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

Dividing 3-Digit by 2-Digit Numbers: The Long Division Algorithm

B2.7 B2.1 A1About 45 minutes

Learning goals

I can divide a 3-digit number by a 2-digit number using long division.
I can estimate each digit of the quotient and adjust it if it's too big or too small.
I can connect the steps of long division to partial quotients and the area model.

Words to know

long division
An algorithm for dividing, done one place value at a time from left to right: divide, multiply, subtract, bring down.

bring down
Move the next digit of the dividend down beside what's left, to keep dividing.

estimate
A close guess. In long division you estimate each digit of the quotient, then check it.

Let's start with a comparison

With partial quotients, 672 ÷ 32 could look like this:

32 × 20 = 640. 672 − 640 = 32 left.
32 × 1 = 32. 32 − 32 = 0 left.
20 + 1 = 21.

Long division does the same thing, but it works one place at a time and writes it more compactly. The 20 groups become a 2 in the tens place.

The four steps

Repeat these steps for each digit:

Divide: How many times does the divisor fit?
Multiply: Multiply that digit by the divisor.
Subtract: Find what's left.
Bring down: Bring down the next digit.

A way to remember: Does Math Seem Boring? No way!

Long division for 672 ÷ 32. The quotient 21 is on top. 32 × 2 = 64 is subtracted from 67, leaving 3. Bring down the 2 to make 32. 32 × 1 = 32, leaving 0.

Worked example

Example 1: Long division

Solve 672 ÷ 32.

32 doesn't fit into 6 (hundreds). So look at 67 tens.
Divide: How many 32s in 67? Try 2: 32 × 2 = 64. Try 3: 32 × 3 = 96, too big. So write 2 above the 7 (tens place).
Multiply: 2 × 32 = 64. Subtract: 67 − 64 = 3 tens left.
Bring down the 2. Now you have 32 ones.
Divide: How many 32s in 32? 1. Write 1 above the 2. Multiply: 1 × 32 = 32. Subtract: 32 − 32 = 0.
Check: 32 × 21 = 640 + 32 = 672. ✓

672 ÷ 32 = 21

Where's the connection?

The 2 in the tens place means 20 groups of 32 = 640. That's the first part of the area model, or the first partial quotient. The 64 in long division is really 64 tens = 640. Long division just leaves out the zeros while you work.

Estimating each digit

With a 2-digit divisor, the hard part is guessing how many times it fits. Round the divisor to the nearest ten to make a first guess. Then multiply to check.

If your product is too big, try one less.
If what's left is as big as or bigger than the divisor, try one more.

Worked example

Example 2: Adjust the estimate

Solve 828 ÷ 23.

Look at 82 tens. Round 23 to 20. 82 ÷ 20 is about 4.
Check: 23 × 4 = 92. That's more than 82, so 4 is too big. Try 3: 23 × 3 = 69. ✓ Write 3 in the tens place.
Subtract: 82 − 69 = 13. Bring down the 8: 138.
How many 23s in 138? Round: 140 ÷ 20 = 7. Check: 23 × 7 = 161, too big. Try 6: 23 × 6 = 138. ✓
Subtract: 138 − 138 = 0.
Check: 23 × 36 = 690 + 138 = 828. ✓

828 ÷ 23 = 36

Long division for 828 ÷ 23. 23 × 3 = 69, and 82 − 69 = 13. Bring down 8 to make 138. 23 × 6 = 138, leaving 0. The quotient is 36.

Worked example

Example 3: A story problem

A cyclist rides 416 km in 13 days, the same distance each day. How far does she ride each day?

We need 416 ÷ 13.
13 doesn't fit into 4. Look at 41 tens. 13 × 3 = 39. ✓ (13 × 4 = 52 is too big.) Write 3.
41 − 39 = 2. Bring down the 6: 26.
13 × 2 = 26. Write 2. 26 − 26 = 0.
Check: 13 × 32 = 390 + 26 = 416. ✓

32 km each day

Long division tips

Write a short list of multiples of the divisor beside your work. It saves lots of guessing.
After every subtraction, the number left must be smaller than the divisor. If not, your digit was too small.
If long division gets confusing, switch to partial quotients. Both methods are correct, and choosing a method that works for you is smart math.

Just for fun

Can't remember Divide, Multiply, Subtract, Bring down? Try a sillier sentence: "Dancing Monkeys Sell Bananas." Then make up your own. The sillier it is, the easier it is to remember!

Hands-on activity

Teach it back

You need: paper, a pencil and a grown-up who is willing to be your student.

Write 945 ÷ 45 on the paper.
Teach your grown-up how to solve it with long division. Say each step out loud: divide, multiply, subtract, bring down.
Now solve the same question with partial quotients or an area model. Show where the 2 in the quotient means 20 groups of 45 (900).
Swap roles: your grown-up makes up a division with a 2-digit divisor, and you check their work with multiplication. (Answer for 945 ÷ 45: 21.)

Math talk: "What does the first digit of the quotient really stand for?"

Your turn

495 ÷ 15 = ?

736 ÷ 23 = ?

924 ÷ 42 = ?

384 ÷ 12 = ?

975 ÷ 25 = ?

567 ÷ 21 = ?

A ski club pays $840 for 24 lift tickets. Each ticket costs the same. How much is one ticket?

More practice: warm-up

26 ÷ 13 = ?

48 ÷ 24 = ?

60 ÷ 15 = ?

75 ÷ 25 = ?

99 ÷ 33 = ?

More practice: on your own

858 ÷ 33 = ?

612 ÷ 18 = ?

952 ÷ 56 = ?

744 ÷ 31 = ?

A school raised $851 and shares it equally among 37 classes. How much does each class get?

Challenge

988 ÷ 52 = ?

Ty solved 945 ÷ 45 and got 2. What did he forget? What is the right answer?

Make up a story for 864 ÷ 24 and solve it.

Check your understanding

Quiz

What is 672 ÷ 32?

What is the first question to ask when solving 828 ÷ 23 with long division?

For 82 ÷ 23, someone tries 4. Since 23 × 4 = 92, what should the digit be?

What is 750 ÷ 25?

Which checks 912 ÷ 24 = 38?

What is 540 ÷ 15?

For parents and teachers

Big idea: B2.7 asks Grade 5 students to divide three-digit numbers by two-digit numbers using algorithms as well as the area model, and to connect them. Long division is one algorithm; partial quotients is another. Both are correct, and your child may use either.

Common difficulties:

Estimating each digit. Round the divisor to the nearest ten for a first guess, then adjust.
Forgetting to bring down a digit, or writing quotient digits in the wrong place.
Not knowing what the digits mean. The first digit of a 2-digit quotient stands for tens of groups.

How to help: You may have learned long division differently; that's fine. Let your child teach you their method and explain why it works. Encourage checking every answer with multiplication. Choosing a strategy and persevering through estimates that need adjusting are A1 process skills.