Grade by Grade
B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning

Adding and Subtracting Fractions, Mixed Numbers and Negative Fractions

B2.5 B2.1 A1About 45 minutes
Learning goals
  • I can add and subtract fractions and mixed numbers, choosing a strategy that fits.
  • I can add and subtract three or more fractions using the lowest common denominator.
  • I can add and subtract negative fractions using what I know about integers.
  • I can estimate first with benchmarks like 0, 1/2 and 1.

Words to know

lowest common denominator
The lowest common multiple of the denominators. For 1/6 and 3/8 it is 24.
mixed number
A whole number and a fraction together, like 4 1/3.
improper fraction
A fraction whose numerator is greater than or equal to its denominator, like 13/3.
negative fraction
A fraction less than zero, like −3/4. It is the opposite of 3/4.
benchmark
A friendly number used to estimate, like 0, 1/2 or 1.

Remember from Grade 7

In Grade 7 you used the lowest common multiple to add and subtract fractions with unlike denominators, and you regrouped mixed numbers, like 5 1/6 − 2 3/4 = 2 5/12. In Grade 8 you will choose strategies for longer questions, and you will add and subtract negative fractions, which follow the same rules as integers.

Let's start with a snowy weekend

It snowed 3 1/2 cm on Friday, 4 2/3 cm on Saturday and 1 3/4 cm on Sunday. How much snow fell in all?

  • Add the whole numbers: 3 + 4 + 1 = 8.
  • Add the fractions with the lowest common denominator, 12: 6/12 + 8/12 + 9/12 = 23/12 = 1 11/12.
  • Total: 8 + 1 11/12 = 9 11/12 cm.

Estimate to check: about 3 1/2 + 4 1/2 + 2 = 10. ✓

Friday: 1/2= 6/12Saturday: 2/3= 8/12Sunday: 3/4= 9/12Fractions total:1 whole11/126 + 8 + 9 = 23 twelfths = 1 11/12. Add the wholes: 3 + 4 + 1 = 8.Total snow: 8 + 1 11/12 = 9 11/12 cm
In twelfths, the fraction parts are 6 + 8 + 9 = 23 twelfths, which is 1 11/12. Add that to the 8 whole centimetres.

Strategies

  1. Common denominators. Use the lowest common multiple of the denominators, then add or subtract the numerators.
  2. Mixed numbers, parts separately. Add or subtract the whole numbers and the fractions separately. Regroup if a fraction part is bigger than 1, or if you need to borrow 1 whole.
  3. Mixed numbers as improper fractions. Change each mixed number to an improper fraction first. This avoids borrowing, which is handy for subtraction.
  4. Negative fractions. Use the integer rules. Adding a negative moves left on the number line, and subtracting is adding the opposite: 1/2 − (−3/8) = 1/2 + 3/8.
−1−3/4−1/2−1/401/41/23/41+1 1/4−3/4 + 1 1/4 = 1/2
A number line from −1 to 1 marked in quarters. Start at −3/4 and move 1 1/4 to the right. You land on 1/2, so −3/4 + 1 1/4 = 1/2.
Worked example

Example 1: Three fractions

Find 2/3 + 5/6 − 1/4.

  1. The lowest common multiple of 3, 6 and 4 is 12.
  2. 8/12 + 10/12 − 3/12 = 15/12.
  3. 15/12 = 1 3/12 = 1 1/4.

1 1/4

Worked example

Example 2: Subtract mixed numbers using improper fractions

Find 4 1/3 − 2 5/6.

  1. 4 1/3 = 13/3 = 26/6, and 2 5/6 = 17/6.
  2. 26/6 − 17/6 = 9/6 = 1 3/6 = 1 1/2.
  3. Check by adding: 2 5/6 + 1 1/2 = 2 5/6 + 1 3/6 = 3 8/6 = 4 2/6 = 4 1/3. ✓

1 1/2

Worked example

Example 3: A negative plus a positive

Find (−3/4) + 1/3.

  1. Common denominator 12: (−9/12) + 4/12.
  2. Like integers: (−9) + 4 = −5, so the sum is −5/12.
  3. Sense check: −3/4 is farther from 0 than 1/3, so the answer is negative. ✓

−5/12

Worked example

Example 4: Subtract a negative

Find 1/2 − (−3/8).

  1. Add the opposite: 1/2 + 3/8.
  2. 4/8 + 3/8 = 7/8.

7/8

Worked example

Example 5: Cutting a board

A board is 2 1/2 m long. Two pieces, 3/4 m and 1 1/3 m, are cut off. How much is left? (Ignore the width of the saw cut.)

  1. Write everything in twelfths: 2 1/2 = 30/12, 3/4 = 9/12, 1 1/3 = 16/12.
  2. 30/12 − 9/12 − 16/12 = 5/12.

5/12 m is left

Estimate first

Round each fraction to the nearest benchmark: 0, 1/2 or 1. For 7/8 + 5/9: 7/8 is close to 1 and 5/9 is close to 1/2, so the sum is about 1 1/2. If your exact answer is far from your estimate, look for a mistake.

Hands-on activity

Measuring cup fractions

You need: measuring cups (1 cup, 1/2 cup, 1/3 cup and 1/4 cup), a bowl of dry rice or oats, a tray to catch spills.

  1. Fill the 1/2 cup and the 1/3 cup and pour both into the 1 cup measure. Is it full? Write the addition.
  2. Do the same with 1/4 cup + 1/3 cup.
  3. Try 1/2 cup + 1/4 cup + 1/3 cup. Does it overflow? By about how much?
  4. Fill the 1 cup measure, then scoop out 1/3 cup. What fraction is left? Write the subtraction.

Model answers: 1. 1/2 + 1/3 = 5/6, a little less than full. 2. 1/4 + 1/3 = 7/12. 3. 1/2 + 1/4 + 1/3 = 6/12 + 3/12 + 4/12 = 13/12 = 1 1/12, so it overflows by 1/12 cup. 4. 1 − 1/3 = 2/3 cup.

Just for fun

Why couldn't 1/3 and 1/4 agree on anything at first?

They had to find some common ground. (It was twelfths.)

Your turn

1

3/5 + 1/4 = ?

2

5/6 − 3/8 = ?

3

2 1/2 + 1 2/3 = ?

4

6 1/4 − 3 2/3 = ?

5

(−1/2) + (−1/3) = ?

6

(−5/8) + 3/4 = ?

7

2/5 − (−1/10) = ?

More practice: warm-up

1

1/3 + 1/3 = ?

2

3/4 − 1/2 = ?

3

1/2 + 1/4 = ?

4

2 1/5 + 1 3/5 = ?

5

(−1/4) + 1/4 = ?

More practice: on your own

1

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

2

3 3/8 − 1 3/4 = ?

3

(−2/3) − 1/4 = ?

4

A recipe uses 1 1/2 cups of flour, 3/4 cup of oats and 2/3 cup of sugar. How many cups of dry ingredients is that?

5

The temperature is −2 1/2°C. It rises 4 3/4 degrees. What is the new temperature?

Challenge

1

Find the missing number: 3/4 + ___ = 1/6.

2

A trail is 5 km long. Ben walks 1 3/4 km before lunch and 2 2/5 km after lunch. How far does he still have to go?

3

Find 1/2 + 1/4 + 1/8 + 1/16. How far is the sum from 1? What pattern do you notice?

Check your understanding

Quiz

1.

What is 2/3 + 1/4?

2.

What is 5 1/4 − 2 3/4?

3.

What is (−1/2) + 1/5?

4.

What is the lowest common denominator for 1/6 + 3/8?

5.

What is 1/3 − (−1/3)?

6.

Which is the best estimate for 7/8 + 5/9?

For parents and teachers

Big idea: Ontario's Grade 8 expectation B2.5 asks students to add and subtract fractions, using appropriate strategies, in various contexts. Grade 7 covered the basic methods; Grade 8 adds choosing strategies, longer expressions and negative fractions (which link to B2.1, rational numbers).

Common difficulties:

  • Adding numerators and denominators (2/3 + 1/4 = 3/7). Use the measuring cups to show why this is wrong.
  • Regrouping mixed numbers. If borrowing is confusing, use improper fractions.
  • Signs with negative fractions. Treat the numerators like integers once the denominators match.

How to help: Cook or bake together and add up ingredient amounts. Ask for an estimate before every exact answer.