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# Divisibility by 9

## Overview and Objective

Many students learn the short-cut to determine if a number is divisible by 9. Specifically, if the digit-sum of the number is divisible by 9, then the number itself is divisible by 9. In this lesson, students will move past the memorization of this rule and will use the number tiles in Polypad to create a visual representation for understanding the underlying mathematics of the divisibility rule for 9.

## Warm-Up

If students have already learned the divisibility rule for 9, invite some students to remind the class of the rule and then do a few examples all together. If students have not learned the divisibility rule for 9 prior to this lesson, introduce the procedure to the students and do a number of examples as a class. Share with students that the goal of the lesson is to understand the mathematical reasoning as to WHY the rule works. Discuss with students the difference between understanding how to use a rule versus understanding the logic and mathematics behind the rule.

## Main Activity

Begin by modeling for students how to use the number tiles to check for divisibility by 9. Click here for a tutorial on using the number tiles. After doing an example together, students will repeat the process for a 3 digit number of their own choosing.

### Teacher Example

Create the number 258 on the canvas using the Number Tiles. Represent 258 as 2(100) + 5(10) +8.

Separate the groups of 100 into groups of 99 + 1, separate the groups of 10 into groups of 9 + 1, and make the 8 the same color as the groups of 1's.

Gather together the purple tiles and then gather together the groups of 99 and 9. Hold down the shift key to select multiple tiles at the same time.

Discuss with students why the groups of 99 and 9 can be labeled "Divisible by 9" but the leftover pieces need to checked.

Do not spend too much time here discussing the mathematics in this example. Rather, move on to students working through an example of their own.

### Student Examples

Invite students to create their own Polypad in which they use a similar process to check for divisibility by 9. Encourage students to select any number they like, however, you may want to caution them that numbers larger than 300-400 may be difficult to represent on one Polypad canvas. Encourage students to do one example of a number that is divisible by 9 and one example of a number that is not divisible by 9. Remind students to save their Polypads so you can view them and share them with the class. Click here to learn how to view students' saved Polypads.

After students finish both examples, encourage them to either discuss with a classmate or put down in writing how the visual models they have created helps them see the underlying mathematics of the 9 divisibility rule.

## Closure

Gather as a class and share a number of student examples as you discuss the underlying mathematics of the divisibility rule. Finding the digit sum of the number is the short-cut process for finding the sum of the left-over purple tiles shown in the original example. The groups of 99 and 9 that are in shown in orange never need to be checked since 99 and 9 are always multiples of 9. Discuss how all the place values in Base 10 are 1 more than a multiple of 9 (9, 99, 999, 9999, and so on), so this process holds true with much larger numbers. Perhaps do something like this for a larger number to demonstrate the abstraction of the idea:

## Support and Extension

For students needing extra support with creating their own visual representation, starting with a 2-digit number would be an appropriate support.

For students looking to push their thinking even further, encourage them to apply these ideas towards developing an understanding for the divisibility rules for 3 and 11.