Talking through Ben Orlin’s exponent problem

Saw this problem last week via a Joshua Bowman retweet:

I thought it would be a good problem to talk through with the boys this morning, and it turned out to be an even better discussion that I was expecting.

Talking through this problem with kids gives you the opportunity to discuss lots of specific ideas in arithmetic. Three that come to mind are:

(i) exponent rules,
(ii) different ways to represent numbers,
(iii) the differences between adding and multiplying.

There’s also a great opportunity to discuss problem solving, and that discussion happens in the 2nd video when my older son says “we don’t really know anything about [the number] 2^{1/3}.”

Here’s the beginning of our discussion. The first idea that my older son has is writing both numbers in the form a^{1/10}.

Before we went down that path, though, I wanted to make sure that my younger son understood a little bit about the two different ways to write numbers as nth roots. So, the discussion here is half about exponent rules and half about our start at solving the problem:

In the last video we found that Orlin’s problem was equivalent to comparing 10 with 2^{10/3} but we sort of got stuck there. Getting stuck at this step was fascinating to me because [to me!] it was so obvious what the next step was. The ensuing discussion taught me a lot about how kids see numbers.

In the next part of the project we explored comparing 10 and 2^{10/3} a bit more. The first new idea in this part of the discussion was to remove a factor of 2. Now we were left comparing 5 and 2^{7/3}. What now?

My older son has a neat observation – if we knew if 2^{1/3} was larger or smaller than 1.25 we’d be able to answer the question. This particular observation comes from understanding a little bit about the relationship between addition and multiplication.

How, though, do we compare 2^{1/3} to 1.25?

So, having solved the problem at the end of the last video, I wanted to show the boys the actual values on the calculator. Unfortunately, I decided to zoom in on the calculator and then not zoom out for the rest of the discussion. Hopefully the words in that discussion are enough to know what we were talking about 🙂 Sorry about the camera goof up – at least the camera mistake was in the shortest video!

Definitely a fun project. I think Orlin’s problem is a great one to talk through with kids learning about exponents. There are so many different ways that conversation can go, and so many opportunities along the way for kids to think about and discussion ideas about numbers.