The product of the girls ages?<br /><br />Have those products ever been equal? We argued this based on the intermediate value theorem.<br />When were those products equal? We numerically approximated to the closest half year.<br /><br /><h2>Some other families</h2><div>J2 asked me to include these, which we'd discussed during a dinner last week.</div><div><br /></div><div>Bill is five years younger than his sister. In seven years, Bill will be 2/3 his sister's age. How old are they now?</div><div><br /></div><div>John is ten years older than his brother Joe. In six years, John will be twice as old as Joe. How old is Joe now?</div><h2>What are your favorites?</h2>If you have any of these types of puzzles, please let us know in the comments!Joshua Greenehttps://plus.google.com/111266546987719295970noreply@blogger.com0