Last weekend we did a fun little project on Fibonacci numbers and pentagons that we found in Zome Geometry.
Fibonacci Spirals and Pentagons
I got a happy surprise this week when I saw that problem #22 on the 2015 was related to this project. Today I thought it would be fun to revisit the project and see if we could use our Zometool set to solve the problem. Although we didn’t get to the solution in a perfectly straight line, I think this is a great project for kids. If for no other reason, it shows how you can make progress on a really difficult problem using some basic ideas in math.
So, we started by looking at the problem on Art of Problem Solving’s website:
Problem 22 from the 2015 AMC 10b
The first step in solving the problem was building the shape out of our Zometool set:
Now that we had all of the pieces of the puzzle, we went to the living room floor to see if we could figure out any ways to approach the problem. My younger son came up with the idea of chopping up the long segments into segments that matched the one with length 1.
Right away we see that the length is 3 plus an extra medium blue. Then my older son has an interesting idea about how to measure the length of a medium blue.
Next we tested my older son’s idea that a medium blue was about 3/5 of a long blue. The great thing about this part of the project is that we ended up having a great talk involving fractions.
Having found one approximate answer in the previous video – the length is 3 3/13 – I thought it would be fun to try to find a different way of thinking about the problem. My younger son came up with the idea of measuring everything using short blues.
To save a little time, I had them build the models out of smalls off camera. Again we had a nice discussion about fractions.
Now we returned to the original problem to see if our two estimates for the answer would help us identify the right answer. The great part is the discussion of decimals (and numbers in general) we had here.
Finally, the last thing that I wanted to do was to try to connect this problem with the golden ratio. This is probably a bit of a stretch for younger kids, but it did spark an interesting thought from my older son – what if the lengths of the medium blue and long blue zome struts were related by the golden ratio?
The last part was a little unexpected, and perhaps not the most exciting way to end the project, but with my son speculating that the long and medium blues were related by the golden ratio, I wanted to show him what that would mean.
One interesting fact about the golden ratio is that it is easy to square. In fact, squaring the golden ratio is the same as adding one to it! That algebraic relationship helps is see that a relationship that we saw in the last video is exactly the same as multiplying by twice the golden ratio. Surprise!
So, a super fun project solving one of the most challenging AMC 10 problems. I think it is nice for kids to see that some simple math ideas can be used to solve some of these difficult problems. Along the way we got to have great conversations about fractions and decimals – so that was a nice bonus in this project. What a lucky break to have one of these contest problems almost match one of our prior projects!