tag:blogger.com,1999:blog-81955873554504362792024-03-06T15:02:56.553-05:00Math to the 7th PowerUnknownnoreply@blogger.comBlogger43125tag:blogger.com,1999:blog-8195587355450436279.post-81050955336686059832018-02-18T16:40:00.002-05:002018-03-08T13:51:30.027-05:00Percent Exploration Stations!<div>
<span style="font-family: "arial";"><span style="white-space: pre-wrap;">When possible, I like to design an exploration activity for the first day of each new unit. The purpose of these explorations is to help students activate their</span></span><span style="font-family: "arial"; white-space: pre-wrap;"> prior knowledge about the topic and also help them build on that prior knowledge with hands on and engaging activities. Below are the instructions for the seven stations that I came up with for our next unit on percent. Here is the link to the document I created for students to record their work at each station: <a href="https://docs.google.com/document/d/15MKOD4vLMgj_MZyilXfVwXwGFu4bEToAXriX4w9HPBg/edit?usp=sharing">Percent Exploration Stations</a></span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b>1. Basketball:</b> Shoot a basketball <b><u>ten</u></b> times.</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"> a) How many times did you make the basket?</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"> b) What percent of your shots did you make?</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"> c) What percent of your shots did you miss?</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b>2. Percent Fail:</b> Look at the posters and explain why the percents used in three of the posters are incorrect. Link to the hundreds of <a href="https://saravanderwerf.com/2017/07/31/math-wall-of-shame-part-3-73-new-images-to-download/">Mail Fail images</a></span><br />
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<b><span style="font-family: "arial" , "helvetica" , sans-serif;">3. Pom Pom Draw:<span style="font-weight: normal;"> Without looking in the bag, randomly select 10 colored pom poms from the bag and record your results. Replace the pom poms in the bag each time before you select another pom pom (with replacement).</span></span></b><br />
<b style="font-weight: normal;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> <u>Red</u> <u>Blue</u></span></b><br />
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<b style="font-weight: normal;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> a) What fraction of the pom poms that you selected are red?</span></b><br />
<b style="font-weight: normal;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> b) What fraction of the pom poms that you selected are blue?</span></b><br />
<b style="font-weight: normal;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> c) What percent of the pom poms that you selected are red?</span></b><br />
<b style="font-weight: normal;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> d)What percent of the pom poms that you selected are blue?</span></b><br />
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<b style="font-weight: normal;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Now take out all of the pom poms from the bag.</span></b><br />
<b style="font-weight: normal;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> e) What percent of all the pom poms are red?</span></b><br />
<b style="font-weight: normal;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> f) What percent of all the pom poms are blue?</span></b><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"> g) Are the percents in c) and e) the same? Why or why not?</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b>4. Go Fish: </b> Play Go Fish with your partner. You are trying to get sets of four cards where each set has a fraction, decimal, percent and area model that all the the same value. Record below the sets that both you and your partner find during the game. If you found the set, write your initials in that row in the table below.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b>5. <a href="http://mail.wecdsb.on.ca/~david_petro/WebSketches/percentguesser/index.html">Percent Guesser:</a></b></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">Use the link posted on Google Classroom to practice finding the amount represented by each given percent. See how close you can get to the exact value. Record your guesses, actual values and the how close you got to the actual percent below and circle your best guess!</span><br />
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<b><span style="font-family: "arial" , "helvetica" , sans-serif;">6. Tarsia Puzzle:<span style="font-weight: normal;"> Work with your partner to put together the triangle pieces to form a hexagon. Find equal values on the sides of two triangles and place them together. Here is the link to the creators of this <a href="http://engaging-math.blogspot.com/2015/04/tarsia-puzzle-fractionsdecimals-and.html">Tarsia puzzle</a> and free downloads.</span></span></b><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b>7. <a href="https://teacher.desmos.com/activitybuilder/custom/57892a0ddb4f8f2714530e47">Desmos Card Sort: Fractions, Decimals, and Percents:</a> </b> Go to student.desmos.com and type in the class code:</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Drag the cards that the same value together to make sets--each set has four cards.</span></div>
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<br />Unknownnoreply@blogger.com4tag:blogger.com,1999:blog-8195587355450436279.post-17314756545984684632017-12-28T14:59:00.000-05:002017-12-28T23:28:32.106-05:00Pass to Leave Class Tear Off Sheet<span style="font-family: inherit;">I think I finally have a workable system for allowing students to leave class for emergencies, while at the same time rewarding students who do not use "leaving class" as an excuse to get out of doing work. When students leave class for a drink of water, etc., they often miss 10-20% of the class period, which over time can have a huge negative impact on learning for students who leave class frequently. In addition, when students leave class they often miss important instructions or information and then need to ask the teacher or other students to explain what they missed when they return, which can also negatively impact the learning for all the other students in the classroom.</span><br />
<span style="font-family: inherit;">I like the idea of a ticket system whereby students can use tickets to leave class when necessary and can also get rewards for unused tickets at the end of each quarter. In the past I have set up this system with students only to have it fall apart at the beginning of the second quarter when I did not have a new set of tickets to hand out and/or did not give students the opportunity to redeem unused tickets.</span><br />
<span style="font-family: inherit;">This year I gave students a tear off sheet for the entire school year that they taped into the back of their math journal. Math journals are required every day and are stored in the classroom, hence students should have ready access to these tickets at all times. There are twelve tickets in total--three for each marking period. I think this is a reasonable amount for the entire school year if students are leaving class only for emergencies.</span><br />
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<span style="font-family: inherit; margin-left: 1em; margin-right: 1em;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_T__X5Db5S5BMTLEdEtaXo_MflQiO9ghMwfREePcZZHUR05RLFQS_YuO6_kYVPeJ147BqIVG-2bMqd3bw7DGMWnFhN-Yjy78I-H_qj3d-Bf0F06pD_SLFGrBlK29WMNNhrq5sbNdA5WOE/s1600/Pass+to+Leave+Class.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="585" data-original-width="971" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_T__X5Db5S5BMTLEdEtaXo_MflQiO9ghMwfREePcZZHUR05RLFQS_YuO6_kYVPeJ147BqIVG-2bMqd3bw7DGMWnFhN-Yjy78I-H_qj3d-Bf0F06pD_SLFGrBlK29WMNNhrq5sbNdA5WOE/s400/Pass+to+Leave+Class.PNG" width="400" /></a></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6SC5ty-ad-8WzgAL7E5l3RL_m0sZnqPRNtSa_Mm01ym7L2bSosc2nIGoEMfr1c1r4HM86aKy3RpMkVXVwv0eFVePiWZx5KmeF9DBjix47_NfE1iQbw2M1hb0Rha2QPy0AP0XGT9VQ027S/s1600/Piggy+Bank.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img border="0" data-original-height="530" data-original-width="511" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6SC5ty-ad-8WzgAL7E5l3RL_m0sZnqPRNtSa_Mm01ym7L2bSosc2nIGoEMfr1c1r4HM86aKy3RpMkVXVwv0eFVePiWZx5KmeF9DBjix47_NfE1iQbw2M1hb0Rha2QPy0AP0XGT9VQ027S/s320/Piggy+Bank.PNG" width="308" /></span></a></div>
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<span style="font-family: inherit;"><b>Instructions on Tear Off Sheet:</b></span></h3>
<span style="font-family: inherit;">"Our math class is very short and our time to learn together is precious. Please use passing time or lunchtime to use the restroom or water fountain or get supplies from your locker. I understand that sometimes there are emergencies and so please save these "tickets" for those circumstances (three per marking period). Please keep this sheet in your math journal so that is available whenever you need to leave the room. After asking for permission to leave the room, tear out one "ticket" and place it in the piggy bank at the sign out table where you must also record your name, destination and time before you leave the room. Any unused "tickets" can be redeemed for 𝝅 Dollars at the end of each marking period (up to three per marking period)."</span><br />
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<span style="font-family: inherit;">In general students have been very cooperative about this system. When students ask to leave class, I usually ask if it is an emergency and then give permission if necessary, reminding them to "use a ticket." For me this is finally a sustainable low maintenance classroom management system that also reduces the number of requests to leave class and the loss of classroom instructional time.</span><br />
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<span style="font-family: inherit;">Here is a link to the document: <a href="https://docs.google.com/document/d/1Saj3a0sTsi2vkx-oNDaTBs_J7YuBHr_5iK5w2665ZPQ/edit?usp=sharing">Pass to Leave Class Tear Off Sheet Blog Version</a></span>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-8195587355450436279.post-50693619371777308472017-11-15T08:58:00.001-05:002017-12-30T11:49:49.032-05:00Two Types of Division<div class="separator" style="clear: both; text-align: center;">
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I am teaching dividing fractions now and wanted to be clear in my own mind about the differences between these two ways of thinking about and describing division problems. I have found myself switching between these two different ways of describing division problems and I really did not understand why. I made the graphic below to help me understand why some problems make more sense with partitive division (sharing) and others with quotative division (measurement). And how the language used to describe a problem changes how you would draw a picture of the problem (model) and the meaning of the quotient.</div>
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Partitive Division</h3>
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<li>Sharing Division</li>
<li>I know the number of groups</li>
<li>How many in each group?</li>
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Quotative Division</h3>
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<li>Measurement Division</li>
<li>I know how much will be in each group</li>
<li>How many groups?</li>
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I find quotative division (measurement division) very helpful when describing fraction division problems. For example, how many 1/2s are in 10 wholes? However, partitive division (sharing division) makes more sense when the dividend is a fraction or mixed number and the divisor is a whole number. For example, 4 3/5 lbs of candy is divided among 6 friends. How much candy does each friend get?</div>
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This <a href="https://cptv.pbslearningmedia.org/resource/mgbh.math.ns.twodiv/two-kinds-of-division/#.WkZXQt-nGUk">video</a> helped me understand the difference between these two different ways of thinking about division problems.</div>
<br />Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-8195587355450436279.post-77114731638041794382017-09-16T15:46:00.001-04:002017-09-16T15:57:17.291-04:00Factors and Multiples Puzzle from NRICH<div class="separator" style="clear: both; text-align: center;">
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This <a href="https://nrich.maths.org/5448">Factors and Multiples Puzzle</a> from NRICH was a perfect way to start the school year with my sixth grade students. It is one of the best low threshold high ceiling tasks I've used and creates opportunities for students to work together and discuss numbers and sets of numbers while they review essential vocabulary in the context of an interesting challenge.</div>
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Here are some things that I did the day before we worked on the puzzle that I think helped students get started right away and work on the puzzle for the entire class period.<br />
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1) I did notice and wonder with square numbers and triangular numbers using these graphics which I recreated based on inspiration from this <a href="https://www.tes.com/teaching-resource/triangular-square-and-cube-numbers-and-quadratic-sequence-posters-11297180">poster</a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhocsJP2-AQnrCbvNhcH2PlKYAD7Qx8bnYCT6tgc4khta6fA8K1qgmSkJt9glVwaqxtrCNGrRri1jApuKAT0EhLkO3yiTbe1ubjEYLDJ3YmK6XcU8_Bqyn5-zukPwAuT8-eg8_HlQhqhsPX/s1600/Square+Numbers+Graphic.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="242" data-original-width="854" height="112" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhocsJP2-AQnrCbvNhcH2PlKYAD7Qx8bnYCT6tgc4khta6fA8K1qgmSkJt9glVwaqxtrCNGrRri1jApuKAT0EhLkO3yiTbe1ubjEYLDJ3YmK6XcU8_Bqyn5-zukPwAuT8-eg8_HlQhqhsPX/s400/Square+Numbers+Graphic.PNG" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2T7iwo-XaiwB17jqf9HHuH3S7iYIzt8hONlZ24YaeSSeMIZLteaIJQv0lowSeQf-71X-V_VJSDvdTazCJOvDAp1mgfCj2jtKLrbCLmQcTtMNGBy_dApUIofXUGUHtmsgdYHf6hUIDE7PN/s1600/Triangular+Numbers+Graphic.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="188" data-original-width="891" height="83" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2T7iwo-XaiwB17jqf9HHuH3S7iYIzt8hONlZ24YaeSSeMIZLteaIJQv0lowSeQf-71X-V_VJSDvdTazCJOvDAp1mgfCj2jtKLrbCLmQcTtMNGBy_dApUIofXUGUHtmsgdYHf6hUIDE7PN/s400/Triangular+Numbers+Graphic.PNG" width="400" /></a></div>
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2) I did some mini-challenges where students, using small whiteboards, wrote down as many numbers as they could think of that meet two conditions. Students loved this!</div>
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<li>Think of numbers that are <b>Odd</b> and <b>Prime</b></li>
<li>Think of numbers that are <b>Even</b> and <b>Multiples of 5</b></li>
<li>Think of numbers that are <b>Less than 20</b> and <b>Factors of 60</b></li>
<li>Think of numbers that are <b>Square Numbers</b> and <b>Multiples of 3</b></li>
<li>Think of numbers that are <b>More than 20</b> and <b>Triangular Numbers</b></li>
<li>Think of numbers that are <b>Prime</b> and <b>Square Numbers</b></li>
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3) I think this is a very good activity for partners. I copied half of the heading cards and number cards on a different color of copy paper so that two sets of partners working at the same table would not mix up their cards. </div>
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4) It was very easy for the students to accidentally disturb some or all of their pieces which caused some frustration. Next year I plan to enlarge the puzzle and copy on card stock and/or laminate. Several different colors of heading and number cards would also be helpful.<br />
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5) When partners believed that they had completed the puzzle correctly, I asked another partnership to look at their puzzle to see if they could find any errors. This on the fly extension helped push the ceiling even higher for early finishers.<br />
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<a href="https://nrich.maths.org/content/id/5448/board.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="board" border="0" src="https://nrich.maths.org/content/id/5448/board.gif" /></a></div>
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<a href="https://nrich.maths.org/content/id/5448/heading%20cards.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="heading cards" border="0" src="https://nrich.maths.org/content/id/5448/heading%20cards.gif" /></a></div>
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<img alt="number cards" src="https://nrich.maths.org/content/id/5448/number%20cards.gif" /><br />
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Here are some photographs of completed puzzles from my classes. There is at least one error on each of these puzzles and I am thinking I could use these photos next year to help introduce the activity and/or use as an extension.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikEj5dZTd3nY0wX8VyFLeeTff0TINKWF1Du_Bp7KyqPURF-77RDpv2Qz8dnHtY2OWQFSAwbNuU-UmPYH2a6njpcmCG1be3d4guGKeNsgfbcAvMLay0pysaGMeJasZkqFroKRQzRzOnR41S/s1600/Puzzle+Solution+%25231.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="659" data-original-width="628" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikEj5dZTd3nY0wX8VyFLeeTff0TINKWF1Du_Bp7KyqPURF-77RDpv2Qz8dnHtY2OWQFSAwbNuU-UmPYH2a6njpcmCG1be3d4guGKeNsgfbcAvMLay0pysaGMeJasZkqFroKRQzRzOnR41S/s320/Puzzle+Solution+%25231.PNG" width="304" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibwP4kRKZtqMuluFiUS0KelKIMYRQnmaM7ZktHLaElPP4yXakeG3rh7i2hg909Xq7d9gi5Oa3X9JtXU8faQHkKw9VtPxoWhPvvvty7lKC_vma7zohw4IwjXpyrRX50Jo7lxyO879WGQw9I/s1600/Puzzle+Solution+%25232.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="497" data-original-width="468" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibwP4kRKZtqMuluFiUS0KelKIMYRQnmaM7ZktHLaElPP4yXakeG3rh7i2hg909Xq7d9gi5Oa3X9JtXU8faQHkKw9VtPxoWhPvvvty7lKC_vma7zohw4IwjXpyrRX50Jo7lxyO879WGQw9I/s320/Puzzle+Solution+%25232.PNG" width="301" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBs-iSVOdhWWpFadL7a7FEQBwiiD1dZXUQMIWLnu6l0tgzJ2alOJ1ANNxny4IIa8OPwV2RaIMzshRFggqObzTkglDOny18Poi9iVM1GE-k4q2TP3digz77rC9wnCNxCzSJlAILjgdz57oD/s1600/Puzzle+Solution+%25233.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="513" data-original-width="539" height="304" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBs-iSVOdhWWpFadL7a7FEQBwiiD1dZXUQMIWLnu6l0tgzJ2alOJ1ANNxny4IIa8OPwV2RaIMzshRFggqObzTkglDOny18Poi9iVM1GE-k4q2TP3digz77rC9wnCNxCzSJlAILjgdz57oD/s320/Puzzle+Solution+%25233.PNG" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQBdEKydQslypQ8a9UHR3lg_-yp6mYJ4sBBh9mlnwn9icym4jX4CbgvMx71gjJ4OLhkjupCJeH6AeYdH9pJlff1101vwwlJWQyYBKhESAs6_mfQtOv-IPixau2mATQg0r1bg8PDQ6PQGXJ/s1600/Puzzle+Solution+%25234.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="429" data-original-width="444" height="309" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQBdEKydQslypQ8a9UHR3lg_-yp6mYJ4sBBh9mlnwn9icym4jX4CbgvMx71gjJ4OLhkjupCJeH6AeYdH9pJlff1101vwwlJWQyYBKhESAs6_mfQtOv-IPixau2mATQg0r1bg8PDQ6PQGXJ/s320/Puzzle+Solution+%25234.PNG" width="320" /></a></div>
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This is a very cool chart that one of my students created on her own to help her logically and methodically determine where heading cards and number cards could or could not be placed in order to solve the puzzle.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiDqz0QbsmS9LvR7q4GeBRwWr6lHGfgPB4OwWFnWGQ5Ri18EbEKPyKdhrL_7Ty1-C8MxiKapQS7m2fjleMghJynHBDEHI_wScMHqZ0HyDTxDtTx6ISaMYz1_XikDXZSmMfPjj6k7_1Wbw-/s1600/Puzzle+Solution+Chart.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="909" data-original-width="1168" height="249" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiDqz0QbsmS9LvR7q4GeBRwWr6lHGfgPB4OwWFnWGQ5Ri18EbEKPyKdhrL_7Ty1-C8MxiKapQS7m2fjleMghJynHBDEHI_wScMHqZ0HyDTxDtTx6ISaMYz1_XikDXZSmMfPjj6k7_1Wbw-/s320/Puzzle+Solution+Chart.PNG" width="320" /></a></div>
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And here is a link to some <a href="https://nrich.maths.org/5448/solution">correct solutions</a> to the puzzle posted on the NRICH website.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-58568219639014879342017-08-20T18:20:00.001-04:002017-08-20T21:40:17.674-04:00Student Survey Questions: Middle School Edition<span style="font-family: "arial"; font-size: 14.6667px; white-space: pre-wrap;">Surveys are a great way for students to practice collecting data and then making sense of that data by create a graph to display the data and finding the measures of central tendency. Here is a list of survey questions that are appropriate and engaging for middle school students. Please share any ideas you have for survey questions and data displays and analysis in the comments!</span><span id="docs-internal-guid-8948843d-9fcb-cb43-c647-5d40d0c272aa"></span><br />
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<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">Here is a link to the document I created for this project. The survey questions below are included in the document for student to select from or get examples in order to write their own survey question.</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><a href="https://docs.google.com/document/d/1eqoZ2M8zRWyc3Ou4U5jWUJjknZyqNZyx0pf2ekCHBxU/edit?usp=sharing">Data and Statistics Project</a></span></span><br />
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<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">1) How many pets do you have?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">2) What time did you go to bed last night?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">3) How many hours of sleep did you get last night? (round to the nearest 1/2 hour)</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">4) How many letters are in your first and last name?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">5) How many books have you read this school year?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">6) How many siblings do you have? (include half-siblings and step brothers and sisters)</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">7) How many movies did you watch in the theater last month?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">8) How many times did you buy your lunch at school last week?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">9) What time did you wake up last Saturday morning?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">10) How many glasses of milk do you drink daily?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">11) What is your shoe size?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">12) How many states have you visited?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">13) How many texts do you send a day?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">14) How many pieces of candy to you eat on Halloween?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">15) If I gave you 34 hot wings right now, how many would you eat?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">16) How many times a week do you do chores?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">17) How many times a week do you eat fast food?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">18) How many digits of pi do you know?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">19) How many times do you watch TV in a week?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">20) How much money do you spend when you go to town?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">21) How many pairs of shoes do you have?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">22) What is your height?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">23) How long does it take you to get ready in the morning? (in minutes)</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">24) How many times a day do you brush your teeth?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">25) How many times a week do you go shopping?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">26) How many hours of sports do you play in a week?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">27) How many minutes do you spend a night on your homework?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">28) What time do you get up in the morning?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">29) How many times have you been to a different country?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">30) How many people do you consider to be your closest friends?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">31) How much money would you need to be given to drink sour milk?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">32) How many sports do you play?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">33) Flip a water bottle 10 times, how many times does it land upright?</span></span><br />
<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">34) How many types of soda do you drink?</span></span><br />
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<span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">I got some ideas for survey questions from f(t) <a href="http://function-of-time.blogspot.com/2012/05/lately.html">here</a>: </span></span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-9101889966330978622017-05-21T18:12:00.001-04:002017-05-21T18:12:57.684-04:00Area of Polygons Discovery Lesson on Dot Paper<div class="separator" style="clear: both; text-align: left;">
I created these pictures to give my students the opportunity to explore the areas of triangles, parallelograms and trapezoids in reference to the area of rectangles. I made copies of these two slides from my lesson presentation for the students to write on and they explored the areas of the shapes by drawing on the dot paper, counting square units, cutting off parts of the polygons and moving them to different places, and making conjectures about how the areas of the figures were related and why. This was a very engaging activity and the students used many different strategies to find the areas of these figures and these explorations formed the foundation for understanding the formulas for finding the areas of these figures. I was very pleased to see that students continued to use ideas from these pictorial models to help them with areas of polygons that were not displayed on dot paper (for example, triangle from left side of parallelogram is moved to the right side to form a rectangle with the same area).</div>
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The students had lots of interesting ideas for finding the area of Triangle F. I was amazed and thrilled when several students, connected the area of the obtuse triangle to the area of a paralleogram, instead of a rectangle, which was much easier for students in the class to see.<br />
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I use this slide to summarize the exploration. I highlighted part of each of the formulas in red to help students see the connection between all of the formulas: base x height. Even the formula for trapezoid can be understood in this way if we talk about finding the average base x height.</div>
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Here is a link to the google presentation, <a href="https://docs.google.com/presentation/d/1dAeTVBcXHgg2BqyKTc9_bNYGTytub2s2pHf0GZUeHuI/edit?usp=sharing">Area of Polygons Exploration</a>, that contains these slides and some others I used to discuss and summarize the formulas.<br />
<br />Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-47626186006891692242017-02-09T12:22:00.000-05:002017-02-09T12:22:50.424-05:00Cliff Jumping and Absolute ValueI love to show this <a href="http://www.redbull.com/en/adventure/stories/1331731012784/highdive-world-record">video</a> of Laso Schaller's world record jump of 58.8 meters in Maggia, Switzerland.<br />
I use it when we start our unit on absolute value and it sparks a nice discussion about the distance above and below zero. I show the video and then ask the students what they would want to know before they jumped off the cliff. They usually come up with some very interesting comments including how deep the water is and how far down you will go after you hit the water. Using these questions I guide the discussion towards the total distance traveled by drawing a diagram on the board with a vertical number line showing positive numbers above zero and negative numbers below zero. I use 192 ft (58.8 m) for the top of the cliff and -48 ft. for the depth of the water needed for the jump. I got these numbers by using the following information about the depth of water needed for a safe jump based on the height of the jump.<br />
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(Depending on time constraints, I could leave off the numeric expression and ask the students to figure out what the depth of the water must be for this jump.)</div>
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This discussion helps provide the mathematical motivation for absolute value and why positive numbers remain positive and negative numbers become positive when we are measuring distance. And then at this point I offer the mathematical notation and definition of absolute value.<br />
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Here are some still shots from the video:<br />
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The video replays the jump several times and shows different angles and information with each replay. For some of the replays, additional information about the jump overlays the video including the height of the top of the cliff as it is compared to the Leaning Tower of Pisa and also a time lapse of the speed and time of the Laso Schaller throughout the jump.<br />
<br />Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-31788099908536601322017-01-29T12:31:00.002-05:002017-01-29T12:39:09.623-05:00A good problem is worth a thousand worksheets<br />
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I got this <a href="http://www.openmiddle.com/dividing-fractions/">problem</a> from openmiddle.com.<br />
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All of my students worked on this problem very enthusiastically for most of the class period. It was truly a low floor high ceiling task because every student in the class was dividing fractions (practicing that skill) and noticing that the quotient was either smaller or larger than previous attempts. Some students used guess and check and were thrilled when they found larger quotients. Other students were more systematice and tried the digits 1,2,3 and 4 and found the largest quotient with these numbers. Others tried 6, 7, 8 and 9 and found the largest quotient with these numbers. And other students noticed that they should use 1, 2, 8 and 9. There was a lot of discussion in the room and students were comparing and discussing their results. Some students noticed that there was more than one way to get a quotient of 36 (or 1/36). And many students were starting to notice and generalize that larger quotients result from larger dividends and/or smaller divisors.<br />
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Low floor high ceiling tasks are also sometimes called low threshhold high ceiling or low entry high <span style="font-family: inherit;">ceiling tasks. <a href="http://nrich.maths.org/7701">NRICH</a> describes a low threshhold high ceiling task like this:</span><br />
<span style="font-family: inherit;">"A LTHC mathematical activity is one which pretty well everyone in the group can begin, and then work on at their own level of engagement, but which has lots of possibilities for the participatns to do much more challenging mathematics."</span><br />
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<span style="font-family: inherit;">I am looking forward to trying <a href="http://www.openmiddle.com/finding-equivalent-ratios/">this problem</a> with my students now that we are working on ratios:</span><br />
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<span style="background-color: white; font-size: 16px;"><span style="font-family: inherit;"><br /></span></span>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-67739318676726025282016-12-31T19:43:00.003-05:002017-01-01T11:56:58.909-05:003D Shapes Exploration<span style="font-family: "arial" , "helvetica" , sans-serif;">I like to set up an exploration of 3-D shapes at the beginning of the unit on volume and surface area. Last spring I set up six stations using the outdoor courtyard of our school and the students enjoyed hands on explorations of 3-D shapes in the sunshine. It was wonderful.</span><br />
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</span> <span style="font-family: "arial" , "helvetica" , sans-serif;">1) Cutting, Coloring and Folding Paper Nets of 3-D Shapes</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">2) Building 3-D Shapes with Magnetix</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">3) Building 3-D Shapes with Zome Tooels (and bubbles!)</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">4) Drawing Nets and 3-D Shapes with Sidewalk Chalk</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">5) Building 3-D Shapes with Toothpicks and Marshmallows</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">6) Exploring with 3-D Models and Removable Nets</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">I also created this graphic organizer that I ask the students to fill out as they work at the stations and explore six 3-D shapes. I ask the students to focus on filling out the first six columns of the graphic organizer during the eploration and then reference it during subsequent lessons on volume and surface area. This is a good introduction to the vocubulary of three dimensional solids. Graphic organizer </span><span style="font-family: "arial" , "helvetica" , sans-serif;">can be downloaded </span><a href="https://docs.google.com/document/d/1snriZpHVqvKFSsP-yogSqrLu8F16sk3pt8eybzfOAE8/edit?usp=sharing" style="font-family: arial, helvetica, sans-serif;">here</a><span style="font-family: "arial" , "helvetica" , sans-serif;">.</span><br />
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</span>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-29393934868320741352016-11-12T22:29:00.001-05:002016-11-12T22:30:00.158-05:00Multiplying Mixed Numbers Area ModelThis area model created some rich mathematical discussions in my sixth grade class<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtbJWfO1aXsvEUmpntS8M6Oco2D3RQBNjKNZOBXkOnrRl3bmPKdVYzbShyphenhyphenAvMqSetAxcZUOHg_yrUf8j4tkzqDGD1ng5LeZZWyP5t6PqV_0KALRqyt20NvAKPOTAYpvtiVPwMlgRV3ELDn/s1600/Multiplying+Mixed+Numbers+Area+Model.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="305" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtbJWfO1aXsvEUmpntS8M6Oco2D3RQBNjKNZOBXkOnrRl3bmPKdVYzbShyphenhyphenAvMqSetAxcZUOHg_yrUf8j4tkzqDGD1ng5LeZZWyP5t6PqV_0KALRqyt20NvAKPOTAYpvtiVPwMlgRV3ELDn/s400/Multiplying+Mixed+Numbers+Area+Model.PNG" width="400" /></a></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-39588380079216860122016-10-03T00:17:00.001-04:002016-10-03T00:27:03.509-04:00Mistakes are the Portals of Discovery<h2>
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<b style="font-size: xx-large;">Mistakes are the Portals of Discovery</b></div>
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<b><span style="font-size: large;">--James Joyce</span></b></div>
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<a href="https://i.ytimg.com/vi/160GmnKOeps/maxresdefault.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Image result for portal" border="0" height="225" src="https://i.ytimg.com/vi/160GmnKOeps/maxresdefault.jpg" width="400" /></a></div>
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I found this quote last summer when I was thinking about and preparing lessons for the first week of school. I knew that I wanted to spend some time talking about growth mindset with my students and emphasize the value of mistakes in the learning process. I don't want my students to be afraid of making mistakes. Instead, I want them to take risks and celebrate mistakes and use them as opportunities to learn new things. I love the words portal and discovery in this quote. The definition of portal is "a doorway, gate, or other entrance, especially a large and elaborate one." I love the idea of my students passing through a portal and discovering new knowledge and insight by making mistakes.<br />
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I liked this quote so much that I decided to finally become a bona fide teacher and order my first gross of custom printed pencils:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiauBx3RfwTEm5nNX956sxPgxVcl5ZkpxL-o9ticqoENnmlC1NRUongXYEcEmK2AX6m3XPMFQLclKteXpojnqk740SeTQlx_hW87ogzpuM7kTAG5lLz9gxlru3odbuU_JWpw6dJhTb3gX9-/s1600/Pencil.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="27" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiauBx3RfwTEm5nNX956sxPgxVcl5ZkpxL-o9ticqoENnmlC1NRUongXYEcEmK2AX6m3XPMFQLclKteXpojnqk740SeTQlx_hW87ogzpuM7kTAG5lLz9gxlru3odbuU_JWpw6dJhTb3gX9-/s400/Pencil.PNG" width="400" /></a></div>
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I had to change the quote slightly because 36 characters and spaces are the maximum allowed on a custom printed pencil from <a href="http://www.forteachersonly.com/">ForTeachersOnly</a>. I also ordered some brain erasers to give out with the pencils throughout the year to recognize students who learn from mistakes and exhibit a growth mindset.<br />
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In class, after we talked about the differences between fixed and growth mindsets, students wrote growth mindset quotes and statements on lightbulbs, speech bubbles, neon starbursts and thought bubbles to decorate a mural with our classroom slogan.</div>
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<span style="font-size: x-large;">Our Classroom Mural:</span></h2>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4YQrkJ1FdE3VcSZxVb8eHJqQ6iJJJ3IofwV8HtYrDeh0FBQy8aRhzVsHLft-yaRRkZG_ruQhpuJ3EYak9BF1um8FIc6z2vIHxoo5kfW4Pkcsc2aRdXzuXsC68AXI9VkDpD7yUDeBgBKtN/s1600/Classroom+Mural.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="51" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4YQrkJ1FdE3VcSZxVb8eHJqQ6iJJJ3IofwV8HtYrDeh0FBQy8aRhzVsHLft-yaRRkZG_ruQhpuJ3EYak9BF1um8FIc6z2vIHxoo5kfW4Pkcsc2aRdXzuXsC68AXI9VkDpD7yUDeBgBKtN/s400/Classroom+Mural.jpeg" width="400" /></a></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-10323736913105849102016-08-16T11:25:00.000-04:002017-01-09T17:26:20.930-05:00Rate Exploration ActivityIt was the first day back after the holiday break and I wanted to plan something active for my students to do. We were in the middle of the chapter on rates and so I was looking for an activity where student timed themselves doing different activities and then converted these rates to different time periods. I found this activity on NCTM Illuminations: <a href="https://illuminations.nctm.org/Lesson.aspx?id=1660">What's Your Rate</a>?<br />
Other than that, I wasn't able to find much online. I decided to take the worksheet from NCTM Illuminations and add a few more activities and add some more challenging conversion questions.<br />
Here is what I came up with:<br />
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As I worked on the tasks and questions for this activity, I realized that this would be an excellent anchor tasks to introduce the concept of rates. In the past I have used a YouTube video called the <a href="https://www.youtube.com/watch?v=wiw7hCJfSRM">The World's Fastest Everything - 2013</a>. The students really enjoy the video, but I think they will get more out of this activity. Students have an intuitive sense of rates and I think it is important to tap into their prior knowledge about rates by exploring activities they have experience with. I love how this activity makes it clear that two things are being measured simultaneously and that those two things together create a rate. I really enjoyed reading this blog posts about rates (<a href="http://function-of-time.blogspot.com/2016/01/what-i-think-rate-is-right-now.html">What I think a rate is now</a> from f(t).Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-60386429316778691212016-08-03T14:44:00.000-04:002016-08-03T14:44:40.361-04:00Look What I Found at Dollar Tree!I found these 3-D Geometric Shapes at Dollar Tree yesterday! I plan to use them during the first week of school to have students create a growth mindset classroom display! I will give each student a face to color and write a growth message on!<br />
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Dodecahedron: 12 faces (pentagons)<br />
Icosahedron: 20 faces (triangles)<br />
Rhombicuboctahedron has: 26 faces (8 triangles and 18 squares)<br />
Small Stellated Dodecahedron: 60 faces (triangles)Unknownnoreply@blogger.com3tag:blogger.com,1999:blog-8195587355450436279.post-31161953611662363562016-07-05T13:07:00.001-04:002016-07-05T17:39:40.017-04:00Coordinate Plane Graphing Pictures Hallway DisplayOne of the teachers I most admire gave me some advice when I was a new teacher. I asked him what is the most important thing a teacher can do to be effective and he said that he gets the best work and most significant learning from his students when he asks them to create something that they will share with other people.<br />
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After learning about negative numbers, my students were ready to learn how to graph ordered pairs in all four quadrants of the Coordinate Plane. After some practice with this skill, I gave the students a list of designs they could create on large pieces of graph paper by graphing points and connecting the dots. I got permission to hang them in the hallway leading into the library media center which allowed all the students and staff in the building to see their work. I think the students learned more about graphing points knowing the final product would be on public display. They worked harder on the designs, found and corrected any points that were graphed incorrectly, and worked collaboratively in small groups to create designs that they were proud to display for the whole school to see.<br />
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Many of the pictures came from graphing instructions available on <a href="http://bellevillebulldogs.tripod.com/id21.html">Mr. Colli's website</a>.<br />
This is the best deal I have found for <a href="http://www.amazon.com/gp/product/B000ON9WZM?psc=1&redirect=true&ref_=oh_aui_detailpage_o03_s00">large graph paper with grid lines</a> on Amazon (TOPS Standard Easel Pads, 3-Hold Punched, 27 x 34 Inch, 1" Grid, White, 50 Sheets/Pad, Cart of 4 Pads (7900))<br />
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NOTE: I have found that most designs will not fit on this graph paper unless you number the grid lines by 2s (. . . -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10 . . .).Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-68602373157548101852016-02-08T17:29:00.001-05:002016-02-16T23:10:16.979-05:00Grocery Store ChallengeThe middle school math curriculum is loaded with real world problems that adults use daily in their jobs and lives. One example is unit cost comparisons. When we study rates, my students learn how to calculate the unit cost of an item and compare it to similar items by comparing the unit costs and deciding which item is the best deal and how much money they can save by purchasing the item with the lowest unit cost. Grocery stores offer helpful information to people hoping to save money when they shop. In order to help my students make mathematical connections to real world problems, I challenge them to find a price tag at the local grocery store showing the highest unit cost in the store. They are amazed to find that some items at the store cost well over $1,000 per lb. I make it a contest and ask the students to take a picture of the item and its accompanying price and unit cost sticker as displayed at the store and send it to me. I thought no one would ever find an item with a higher unit cost than saffron at $11,354.67 per lb.<br />
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However, this year a student found an item in the cosmetics aisle with a higher unit cost!</div>
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<a href="http://ecx.images-amazon.com/images/I/61WoCath-aL._SX522_.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="CoverGirl 230 Ink It By Perfect Point Plus Eyeliner, Black Ink, 0.006 Ounce" border="0" src="http://ecx.images-amazon.com/images/I/61WoCath-aL._SX522_.jpg" /></a></div>
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<span style="background-color: white; color: #111111; font-family: "arial" , sans-serif; font-size: 20px; line-height: 26px;">Cover Girl Ink It By Perfect Point Plus Eyeliner</span></div>
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I also like to do a Unit Cost Carnival in class where students bring in two items and set up a "booth" with the items displayed and information about the cost and weight of the items. Students circulate around the classroom and at each booth they calculate the unit cost of the two (or more) items and use that information to figure out which item is a better deal.</div>
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Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-8195587355450436279.post-55106449899825654502015-12-21T21:18:00.003-05:002015-12-21T21:18:19.100-05:00Holiday Fractal Design Project<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuF3IIM7MiwXbWsgea4xtKnZkg9hJT0JnQnYixCK05P2NqjkbLqBUgQhELPtVDqZP4JstoaWp2QZoJS-8hZQHyz2-_IVBPiZmhxRYF29GBTpu7VQjd2xj3X9ba_-lvUOB8h8y7S8DlAwC6/s1600/Fractal+Project+7.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="395" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuF3IIM7MiwXbWsgea4xtKnZkg9hJT0JnQnYixCK05P2NqjkbLqBUgQhELPtVDqZP4JstoaWp2QZoJS-8hZQHyz2-_IVBPiZmhxRYF29GBTpu7VQjd2xj3X9ba_-lvUOB8h8y7S8DlAwC6/s400/Fractal+Project+7.PNG" width="400" /></a></div>
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"A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales.They are created by repeating a simple process over and over in an ongoing feedback loop." This definition of fractals comes from the <a href="http://fractalfoundation.org/resources/what-are-fractals/">Fractal Foundation</a>.<br />
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This is one of my favorite math activities to do before the holiday break. I like this simple project because it gives students a hands on experience with self-similar pattern making and also after completing only 2 recursions, the students already get a very strong sense of how dramatically the scale has changed.<br />
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This project is open-ended enough to allow for colors and designs from many different cultures and the students enjoy coming up with their own designs. And I love the calm mindful work of children coloring their repeating patterns and experiencing mathematics.<br />
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Here are the steps for the project with illustrations below:<br />
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1) Cut out a 27x27 grid from graph paper and draw lines making 3 rows and 3 columns so that each square is a 9x9 grid<br />
2) Draw a simple, colorful design for the four corner squares and the middle square.<br />
3) With the four remaining 9x9 squares, draw lines making 3 rows and columns so that each square is a 3x3 grid.<br />
4) Shrink the five original designs by 1/9 and draw them in the five 3x3 squares that correspond to their original locations in the 27x27 grid.<br />
5) In the four remaining 3x3 squares, draw lines making 3 rows and columns where each square is a 1x1 grid.<br />
6) Shrink the five original designs by 1/9 again and draw them in the five 1x1 squares that correspond to their locations in the original 27x27 grid (and also in the 9x9 grids).<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZGE9LgeW9x1Pq489MkUDf5TIQPu9MFGe1lJAhtxB30e7i3PZJceluwercTXgZJlScsOeQHLgeS2socB_B4_wtjAUtQb-MI_Z3gbItmfCGI1NICyTNMrDkpJ63cmrRLHz64S_oIF-cwLtk/s1600/Fractal+Project+1.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="387" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZGE9LgeW9x1Pq489MkUDf5TIQPu9MFGe1lJAhtxB30e7i3PZJceluwercTXgZJlScsOeQHLgeS2socB_B4_wtjAUtQb-MI_Z3gbItmfCGI1NICyTNMrDkpJ63cmrRLHz64S_oIF-cwLtk/s400/Fractal+Project+1.PNG" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAYkbzi0wr37j3mS4ZxxiN9SAH_rAEQOl8awnfgcJKQN06WLbfyC6GG4_9It75_AGtY5bUmmLAvHKHZczkCoQVyziT32Qd4nr7jScTg2IGpjN3dn0k0wjej2Fja4oKFU_bB4ai5Ly7PMXI/s1600/Fractal+Project+2.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="391" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAYkbzi0wr37j3mS4ZxxiN9SAH_rAEQOl8awnfgcJKQN06WLbfyC6GG4_9It75_AGtY5bUmmLAvHKHZczkCoQVyziT32Qd4nr7jScTg2IGpjN3dn0k0wjej2Fja4oKFU_bB4ai5Ly7PMXI/s400/Fractal+Project+2.PNG" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBvHyIv5eWaDMMz1dlbwgOw27XIYJMo0o63fCKjHOSIfYa-xOMptB8FvWVcNV0iMzRAM_RHQqr2n0YFVrwyuTiyBDiruC-pHCegwb0oUkGAqXuqKIo-9S5aGAZiN_6e3V1A7ue8U0lRGFm/s1600/Fractal+Project+5.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="385" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBvHyIv5eWaDMMz1dlbwgOw27XIYJMo0o63fCKjHOSIfYa-xOMptB8FvWVcNV0iMzRAM_RHQqr2n0YFVrwyuTiyBDiruC-pHCegwb0oUkGAqXuqKIo-9S5aGAZiN_6e3V1A7ue8U0lRGFm/s400/Fractal+Project+5.PNG" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDO5DnCWIsIo4Jw6rTFcavr4WGPKefNAwz39RwqovkvJO7HabvQspVYlM2pauAC5DARQtQb62zi5pT8SqQKoky05VmziDDQNgjrfihSpHNZTN66BQbiyytB8c7epu-ZcHyWP2CNMx7YrVo/s1600/Fractal+Project+6.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="390" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDO5DnCWIsIo4Jw6rTFcavr4WGPKefNAwz39RwqovkvJO7HabvQspVYlM2pauAC5DARQtQb62zi5pT8SqQKoky05VmziDDQNgjrfihSpHNZTN66BQbiyytB8c7epu-ZcHyWP2CNMx7YrVo/s400/Fractal+Project+6.PNG" width="400" /></a></div>
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Below are more designs that were completed by students.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRUEso8lxQeQ6_GvwvdK6432Z5Jw47BXDAItMx3DHw1mFwUbOqDy1Nh0xqIQBRKV118zbwmo46hL7WmNKKG05kVY-kGGTKha6AAptfL68YslBmjnspvyWgplMGU9gaBz80Y3Y94_lP5N27/s1600/Fractal+Project+8.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="396" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRUEso8lxQeQ6_GvwvdK6432Z5Jw47BXDAItMx3DHw1mFwUbOqDy1Nh0xqIQBRKV118zbwmo46hL7WmNKKG05kVY-kGGTKha6AAptfL68YslBmjnspvyWgplMGU9gaBz80Y3Y94_lP5N27/s400/Fractal+Project+8.PNG" width="400" /></a></div>
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<br />Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-44774693532105325562015-11-11T10:04:00.001-05:002017-08-20T17:59:18.106-04:00Anchor Task Graphic Organizer<div class="separator" style="clear: both; text-align: center;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPlNrnANQJA4EfsqvyR3GxR2elNvWoYUOnfHjmLSBcS1EedM9IUW0gVYKe29MXT3cFAHmai9pKjzCKeGQXukbhG9dBWIxWeOKBmaRIJr922IgPfNzKePfbBgtrArQoVDowW5JF_jd01ang/s1600/George+Polya+Quote.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="682" data-original-width="1287" height="169" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPlNrnANQJA4EfsqvyR3GxR2elNvWoYUOnfHjmLSBcS1EedM9IUW0gVYKe29MXT3cFAHmai9pKjzCKeGQXukbhG9dBWIxWeOKBmaRIJr922IgPfNzKePfbBgtrArQoVDowW5JF_jd01ang/s320/George+Polya+Quote.PNG" width="320" /></a></div>
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Recently I made a poster of this quote from George Polya and displayed it in the front of my classroom. To go along with this, I created the graphic organizer below and used it to introduce anchor tasks for a new topic. The students wrote the problem in the middle and then tried to figure out four ways to represent and solve the problem. This is new for my students and so I have been giving them prompts for each of the rectangles like: draw a model, write a word problem, explain in words, computation, etc.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaeey_WLtYfDTpzsRq1QOBfKGTZ0mC-5m3xmNcWiJiaJKIAdcZF_17DL5Fb0Ko8VFyEk_f6Ogszg20rZ7a7WFTxMToQP2n1Q2Kzosym71oAFENgFqulhqNFSrEN3id46MflkKBHwPNKXko/s1600/Graphic+Organizer+Snip.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="260" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaeey_WLtYfDTpzsRq1QOBfKGTZ0mC-5m3xmNcWiJiaJKIAdcZF_17DL5Fb0Ko8VFyEk_f6Ogszg20rZ7a7WFTxMToQP2n1Q2Kzosym71oAFENgFqulhqNFSrEN3id46MflkKBHwPNKXko/s400/Graphic+Organizer+Snip.PNG" width="400" /></a></div>
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<a href="https://docs.google.com/document/d/1FJlvB1K8zUO7LlvMHALn81SG2cpPgBrwY0h77xX3_kQ/edit?usp=sharing">Anchor Task Graphic Organizer (Full Page)</a><br />
<a href="https://docs.google.com/document/d/1H46KmSeCDJXo0zC27igd79boy7SHhydbmjfWcFddIkQ/edit?usp=sharing">Anchor Task Graphic Organizer (Half Page)</a><br />
<a href="https://docs.google.com/presentation/d/1DzodjCf6cqLElDr4UFNJCNat14WUHZujUqtadWW_wAk/edit?usp=sharing">Anchor Task Graphic Organizer</a> (in Google Slides)<br />
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Shown below are examples of student work on three anchor tasks:<br />
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1) Dividing Fractions (2 divided by 1/4)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvxe80pFzeKDqx3yuEt5hhRDZ_vw_XfsfMZN0pH9iMwxy5pBYuChxrYA6Xj33Vh7am66xDpceHa4so4yB9v0CdHBydO72rtJ5FLb3WsWp8VXB1QEW4MfuaQANoI9BjBrW5JAH_kprftl6i/s1600/Fraction+Division+Student+Work+1.bmp" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="285" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvxe80pFzeKDqx3yuEt5hhRDZ_vw_XfsfMZN0pH9iMwxy5pBYuChxrYA6Xj33Vh7am66xDpceHa4so4yB9v0CdHBydO72rtJ5FLb3WsWp8VXB1QEW4MfuaQANoI9BjBrW5JAH_kprftl6i/s400/Fraction+Division+Student+Work+1.bmp" width="400" /></a></div>
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<span style="text-align: left;">I love the exploration that this student did to figure out that as the divisor decreases, the quotient increases.</span></div>
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This student was exploring many ways to do the computation.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0Lhn2iQbf-peKSlFU_85S_cb1Krp5uwr60bUwWzhFxhUu9OHA2BKN0apopGvPHjgZIdtQamtz8yXq9YLsk8gP8j1po3KAsC5jmnfrn5PXvPDnUyesqhwX0q-600FH3tfBg1cLuQ8w-sa0/s1600/Fraction+Division+Student+Work+3.bmp" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="298" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0Lhn2iQbf-peKSlFU_85S_cb1Krp5uwr60bUwWzhFxhUu9OHA2BKN0apopGvPHjgZIdtQamtz8yXq9YLsk8gP8j1po3KAsC5jmnfrn5PXvPDnUyesqhwX0q-600FH3tfBg1cLuQ8w-sa0/s400/Fraction+Division+Student+Work+3.bmp" width="400" /></a></div>
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This student added some ideas to their graphic organizer from another student in the group and gave them credit. I also like how this student wrote out their thinking about how the quotient changes with different divisors.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhm4FCN4pg5lakecsSti7s_u1VJ9dOBk6Hunf0c7fLEz2wx5HycM4Av_Vd7WqAAM8dSmVxxamAowTdFfrbu-fmyJNlKPLoVzJaBbcM178qrIvp3sfHS-WmZu5ZAlpFY6G0oPZcaCQYDYAep/s1600/Fraction+Division+Student+Work+4.bmp" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="285" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhm4FCN4pg5lakecsSti7s_u1VJ9dOBk6Hunf0c7fLEz2wx5HycM4Av_Vd7WqAAM8dSmVxxamAowTdFfrbu-fmyJNlKPLoVzJaBbcM178qrIvp3sfHS-WmZu5ZAlpFY6G0oPZcaCQYDYAep/s400/Fraction+Division+Student+Work+4.bmp" width="400" /></a></div>
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This student shows a strong understanding of the meaning of reciprocal by explaining that since 4 1/4s go into 1, then you simply multiply by the reciprocal to find out how many 1/4s go into 2.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmKlasIZAxny4GazFlEYtqwxiFVZQoHbiB2LU-X90y5yeZyqpaCL0gu4c6CZjsSijZQh-KJBuGOeo3xImHUl26UIC7xzJ8_ytOIhyphenhyphenhRsYX4UWkPlSGONVczwPcN6epV5i_SmdVI89nPiV0/s1600/Fraction+Division+Student+Work+5.bmp" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="306" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmKlasIZAxny4GazFlEYtqwxiFVZQoHbiB2LU-X90y5yeZyqpaCL0gu4c6CZjsSijZQh-KJBuGOeo3xImHUl26UIC7xzJ8_ytOIhyphenhyphenhRsYX4UWkPlSGONVczwPcN6epV5i_SmdVI89nPiV0/s400/Fraction+Division+Student+Work+5.bmp" width="400" /></a></div>
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This student tried so many ways to understand fraction division (models, decimals, percents, etc). What I was most impressed by was the reflection that occurred throughout the class period during small group and whole group discussion and is shown clearly in the work. Student wrote "changed my thinking after discussion" and then indicated on the graphic organizer where he needed to make revisions.</div>
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2) Absolute Value (Distance from zero--Find the total distance to touch the rock)<br />
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3) Multiplying Decimals (4 x 0.3 and 0.3 x 0.5)<br />
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<br />Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-8195587355450436279.post-89791916066519367182015-10-05T00:34:00.001-04:002015-10-05T00:36:36.375-04:00Teaching Effective Math Lessons<div style="text-align: center;">
Teaching Effective Math Lessons</div>
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by Dr. Yeap Ban Har</div>
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I was very fortunate to have the opportunity to attend a district math training with Dr. Yeap Ban Har. Here is a summary of some of the things that I learned:</div>
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1) Anchor Tasks are used at the beginning of the lesson to engage students in the new learning for the day. An anchor task should be an interesting and rich problem that allows students to explore the mathematical concepts and ideas that are the most integral part of the lesson. This part of the class period should last about 15 minutes which gives students the processing time needed to allow for new learning. </div>
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2) Dr. Yeap encourages his students to find as many ways as possible to solve math problems. He tells his students that it is a clever day when they are able to find more than two ways to solve a problem. He rewards his students when they have 20 or more clever days in one month.</div>
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<img height="170" src="https://s-media-cache-ak0.pinimg.com/236x/8e/a0/98/8ea098d6c0b12459b8619a3a7453e6ba.jpg" width="320" /></div>
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3) Anchor Task Example: Instead of how can we solve x + 6 = 9, Dr. Yeap said:</div>
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<li>Can you guess what number I am thinking of?</li>
<li>If I increase that number by six then the result will be nine.</li>
<li>Can we figure out this number?</li>
<li>How can we show it mathematically?</li>
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<li>Equation</li>
<li>Guess and Check (substitution)</li>
<li>Balance Scale</li>
<li>Mental Math</li>
<li>Number Line</li>
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4) Students are encouraged to write in their journals throughout the lesson to work out their mathematical thinking and at the end of the anchor task, students are asked to: </div>
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<li>Explain in their own words the problem they were solving</li>
<li>Show one method to solve it and</li>
<li>Give the answer to the problem.</li>
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<img alt="Image result for composition books" 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" 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5) Dr. Yeap reads his students' journals frequently. He has a system where he is able to read all of his students' math journals once a week. He has two stamps for checking and commenting in the journals:</div>
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<ul>
<li>I like this journal entry because it is creative</li>
<li>I like this journal entry because it shows initiative.</li>
</ul>
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6) He used a very large pencil pointer. He uses this in his classes to constantly remind the students to WRITE in their journals!<br />
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<a href="http://dovanusalis.lt/image/data/Dovanu%20salis/Prekiu_katalogas/Gigantiskas_piestukas/gigantiskas-piestukas-4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://dovanusalis.lt/image/data/Dovanu%20salis/Prekiu_katalogas/Gigantiskas_piestukas/gigantiskas-piestukas-4.jpg" height="200" style="text-align: center;" width="200" /></a></div>
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7) Questions Dr. Yeap asks throughout the lesson:</div>
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<ul>
<li>Can you imagine that? (He uses this question frequently in his teaching which encourages the students to see the problem in their minds.)</li>
<li>Does anyone want to challenge this?</li>
<li>Do you think this method works all of the time?</li>
<li>Do you accept that?</li>
<li>Is my diagram reasonable?</li>
<li>How is it the same? How is it different?</li>
<li>Do you notice anything?</li>
</ul>
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8) Theories of Piaget and Vygotsky were mentioned frequently:</div>
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<a href="https://s-media-cache-ak0.pinimg.com/736x/f5/10/27/f51027ea7496b053946863f94721c519.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Multiplicity in Materials: Reggio Inspiration and the Knowledge of Jean Piaget: " border="0" height="320" src="https://s-media-cache-ak0.pinimg.com/736x/f5/10/27/f51027ea7496b053946863f94721c519.jpg" width="212" /></a><img height="320" src="https://s-media-cache-ak0.pinimg.com/236x/28/a0/ff/28a0ff532d3e514dd5ab659c95953703.jpg" width="227" /></div>
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<img alt="Vygotsky.gif" src="http://edpsychtemple.wikispaces.com/file/view/Vygotsky.gif/202166934/491x370/Vygotsky.gif" height="240" width="320" /></div>
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9) Differentiation: When students finish their work early he has three possible assignments for them:</div>
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<ul>
<li>Write a story for the equation or problem that was solved</li>
<li>Write a note in their journal explaining how to solve the problem that they worked on class to a student that was absent. He will then copy the note and give it to the absent student upon their return to class.</li>
<li>Can you come up with a new method, an original method, one that no one on earth has ever used before.? If so, you can name it after yourself!</li>
</ul>
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10) Homework: Pick five problems that you don't think that you can solve (from the exercises page at the end of the section) and then do those five for homework. If they are all too easy, write a paragraph about why they are all too easy and then write three equations that are difficult for you and solve them.</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-23024254359475667192015-08-22T22:17:00.001-04:002015-08-22T22:17:38.768-04:00Math App ReviewsThis year I have some bulletin board space available for students to post reviews of math apps and website that they have tried. I created the following <a href="https://drive.google.com/file/d/0B8hsPQ_IEyPIMWZsb05RQk45cEU/view?usp=sharing">template</a> for them to write and post their reviews in the classroom.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvEfYTD80XkCsGRKL2ixMc9pqJTx8SHeuRmc17luteaRD9EBegxl4-52n1-dAGfNiMR6X9GvPqHAVABGwIWZlgvI9z4NY7-qdjsvLtvtoMDwCd8IM92eml1QtpeW_3EVfTT5JcRLQt4tUQ/s1600/Math+App+Review.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvEfYTD80XkCsGRKL2ixMc9pqJTx8SHeuRmc17luteaRD9EBegxl4-52n1-dAGfNiMR6X9GvPqHAVABGwIWZlgvI9z4NY7-qdjsvLtvtoMDwCd8IM92eml1QtpeW_3EVfTT5JcRLQt4tUQ/s640/Math+App+Review.PNG" width="401" /></a></div>
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There are so many apps and website for students to practice procedural mathematics. I am hoping that this opportunity will encourage students to try new apps and websites and share the ones they like with their classmates.Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-25138629113433054322015-08-12T17:24:00.002-04:002015-08-12T17:27:32.146-04:00First Day of School Handshake!On the first day of school my two goals are to make every child feel welcome and excited about learning math in my classroom. In the book <i>The First Days of School: How to be an Effective Teacher</i>, Dr. Wong recommends standing at the door and shaking each child's hand as they enter the classroom. So I stand outside Room 108, look each student in the eye and give them a firm handshake and a big smile as they come in and join the class.<br />
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And then once the students are seated, I want to generate some enthusiasm for mathematical problem solving by giving my students an opportunity to do some interesting math on the very first day of school by posing an intriguing and mathematically rich problem for the students to consider and solve. The handshake problem is a fun way to combine these two first day of school goals--it goes nicely with the idea of meeting and greeting new people, it is pretty easy to set up and explain the problem, it is accessible for most students, there are multiple ways to solve this problem and it contains some intriguing and elegant mathematical patterns. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTqaKLS-biJ3GHlPg-k1BYnV2SiKwjEz8o_7q49_kpcn3Cjq-QPtItbUleNjbEEe9hfpwug62rkLuCunFeq5A32pW1g3gLRKcYBXKkPdbfoyJkskEjX5EsVxkh24RmCKk94eoqOiHMRkyw/s1600/Handshakes.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="223" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTqaKLS-biJ3GHlPg-k1BYnV2SiKwjEz8o_7q49_kpcn3Cjq-QPtItbUleNjbEEe9hfpwug62rkLuCunFeq5A32pW1g3gLRKcYBXKkPdbfoyJkskEjX5EsVxkh24RmCKk94eoqOiHMRkyw/s400/Handshakes.PNG" width="400" /></a></div>
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After I shake every student's hand at the door, I ask the students to get up and shake the hand of every other person in the room. I play James Taylor's version of <i><a href="https://www.youtube.com/watch?v=j7E8dC2g_XA">Getting Know You</a></i> while students walk around the room shaking each others' hands. <a href="https://www.youtube.com/watch?v=ZN3E9Yyd1HA"><i>Thank You For Being a Friend</i></a> would also be good background music to play. Then I ask "I wonder how many handshakes it took for every person in this room to shake hands with each person in the room."</div>
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Here are some ideas I would like to try this year to improve my presentation of the problem and to get students more engaged in the problem by asking them what they notice and wonder about this activity. I will have a slide with a photo of each child on the screen and I will get the discussion started by asking several students if they shook hands with all the students in the room. I could also ask a child if they shook hands with another particular child and I will start to draw lines on the slide connecting any two students that acknowledge that they did shake hands with each other. This would be a great time to discuss the question about how we count handshakes: if two people shake hands does this count as one handshake or two handshakes? After I have drawn several lines connecting various students in the classroom, I will ask some general questions to get the students thinking about this problem: What are you wondering about? What mathematical questions could we ask about this activity? What is the problem? What do you find interesting about this problem? What do you notice about this problem?, etc. Hopefully through this discussion, the students will come up with the mathematical question posed by the <a href="http://illuminations.nctm.org/lesson.aspx?id=2112">classic handshake problem</a>: "if you have a room full of people and everyone shakes hands, how many total handshake are there?" and suggest some possible ways to think about and solve this problem. Then I will give the students some time to discuss this problem in small groups. Other possibilities for class discussion, either before or after the students work on the problem in small groups, could include showing a slide of all the students, but this time the photos would be shown in a circle instead of in rows. In addition, I could show a slide with one student and ask how many handshakes?, two students: how many handshakes?, three students: how many handshakes?, etc. and then discuss possible patterns that emerge.</div>
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I plan to use this problem for the first Problem of the Week (POTW) for the 2015-2016 school year:) And I am hoping that it will get the class off to a good start in terms of working on challenging math problems in small groups, discussing and sharing our mathematical thinking with others and finding multiple ways to solve problems.</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-78415402177191565782015-07-10T15:00:00.002-04:002015-07-11T11:48:48.490-04:00Activities for Exploring Number Sets1) Who Has a Number Bingo Board:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnDVO0sNuc5qgVBjflYj0B-DxmOyyQkBQJWvlHEWSASWiOR-gr_C-nk86BqqVuJxzqoTKkq_BajIF6zR1W25u2YUwbYWbiQ5HgBI6tZHmn0H2HVWSwtekGv2s_p28-eZCs_keddVWE3V5g/s1600/Number+Bingo+Board.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnDVO0sNuc5qgVBjflYj0B-DxmOyyQkBQJWvlHEWSASWiOR-gr_C-nk86BqqVuJxzqoTKkq_BajIF6zR1W25u2YUwbYWbiQ5HgBI6tZHmn0H2HVWSwtekGv2s_p28-eZCs_keddVWE3V5g/s640/Number+Bingo+Board.PNG" width="553" /></a></div>
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Who Has a Number Bingo Board and Vocabulary Review (below) available <a href="https://drive.google.com/file/d/0B8hsPQ_IEyPIa212a3BaNnlfakE/view?usp=sharing">here</a>.<br />
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2) Students work in groups to review and learn vocabulary associated with number sets. (I plan to do the vocabulary review first to help student activate prior knowledge before playing the Bingo game above.)<br />
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3) Number Sets Search found on <a href="http://www.resourceaholic.com/2015/07/eot.html">End of Year Resources</a> post from Resourceaholic which came from <a href="http://www.cleavebooks.co.uk/trol/indexsc.htm">here</a>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4L4T3GDrdCdiPbr0WXScNF0N9pCUlGQV39IKGD64wNmBv6vvIvhcEPhneW1EspIemiGk5v7w7mIHjpp6eJ_NpT0rVYwMszJ0kawY8nUmPYYngrEob2XDLSQpCDf1y4T1bLdK532td-4B6/s1600/Number+Sets+Search.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4L4T3GDrdCdiPbr0WXScNF0N9pCUlGQV39IKGD64wNmBv6vvIvhcEPhneW1EspIemiGk5v7w7mIHjpp6eJ_NpT0rVYwMszJ0kawY8nUmPYYngrEob2XDLSQpCDf1y4T1bLdK532td-4B6/s640/Number+Sets+Search.PNG" width="540" /></a></div>
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4) Number Experts: a <a href="http://www.resourceaholic.com/p/bitesize-gems.html">Bitesize Gem</a> from <a href="http://resourceaholic.com/">Resourceaholic.com</a>: Assign a number to each student. They become an expert in that number, creating a display and giving a presentation about its interesting properties.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp3jvciQDrcC8NGihyGu4gt-FXf-ydt0dlZM2C66AoQXdLLTwOc0Qp-NxnswSXH2xAFu0qkSx4DV7MUTjng6pIccUm197_4NOsRV_WxUcAjlFTwT8jY9VfZxuGcoaZSeQtijGH_adRvWIP/s1600/Number+Expert.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp3jvciQDrcC8NGihyGu4gt-FXf-ydt0dlZM2C66AoQXdLLTwOc0Qp-NxnswSXH2xAFu0qkSx4DV7MUTjng6pIccUm197_4NOsRV_WxUcAjlFTwT8jY9VfZxuGcoaZSeQtijGH_adRvWIP/s320/Number+Expert.PNG" width="238" /></a></div>
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5) <a href="http://findthefactors.com/">Find the Factors</a>: I think these puzzles are brilliant and plan to use these during the first week of school to help my students practice their multiplication facts.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipoypeuWqOf2ECkrfJmLS7NYJvsK21DiOJKM1WB36fEItro47oO0NlTQ4_7B5ViXpOuZE3lsKJqUTXBiulhK4YAG8Ujn9Vgb-4hGMZZxe3wYmgGRWzXCsIyL1_LDrdmCDo3SnYfHQlsZJ5/s1600/Find+the+Factors+Puzzle.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipoypeuWqOf2ECkrfJmLS7NYJvsK21DiOJKM1WB36fEItro47oO0NlTQ4_7B5ViXpOuZE3lsKJqUTXBiulhK4YAG8Ujn9Vgb-4hGMZZxe3wYmgGRWzXCsIyL1_LDrdmCDo3SnYfHQlsZJ5/s320/Find+the+Factors+Puzzle.PNG" width="312" /></a></div>
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6) Stand Up Sit Down (<a href="http://mathforlove.com/2015/05/quick-physical-games-for-the-math-classroom/">Quick Physical Games for the Math Classroom</a>)</div>
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<ul>
<li>Stand when the number is prime; sit if it is composite</li>
<li>Stand when the number is even; sit if it is odd</li>
<li>Stand when the number is a multiple of 3; sit if it is not a multiple of 3</li>
<li>Stand when the number is a factor of 24; sit if it is not a factor of 24</li>
<li>Stand when the number is a perfect square; sit if it is not a perfect square</li>
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Try to "trick" the students by standing up or sitting down when they should be doing the opposite.</div>
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6) <a href="http://mathtothe7thpower.blogspot.com/2012/09/i-have-who-has-games-for-math.html">I Have Who Has Games</a>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-27029167239581389702015-07-07T16:16:00.002-04:002015-07-07T16:18:54.164-04:00Deposit and Withdrawal CardsWhen we study negative numbers, we have the opportunity to learn about real world applications of this mathematical concept. In general, sixth grade students are unfamiliar with banks and how bank accounts work. The <a href="http://mathtothe7thpower.blogspot.com/2015/03/the-bank-account-game.html">Bank Account Game</a> is a fun way for students to explore these ideas and familiarize themselves with the vocabulary used in basic personal finance. Before we play the game in class, I ask the students to make two deposit cards and two withdrawal cards that we can use for the game. Making the cards themselves is a valuable learning experience and makes the game more fun for the students to play.<br />
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Here are some examples of student made cards:<br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-41584624538216294302015-07-06T14:38:00.000-04:002015-07-07T10:47:15.887-04:00Open Questions and Parallel Tasks for Sixth Grade Math<div class="separator" style="clear: both; text-align: center;">
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This book is a wonderful resource for good math questions. It contains questions:<br />
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<li>that are perplexing so that students are engaged and want to use mathematics to explore the topic</li>
<li>that have multiple entry points for students of varying abilities</li>
<li>that are rich enough so that the exploration can drive the entire lesson</li>
<li>that allow students to solve a problem in many different ways and then explain and prove their results.</li>
</ul>
<br />
As I read this book it really made me think about what makes a good math question and how I could adapt and improve anchor questions that I have used in order to make them more open ended and accessible to all students. I decided to go through the book and pull out all the questions that I could use in my sixth grade math classroom and organize them by the topics that we cover, in the order that we cover them, giving me ready access to some model questions that I can use or adapt throughout the school year. This was a very valuable process for me because by reviewing all the questions, especially those aligned with the sixth grade common core standards, I was able to more clearly understand what makes a good question and how to create these type of questions for other topics we cover.<br />
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As the TIMSS video study found, Japanese teachers use a structured problem solving format for most of their lessons where the teacher poses a complex thought-provoking problem, students struggle with the problem, various students present ideas or solutions to the class and then the teacher leads a class discussion using the various solution methods provided by the students and summarizing the class' conclusions (from The Teaching Gap by James Hiebert and James Stigler and CT Regional School District #10 Math Program <a href="http://www.region10ct.org/math/region10mathsitefaq/whatisjaplesson.html">FAQ</a> Page). This lesson format has proven to be an extremely effective method of teaching mathematics and a model I have tried to use in my classroom. But finding rich problems that create the set up for a student centered lesson for each of the topics covered in our curriculum can be a challenge. Japanese educators are part of a teaching community that uses "lesson study" as a way for teachers to share and refine the problems and questions they pose at the beginning of a lesson that create the conditions in the classroom where students are engaged in deep mathematical thinking. This book is one resource that teachers in the US can use to find good thought provoking questions.<br />
<br />
Here are a few of my favorite open questions and parallel tasks from the book:<br />
<br />
1) Percent: <span style="font-family: 'Times New Roman', serif;">72 is _____% of
______</span><br />
<span style="font-family: 'Times New Roman', serif;"><br /></span>
<span style="font-family: 'Times New Roman', serif; line-height: 115%;">2) Percent: Choose something you have been wanting to buy that costs more than $50.
Imagine you have $30 saved. What discount does the store need to offer before
you can afford it?</span><br />
<span style="font-family: 'Times New Roman', serif; line-height: 115%;"><br /></span>
<span style="font-family: 'Times New Roman', serif;">3) Algebraic Expressions: </span><span style="font-family: 'Times New Roman', serif; line-height: 115%;">An expression involving the variable </span><i style="font-family: 'Times New Roman', serif; line-height: 115%;">k </i><span style="font-family: 'Times New Roman', serif; line-height: 115%;">has the value 10
when </span><i style="font-family: 'Times New Roman', serif; line-height: 115%;">k</i><span style="font-family: 'Times New Roman', serif; line-height: 115%;"> = 4. What could the expression be?</span><br />
<span style="font-family: 'Times New Roman', serif; line-height: 115%;"><br /></span>
<span style="font-family: 'Times New Roman', serif; line-height: 115%;">4) Algebraic Expressions: </span><span style="font-family: 'Times New Roman', serif;">Order these values from
least to greatest. Will your order be the same no matter what the value of n
is? Explain.</span><br />
<ul type="disc">
<li class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">Choice 1: <i>n</i>/2 3<i>n</i> <i>n</i>^2
3<i>n</i> + 1 10 - <i>n</i><o:p></o:p></span></li>
<li class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">Choice 2: 4<i>n</i> 3<i>n</i> 10<i>n</i>
3<i>n</i> + 1 5<i>n</i> + 2 -<i>n</i></span></li>
</ul>
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: 'Times New Roman', serif;"><o:p></o:p></span></div>
5) Geometry: <span style="font-family: 'Times New Roman', serif; line-height: 115%;">You start with a parallelogram. You increase its height by the same
amount as you decrease its base length. How does the area change?</span><br />
<span style="font-family: 'Times New Roman', serif; line-height: 115%;"><br /></span>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-8195587355450436279.post-28129545059725857812015-06-25T22:23:00.003-04:002015-06-29T09:45:45.005-04:00Math T-Shirt FridayOn Friday, teachers at our school can "dress down" and wear jeans. I have used this opportunity to also wear Math T-Shirts. Over the last several years I have been collecting T-shirts that have a math theme and my goal is to have 40+ T-shirts so that I can wear a different one every week! Some of my favorite shirts came from <a href="http://woot.com/">Woot.com</a> and <a href="http://www.savers.com/">Savers</a> (thrift shop) and <a href="http://www.cafepress.com/">CafePress</a>.<br />
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-8195587355450436279.post-14581222661992263532015-03-28T16:39:00.004-04:002015-07-07T16:27:11.935-04:00The Bank Account GameWhen I first started teaching high school math, a colleague made a simple card game that the students LOVED to play. It was kind of like the Game of Life without the board and the little cars with removable blue and pink pegs.<br />
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Each student had a blank check register and they would keep track of deposits and withdrawals to their checking account by recording these amounts in the deposit or withdrawal column of the check register based on the cards that they drew and then add or subtract to find the current balance in their account. In this game, each player had a job that earned them an annual salary that was paid out to all the players every few rotations. Hence the students had practice keeping an accurate checking account balance but never had to worry about bouncing a check or having a negative balance. The student who had the highest balance in their account at the end of the game was the winner.<br />
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Throughout the school year, I found myself going to her regularly to borrow her card game for a few days and always wanted to make my own set of cards, but never found the time.<br />
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Five years later, when I started teaching sixth grade, I realized that this game structure would be a great way for my students to practice working with negative numbers. I still didn't have the time to make a set of card myself, so I got my students involved by asking them to make two deposit cards and two withdrawal cards. I told the students to only use increments of $10 with a maximum of $50 for each deposit or withdrawal. These friendly numbers allowed the students to play the game using mostly mental math by referencing a number line drawn in increments of 10. They loved playing the game with cards they had created themselves and now I have an abundance of student made cards--enough for several groups to be playing at the same time. Here are some examples that I showed them.<br />
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And find some examples of student made deposit and withdrawal cards <a href="http://mathtothe7thpower.blogspot.com/2015/07/deposit-and-withdrawal-cards.html">here</a>.</div>
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The goal of the game was not to practice integer computation following a set of memorized rules, but instead for them to get a feeling for moving along the number line especially when the balance changes from positive to negative or vice versa.<br />
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I encouraged the students to draw number lines to work with during the game. Next year I will provide large number lines in sleeve protectors that they can draw on with dry erase markers as they calculate the balance in their account.<br />
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Here is the link to the <a href="https://docs.google.com/document/d/1e6mlVz8cO0JQyd2s1CkNx9tiWCiJDxQgsLkGUBvw1ZM/edit?usp=sharing">Bank Account Game Template</a>.</div>
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<br />Unknownnoreply@blogger.com1