Estimating with Fractions

An important part of number sense is to have a concept of the size of fractions independent of doing calculations. For example, a student with a good sense of number will know immediately that 11/21 is more than 1/2 because 11 is over half the available 21 pieces.

Using Materials

Problem: Show 4/9 using the two different-coloured circle pieces and discuss whether  it is more or less than 1/2.

Each student rotates the circles until one colour circle shows as near to 4/9 as they can. Discuss why it is 4/9.Look at the other colour and discuss what fraction it is.

(Answer: 5/9 because there are 9/9 in 1 whole.)

Examples: Show these fractions on the circles and work out the other fraction that adds up to 1 whole: 3/7 4/5 5/9 5/11 11/21...

Using Imaging

Problem: “I have made 15/31, but you cannot see it. Describe in words what this fraction looks like.”

(Possible answer: the fraction is just a little bit less than 1/2.)

“What is the other fraction?” (Answer: 16/31.)

Examples: Describe these fractions without using the circles and work out the other fraction that adds up to 1 whole: 5/11 2/5 4/7 8/17 6/13

Using Number Properties

Problem: “Maurice eats 2/5 of a cake, and Norris eats 3/7 of a same sized cake. In total, do they eat more or less than 1 cake?”

Discuss the answer. (Answer: Both are a little less than 1/2 so the total is less than 1 whole.)

Examples: Without calculating, determine whether these are more or less than 1:

2/5 + 12/23 17/33 + 6/11 51/100 + 6/12 24/48 + 29/60 24/47 + 61/120 ...

Examples: Is 1 – 21/43 more or less than 1/2? Is 11/45 more or less than 1/4? Is 3/4 – 11/40 more or less than 1/2? Is 3/4 + 15/62 more or less than 1?

Hard example: Is 3/7 + 14/27 more or less than 1? (Answer: Here the gap between 3/7 and 1/2 is more than the gap between 14/27 and 1/2. So 3/7 + 14/27 < 1.)

Understanding Number Properties:

Make up two fractions that both have  denominators greater than 30 and that add up to just a little bit less than 1.