tag:blogger.com,1999:blog-5964889903484807623Wed, 18 Jun 2014 16:41:21 +0000pedagogyalgebraanecdoteslessonsproblem solvingGeoGebraassessmentinquiryQuadraticsWCYDWTlinear equationscurriculumprojectquestionsAre You Kidding Me?homeschoolingtechnologyCCSSexamviewprojectile motionstandards based gradingfractionsgeometryvertical motionToolsappletsdesmosfactoringgrade7integratedquadratics unithabits of mindproofslopeworksheets15.02.03 Acts6.07.09.0Homeschooling pedagogyTranslating expressionsTrianglesWeb 2.0advanced studentsconesdilationseaster eggelectricityexponentsfeedbackformulafunctionsgraphinghomeschoolhypothesis wreckinglinear functionsmath 8misconceptionsmixturemodelingparametric equationspblpedagogy homeschoolingperpendicular bisectorspresentationprofessional developmentpropertiesrate of changereflectionrow gamerubricself assessmentsmartboardsolidssystems of equationstwitterunitsvideovolumewolfram alphawork problemsQuestions?Trying to make it matter.http://coxmath.blogspot.com/noreply@blogger.com (David Cox)Blogger227125tag:blogger.com,1999:blog-5964889903484807623.post-3942946854648283372Fri, 09 May 2014 18:55:00 +00002014-05-09T11:55:33.352-07:00conesdesmospedagogyvolumewolfram alphaWell, since you asked...We've been looking at the volume of prisms, cylinders and cones this week. I had students working on a project where they had to build one of each with equal heights and widths/diameters. The idea is to explore the volume of each and see how the eventual formulas will relate to one another. <br /><br />Then, Jacob traces can on his paper and cuts out the circle. He cuts a radius and begins rolling the paper (as if he's cut out different sized sectors) to make different cones. He comes up and says, "Mr. Cox, I think the cone that is almost flat has the highest volume because the tighter I roll the paper, the less stuff I can fit in it."<br /><br />Me: What if the circle is flat? What's the volume then? <br /><br />J: There isn't any volume.<br /><br />Me: So then when does the cone go from 'flat' to having the most possible volume?<br /><br />J: ...<br /><br />Me: ...<br /><br />J: What do you mean?<br /><br />Me: Maybe there's some kind of sweet spot where the volume gets bigger then starts to get smaller. <br /><br />J: Let me think about that.<br /><br />At this point, I was with Jacob. I didn't really know what the volume did as the cone changed. But we were both interested.<br /><br />The next day, Jacob comes in and says, "Mr. Cox, I thought about what you were saying and I think you're right, there has to be some kind of sweet spot."<br /><br />So, we sit down and go to work.<br /><br />I'm thinking about how to model this thing and Jacob enlisted the help of a friend to gather data. They're cutting sectors from a circle and making cones. Jacob has dibs on 30, 60, 90, ... degrees and Armando has 15, 45, 75, ...<br /><br />Our first bit of trouble came when Jacob said, "I can find the radius of the cone's base, but I'm having trouble getting the height because of this..."<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-ZkdoVLavv-o/U20Uubn5TXI/AAAAAAAABfg/-JpU32sXKSw/s1600/height+measuring.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-ZkdoVLavv-o/U20Uubn5TXI/AAAAAAAABfg/-JpU32sXKSw/s1600/height+measuring.JPG" height="240" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Wish these rulers came with a bubble level. </td></tr></tbody></table><br /><div style="text-align: center;"></div>But, we figured out that the Pythagorean Theorem was a nice work around. <br /><br />Now, does our data match the model?<br /><br />It took a while, and thanks to CalcDave for cleaning things up, but this is a pretty cool function.<br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-ygZQ67MNvbM/U20XjdE0DaI/AAAAAAAABfs/SvS6anByOuE/s1600/cone+volume+model.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-ygZQ67MNvbM/U20XjdE0DaI/AAAAAAAABfs/SvS6anByOuE/s1600/cone+volume+model.png" height="241" width="640" /></a></div> Desmos graph is <a href="https://www.desmos.com/calculator/lrgxqyexk4">here</a>.<br /><br /><br />We're estimating the maximum to be about 66 degrees. And because my calculus is a little rusty, I'm thankful for the folks at WolframAlpha.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-B4dcycE_9Fc/U20XjQqz19I/AAAAAAAABf4/pXX4FNlEMFI/s1600/WA+Maximum.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-B4dcycE_9Fc/U20XjQqz19I/AAAAAAAABf4/pXX4FNlEMFI/s1600/WA+Maximum.png" height="640" width="584" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">This particular function is using a circle with radius = 3.1. </td></tr></tbody></table><br />The function is a little dense at this point, but Jacob was dialed in as we talked about it. The idea that these crazy expressions really just amounted to <b>V<sub>cone</sub> = ⅓πr<sup>2</sup>h</b> blew him away.<br /><br /><br />http://coxmath.blogspot.com/2014/05/well-since-you-asked.htmlnoreply@blogger.com (David Cox)1tag:blogger.com,1999:blog-5964889903484807623.post-3496910230572973059Tue, 06 May 2014 19:13:00 +00002014-05-06T12:13:09.988-07:00dilationspedagogysolidsFull CircleIt was one of those moments when I was trying to explain something to them and they ended up explaining something to me.<br /><br />We're in the middle of a unit on volume and exploring prisms, cylinders and cones. I was inspired by <a href="http://www.jamestanton.com/?p=947">James Tanton's</a> ability to explain things by getting at their essence. As if to say, "we can call a cylinder a 'cylinder' but it's just a prism made of circles--or a cone can be called a 'cone' but is it really any different than a pyramid?" <br /><br />It was one of those, sitting around a campfire moments. We're using stacks of paper and stacks of CDs to demonstrate why calculating the base area is critical because the rest of the solid is just like a stack of that area and no matter where we slice the solid, we get the same shape--over and over again.<br /><br />Then comes the question about the cone.<br /><br />The base is a circle but when you slice it, you get a...circle? Wait, but it's a different circle. Waitaminit. What about a pyramid? Triangle base and when you slice it, you get a triangle. But a different one.<br /><br />Are the triangles related?<br /><br />"They're similar. Hey wait, this is a dilation." <br /><br />And the tip of the pyramid is the center of dilation. <br /><br />We did dilations in Unit 1. This was a callback I didn't anticipate: A pyramid is like a 3D representation of a dilation. <br /><br />Thanks, kids. I'd never thought of it that way before. <br /><br /><br /><br /><br />http://coxmath.blogspot.com/2014/05/full-circle.htmlnoreply@blogger.com (David Cox)2tag:blogger.com,1999:blog-5964889903484807623.post-5604451985917159672Tue, 06 May 2014 16:28:00 +00002014-05-06T09:28:04.889-07:00I'm Taking You With MeI've applied for one of our district coaching positions. There are still a lot of details to work out, so I'm not quite sure I'm ready to leave the classroom. One thing I am sure of, though, is that I'd like the interview panel (assuming I get an interview) to understand how amazing you all are. <br /><div><br /></div><div>I tweeted this form, but I'll leave it here as well. If you have a minute, I'd love your help. </div><div><br /></div><div>Feel free to exclude any information you're not interested in sharing. </div><div><br /></div><div>Thanks a bunch.<br /><iframe frameborder="0" height="500" marginheight="0" marginwidth="0" src="https://docs.google.com/forms/d/13V5R87MUXFpRXJDhCk8qLXf8i0G0TktQziWjYl5nog4/viewform?embedded=true" width="760">Loading...</iframe><br /></div>http://coxmath.blogspot.com/2014/05/im-taking-you-with-me.htmlnoreply@blogger.com (David Cox)0tag:blogger.com,1999:blog-5964889903484807623.post-5393420572186800548Fri, 02 May 2014 22:04:00 +00002014-05-02T15:04:18.697-07:00hypothesis wreckingpedagogyproblem solvingWhen It Can't Be WreckedWe're getting some mileage out of <a href="http://coxmath.blogspot.com/2014/04/hypothesis-wrecking-and-diagonal-problem.html">this</a> lately. Today, I have a new problem to add to the pile of those that foster the <a href="http://coxmath.blogspot.com/2014/04/fostering-hypothesis-wrecking-mindset.html">process of hypothesis wrecking.</a><br /><br />I posed the question with a rubric.<br /><br /><span style="font-size: large;"><b>Can a unit fraction always be written as the sum of two unique unit fractions?</b></span><br /><span style="font-size: large;"><b><br /></b></span><b>Rubric</b><br />5: Precise proof that demonstrates all cases (abstract, general rule)<br />4: Reasonable argument that demonstrates some cases (numeric, gives examples)<br />3: Gut level or weak argument <span class="Apple-tab-span" style="white-space: pre;"> </span><br />2: Does not present an argument<span class="Apple-tab-span" style="white-space: pre;"> </span><br /><br />1: No evidence of understanding<br /><br />Students played around with a few unit fractions and after a few minutes we had a couple of them. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-iGWVTcDJ79o/U2QQJBt945I/AAAAAAAABew/yr8S036V440/s1600/unit+fractions+1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-iGWVTcDJ79o/U2QQJBt945I/AAAAAAAABew/yr8S036V440/s1600/unit+fractions+1.png" height="82" width="400" /></a></div><br />Shortly, we had a student come up with an hypothesis:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-048sRzfHQvE/U2QQwKc9K0I/AAAAAAAABe4/NVlEPmIuTFI/s1600/unit+fractions+2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-048sRzfHQvE/U2QQwKc9K0I/AAAAAAAABe4/NVlEPmIuTFI/s1600/unit+fractions+2.png" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">which was soon followed by another student example:</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/--P1zyloa578/U2QQwn7r9pI/AAAAAAAABfA/JVa2pR1Szh0/s1600/unit+fractions+3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/--P1zyloa578/U2QQwn7r9pI/AAAAAAAABfA/JVa2pR1Szh0/s1600/unit+fractions+3.png" /></a></div><br />Uh-oh, that doesn't fit the pattern. <br /><br />"Does this example wreck our hypothesis?"<br /><br />This led to a nice conversation on whether this new example and our hypothesis can coexist. It was interesting to see how many students initially thought the hypothesis was wrecked. <br /><br />We tested a few more examples and shared results--all confirming our hypothesis.<br /><br />Then I asked, "So where does this put us on the rubric?"<br /><br />And a student asks, "What has to happen for a 4 to become a 5?"<br /><br />In other words, <b>when does a numeric (quantitative) argument become abstract <span style="font-size: x-small;"><sub>[1]</sub></span></b>?<br /><br />Had to pause. This one is worth it. So we discussed simple example:<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-RGoFwX1zu0s/U2QTyMD9yqI/AAAAAAAABfM/NMmeAhEhKnM/s1600/unit+fractions+4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-RGoFwX1zu0s/U2QTyMD9yqI/AAAAAAAABfM/NMmeAhEhKnM/s1600/unit+fractions+4.png" height="280" width="640" /></a></div><br /><br />I quickly came up with the question and answers 1, 3, and 4. At lunch I added 2, which really added to the conversation for 6th period. <br /><br />Which answer provides the stronger argument? Most saw 4 as the strongest and agreed 1 was the weakest. But very few saw 2 on the same level as 4. Then one student says, "I see that 2 and 4 are similar but 4 is just kinda strung out." <br /><br />Yep, the kid has a feel for brute force vs. elegance. Love it. <br /><br />By the end, we agreed that 2 and 4 were more abstract and 3 was more quantitative. What about 1? <br /><br />Well, 1 was what they would've considered a great answer a few months ago. <br /><br /><span style="font-size: x-small;">[1] This is what prompted my <a href="https://twitter.com/dcox21/status/462315590321852416">question</a> about SMP 2 on Twitter. </span>http://coxmath.blogspot.com/2014/05/when-it-cant-be-wrecked.htmlnoreply@blogger.com (David Cox)0tag:blogger.com,1999:blog-5964889903484807623.post-178596013157925776Wed, 16 Apr 2014 18:55:00 +00002014-04-16T12:04:21.293-07:00pedagogyproblem solvingDirty TrianglesI've been out for a couple of days--let's just say that I can think of better ways to drop 10 pounds--so, I'm in a really special frame of mind today. While I was out, I left a few distance/rate/time problems for students to solve. Upon my return, I was asking students about the problems and many students had similar responses.<br /><br />S: "This is easy, you just use the <b>Dirt Triangle</b>."<br /><br />Me: "The what?"<br /><br />S: "The <b>Dirt Triangle</b>."<br /><br />Me: "Hmm. I don't know what that is."<br /><br />S: "Look, Mr. Cox it's like this...<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-jlEQg_RH64g/U07Hm3jZ5RI/AAAAAAAABd8/aVMB7J-f8v8/s1600/dirt.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-jlEQg_RH64g/U07Hm3jZ5RI/AAAAAAAABd8/aVMB7J-f8v8/s1600/dirt.png" height="283" width="320" /></a></div><br /><br />"...You cover up the one you're looking for and if the other two are next to each other, you multiply. If one is above the other, you divide."<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-8R5l7lFbTTA/U07IulcasYI/AAAAAAAABeM/YtZYx4og-So/s1600/d+over+t.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-8R5l7lFbTTA/U07IulcasYI/AAAAAAAABeM/YtZYx4og-So/s1600/d+over+t.png" height="305" width="320" /></a></div><br /><br /><br />Me: "Really? That's strange. I never learned the <b>Dirt Triangle</b>. I learned...<br /><br /><br /><span style="font-size: large;"><b>The Turd Triangle</b></span><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-t0ARzB28ZnU/U07HpqTIb-I/AAAAAAAABeE/xqPUvMxlyHA/s1600/turd.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-t0ARzB28ZnU/U07HpqTIb-I/AAAAAAAABeE/xqPUvMxlyHA/s1600/turd.png" height="279" width="320" /></a></div><br /><br />S<sub>1</sub>: "No, that won't work. That's not what he<sub><span style="font-size: x-small;">[1]</span></sub> told us."<br /><br />S<sub>2</sub>: "He said it didn't matter how we wrote it."<br /><br />Me: "So which is it; does one work or are they the same? Make your case and be ready to defend it."<br /><br /><br />Helping students develop a turd detector one day at a time.<br /><br /><br /><br /><span style="font-size: x-small;">[1] Students picked up the triangle in another class. They said that the formulas were given early on and explained. However, many were still missing problems so the triangle was introduced. </span>http://coxmath.blogspot.com/2014/04/dirty-triangles.htmlnoreply@blogger.com (David Cox)4tag:blogger.com,1999:blog-5964889903484807623.post-8171050139634794860Mon, 07 Apr 2014 23:06:00 +00002014-04-07T16:06:03.140-07:00habits of mindproblem solvingFostering the Hypothesis Wrecking MindsetHypothesis wrecking is not natural. I think a few of you have summed it up quite well.<br /><br /><a href="https://twitter.com/Mythagon?original_referer=https%3A%2F%2Ftwitter.com%2Fi%2Fnotifications&tw_i=452525091776368640&tw_p=tweetembed">Ashli Black:</a><br /><blockquote class="twitter-tweet" lang="en"><a href="https://twitter.com/dcox21">@dcox21</a> <a href="https://twitter.com/cheesemonkeysf">@cheesemonkeysf</a> I love the 'see if you can wreak' it mentality. It's not something ppl naturally do and so important<br />— Ashli (@Mythagon) <a href="https://twitter.com/Mythagon/statuses/452525091776368640">April 5, 2014</a></blockquote><br /><br /><a href="http://blog.mrmeyer.com/2014/oh-you-think-you-have-a-rule-see-if-you-can-wreck-it/">Dan Meyer:</a><br /><blockquote class="tr_bq"><span style="font-size: x-small;"><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;">The fact that </span><em style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px; margin: 0px; padding: 0px;">you</em><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;"> </span><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;">are supposed to wreck</span><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;"> </span><em style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px; margin: 0px; padding: 0px;">your own</em><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;"> </span><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;">conjecture. Your conjecture isn't something you're supposed to</span><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;"> </span><em style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px; margin: 0px; padding: 0px;">protect</em><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;"> </span><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;">from your peers and your teacher as though it were an extension of your ego. It's </span><em style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px; margin: 0px; padding: 0px;">supposed</em><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;"> </span><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;">to get wrecked. That's okay! In fact,</span><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;"> </span><em style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px; margin: 0px; padding: 0px;">you're</em><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;"> </span><span style="background-color: white; color: #333333; font-family: verdana, tahoma, arial, sans-serif; line-height: 19.45599937438965px;">supposed to wreck it.</span></span></blockquote><br />Kirsten (1st Period):<br /><blockquote class="tr_bq">It's easier said than done. </blockquote><blockquote class="tr_bq"> </blockquote>We've grown accustomed to math that does the following:<br /><br />1. Teacher asks question<br />2. Student answers question<br />3. Teacher evaluates answer while student moves on about her day<br /><br /><br />Hypothesis wrecking requires a different model -- one that asks students to take a look in the mirror and give constant self-evaluation. It also depends on problems that lend themselves to establishing this mindset. These aren't always easy to find, though. I've found that the problems with a really simple prompt tend to work best. Here is a list of problems I've used:<br /><br />1. <a href="http://www.geogebratube.org/student/m1992">The Diagonal Problem</a><br /> <br />2. <a href="http://nrich.maths.org/1867">Pick's Theorem</a><br /> <br />3. <a href="http://sites.davidson.edu/mathmovement/doodling-with-euler/">Doodle Math</a><br /><br />4. <a href="http://connectedmath.msu.edu/CD/Grade6/Locker/">The Locker Problem</a><br /><br />5. <a href="http://nrich.maths.org/6708&part=solution">The Handshake Problem</a><br /><br />6. <a href="http://map.mathshell.org/materials/lessons.php">Tilted Squares</a> (or Pythagorean Theorem disguised)<br /> I don't actually use this lesson, but I liked how the "tilt" of the square was defined as x/y which makes data gathering quite nice.<br /><br />7. <a href="https://drive.google.com/file/d/0B3qH9ejRh_g0RzhwTVk0WUkyMk0/edit?usp=sharing">How many ways?</a><br /><br />8. Pile Pattern Problems<br /> a. <a href="https://twitter.com/fawnpnguyen">Fawn's</a> <a href="http://www.visualpatterns.org/">Visual Patterns</a><br /> b. <a href="http://geogebratube.org/student/b91447">GeoGebra Book</a> we're working on<br /><br />9. <a href="https://twitter.com/woutgeo">Avery's</a> <a href="https://drive.google.com/a/portervilleschools.org/?tab=mo#folders/0B3qH9ejRh_g0YzlmODMzMWUtMjExMC00NjBmLWIyNDktNTEyMDFhNDk5YjJi">Edges, Vertices, and Faces</a><br /><br />10. <a href="http://nrich.maths.org/2322">The Painted Cube</a><br /><br />11. <a href="http://www.lulu.com/shop/james-tanton/math-without-words/ebook/product-17439362.html">Math Without Words</a><br /><br />12. Add the numbers 1 to n.<br /><br /><br /><br /><br /><br /><blockquote class="tr_bq"> </blockquote>http://coxmath.blogspot.com/2014/04/fostering-hypothesis-wrecking-mindset.htmlnoreply@blogger.com (David Cox)3tag:blogger.com,1999:blog-5964889903484807623.post-7820483544635659242Fri, 04 Apr 2014 18:59:00 +00002014-04-04T11:59:45.749-07:00pedagogyperpendicular bisectorssystems of equationsInsteadYou know what?<br /><br />Instead of having to teach things like perpendicular bisectors and systems of equations, I just wish we could do things like<a href="https://www.jasondavies.com/maps/voronoi/airports/"> this.</a><br /><br /><div class="separator" style="clear: both; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-k-tKSyjrC7E/Uz8AthwwyrI/AAAAAAAABdg/6jEWiFkzhOQ/s1600/Airports.jpg" height="520" width="640" /></div><br />http://coxmath.blogspot.com/2014/04/instead.htmlnoreply@blogger.com (David Cox)4tag:blogger.com,1999:blog-5964889903484807623.post-3112469284640633620Fri, 04 Apr 2014 17:58:00 +00002014-04-04T11:00:57.174-07:00pedagogyproblem solvingHypothesis Wrecking and the Diagonal ProblemWe've been doing more problems lately where students can gather data and look for patterns. Today's installment is via the <a href="http://www.geogebratube.org/student/m1992">Diagonal Problem</a> which I think I first saw via <a href="http://function-of-time.blogspot.com/2011/08/good-problems-follow-that-diagonal.html">Kate</a>.<br /><div><br /></div><div>I'm noticing that more kids are gaining confidence in looking for patterns, forming hypotheses and then seeing if they can make the hypothesis fail. The phrase that seems to be gaining ground when it comes to hypothesis testing is "wreck it"-as in "Oh, you think you have a rule? See if you can wreck it."</div><div><br /></div><div>This diagonal problem is nice because a lot of students seem to zero in on special cases. For example, an n x n (or I just call them squares) rectangle has a diagonal that passes through n squares. There have also been some nice attempts at nailing down rules for odd x odd and even x even rectangles. We're finding that special cases don't lead us right to a general rule, but the information can be useful. </div><div><br /></div><div><br /></div><div>I've put together a flow chart that seems to be helpful. </div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-qlRQczKqJB0/Uz7xcOAL_oI/AAAAAAAABdQ/Jn4D_TPQzJE/s1600/Hypothesis+Wrecking.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-qlRQczKqJB0/Uz7xcOAL_oI/AAAAAAAABdQ/Jn4D_TPQzJE/s1600/Hypothesis+Wrecking.jpg" height="532" width="640" /></a></div><div></div><div>Some students get caught in the <b>Do research-->do you see a pattern?--> Do research</b> loop others are making it to the hypothesis before being kicked back to research. All are having to come face to face with their impatience. Some are owning it. </div><div><br /></div><div>There are a lot of mistakes being made. There's some frustration. There's arguing. There's collaboration. </div><div><br /></div><div>There's learning. </div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div>http://coxmath.blogspot.com/2014/04/hypothesis-wrecking-and-diagonal-problem.htmlnoreply@blogger.com (David Cox)0tag:blogger.com,1999:blog-5964889903484807623.post-6466968496535591692Fri, 28 Mar 2014 22:38:00 +00002014-03-28T15:38:43.143-07:00algebraCCSSdesmoslinear functionsmath 8Desmos Art: Fries<div><iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/7ThEbWfw32o" width="560"></iframe></div><div><span style="font-size: large;"><b><br /></b></span><span style="font-size: large;"><b>The Assignment</b></span></div><div><br /></div><div>1. Draw a design using only straight lines. </div><div>2. Snap a photo of your drawing. </div><div>3. Import your photo into <a href="http://desmos.com/">Desmos</a>. </div><div>4. See if you can duplicate your drawing using linear functions. </div><div><br /></div><div><b><span style="font-size: large;">Assessment</span></b></div><div><br /></div><div>We looked for two things: challenge and precision. We got anything from a simple right triangle to a box of fries. Anything with 3 or more lines intersecting at the same point proved to be very challenging; 2 lines fairly challenging; no intersections...not so much. </div><div><br /></div><div>Precision was key. They could fool themselves as long as the grid, axes and labels were turned on. But once those things went away, it was just lines drawn (with pencil or graphs) by students. </div><div><br /></div><div>I created a filter that contained the words "shared a graph with you" that dumped the graphs students emailed directly into a folder labeled "Desmos Art." I used <a href="http://www.techsmith.com/snagit.html">SnagIt </a>to take screenshots and highlight areas of the graphs that needed feedback and sent the screenshots back to students. It was a pretty nice workflow, actually. <br /><br /><b><span style="font-size: large;">Student Feedback</span></b></div><div>Most students noted that they started out thinking this assignment was really hard--that they couldn't do it. Then they got their first line to match. The second line was easier then the first; third easier than the second , and so on. Domain restrictions turned into range restrictions for vertical lines and some learned really quickly that it was easier to restrict the range for really steep lines. <br /><br />The perseverance I saw in students makes this one a keeper. </div><div><br /></div><div><br /></div><div><br /></div>http://coxmath.blogspot.com/2014/03/desmos-art-fries.htmlnoreply@blogger.com (David Cox)2tag:blogger.com,1999:blog-5964889903484807623.post-7929081888327521442Wed, 26 Feb 2014 23:01:00 +00002014-02-26T15:01:31.389-08:00desmosfunctionsThe Effects of Immediate FeedbackA real big shout out to <a href="http://desmos.com/">Desmos</a>, <a href="http://blog.mrmeyer.com/">Dan</a> and <a href="http://christopherdanielson.wordpress.com/">Christopher</a> for their work on the <a href="http://class.desmos.com/carnival">Function Carnival</a>.<br /><br /><br /> <iframe allowfullscreen="" frameborder="0" height="360" src="//www.youtube.com/embed/Gze55bRVqUM?rel=0" width="480"></iframe>http://coxmath.blogspot.com/2014/02/the-effects-of-immediate-feedback.htmlnoreply@blogger.com (David Cox)0tag:blogger.com,1999:blog-5964889903484807623.post-8628627520670697699Wed, 26 Feb 2014 00:12:00 +00002014-02-25T16:13:00.505-08:00Science: Completing the Square<a href="http://www.youtube.com/watch?v=vKA4w2O61Xo&list=UUHnyfMqiRRG1u-2MsSQLbXA&feature=c4-overview&edufilter=r3WxLadeYrtfg780KwZwqQ&safe=active">Veritasium</a><u>: </u><br /><blockquote class="tr_bq">"If you think that something is true, you should try as hard as you can to disprove it. Only then, can you really get at the truth and not fool yourself."</blockquote>This video could not have come at a better time. In fact, Derek sums up in about 4:40 what's taken me a semester to convey to my students. <br /><br />Last week's quiz asked students to investigate a parabola given these three points. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-Bq3WVE-gdSI/Uw0rpkP9f7I/AAAAAAAABcc/ouMHcKOXrAo/s1600/prompt.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-Bq3WVE-gdSI/Uw0rpkP9f7I/AAAAAAAABcc/ouMHcKOXrAo/s1600/prompt.jpg" height="320" width="240" /></a></div>Naturally, many assumed that (-5, -3) was the vertex. However, upon close examination, they should have realized that the rates of change between the points wouldn't allow that. So, the next day, we set out on an exploration. <br /><br />"Ok, open up Desmos, plot the points and find a quadratic that fits these points."<br /><br />It didn't take long before a student came up with <b>y = x<sup>2</sup> + 8x + 12</b>. We all verified it and then I suggested we try something else. <br /><br />"Enter the function <b>y = a(x - h)<sup>2</sup> + k</b> and make sliders for a, h and k. Now find a function that fits."<br /><br />Soon we had <b>y = (x + 4)<sup>2</sup> - 4</b>. Students loved this form because of the obvious horizontal and vertical shifting that was going on. <br /><br /><span style="font-size: large;"><b>Problem</b></span><br />Is <b>y = x<sup>2</sup> + 8x + 12 </b>the exact same function as <b>y = (x + 4)<sup>2</sup> - 4? </b>If so, is there a way we can take a quadratic in standard form and re-write it in this magical form? <br /><br /><span style="font-size: large;"><b>Research</b></span><br />We spent the better part of a period graphing functions in standard form and then matching them in vertex form. <br /><br />"Now, look at all the different functions you have. Write down what you <i>think</i> is going on here. <br /><br /><span style="font-size: large;"><b>Hypotheses</b></span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-9ijViP44RSY/Uw0wuYoOPNI/AAAAAAAABcs/A--ejGjnP2s/s1600/hyp.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-9ijViP44RSY/Uw0wuYoOPNI/AAAAAAAABcs/A--ejGjnP2s/s1600/hyp.jpg" height="320" width="152" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Note: To this point, all quadratics have been a =1. </td></tr></tbody></table><b><br /></b><b><br /></b>Their job over the next day is to determine which hypotheses (if any) they'd like to accept. <br /><br />I'll keep you posted. <br /><br />http://coxmath.blogspot.com/2014/02/science-completing-square.htmlnoreply@blogger.com (David Cox)1tag:blogger.com,1999:blog-5964889903484807623.post-4211493627760816660Thu, 05 Dec 2013 23:12:00 +00002013-12-06T10:22:03.380-08:00factoringinquirypedagogyquestionsI Like TrianglesLast night, I asked if anyone could point me back to this fantastic<a href="http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/"> animated factorization visualization.</a> (h/t <a href="https://twitter.com/calcdave">@calcdave</a>)<br /><br />Now, I'm kicking myself for not thinking to use this in the first weeks of the school year. Talk about some <a href="http://blog.mrmeyer.com/?p=18252">Fake World</a> math doing a number on pseudo engagement strategies. <br /><br />I started the animation at the end of each period and walked out to greet students as they walked in. Once everyone got settled, I walked back in the room and each time the dots would circle up, I'd yell, "PRIME!"<br /><br />"Alright, I'm up 1-0. PRIME!, man I'm smoking you guys."<br /><br />Kids caught on really quick and started looking for the circled numbers. In fact, it took many students a while to realize that the applet literally said "prime." <br /><br />We started out by looking at the patterns and how each number was represented visually.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-kgP71hySp2s/UqECS7mBP_I/AAAAAAAABZw/_iqhKq-9ATA/s1600/105.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="380" src="http://2.bp.blogspot.com/-kgP71hySp2s/UqECS7mBP_I/AAAAAAAABZw/_iqhKq-9ATA/s400/105.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">But, next I said, "You know what, I really like triangles. What is the smallest number that will give us a triangle?"</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This one's easy. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-UDs8C2u7mz0/UqECWDVllxI/AAAAAAAABas/5JnQ-W1Sfjw/s1600/3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="298" src="http://1.bp.blogspot.com/-UDs8C2u7mz0/UqECWDVllxI/AAAAAAAABas/5JnQ-W1Sfjw/s320/3.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Alright, what's the next number that will give us nothing but triangles? Write your guesses on your easel."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Guesses were about 50-50 between 6 and 9. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-1wNjiZNPWQo/UqECUyRc4nI/AAAAAAAABao/EAaaQdZNxos/s1600/9.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="256" src="http://1.bp.blogspot.com/-1wNjiZNPWQo/UqECUyRc4nI/AAAAAAAABao/EAaaQdZNxos/s320/9.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Alright, what about the next one?" </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Still guesses were a little sporadic. But by the time we got to 81, most students thought they figured out a pattern. From 243 on, we were at about 100%. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-QOt_6sN8JBM/UqECUvAzsFI/AAAAAAAABag/lD5RF_bKP-Q/s1600/81.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="280" src="http://4.bp.blogspot.com/-QOt_6sN8JBM/UqECUvAzsFI/AAAAAAAABag/lD5RF_bKP-Q/s320/81.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div>After about 10 minutes of doing this and discussing our results, I put up this slide.<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-EUoZHwndQJ4/UqEH4C0-h2I/AAAAAAAABa4/sRegfDiihl8/s1600/6561.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="352" src="http://1.bp.blogspot.com/-EUoZHwndQJ4/UqEH4C0-h2I/AAAAAAAABa4/sRegfDiihl8/s640/6561.jpg" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">There was a nice discussion on clarifying our question. Three different student offerings illustrated the idea of First Idea; Best Idea. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Student 1: At what stage is each triangle?</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Student 2: How many triangles are in the green circle?</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Student 3: How many dots are there in each circle? *Boom*</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Now get to it and be prepared to justify your answer. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The first student said there were 9, 27, 81 and 243 dots. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Ok, great. So how did you do that?"</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Well, the green circle has 9 dots, then I multiplied by 3 to get the red. Multiplied by 3 again to get the blue and then by 3 again to get the black."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Alright, so let's press on this idea a little."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I know what question I want to ask, but I just bit my tongue until a student speaks up.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"How can you be sure that there are 9 dots in the green circle?" *There it is*</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"I estimated."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Here's where it gets good. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"> From across the room, I hear, "It looks like there are 9 dots in the green circle, but we have to look past that."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Wait, what?</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Yeah, we can't trust the picture because the dots are too small. We know there are 6,561 dots on the whole page and there are three black circles of dots. We have to start there."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">So, Dave, keep this link handy, I'm sure I'll be asking for it again next year. </div><br />http://coxmath.blogspot.com/2013/12/i-like-triangles.htmlnoreply@blogger.com (David Cox)3tag:blogger.com,1999:blog-5964889903484807623.post-6442036777137747016Tue, 26 Nov 2013 16:30:00 +00002013-11-26T08:30:02.181-08:00algebralinear equationspedagogyquestionsFirst Idea; Best Idea......and the Worst Idea<br /><br /><a href="http://coxmath.blogspot.com/2010/07/creating-culture-of-questions.html">Creating a Culture of Questions</a> was, by far, the most popular post on this blog until someone somewhere starting linking to the post on Exponent Rules. <br /><br />I think a natural follow up to the Culture piece would be with regards to establishing a classroom culture where feedback is given and accepted. <br /><br /><br /><b><span style="font-size: large;">The First Idea is the Best Idea and the Worst Idea</span></b><br /><br />The first time students hear this, I usually get, "Gosh, that's mean."<br /><br />But we discuss how the first person who puts forth an idea holds the best idea as there is nothing to which we can compare it. But using the same logic, this idea should be the worst. This assumes the flow of ideas that should follow. <br /><br />I think this encourages two important things:<br /><br />1. "If I go first, it doesn't matter that my idea isn't fully formed." This student has established a floor on which each other student can stand and/or build.<br /><br />2. "I can take someone's idea and help them make it better." The real work is done by the first follower. This student chips away at any imperfections and helps the first student refine her idea. Subsequent students then follow suit. <br /><br /><br /><b><span style="font-size: large;">What's this look like?</span></b><br /><br />Yesterday, we trying to determine the equation between the points below and students wanted the y-intercept.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-zHGEOHWx5tw/Uo_lFo6pmNI/AAAAAAAABZY/E61I_pL6Ovo/s1600/graph.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="268" src="http://4.bp.blogspot.com/-zHGEOHWx5tw/Uo_lFo6pmNI/AAAAAAAABZY/E61I_pL6Ovo/s400/graph.jpg" width="400" /></a></div><br />Students were using what they knew about slope to find other points and had to wrestle with the fact this particular line doesn't have a lattice point for a y-intercept. Once we were finished, I asked students to write down any questions they had.<br /><br />Student 1: "I have a comment."<br /><br />"Ok, what is it?"<br /><br />Student 1: "No matter which points we choose, the slope simplifies to the same thing."<br /><br />"Can you turn your observation into a question?"<br /><br />Student 1: "Will that happen all the time?"<br /><br />Now here is where it happens.<br /><br />"I can misunderstand [Student 1]'s question, can we make this more precise?"<br /><br />Student 2: "Will the slopes always simplify to the same thing?"<br /><br />Student 3: "Will the slopes between two points always simplify to the same thing?"<br /><br />"Are we only using two points?"<br /><br />Student 4: "Will the slopes between three points always simplify to the same thing?"<br /><br />Student 5: "Will the slopes between any two pairs of points always simplify to the same thing?"<br /><br />Student 6: "Are the slopes between any two pairs of points always equal?"<br /><br />"Are we really talking about any 4 points here?"<br /><br />Student 7: "Are the slopes between any two pairs of points on a line always equal?"<br /><br /><br /><br /><br />http://coxmath.blogspot.com/2013/11/first-idea-best-idea.htmlnoreply@blogger.com (David Cox)3tag:blogger.com,1999:blog-5964889903484807623.post-4551328872912182142Fri, 22 Nov 2013 19:15:00 +00002013-11-22T11:15:51.463-08:00assessmentpblproblem solvingThe Farming [Thing]I called this a <a href="http://coxmath.blogspot.com/2010/03/farming-project.html">Project</a>. It's not. It's more of a problem-y kind of performance task learning opportunity assessment <strike>of</strike> <strike> for</strike> <strike>of</strike> for? learning that hits close to home. Literally. We<a href="http://en.wikipedia.org/wiki/San_Joaquin_Valley#Agriculture"> live in</a> a huge agricultural area and kids don't know what an acre is. Anything that gives students a chance to wrestle with the fact that a piece of land can't have dimensions of 20 acres x 20 acres, is a win. Anything that allows me to answer the question "What's an acre-foot?" by doing <a href="http://lmgtfy.com/?q=what+is+an+acre-foot%3F">this</a>, is a win. <br /><br />In this <strike>project </strike> problem's first iteration, I was focused on the skills of equation writing, line graphing and solving mixture and work problems. <br /><br />In the second iteration, I was less focused on the skills and more interested in having students explain what each component of an equation represented, why we'd want that equation and how graphing inequalities made sense. We got to discuss why understanding the problem makes sense--kids tried to hire crews to prune cotton. For you city-slickers out there--you don't prune cotton. It doesn't grow on trees. Students had to sign up via Google form to interview with me as they finished a task. I did something north of 175 interviews for one class that year. <br /><br />This year, I've changed it a bit more. They are no longer tasks, they're constraints. There are fewer of them and they don't specifically tell kids what to do. Before, I told them to create inequalities and graph them. Now, I'm removing some of the scaffold. They get to decide what tools they want to use. Before, I did this project after we had done systems, mixture and work problems. This time, we have only done systems. They're going to have to work through the mixture/work stuff. <br /><br />That's been the highlight--the mixture problems. I have a few students who went straight for that constraint and have been on a mission to figure out how to make sense of it. <br /><br />Today, one boy asked, "Mr. Cox, how accurate to I need to be? I'm accurate to the trillionth, but I can't get it to be exactly 36%."<br /><br />I said, "How accurate do you think you need to be? We're killing weeds, not sending someone to space."<br /><br />So, with all that, here's the <a href="http://www.geogebratube.org/material/show/id/19311">updated version</a> complete with dynamic answer key. <br /><br />http://coxmath.blogspot.com/2013/11/the-farming-thing.htmlnoreply@blogger.com (David Cox)0tag:blogger.com,1999:blog-5964889903484807623.post-4453116561316873832Wed, 06 Nov 2013 16:54:00 +00002013-11-06T08:54:20.285-08:00The Real FlipIf we can get students to flip their thinking from this:<br /><br /><br /><br /><div style="text-align: center;"><span style="font-size: x-large;">If I know the rules, then I can do the math. </span></div><br /><br /><br /><br />to this:<br /><br /><br /><br /><div style="text-align: center;"><span style="font-size: x-large;">If I do the math, I can know the rules.</span></div><br /><br /><br />Then we've won.http://coxmath.blogspot.com/2013/11/the-real-flip.htmlnoreply@blogger.com (David Cox)2tag:blogger.com,1999:blog-5964889903484807623.post-2178606726876974092Fri, 01 Nov 2013 22:03:00 +00002013-11-01T15:03:11.300-07:00assessmentCCSSpedagogyI'm Bringing Multiple Choice BackSo here's the idea:<div><br /></div><div>One problem with multiple paths to solution. Students connect as many skills as they can to the problem. I listed eight possible skills two of which wouldn't necessarily apply to the problem. Students had to assess themselves on the skills they demonstrated. </div><div><br /></div><div>Question of the day: "Mr. Cox, is it possible to use all of these skills?"</div><div><br /></div><div>Answer to Question of the day: "It's possible that some of the skills don't apply."</div><div><br /></div><div>For this first iteration, I used the standard Ticket Problem. </div><div><br /></div><div>Below are samples of student work. </div><div><br /></div><div>As an exercise for the reader:</div><div><br /></div><div>1) What are your thoughts on this process? </div><div>2) How did each student do?</div><div><br /></div><div>Let me know in the comments. </div><div><br /></div><div style="text-align: center;"><span style="font-size: large;"><b>Student A</b></span></div><div><a href="http://3.bp.blogspot.com/-J2HcW5IM1MU/UnQj7CzVwbI/AAAAAAAABY8/GuxHN3TBN_g/s1600/Assessment+4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="568" src="http://3.bp.blogspot.com/-J2HcW5IM1MU/UnQj7CzVwbI/AAAAAAAABY8/GuxHN3TBN_g/s640/Assessment+4.jpg" width="640" /></a></div><div><span style="font-size: large;"><b><br /></b></span></div><div><span style="font-size: large;"><b><br /></b></span></div><div style="text-align: center;"><span style="font-size: large;"><b>Student B</b></span></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-nrkYgLiFyzw/UnQj3m5uXQI/AAAAAAAABY0/JacQgBiuVlE/s1600/Assessment+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="582" src="http://1.bp.blogspot.com/-nrkYgLiFyzw/UnQj3m5uXQI/AAAAAAAABY0/JacQgBiuVlE/s640/Assessment+3.jpg" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div><span style="font-size: large;"><b><br /></b></span></div><div><span style="font-size: large;"><b><br /></b></span></div><div style="text-align: center;"><span style="font-size: large;"><b>Student C</b></span></div><div class="separator" style="clear: both; text-align: center;"></div><div><a href="http://2.bp.blogspot.com/-F_vgeiJj34U/UnQjygfVYsI/AAAAAAAABYs/60MtWdpV2iU/s1600/Assessment+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="640" src="http://2.bp.blogspot.com/-F_vgeiJj34U/UnQjygfVYsI/AAAAAAAABYs/60MtWdpV2iU/s640/Assessment+2.jpg" width="638" /></a></div><div><span style="font-size: large;"><b><br /></b></span></div><div style="text-align: center;"><span style="font-size: large;"><b>Student D</b></span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-IBFtkpP3DiA/UnQesAHh-gI/AAAAAAAABYY/gxqYc1ZVF-w/s1600/Assessment+Photo.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://3.bp.blogspot.com/-IBFtkpP3DiA/UnQesAHh-gI/AAAAAAAABYY/gxqYc1ZVF-w/s640/Assessment+Photo.JPG" width="502" /></a></div><div><span style="font-size: large;"><b><br /></b></span></div>http://coxmath.blogspot.com/2013/11/im-bringing-multiple-choice-back.htmlnoreply@blogger.com (David Cox)3tag:blogger.com,1999:blog-5964889903484807623.post-9010108295048046276Sun, 20 Oct 2013 19:27:00 +00002013-10-20T12:27:00.637-07:00anecdotesCCSSpedagogyproblem solvingWhen Perseverance Pays OffOur high schools are committed to taking the integrated path with the first three courses and since the middle schools will be teaching these courses, I'm part of the team building the units. I've been piloting the curriculum by the folks from <a href="http://www.mathematicsvisionproject.org/secondary-one-mathematics.html">Utah</a> and, for the most part, I like it. I'm particularly enjoying the learning cycle they employ: Develop Understanding, Solidify Understanding and Practice Understanding, mostly because it's pretty easy to discuss with the majority of teachers. <br /><br />This task was at the beginning of a learning cycle. <br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Xsqz9Cy47n4/UmGHERkEX6I/AAAAAAAABXs/QVZTu54tUq8/s1600/Shopping+for+cats+and+dogs.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://2.bp.blogspot.com/-Xsqz9Cy47n4/UmGHERkEX6I/AAAAAAAABXs/QVZTu54tUq8/s640/Shopping+for+cats+and+dogs.jpg" width="571" /></a></div>Source: <a href="http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec1_mod2_systems_se_71713.pdf">Mathematics Vision Project</a><br /><br />I had a student, H, come to me before class and say, "Mr. Cox, I spent like three hours on problem 3 last night. I couldn't quite get it."<br /><br />During class, students worked with their groups and started presenting solutions. As I approach H's group, she gives a high-five to the student next to her.<br /><br />Me: "What's that about?"<br /><br />H: "We figured it out! I get it now." Then she shows me her solution.<br /><br />"Feels nice, huh?"<br /><br />"Yeah, I think I'm gonna cry."<br /><br />Me too, H. Me too. http://coxmath.blogspot.com/2013/10/when-perseverance-pays-off.htmlnoreply@blogger.com (David Cox)0tag:blogger.com,1999:blog-5964889903484807623.post-7437747291448883314Fri, 18 Oct 2013 19:03:00 +00002013-10-18T12:03:42.728-07:00assessmentCCSSproblem solvingrubricself assessmentThe Student RubricWe are currently working on a performance task where students have to gather data, apply a line of best fit, determine a rate and then make a prediction. It's been a task to help students shift their thinking from right/wrong to more/less. In other words, I don't want them to see their understanding as binary--I get it; I don't get it. I want them to see their understanding as something that falls on a continuum. <br /><br />When doing something like finding a line of best fit, I think it's less important to discuss <b>what </b>the line looks like and more important to discuss <b>why </b>a particular line is <i>best.</i> This leads us to the descriptors we've been using to discuss both sides of the same coin:<br /><b><br /></b><b><span style="font-size: large;">Concept and Precision</span></b><br /><br />5: Strong concept; Precise<br /><br />4: Strong concept; Somewhat precise<br /><br />3: Problem with concept; Somewhat precise<br /><br />2: Problem with concept; Lacks precision<br /><br />1: No attempt<br /><br />Through a few discussions with different classes, the top three descriptors have evolved into something like this.<br /><br />5: Precise answer with precise method<br /><br />4: Estimate backed by reason<br /><br />3: Estimate<br /><br />Then I walked by a student and noticed the self-assessment she was doing. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-0rhMg5kiiPQ/UmGFiV8qRcI/AAAAAAAABXk/6MCiqUcY1bw/s1600/student+rubric.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://4.bp.blogspot.com/-0rhMg5kiiPQ/UmGFiV8qRcI/AAAAAAAABXk/6MCiqUcY1bw/s640/student+rubric.jpg" width="452" /></a></div><br />How's that for kid friendly?http://coxmath.blogspot.com/2013/10/the-student-rubric.htmlnoreply@blogger.com (David Cox)2tag:blogger.com,1999:blog-5964889903484807623.post-2694928508309807647Wed, 17 Jul 2013 17:45:00 +00002013-07-17T10:45:24.266-07:00anecdotescurriculumpedagogyAdventures in Pedagogy: Four ZeroAidan was having trouble with subtracting numbers that required <s>renaming</s> <s>renaming</s><s>borrowing</s>, umm, taking away more than you have. He was simply taking the bottom smaller digit from the larger regardless of which one was <s>the subtrahend or minuend</s> on top or bottom.<br /><br />He picked up some sort of rule along the way and was obviously misusing it. So he decided to get a little creative. <br /><br /><a target="_blank" href="http://youtu.be/-IXdw04D6-E">Take a look.</a><br /><br /><br />- Posted using BlogPress from my iPhone<br />http://coxmath.blogspot.com/2013/07/adventures-in-pedagogy-four-zero.htmlnoreply@blogger.com (David Cox)5tag:blogger.com,1999:blog-5964889903484807623.post-4205340247953446760Thu, 23 May 2013 14:30:00 +00002013-05-23T07:30:02.685-07:00The Dataor<b> How Fast Does Google Think We Drive?</b><br /><br />I picked this question up on Twitter a while back and really liked it. So, after we worked on making a <a href="http://coxmath.blogspot.com/2013/05/the-plan.html">plan</a>, I thought it would be good to look at data. Lots of data. And then coming up with ways to make sense out of it.<br /><br /><b><span style="font-size: large;">Process</span></b><br /><br />So, we took to Google. Students went to Maps, entered starting location and destination (I didn't realize that I'd need to explain that you can't drive from California to China, but, whatever.) and then entered their data into a Google Form. We did this over five different classes and got a lot of "<a href="https://docs.google.com/spreadsheet/ccc?key=0AnqH9ejRh_g0dEprN1V1MUxEbkc2cGNLMFlXQmR0bXc&usp=sharing">stuff</a>" to sift through. <br /><br />We dumped the data into GeoGebra and then took a look at a <a href="https://docs.google.com/file/d/0B3qH9ejRh_g0eXJIZnZZQzJyZXM/edit?usp=sharing">few different perspectives.</a> It's interesting to see how the data changes as we look at trips of different distances. <br /><br />The applet below will really give you a good picture.<br /><br /><iframe height="500px" scrolling="no" src="http://www.geogebratube.org/material/iframe/id/28760/width/800/height/500/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/false" style="border: 0px;" width="800px"> </iframe><br /><br /><br /><span style="font-size: large;"><b>Takeaways</b></span><br /><br /><br /><ul><li>Unit rates are valuable. </li><li>When points don't line up perfectly, sometimes we can use a line to help us answer questions. </li><li>As soon as we have a line we like, the actual data points can kind of get in the way. </li><li>You can't drive to China. </li></ul><br /><br />These are 7th graders and they have some experience with linear relationships. However, that experience has been limited to "the number in front of the x is the slope and the other number is the y-intercept" kind of stuff. It really threw some kids that their line of "best fit" may not have been the same as everyone else's. We are doing this very informally at this time. <br /><br /><span style="font-size: large;"><b>Questions</b></span><br /><br />I know that the formal process for determining a linear regression is pretty involved, but does it have to be for a proportional relationship? That is, if we know a relationship (like distance : time) is a proportion but the data doesn't line up exactly, is it appropriate to simply average the distance:time ratios to determine a "rate of best fit?"<br /><br />When informally drawing a line of best fit for a proportional relationship, should (0,0) always be the starting point?<br /><br />http://coxmath.blogspot.com/2013/05/the-data.htmlnoreply@blogger.com (David Cox)4tag:blogger.com,1999:blog-5964889903484807623.post-4141968628059429759Wed, 22 May 2013 16:38:00 +00002013-05-22T09:38:57.984-07:003 ActselectricitymodelingQuestions? Yeah, I've got questions.or Modeling Problems<div><br /></div><div><br /></div><div>My electric bill is a mystery. I started looking into how the bill is actually calculated and found some interesting stuff. </div><div><br /></div><div><blockquote class="twitter-tweet">So Cal Edison coming through with some lesson potential. <a href="http://t.co/lLR0qYzVxH" title="http://twitter.com/dcox21/status/333267878218457088/photo/1">twitter.com/dcox21/status/…</a><br />— David Cox (@dcox21) <a href="https://twitter.com/dcox21/status/333267878218457088">May 11, 2013</a></blockquote></div><div><div><blockquote class="twitter-tweet" data-conversation="none">@<a href="https://twitter.com/dcox21">dcox21</a> oh you Californians and your complicated pricing schemes.<br />— Dan Anderson (@dandersod) <a href="https://twitter.com/dandersod/status/333280017717030912">May 11, 2013</a></blockquote></div><div><blockquote class="twitter-tweet" data-conversation="none">@<a href="https://twitter.com/dandersod">dandersod</a> I prefer to call it "social engineering."<br />— David Cox (@dcox21) <a href="https://twitter.com/dcox21/status/333292262270312448">May 11, 2013</a></blockquote></div><div><blockquote class="twitter-tweet" data-conversation="none">@<a href="https://twitter.com/dcox21">dcox21</a> Hi David, please feel free to contact us at CCOeChannel@sce.com or call us at 800-655-4555 if you have questions.<br />— SCE (@SCE) <a href="https://twitter.com/SCE/status/333464025302171649">May 12, 2013</a></blockquote></div><div><script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script></div></div><div><br /></div><div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I don' think SCE appreciated the "social engineering" comment. </div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">So, I decided to turn this into a <a href="http://www.101qs.com/2229-electric-bill">3 Act lesson</a>. Except, their price doesn't fit my model that I modeled after their model.</div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-AIcHwqkSe-0/UZzw4hGzBjI/AAAAAAAABU4/W82A4cfjQ_g/s1600/Model+Trouble.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="350" src="http://3.bp.blogspot.com/-AIcHwqkSe-0/UZzw4hGzBjI/AAAAAAAABU4/W82A4cfjQ_g/s640/Model+Trouble.jpg" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div><br /></div><div>What am I missing? </div></div>http://coxmath.blogspot.com/2013/05/questions-yeah-ive-got-questions.htmlnoreply@blogger.com (David Cox)2tag:blogger.com,1999:blog-5964889903484807623.post-5702105727833507444Thu, 02 May 2013 19:34:00 +00002013-05-02T14:16:10.399-07:00integratedproblem solvingThe PlanI blogged about the <a href="http://coxmath.blogspot.com/2013/05/middle-school-modeling-integrated.html">template</a> I'm using. Most of the activities we have done have focused on a particular piece. We did two quick activities focusing on making a plan. Before sending students outside, they had to submit their plan for peer review. If another group could read their plan and understand what was going to be done, then I signed off on it. <br /><span style="font-size: large;"><b><br /></b></span><span style="font-size: large;"><b>Day 1: How Far?</b></span><br /><b><br /></b><b>Question:</b> How far is it from the first tree to the last tree (ie. point A to point B)? <br /><div><br /><b>Rules:</b> You can take a pencil, paper and clipboard outside with you. Nothing else. </div><br /><br /><div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><br /> <a href="http://1.bp.blogspot.com/-47wmBu3PNV8/UYGHWCrRrZI/AAAAAAAABRc/owhWJdKJ0uA/s1600/sequoia+plaza.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="271" src="http://1.bp.blogspot.com/-47wmBu3PNV8/UYGHWCrRrZI/AAAAAAAABRc/owhWJdKJ0uA/s400/sequoia+plaza.jpg" width="400" /></a><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br />Different groups were able to tell me the distance from one tree to the other using units like:<br /><ul><li>Jose's feet</li><br /><li>Jasmines (not her feet, but her)</li><br /><li>Clipboards</li><br /><li>Brandon's longest stride</li></ul>A few groups made adjustments to their plans once they got outside and saw how tedious it would be to try to walk a heel-to-toe straight line. We had quite a few groups decide to measure the distance from the first tree to the second and then just multiply. This led to a couple of really good conversations that went something like this:<br /><br /><div class="separator" style="clear: both; text-align: left;"><br /></div>Student: "Mr. Cox, we are going to measure from the first to the second then multiply by the number of spaces."<br /><br /><div class="separator" style="clear: both; text-align: left;">Me: "Will that work?"</div><br /><div class="separator" style="clear: both; text-align: left;">"Yeah. Because the spaces are the same."</div><br /><div class="separator" style="clear: both; text-align: left;">"How can you be sure? "</div><br /><div class="separator" style="clear: both; text-align: left;">"Because look at them..."</div><br /><div class="separator" style="clear: both; text-align: left;">"Yeah, I want them to be the same too. That'd be really helpful, huh?"</div><br /><div class="separator" style="clear: both; text-align: left;">Now they have dilemma: do they go and measure the distance between each tree or just measure the entire distance from the first to the last? (wait, that's the same thing...which makes it a doublemma) </div><br /><div class="separator" style="clear: both; text-align: left;">Oh, you want my help? Lemme show you how Google Earth can help you out here. </div><br /><div class="separator" style="clear: both; text-align: left;"><br /></div><br /><b style="font-size: x-large;">Day 2: How high?</b><br /><div class="separator" style="clear: both; text-align: left;"><b><br /></b></div><div class="separator" style="clear: both; text-align: left;"><b>Question</b>: Come up with two different methods for finding the height of the building. </div><b><br /></b><b>Rules</b>: Don't climb up there. </div><br /><a href="https://picasaweb.google.com/100113176767768609240/CoxBoys#5873452300848404610"><img border="0" height="400" src="https://lh3.googleusercontent.com/-69MwxxibHsI/UYKxv61FGII/AAAAAAAABR8/u9Yfhr2LPBA/s400/7.jpg" style="margin: 5px;" width="298" /></a><br /><br />Some of the methods:<br /><ul><li>Ask Chuck. (Turns out Chuck, our custodian, had a copy of the elevations.)</li><br /><li>Take a picture of Cameron next to the building and see how many Camerons to the top. </li><br /><li>All kinds of crazy uses of a meter stick. </li><br /><li>Count how many bricks in a foot and then count the total bricks. (3 bricks and spaces = 1 foot.)</li></ul>"Ask Chuck" allowed us to discuss the importance of trustworthy sources of information. And Chuck is awesome. He'd throw out all kinds of crazy numbers and see if kids would bite.<br /><b style="font-size: x-large;"><br /></b><b style="font-size: x-large;">Takeaways</b><br /><ul><br /><li>A well thought out plan makes jobs easier </li><br /><li>Sometimes we need to adjust our plans</li><br /><li>Assumptions need to be investigated</li><br /><li>We can use some tools in ways we've never imagined (eg. cell phone camera, Google Earth)</li><br /><li>Some sources aren't trustworthy</li></ul><br /><br /><br />http://coxmath.blogspot.com/2013/05/the-plan.htmlnoreply@blogger.com (David Cox)3tag:blogger.com,1999:blog-5964889903484807623.post-3589897158350418542Wed, 01 May 2013 22:12:00 +00002013-05-02T12:34:39.659-07:00integratedproblem solvingMiddle School Modeling: Integrated Math/Science<br />or<span style="font-size: large; font-weight: bold;"> </span><span style="font-weight: bold;">My Apologies to the Scientists, Polya and All the Modeling Teachers Out There</span><br /><span style="font-weight: bold;"><br /></span>I decided to go with a process rather than specific content in this class. I know stuff is going to be on the test and we need to cover it. But, I also know that my students will one day leave and go be anything <i>but </i>a scientist or a mathematician. <br />So I settled on asking students to question, think, plan, model/analyze and tell people about what they did. That's it.<br />Everything we did this semester followed this template. I found the following questions/directives to be helpful when turning students loose on a problem. <br /><br /><span style="font-size: large;"><b>1. What's the problem?</b></span><br />I think we call this "inquiry", but I really don't know anymore. Does it count if I give the question?<br /><br /><b><span style="font-size: large;">2. What do you think the answer's going to be?</span></b><br />Props to Dan for making a guess be an explicit part of the lesson plan. Something I should've been doing 10 years ago but somehow didn't.<br /><br /><span style="font-size: large;"><b>3. What smaller questions will you need to answer first?</b></span><br />This is tough. Students live in circular argumentation. I mean, c'mon kid, give me at least a spiral argument once in a while. The name of <a href="http://coxmath.blogspot.com/">this blog</a> should mean I have some grasp on the importance of questions, but I've never explicitly asked students to break larger questions into smaller manageable questions nor realized how badly students need help with this.<br /><br /><b><span style="font-size: large;">4. What's the plan for answering the smaller questions?</span></b><br />Two big take away here for students:<br />1. a good plan = good data = good analysis<br />2. plans change<br /><br /><b><span style="font-size: large;">5. Go do the plan. (ie. get your data)</span></b><br />See #4<br /><br /><span style="font-size: large;"><b>6. Make sense out of the data.</b></span><br />This was the sweet spot. How can math be used to turn data into an answer? Kids are getting the hang of this and it's fun to watch. <br /><br /><span style="font-size: large;"><b>7. Answer your question. </b></span><br />Cross check the answer with the guess.<br /><br /><span style="font-size: large;"><b>8. Tell someone about it. </b></span><br />I use the word "presentation" very loosely here. This was anything from a write-up to a group presentation to an informal interview after an activity. <br /><br /><br />None of this is new. But, for some reason, it seemed new. The first few activities we did would focus on a particular piece (I'll blog about these--this year. Promise.). The challenging part was to keep from over-planning. Not because I'm that kind of teacher, but because the more I planned, the less students had to. And, well, #4. Oh, and time. It takes a lot more time to have students make the plan and we have bells. <br /><br />http://coxmath.blogspot.com/2013/05/middle-school-modeling-integrated.htmlnoreply@blogger.com (David Cox)5tag:blogger.com,1999:blog-5964889903484807623.post-34906220969992920Tue, 26 Feb 2013 17:16:00 +00002013-02-26T09:22:23.130-08:00CCSScurriculumpedagogyunitsCCSS 8: Unit BuildingAt the end of February, representatives from grade levels K-8 spent two days unpacking the CCSS and clustering them into units of study. My previous experience with unpacking standards became a process of identifying "essential" standards which assumed the existence of non-essential standards. Those standards that didn't make the cut were ultimately ignored in favor of those that were <strike>most heavily tested</strike> essential. <br />We have the same <a href="http://www.wested.org/cs/we/print/docs/we/home.htm">provider</a> leading these new sessions, so I was a little worried we would end up looking for content to cut rather than incorporate. So far, that hasn't been the case. Obviously, there are certain topics that will require more focus (eg. linear relationships as opposed to exponents) but the goal has been to see how and where these supporting standards fit with the focus standards.<br />Our team has come up with the following units of study. We haven't reached the point where I can discuss specific activities/tasks, but I'd like some feedback on the pedagogy that motivated the clustering and sequencing.<br /><h2><br /><b>Transformational Geometry</b></h2><br />Use the coordinate plane to discuss transformations, congruence and similarity. Use dilations as an application of the ratios/proportions work done in grades 6 and 7. Use a graph as a tool to describe proportional relationships.<br /><h2><br /><b>Data Analysis</b></h2><br />Use bivariate data to create scatter plots which can then be the jumping off point for informal line of best fit (where the line may have an initial value other than zero) and an introduction for a future defining of function/non-function. <br /><h2><br /><b>Linear Relationships</b></h2><br />Graphing, graphing and more graphing. Take the informal line-of-best-fit and formalize the definition. Allow math to be it's own context. Graph systems of equations and look for common point (read: solution to system).<br /><h2><br /><b>Equations</b></h2><br />Use the work done in graphing systems to motivate more abstract symbolic manipulation required for solving linear equations.<br /><h2><br /><b>Exponents</b></h2><br />This was a tough one. Expressions with integer exponents and scientific notation seemed like an island unto themselves. We are still working on finding a place that fits nicely for these ideas.<br /><h2><br /><b>Functions</b></h2><br />Use the informal introduction to functions and formally define a function. Look at linear and non-linear functions. Compare functions using different representations (ie. graph vs. table vs. equation vs. verbal).<br /><h2><br /><b>Pythagorean Theorem</b></h2><br />I think this one speaks for itself.<br /><h2><br /><b>3D Geometry</b></h2><br />Problem solving involving the volume of cones, cylinders and spheres.<br />We are trying to move from the concrete/informal to the abstract/formal while allowing students to explore these ideas while creating their own formal definitions. I'm particularly interested in the sequence that runs from Data Analysis to Functions (note: Exponents look to be a unit that can be dropped in and our school calendar lends itself to having that unit kick off the second semester) as it may receive the most push-back from our high school colleagues. Traditional textbooks usually go the route of<br />Functions-->Equations-->Graphing-->Applications so we're going to have to have solid rationale. <br />No one pushes back better than you all. I'm counting on that.<br />http://coxmath.blogspot.com/2013/02/ccss-8-unit-building.htmlnoreply@blogger.com (David Cox)6tag:blogger.com,1999:blog-5964889903484807623.post-8632486296383134144Mon, 14 Jan 2013 18:15:00 +00002013-01-14T10:15:47.706-08:00Do you see what I see?Who knew that Volkswagen could nail the state of Math Ed in :32? <div><br /></div><div><iframe allowfullscreen="allowfullscreen" frameborder="0" height="315" src="http://www.youtube.com/embed/FjTQV6CjAPE" width="560"></iframe></div>http://coxmath.blogspot.com/2013/01/do-you-see-what-i-see.htmlnoreply@blogger.com (David Cox)5