B. Number · Operations: Numbers to 100 000, Decimals, Fractions, Ratios and Rates

Dividing 3-Digit by 2-Digit Numbers: Area Model and Partial Quotients

B2.7 B2.1About 45 minutes

Learning goals

I can make a list of friendly multiples of a 2-digit divisor.
I can use an area model to divide a 3-digit number by a 2-digit number.
I can divide using partial quotients and connect it to the area model.
I can check a division with multiplication.

Words to know

dividend
The number being divided. In 312 ÷ 24 = 13, the dividend is 312.

divisor
The number you divide by. In 312 ÷ 24 = 13, the divisor is 24.

quotient
The answer to a division. In 312 ÷ 24 = 13, the quotient is 13.

partial quotient
One part of the answer. You add the partial quotients to get the whole quotient.

multiple
The product of a number and a whole number. 24, 48 and 240 are multiples of 24.

Remember from Grade 4

In Grade 4 you divided 2- and 3-digit numbers by 1-digit numbers. You used partial quotients: take away friendly groups, like 20 groups of 6, keep track, then add up the groups. In Grade 5 you'll divide by 2-digit numbers. The idea is the same!

Let's start with a field trip

312 students are going to a science centre. Each bus holds 24 students. How many buses are needed?

We need 312 ÷ 24. Think: how many groups of 24 fit in 312?

Start with a friendly number of groups: 10 buses hold 24 × 10 = 240 students. That leaves 312 − 240 = 72 students. 3 more buses hold 24 × 3 = 72. Nobody is left!

10 + 3 = 13 buses.

A row of yellow school buses parked side by side in a parking lot. — A big field trip needs many buses. Dividing the number of students by the seats on each bus tells how many buses are needed. AI-generated image

Step 1: Make a list of friendly multiples

Before dividing by a 2-digit number, it helps to list a few easy multiples of the divisor. You can find them by doubling and by multiplying by 10.

Groups	× 24	How I found it
1	24
2	48	double 24
5	120	half of 240
10	240	24 × 10
20	480	double 240

Friendly multiples of 24. You can use these to take away big groups quickly.

Step 2: The area model for division

In division, you know the total area (the dividend) and one side (the divisor). You need the other side (the quotient).

Build the rectangle in parts. Each part uses a friendly multiple. Add the side lengths of the parts to get the quotient.

An area model for 312 ÷ 24. The height is 24. The first part is 10 wide (area 240) and the second part is 3 wide (area 72). 240 + 72 = 312, so the width is 10 + 3 = 13.

Worked example

Example 1: Use the area model

Solve 588 ÷ 14.

Friendly multiples of 14: 14 × 10 = 140, 14 × 20 = 280, 14 × 40 = 560.
First part: 14 × 40 = 560. 588 − 560 = 28 left.
Second part: 14 × 2 = 28. 28 − 28 = 0 left.
Add the widths: 40 + 2 = 42.
Check: 14 × 42 = 560 + 28 = 588. ✓

588 ÷ 14 = 42

An area model for 588 ÷ 14. Height 14. Parts: 40 wide (area 560) and 2 wide (area 28). The quotient is 40 + 2 = 42.

Partial quotients: the same thing in a list

Partial quotients is the area model written as a list. Each row takes away one part of the rectangle.

Worked example

Example 2: Partial quotients

Solve 864 ÷ 36.

36 × 20 = 720. Take it away: 864 − 720 = 144 left. (Partial quotient: 20)
How many 36s in 144? 36 × 2 = 72, so 36 × 4 = 144. Take it away: 144 − 144 = 0. (Partial quotient: 4)
Add the partial quotients: 20 + 4 = 24.
Check: 36 × 24 = 720 + 144 = 864. ✓

864 ÷ 36 = 24

Take away	Groups of 36	What's left
Start		864
36 × 20 = 720	20	864 − 720 = 144
36 × 4 = 144	4	144 − 144 = 0
Total	24

Partial quotients for 864 ÷ 36. Each row is one part of the area model.

Division tips

There is no single right first step. Taking away 10 groups and then 10 more is fine, just slower than taking 20 at once.
If you take away too much (more than what's left), use a smaller number of groups.
Always check by multiplying the quotient by the divisor.

Just for fun

Riddle: Why are division problems never lonely?

Because they are always sharing!

Hands-on activity

Share the beans

You need: a bag of dried beans, macaroni or other small items (keep small items away from young children), cups, and paper.

Count out 156 items. (Make 15 piles of 10 and 6 more to count faster.)
Use cups of 12 to find 156 ÷ 12. First fill 10 cups of 12 (that's 120). How many items are left? How many more cups of 12 can you fill?
Write it as partial quotients: 10 + 3 = 13. Check: 12 × 13 = 156.
Draw the area model for what you did.

Math talk: "Why is it faster to take away 10 groups at a time?"

Your turn

168 ÷ 12 = ?

275 ÷ 25 = ?

390 ÷ 15 = ?

448 ÷ 16 = ?

851 ÷ 37 = ?

220 ÷ 11 = ?

576 ÷ 24 = ?

More practice: warm-up

120 ÷ 12 = ?

240 ÷ 12 = ?

300 ÷ 15 = ?

500 ÷ 25 = ?

440 ÷ 11 = ?

More practice: on your own

252 ÷ 18 = ?

714 ÷ 21 = ?

972 ÷ 36 = ?

384 apples are packed in bags of 16. How many bags?

660 ÷ 55 = ?

Challenge

999 ÷ 27 = ?

Find the missing number: □ ÷ 23 = 17.

Noah says 805 ÷ 35 = 22. Check with multiplication. Is he right?

Check your understanding

Quiz

What is 360 ÷ 12?

For 735 ÷ 35, which first step takes away the most without going over?

What is 504 ÷ 24?

Which checks 456 ÷ 19 = 24?

What is 625 ÷ 25?

288 eggs go into cartons of 12. How many cartons?

For parents and teachers

Big idea: The Grade 5 curriculum (B2.7) asks students to divide three-digit numbers by two-digit numbers using the area model and algorithms, and to connect the methods. This lesson uses the area model and partial quotients; the next lesson connects them to the standard (long division) algorithm.

Common difficulties:

Not knowing where to start. A quick list of multiples (×1, ×2, ×5, ×10, ×20) gives safe choices.
Taking away too much. If the amount is bigger than what's left, try fewer groups.
Forgetting to add the partial quotients at the end.

How to help: Use real sharing and packing situations: cartons of 12, teams of 15, buses of 24. Always ask "How can you check?" and have your child multiply.