A nice geometry and algebra problem

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Dan Meyer recently posted a nice geometry and algebra problem on his blog for discussion. Here is the original text of the problem:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

A lot has already been written on that problem on his blog and others. I wanted to post solutions to two interesting extensions of the problem in case anyone was interested.

The first solution is for dividing the line into an n-sided regular polygon and a circle:

The other solution is for dividing the line into two different n-sided regular polygons:

It’s worth noting that most people (me included at first) seem to solve these problems using quadratic equations, but that is not necessary (which is convenient since my students are not yet able to solve quadratics). There is nice way to solve them by just setting up a simple fraction based on the perimeters and setting the areas equal.

It was fun to check the limit as n went to infinity to see that I did end up with the circle solutions as well.

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My favorite algebra problem

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I gave my 7th graders a quiz today on solving equations. It was fun (admittedly in a slightly evil way, but not really) to see their frustration with the final problem, which over the years has quickly become my favorite algebra problem. Of course I don’t put it on the quiz to be mean, but to teach important lesson about reasoning and life. Here is the question, always coming after a series of much more complicated ones that they usually have no trouble with:

x + 1 = x -1

 

Practice apps update

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I’ve created a whole bunch of new math practice programs since last I posted about them. I thought it might be useful to anyone who cares to put publish a link where you can find almost all of them in one place. They can be gotten to through the links in the ThingLink image, or from the explicit list underneath. As usual, these are mainly just for basic skills practice (and some harder skills). Let me know if you find any of them useful:

https://sites.google.com/a/karlstad.engelska.se/math-grading-criteria/practice-math-apps

Fun with Desmos

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A new activity for my students. I put up the following image which I made on Desmos and let them try and figure out how to graph it with linear equations on Desmos on the iPads:

ymmiuqojan

Group discussion with post-it notes

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I tried a new idea today that I borrowed from Andrew Knauft, found vid Dan Meyer’s blogg.

I divided my 7th graders into groups of 4 and asked each group to decide which of the number in the set {9, 16, 25, 43} didn’t belong and why. They stuck the post-it on the whiteboard when done and then took another group’s note. They then had to say whether or not they agreed with the reason the other group gave for their decision.

At least that was the plan. The reality was that three of the groups had exactly the same number and reason (43 because it is the only prime number), a fourth group had 43 because it is the only not perfect square. Only the fifth group had anything different.

So then I pulled out another harder problem and set them to work on it. I showed the following picture with this question

“Each of the four cards below has a solid color on one side and a number on the other side. What is the smallest number of cards you need to turn over to decide whether or not the following statement is true: if a card has an even number on one side then the other side is red?”

colorcards

That gave rise to a lot more lively discussion and a lot of disagreement between the different groups on what the right answer was. It was a very fun lesson.

Practice rate problems program

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I just finished my most ambitious basic skill practice app – for practicing rate problems (hastighet). It has everything from very simple problems to very hard problems. However, I have not tests all the different problem types yet, so anyone reading this who wants to test them and get back to me with any errors they find would be very appreciated. Here is the link:

 

http://hem.bredband.net/taub/rate.html

ThingLink with Math Apps

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I have created another ThingLink image (once again inspired by MathyCathy) where I have collected all my math apps (plus one I didn’t write cause it fit right in). I thought this might be an easier way to organize the practice apps and make them easier to find and keep track of. I will update the image as I create or find more apps. Here is a link:

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