tag:blogger.com,1999:blog-36451334814075133402024-03-13T01:01:08.345-07:00median Don Steward Magic Squaresdon stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.comBlogger28125tag:blogger.com,1999:blog-3645133481407513340.post-56518029450689228532017-02-11T09:30:00.002-08:002017-02-11T09:30:56.457-08:00magic square vector space<div class="separator" style="clear: both; text-align: center;">
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<br />don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-88853108032045397682017-02-11T07:01:00.000-08:002017-02-11T07:01:06.986-08:00Somerset sessionthanks for a great session in Somerset<br />
magic square training...<br />
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the <a href="https://drive.google.com/open?id=0B5Rl8vLrghq_aFNtdVo1NnlOdkU">powerpoint</a>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-84942365700975304782011-08-22T05:57:00.000-07:002011-08-22T05:57:57.149-07:00proofs 3Prove that any three sets of three numbers will form a 3 by 3 magic square providing:<br />
<ul><li>each set of three must be in a linear sequence with the same difference 'a' (e.g. 1, 7, 13 & 10, 16, 22 & 19, 25, 31 all with a common difference of 6) </li>
<li>the first terms in the three sequences must also be in a linear sequence, difference 'b' (1 , 10 and 19 in the above example)</li>
</ul><div class="separator" style="clear: both; text-align: center;"></div>this is why any 3 by 3 block from a calendar (with a = 1 and b = 7) will form a magic square<br />
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for any magic square<br />
<ul><li>use the fact that the corners are the average of the two outer middles not in line with the corner (proved elsewhere) to establish that the lowest and highest numbers must be in a middle edge cell position</li>
<li>use the line total property to establish that the four corner numbers sum to a multiple of 4</li>
<li>what happens when (using the terminology above) b = 2a? try to explain why this happens</li>
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don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-71558351217612186152011-08-22T05:21:00.000-07:002011-08-22T05:22:13.748-07:00proofs 2the 'Lo Shu'<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxxLLGJwfg0TQOkkxF6PCmZotfWlFmYQglKXD1waM7KB1mGd_sVqzGOV1hD06b9WVpo4K46HF0HZtL42XOrGMd-fzDhkk-nLnVr9v4JF_vm2IP7GUTuoGJpx7EcfumOKZIdQzcIiLMjX0/s1600/work.2514.11.flat%252C550x550%252C075%252Cf.lo-shu-turtle.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="177" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxxLLGJwfg0TQOkkxF6PCmZotfWlFmYQglKXD1waM7KB1mGd_sVqzGOV1hD06b9WVpo4K46HF0HZtL42XOrGMd-fzDhkk-nLnVr9v4JF_vm2IP7GUTuoGJpx7EcfumOKZIdQzcIiLMjX0/s200/work.2514.11.flat%252C550x550%252C075%252Cf.lo-shu-turtle.jpg" width="200" /></a></div>this magic square consists of all the digits, from 1 to 9 inclusive<br />
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although there are a total of 8 rotations and reflections of it there is essentially one such magic square<br />
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what do 1 to 9 add up to?<br />
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what must each line total be?<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEie9-nUNMUT2NzchqQgQi1bjTsWs73a1OFd3TB4rSPyoRcyXXFbeVgtfcLXSiygdhc5LfKV9LRQhgi1VH9u2b2Yq2LixjuUUo3b6FxSA5yq4vCw2JcA5lTFg0rYCKCB6WQkIp9Mw6LZhiw/s1600/1000px-Lo_Shu_3x3_magic_square.svg.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="186" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEie9-nUNMUT2NzchqQgQi1bjTsWs73a1OFd3TB4rSPyoRcyXXFbeVgtfcLXSiygdhc5LfKV9LRQhgi1VH9u2b2Yq2LixjuUUo3b6FxSA5yq4vCw2JcA5lTFg0rYCKCB6WQkIp9Mw6LZhiw/s200/1000px-Lo_Shu_3x3_magic_square.svg.png" width="200" /></a></div>think of some (good) reasons why the central number must be 5<br />
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by considering what numbers can be in the same line as 9, prove that 9 cannot occupy a corner position (and neither can 1)<br />
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by considering sums of odd and even numbers, prove that the corner numbers must all be even<br />
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given that 9 must be in a middle edge position, where must 8 go?don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-45560423992250016682011-08-21T14:25:00.001-07:002011-08-21T14:25:30.165-07:00proofs 1<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb4V8CiJJm-S7eDco_k-ed0-5WcCtCdAvi8AHpcSywYHMyXQq50v_g3W4zVOb-QT2N8tbee0QOWd4FRPm7IT4RJ7olrzWMLUYI7B_ucRdve0-gcjKZvtBOIjuhWWurevS8tEPTJhSCzkA/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="317" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb4V8CiJJm-S7eDco_k-ed0-5WcCtCdAvi8AHpcSywYHMyXQq50v_g3W4zVOb-QT2N8tbee0QOWd4FRPm7IT4RJ7olrzWMLUYI7B_ucRdve0-gcjKZvtBOIjuhWWurevS8tEPTJhSCzkA/s400/Picture1.png" width="400" /></a></div><br />
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don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-24749155666820118412011-08-21T12:22:00.000-07:002011-08-21T12:22:06.018-07:00quadratic equations<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUUyl3K2VFJ3jpgUcIdKrytCodfzdeEGRO__po9FHUjDmwotR7Y4QMltlsA0PMiWCuC4fvP67q0of7M95i6Ed-iFAty9cCVzckIukoar7LexFfJ9LJ9P3-OqMT6VxZOZdf6lbUjStrcK0/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUUyl3K2VFJ3jpgUcIdKrytCodfzdeEGRO__po9FHUjDmwotR7Y4QMltlsA0PMiWCuC4fvP67q0of7M95i6Ed-iFAty9cCVzckIukoar7LexFfJ9LJ9P3-OqMT6VxZOZdf6lbUjStrcK0/s320/Picture1.png" width="320" /></a></div><br />
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don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-6054952659417249182011-08-21T12:17:00.001-07:002011-08-21T12:17:39.080-07:00simultaneous equations 2<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKEXTGAJTeTnOmVSecXAqWLvRNxZ2yFJoYM-CzybVOux4wr5TRCHx07t_hTgw8aXllHAQNQ__Vo2z1hxYpYdfqfLvcoInLT6NcMxGfKwwj6sIFepOPIfcn5PDoy2XkRtTMZqSzHaZLE7A/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="290" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKEXTGAJTeTnOmVSecXAqWLvRNxZ2yFJoYM-CzybVOux4wr5TRCHx07t_hTgw8aXllHAQNQ__Vo2z1hxYpYdfqfLvcoInLT6NcMxGfKwwj6sIFepOPIfcn5PDoy2XkRtTMZqSzHaZLE7A/s400/Picture1.png" width="400" /></a></div><br />
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don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-90126517340737068272011-08-21T12:10:00.000-07:002011-08-21T12:10:13.740-07:00simultaneous equations 1<div class="separator" style="clear: both; text-align: center;"></div><br />
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don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-30908656068645636382011-08-04T07:11:00.000-07:002011-08-04T07:11:22.414-07:00equations with a letter on both sides<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit4wnjEg8ugG-p6TadnyjX6fpe1KiqG6OiyUz9Imq1OOhxjkWVEtJKtgTz9g2iD01e567S0PevAgFURDgZLmeKloR6mnHik7U5nw8UOFs7aKFFK5NeTIM4kowp_N2ixHG5YnGv0B1EgHs/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="375" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit4wnjEg8ugG-p6TadnyjX6fpe1KiqG6OiyUz9Imq1OOhxjkWVEtJKtgTz9g2iD01e567S0PevAgFURDgZLmeKloR6mnHik7U5nw8UOFs7aKFFK5NeTIM4kowp_N2ixHG5YnGv0B1EgHs/s400/Picture1.png" width="400" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;"></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-19668023435620083592011-07-07T15:37:00.000-07:002011-07-07T15:37:02.029-07:00linear functions<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_upoV-iS_3dp16f6pSz58ddwilPt8mj02cHYB7GC-HEISOKClrN8fE6EJ9HS44Ne7GYDqG2JwdFzS1xj0ZQ29G_FHJ7Aci8IQow5lMkTwEFyc0IDvKf2ncDSjw4T94t1WuTFSfIV3ISk/s1600/1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_upoV-iS_3dp16f6pSz58ddwilPt8mj02cHYB7GC-HEISOKClrN8fE6EJ9HS44Ne7GYDqG2JwdFzS1xj0ZQ29G_FHJ7Aci8IQow5lMkTwEFyc0IDvKf2ncDSjw4T94t1WuTFSfIV3ISk/s400/1.png" width="400" /></a></div><br />
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find two reasons why the bottom right number must be 4 then complete the remainder... <br />
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students could go on to explore what happens when 3 (different) numbers are placed anywhere on the grid (but not in a line)...don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-55412386234196705382011-05-14T20:54:00.000-07:002014-09-29T12:10:47.806-07:00with negative numbers<div class="separator" style="clear: both; text-align: center;">
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don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-67527063947228523262011-05-14T00:35:00.000-07:002011-05-14T00:35:49.777-07:00geomagic squares - 3D<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeG8vs2Ub1TOunJ-6C8GZoGx7S386w5VTxiGp2k2xjb-hs3pP3M6q0LD6bOjp2lwIrm4uJcyj9l6-nEN2ZhAP4OvHrUgnIo0DN0DY0Kp4KrHpDp3ia9AO4kbEwKaFsIMvPTM2_rYzcP9A/s1600/6.png.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeG8vs2Ub1TOunJ-6C8GZoGx7S386w5VTxiGp2k2xjb-hs3pP3M6q0LD6bOjp2lwIrm4uJcyj9l6-nEN2ZhAP4OvHrUgnIo0DN0DY0Kp4KrHpDp3ia9AO4kbEwKaFsIMvPTM2_rYzcP9A/s400/6.png.jpeg" width="391" /></a></div>more of Lee Sallows's workdon stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-80498679128973332002011-05-11T09:26:00.000-07:002011-05-17T06:01:32.174-07:00geomagic squaresLee Sallows's ever so impressive work on making geometric magic squares was described in the <a href="http://www.guardian.co.uk/science/2011/apr/03/magic-squares-geomagic-lee-sallows">Guardian</a> recently and many examples can be found on Lee's <a href="http://www.geomagicsquares.com/">website</a><br />
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the shapes combine to make the same overall rectangle (or other shape)<br />
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one way to use these with students is to present them with the magic square shapes and ask them what the totals are (as shown below), possibly with some of the shapes missing<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCIKPWdkLRVbbHWFFV7fMuuZbXy3BE9UFIla5WTUFfHOg4JP3cLswSEk6hNpBcQxj663c65J_Nb7f4N7ztxVH_S47bOl5tVfHmnx72qUluM-_hI2QxHLoPwhoYXE_Ldp5hKpLikAJ3aAY/s1600/md-201101-square.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCIKPWdkLRVbbHWFFV7fMuuZbXy3BE9UFIla5WTUFfHOg4JP3cLswSEk6hNpBcQxj663c65J_Nb7f4N7ztxVH_S47bOl5tVfHmnx72qUluM-_hI2QxHLoPwhoYXE_Ldp5hKpLikAJ3aAY/s200/md-201101-square.jpg" width="200" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ0w32Cqna7K-DaPz2TpAbbrSQXVTXCStLrkrDh8UjRbEnMLu9YblnM7xvKF5uhv3vHiKJZBAfVObfWcBDK1dt6bFN_BjM6yZ5QCS5h5AwexnY6RWNIx5qhPYk7r-r4pONc-kr46nX5xc/s1600/zGeoMagic+2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ0w32Cqna7K-DaPz2TpAbbrSQXVTXCStLrkrDh8UjRbEnMLu9YblnM7xvKF5uhv3vHiKJZBAfVObfWcBDK1dt6bFN_BjM6yZ5QCS5h5AwexnY6RWNIx5qhPYk7r-r4pONc-kr46nX5xc/s200/zGeoMagic+2.jpg" width="196" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;"></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-40552584259700616182011-05-11T09:16:00.000-07:002011-05-14T00:39:57.437-07:00multiplication<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaVTGOL58L9p9s1J92z37KGgj8BmKpp4qTgKKG_htxEiPhZd_xLmUhOEE7uIHY1XKQGKdpD7YhvDOXZEn6AGv9XunrJ2BjM6Xz6nY0TYFj4g2KujhmaJuk4a3dOKppXP0IN4FBTfK2n7Y/s1600/1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="313" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaVTGOL58L9p9s1J92z37KGgj8BmKpp4qTgKKG_htxEiPhZd_xLmUhOEE7uIHY1XKQGKdpD7YhvDOXZEn6AGv9XunrJ2BjM6Xz6nY0TYFj4g2KujhmaJuk4a3dOKppXP0IN4FBTfK2n7Y/s320/1.png" width="320" /></a></div>multiply across the rows:<br />
2 x 7 x 6 etc<br />
then add these<br />
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now multiply down the columns:<br />
2 x 9 x 4 etc<br />
then add these<br />
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does this work for other magic squares?<br />
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using a calculator, square the rows as 3-digit numbers and add them:<br />
276^2 + 951^2 + 438^2<br />
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then do the same thing with the columns, reading downwards:<br />
294^2 + 753^2 + 618^2<br />
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does this work for other magic squares?don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-4866255762102250812011-05-11T09:04:00.000-07:002011-07-04T03:51:52.381-07:00decimal addition and subtraction practice<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQj15ZH-dbvZkT5pcP6U5eOuh4Ly9fkRz_PQ8IoEMpJSY8MxmutsG835vXF_PzpxdBfd3TjgLQ_CX4yAtcGfryph05Xy2r9Q-PEmTuxUFisaajqPJ-H4PxYArUrmKNF_BUrS6E5zXrpEs/s1600/1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQj15ZH-dbvZkT5pcP6U5eOuh4Ly9fkRz_PQ8IoEMpJSY8MxmutsG835vXF_PzpxdBfd3TjgLQ_CX4yAtcGfryph05Xy2r9Q-PEmTuxUFisaajqPJ-H4PxYArUrmKNF_BUrS6E5zXrpEs/s400/1.png" width="400" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;"></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-44336852892462420702011-05-11T07:14:00.000-07:002011-05-11T07:14:59.344-07:00fraction practice<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLYEAWdJwWAdR3Y_NUrSvsVGmI8Ry94OaW-gQoXzlyVlC36No5QBjiXwYtFQyj5k27hbrERnQ_0N8ob06mhgdY_RvOwNHthCE8Ke6CpGflVX-SJ9u6bKo_wSTI6pIq3OOtfbLEw_Cg2hk/s1600/2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLYEAWdJwWAdR3Y_NUrSvsVGmI8Ry94OaW-gQoXzlyVlC36No5QBjiXwYtFQyj5k27hbrERnQ_0N8ob06mhgdY_RvOwNHthCE8Ke6CpGflVX-SJ9u6bKo_wSTI6pIq3OOtfbLEw_Cg2hk/s400/2.png" width="400" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpn_O4OWGOPxBsnt8JEsqXprKc9ggnbhG9Hnjv4aEUnk5mMT0PIfs5Am0T2iAzguUPQ7FZXqmHHUp1Cj4qM3S33ZkEtUJpj551nAT4Q1txUq-k44QcKngFLoZSkCl73GiJzZgb7zpQfGg/s1600/5.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpn_O4OWGOPxBsnt8JEsqXprKc9ggnbhG9Hnjv4aEUnk5mMT0PIfs5Am0T2iAzguUPQ7FZXqmHHUp1Cj4qM3S33ZkEtUJpj551nAT4Q1txUq-k44QcKngFLoZSkCl73GiJzZgb7zpQfGg/s400/5.png" width="400" /></a></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-3645133481407513340.post-20658263591773355192011-05-11T05:26:00.000-07:002011-05-11T05:26:09.423-07:00rotations and reflections<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfjo6tuXVdoD-4802l9S4nrnh7yte6yssvDcyyWRtGMP3ZLVwfN8O6k3YtJP1f9VYeEcrI9igH7P849gWDtOqoIJnY8c7m4WWTiOLRH7sZ0gbIKf1WWcmTbEYeXRlEYKv6eXET9oJXZgA/s1600/1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="391" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfjo6tuXVdoD-4802l9S4nrnh7yte6yssvDcyyWRtGMP3ZLVwfN8O6k3YtJP1f9VYeEcrI9igH7P849gWDtOqoIJnY8c7m4WWTiOLRH7sZ0gbIKf1WWcmTbEYeXRlEYKv6eXET9oJXZgA/s400/1.png" width="400" /></a></div>find all the ways that the 1 to 9 magic square can be rearranged (and still be a magic square) - using rotations and reflectionsdon stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0