Multiples and Divisibility Rules

There is nothing more basic about number theory than counting by a multiple of a number. We first learned how to count by multiples of 1 (1,2,3,4....).

Then we graduated to counting by 2s (2,4,6,8,10,..). It is surprising how well we do through 10 and then slow down at 12, 14, 16, ....

Multiples of five we do well through 100, then it slows down. Of course, there are the tried and true multiples of 10, which we do well up to 100 as well.

This week we gave them the vocabulary word "multiple" instead of counting numbers. Let's use it from now on. Most schools use the 100s table but I have found that this is the reason the children slow down after 100. We use a 500s table and even up to 1000 on the reverse side of the page for those who want to explore beyond.

I had them use highlighters (you can use a light marker as well; just make sure you can see the number under the marker) to identify odd numbers, multiples of 2, 5, 10, and then 3,4,6,7,8,9,11,12, and 13. Of course, there is a page to chose your own multiple. 

At the beginning of exploring the non-vertical patterns in the multiples of 3, they should count off 1,2,3 (highlight), then when they see patterns use them. However, since they are human and not aliens, they will make mistakes. Every so often, have them check their accuracy with the divisibility rules. For example, for 3, each number must have a digital root of 3, 6, or 9. For 2, the numbers must end in 0,2,4,6,or 8. For 5, end in 0 or 5. For 10, end in 0. For multiples of 6, they have to be even AND have a digital root of 3, 6, and 9. For 9, it must have a digital root of just 9.

For numbers like 7, 11, 13 and beyond, I explored these divisibility rules with the 4th and 5th graders.

Enjoy exploring "multiples" and checking if those numbers are "divisible" by that multiple. Of course, these words are roots to multiplication and division, but it is definitely not too early for them to see that these concepts are not hard since it is all about counting (adding like numbers) and looking for patterns.

The pdf is attached.