[note: due to a little time pressure and tweaking my back, this post is not edited, or even proof read, very well]
Saw this tweet from David Wees this afternoon:
Since I hurt my back moving snow around yesterday I was looking for something that I could do sitting down, and this exercise looked like it would be pretty fun – and it delivered much more than I was expecting!
We started off talking through what we’d be looking at for this project and looked at the first equation via snap cubes. I like the fact that my son talked about 3×5 being the same as 5×3. I also found it interesting that he didn’t naturally think that n = 1 would be a good value to look at.
The next equation produced some really interesting ideas from him. At first he thought that since three terms were being added together, it might be good to make a 3 dimensional shape here rather than the rectangles from the last equation.
After making the cube, he notices that we got 8 for the n = 1 case last time! That observation leads to him noticing that we get the same number from both equations for n = 2. However, he still wants to make a 3d shape. That idea leads to a little struggle trying to figure out what to build. Had I anticipated this struggle I would have adjusted the camera, but luckily he talks through the build for most of the time.
Eventually he finds his way to constructing three separate rows to represent the three terms in the second equation.
I broke the last video around the 5:00 mark and we started a new video checking out the n = 3 case for the 2nd equation. He suspects that it will be equal to n = 3 for the first equation. It does. He then goes back to looking at the equations to see why they are the same.
He uses the term “add up the equations” to describe what he wants to do. What he really means is that he wants to multiply out the first equation. I love his surprise when he finds that the two equations are indeed the same. I didn’t expect the path to seeing these equations were the same would be through the algebra.
Next we moved on to looking at the third equation. He wants to use the idea of “adding up the equations” here, too, but he gets tripped up by thinking that unfortunately (or maybe fortunately). The difficulty with algebra sends him back to checking out the values with the snap cubes and he finds that the cases of n = 1, 2, and 3 all match the values in the equations we found previously:
I cut the last video around the 5:00 minute mark again when he was having trouble with the algebra. In the last part we looked a little more carefully at the algebra. He again things that . We plug in some numbers to see that these equations do not seem to be equal. Eventually we find what was going wrong. As I said above, I really didn’t expect this exercise to go back to the algebra, but that’s where it did go and it turned out to be a really fun conversation.