tag:blogger.com,1999:blog-51344188017033499752024-03-05T01:37:02.975-05:00Math Arguments180 Days of Ideas for Discussion in Math Class.
(as of 9July2014, we're in overtime!)Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.comBlogger534125tag:blogger.com,1999:blog-5134418801703349975.post-41637021768669573802020-07-01T12:39:00.000-04:002020-08-04T18:18:23.062-04:00524: Is a HotDog a sandwich?Does the topology matter? If I were to completely separate the two halves, would it make a difference?
Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-61218821218349035112020-01-31T14:14:00.000-05:002020-01-31T14:14:12.650-05:00False Negatives and Cancer.In the voiceover, Cologuard claims it finds 92% of colon cancers. The ad also provides a generously large and easy legible banner stating that false positives and false negative results can occur. <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6bPExw0E_aOOaIYo52afkcRq5ZF0n-7xR4ajI_aISHutsPfqa9eL9yagD6r13EuNamH3AiUoUpzwB76sxYWO0O3Pg7pO2f2SzZaPeoKgglvqly7-EM4f3fx5S8w1hQPTxJgKEiHkuTT10/s1600/cologuard1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="467" data-original-width="635" height="293" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6bPExw0E_aOOaIYo52afkcRq5ZF0n-7xR4ajI_aISHutsPfqa9eL9yagD6r13EuNamH3AiUoUpzwB76sxYWO0O3Pg7pO2f2SzZaPeoKgglvqly7-EM4f3fx5S8w1hQPTxJgKEiHkuTT10/s400/cologuard1.png" width="400" /></a></div>
<br />
The next banner gives more detail.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiReuVxedfM_jMwrE9fQofGhn6ndqtyMIdrwq0LJdFYerTX2gU7h2fHvC1nO2uZ9hOt1ljth8RUQ1sdHvLWIrJsNghYZyrx_NMFMTHm4CzZcHm9vCXzG041AhKDE2QFOfFlpOefsnZE0pI1/s1600/cologuard2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="457" data-original-width="603" height="302" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiReuVxedfM_jMwrE9fQofGhn6ndqtyMIdrwq0LJdFYerTX2gU7h2fHvC1nO2uZ9hOt1ljth8RUQ1sdHvLWIrJsNghYZyrx_NMFMTHm4CzZcHm9vCXzG041AhKDE2QFOfFlpOefsnZE0pI1/s400/cologuard2.png" width="400" /></a></div>
<br />
If your grandparent gets a "Yes" response, what is the probability that they do have colon cancer? If they get a "No", what are the chances they don't have it?<br />
<br />
https://www.ispot.tv/ad/op0r/cologuard-finding-things<br />
<br />
If you've never paid much attention to advertisements before, this is pretty amazing. Usually the fine print is illegible even on a 62" ultra hi-definition tv paused at just the right moment.<br />
<br />
So ... thank you Cologuard for being honest and up-front.<br />
<br />Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-51692973620529895592019-12-19T11:23:00.000-05:002019-12-19T11:23:06.902-05:00522: Where to start with factoringYou are asked to find the roots and factors of the following polynomial function:<br />
f(x) = x<sup>4</sup> – 2x<sup>3</sup> – 18x<sup>2</sup> + 6x + 45<br />
<br />
By the rational root theorem, possible rational roots are<br />
± 1, 3, 5, 9, 15, or 45.<br />
<br />
In order to minimize your effort, you know that you should begin with the possibility that is most likely to be a root. Which one is most likely?Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-85208030790547052982019-12-17T18:55:00.001-05:002019-12-17T18:55:15.337-05:00521: When is an outlier still an outlier?In a set of data, can the outlier also be the mode or does it stop being an outlier?Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-40794075766266242172019-12-17T18:53:00.002-05:002019-12-17T18:53:27.103-05:00520: Dan or Mike? Who's Better at Doodle Jump?Every year I give this graphic comparing the Doodle Jump stats for @ddmeyer and his friend Mike.<br />
<br />
<br />
Give a couple of reasons why we could consider Dan the better player.<br />
Give a couple of reasons why we could consider Mike the better player.<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiSD5lavHeV5-HLoVJul4-5DWOnSOGNxRnINZjHQwhIVoIPGzoIe_H2cIrNnMg_n-1bFcx2L1BcnLJqDvxQvfoTd_BBiotgSRJfafR-OfAZFIJLVU_RhKSoWXX9oPMVkHgKNuRHBndDAZx/s1600/whos+better+at+doodle+jump.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="375" data-original-width="500" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiSD5lavHeV5-HLoVJul4-5DWOnSOGNxRnINZjHQwhIVoIPGzoIe_H2cIrNnMg_n-1bFcx2L1BcnLJqDvxQvfoTd_BBiotgSRJfafR-OfAZFIJLVU_RhKSoWXX9oPMVkHgKNuRHBndDAZx/s1600/whos+better+at+doodle+jump.jpg" /></a></div>
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Measures of Central Tendency.Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-37697111329063197142019-04-16T10:18:00.000-04:002019-12-17T18:54:01.327-05:00519: ConcavityI have a question that will help me understand the fundamental definition of concavity. If we have y=x^4, can we say that is concave up from (-inf, +inf) or do we need to say (-inf,0)U(0,+inf)?
Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com1tag:blogger.com,1999:blog-5134418801703349975.post-64275077406185254082016-12-21T13:27:00.001-05:002016-12-21T13:27:16.934-05:00518: Interpret your results.Say you had data that told you that<br />
The proportion of all US public school students whose families are low-income:<br />
In 1989, less than ⅓<br />
In 2013, more than ½
<br />
<br />
Interpret this in a positive way.<br />
Interpret this in a negative way.<br />
<br />
<br />
<blockquote class="twitter-tweet" data-lang="en">
<div dir="ltr" lang="en">
Proportion of all US public school students whose families are low-income: In 1989, <1/3. In 2013, >1/2.</div>
— Alfie Kohn (@alfiekohn) <a href="https://twitter.com/alfiekohn/status/811570690025553920">December 21, 2016</a></blockquote>
<script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-2811195485541003392016-12-17T10:47:00.000-05:002016-12-17T10:50:54.865-05:00517: What is the function?<div class="separator" style="clear: both; text-align: center;">
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<br />
Algebra 2: <br />
Write a function for this graph ...<br />
<br />
Estimation:<br />
What are some possible values for a,b,c,d?<br />
If I told you it also went through (0,54), what might the leading coefficient be?<br />
<br />
I would not use the phrase "in factored form", if I were using this in a review or summative assessment since students should be selecting the form that's most appropriate. I might use it if we are in the middle of learning about functions for the first time and we hadn't really made the case for the utility of various forms of the equation. <br />
<br />
source: <br />
<blockquote class="twitter-tweet" data-lang="en">
<div dir="ltr" lang="en">
How would your Ss write a function in factored form for this graph? Nice #MP7 #CthenC problem from Bossé #ncctm16 pic.twitter.com/DoUZUzNR7L</div>
— Jennifer Wilson (@jwilson828) <a href="https://twitter.com/jwilson828/status/792029970453454852">October 28, 2016</a></blockquote>
<script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>
Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-36245340743281488622016-12-12T09:36:00.004-05:002016-12-12T09:36:46.280-05:00516: DataIs money a discrete variable or a continuous one?<br />
<br />Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com1tag:blogger.com,1999:blog-5134418801703349975.post-4541474567894892042016-08-18T09:50:00.000-04:002016-08-18T09:50:01.260-04:00515: Pascal ProofAn interesting Question ... is an algebraic method the only way to do this? <br />
<blockquote class="twitter-tweet" data-lang="en">
<div dir="ltr" lang="en">
Prove that anywhere on Pascal's triangle, products of numbers in yellow and in orange are equal.<br />Via <a href="https://twitter.com/HopeHana">@HopeHana</a> <a href="https://t.co/zgs3d0wenJ">pic.twitter.com/zgs3d0wenJ</a></div>
— ⋁△ ᴹᴬᵀᴴcafé ☕ (@Five_Triangles) <a href="https://twitter.com/Five_Triangles/status/766266668637184000">August 18, 2016</a></blockquote>
<script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-60548168000487670492016-06-16T11:17:00.003-04:002016-06-16T11:17:42.271-04:00514: Probability GameTemporary return! A quick probability question ...<br />
<br />
Player A's score is determined by taking the highest of 3 dice.<br />
Player B's is determined by taking second-highest of 8.<br />
Who wins more games?
<br />
<blockquote class="twitter-tweet" data-lang="en">
<div dir="ltr" lang="en">
Player A's score is determined by taking the highest of 3 dice.<br />Player B's is determined by taking second-highest of 8.<br />Who wins more games?</div>
— Ben Orlin (@benorlin) <a href="https://twitter.com/benorlin/status/743416545435619328">June 16, 2016</a></blockquote>
<script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com2tag:blogger.com,1999:blog-5134418801703349975.post-4306251468876546892015-12-17T17:58:00.002-05:002015-12-17T17:58:45.267-05:00513: Compound AreaCan you come up with other, better, divisions?<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhT3vRydIugOcSiIstoavuwTCEQaOIHuIfIs4sQRwzyBK0EGAclT7sQlgZ_h4OXvnArUZy_lvluKWBZV3Q6YiokHQ-Vgj0uS4F0QqHpRbYQAVQhH-DchQTiP9ogL0mlaXWH-3v7pBXLuGS6/s1600/compoundarea1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="241" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhT3vRydIugOcSiIstoavuwTCEQaOIHuIfIs4sQRwzyBK0EGAclT7sQlgZ_h4OXvnArUZy_lvluKWBZV3Q6YiokHQ-Vgj0uS4F0QqHpRbYQAVQhH-DchQTiP9ogL0mlaXWH-3v7pBXLuGS6/s320/compoundarea1.png" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2O1vU9gSQUrPxtVlSI41l3eqW72zUYhIgqG7MAl4BWH4pQVw4HlYwshDoI73TDhYCf7OxYkxQcDDKvklxr-eozCb3z0TbUIj_SgjWQxPJWJe009XEdn5-HzwBYobotXH4Aao232FNCoOJ/s1600/compoundarea2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="241" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2O1vU9gSQUrPxtVlSI41l3eqW72zUYhIgqG7MAl4BWH4pQVw4HlYwshDoI73TDhYCf7OxYkxQcDDKvklxr-eozCb3z0TbUIj_SgjWQxPJWJe009XEdn5-HzwBYobotXH4Aao232FNCoOJ/s320/compoundarea2.png" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMvdi9xxTA6kVoYaDOvBa7XWyUmMBBpTSYevu7KEH3hhtsPcLd6LEA2OKJlYvjNj8_nTd0NAf1hLFAei87nvVbZdAdlwsPaw7155oe_lE4hkkNDm5ipP_nXdz1ntejwFqwhrM0P6ucdpIg/s1600/compoundarea3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="244" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMvdi9xxTA6kVoYaDOvBa7XWyUmMBBpTSYevu7KEH3hhtsPcLd6LEA2OKJlYvjNj8_nTd0NAf1hLFAei87nvVbZdAdlwsPaw7155oe_lE4hkkNDm5ipP_nXdz1ntejwFqwhrM0P6ucdpIg/s320/compoundarea3.png" width="320" /></a></div>
<br />
<br />
<a href="http://donsteward.blogspot.co.uk/2015/12/compound-rectangular-shapes.html">source</a>.Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-6666987073199535742015-12-16T17:27:00.001-05:002015-12-16T17:27:51.468-05:00512: Factory Ratios 3We started with this:<br />
<div style="border: 1px solid red; margin: 10px; padding: 15px;">
In a factory, the ratio of men to women is 2:3. The ratio of right-handed men to left-handed men is 7:3. The ratio of right-handed women to left-handed women is 11:1. What fraction of the factory workforce is right-handed? <br />
And then we extended with: What is the fewest number of employees possible in this building?</div>
<br />
If the ugly-sweater party was held in that factory with the minimum number of workers possible and the ratio of red to green ugly sweaters was 7:13, how likely is it that a left-handed man wore a red ugly sweater?<br />
<br />
Source:
<br />
<blockquote class="twitter-tweet" lang="en">
<div dir="ltr" lang="en">
New GCSE Maths question:
Ratio and proportion is the buzz thing!
How are your students going to tackle it? <a href="https://t.co/RyPsmc9z4m">pic.twitter.com/RyPsmc9z4m</a></div>
— m4ths.com (@m4thsdotcom) <a href="https://twitter.com/m4thsdotcom/status/677196776319950852">December 16, 2015</a></blockquote>
<script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>
Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-43641973439568015892015-12-16T17:14:00.002-05:002015-12-16T17:14:52.218-05:00511: Factory Ratios 2We started with this:<br />
<div style="border: 1px solid red; margin: 10px; padding: 15px;">
In a factory, the ratio of men to women is 2:3.<br />
The ratio of right-handed men to left-handed men is 7:3<br />
The ratio of right-handed women to left-handed women is 11:1<br />
What fraction of the factory workforce is right-handed?
</div>
But let's extend things a bit.<br />
<br />
<span style="color: yellow;"><i><b>What is the fewest number of employees possible in this building?</b></i></span><br />
<br />
Source:
<br />
<blockquote class="twitter-tweet" lang="en">
<div dir="ltr" lang="en">
New GCSE Maths question:
Ratio and proportion is the buzz thing!
How are your students going to tackle it? <a href="https://t.co/RyPsmc9z4m">pic.twitter.com/RyPsmc9z4m</a></div>
— m4ths.com (@m4thsdotcom) <a href="https://twitter.com/m4thsdotcom/status/677196776319950852">December 16, 2015</a></blockquote>
<script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>
Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com2tag:blogger.com,1999:blog-5134418801703349975.post-62529142735510178132015-12-16T17:10:00.001-05:002015-12-16T17:10:16.964-05:00510: Factory Ratios 1In a factory, the ratio of men to women is 2:3.<br />
The ratio of right-handed men to left-handed men is 7:3<br />
The ratio of right-handed women to left-handed women is 11:1<br />
<br />
What fraction of the factory workforce is right-handed?<br />
<br />
<br />
Source:
<br />
<blockquote class="twitter-tweet" lang="en">
<div dir="ltr" lang="en">
New GCSE Maths question:
Ratio and proportion is the buzz thing!
How are your students going to tackle it? <a href="https://t.co/RyPsmc9z4m">pic.twitter.com/RyPsmc9z4m</a></div>
— m4ths.com (@m4thsdotcom) <a href="https://twitter.com/m4thsdotcom/status/677196776319950852">December 16, 2015</a></blockquote>
<script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>
Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-5318927939964746732015-11-28T17:06:00.002-05:002015-11-28T17:06:43.677-05:00509: Circles 2Several days ago, I posted <a href="http://matharguments180.blogspot.com/2015/11/503-circles.html">a slightly different question about tangential circles</a> and the spaces in between. Compare this question, from the 2004 SAT Practice Test to <a href="http://matharguments180.blogspot.com/2015/11/503-circles.html">that question from Emma Bell.</a><br />
<ul>
<li>Which question seems harder? What are the difficult aspects of each?</li>
<li>Do we have to specify the angle APB?</li>
<li>How are these two questions different in terms of the knowledge they require for solving?</li>
<li>Is this question made harder by the "how many times" part? Does that phrasing make the question unnatural? </li>
</ul>
<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-iM03NVG5zzy4p3tQ9k2951OtjH-XQ5aOvz1mq4fgnFxl3rb760YpaBuOiShkZ9pNYT1-29J8kSpNRm7eoBZBban_qNb_cMb-1jGjiW8CL1MFx50ULCQLBEKjcrUVgVGJTL_jLI5ma4Oy/s1600/circles2.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="321" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-iM03NVG5zzy4p3tQ9k2951OtjH-XQ5aOvz1mq4fgnFxl3rb760YpaBuOiShkZ9pNYT1-29J8kSpNRm7eoBZBban_qNb_cMb-1jGjiW8CL1MFx50ULCQLBEKjcrUVgVGJTL_jLI5ma4Oy/s400/circles2.PNG" width="400" /></a></div>
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source: ETS, 2004Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com1tag:blogger.com,1999:blog-5134418801703349975.post-32851799142292051472015-11-28T15:01:00.000-05:002015-11-28T15:01:24.313-05:00508: Shorter Path<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikYnHlBG1I3de14WxVbeG-wSStyujWSQrRf_M75e2_mQ5QSElqofFwQtvHKw5v_c0CQnNCsbivVWtGh8Bl5dXpALC-piLEJ_GB_ME7N06HvNrTY0uRwiL91W6Q8bfVexn6YHmAr9cUgKRh/s1600/shorter+path+from+aion.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikYnHlBG1I3de14WxVbeG-wSStyujWSQrRf_M75e2_mQ5QSElqofFwQtvHKw5v_c0CQnNCsbivVWtGh8Bl5dXpALC-piLEJ_GB_ME7N06HvNrTY0uRwiL91W6Q8bfVexn6YHmAr9cUgKRh/s320/shorter+path+from+aion.jpg" width="550" /></a></div>
<br />
What car will have the shorter path?<br />
How could you tell for certain?<br />
<br />
source: Justin Aion Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-71653806524132518612015-11-22T11:18:00.000-05:002015-11-22T11:18:07.132-05:00507: Orderly ProbabilityScenario 1: Which winning number group is more likely to occur?<br />
<br />
<div style="text-align: center;">
1-2-3-4-5-6 OR 4-8-15-16-23-42</div>
<br />
Scenario 2: Which winning number group is more likely to occur if the numbers are drawn in any order and THEN put into ascending order by the presenter?<br />
<br />
<div style="text-align: center;">
1-2-3-4-5-6 OR 4-8-15-16-23-42</div>
<br />
In which of the above two scenarios is getting the winning numbers more likely?<br />
<br />
source: Jeff SuzukiCurmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-66041768190905147012015-11-22T11:06:00.005-05:002015-11-22T11:22:27.379-05:00506: The Conical Tank<div class="separator" style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4VujOrVLdySxGinmqQwGWA_eP_lbgvtCFbDpr9OT3auaJq2ckMtyTY-c_pBBEMeDendboaegu4as9sma7tt4yi-u_C15iCqQHXnrUF_9ufaZGpr_cKiALsgGGpQsgRwJK_wsDevwuLC_N/s1600/calccone.PNG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4VujOrVLdySxGinmqQwGWA_eP_lbgvtCFbDpr9OT3auaJq2ckMtyTY-c_pBBEMeDendboaegu4as9sma7tt4yi-u_C15iCqQHXnrUF_9ufaZGpr_cKiALsgGGpQsgRwJK_wsDevwuLC_N/s320/calccone.PNG" width="304" /></a></div>
The last of the three related-rate geogebra problems from Kate Nowak. It's the related rate problem from calculus: the conical tank being filled with water.<br />
<br />
Adjust the slider and... wait, what is changing and how?<br />
<br />
For every click of the slider:<br />
Is the depth increasing at a constant rate? <br />
Is the radius increasing at a constant rate? <br />
Is the volume increasing at a constant rate? <br />
How can you tell?<br />
<ul>
<li>Where or how, in the RealWorld<sup>tm</sup>, could we see the constant increase in volume?</li>
<li>Where or how, in the RealWorld<sup>tm</sup>, could we see the constant increase in radius, or depth?</li>
</ul>
<br />
If you want to play with the animation, <a href="http://bit.ly/calccone">Conical Tank Problem</a>. source: <a href="https://twitter.com/k8nowak">@k8nowak</a>Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-22895624865129322802015-11-22T10:58:00.001-05:002015-11-22T11:22:35.851-05:00505: The Balloon Problem<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_OjbJL4IHxKSe6TiCHxHrh-9raPuggZrE6-1hnnkX21F_tRSX3yTZ8ue4tYVb80D1hV45Pn8m8_3VORShZxKL7NFFJKRk8GOJy7jjA0ZOmAINfleeFM24Mn0rj9LcbtcPA8SGinKggBhR/s1600/calcballoon.PNG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="186" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_OjbJL4IHxKSe6TiCHxHrh-9raPuggZrE6-1hnnkX21F_tRSX3yTZ8ue4tYVb80D1hV45Pn8m8_3VORShZxKL7NFFJKRk8GOJy7jjA0ZOmAINfleeFM24Mn0rj9LcbtcPA8SGinKggBhR/s320/calcballoon.PNG" width="320" /></a></div>
We've all seen this problem, but many of our students haven't. It's the related rate problem from calculus: the balloon being filled with air.<br />
<br />
There are two questions being demonstrated here.<br />
(1) "If the volume increases at a constant rate, what is happening to the radius?" and<br />
(2) "If the radius increases at a constant rate, what is happening to the volume?"<br />
<br />
The first question is to figure out which situation is modeled in red and which in blue.<br />
Then we can ask: <br />
<ul>
<li>Does the radius increase at a constant speed in both models? How can you tell?</li>
<li>Does the volume increase at a constant speed in both models? How can you tell?</li>
<li>Where or how, in the RealWorld<sup>tm</sup>, could we see the constant increase in volume?</li>
<li>Where or how, in the RealWorld<sup>tm</sup>, could we see the constant increase in radius?</li>
</ul>
<br />
If you want to play with the animation, <a href="http://bit.ly/calcballoon">Balloon Problem</a>. source: <a href="https://twitter.com/k8nowak">@k8nowak</a>Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-52819193223630996572015-11-06T18:08:00.002-05:002015-11-22T11:22:43.170-05:00504: The Ladder Problem<div class="" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgC1B3nlEdkmTwSOEU6WlqPmiFWHVaAcUddGvzZtHAxok-1sOEmg0flQd2L0FNRG3XIwkuBf603BTMkkYoc4STgyRODtAeZA7B8G-7HLduwQRByO77wTxXFcxcIcwdyhbN3muOTH2Io8Vy0/s1600/calclad.PNG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgC1B3nlEdkmTwSOEU6WlqPmiFWHVaAcUddGvzZtHAxok-1sOEmg0flQd2L0FNRG3XIwkuBf603BTMkkYoc4STgyRODtAeZA7B8G-7HLduwQRByO77wTxXFcxcIcwdyhbN3muOTH2Io8Vy0/s320/calclad.PNG" width="320" /></a></div>
<br />
We've all seen this problem, but many of our students haven't.<br />
<br />
It's the related rate problem from calculus: the ladder sliding down the wall.<br />
<br />
The "official" question?<br />
<br />
How fast is the ladder's top sliding down the wall if the bottom is being pulled out at a rate of 1 ft/sec?<br />
<br />
We can ask a few questions of kids at any level, though, based on the given that the bottom of the ladder is being pulled to the left at 1 foot per sec.<br />
<br />
<ul>
<li>Does the top drop at a constant speed?</li>
<li>Does the top drop a distance equal to the horizontal movement?</li>
<li>When is the speed of the top greater than 1, less than 1, and equal to 1?</li>
<li>If this is a 25 foot ladder, with the bottom 7 feet out from the base of the wall, and the top drops 4 feet ... how far out does the bottom of the ladder have to go?</li>
</ul>
<br />
If you want to play with the animation, <a href="http://bit.ly/calclad">Ladder Problem</a>. source: <a href="https://twitter.com/k8nowak">@k8nowak</a>Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com1tag:blogger.com,1999:blog-5134418801703349975.post-57810139179059453452015-11-01T10:31:00.001-05:002015-11-01T10:31:34.798-05:00503: Circles<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyNcO3gLuVqKLkfxpthe7Js4RfX_razibHXSdZpC8p-xblSFddbd_uxOwpHAcWa92K67Exnzzyr9zSIkzup8BvI04E3C86QcgZrKxhIkXbLcXFtvPUA3K9HR3sg3W7NQ9e-t3KqpMTnEQ9/s1600/CSekLLSWUAA2dw7.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyNcO3gLuVqKLkfxpthe7Js4RfX_razibHXSdZpC8p-xblSFddbd_uxOwpHAcWa92K67Exnzzyr9zSIkzup8BvI04E3C86QcgZrKxhIkXbLcXFtvPUA3K9HR3sg3W7NQ9e-t3KqpMTnEQ9/s400/CSekLLSWUAA2dw7.jpg" width="400" /></a></div>
<br />
This is a straightforward question. I'd like to make all of you students into teachers for a minute ... Let's create a test question !<br />
<ul>
<li>Do we have to specify angle AOB?</li>
<li>Is there a better way to say something without actually saying it?</li>
<li>What other instructions and given information could we provide that would lead to the same answer? </li>
<li>What is the best question here?</li>
</ul>
Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-79818317881035837622015-11-01T06:57:00.000-05:002015-11-01T07:01:16.759-05:00502: Powerful QuestionIt's not included in the PEMDAS Order of Operations ...<br />
<br />
Should $a^{b^c} = ({a^b})^c$ or should it be $a^{b^c} = a^{(b^c)}$ ??<br />
<br />
Does $3^{2^0}$ equal 1 or 3?<br />
<br />
Let's just consider easy numbers {1, 2, 3, 4} so we can explore. What's the probability that the two methods arrive at the same answer?<br />
<br />
For the record, $a^{b^c} = a^{(b^c)}$ is the accepted order of operations here.Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com2tag:blogger.com,1999:blog-5134418801703349975.post-31176935913482375422015-10-24T11:27:00.000-04:002015-10-24T11:27:03.602-04:00501: Learning timeRead this post on <a href="http://www.justintarte.com/2015/10/learning-time-loss-why-bell-to-bell.html">lost learning time</a>.<br />
<br />
Here's his basic point:<br />
<blockquote class="tr_bq">
<div style="text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI4ANjp-JBBPr4jw5veRiUAiXcsqBUo1XyNUVb2C-ybAHcSC0v_vRn1QBUqdau2Lp189MzAFjkXmelRbv14rkVQYKeDunjhHiGh7TrBy6i4mEAJYLdgAAzPppSqMl0GV-2483XfLWBiQs/s1600/instructional+time.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI4ANjp-JBBPr4jw5veRiUAiXcsqBUo1XyNUVb2C-ybAHcSC0v_vRn1QBUqdau2Lp189MzAFjkXmelRbv14rkVQYKeDunjhHiGh7TrBy6i4mEAJYLdgAAzPppSqMl0GV-2483XfLWBiQs/s200/instructional+time.jpg" width="200" /></a> ... roughly 10 of 60 minutes are not spent learning. <br />
In one day, roughly 50 minutes.<br />
In a 5 day week, 250 minutes.</div>
In a typical school year, 8,700 minutes not utilized for learning.<br />
<br />
Now, let's be realistic and <b><u>cut that number in half</u></b>
because we all know there are assemblies and other events that cut into
learning time throughout the school year. That leaves us with 4,350
minutes of time not spent learning.
</blockquote>
<ol>
<li>Do you agree with his analysis?</li>
<li>How about his calculations?</li>
<li>What should be done? </li>
</ol>
Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0tag:blogger.com,1999:blog-5134418801703349975.post-76572567984101401922015-10-19T20:31:00.000-04:002015-10-19T20:31:15.413-04:00500: Secret SalaryMy employer has nine workers. The nine of us want to determine what
our average salary is, but none of us wants to divulge his own salary.
Can we find the average without doing so?<br />
<br />
<a href="http://www.futilitycloset.com/2015/10/17/earnings-report/">source</a>. Curmudgeonhttp://www.blogger.com/profile/04323026187622872114noreply@blogger.com0