So I asked them to determine how many toothpicks I would need to make as large a triangle I could on the floor of the classroom. Here is all of the important information I gave the kids:
My classroom is 30 feet by 20 feet.
That's it. And with that, my students had everything they needed to figure this out. Well...almost everything. They needed the length of a toothpick. I didn't give them this. But I did give them a handful of toothpicks and a ruler. They found the toothpicks to be 2.5 inches long.
So where did they struggle? Everywhere.
First, many weren't sure about how the triangle should be oriented. I saw several diagrams that looked like this::
I asked them how they knew that this would be the biggest triangle. After a lot of shrugs, I asked them to consider what would happen if I rotated the triangle so that it lay on the bottom of the rectangle:
And couldn't you then make that triangle bigger?
Then I went to another group. They wanted to know the distance across the room diagonally (from corner to corner). They wanted to measure it with a ruler, but this would have been a little distracting to the rest of the class. So I quickly grabbed a calculator, did a little Pythagorean Theorem, and said, "It's 39 feet." I cringed at the thought of what they were going to do with that information, but moved on to the next group.
This new group wanted to know the length from the corner to the midpoint of the opposite wall. Again, a little PT and..."It's 29 feet and 2 inches." I thought, that's not going to work. And they're going to hate me when they realize that I knew this wasn't going to work and didn't point it out to them.
Next, a girl asked me for masking tape. I had no idea what she was going to do with it, but it sounded exciting. So I said, "Yes! Tape! Coming right up! You're not going to use a lot, right? No? Great. Have at it." I walked away.
I come back to the other group who wanted the diagonal. Sure enough, they have this drawn on their paper:
So I ask, "What kind of triangle is that?" After some fumbling for answers, they come up with "right triangle". "Right, and what kind of triangle are we making?" Again, after some discussion, they realize that they need an equilateral triangle.
After receiving a few dirty looks, I moved on and checked in on the other group who wanted the midpoint.
I have a similar discussion with this group about how this isn't an equilateral triangle. I move on...
The girl who wanted the tape is making a 1 square foot box outline on the floor. She is asking for toothpicks to place within the box. I'm scratching my head at this point, still hopeful that this is going somewhere. I ask, "What are you going to do with this information?" She explains how she's going to use proportions to find the number of toothpicks in the large triangle. At first this sounded awesome, but then she started to talk about comparing the area of the small box to the entire room. I explained that this comparison may not be an equal proportion, and that she might be able to use this method by comparing areas of triangles instead. As cool as it would have been, this group eventually abandoned the proportion method.
The day goes on, and there are plenty of errors to correct. One group thinks that there are only 12 little triangles along the bottom of the big triangle. I asked them if that makes sense and I got the blank stares associated with the though process of "Crap. That doesn't make sense. I just wasted three minutes of my life doing this the wrong way." I asked them how they came up with 12...here's what they did:
30 feet / 2.5 inches = 12 toothpicks
Aaahhh! These are two different units of measurement! I pointed this out and moved on.
Another groups found that there are 144 little triangles along the bottom of the big triangle. And with this knowledge they attempted to find the number of triangles in the entire structure. They completely disregarded the 25 foot length of the room and didn't bother to check to see if the triangle would actually fit in that direction. I pointed this out and helped them see that the height of the large triangle was dependent on the height of one little triangle. And with that, the period was over....
The next day more groups got to this same point. But none of them knew how to find the height of this small triangle. Did they completely forget that I gave them toothpicks and rulers? Why didn't they think to measure it?
(It's probably important that I mention that these students have not had any exposure to the Pythagorean Theorem. That doesn't happen until 8th grade.)
I drew a triangle made of toothpicks on the board and pointed out that many of them have already found the length of a toothpick.
I asked them how they found the length of one toothpick. "We measured it." Right! So what's to stop you from putting some toothpicks together and measuring the height? Some heads started to nod, and after some assembling of toothpicks and measuring, we got a height.
Finally, they were getting close. They found the actual number of layers in the triangle and they started to calculate the number of little triangles. Problem is, there are a lot of layers...well over 100. And I know from experience that my kids are prone to making mistakes when they are making a lot of calculations. One missed button on the calculator ruins the whole thing. But I let it go, hoping someone would come up with a better way to do this.
The next day, I presented this problem on the board as a warm-up:
Students quickly realized that they just needed to add the numbers from 1 to 20.
And of course, they added them from left to right. But a few knew of a trick with the commutative property that would make it easier.
One explained how you could add the first and last number to get 21, then do the same thing with 2 and 19, 3 and 18, and so on, until you have ten sets of 21. Therefore, the answer is 10x21=210. (Supposedly, Gauss came up with this method as a kid in school. He sounds like an annoying little know-it-all.)
I gave the students the remaining class time and all weekend to finish their work. Tomorrow I get their solutions. Let's hope it all comes together.
This was a very difficult task, and at times, I regretted giving this to them. I was often worried that I didn't scaffold them enough ahead of time. But in the end, it all worked out Each step was something of a hurdle, and with a lot of good discussions, we somehow managed to get through it all.
You're still here? Great. I'm thinking about actually building this thing in my classroom. I went on amazon.com to order toothpicks, and apparently, I can choose between new and used toothpicks. How exciting!