tag:blogger.com,1999:blog-4132131476925321902024-03-17T21:02:01.516-06:00Real teaching means real learningComing together to create a real learning environment for studentsDavehttp://www.blogger.com/profile/13790571271386459698noreply@blogger.comBlogger276125tag:blogger.com,1999:blog-413213147692532190.post-78607746810095536072019-10-31T07:08:00.003-06:002021-02-04T13:31:04.097-07:00Pumpkin Teeth<div dir="ltr" style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheM23bkmBJpOnEOm4GR8eD3crR6MuH71FuqWYsoNE3CUY8n8DmNLrgKvAGYWspQejZTeuBx2-p5Ck6EZiemDDWuyzY8991OJD5xDf99SaCVLhUG8HAALWYg1_gdipofVVWWp740cKlHN68/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="379" data-original-width="484" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheM23bkmBJpOnEOm4GR8eD3crR6MuH71FuqWYsoNE3CUY8n8DmNLrgKvAGYWspQejZTeuBx2-p5Ck6EZiemDDWuyzY8991OJD5xDf99SaCVLhUG8HAALWYg1_gdipofVVWWp740cKlHN68/" width="306" /></a></div><br /><br /></span></div><div dir="ltr" style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">Objective: To assist students with number sense.</span></div>
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<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">Materials: </span><a data-saferedirecturl="https://www.google.com/url?q=https://docs.google.com/document/d/192oFzyYKOeehXIbzUMn2uUAstbdXskzdkvu_XE65mK8/edit?usp%3Dsharing&source=gmail&ust=1572613909069000&usg=AFQjCNH9AlA9ituWmiMbnoCAM8AUujjokw" href="https://docs.google.com/document/d/192oFzyYKOeehXIbzUMn2uUAstbdXskzdkvu_XE65mK8/edit?usp=sharing" style="color: #1155cc; text-decoration-line: none;" target="_blank"><span style="background-color: transparent; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">Pumpkin Templates</span></a><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">, counters or small blocks, growth mindset</span><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><br /></span></div>
<div dir="ltr" style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">The story:</span></div>
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<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">Pumpkins have been cut with different amounts of teeth, sometimes equal on the top and bottom but usually a different amount of teeth on the top and bottom.</span></div>
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<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14pt; font-weight: 700; vertical-align: baseline; white-space: pre-wrap;">Grade 1 - 2: </span><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14pt; vertical-align: baseline; white-space: pre-wrap;">Print of pages 1 and 2 of Pumpkin Template</span></div>
<ol style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; margin-bottom: 0px; margin-top: 0px;">
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Provide students each with 4 blocks (or counters) and have them put them inside the large pumpkin mouth (page 1 of template). Ask students to stand up and walk around and do a gallery walk of different ways to arrange 5 teeth into the pumpkin.</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Draw 5 pumpkins on the board and ask students if they can determine all the ways to arrange 4 teeth. (5 different ways; 0 teeth on top, 1 on top, 2 on top, 3 on top or all 4 teeth on top) </span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Have students flip over their sheet (page 2 of template), and give students 1 more block (a total of 5 blocks each), and ask them to show all the different ways to arrange 5 blocks.</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Complete a gallery walk again.</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Ask if we, as a class, figured out all the ways to arrange 5 teeth?</span></div>
</li>
</ol>
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<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14pt; font-weight: 700; vertical-align: baseline; white-space: pre-wrap;">Grade 3 - 5:</span><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14pt; vertical-align: baseline; white-space: pre-wrap;"> Print off pages 2 and 3 of Pumpkin Template</span></div>
<div dir="ltr" style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Might want to use part of the previous ideas and then….</span><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"> Give students the second page template, and a handful of blocks...</span></div>
<ol style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; margin-bottom: 0px; margin-top: 0px;">
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Using page 2, Ask students how many different ways are there to arrange:</span></div>
</li>
<ol style="margin-bottom: 0px; margin-top: 0px;">
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">1 tooth? (2 ways) </span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">2 teeth? (3 ways)</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">5 teeth? (6 ways)</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">10 teeth?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">What if there were 50 teeth?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Is there a pattern?</span></div>
</li>
</ol>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Using page 3, provide students each with 5 blocks (or counters) and have them put them inside </span><span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; font-weight: 700; vertical-align: baseline;">either of the pumpkins</span><span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">, all in one pumpkin, or some in one and some in other.. Ask students to stand up and walk around and do a gallery walk of different ways to arrange 5 teeth into the </span><span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; font-weight: 700; vertical-align: baseline;">2 pumpkins</span><span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">.</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Using page 2, Ask students if they can determine how many different ways to arrange:</span></div>
</li>
<ol style="margin-bottom: 0px; margin-top: 0px;">
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">5 teeth in 2 pumpkins?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">6 teeth in 2 pumpkins?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">7 teeth in 2 pumpkins?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">What about 50 teeth in 2 pumpkins?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: lower-alpha; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Is there a pattern?</span></div>
</li>
</ol>
</ol>
<br style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small;" />
<div dir="ltr" style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14pt; font-weight: 700; vertical-align: baseline; white-space: pre-wrap;">Grades 6+:</span><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14pt; vertical-align: baseline; white-space: pre-wrap;"> Print off pages 2 and 3 of Pumpkin Template:</span></div>
<div dir="ltr" style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 11pt; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Might want to use part of the previous ideas and then…</span></div>
<ol style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: small; margin-bottom: 0px; margin-top: 0px;">
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">How many ways can you arrange 5 teeth in 3 pumpkins?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">How many ways can you arrange 7 teeth in 4 pumpkins?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">Is there a pattern?</span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; list-style-type: decimal; margin-left: 15px; vertical-align: baseline; white-space: pre-wrap;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">How many ways can you arrange</span><span style="background-color: transparent; font-size: 11pt; font-style: italic; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;"> n</span><span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;"> teeth in </span><span style="background-color: transparent; font-size: 11pt; font-style: italic; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">m </span><span style="background-color: transparent; font-size: 11pt; font-variant-east-asian: normal; font-variant-numeric: normal; vertical-align: baseline;">pumpkins?</span></div>
</li>
</ol>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com12tag:blogger.com,1999:blog-413213147692532190.post-12146924835915420782019-10-28T09:39:00.002-06:002019-10-28T09:44:19.785-06:00Halloween MathAs we approach Halloween here are some "Scary" problems your students might find interesting:<br />
<br />
<h2>
Elementary: </h2>
3 different monsters were out for halloween. Together they had 12 feet. If each monster has a different number of feet, how many feet does each monster have?<br />
<h3>
Extensions:</h3>
<ul>
<li>What could be the smallest total number of feet?</li>
<li>If each monster has 2 more eyes than feet, what would be the total number of eyes? Is there a total number of eyes that you cannot make?</li>
</ul>
<br />
<h2>
Middle School: </h2>
<a href="https://www.zorbitsmath.com/" target="_blank">Zorbit's</a> released some amazing pattern chats around Halloween. Here are some of them: (For more halloween activities check out <a href="https://www.zorbitsmath.com/halloween-math-talks/?utm_source=email&utm_medium=pardot&utm_campaign=halloweenmathtalks" target="_blank">Zorbit's activities here</a>)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7Yn1vFyZD_dwCh9CCGnRseRVYVE8Gbo87dVENQud_pegOisSBIdyoU8l-m_vxYu0zbT8fEzfNzfo0JOGwp7RxUIKqZVZfWwg025LDihShci_2vtPwKz7lrAHJJ4cuv5KDYGPPW6lkBp8a/s1600/Capture.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="546" data-original-width="964" height="362" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7Yn1vFyZD_dwCh9CCGnRseRVYVE8Gbo87dVENQud_pegOisSBIdyoU8l-m_vxYu0zbT8fEzfNzfo0JOGwp7RxUIKqZVZfWwg025LDihShci_2vtPwKz7lrAHJJ4cuv5KDYGPPW6lkBp8a/s640/Capture.PNG" width="640" /></a></div>
<br />
<h3>
Extension:</h3>
<ul>
<li>Could you rearrange the pictures above to make it easier to see the pattern? </li>
<li>Can you make a pattern that doesn't increase at a constant rate? How would you describe your pattern?</li>
</ul>
<br />
<h2>
High school:</h2>
<a href="http://realteachingmeansreallearning.blogspot.com/2014/04/zombies-meet-mathematics.html" target="_blank">World War Z and Mathematics</a>Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com2tag:blogger.com,1999:blog-413213147692532190.post-7010937384444592752019-06-12T10:05:00.003-06:002019-06-12T10:09:22.205-06:00Monster Soup!<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt; text-align: center;">
<span style="font-family: inherit; vertical-align: baseline; white-space: pre-wrap;">Monster Multiplication Game</span></div>
<span style="font-family: inherit;"><span id="docs-internal-guid-db96df48-7fff-f67b-de48-9e0c772defaf"><br /></span>
</span><br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: inherit; margin-left: 1em; margin-right: 1em; vertical-align: baseline;"><img height="205" src="https://lh6.googleusercontent.com/VvBKmmp_6DkXkhV-SRop6KyYDc8b45crNcGrEo1FPBklXn_OfRkPrA4Nx34EyUDKvV2USi9miLKI-mB4IcAcGuR8nc6ngwppzd_MIxVusMm9qXWWsvbUE9oNCxkysE6WpskjSqOd" style="border: none;" width="246" /></span></div>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="vertical-align: baseline; white-space: pre-wrap;"><span id="docs-internal-guid-228edd69-7fff-8510-601a-3f9d315b05f2" style="font-family: inherit;"></span></span></div>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-family: inherit; vertical-align: baseline; white-space: pre-wrap;"><br /></span></div>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-family: inherit; vertical-align: baseline; white-space: pre-wrap;">Scenario: We are cooking monsters today!!! Some monsters have 1 eyes, others have 5 eyes, and there are even other crazy monsters. We want to get as close to the number of eyes the recipe calls for as possible.</span></div>
<span style="font-family: inherit;"><br /></span>
<br />
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-family: inherit; vertical-align: baseline; white-space: pre-wrap;">Do example of “1 eyed monsters” with students. Have one student up to the front to demonstrate with you.</span></div>
<span style="font-family: inherit;"><br /></span>
<br />
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-family: inherit;"><span style="vertical-align: baseline; white-space: pre-wrap;">Activity: Roll two 10 sided dice. Determine which roll you want, and that is the number of monsters you will add to the recipe. Repeat 4 more times. (</span><span style="font-weight: 700; vertical-align: baseline; white-space: pre-wrap;">Total of 5 rolls of 2 dice, each time picking one of the 10-sided dice to represent the number of monsters you add to the pot</span><span style="vertical-align: baseline; white-space: pre-wrap;">)</span></span></div>
<span style="font-family: inherit;"><br /></span>
<br />
<span style="font-family: inherit;">Provide each student with a recording sheet and place them in </span><span style="font-family: inherit; vertical-align: baseline; white-space: pre-wrap;">the</span><span style="font-family: inherit; vertical-align: baseline; white-space: pre-wrap;"> appropriate station to practice the fluency skill they need:</span><br />
<span style="font-family: inherit;"><br /></span>
<br />
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-family: inherit; vertical-align: baseline; white-space: pre-wrap;">A possible progression could be: (Dr. Nicki Newton progression of multiplication)</span></div>
<span style="font-family: inherit;"><br /></span>
<br />
<ol style="margin-bottom: 0pt; margin-top: 0pt;">
<li><span style="font-family: inherit; white-space: pre-wrap;">1 eyed monsters - 30 eyes or 10 eyed monsters - 300 eyes or 5 eyed monsters - 150 eyes</span></li>
<li><span style="font-family: inherit;">2 eyed monsters - 60 eyes or 4 eyed monsters - 120 eyes or 8 eyed monsters - 240 eye</span></li>
<li><span style="font-family: inherit;">3 eyed monsters - 90 eyes or 6 eyed monsters - 180 eyes or 9 eyed monsters - 270 eyes</span></li>
<li><span style="font-family: inherit;">7 eyed monsters - 210 eyes</span></li>
</ol>
<div>
<span style="font-family: "comic sans ms";"><span style="font-family: inherit; white-space: pre-wrap;"><br /></span></span></div>
<div>
<span style="font-family: "comic sans ms";"><span style="font-family: inherit; white-space: pre-wrap;"><a href="https://docs.google.com/document/d/1_FI58inoEcgClj62S26okPYpd0b6u8awCL2vs_0qV1I/edit?usp=sharing" target="_blank">For all resources click here</a>.</span></span></div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com4tag:blogger.com,1999:blog-413213147692532190.post-78033941228179289472019-05-23T14:48:00.001-06:002019-05-23T15:07:45.300-06:00Wacky Quadrilaterals <div class="separator" style="clear: both; text-align: center;">
</div>
<div style="margin-left: 1em; margin-right: 1em;">
<span style="font-family: inherit;"><img alt="Image result for quadrilateral" class="irc_mi" data-iml="1558643979050" height="221" src="https://i.ytimg.com/vi/CkbClyVxRIU/maxresdefault.jpg" style="margin-top: 0px;" width="400" /></span></div>
<div style="margin-left: 1em; margin-right: 1em;">
<span style="font-family: inherit;"><br /></span></div>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">What happens when you make a quadrilateral (a four sided object) and connect the midpoints of each side? What if you did this over and over again?
</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Here is an activity that investigates that!</span><br />
<span style="font-family: inherit;"><br /></span>
<br />
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt; text-align: center;">
<span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: underline; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: inherit;">Wacky Quadrilaterals</span></span></div>
<b id="docs-internal-guid-43446dc8-7fff-48e9-635e-ffa10a28b10b" style="font-weight: normal;"><span style="font-family: inherit;"><br /></span></b>
<br />
<ol style="margin-bottom: 0pt; margin-top: 0pt;">
<li dir="ltr" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline; white-space: pre;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-family: inherit;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Draw any </span><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: underline; vertical-align: baseline; white-space: pre-wrap;">quadrilateral </span><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">that takes up most of the page. Measure the interior angles and add them all together and record this. Measure and record the perimeter</span></span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline; white-space: pre;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: inherit;">Measure and determine the midpoint of each line. Connect the midpoints and make a new quadrilateral. (Iteration 1) Measure the interior angles of the new quadrilateral and perimeter.</span></span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline; white-space: pre;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: inherit;">Repeat step 2, and fill in the chart with the sum of the new angles and perimeter.</span></span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline; white-space: pre;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: inherit;">Estimate what the sum of the angles and perimeter will be the for 10th shape. (Do not make the 10th shape, simply estimate based on the pattern).</span></span></div>
</li>
<li dir="ltr" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; list-style-type: decimal; text-decoration: none; vertical-align: baseline; white-space: pre;"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: inherit;">Color in your shape to make a creative design.</span></span></div>
</li>
</ol>
<br />
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: inherit;">What do you notice? Wonder? What would be the sum of the interior angles of iteration 10? Perimeter?</span></span></div>
<span style="font-family: inherit;"><br /></span>
<a href="https://docs.google.com/document/d/1N6zefKGgqWqvp0wy9uH_532K93vdJLkWMVvK0v1eHuY/edit?usp=sharing" target="_blank">Here is a recording sheet.</a><br />
<br />
A possible example:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLGfljqe8ChZEioqXCwdozkPB3TmEXjcnLYLg1Bp6pyl-qp3knkaXQc_xaA4p4xixcz-QWK6rSvb8B9vNwuXhHWSP7Up-9QRy_cKgDK4w0q7g7dnvwWgntrItDoVRbBh9EpHtKlJfvtfNd/s1600/20190523_144634.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1433" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLGfljqe8ChZEioqXCwdozkPB3TmEXjcnLYLg1Bp6pyl-qp3knkaXQc_xaA4p4xixcz-QWK6rSvb8B9vNwuXhHWSP7Up-9QRy_cKgDK4w0q7g7dnvwWgntrItDoVRbBh9EpHtKlJfvtfNd/s400/20190523_144634.jpg" width="358" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br />
<span style="font-family: inherit;">Extension:</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Simply show this animation and ask "What is this visually proving?"</span><br />
<span style="font-family: inherit;"><img alt="Image result for quadrilateral" src="https://www.mathwarehouse.com/animated-gifs/images/exterior-angles-polygon-animated-demonstration.gif" /></span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;"><br /></span>
A possible video to show is here:<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/V1gT2f3Fe44/0.jpg" src="https://www.youtube.com/embed/V1gT2f3Fe44?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
<br />
<br />Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com2tag:blogger.com,1999:blog-413213147692532190.post-50825616527513962532019-03-29T13:59:00.000-06:002019-03-29T18:58:23.537-06:00How often do I do math?I have seen English teachers sit around and discuss the books they are currently reading, or Social Studies teachers debate current issues and the impact they may have on society. I have seen CTS teachers talk to their students about the home projects they are working on; whether it be a new woodworking project, an automotive problem they are trying to solve, or even how one is trying to code an arduino board to allow for more functionality within their home. As I visit and meet more teachers, I am constantly hearing about teachers being 'students' of their own subject area outside the walls of their classroom.<br />
<br />
This then caused me to reflect, which I will ask you to do as well, on the question "How often do I sit down and work on mathematical problems outside my own classroom?"<br />
<br />
When I first asked this question, I sadly had to respond with "rarely or never". At the time, I would ask my students to try multiple questions daily, learn new ideas, consolidate older information and ultimately be problem solvers around questions they have never seen before; sadly I modeled none of this outside the walls of my classroom.<br />
<br />
Perseverance, resiliency, creativity, and critical thinking is what I expected of my students on a daily basis around mathematics, but until I embraced these practices in my own life I didn't actually know what if felt like to be stuck in a problem without knowing what to do.<br />
<br />
"What do you do when you don't know what to do in a math problem?" I asked this question to 800 Grade 4 - 12 students and the number one answer (by over 80% of the respondents) was "ask the teacher". This was startling!! I couldn't arm my students with authentic problem solving strategies until I actually put myself in their shoes. Once I tried working on problems that caused me to stop and ask "what should I do now?", I was able to understand that global problem solving strategies was missing in my own math classes.<br />
<br />
Originally, I would teach "When working on a problem from unit X, try these strategies. On unit Y, try these.." and so on. The issue is that I wasn't teaching true problem solving but instead strategies specific to certain domains. After trying math on my own time, and at my own level, I quickly learned that some of the best strategies include, but not limited to, are:<br />
<ul>
<li>Visualize the problem; draw it out.</li>
<li>Guess and check; change guess slightly and see how it changes the result.</li>
<li>Approach it logically; Use "if then" statements to simplify information.</li>
<li>Identify a pattern; change a number, a sign, or something critical and see how it changes the problem.</li>
<li>Work backwards; if we can hypothesize the result, what else would have to be true?</li>
<li>Solve an easier problem; simplify the problem into one that is easier to work with and see if you can identify anything new.</li>
</ul>
<div>
My challenge for myself now, and I am extending it you as well, is to try a math problem once a week. Ensure the problem isn't one that you can solve in seconds, or even minutes. Try and find one that makes you reflect on "What do you do when you don't know what to do in a math problem?"</div>
<div>
<br /></div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com6tag:blogger.com,1999:blog-413213147692532190.post-87422852407684768442018-12-14T07:56:00.001-07:002018-12-14T07:59:13.920-07:00Triangle in a circle Show this first animation and ask, what do you notice? What do you wonder?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXOprQ8xXNTMW4CQZ-KiuxRpjUiEIvcuscmuZQdm3WrjSHOSAsqp_cttyWdyu46EVMtVhPV0rSBI17NvKvgJQRUMqyvRWIiJ46kPeVQVIILreDk8H7Q4FbG4vGNffLTxf-pBthiMLUNT3f/s1600/TriangleInCircleNoNumb.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="418" data-original-width="804" height="331" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXOprQ8xXNTMW4CQZ-KiuxRpjUiEIvcuscmuZQdm3WrjSHOSAsqp_cttyWdyu46EVMtVhPV0rSBI17NvKvgJQRUMqyvRWIiJ46kPeVQVIILreDk8H7Q4FbG4vGNffLTxf-pBthiMLUNT3f/s640/TriangleInCircleNoNumb.gif" width="640" /></a></div>
<br />
Then ask the following: If the diameter of the circle is 8 cm. What (or when) is the largest area? What are the dimensions of this triangle? <br />
<br />
After students solve that pose, When is the area exactly half the largest? What are the dimensions of that triangle? What is the angle?<br />
<br />
<br />
After some work time and playing around with the above problem, show them this animated version<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyKPfKhAe12TbA8tQjk83CwtH5roMFpYCYj_0PxUaJjM4tVFT-Q5cAnmm98fOuBQ1oqaMbTs5IGvia7qOfCwZ-DMRvcQJkB-gORkMyDdXnvDRSA3xjuy6cQgY5T9QuMH7aQ7xVBEmQgV2e/s1600/TriangleInCircle.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="418" data-original-width="804" height="332" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyKPfKhAe12TbA8tQjk83CwtH5roMFpYCYj_0PxUaJjM4tVFT-Q5cAnmm98fOuBQ1oqaMbTs5IGvia7qOfCwZ-DMRvcQJkB-gORkMyDdXnvDRSA3xjuy6cQgY5T9QuMH7aQ7xVBEmQgV2e/s640/TriangleInCircle.gif" width="640" /></a></div>
<br />Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com14tag:blogger.com,1999:blog-413213147692532190.post-10798818782656797862018-12-05T22:04:00.001-07:002018-12-05T22:12:22.306-07:00Trig Ratios animated<div class="separator" style="clear: both; text-align: left;">
Two different triangles are animated. Show your class this animation and simply ask:</div>
<div class="separator" style="clear: both; text-align: left;">
</div>
<ul>
<li>What do you notice?</li>
<li>What do you wonder?</li>
</ul>
<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpdK6MZdpbLaeSnU1VWmcg5-126-LAsyGseDDc5hHg5bMpzJfMfVy4O1QcZZFB-7LmvuitdQ7GLLjxkfEU3Whh-bPSVOe-wY65CmK0yix37-eDOuKMuHL7KmEE3AO4U-VaJSx7y1KYAjAM/s1600/TrigRatioNoNumb.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="343" data-original-width="629" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpdK6MZdpbLaeSnU1VWmcg5-126-LAsyGseDDc5hHg5bMpzJfMfVy4O1QcZZFB-7LmvuitdQ7GLLjxkfEU3Whh-bPSVOe-wY65CmK0yix37-eDOuKMuHL7KmEE3AO4U-VaJSx7y1KYAjAM/s1600/TrigRatioNoNumb.gif" /></a></div>
<br />
<br />
After some conversations show them this one, with the side lengths being shown and ask again:<br />
<br />
<ul>
<li>What do you notice?</li>
<li>What do you wonder?</li>
</ul>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH42nCWqLZG2u0uYab8h4FR8IYl3vpiLwFCBupH41yFUoLQ10Adn05S9rRauc4iA-USgqCGhP2V_VphwAaQCin9Bnt27i2PZgqqyAxhFybD2dSkIqh-o29VuxLvx_liBfXXblnpO3KnQix/s1600/TrigRatioNumb.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="326" data-original-width="666" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH42nCWqLZG2u0uYab8h4FR8IYl3vpiLwFCBupH41yFUoLQ10Adn05S9rRauc4iA-USgqCGhP2V_VphwAaQCin9Bnt27i2PZgqqyAxhFybD2dSkIqh-o29VuxLvx_liBfXXblnpO3KnQix/s1600/TrigRatioNumb.gif" /></a></div>
<br />
<br />
Lastly, show them this image and ask them "How could we analyze this?"<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0lbHDeouDRf5a_sAaQuqqqzUE4J94AC61s-4jlV4I1OJseAQkjSwcZE6NqXTeZPSz9R0cGCXrL0UMfECWZQT7Yaaqi_hjzkJxdVeBdb7_2tr64v0N_VMotrOSXSbL2kqd7Aq15DAr_vzS/s1600/Capture.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="394" data-original-width="730" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0lbHDeouDRf5a_sAaQuqqqzUE4J94AC61s-4jlV4I1OJseAQkjSwcZE6NqXTeZPSz9R0cGCXrL0UMfECWZQT7Yaaqi_hjzkJxdVeBdb7_2tr64v0N_VMotrOSXSbL2kqd7Aq15DAr_vzS/s1600/Capture.PNG" /></a></div>
<br />
Some possible routes:<br />
<br />
<ul>
<li>Prove the sides hold true for Pythagorean theorem</li>
<li>What angle(s) would be in the triangle?</li>
<li>How are these 2 triangles similar? How are they different?</li>
</ul>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-45267784094832920612018-11-28T05:58:00.000-07:002018-11-28T05:58:01.808-07:00Area and perimeter of a triangleAnother way to possibly introduce, or reinforce, the area of a triangle and perimeter could be to show this gif:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXYTsVnptqTiYwueiQgPP3HZxiUqLzUaqL-8b5WwkhqneMGxNONC5b5g_9931K0AtGaHykEG9hKQG-cZdGmq1byOw4xm98ElxHpwtRTsZ8hPVSd7uPlLmlbdfad9BiTF2VJLgkAqhZi6dv/s1600/AreaNoNum.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="424" data-original-width="990" height="273" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXYTsVnptqTiYwueiQgPP3HZxiUqLzUaqL-8b5WwkhqneMGxNONC5b5g_9931K0AtGaHykEG9hKQG-cZdGmq1byOw4xm98ElxHpwtRTsZ8hPVSd7uPlLmlbdfad9BiTF2VJLgkAqhZi6dv/s640/AreaNoNum.gif" width="640" /></a></div>
<br />
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Ask, "What do you notice? What do you wonder?". After some discussion pose, if not already asked, the following questions:<br />
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<ul>
<li>When is the area the least/greatest?</li>
<li>When is the perimeter the least/greatest?</li>
</ul>
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<div>
After some discussion show this gif.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglLeq1829rf7Nk8Gz9JuQZvcQCC44IqT1a1AmTy7psmZtH9X8u7gbi6nh8nkvQTAE8dcCYL28BOBNc0d4rggRheefW1I819eipDCNbUzsKw2IU18BcZ57M4VhcFxw-xu9KBiSrLzzIBh79/s1600/areanonum2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="431" data-original-width="993" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglLeq1829rf7Nk8Gz9JuQZvcQCC44IqT1a1AmTy7psmZtH9X8u7gbi6nh8nkvQTAE8dcCYL28BOBNc0d4rggRheefW1I819eipDCNbUzsKw2IU18BcZ57M4VhcFxw-xu9KBiSrLzzIBh79/s640/areanonum2.gif" width="640" /></a></div>
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Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com1tag:blogger.com,1999:blog-413213147692532190.post-77933898599008008072018-09-04T21:12:00.000-06:002018-09-04T21:12:23.504-06:00Zombies and number talks....<img alt="Image result for the walking dead our world" height="112" src="https://images.amcnetworks.com/amc.com/wp-content/uploads/2017/08/TWD_NextGames_OurWorld_TRAILER_REV.jpg" width="200" /><div>
<br /></div>
<div>
Recently, during a walk I was playing The Walking Dead: The Zombies, and an awesome mathematical experience occurred. <div>
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<div>
I noticed that my team had slaughtered 497 zombies out of the 800 required to complete a challenge. Sadly, I commented to my friend, after killing 3 zombies, "<i>Well at least we have 500....out of 800!</i>", to which she asked "<i>What percentage is that</i>?"</div>
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I smiled, and replied "<i>How would we do that?"</i></div>
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<i><br /></i></div>
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As she grabbed for her phone, I asked her to humor me and try it mentally....and as such began a wonderful minute of beautiful number play:</div>
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"<i>Well I would make it 5 divided by 8.....then...." </i> </div>
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At this moment she begins to draw out 5 under a divisor sign and an imaginary 8 to the left of it, and continues with...</div>
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<div>
<i>"Ok 8 goes into 5....oh...this doesn't work....Why can't I just use my phone?"</i></div>
<div>
<i><br /></i></div>
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After smiling, I asked her, "<i>And how else might you say '5 divided by 8'?"</i></div>
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Revisiting middle school in her head, she exclaimed, "<i>5 eighths. but that does not help"</i></div>
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<i><br /></i></div>
<div>
<i>"Yeah eights are a pain, is there an easier grouping we could from eighths?"</i></div>
<div>
<i><br /></i></div>
<div>
Seconds pass, and she admits she does not understand my question. I offer assistance with,</div>
<div>
<br /></div>
<div>
"<i>Ok, eights is the denomination we are working with, but are tricky, so think about this....If I gave you unlimited eighths....what other fractions could you make?"</i></div>
<div>
<i><br /></i></div>
<div>
Eyebrow touch for a moment, and then she replies with "<i>I could make quarters with 2 of them...."</i></div>
<div>
<i><br /></i></div>
<div>
At this point, I could not help from smiling as I knew to continue the journey of numerical delight all I had to do was ask, "<i>So how many quarters is 5 eighths?" </i>and so I did.</div>
<div>
<br /></div>
<div>
Tilting her head to the side, I could almost see the imaginary manipulatives going through her mind as she talked aloud, "<i>Well 2 eighths is 1 quarter, and so 4 eighths is 2 quarters, and then 6...oh....2 and half eighths?"</i></div>
<div>
<i><br /></i></div>
<div>
I decided to ignore my Grade 6 teacher's rule of "NO DECIMALS IN FRACTIONS" (Which has never made sense!), and simply say "<i>Hmmm...Interesting so what percentage would be 2 and half quarters?"</i></div>
<div>
<i><br /></i></div>
<div>
She had already started.."<i>2 quarters is 50...half a quarter is... 12....no 12.5.....and so 50 plus 12.5 is 60....62.5 percent?"</i></div>
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<i><br /></i></div>
<div>
I cheered, awarded her with a high 5, and asked "<i>If it took us 2 days to do 5 eights and we have only 1 day left, are we going to make it?", </i>to which he exclaimed <i>"I don't love zombies enough..."</i></div>
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As I reflect on this experience, this is what I want my math classes to feel like for my own daughters. Moments of exploration, followed with choices from different mathematical tools, and ended with delight as they solve problems that make sense. </div>
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<br /></div>
<div>
PS: This would have lost all of its joy if I simply entertained the idea that dividing 5 by 8 mentally, is best done with the traditional algorithm</div>
</div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com2tag:blogger.com,1999:blog-413213147692532190.post-76154403965743132882018-08-27T05:00:00.000-06:002018-08-27T10:25:33.598-06:00Happy and Sad Numbers<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
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<img alt="Image result for happy sad numbers" src="https://tapoueh.org/img/old/happy-numbers.png" /><br />
A happy number is defined by the following process: <br />
<br />
Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).<a href="https://en.wikipedia.org/wiki/Happy_number#cite_note-1" style="font-family: arial; font-size: 8.4pt; text-decoration-line: none; white-space: pre-wrap;">[1]</a></div>
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For example: 19 is happy because:</div>
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<span style="font-family: "arial"; font-size: 11pt; margin-left: 1em; margin-right: 1em; vertical-align: baseline; white-space: pre-wrap;"><img alt="4 lines . Line 1, 19 long right-arrow 1 squared plus 9 squared equals 82 . Line 2, 82 long right-arrow 8 squared plus 2 squared equals 68 . Line 3, 68 long right-arrow 6 squared plus 8 squared equals 100 . Line 4, 100 long right-arrow 1 squared plus 0 squared plus 0 squared equals 1" height="129" src="https://lh6.googleusercontent.com/FAM968CXkk8RLvcYNY-CJCmd9LxWz0WxqfZzZKYB_9sXMCZ1t61mMqopttSpOHqkh8IXRZPN4b0moPgvCqOWUAjIxIRUKpTtLbYPS0jBB8f8-GWg95xTW72jI4zLRUXdpxm_EsvU" style="border: none; transform: rotate(0rad);" width="303" /></span></div>
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4 is sad because, following the process above, the pattern is: <br />
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<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span></div>
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<span id="docs-internal-guid-d49f1acc-7fff-f8f3-ce3e-9b2bfc7371fd"><span style="background-color: white; color: #0b0080; font-family: "arial"; font-size: 8.4pt; vertical-align: super; white-space: pre-wrap;"><a href="https://en.wikipedia.org/wiki/Happy_number#cite_note-1" style="text-decoration-line: none;"><img alt="4 long right-arrow 16 long right-arrow 37 long right-arrow 58 long right-arrow 89 long right-arrow 145 long right-arrow 42 long right-arrow 20 long right-arrow 4 long right-arrow period period period" height="17" src="https://lh3.googleusercontent.com/NTKM9GwKmgTvbQxM9T4JPhufi5VVqxv2x-6aC-2wQq1ukTIRfFFXF4VHeQzocGSEpYmM6DZm-RHwfnHrY-4zpDZuaz70A6RD07Utl-tCV9iCubLDbbo7FN3xmcG7Pyt4R5sJ8b2y" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="624" /></a></span></span></div>
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After introducing this to the class some questions you may ask could be:<br />
<div>
<ul>
<li>What other numbers are happy? Sad? </li>
<li>Are there any certain numbers that are easy to determine if they are happy/sad? How do you know? </li>
<li>Would there be an infinite or finite number of happy numbers? </li>
<li>True or false: If a number is happy (sad), then all numbers of its sequence is happy (sad). </li>
</ul>
<br />
Further Extensions:<br />
<br />
<ul>
<li>A happy prime is a number that is both happy and prime. Determine any happy prime numbers. What would be common among all happy primes?</li>
<li>Change the base (from 10 to 2, or any number) are the numbers still happy/sad, or does it change how the "feeling" of the number? </li>
<li>Try cubing the digits instead of squaring. Are the numbers still happy/sad, or does the number change "feelings"? </li>
<li>Computer science challenge: Could you write a code or algorithm that determines if the number is happy or sad?</li>
</ul>
</div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-70245211444025330522018-06-18T09:33:00.000-06:002018-06-18T09:33:55.340-06:00Dice ChatsAnother way to bring conversations alive in your classroom is with "Dice Chats"!! These are similar to Dan Finkel's <a href="https://mathforlove.com/lesson/unit-chats/" target="_blank">Unit Chats</a>.<br />
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<br /></div>
<div>
Simply show these images and ask your students "How many do you see?"</div>
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The students could:</div>
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<ul>
<li>Count the total dots</li>
<li>Number of dice</li>
<li>Number of colors</li>
<li>Number of dots on a certain dice</li>
<li>Number of dots on a certain part of the pattern</li>
<li>BEYOND!</li>
</ul>
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</div>
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I would provide students white boards (or have them at non permanent table tops) and have them write down how they know.</div>
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After 2 or 3 minutes, simply start the conversations and reply to every answer "How do you know?".</div>
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Remember, during this time try to ask more questions than give answers.</div>
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Here is a slideshow showing different possible pictures.</div>
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<iframe allowfullscreen="true" frameborder="0" height="299" mozallowfullscreen="true" src="https://docs.google.com/presentation/d/e/2PACX-1vS64XJVmUCXs4WAYhQRQvdRWJdtrlyJWJlXl7Bh6qlx7p8DsUybbkMcxvf9rCUPMB3ZZxi5cNPl4c_4/embed?start=false&loop=false&delayms=3000" webkitallowfullscreen="true" width="480"></iframe>Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com14tag:blogger.com,1999:blog-413213147692532190.post-36438770398415031382018-05-16T06:00:00.000-06:002018-05-17T14:17:31.769-06:00Differentiating Assessment<blockquote class="tr_bq">
<i>“The primary purpose of assessment is to improve student learning”</i> -Anne Davies, PHD.</blockquote>
The above quote sums up why I challenged my assessment practices. In 2013, as a High School Math teacher, here is the model I used:<div>
<ul>
<li>Teach concept </li>
<li>Various quizzes during the learning journey.</li>
<li>Summative unit exam.</li>
<li>Repeat </li>
</ul>
After 8 years of teaching in this model, I realized there was an issue...students who entered my class with passion towards mathematics, were leaving the class beaten down, sometimes dropping out, and ultimately not having the desire to learn more math. <br /><br />Originally, I thought this was normal!! When I went to high school, math classes always ended with less students than it started with. As a student, I remember daily expectations of having to do the odd (or even) numbered questions on page X, multiple worksheets, and having to prepare for weekly quizzes or tests. This was my normality. This was the machine I wanted to perpetuate when I entered teaching.<br /><br />Why?<br /><br />Because this worked for me. I am intrinsically motivated by mathematics, and I find prime, fibonacci, and complex numbers inherently interesting...because they are!! However, too many people have not had the chance to struggle, discover, and play with these awesome, and other, mathematical ideas.<br /><br />In 2013, as an educator, I saw the true problem...my assessment style was more about ranking, sorting and grading, not at all about learning. Even furthermore, I was more focused on preparing students for AP, or diploma exams, instead of creating an environment which allowed students to bring their passions and interests in, next to their pencils and paper. <br /><br /><img src="https://lh6.googleusercontent.com/5JtQHphB3wZ04gZWCqTZtVAXhD0rGBfp4dLFQ9-q04fFOPxijbqPX2EfBsMIMgY5vVXr4plY74JsB49DM1h19xsl4WUyvRaa8Dc0AwVbrFiPeAigpKH4VUrHe4E7OEgffzUVOtc2" /><br /><br />Similar to the comic, my grades were focused on what was easy to test, grade, and report on, instead of what was important.<br /><br />This had to change. If I was differentiating my instruction, why was I focusing on standardizing my assessment? <br /><br />In 2013, I made a stand: <br /><blockquote class="tr_bq">
I will only assess in a way that increases learning; if my assessment isn’t increasing learning then the assessment needs to change. </blockquote>
In this year, my late friend Joe Bower reminded me “the word ‘assessment’ comes from the latin word ‘Assidere’: to sit beside”; an action that was rarely taken when I was assessing my students.<br /><br />Here is my journey, and the steps I took, to explore what it mean to provide differentiated assessment.<br /><br /><b><u>1. Manageable outcomes:</u></b></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcoCfl5m5XRvZo0LUM_avm13kcsGV3Q7SJrD_gkMoDlLVm34AF4ZC4bKRowsXOG6PBcMhBzCrWH8P5OpyeYrikq7vzbfSXkh_b8gy54X96OKD3Jss_nMyI7ogNMQxef4OvTjCJjWXm1ZxQ/s1600/BigRocks-sm-greenertrees-net.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" data-original-height="789" data-original-width="1024" height="153" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcoCfl5m5XRvZo0LUM_avm13kcsGV3Q7SJrD_gkMoDlLVm34AF4ZC4bKRowsXOG6PBcMhBzCrWH8P5OpyeYrikq7vzbfSXkh_b8gy54X96OKD3Jss_nMyI7ogNMQxef4OvTjCJjWXm1ZxQ/s200/BigRocks-sm-greenertrees-net.jpg" width="200" /></a></div>
<div>
In consultation with University professors, colleagues, and teachers across the province, I looked at every course outcome through the “Rock, Sand, Water” analogy: If you plan for the rocks first, then sand and then water, it will all fit, however if you simply plan a course to cover all outcomes equally, all the outcomes will rarely fit.</div>
<div>
<br />During this process I combined parts of one outcome with another, broke up some outcomes into smaller chunks, and then I created a list of “Rock, Sand and Water” objectives:<br /><br /><div class="separator" style="clear: both; text-align: center;">
<a href="https://lh5.googleusercontent.com/8i5MO70p6EcSxPQPQNUl19SDTy1VzqwHbXsTLQvEu54xq-GuuTkK0N77sykBOWH_qmHQA2wGE5CujG7u0-zt7AOTtrIdYNhO5VTJe8rww3Qa4GAvnhLqhU0i6wI_EgL454MoRpOK" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="240" src="https://lh5.googleusercontent.com/8i5MO70p6EcSxPQPQNUl19SDTy1VzqwHbXsTLQvEu54xq-GuuTkK0N77sykBOWH_qmHQA2wGE5CujG7u0-zt7AOTtrIdYNhO5VTJe8rww3Qa4GAvnhLqhU0i6wI_EgL454MoRpOK" width="320" /></a></div>
<ul>
<li>Rock outcomes (outcomes that pass the endurance, leverage and readiness test) - Expect ALL students to master. </li>
<li>Sand outcomes - Expect MOST students to master. </li>
<li>Water outcomes - Expect SOME of my students to master. </li>
</ul>
I then ensured that these decisions reflected in my long range outcomes, course outlines, and daily plans. I planned my courses in a way to ensure that the essential learning outcomes were weaved throughout the entire year, while less essential outcomes were covered through the lens of a higher leverage outcome. Of course, I still taught all the outcomes, but I decided to only report on the essential ones, regardless of how difficult it might be to do so.<br /><br /><b><u>2. Change the tests</u></b><br />After having a smaller list of outcomes to report on, I decided to ensure my summative assessments matched this philosophy. Instead of giving tests grouped by question type, I grouped questions based on outcome. Any assessment that covered more than 1 outcome would be given back to students with more than 1 grade. Each grade represented the learning of the student on a specific outcome; no longer did I average 2 or 3 outcomes into one mark and call it “Unit X Test”. I then changed the categories on the online reporting program to “outcomes” instead of “Quizzes, tests, homework, etc”. Every mark, on a specific outcome, was reported in the corresponding “outcome” category.<br /><br /><b><u>3. Ensure learning is the focus on every assessment</u></b><br />During this time, my summative assessments where 1 part multiple choice, 1 part numerical response and 1 part written. Simply sorted by outcomes not by question type.<br /><br />I quickly realized that when my students answered a multiple choice question wrong (or even when they guessed right) I was clueless as to how to support them from their current understanding to mastery. If I wanted learning to be the primary focus, I could not administer multiple choice exams. <br /><br />In 2014-2015 I moved to an entire written response assessment strategy grouped by outcomes. Instead of 1 part multiple choice, 1 part numerical response, and 1 part written, I only assessed with questions that forced students to make their understanding visible. It was in this year, I truly started to sit next to my students and provide them written and verbal <a href="http://www.ascd.org/publications/educational-leadership/sept12/vol70/num01/Seven-Keys-to-Effective-Feedback.aspx">feedback </a>that pushed their learning forward instead of simply saying “here are the X questions you answered incorrectly and here are the correct answers”. My feedback was focused on learning, not on which questions did they answer wrong. <br /><br />I was writing grades and comments on everything my students handed in. This was the inherent problem; I was giving both grades and comments. <br /><br />Every time I handed back an assessment with a <a href="https://www.alfiekohn.org/article/case-grades/">mark</a>, I quickly noticed that students focused on their individual grade, their friend’s grade; and how they ranked within their peer group; and most completely ignored the comments. Students were not asking “How do I understand this better?” but instead “How do I get an A (or 90% or excellent)?” <br /><br />I had invested a lot of my time into giving useful and effective feedback however these comments were being overshadowed by mark. Grades were the commodity of my classroom not learning. This had to change.<br /><br /><b><u>4. Challenge the grading system</u></b></div>
<div>
THIS IS THE GAME CHANGER!! <br /><br />Simply put, I stopped writing grades, learning levels, or any other ranking system on student work. Instead I only provided feedback and asked questions that pushed learning forward. Even if a student demonstrated “mastery” of an outcome, I would still provide feedback or leave them with a question that pushed them beyond the scope of the outcome.<br /><br />This was the most profound transformation I have ever experienced in my entire career. <br /><br />Students truly became engaged in their learning, not their grade (or ranking). As well, I was able to truly push my students forward when they made mistakes. When I looked at the work of my students, I simply focused on 3 essential questions:<br /><br /><div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRpG-HX9ZKt-zPTTUu5aH-AN_lptXkPhmHZwKviAuUxs8OeAZRyiH7L9wxefT2X7djp5UdQtkj_xADwA_hbruttcQxbX0yy62k5lRJqoKPKw6TrnzCO_phLUpScIgl52PLO5evro2d-1Zg/s1600/Capture.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="820" data-original-width="1467" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRpG-HX9ZKt-zPTTUu5aH-AN_lptXkPhmHZwKviAuUxs8OeAZRyiH7L9wxefT2X7djp5UdQtkj_xADwA_hbruttcQxbX0yy62k5lRJqoKPKw6TrnzCO_phLUpScIgl52PLO5evro2d-1Zg/s320/Capture.PNG" width="320" /></a></div>
<br /><ul>
<li>What does success on the essential learning outcome (rock outcome) look like? </li>
<li>Where is the student now? </li>
<li>How do we close gap? </li>
</ul>
<div>
<br /></div>
This ensured that the feedback I was providing to students was truly “learning focused”. Every written comment was also input into our online reporting system.<br /><br />This meant that when a parent, or a student, logged into our online reporting tool, they didn’t see grades, but instead comments for any outcome. Instead of seeing “80%” on an outcome, parents (and students) would see what they need to work on to close the gap...without the support of a grade. <br /><br />Even the conversations I was having with parents were learning focused and not grade focused. Incredible shift! At the beginning of the year, parents were apprehensive of not receiving any marks as feedback, however when parents saw the products their children were bringing home from my courses, parents were quick to become allies of this new model of assessment.<br /><br /><b><u>5. Differentiate the assessments</u></b></div>
<div>
I finally started a “Differentiated Assessment” model. After teaching outcome X, I would have an assessment on the outcome, however, I started to use this time to also assess each students’ understanding on a previous outcome. For example, students who were also being retaught ideas on outcome 2 would have questions around outcome 2 on their sheet, the ones who were working on outcome 3 would see questions around this outcome...truly every single assessment was tiered to the individual student and what he/she has been working on in the previous weeks.<br /><br />The assessment Jimmy and Jane received on this day, would only match if they were working on identical material with identical errors and misconceptions; which was rarely the case. Even when designing the questions from previous outcomes, the questions were focused around the feedback the learner had received on their last assessment. For example:<br /><ul>
<li>Jane might be tasked to demonstrate understanding of a specific part of a certain essential outcome, because I saw only a minor learning gap when I previously assessed her on this outcome. </li>
<li>Jimmy, however, might have more questions around the same essential outcome, because when I assessed him previously I saw major learning gaps. </li>
</ul>
This is when learning became the focus of every single assessment I gave. I can honestly say that every assessment had learning as the only priority!<br /><br />Looking back, I have always believed that every child can learn math to the highest levels, but only in the past 3 years did I take a differentiated approach to what happens when they don’t.</div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com31tag:blogger.com,1999:blog-413213147692532190.post-56648529314277433892018-01-29T16:39:00.002-07:002018-01-29T16:39:25.465-07:00Folding Paper Activity<span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: medium;"><b>How high will a piece of paper get if you fold it 3 times in half? 7? 10? 20? How long until you get to space??</b></span><br />
<div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">
<span style="font-size: medium;"><br /></span></div>
<div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">
<span style="font-size: medium;">Check it out digitally with "<a data-saferedirecturl="https://www.google.com/url?hl=en&q=http://neal.fun/paper/&source=gmail&ust=1517319983783000&usg=AFQjCNEbnv8_2b6zSbe_GD4G_tDb2Jgkjw" href="http://neal.fun/paper/" style="color: #1155cc;" target="_blank">Folding Paper digitally</a>" Check out possible uses below</span></div>
<div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">
<span style="color: blue;"><br /></span></div>
<div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">
<span style="color: blue;"><b>Early grades:</b> Ask the question I did, wonder, try it, wonder more, and then to the site...wonder before you click each time</span>.</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;">
<span style="color: red;"><b>Middle grades:</b> Ask the question I did, wonder, try it, wonder more, see if you can create an actual equation for the height vs times folded..and then to the site...wonder before you click each time. Ask then, how big would the original sheet have to be to be able to stand on the stack after X folds?</span></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;">
<span style="color: #9900ff;"><b>High Grades</b>: Ask the question I did, wonder, try, wonder more, ask why type of relation would be best to model this type of growth? How would you change the question so it grew as linear, quadratic (would be fun to explore this), sinusoidal, etc, functions? and then to the site...wonder before you click each time...Ask then, how big would the original sheet have to be to be able to stand on the stack after X folds?</span></div>
<div>
<span style="color: #9900ff;"><br /></span></div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-5827943827018919932017-11-26T11:52:00.001-07:002017-11-26T11:52:16.007-07:00The flaws of (some) textbooks<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Recently, I was asked "Why do you dislike Textbooks?" and upon reflection, here are the issues I see:</span></div>
<b id="docs-internal-guid-17ff0f1a-f9aa-9813-af87-eb9244992629" style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Problem 1: </span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Textbooks assume you need to be taught and shown how to solve a problem before you are given the problem.</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">This is absurd to me. When I, and probably most, encounter something I do not know how to do, the first thing I rarely do is look for an instructional video on how to complete the task. </span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Ironically, the first thing I do is actually play with the problem and see how far I can get without any assistance. That is right; I play! This play cannot happen if my hand is being held and shown how to complete the task. Learners, specifically children, are not afraid to be wrong, take chances, and try to truly problem solve, yet a textbook is designed around the idea that a child loves to be told what to do.</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">This ruins the fun!!! I say again, this ruins the fun! It is comparable to turning on a movie and someone telling you "The main character dies at the end!"; Joy lost!</span></div>
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<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Problem 2: </span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> Pseudo - context</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In almost every textbook I have seen there is always some sort of situation that can only exist in "textbook land", a magical place where the following is true:</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-left: 11pt; margin-right: 11pt; margin-top: 0pt; text-align: center;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><img height="371" src="https://lh6.googleusercontent.com/v9R_AgEOOe2zZmgTawgPEK_9KldTnjDwVSr1DVTgyqzLYu0yO2QO8CD84OSUDwr1d0p2n9yZHYnAh8TMiZhxdfFeFJAkuzH0ZNkJ0oewPWX9VxBZ0p8ZpA9UjcPUVaCgG00wtAFF" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="427" /></span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Textbooks usually tell students all the information, in the order the need it, and then call it "Problem Solving". My favourite question and answer was when I read the following question from a textbook:</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: italic; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Jason weighed a fish, and found out that if you took the weight of the fish and added it to half the weight, the result is 20 lbs. How much does the fish weigh?</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The best answer was from a student who exclaimed:</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: italic; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Ask Jason he weighed the damn thing!</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Brilliant response to a horrible question!</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Problem 3:</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> They "unitize" learning.</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Again, a common problem I see with textbooks, is they assume you need to learn A, then B, then C, to master idea D. In my experience, creating these disjoint learning situations, or what I call "silos of learning" causes problems for students. </span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">A common practice in textbooks are "Chapter Tests", which means the pages after these tests have rarely little or nothing to do with the previous pages. In essence, the learning that happened yesterday will have nothing to do with tomorrow. </span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">A great practice is to weave essential learning outcomes throughout your entire course. This contradicts the textbook. If you feel a certain outcome is important for all to master, I would hope that your students work with that idea throughout the entire course and not just for a finite time (week, or month) and then move on and never relate new learning to previous learning. </span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Disclaimer:</span><span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> Does this mean I don't think textbooks belong in schools? NO!</span></div>
<b style="font-weight: normal;"><br /></b>
<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">This means educators have to be aware of the shortfalls of textbooks. The biggest idea we always have to remember is that these resources were created, usually in an office, to be sold across an entire continent or country. They are not designed for "your" kids; or really anyone's kids for that matter. </span></div>
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<div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: Arial; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Textboo</span><span style="font-family: Arial; font-size: 11pt; white-space: pre-wrap;">ks should be used similar to encyclopedias in the classroom. Reference material. If a student is struggling with a concept, give them a textbook, show them a certain page and advise him/her to complete some (not all) questions. When completed, have a conversation, and then ask him/her to return the textbook to the classroom shelf.</span></div>
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Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-67711555683174472542017-11-23T20:24:00.003-07:002017-11-23T20:25:42.800-07:00Visualization in Math<iframe src="https://docs.google.com/presentation/d/e/2PACX-1vR_UUupIpjZd2PtcpctVMAB9jotcyH4EoAIDmMBu0xZGVZTpVPQMFaFBKY3Mx8H2l-00B82XzbFp1TK/embed?start=false&loop=false&delayms=3000" frameborder="0" width="480" height="299" allowfullscreen="true" mozallowfullscreen="true" webkitallowfullscreen="true"></iframe>Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com1tag:blogger.com,1999:blog-413213147692532190.post-45332323016168606962017-11-18T11:18:00.001-07:002017-11-18T11:18:38.550-07:005 things I was NOT doing as a Math Teacher... that I wish I didRecently, I heard a Harvard researcher state,<br />
<blockquote class="tr_bq">
<i>Sometimes the problem is not in what you are doing, but instead in what you are NOT doing.</i></blockquote>
This made me reflect upon my first 8 years of teaching math. What was I NOT doing that may have increased learning? If I could go back in time, here are 5 things I wish I did more (or even at all), in no specific order:<br />
<div>
<br /></div>
<div>
<ul>
<li>I never gave pictures to students and asked them, "Where should the origin be placed to best understand, or work with, this image?". Instead I would always supply images to my students with a Cartesian coordinate already drawn on. If I could go back, I would make time for students to discuss and debate on where is the "best place" for the origin to be placed on an image to solve the problem given.</li>
</ul>
<div>
<br /></div>
<ul>
<li>I never explained what "simplify" truly meant. I would give students loads of questions and I would write "simplify" as a directing word. In my classes "simplify" meant: Add, subtract, factor, expand, combine like terms, rationalize denominator, etc. I wish I showed students where, and ultimately why, each form may be simpler than other forms; however, change the question and then a different form may be actually simpler.</li>
</ul>
<div>
<br /></div>
<ul>
<li>I never asked students to actually measure needed quantities to solve problems. I usually gave students problems with all the information needed, even in the order they needed it, and then asked the question. If I could go back, I would have started asking questions such as "To solve this problem, what would we need to measure and/or determine?". I think it is important that students know how to measure, but more importantly they know what is worth measuring.</li>
</ul>
<div>
<br /></div>
<ul>
<li>I never allowed students to be individuals, not only in the instruction process, but also the assessment process. Most of my tests required students to learn the required material by the same day, and then even asked my students to demonstrate learning the same way. If I could go back, I would allow students to demonstrate learning when they have mastered the material, regardless of the speed and pace of the other students. In addition, I would also have asked students to relate their learning, when possible, to their passions and interests.</li>
</ul>
<div>
<br /></div>
<ul>
<li>I never built my course to allow connections to be built between essential learning outcomes. Instead, I built my course in units where I would teach outcomes as disjointed ideas and rarely make connections between each unit; I created silos of learning throughout the year. If I could go back, I would actually remove all notions of "units" in my course and instead weave big ideas throughout my entire course. Instead of teaching a big idea in September, and then only discuss it again during our "final exam review", I would ensure big ideas spanned the entire length of the course.</li>
</ul>
<div>
What are things, thinking back, did you NOT do?</div>
</div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-32958878222653470062016-05-06T07:00:00.000-06:002016-05-06T15:41:49.915-06:00Number Talks<span style="font-family: inherit;">How do you foster numeracy in a math class?</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Very simply; Once a day, for no more than 15 minutes, complete a <a href="http://www.mathsolutions.com/documents/numbertalks_sparrish.pdf" target="_blank">Number Talk</a>.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">What is a Number Talk?</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Simply put, a Number Talk is a "naked number question" where students must use mental math to arrive at the answer. This tasks removes the myth of "there is only one way", or "there are better ways than others to do math" and instead ensures that all students are aware that each of them have some sort of mathematical insight to offer everyone else in the class.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Here is an example...</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">In Grade 2, Jennifer Smith put this up on the board and asked "How many dots are there?". </span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEib9dwH1F_SF5648GE3uuMqj0Jq4zJFpz74230WOfy6R5m2K_50aigxdqLpEYBpHVWL9Q2bwBaETWVdWSRg-hHE8mGghvt7x3uM147ahcve_vqHFyUYFZ4acZSTaYIuMybQNPPJI9yVIv8i/s1600/unnamed.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEib9dwH1F_SF5648GE3uuMqj0Jq4zJFpz74230WOfy6R5m2K_50aigxdqLpEYBpHVWL9Q2bwBaETWVdWSRg-hHE8mGghvt7x3uM147ahcve_vqHFyUYFZ4acZSTaYIuMybQNPPJI9yVIv8i/s320/unnamed.png" width="239" /></span></a></div>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;"> After most of the students said "7", she asked "How did you count them?", and this begins the Number Talk. I will let her share her story:</span><br />
<div style="background-color: white; color: #222222;">
<span style="font-family: inherit;"><br /></span></div>
<blockquote class="tr_bq">
<span style="font-family: inherit;"> I was surprised that they had this many different ways of counting the dots and my class had no trouble explaining their thinking. I even had a girl; say 5. I was careful and said come and show us and she pointed to the middle 5 and then said oh ya and 2 is 7.</span></blockquote>
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<span style="font-family: inherit; margin-left: 1em; margin-right: 1em;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8TLZHjgq3WyeIbht4ecMEUjLnDy-HKmGhARUj0r_ayzEgqv2xy7AOsV0fRxN3cj1Rns-TY1Ktcjvwz5IMNoIvK_qUUicqDyA3GyItOojEtbUPyMm6o08Q_eqmYurNPfA0cc1EP-qSCG_o/s1600/a5f87a4b-344b-4047-845c-7e89d9ad88e5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8TLZHjgq3WyeIbht4ecMEUjLnDy-HKmGhARUj0r_ayzEgqv2xy7AOsV0fRxN3cj1Rns-TY1Ktcjvwz5IMNoIvK_qUUicqDyA3GyItOojEtbUPyMm6o08Q_eqmYurNPfA0cc1EP-qSCG_o/s320/a5f87a4b-344b-4047-845c-7e89d9ad88e5.png" width="239" /></a></span></div>
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<span style="background-color: white; color: #222222;">Now the cool part!!! 2 girls even asked to stay in during lunch and continue to count the number of ways to count to 7, (This is Grade 12 Math outcome!!)</span></div>
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<span style="background-color: white; color: #222222;">Number Talks are a great way to get students talking, explaining, reasoning and ultimatley arriving at a deep conceptual understanding of how numbers work. If you are interested in knowing more I would suggest you read the following book:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSwpsI49AAeYYh8KwK0C5XUOZDX0D5Qw5W9_XTNXBf609uhDIzUZG1rDhbCcW4WurbxLjLob_lupbDB8Sot0sonftAhN4SzbTlkrtQagHlt567WiJvPaZ5BK3q3_fq4GPvbDmWQQiRGtlh/s1600/Capture.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSwpsI49AAeYYh8KwK0C5XUOZDX0D5Qw5W9_XTNXBf609uhDIzUZG1rDhbCcW4WurbxLjLob_lupbDB8Sot0sonftAhN4SzbTlkrtQagHlt567WiJvPaZ5BK3q3_fq4GPvbDmWQQiRGtlh/s320/Capture.PNG" width="252" /></a></div>
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<span style="background-color: white; color: #222222;">"<a href="https://www.stenhouse.com/content/making-number-talks-matter" target="_blank">Making Number Talks Matter</a>"</span></div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-71050569666056916732016-03-04T08:53:00.000-07:002016-03-04T08:53:37.005-07:00Is Streaming an Intervention Technique?<a href="https://en.wikipedia.org/wiki/Tracking_(education)" target="_blank">Streaming, or tracking,</a> students occurs quite regularly around the world. This means that, at some point in their K-12 education, they are grouped by ability or intelligence. Many educators and parents support this idea. The philosophy, behind this practice, is that it allows for teachers to teach groups of similar intelligence levels. Also many believe that the high end students, through streaming, can be enriched on topics beyond the course.<br />
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This streaming occurs at different ages, or grades, in different countries. In Alberta, streaming usually occurs at grade 10, while in USA this practice starts in middle school. Finland, an international leader in education, has outlawed streaming entirely. This then begs the question, "What is best practice for streaming students?"<br />
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Some educators, who support streaming, say that this practice allows them to teach to "like-minded" individuals and the need for scaffolding diminishes as there is a homogeneous group in front of them. The material, consequently, is presented in such a way for the "average" of the group to understand; using the logic that all students are more or less the same. <br />
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This, unfortunately, contradicts almost all research around the <a href="https://www.youtube.com/watch?v=4eBmyttcfU4" target="_blank">growth of individual students</a>. No matter how well you group students based on ability, there will always be some students who may find a certain topic easy, and other topics more difficult. Teaching to the "middle" will actually cause some students to struggle while preventing others from being enriched. I also fear, in an education system where streaming is prevalent, the practice might become, "If you are not understanding, you must simply be in the wrong stream".<br />
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In a mixed-ability class, the teacher is forced to create material that all levels can benefit from, where the top students are challenged while the weaker students are comforted. This results with all students learning at the highest of their levels and a philosophy of teaching that "all can succeed". <br />
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Also, when students are streamed, certain stereotypes occur. Early in my career, this was a regular occurrence for myself. I remember thinking while teaching a dash-2 course (a second stream in Alberta) "Well this is the lower stream so they won't be able to handle this..." or "We won't have time to do that project, as my students will take longer to learn...". This ideology is harmful to students.<br />
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In 1960, Rosenthal and Jacobson conducted an experiment to look at the impact of teacher expectations. Students were randomly placed in two groups, regardless of ability or talent, and these groups were labelled as "smart" and "weak" for 2 teachers. After a certain period, they determined that actually the "smart" class had scored at higher levels on IQ tests, while the "weak" class struggled with many concepts taught. Expectations from teachers was the difference. <br />
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This stereotype, once placed, becomes almost unbreakable. In England, where they stream by age 4, it has been shown that 88% of children remain in the same groupings until they leave school. This should be alarming!! A label, we give to a person who should be playing with blocks, dolls, and laughing, will determine his/her success for life! This, again, contradicts almost all research around child development and learning.<br />
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This label is not limiting only the weak students but also the "smart" ones as well. Carol Dweck, found that the moment students are streamed, by ability or intelligence, the students who were most negatively affected were those going into the top rank. Their positive growth-mind-set thinking reduced almost instantly, and they became fearful of making mistakes and consequently avoided more challenging work. This was especially prevalent in high achieving girls.<br />
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Students and parents usually support streaming due to the fact that this practice can allow schools to prepare students more appropriately for their future. This argument is actually flawed. <a href="https://ed.stanford.edu/faculty/joboaler" target="_blank">Jo Boaler,</a> followed students from two different school experiences. The first group came from a school where they organized students heavily by ability, while the second school mixed all abilities together. She found that the students who experienced mixed-ability grouping, despite growing up in one of the poorest areas in the country, were now in more professional jobs than those who had experienced streaming. <br />
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What is even more interesting was the attitude of the students who learned in mixed group settings. At first some of the brightest students were aphrenrsive around the fact that they would be constantly explaining their ideas to the rest of the class. However after one year this changed as they were quick to realize that this practice actually helped them understand the concepts being taught.<br />
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What is the solution?<br />
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Simply don't stream students based on prior knowledge or ability. <br />
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A great practice I have seen comes from a colleague of mine, <a href="https://twitter.com/jnlmauro" target="_blank">Jonathan Mauro.</a> He is the department head of a high school physical education group and has started a creative way of streaming; he lets the students decide. Instead of saying, "this group has high physical literacy, and this group has low physical literacy", he provides the students options of engaging in physical literacy. 3 full classes are scheduled during the same time slot and then the 3 teachers teach each unit using a different sport or activity to address the PE curriculum. Meaning, one student can learn through volleyball, while a different student can learn through badminton. Regardless of previous ability or experience, each student is free to choose which activity would be most engaging to them.Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-19179453019418673212016-03-01T21:16:00.000-07:002016-03-01T21:21:10.728-07:00More to the PISA scoresMany people try and use <a href="http://www.oecd.org/pisa/" target="_blank">PISA test</a> data to infer that there is a problem with Math education in their area. I have seen this argument used in many blogs, papers, and social media outlets.<br />
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Initially, some people will show fancy graphs and try to convince others that the drop of PISA results has actually been caused by a change in curriculum. Later they will conclude that unless we dig up some old curriculum these scores will continue to drop in the area of mathematics.<br />
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First, any arguments, at best, truly show that there is only a correlation between the type of math curriculum their area has and a drop in PISA data. All arguments (which I have seen anyways) always fails to show causation. What is the difference?<br />
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Correlation is when two or more things or events occur near or around the same time. These things might be associated with each other, but are not necessarily connected to each other by a cause/effect relationship. <br />
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An easy example is when people get a cold during the winter months they usually end up with a runny nose and a sore throat. These two events are correlated but we cannot conclude that a runny nose will actually cause a sore throat to occur. <br />
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Most arguments around PISA, tend to show some sort of data analysis and link it to a change in curriculum, and again this would be correlation, at best. However, we can take a closer look at the data ourselves...<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH6JruX5wO7un0NU5k5-nAQFivtK145GjO5K4BqGP3dftpgnKQy_nNprjzyLbyfYA0SPR-K8o4MFpk-prsnTW5D_qZESZK0NsPyUUXIxW0s0LM8MRb7352WvZncpQpOFCfp28bm1jPS-qz/s1600/Screen+Shot+2016-03-01+at+8.23.39+PM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="262" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH6JruX5wO7un0NU5k5-nAQFivtK145GjO5K4BqGP3dftpgnKQy_nNprjzyLbyfYA0SPR-K8o4MFpk-prsnTW5D_qZESZK0NsPyUUXIxW0s0LM8MRb7352WvZncpQpOFCfp28bm1jPS-qz/s400/Screen+Shot+2016-03-01+at+8.23.39+PM.png" width="400" /></a><br />
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If we look at the Canadian results of PISA by province, we see that every single province dropped from 2009 to 2012, other than Quebec and Saskatchewan. On the international level, Netherlands, Belgium, Australia, Denmark, New Zealand, France, and many other countries fell at similar rates to Canada. <br />
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If there is a Math Crisis in your zone, then there must also be a Math Crisis around the entire planet? Does this sound likely? I would hope not!<br />
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Some "back to the basics" folk will ignore the fact that they are implying here is an International Math Crisis and simply tell us that:<br />
<blockquote class="tr_bq">
<i> PISA tells how well the math curriculum is in a certain area compared to other parts of the world.</i></blockquote>
Well here are some stats that show how false that statement is:<br />
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Shanghai had the highest score on the 2012 PISA test with an average of 600.24, while Australia had a much smaller average of 515.01.<br />
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What can you conclude? That Australia's math curriculum is much weaker than Shanghai's?<br />
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If we look at students who were born in China and moved to Australia before ever entering school, their average on the 2012 PISA test was 614.77... 14 POINTS HIGHER THAN SHANGHAI!!! <br />
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This must mean that Australia's math curriculum is superior to Shanghai's? See the problem?<br />
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Using PISA scores, alone, to determine the quality of education in a province, state, region, or country is similar to judging someone's ability to drive simply by only watching them parallel park. While this single test can make many great observations around education on a global scale, we need to realize that it is exactly that: a single test. <br />
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There are much more variables at play in an Education system than simply the results on a test that some countries value more than others. If you attend school in Scotland and are called to write PISA, you will be forced to watch champions winning Gold Medals for their country and informed that you have the ability to "bring home the Gold for Scotland". <br />
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If you attend school in China and are called to write PISA, your name will be broadcasted and people will cheer you on as you walk into the testing room. The test you will write will be similar to the test preparation you have received over the previous months.<br />
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If you attend school in Canada and you are called to write PISA, you will be quietly removed from a class, brought to a room to write a test you know nothing about and have had no formal preparation for.<br />
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So I ask again, does a drop in PISA scores really mean a math crisis? I think not!<br />
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Lastly, we need to understand that a lot of arguments against current math practices are actually <a href="http://realteachingmeansreallearning.blogspot.ca/2016/02/math-wars-confusing-curriculum-and.html" target="_blank">attacking teachers and not curriculum.</a><br />
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Further Reading:<br />
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Sam Sellar- <a href="https://books.google.ca/books/about/Globalizing_Educational_Accountabilities.html?id=u5FGCgAAQBAJ&source=kp_cover&hl=en" target="_blank">Globalizing Educational Accountabilities</a><br />
<a href="http://www.oecd.org/pisa/keyfindings/pisa-2012-results.htm" target="_blank">PISA Key findings</a>Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-41715560530320562492016-02-11T10:26:00.001-07:002016-02-11T10:28:28.927-07:00Where do the misfits go?<iframe allowfullscreen="true" frameborder="0" height="389" mozallowfullscreen="true" src="https://docs.google.com/presentation/d/13FiUMvlUvZyH6AWCMHC6MwyVNhyHoP_QJFI6HmVH8sk/embed?start=false&loop=false&delayms=30000" webkitallowfullscreen="true" width="640"></iframe>Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com1tag:blogger.com,1999:blog-413213147692532190.post-29246790636378433112016-02-10T07:07:00.000-07:002016-02-10T07:10:07.973-07:00Algebra Day 1Some points after reading <a href="http://www.amazon.ca/Connecting-Mathematical-Ideas-Teaching-Learning/dp/0325006709" target="_blank">Connecting Math Ideas</a> by <a href="https://twitter.com/joboaler?lang=en" target="_blank">Jo Boaler</a> on creating algebra. Also a great activity to have with students, or yourself.<br />
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Tell your students the following everyday<br />
<ul>
<li>Addition, subtraction, multiplication, and division is part of math but math is so much more! While algebra is a focus in many classrooms, don't forget about geometry, probability, or data analysis.</li>
<li>You can't make understanding an algorithmic process. Students need to experience confusion, misunderstanding, and failure as part of the understanding process.</li>
<li>Being good at math is not something that is easily explainable. Accuracy and speed in procedural mathematics is only one way. Other ways include being open minded, logical, discussing conjectures, and trying to answer "why?". Everyone can be good at math and we need people who can do math in those different ways.</li>
<li>While a problem may only have one correct answer, it may have many different correct solutions. While some solutions are more efficient, or effective than others, it is more important that everyone understands at least one way of arriving at the correct answer. Math must make sense.</li>
<li>We can get better at certain skills of math by practice, however talking and discussing with your fellow students will allow you to understand mathematical ideas deeper.</li>
</ul>
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When developing algebraic thinking with students, we shouldn't force students do what is the most efficient (as sometimes this is objective anyways), but to do what makes sense. When presented with a set of data, or a pattern, Thorton (2001) argues that it is less important that students be able to find the algebraic rule than they recognize a rule can be represented in equivalent algebraic expressions. When "finding <b>the</b> rule becomes the focus" most mathematical thinking is lost.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtGijTcwcUR6Eo7Hy2wjTp7Sbei-PI_mRfeW4Vkf9DE2CfP3S9yOST_VPoBMHRlmwWyCuAPfOyQPQ26HlNeP-rKU6p0KisjQ9Fcghcxb4wfC8hLLWBDjmRctWDOt4mqP5hSGZnFeyLw-eK/s1600/Capture.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="187" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtGijTcwcUR6Eo7Hy2wjTp7Sbei-PI_mRfeW4Vkf9DE2CfP3S9yOST_VPoBMHRlmwWyCuAPfOyQPQ26HlNeP-rKU6p0KisjQ9Fcghcxb4wfC8hLLWBDjmRctWDOt4mqP5hSGZnFeyLw-eK/s200/Capture.PNG" width="200" /></a>Lets, for example, look at the following grid. I want you to tell me how many blocks are coloured in this 10x10 grid without counting the blocks individually. It is crucial you do this without writing, talking to a partner or counting, as it will force your mind to make generalizations which will be the basis of deeper learning to come. </div>
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<i>As a teacher we should first ask students now to share answers, not strategies, with an elbow partner. This would then force some to reevaluate their strategy and possibly pick up any common errors he/she may have made.</i></div>
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Now how did you do it? Here are some strategies in Arithmetic form:</div>
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4x10-4</div>
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10+9+9+8</div>
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10+10+8+8</div>
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10x10-8x8</div>
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4x8+4</div>
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As a reader, are you able to explain how each of the expressions above arrive at the same answer? An important task in furthering one's mind into algebraic thinking.</div>
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<i>Imagine if this was done in a classroom. Jason stands up and explains why he simply went 4x10-4. There might be some smiles, nods, confused looks. Some students will have seen the same answer provided a different way. Does this happen on a worksheet?</i></div>
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Of course now, you have realized that the correct answer is 36. Would then the expression "30+6" be appropriate? This is where conceptual differs from procedural. Yes 30+6 equals 36 but unless a child can create meaning to the 30 and to the 6, I would disagree that 30+6 would be an appropriate response in this context. This illustrates the idea of solution(30+6) vs answer (36). </div>
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What if the grid was 8x8, how would the above expressions change? Would a number change in every expression or only some? This idea of visualizing a problem and solving it, is crucial to the advancement of one's knowledge. </div>
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Going back to your strategy and stretching the square to an unknown length, could you create a verbal description of what would you do?</div>
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Now the algebra begins..but first...</div>
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Noss, Healy, and Hoyles (1997) point out that somewhere we stop seeing algebra as a tool but instead of the end point of a problem. The confusion starts when we see problems as a way to practice algebraic skills instead of using algebra to explore problems.</div>
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When bringing in letters into these expressions we must remember some powerful pitfalls:</div>
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<ul>
<li>The largest misconception is that letters are labels or initials. (pg 25)</li>
<li>Changing the variable to a different letter changes the solution. (Let students pick the letter, and encourage different students with different letters)</li>
<li>The equal sign simply means "gets the answer". (Try writing 5=2+3 as often as 2+3=5 especially in earlier grades)</li>
</ul>
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Taking these points into account, create a variable for the side length of the square. Could you create an expression using your variable to show the amount of squares which would be shaded? Knowing what you have done so far, how could you test your expression?</div>
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<div>
<i>In a class at this point it would be helpful to ask the class "What will be staying the same? What will be changing? As we bridge the gap from numbers to a variable". For example if the strategy you are using is 10*10-8*8, then the multiplication and subtraction will remain the same while 10 and 8 will change to x and x-2 respectfully. I would not tell them this, but instead question it.</i></div>
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If you followed all of these tasks then you have now done the problem in 3 different approaches: Arithmetic, Verbal, and Algebraic.</div>
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The crucial part of this example, or if this lesson was to be developed into a classroom, is to ask questions and not simply give answers. Remind them we are not seeking the solution, but a solution that makes sense to them. Embrace students' wrong answers since learners who are given competing ideas, engage in cognitive conflict and such conflict promotes learning more than the passive reception of ideas that are always correct and seem straightforward (Fredricks, Blumenfeld, and Paris 2004).</div>
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Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0tag:blogger.com,1999:blog-413213147692532190.post-39967722190151593692016-02-09T11:27:00.000-07:002016-02-09T12:56:39.760-07:00PBL on Math 30-2 Research ProjectHere is my attempt to create a PBL for a High School Math class. In WCNP, in Math 30-2 and 20-2 there is an actual outcome around researching a topic and relating it to math. Below you will find all resources and timelines. Please feel free to use, change, alter, as you see fit.<br />
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<a href="https://docs.google.com/document/d/1cyiAS5c599tQ6DTKIzmG9hVM4j382nTcM_zOihsHCLY/edit?usp=sharing" target="_blank">PBL Math Research Project</a><br />
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<iframe height="1200" src="https://docs.google.com/document/d/1cyiAS5c599tQ6DTKIzmG9hVM4j382nTcM_zOihsHCLY/pub?embedded=true" width="750,"></iframe>Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com1tag:blogger.com,1999:blog-413213147692532190.post-74324298092787547022016-02-05T07:00:00.000-07:002016-02-05T07:00:16.539-07:00Creating Discourse-Friendly ClassroomsSome great simple things to create a discourse-friendly classroom from<br />
"<u>Literacy Strategies for Improving Mathematics Instruction</u>" (2005)<br />
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<ul>
<li>Arrange desks so that students can easily turn to see each other.</li>
<li>Encourage students to direct questions and explanations to the class, rather than the teacher.</li>
<li>When recording ideas, use the students' words as much as possible.</li>
<li>Try not to repeat or paraphrase everything students say. This teaches the other students that they can simply listen to you. Ask the student to repeat louder if need be.</li>
<li>Remind students that a conversation has both listening and speaking skills.</li>
<li>Stand in a variety of spots in the classroom.</li>
<li>Remember, students listen harder when a peer speaks than when an adult does!</li>
<li>Give students time to think.</li>
<li>Arrange lessons so that students have a product to share as they explain their thinking. </li>
</ul>
<div>
Even better is if the student takes the lead in the class. This is done by </div>
<div>
<ul>
<li>Asking open questions to stimulate thinking. "Is this logical?" "What do you wonder about?"</li>
<li>Honor ideas even if they are incorrect.</li>
<li>Encourage arguments between students.</li>
<li>Confusion is ok! Make sure students know that you want them to be confused, and that you will let them be this way.</li>
<li>Tangents are great teachable moments.</li>
<li>When a student brings up an idea ask the rest of the class if they have any questions or ideas.</li>
<li>Counter questions with questions not explanations. </li>
<li>Even with a correct answer, ask if there is any another way this can be done, or if there are improvements to be made.</li>
</ul>
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A great reminder!</div>
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Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com1tag:blogger.com,1999:blog-413213147692532190.post-90766895036182720062016-02-04T05:40:00.000-07:002016-04-08T14:42:14.808-06:00Math wars confusing curriculum and pedagogyRecently, in the media there has been a lot of false statements around the <a href="http://www.wncp.ca/1373.aspx" target="_blank">WNCP</a> curriculum, and consequently these attacks fall on the hands of teachers. Before I continue I will explain two important words in the educational world.<br />
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<a href="https://education.alberta.ca/media/464621/k_to_9_math_pos.pdf" target="_blank">Curriculum</a>: This is simply WHAT a teacher needs to teach. For example, a Grade 3 student needs to be able to recall and understand up to 5 x 5.<br />
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Pedagogy: Is HOW the teacher teaches. For example, using direct instruction, peer coaching, PBL, etc.<br />
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Contrary to social media, presentations, and other means of critiquing the curriculum, the curriculum does not say:<br />
<ul>
<li>How teachers need to teach the outcomes.</li>
</ul>
<ul>
<li>Discovery learning is a must. In fact the word discovery <b>does not appear once </b>in the entire document.</li>
</ul>
<ul>
<li>"21st Century skills development", and "experiential learning". N<b>either of these phrases </b><b>appear once</b> in the entire document.</li>
</ul>
<ul>
<li>Students should not be memorizing their basic facts.</li>
</ul>
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Keep in mind that the curriculum is the WHAT not the HOW nor even the WITH WHAT.</div>
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Recently, an event, around public education, was held in Calgary at a private, gated school. The event was designed to inform parents of "the best practices in math". While the event was designed to encourage change at the government level, make no mistake this was a blatant attack on math teachers in Alberta, and other teachers in provinces following the same curriculum.<br />
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Here is one of their recommendations:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPuBk9kHwHxFKXgZxN1XZKWnXaFZZqX1bqWORXAaUM_kKo3fyz_Ok8sgKCCVL65OJJ533kx6eM58MJs3yBrSSD6A1ZOM04HV0yPDuJl1Nw4N0EQ48vdq7bikr_8dALiP3ucNTduJLi1FWc/s1600/DirectInstruction.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPuBk9kHwHxFKXgZxN1XZKWnXaFZZqX1bqWORXAaUM_kKo3fyz_Ok8sgKCCVL65OJJ533kx6eM58MJs3yBrSSD6A1ZOM04HV0yPDuJl1Nw4N0EQ48vdq7bikr_8dALiP3ucNTduJLi1FWc/s400/DirectInstruction.PNG" width="400" /></a></div>
Now as you can see "Teaching Strategy" does not fall under curriculum but actually pedagogy. The irony is that I am sure all teachers, at some point, do some direct instruction. I have had the honor of being in many math classrooms around the province and I can attest that teachers have balance. This slide almost paints the picture that teachers are simply sitting around hoping that students will learn math through osmosis. Also, there is no "imposing of one model of instruction". Enter a classroom and you will see that teachers truly implement strategies which are the most beneficial based on the classroom make-up and the outcome(s) being taught.<br />
<br />
Next we have "Some things to watch out for"<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwagFRjDtw8_onQYkAVjtbvpVSVlLR16U2paqpAfK4MgbrXARtpKazkGtgOkJaIFOZLQXQEddxOVapwRxRiNrIZIdXLBDb-gSaqdc7-w51tTZg6J1dxceOkMNVFufj2ku-ETL3fqyyN4oZ/s1600/CZ29f23WkAEEKlf.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwagFRjDtw8_onQYkAVjtbvpVSVlLR16U2paqpAfK4MgbrXARtpKazkGtgOkJaIFOZLQXQEddxOVapwRxRiNrIZIdXLBDb-gSaqdc7-w51tTZg6J1dxceOkMNVFufj2ku-ETL3fqyyN4oZ/s400/CZ29f23WkAEEKlf.jpg" width="400" /></a></div>
Of course some of these points make sense. We should be wary of many phrases as their intent could be misguided. Also, in reference to the second point, no one is arguing <b>against </b>memorization and procedures. <br />
<br />
For their third point, I have not met one person who suggests that students should not memorize their math facts. The difference is, however, students should memorize these facts out of application and use, not out of necessity.<br />
<br />
This means, show students math in a context and for a purpose, and the memorization will occur. Have students roll dice, play cards, board games, car games, etc, as most (if not all) games have some link to reasoning, logical thinking, and mental mathematics. <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFPMvhHQE1ISW5dug4GLQtFPVFJiqtpD04LmCh8pVu3LH_DqAzjsldtZsUxHC7IdV6AO3nT3cmkrtyI_ArJk1KKuVRQ7gU2dE9lsBoNXzLBV2dz1YVoFLTqihILn42b2NGK-ShKPt0_VRW/s1600/memorizing+understanding.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFPMvhHQE1ISW5dug4GLQtFPVFJiqtpD04LmCh8pVu3LH_DqAzjsldtZsUxHC7IdV6AO3nT3cmkrtyI_ArJk1KKuVRQ7gU2dE9lsBoNXzLBV2dz1YVoFLTqihILn42b2NGK-ShKPt0_VRW/s320/memorizing+understanding.png" style="cursor: move;" width="214" /></a><br />
<br />
"Understanding is not more important than skill"- This is again in reference to the actual art of teaching in the classroom. I have yet to meet one teacher who denies that skill is useful, but let's remember if we only focus on skill then learning can be <a href="http://realteachingmeansreallearning.blogspot.ca/2012/08/how-my-dog-taught-me-about-math.html" target="_blank">disguised with simple memorization.</a><br />
<br />
So why is there such an attack on the curriculum?<br />
<br />
I believe because some are confusing the terms curriculum and pedagogy. Also, because we have a generation (parents) who learned math through memorizing algorithms and are confused around why their own children are not coming home with the same algorithms. Recently, <a href="https://www.washingtonpost.com/local/education/many-parents-hated-common-core-math-at-first-before-figuring-it-out/2016/01/31/33c1b59e-c6bd-11e5-9693-933a4d31bcc8_story.html" target="_blank">some parents are now seeing the benefit of the change.</a><br />
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Also there has been use of the drop in PISA scores, however there has been no <a href="http://realteachingmeansreallearning.blogspot.ca/2014/01/problems-with-pisa-arguement.html" target="_blank">actual evidence that this drop has been caused by curricular changes.</a><br />
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The confusion, for a child, might start when a child is learning one way at school, and then coming <a href="http://realteachingmeansreallearning.blogspot.ca/2014/03/case-for-new-math.html" target="_blank">home to hear that the strategy is not righ</a>t. We must also realize that teachers are trained professionals around education. These professionals implement effective instruction based on the individual needs of the students. It is unfortunate that some want to see the art of teaching go to a procedural task of "tell students what to do, ask students to imitate the learning, repeat". <br />
<br />
If you have a question around the math your child is learning, phone the teacher. Social media, news, and other hands not in K-12 education, have a way of distorting the truth. Keep in mind that teachers are trained to teach your child math in a way that is meaningful, and creating a passion towards numbers.<br />
<br />
I remember back when I was in school and how there was an immense number of people whom hated math. It seemed as if math was the number one hated subject in school. (No research simply guessing here). Isn't it time this changes? Isn't it time we cultivate passion and number sense?<br />
<br />
<a href="http://realteachingmeansreallearning.blogspot.ca/2014/01/math-in-new-setting.html" target="_blank">Math class needed a chang</a>e, and this change is healthy. There is now balance. Before there was a focus to teach it one way and all students were required to learn that one way. Finally, alternative efficient strategies are not only accepted but encouraged! We are allowing students to not only learn math, but actually like it!Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com6tag:blogger.com,1999:blog-413213147692532190.post-63751993798146627302016-02-03T13:23:00.000-07:002016-02-03T16:43:37.779-07:00When is the world going to be full?<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgO8yfmzcHfkfPcI41K7PC-2arVSmGFrXO38odV4MLV0WJYdja3cxC-r0aTotxBSvTxG6t4J_9K-EKw7e9usOqNIzjwkzLN6SMdyXW2j1UGvCfJWPSIqbWtvPv-JWuV9mvBrtkbgnBdM-OU/s1600/population-growth-shows-family-reproduction-expecting-representing-newborn-44556884.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="260" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgO8yfmzcHfkfPcI41K7PC-2arVSmGFrXO38odV4MLV0WJYdja3cxC-r0aTotxBSvTxG6t4J_9K-EKw7e9usOqNIzjwkzLN6SMdyXW2j1UGvCfJWPSIqbWtvPv-JWuV9mvBrtkbgnBdM-OU/s320/population-growth-shows-family-reproduction-expecting-representing-newborn-44556884.jpg" width="320" /></a>A possible project which could be used in a math class. I have designed this to fit in the Alberta Math 30-2 curriculum. Feel free to change or alter as you see fit.<br />
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Pose the question:<br />
<br />
Introduction: When is the world going to be full?<br />
Give students 5 minutes to talk in groups. After, have a debrief and share possible hypotheses. <br />
<br />
Next, ask for possible information we will need to answer this question more accurately. Some other possible questions might be<br />
<br />
<ul>
<li>How many people can fit on the plant?</li>
<li>How many people can the plant sustain? (Different question than the first)</li>
<li>What is our current growth rate?</li>
<li>What is our current population?</li>
</ul>
<div>
Some of these questions will be easy to solve, while others might be more difficult. I would ask students, in groups, to research the answers to the questions they asked. Some of these will have an answer all will agree with, such as "What is our current population?", while other questions might have a range of answers depending on the website found, such as "How many people can the planet sustain?" Create answers that everyone in the class can agree with, or even commons answers in different groups as long as each student in each group agrees.</div>
<div>
<br /></div>
<div>
Body:</div>
<div>
Creating an extrapolation of our population.</div>
<div>
<br /></div>
<div>
You can google our current growth rate, however this number, most likely, has no meaning or even understanding of how it was derived. As a class, possible growth rates will be explored. Below is a chart of recent population numbers on our planet. (You might want to use different years, or more years).</div>
<div>
<b id="docs-internal-guid-c405fadb-a8b7-e3e2-c69d-de832618e992" style="font-weight: normal;"><br /></b></div>
<br />
<div dir="ltr" style="margin-left: 0pt;">
<table style="border-collapse: collapse; border: none;"><colgroup><col width="80"></col><col width="155"></col></colgroup><tbody>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Year</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Population (Billion)</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">2016</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">7.4</span></div>
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<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">2015</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">7.3</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">2010</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6.9</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">2005</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6.5</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">2000</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6.1</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1995</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">5.7</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1990</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">5.3</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1985</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4.9</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1980</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4.4</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1975</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4.0</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1970</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">3.7</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1965</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">3.3</span></div>
</td></tr>
<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1960</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">3</span></div>
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<tr style="height: 0px;"><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1955</span></div>
</td><td style="border-bottom: solid #000000 1px; border-left: solid #000000 1px; border-right: solid #000000 1px; border-top: solid #000000 1px; padding: 7px 7px 7px 7px; vertical-align: top;"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: transparent; color: black; font-family: "arial"; font-size: 14.666666666666666px; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">2.8</span></div>
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</tbody></table>
</div>
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Now the question becomes, "What sort of data is this?" This is when I would pause and teach exponential, sinusoidal, linear, quadratic, cubic, and logarithmic functions. Using this data, I would create an equation of each type of function. Here is a graph of all the functions displayed together...(I used 1970 as year 0. This occurred as, after deciding this, I needed more data points. I also added a logistic curve to show what would happen if population growth become 0)<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjah0Sz_CIAodBb_Xh3qrmXknj0KOE5OMwWYEBm1rj0a4pkdllPWBeoaftQEVeCWs5hRYDBUHk3CF9bVbU_inRFG4Wbf76rx79oSkleNuGjB1L7B5cznYD2e6Q3QZvPu7xtm-sUZBN_455l/s1600/Screen+Shot+2016-02-03+at+1.01.51+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="305" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjah0Sz_CIAodBb_Xh3qrmXknj0KOE5OMwWYEBm1rj0a4pkdllPWBeoaftQEVeCWs5hRYDBUHk3CF9bVbU_inRFG4Wbf76rx79oSkleNuGjB1L7B5cznYD2e6Q3QZvPu7xtm-sUZBN_455l/s640/Screen+Shot+2016-02-03+at+1.01.51+PM.png" width="640" /></a></div>
</div>
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<br /></div>
<div>
Now is the discussion time.....</div>
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<br /></div>
<div>
Ask some general questions such as:</div>
<div>
<ul>
<li>Any general thoughts?</li>
<li>Any similarities?</li>
<li>Could you create a possible scenario in which each graph would be accurate?</li>
</ul>
<div>
You could also embed the critical learning of each function into this: For example you could ask:</div>
<div>
<ul>
<li>What is the amplitude, period, and median value for the sine function? What does that mean in this context?</li>
<li>What are the zeros of the cubic function, what does this mean in this context?</li>
<li>What is the growth rate of the exponential function? What does this mean in this context?</li>
</ul>
</div>
<div>
And any other questions which may come to mind. Now you may want to remove any functions in which the class agrees might be inaccurate. Such as the cubic, sinusoidal, and/or linear.</div>
</div>
<div>
<br /></div>
<div>
Using the remaining graphs determine when the Earth would be full, using the number the class researched earlier. The answers might be earlier, or later, than the class first hypothesized. </div>
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<br /></div>
<div>
Conclusion:</div>
<div>
Have the discussion "Should population growth be addressed? Why or why not?"</div>
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<br /></div>
<div>
Do not rush this project, take your time. Stop, throughout, to teach certain skills, or concepts. Also allow students to research on their own and pose their own questions throughout. </div>
Dave Martinhttp://www.blogger.com/profile/16805828337891203825noreply@blogger.com0