Two books that I secretly used to build number sense

[sorry this is written quickly and not really edited – I wrote it while my son was at a school event, so limited time.]

Saw this really great post today from Geoff Krall (via a Tracy Johnston Zager retweet):

It made me think of the two books that I’ve gone back to every now and then to help build a little number sense –


I guess I’ve used one of the books so much that it is no longer recognizable!! Here’s where you can buy them:

Art of Problem Solving’s Introduction to Number Theory

Art of Problem Solving’s Introduction to Counting & Probability

For me the point of these books was never about learning number theory or probability, though I’m sure the boys picked up a few things here and there. Instead the point was finding ways to build number sense by talking through either (i) some interesting properties of numbers, and (ii) some neat counting problems.

Here’s one example project from number theory:

Using Divisibility Rules to Build Number Sense

The last part of this project uses a really neat divisibility rule for 7 idea that I found on Tanya Khovanova’s website.

One other project mentioned in this blog is learning to do arithmetic in binary using duplo blocks. Here’s one of our addition projects which I thought was a great way for kids to see numbers in a slightly different light:

We were even able to use some of the ideas we learned in binary to reinforce some ideas about decimals:

Writing 1/3 in binary

Here are a few number sense examples inspired by ideas in the Introduction to Counting and Probability book:

Counting Arrangements around a Table

The hockey stick theorem and some fun geometry in Pascal’s Triangle

The discussions that we’ve had over the years about Pascal’s triangle sometimes let the kids find Pascal’s triangle in surprising places – :

Talking about Pólya’s Urn with kids – inspired by Jim Propp’s blog post

Again, the point of using these books for me wasn’t to teach number theory or probability, but rather just to find some fun problems that would hopefully help to build some number sense. The idea from Geoff Krall’s post that really reinforced this idea for me was this one:

Also, we’re not talking about shutting everything else down classroom-wise, lest you’re worried about losing precious class time. While coverage is overrated, let’s put that aside for now, shall we? We’re talking 10-20 minute activities and discussion here, maybe a couple times a week.

The nice thing about these two books from Art of Problem Solving is that they are full of neat problems that you can use for 10-20 minutes here and there for a little non-standard (and hopefully fun!) number sense building 🙂