This task was inspired by Dan Meyer's Toothpick task.

Act 1
How many toothpicks are there?

Toothpicks Act 1 no counter time lapse from Nathan Kraft on Vimeo.



Act 2
What information do you need?

The goal was to make the largest triangle possible in the classroom using the pattern in the video

Lined up, end-to-end, you can fit 15 toothpicks alongside a meter stick.
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The room dimensions are 30 feet 1 inch by 27 feet 10 inches. You don't have to tell the students which wall the base of the triangle is against. They can figure this out on their own. However, if you'd like to take away that ambiguity, you can show them this:
room dimensions 2.png


Act 3

Triangle of Toothpicks Act 3 from Nathan Kraft on Vimeo.



Sequels
1. How long did it take to make this?
christopher's question.png
I would suggest giving students toothpicks to time how long it takes to make a smaller version. If you watch the act 3 video, there are 753 frames during the time lapse, where each frame represents 1 minute. But this total time will not match your students' time because I had many helpers throughout its construction. The counter in the video will actually speed up when there are more people working.

2. How far away from the wall will the tip of the triangle be?
Picture1.png

The answer to this is very dependent on how spread out the layers are, and that is dependent on the people building the triangle. On average, the height of a layer was 6.04 centimeters. This is a bit larger that my theoretical layer width of 5.77 centimeters. I calculated that by using a toothpick length of 6.67 centimeters. Here is the actual distance from the wall: gap distance.

3. How large of a square could be made with the same number of toothpicks?

4. How many toothpicks would be needed to cover the entire floor, assuming you don't limit yourself to making an equilateral triangle?

5. How much did those toothpicks cost?
cost.jpg

6. Can you write a formula that tells you the number of toothpicks given the number of layers?
Here's the one I came up with:
formula.png

7. How did I decide where to put the tape in the beginning of the video?
robert's question.png
tape.png
The tape was placed in such a way that an equilateral triangle would be created. That didn't really happen as the gaps in the layers were a little bit bigger than expected (see #2). I used some trigonometry to figure out where to place the tape. Here is my work:
SAM_0614.JPG
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8. How good of an equilateral triangle can you make?
In order to make this beast, I had to have some sort of triangle building test for the students, so I had each create a triangle with four layers and measure their equilateral-ness. This is similar to another task that Dan Meyer came up with (found here).
equilateralness.jpg

For fun

Toothpick Angel from Nathan Kraft on Vimeo.



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