# Using divisibility rules to build number sense

Yesterday we played around with Jo Boaler’s Number Talks idea and that conversation got me thinking about other ways to build number sense. We’ve done a couple of different projects over the last few years that were at least partially motivated by building number sense. Probably the most fun was a talking about counting in bases other than 10 – even building binary adding “machines” out of duplo blocks! We’ve spent so much time counting in other bases over the years, though, so I wanted to try out a different idea. The idea I settled on was divisibility rules.

We’ve have talked a little bit about divisibility rules in the past. Chapter 3 of Art of Problem Solving’s Prealgebra book, for example, has a brief section on these rules. Like many topics in math I’m sure that not all of the details sunk in the first time through, so going back through some of the divisibility rules would be both worthwhile and fun. As an added bonus, my younger son really likes the divisibility rule for 7 for some reason.

We started with divisibility rules for 5 and 10. The goal was just to get them talking and thinking about numbers.

Next came the divisibility rules for 2, 4, and 8. Here we build a little bit on the place value ideas we talked about with the divisibility rule for 5. That idea gives us a slightly more precise way of talking about the divisibility rule for 2. Then we extend that rule to 4 and 8:

Next come the divisibility rules for 3 and 9. Developing these rules uses the a slightly different rule that the the one we used in the two previous videos. The difference here is that 10, 100, 1000, and etc are not divisible by 3 or 9.

I’m not intending to be rigorous here, so the important step of introducing remainders is informal. Looking at the remainders helps us understand how to build up the divisibility rule in these two cases:

Now comes one of my favorites – the divisibility rule for 11! I remember seeing this for the first time in 10th grade and being absolutely amazed. To help understand this rule I have to introduce the concept of negative remainders, which probably seems like a pretty strange concept when you see it for the first time, but will become an important concept much later when we study number theory:

Finally, as I mentioned at the beginning, the divisibility rule for 7 is one of my younger son’s favorite bits of math. Going through the construction of this graph is definitely worth a Family Math day all to itself. For now we have to be satisfied with using the picture from this Tanya Khovanova blog post:

Tanya Khovanova discusses the divisibility rule for 7

We wrap up the talk by walking through a few examples using her picture:

This seemed like a pretty fun activity. The number talks seem to help build understanding of arithmetic and place values. Talking about divisibility rules involves a lot of arithmetic, but the arithmetic is not the main focus. Instead this idea spends a bit more time on the ideas of place value and remainder. Feels like a pretty nice follow up or extension of the number talks idea.