It seemed easy to me – 3D geometry problem solving

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I find it interesting (and a bit depressing) that I still sometimes am surprised but which problems my students find hard. I recently made up a 3D geometry problems solving exercise that I thought was kind of interesting and not too challenging. I mean, I knew some students would have trouble with it, but I fully expected the majority of the class to get it, especially since it was group work. Boy was I wrong.

To be fair, the actual math content of the problem was not difficult for them. It was being organized enough to find a way to see the pattern that they had trouble with. The ones who did solve it made a table starting with smaller numbers and quickly saw the pattern.

Here is the problem. I would be interested in any comments any readers might have about it:

You have a large number of small identical cubes in front of you. The surface area of all of these cubes together is 23328 cm2. You start playing with the small cubes and realize you can put them together to build one giant cube with no small cubes left over. The surface area of the big cube is 1944 cm2.

  1. How many small cubes do you have?

  2. What is the area of one face of one of the small cubes?

  3. How long are the edges of each small cube?

  4. What is the volume of the big cube?

  5. What is the volume of each small cube?
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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

 

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.

Math video prompts

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In case any of you don’t already follow Dan Meyer, I highly recommend his math teacher blog – best around: http://blog.mrmeyer.com/

In line with his theories about trying to create video or picture prompts to stimulate natural questions that require math to really answer, I have a couple ideas. The problem is I’m not really good at video editing and suspect I might not get around to actually making them.

So this post is a request for anyone interested in seeing if they can/want to produce one or two short videos.

  1. This one is of two people running around a track. I think it should be a standard 400 meter running track so that distances are known, though the 100 meter marks should maybe be added to the videos. Two runners/joggers start facing each other a short distance from each other. When they pass each other they smack hands and at that moment a timer starts on the screen. The video is stopped after a short time (before they meet on the other half). The question you are hoping for is where will they pass each other again? The timing information and known distances on the track should provide enough information for estimates. The rest of the video is then shown afterwards to show the answer.
  2. In this one you need a full class of students to help out. The students are standing in a line outside a room, each is holding a big bag of presents. One student is in the room. The first student in line walks in and shakes hands with the student there, and they give each other a present from their bags. They put their presents on a big table maybe (or maybe just keep them in a pile at their feet). The next student comes in and does the same (shakes hands and exchanges gifts) with EVERY student already in the room. The video would show the first two or three students doing this, then pan down the line of students. The exact number in line should be a bit unclear. The questions to answer here are how many presents total (which is why they should maybe be on a big table) and how many handshakes total (for those who don’t know, this is showing an easy connection between these ideas and which later is used to easily motivate the Gauss formula for adding up sequences of integers easily).

Any takers?

A fun little problem

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Lilly is working on a classic math problem:

A hot tub can be filled with one of two water faucets. The first alone would fill the tub in 2 hours and the second one alone would fill the tub in 4 hours. How long does it take to fill the tub with both of them turned on at the same time?

Lilly thinks this problem is a little too easy and so starts to play around with different ways of solving it. Being a big fan of geometry, she draws a right triangle, with one leg being 2 and the other leg being 4 to represent the time for each faucet.

triangle 1

She then inscribes a square in the triangle so that two sides align with the legs and one corner lies on the hypotenuse:

triangle2

Lilly then notices that the length of the side of the square is actually the solution to the problem!

  1. Is she right?
  2. Does this work for other numbers? (different times for the faucets to fill the tub)
  3. Does this work for all numbers?

Math Apps

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I thought I might try posting again. I’ve been working on writing more online math apps for students to practice with in class (on iPads) and at home. I started with Google Apps Script, but the limitations there were just getting too annoying. Then I remembered that my home provider gave me a little space so I recently switched to normal JavaScript pages – much, much easier without the limitations.

I was really happy when I came up with a good way to practice finding area with an app. This was challenging since you can’t really hold a ruler up to your screen, but giving the measurements either makes the problem too easy or too hard. I finally figured out a way to simulate measurement in the app. That was nice. Of course, technology is still generally annoying, so I’m not entirely happy with the result. Some of the line drawing won’t work in some versions of Chrome apparently. And I wrote it initially as a Google Apps Script, which meant there was no hope to get it to work for touch devices (they don’t currently support touch events, only mouse events). I tried switching to normal JavaScript and adding touch support, but it was more complicated than I thought. The result is something half functional on an iPad. I’m considering making a real iPad app for this one. Seems useful enough.

Anyway, for those who are interested in testing some of them, here are links to some of the stuff I’ve done (some more recently than others). None of them have any “frills”. They are not pretty, but are hopefully useful:

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