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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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.

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

Basic Skills Practice

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Here is an updated list of some of the html5 apps I’ve made for practicing basic math skills. The first list has nice addresses:

These I haven’t made short addresses for:

A little chaotic in the listing, but I thought some of you might find some of them helpful. Nothing fancy, just a source of generating basic practice problems.

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?

Geogebra

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I’ve been trying to get a bit better with geogebra and have started putting some work on geogebra tube. Here is a link to one of my experiments:
Triangle Numbers

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