Grade by Grade
B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Adding and Subtracting Mixed Numbers

B2.5 A1About 45 minutes
Learning goals
  • I can add mixed numbers by adding the wholes and the fractions.
  • I can regroup when the fractions add to more than 1.
  • I can subtract mixed numbers, regrouping a whole when I need to.
  • I can solve problems with mixed numbers in measurement contexts.

Words to know

mixed number
A number with a whole number part and a fraction part, like 2 3/4.
improper fraction
A fraction whose numerator is greater than or equal to its denominator, like 11/4.
regroup
Trade 1 whole for an equal number of fraction pieces (1 = 4/4), or trade fraction pieces for a whole.

Remember from Grade 5

In Grade 5 you changed between improper fractions and mixed numbers: 7/4 = 1 3/4. In the last lesson you added fractions with unlike denominators. Now you'll put both ideas together.

Let's start with snowfall

On Monday 2 1/2 cm of snow fell. On Tuesday 3 3/4 cm fell. How much snow is that?

  • Add the wholes: 2 + 3 = 5.
  • Add the fractions: 1/2 + 3/4 = 2/4 + 3/4 = 5/4.
  • 5/4 is more than 1! 5/4 = 1 1/4. So the total is 5 + 1 1/4 = 6 1/4 cm.
2 1/2 cm + 3 3/4 cm of snow2 1/23 3/46 1/41/2 + 3/4 = 5/4 = 1 1/44/4 = 1
Add the wholes (2 + 3 = 5) and the parts (1/2 + 3/4 = 5/4). Four of the quarters regroup into 1 whole, so the total is 6 1/4 cm.

Adding mixed numbers

  1. Make the fractions have a common denominator.
  2. Add the wholes. Add the fractions.
  3. If the fraction part is improper, regroup it into a whole.

Subtracting mixed numbers

  1. Make the fractions have a common denominator.
  2. If the first fraction is too small to subtract from, regroup: take 1 whole and turn it into fraction pieces.
  3. Subtract the fractions. Subtract the wholes.

Another way: change both mixed numbers into improper fractions, then add or subtract. This always works, but the numbers can get big.

Mixed numberRegroupedWhy
4 1/33 4/3Take 1 whole from 4. 1 = 3/3, and 3/3 + 1/3 = 4/3.
5 2/84 10/81 = 8/8, and 8/8 + 2/8 = 10/8.
32 5/51 = 5/5.
Regrouping a whole into fraction pieces. The value does not change.
Regroup 4 1/3 as 3 4/34 1/33 4/33/3 + 1/3 = 4/3cut 1 whole into thirds
One whole is cut into 3 thirds. Now there are 3 wholes and 4 thirds: the same amount, ready for subtracting.
Worked example

Example 1: Add without regrouping

2 1/2 + 1 1/3 = ?

  1. Common denominator 6: 1/2 = 3/6 and 1/3 = 2/6.
  2. Wholes: 2 + 1 = 3. Fractions: 3/6 + 2/6 = 5/6.

2 1/2 + 1 1/3 = 3 5/6

Worked example

Example 2: Add with regrouping

1 3/4 + 2 1/2 = ?

  1. 1/2 = 2/4.
  2. Wholes: 1 + 2 = 3. Fractions: 3/4 + 2/4 = 5/4.
  3. 5/4 = 1 1/4. So 3 + 1 1/4 = 4 1/4.

1 3/4 + 2 1/2 = 4 1/4

Worked example

Example 3: Subtract with regrouping

4 1/3 − 1 3/4 = ?

  1. Common denominator 12: 4 1/3 = 4 4/12 and 1 3/4 = 1 9/12.
  2. You can't take 9/12 from 4/12. Regroup: 4 4/12 = 3 16/12.
  3. Fractions: 16/12 − 9/12 = 7/12. Wholes: 3 − 1 = 2.
  4. Check: 1 3/4 + 2 7/12 = 1 9/12 + 2 7/12 = 3 16/12 = 4 4/12 = 4 1/3. ✓

4 1/3 − 1 3/4 = 2 7/12

Two ways to solve

If a strategy gets confusing, try the other one. You can always change mixed numbers to improper fractions: 1 3/4 + 2 1/2 = 7/4 + 10/4 = 17/4 = 4 1/4. Same answer! Choosing a strategy that works for you is what good problem solvers do.

Just for fun

Why did 4 1/3 break one of its wholes into thirds?

It needed change, just like trading a loonie for coins!

Hands-on activity

Measuring with mixed numbers

You need: measuring cups (1 cup, 1/2 cup, 1/4 cup), a large bowl, a large clear measuring cup with markings, and water (or dry rice).

  1. Measure 1 1/2 cups of water into the bowl (one 1-cup and one 1/2-cup).
  2. Add 3/4 cup more (three 1/4 cups).
  3. Predict the total: 1 1/2 + 3/4 = 1 2/4 + 3/4 = 1 5/4 = 2 1/4 cups.
  4. Pour it all into the clear measuring cup (in two pours if needed) and check.
  5. Now scoop out 1 1/4 cups. Predict how much is left, then measure. (1 cup)

Math talk: "Where did you need to regroup?"

Your turn

1

1 1/4 + 2 1/2 = ?

2

3 2/3 + 1 1/6 = ?

3

5 3/4 − 2 1/4 = ?

4

2 1/2 + 1 3/4 = ?

5

4 1/2 − 1 1/3 = ?

6

3 − 1 2/5 = ?

More practice: warm-up

1

Write 1 1/2 as an improper fraction.

2

Write 7/4 as a mixed number.

3

2 1/5 + 1 2/5 = ?

4

3 3/4 − 1 1/4 = ?

5

1 + 2/3 = ?

More practice: on your own

1

2 2/3 + 1 1/2 = ?

2

5 1/4 − 2 1/2 = ?

3

6 − 2 3/8 = ?

4

1 5/6 + 2 3/4 = ?

5

A recipe needs 2 1/4 cups of flour. You have 3 2/3 cups. How much will be left?

Challenge

1

7 1/5 − 3 1/2 = ?

2

A board is 4 m long. A builder uses 1 3/4 m for one shelf and 1 2/3 m for another. How much board is left?

3

Find the missing number: 2 3/4 + ? = 5 1/3.

Check your understanding

Quiz

1.

What is 1 1/3 + 2 1/3?

2.

What is 2 3/4 + 1 1/2?

3.

What is 4 − 1 1/4?

4.

What is 5 1/6 − 2 1/3?

5.

Write 9/4 as a mixed number.

6.

What is a good first step for 3 1/2 − 1 2/3?

For parents and teachers

Big idea: This lesson continues B2.5 (adding and subtracting fractions with like and unlike denominators) with mixed numbers in measurement contexts. Students may add wholes and fractions separately, or convert to improper fractions. Both are correct.

Common difficulties:

  • Leaving an improper fraction in the answer (3 5/4 instead of 4 1/4).
  • Subtracting the smaller fraction from the bigger one when the first fraction is smaller (in 4 1/3 − 1 3/4, doing 3/4 − 1/3). The fix is regrouping a whole.
  • Regrouping 1 whole as 10 pieces out of habit from decimals. 1 = 12/12 for twelfths, 1 = 4/4 for fourths.

How to help: Cook or bake together and double or combine amounts. Praise your child for checking with the inverse operation.