Divisibility Exploration

In this meeting hosted at 3rd Avenue VFW by teachers at MDC Brooklyn, we continued recent explorations into multiplication and factors. In this meeting, we looked at divisibility rules. After a pair/share and introductions, I asked the group to look at multiples of 9 and share what patterns they noticed. We shared in small groups then talked about a few things people noticed.

Mr. Hill shared a trick for finding multiples of 9.

We then moved into divisibility and considered this question: Of the numbers 2 through 10, which will divide 270 evenly without a remainder?

We then had a conversation about what the word “divisible” means. What does it mean for a number to be divisible by another number? Is 12 divisible by 5? Some people said yes, you can divide 12 into 5 pieces. They won’t be divided into whole number pieces, but they are divided. 12 divided by 5 is 2.4. Other said that 12 is not divisible by 5 because you don’t get a whole number answer. A suggestion was made that if we think about groups, then it depends on the context. You can equally divide $12 among 5 people, but you can’t divide 12 people into 5 groups equally. I asked the group to think about dividing people into groups for the next divisibility activity. We want to think about when it’s possible to divide people into different numbers of groups without a remainder.

Looking back at what you did, did you see any ways to look the number 270 and know what it is divisible by without actually doing the calculation?” I then starred numbers where participants didn’t have to do the actual divisibility.

I then introduced the divisibility rules from 2-10 and asked the group to check our work on 270:

2: If a number is even, then it’s divisible by 2.

3: If the sum of the digits in a number are divisible by 3, then the number is divisible by 3.

4: If the number ends in 00 or if the last two digits are divisible by 4, then the number is divisible by 4.

5: If the number ends in 0 or 5, then the number is divisible by 5.

6: If the number is even and the sum of the digits is divisible by 3, then the number is divisible by 6.

7: (I don’t know of a simple and efficient way to check the divisibility of 7.)

8: If the number ends in 000 or if the last three digits are divisible by 8, then the number is divisible by 8.

9: If the sum of the digits in a number is divisible by 9, then the number is divisible by 9.

10: If a number ends in 0, then the number is divisible by 10.

(At this point, Spencer wondered if it would be a good idea for us to have students try to figure out the divisibility rules themselves rather than just give them the rules. It was a good point and made me wonder what this meeting would have been like if we had worked to create our own divisibility rules for 2-10. My plan was to give the divisibility rules for 2-10 and then pose the problem of creating rules for larger numbers. We probably wouldn’t have gotten to the larger numbers, but it could have made for an engaging meeting. Oh well. Maybe we’ll get another opportunity to try it this way, or someone will try it with a class of students.)

I then gave everyone time to work on their own. I set a timer for 10 minutes. After the timer went off, I asked people to work in small groups on any part of the divisibility exploration packet (linked above) or questions that came up from our earlier discussion.

With just a few minutes left at the end of the meeting, two groups presented some ideas in process:

Jana and Mark explained a chart showing an exploration of the divisibility of 8 and 15. Along with Spencer, they found that the divisibility rules for 2 and 4 are not adequate for determining the divisibility of 8. However, the divisibility rules for 3 and 5 are adequate for determining the divisibility of 15. Their chart gives some clues why this might be true.

With just a few minutes to spare, this group shared how they started looking at the prime factorization of 8, 15 and then 30 to see what might be happening here. They noticed that the prime factorization trees for 8 and 30 have more than one level. Is that why the 2 and 4 divisibility rules aren’t adequate for 8? Are the divisibility rules for 2 and 15 adequate for developing divisibility rules for 30?

With a just a few minutes before we ran out of time, Todd and Gregory  shared some of their thinking. They pointed out that in order for a number to be divisible by 8, it has to be divisible by 2³. If a number is divisible by 2 or 4, that only means that it is divisible by 2².

Todd and Gregory also on a divisibility rule for 7. Their method is based on the partial quotient method of division.

Here’s another trick for determining the divisibility of 7 (shared by our friend, Benjamin Dickman). I’m not sure it’s simple or efficient, but it works. Why?!

Peel off last digit; double it; subtract from remaining digits. If result is divisible by 7, then the original # is divisible by 7. Repeated peeling-doubling-subtraction as necessary.

Example: 8638. So, 863-16=847. Repeating: 84-14=70. Divisible by 7! So, 8638 is divisible by 7.

— Benjamin Dickman (@benjamindickman) January 13, 2018