# Fraction Benchmarks

Purpose

In this unit we develop important reference points or benchmarks for zero, one half and one. We use these benchmarks to help compare the relative sizes of fractions, through estimating, ordering and placing them on a number line.

Achievement Objectives
NA3-5: Know fractions and percentages in everyday use.
Specific Learning Outcomes
• State which of two fractions is larger.
• Explain why a fraction is close to 0, 1/2 or 1.
Description of Mathematics

This unit builds on the following key conceptual understandings about fractional parts.

• Fractional parts are equal sized parts of a whole or a unit. A whole or unit can be an object or a collection of things and is counted as "1".
• Fractional parts are named by the number of equal parts that are needed to make a unit. For example, fifths require five equal parts to make a unit.
• The more fractional parts used to make a whole, the smaller the parts. For example, tenths are smaller than sixths.
• A fraction tells us about the relationship between the part and the whole. A fraction does not say anything about the size of the whole or the size of the parts.

An understanding of fractional parts is important but students also need to develop sense for the size of fractions. This unit helps students develop an intuitive feel for zero, one-half and one. Understanding why a fraction is close to zero, half or one helps students develop a number sense of fractions.

Required Resource Materials
• Fractions recorded on squares of paper
• Number lines marked with a 0 and 1
Activity

Session 1

In this session students begin to develop benchmarks for zero, half and one.

1. Write the following fractions on the board:
1/20, 51/100, 10/9, 13/12, 2/40, 99/100, 103/100
As the difficulty of this task depends on the fractions begin with fractions that are clearly close to zero, half or one.
2. Ask the students in pairs to sort the fractions into three groups: those close to 0, close to 1/2 and close to 1.
3. As the students sort the fractions ask them to explain their decisions.
Why do you think 51/100 is close to half? How much more than a half is it? Why do you think 1/20 is close to zero? How much more than zero? Why is 103/100 close to one? Is it more or less than one? How much more?
4. As the students explain their decisions encourage them to consider the size of the fractional parts and how many of these parts are in the fraction. For example, "99/100 is 100 parts and we have 99 of them. If we had one more it would be 100/100 or 1 so 99/100 is very close to 1".
5. Repeat with another list of fractions. This time use fractions, which are further away from zero, half, and one:
1/10, 5/6, 5/9. 4/9, 17/20, 13/20, 2/20, 9/20, 1/5
Once more encourage the students to explain their decision for each fraction.
6. Add 1/4 and 6/8 to the list of fractions. Ask the students which group they fit it. Ensure that the students understand why these fractions are exactly in between the benchmarks.

Session 2

In this session the students continue to develop their sense of the size of fractions in relation to the benchmarks of zero, half and one by coming up with fractions rather than sorting them.

1. Ask the students to name a fraction that is close to one but not more than one. Record this on the board. For example:
5/6
2. Next ask them to name another fraction that is closer to one than that. Record on the board:
5/6         7/8
3. Ask the student to explain why the second fraction is closer to 1. One possible explanation is "7/8 is closer to one because eighths are smaller parts than sixths and 7/8 is 1/8 smaller than 1 and 5/6 is 1/6 smaller than one. As 1/8 is smaller than 1/6 7/8 is closer to one."
4. Continue for several more fractions with each fraction being closer to 1 than the previous fraction.
5. Repeat with fractions that are close to 0.
6. As the students nominate fractions encourage them to give explanations that focus on the relative size of the fractional parts.
7. Ask the students to work in pairs. Direct one of the pair to record a fraction that is close to but under 1/2 on a piece of paper. The other student then records a fraction that is closer to 1/2 and explains why it is closer. Encourage the pairs to continue to record fractions that are progressively closer to 1/2.
8. As the pairs work circulate checking that they are expressing an understanding of the relative size of the fractional parts.

Session 3

In this session students estimate the size of fractions.

1. Draw the representation below on the board. Ask the students to each write down a fraction that they think is a good estimate for the fraction shown. Ask for volunteers to record their estimate on the board. As they record estimates ask each student to share their reasoning. Listen without judgment to the estimate and then discuss with them why any particular estimate might be a good one. There is will no single correct answer but the estimates need to be reasonable. If the students are having difficulty, encourage them to reflect on the benchmarks developed in the previous sessions.
2. Repeat with some of following shapes and number lines.  3. Ask the students to work in pairs. Direct one of the pair to draw a picture of a fraction and the other student to give an estimate with an explanation. Repeat with the students taking turns drawing and giving estimates.

Session 4

In this activity students identify which fraction of a pair is larger. The comparisons rely on an understanding of the top and bottom number in fractions and on the relative sizes of the fractional parts. Equivalent fractions are not directly introduced in this unit but if they are mentioned by students they should be discussed.

1. Write the following two fractions on the board and ask the students to tell you which is larger.
4/5 or 4/9
Encourage explanations that show that the students understand that the fractions have the same number of parts but that the parts are different sizes. In this example 4/9 is smaller than 4/5 because ninths are smaller than fifths.
2. Write the following two fractions on the board and ask the students to tell you which is larger.
4/7 or 2/7.
Encourage explanations that show that the students understand that both these fractions have the same size parts and therefore 4 of these parts is larger than 2 of these parts.
3. Repeat with 10/9 and 9/10.
In this example the students can draw on their understanding of the benchmark of 1 and notice that 10/9 is larger than 1 and 9/10 is smaller than 1.
4. Give the students a number of paired fractions and ask them to make decisions about which is larger.
7/9 or 6/9
6/7 or 6/9
3/8 or 4/7
9/8 or 4/3 etc

Session 5

In this session the students draw on their conceptual understanding of fraction benchmarks (0, 1/2, 1) and their understanding of the relative sizes of fractional parts to line fractions up on a number line.

1. Ask students in pairs to draw five fractions from a "hat". Their task is to put the fractions in order and also to line the fractions up on a number line that is marked 0, 1/2, 1 and 2. 2. Ask the students to write a description of how they decided on the order for the fractions and where to place them on the number line. To place the fractions on the number line, students must also make estimates of fraction size in addition to simply ordering the fractions.
3. Ask the pairs to join with another pair to see if they agree with one another’s order and placement of fractions.