Discover Mathematics P5: Term 2, Unit 7, Lessons 1–3
GoalUse the benchmarks 0, 1/2 and 1 to estimate sums and differences of fractions before calculating.
New words
benchmark · an easy number to compare with: 0, 1/2 or 1estimate · a quick answer that is close to the exact onenumerator · the top number of a fraction; it says how many parts we have
Explanation
To see which benchmark a fraction is near, compare its numerator with the denominator. If the numerator is tiny (1/10), the fraction is close to 0. If the numerator is about half of the denominator (6/11, since half of 11 is 5.5), the fraction is close to 1/2. If the numerator is almost as big as the denominator (9/10), the fraction is close to 1. To estimate a sum or difference, replace each fraction by its benchmark and calculate with the benchmarks. For example 5/8 + 3/7 is about 1/2 + 1/2 = 1. An estimate also tells us whether an exact answer is sensible.
Fraction
Numerator compared with the denominator
Benchmark
1/8
tiny
0
5/12
about half of 12
1/2
7/8
almost 8
1
Worked examples
Name the benchmark for 1/8, 5/12 and 11/12. 1/8 is close to 0. Half of 12 is 6, and 5 is close to 6, so 5/12 is close to 1/2. 11 is almost 12, so 11/12 is close to 1.
Estimate 5/8 + 3/7. 5/8 is close to 1/2 (half of 8 is 4) and 3/7 is close to 1/2 (half of 7 is 3.5). The estimate is 1/2 + 1/2 = 1. The exact sum, 59/56, is a little over 1.
Estimate 7/8 − 4/9. 7/8 is close to 1 and 4/9 is close to 1/2. The estimate is 1 − 1/2 = 1/2. The exact answer, 31/72, is close to 1/2.
Class activity
Fraction line-up. Draw a line from 0 to 1 with 1/2 in the middle. Write fractions on cards (1/10, 6/11, 9/10, 3/8, 7/12, 1/9, 11/12) and place each card near the benchmark it is closest to. Then pick two cards and estimate their sum before adding.
Practice
1
Give the nearest benchmark (0, 1/2 or 1) for 1/10, 6/11, 9/10 and 1/12.
0, 1/2, 1 and 0.
2
Estimate 11/12 + 5/9 with benchmarks.
11/12 is close to 1 and 5/9 is close to 1/2. The estimate is 1 + 1/2 = 1 1/2.
3
Estimate 9/10 − 3/7.
9/10 is close to 1 and 3/7 is close to 1/2. The estimate is 1/2.
4
Seif ate 1/9 of a pizza and Mina ate 4/7 of it. About how much did they eat in all?
1/9 is close to 0 and 4/7 is close to 1/2 (half of 7 is 3.5). They ate about 1/2 of the pizza.
5
Jana says 5/6 + 7/8 is about 2. Is she right? Explain.
Yes. Both fractions are close to 1, so the sum is about 1 + 1 = 2.
