Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 3 of this unit) Write 2/3 and 3/4 as fractions over a common denominator of 12.
Answer
8/12 and 9/12. 12 ÷ 3 = 4, and 2 × 4 = 8; 12 ÷ 4 = 3, and 3 × 3 = 9. - (Lesson 2 of this unit) Write 1 3/5 as an improper fraction.
Answer
8/5. 1 whole is 5 fifths, and 3 more makes 8. - (Lesson 1 of this unit) Write 10/15 in simplest form.
Answer
2/3. Divide the top and the bottom by 5.
Words used in this lesson
- common denominator
- a bottom number that both fractions can be written over. For example: 12 for quarters and sixths.
- sum
- the result of adding. For example: the sum of 1/2 and 1/3 is 5/6.
- difference
- the result of subtracting; how much more one is than the other. For example: the difference between 3/4 and 1/4 is 1/2.
- regroup
- cut one whole into parts so there are enough to take away. For example: 5 1/3 can be regrouped as 4 4/3.
- simplest form
- a fraction where only 1 divides both top and bottom. For example: 3/6 simplifies to 1/2.
Same-size parts
Ngozi adds half a cake and a third of a cake and writes 1/2 + 1/3 = 2/5. Predict first: is 2/5 of a cake more than a half-cake, or less? Then test it: set the first fraction to 1/2 and the second to 1/3, and the figure recuts both pieces into parts of the same size and joins them.
1/4 = 3/12 and 1/6 = 2/12, so the sum is 3/12 + 2/12 = 5/12.
Show me the steps
- 1/2 + 1/3: over 6 it is 3/6 + 2/6 = 5/6.
- 2/3 + 1/4: over 12 it is 8/12 + 3/12 = 11/12.
- 3/4 − 1/6: over 12 it is 9/12 − 2/12 = 7/12.
- 5/6 + 1/6: over 6 it is 5/6 + 1/6 = 6/6 = 1.
- 3/8 + 1/4: over 8 it is 3/8 + 2/8 = 5/8, already in simplest form.
- 2/3 − 1/6: over 6 it is 4/6 − 1/6 = 3/6, which simplifies to 1/2.
Only parts of the same size can be added or subtracted. Write both fractions over a common denominator, then add or subtract the tops (the numerators). The bottom (the denominator) stays the same, because it names the size of the part, and adding does not change the size of a part.
Then simplify, and if the answer is more than 1, write it as a mixed number. With mixed numbers, you can add the whole numbers and the fractions separately, or turn both into improper fractions first.
Subtracting works the same way: take the count of the smaller from the count of the larger.
Back to Ngozi: 1/2 = 3/6 and 1/3 = 2/6, so the total is 5/6 of a cake. 2/5 is less than 1/2, so it could not be a sum that includes a half.
A common mistake: “Add the tops and add the bottoms.” When the bottoms are the same, adding them as well doubles the number of parts and keeps the count, so the answer is half the right one (the second figure). When the bottoms are different, tops-plus-bottoms always lands between the two fractions (1/2 and 1/3 give 2/5), so it is never more than the larger one and cannot be their sum.
Keep the bottom, add the tops
When the bottoms already match, only the tops change. Move all three sliders and compare the two answers.
3/8 + 3/8: both count eighths, so 3 + 3 = 6 eighths, which is 3/4. Adding the bottoms as well gives 6/16 = 3/8, only half as much.
Show me the steps
- 3/8 + 3/8 = 6/8 = 3/4. The wrong way gives 6/16 = 3/8, half as much.
- 2/5 + 1/5 = 3/5. The wrong way gives 3/10.
- 7/10 + 6/10 = 13/10, which is 1 3/10. The wrong way gives 13/20.
- 5/12 + 7/12 = 12/12 = 1. The wrong way gives 12/24 = 1/2.
- 1/4 + 2/4 = 3/4. The wrong way gives 3/8.
Worked examples
Work out 2/3 + 3/10.
- The lowest common multiple of 3 and 10 is 30.
- 2/3 = 20/30 and 3/10 = 9/30.
- 20/30 + 9/30 = 29/30, which is already in simplest form.
Work out 1 2/5 + 2 3/4.
- Whole numbers first: 1 + 2 = 3.
- Fractions over 20: 2/5 = 8/20 and 3/4 = 15/20, which add to 23/20 = 1 3/20.
- Put them together: 3 + 1 3/20 = 4 3/20.
Work out 5 1/3 − 2 3/4.
- Fractions over 12: 5 4/12 − 2 9/12. But 4/12 is less than 9/12, so there is not enough to take away.
- Take 1 whole from the 5 and cut it into twelfths: 5 4/12 = 4 16/12.
- 4 16/12 − 2 9/12: wholes 4 − 2 = 2, twelfths 16 − 9 = 7. 2 7/12.
Your go. The first step is given. Work out 5/6 − 3/8.
- The lowest common multiple of 6 and 8 is 24.
- Now write both over 24 yourself, subtract, then check.
Check the answer
5/6 = 20/24 and 3/8 = 9/24, so 20/24 − 9/24 = 11/24. It is already in simplest form.
Check before you go on
1. What is 1/4 + 2/5?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. 4/7 + 5/7 is more than 1, so write it as a mixed number. What is the top number of the fraction part?
3. 3/4 + 5/6 = ☐/12. What number goes in the box?
Level 2 · Building
4. 2 2/3 + 1 3/4 = ☐/12. Write the sum as an improper fraction over 12. What number goes in the box?
5. A plank is 3 1/4 metres long and Kofi cuts off 1 3/4 metres. How many quarter-metres are left?
6. 1/2 + ☐ = 7/8. Write the missing fraction over 8. What is its top number?
Level 3 · Stretch
7. Three friends share a pizza. One eats 1/3, one eats 1/4 and one eats 1/6. How many twelfths of the pizza are left?
8. Two different fractions, each with a top of 1, add to make 1/2. One of them is 1/3. What is the bottom number of the other?
Something to think about
Add 1/6 + 1/4 + 1/12. The bottom numbers are 6, 4 and 12, but the answer has a much smaller bottom number. What is the answer, and why does the bottom get smaller?
One way to think about it
Over 12 the sum is 2/12 + 3/12 + 1/12 = 6/12. Six twelfths is half a whole, so the simplest form is 1/2. The 12 only says what size of part we counted in: after counting, the six parts can be regrouped into one larger part, a half.
Exit ticket
9. What is 3/5 + 3/10?