Teaching Divisibility Rules in Middle School Math

Do you teach divisibility rules in your math class?
I've always enjoyed teaching divisibility rules and my 6th grade math students have always seemed to have fun using them!

I've read different opinions about whether or not teaching divisibility rules should be a focus in math class, because they may be viewed as 'tricks.'
However, I think understanding and using them in middle school helps students develop number sense and number fluency.

Rather than being taught as a 'unit,' I think divisibility rules should be introduced and then referred to again and again in any applicable situation throughout the year. To make the continuous revisiting easier for students, I've always liked to have a resource for them to refer to throughout the year. We used to create fold-it-ups, but then I moved to using a Doodle Notes resource or a Math Wheel.

​Any reference sheet is helpful so that when you ask a divisibility question, students can grab it (or look at it on a wall) to quickly refresh their memories, if needed.

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Where can students apply divisibility rules?
There are several math concepts where students can apply the divisibility rules:

1) Working with division facts that are beyond the 'basics.' Like 51 divided by 3, for example. Students so often believe 51 is prime, but if they take a second to test the rule for 3, they can quickly see that 51 is in fact a composite number.

2) Prime factorization. When determining the prime factorization of a number like 51, 57 or 87, using the divisibility rules can be very helpful!

3) GCF. To find GCF, students need to determine what each number can be divided by, so the divisibility rules are quite helpful here.

4) FACTORING. If you teach divisibility rules in elementary school, you might not be thinking of this eventual application. However, the more frequently math students work with divisibility rules, the more their number fluency improves, and the easier factoring will be for them.

Teaching divisibility rules
As I mentioned, I think divisibility rules are learned and retained most effectively when they are introduced and then referred to on a regular basis.

It's also helpful when students have a resource to refer to if they need a quick reminder, so my favorite way to teach the rules are to use the Doodle Notes or the Math Wheel. 

These methods are similar because they both incorporate coloring, doodling, and visual memory triggers. I also love these because they:
1) Increase focus and retention during instruction, by incorporating coloring and doodling (this activates both hemispheres of the brain).
2) Are year-long resources students can keep in their interactive notebooks.

​What are the divisibility rules?
In case you're teaching divisibility rules for the first time (or haven't taught them in a long time), I've included the divisibility rules for 2, 3, 4, 5, 6, 9, and 10 in this post. There are rules for 7 and 8 as well, but I don't normally teach these, as they're a bit more complicated....maybe a little less useful.

 2: if a number ends in a 0, 2, 4, 6, 8, it is divisible by 2

• ​The number 358 ends in 8, so 358 is divisible by 2​​

 3: if the sum of the digits in a number is divisible by 3, the number is divisible by 3

• The sum of the digits in the number 4,563 is 18. 18 is divisible by 3, so 4,563 is divisible by 3.

4: if the last 2 digits of a number are divisible by 4, the number is divisible by 4

• In the number 248, the last 2 digits (48) are divisible by 4, so 248 is divisible by 4.

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5: if a number ends in 0 or 5, it is divisible by 5

• The number 255 ends in 5, so 255 is divisible by 5

6: if a number is divisible by 2 AND 3, it is divisible by 6

• ​The sum of the digits in 246 is 12. 12 is divisible by 3, so 246 is divisible by 3

• 246 ends in 6, so 246 is divisible by 2
• 246 is divisible by 2 AND 3, so 246 is divisible by 6.

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​9: if the sum of the digits in a number is divisible by 9, the number is divisible by 9​

• The sum of the digits in the number 4,563 is 18. 18 is divisible by 9, so 4,563 is divisible by 9.

10: if a number ends in 0, it is divisible by 10​

• The number 4,040 ends in 0, so 4,040 divisible by 10

Note
Finding the sum for checking 3 and 9 sometimes led some students to start adding the digits to check for every number. I found the use of visuals with the Doodle Notes or the Math Wheel reduced this tendency, as compared to when we used the fold-it ups.

What's your favorite way to teach (or use) divisibility rules?

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