Exploring Rules of Divisibility through Patterns in the 100-chart and Digit Sums of Multiples

Rules of Divisibility. How did we decide what they were? That’s where we are headed in our math journey at the moment. I’ve talk about going back, reviewing, revisiting, or doing an activity more than once. This repetition brings depth to our math journey. We are doing just that again in preparation for reducing fractions by factorization.

This week to discover where these rules come from, we are doing an exercise that we first explored in grade one with Quality of Numbers. We’ve done something similar in grade two with the introduction of multiplication, and we did a version last year and this year when we explored the Sieve of Eratosthenes.

Before we started this exercise, we reviewed what a multiple was, and we then discussed it’s relationship to divisibility. We also are used to dividing regardless if something fits evenly, so we discussed that when we are looking for rules of divisibility, we are looking for numbers to fit evenly, without remainders or fractions, into another number.

To begin, we started out with 2’s. We turned over every number that was not a multiple of two. At first, we are turning over one block at a time. Pretty soon though, especially since we’ve done this before, we realized there is a pattern. At that point, we turn whole columns at a time. When we finish, we do a short ‘notice and wonder.’ 

If you’ve used any of my curricula, you’ll be familiar with this concept. I first discovered it when I discovered Gattegno’s work with math education. You can read an example in my post “Notice and Wonder” with Cuisnaire Rods for Homeschool Math. It’s when we take turns making observations about what we notice, and then wonder what else might happen. For multiples of 2’s, our biggest observation that was all our numbers were even, which meant that all number that are even, are divisible by 2.

Afterwards, we moved onto 4’s, because of their connection to 2’s. I left the 2’s a different color so the 2’s and 4’s could be easily compared. Every other 2 is a multiple of 4. There are two 2’s in 4, so if a number can be halved twice, then the number is divisible by 4. The divisibility rule is that if the last two digits of a number are divisible by 4, then the number is divisible by 4. We will go over this, but halving a half is easier at this point.

Then we moved onto the 8’s. Why 8’s? What happen to 3’s, 5’s, and 6’s? We moved onto to 8’s because of their relationship to 2’s and 4’s. If we half 8, we get 4, and if we half 4 we get 2, so if we can half a number three times, then that number is divisible by 8. In the picture, I’ve left the 4’s a different color so we can see that 8’s are every other 4.

From here, we moved quickly through 5’s and 10’s. She was already familiar with the fact that multiples of 5 end in ‘5’ and ‘0,’ and multiples of 10 end in ‘0,’ so we didn’t have to spend much time on this.

From here we will look at 3’s, 6’s, and 9’s. For this we will be reviewing our work within my Number Sums unit, particularly the number sums of multiples. The rule for divisibility of 3 is that if the sum of the digits of a number are divisible by 3, the number is divisible by 3. Nine has a similar rule. Hopefully this will become obvious when we go back to explore digit sums and spirals from my Number Sums unit

And from here, we’ll move onto prime factorization to reduce fractions. Why I’m using prime factorization? Mostly because that’s what we do in algebra, so I want to set her up for how it’s done with numbers first. You can find my tutorial on making your own multiplication chart in my post Making a 100-chart for Math.

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