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

Adding and Subtracting Fractions with Unlike Denominators

B2.5 A1About 45 minutes
Learning goals
  • I can find a common denominator for two fractions.
  • I can add and subtract fractions with unlike denominators using equivalent fractions.
  • I can show fraction addition and subtraction with fraction strips or a number line.

Words to know

denominator
The bottom number of a fraction. It tells how many equal parts make one whole.
numerator
The top number of a fraction. It tells how many parts you have.
like denominators
Fractions with the same denominator, like 2/7 and 3/7.
unlike denominators
Fractions with different denominators, like 1/2 and 1/3.
common denominator
A denominator that two or more fractions can share. 6 is a common denominator for 1/2 and 1/3.
equivalent fractions
Fractions that name the same amount, like 1/2 = 3/6.

Remember from Grade 5

In Grade 5 you added and subtracted fractions with like denominators: 2/8 + 3/8 = 5/8. The pieces were the same size, so you just counted them. You also made equivalent fractions, like 1/2 = 2/4 = 3/6.

Let's start with pizza

You eat 1/2 of a pizza at lunch. Your cousin eats 1/3 of a same-size pizza. How much pizza did you eat together?

You can't just count pieces, because halves and thirds are different sizes. 1 half + 1 third is not "2 of something".

The fix: cut both pizzas into sixths. 1/2 = 3/6 and 1/3 = 2/6. Now the pieces match: 3/6 + 2/6 = 5/6.

1/21/31/2 + 1/31/61/61/61/61/61/63/6 + 2/65 of the 6 sixths are shaded: 5/6
Top: 1/2 and 1/3 placed end to end on a strip. Bottom: the same lengths shown in sixths. 1/2 = 3/6 and 1/3 = 2/6, so together they make 5/6.

The steps

  1. Find a common denominator. Look for a number that both denominators divide into. One way that always works: multiply the denominators. Often there is a smaller one (the smallest is the least common multiple).
  2. Make equivalent fractions with that denominator. Whatever you multiply the denominator by, multiply the numerator by the same number.
  3. Add or subtract the numerators. Keep the denominator.
  4. Simplify if you can, and change an improper fraction to a mixed number.
Worked example

Example 1: Add halves and thirds

1/2 + 1/3 = ?

  1. Common denominator: 2 × 3 = 6.
  2. 1/2 = 3/6 (multiply top and bottom by 3). 1/3 = 2/6 (multiply top and bottom by 2).
  3. 3/6 + 2/6 = 5/6.

1/2 + 1/3 = 5/6

Worked example

Example 2: Subtract

3/4 − 1/6 = ?

  1. Multiples of 4: 4, 8, 12, 16. Multiples of 6: 6, 12, 18. The smallest common denominator is 12.
  2. 3/4 = 9/12 (× 3). 1/6 = 2/12 (× 2).
  3. 9/12 − 2/12 = 7/12.

3/4 − 1/6 = 7/12

3/4 − 1/6 in twelfths3/41/69/12 − 2/127/12 lefttake away 2/12
Fourths and sixths both fit into twelfths: 3/4 = 9/12 and 1/6 = 2/12. Cross off 2 of the 9 twelfths and 7/12 are left.
Worked example

Example 3: An answer bigger than 1

2/3 + 3/4 = ?

  1. Common denominator: 12.
  2. 2/3 = 8/12. 3/4 = 9/12.
  3. 8/12 + 9/12 = 17/12.
  4. 17/12 is 12/12 + 5/12 = 1 5/12.
  5. Does it make sense? 2/3 and 3/4 are each more than 1/2, so the sum should be more than 1. ✓

2/3 + 3/4 = 17/12 = 1 5/12

Watch out

Never add the denominators. 1/2 + 1/3 is not 2/5. Check: 2/5 is less than 1/2, but you added something to 1/2! The denominator names the size of the pieces, and the size doesn't change when you put pieces together.

Just for fun

What did 1/2 say to 1/3 at the pizza party?

"Let's both turn into sixths so we can finally hang out!"

Hands-on activity

Measuring cup fractions

You need: a set of measuring cups (1 cup, 1/2 cup, 1/3 cup, 1/4 cup), a large clear measuring cup with markings, water and a towel.

  1. Pour 1/2 cup and then 1/4 cup of water into the big measuring cup. Predict the total first. Then read the markings. (3/4 cup)
  2. Empty it. Now pour 1/2 cup and 1/3 cup. Predict, then check. (5/6 cup, a little less than 1 cup)
  3. Fill the 1/2 cup. Pour from it into the 1/3 cup until the 1/3 cup is full. How much is left in the 1/2 cup? (1/2 − 1/3 = 1/6 cup)

Math talk: "How did you change the fractions so you could add them?"

Your turn

1

1/2 + 1/4 = ?

2

1/3 + 1/6 = ?

3

2/5 + 3/10 = ?

4

3/4 − 1/2 = ?

5

5/6 − 1/3 = ?

6

1/4 + 1/3 = ?

7

7/8 − 3/4 = ?

More practice: warm-up

1

2/7 + 3/7 = ?

2

5/9 − 2/9 = ?

3

1/2 = ?/8

4

2/3 = ?/12

5

3/10 + 4/10 = ?

More practice: on your own

1

2/3 + 1/4 = ?

2

4/5 − 1/2 = ?

3

5/6 + 3/4 = ?

4

7/10 − 2/5 = ?

5

A muffin recipe uses 3/4 cup of flour and 2/3 cup of oats. How many cups is that altogether?

Challenge

1

A snowshoe trail is 5/6 km long. Jess has walked 3/8 km. How far does she have left?

2

Jay says 1/2 + 1/3 = 2/5. Explain his mistake.

3

In a garden, 1/4 is tomatoes, 1/3 is beans, and the rest is flowers. What fraction is flowers?

Check your understanding

Quiz

1.

What is 1/2 + 1/5?

2.

Which is a common denominator for 1/4 and 1/6?

3.

What is 2/3 − 1/2?

4.

What is 3/4 + 3/8?

5.

What is 5/6 − 1/4?

6.

Mia drinks 1/3 L of milk, then 1/2 L more. How much did she drink?

For parents and teachers

Big idea: B2.5 asks Grade 6 students to add and subtract fractions with like and unlike denominators, using appropriate tools, in various contexts. Grade 5 covered like denominators only, so unlike denominators are the new step. Students can use any common denominator; the smallest one keeps numbers small but is not required.

Common difficulties:

  • Adding the denominators (1/2 + 1/3 = 2/5).
  • Changing the denominator but not the numerator (writing 1/3 as 1/6).
  • Not checking if the answer makes sense. Compare each fraction with 1/2 and 1 to estimate.

How to help: Use measuring cups while baking. Ask "Will it be more or less than 1 cup?" before adding. Let your child use paper strips or drawings as long as they need them; tools are part of the curriculum, not a crutch. Trying again after a mistake builds the A1 skill of perseverance.