## Fraction Benchmarks

Purpose

In this unit students develop important reference points or benchmarks for zero, one half and one. They use these benchmarks to help compare the relative sizes of fractions, through estimating, ordering and placing fractions 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 one (whole 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, or about the relationship between two parts.

An understanding of fractional parts supports students to develop sense for the size of fractions. This unit helps students develop an intuitive feel for zero, one-half and one, as useful benchmarks for ordering fractions. Understanding why a fraction is close to zero, one half or one helps students develop a number sense for fractions.

Students can be supported through the learning opportunities in this unit by differentiating the complexity of the tasks, and by adapting the contexts. Ways to support students include:

• providing a physical model, particularly fraction strips (length model) or regions (area model) (cubes, counters, etc.) and fraction strips, so students can see the relative sizes of fractions
• connecting lengths from zero with the number line, and recognising that the space between zero and one is always visible on a number line for whole numbers
• discussing mathematical vocabulary and symbols, particularly the role of numerator as a count, and the denominator as giving the size of the parts counted
• encouraging students to work collaboratively and share their ideas
• altering the complexity of the problems by simplifying the difficulty of the fractions and whole numbers that are used. Begin with fractions, such as halves, quarters, thirds, and fifths that students may be most familiar with.

The contexts for this unit are purely mathematical but can be adapted to suit the interests and cultural backgrounds of your students. Fractions can be applied to a wide variety of problem contexts, including making and sharing food, constructing items, travelling distances, working with money, and sharing earnings. The concept of equal shares and measures is common in collaborative settings. Students might also appreciate challenges introduced through competitive games or through stories.

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 that are further away from the zero, one half, and one benchmarks so students need to think more carefully about their decisions:
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. Possibly model the location of each fraction with fraction strips to check.

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
How do you know that fraction is close to one?
2. Next ask them to name another fraction that is closer to one than that. Record on the board:
5/6         7/8
How do you know that 7/8 is closer to one than 5/6?
Students might comment on how much needs to be added to each fraction to make it equal to one. "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."
3. Continue for several more fractions with each fraction being closer to 1 than the previous fraction.
Which fraction is closer to one 99/100 or 999/1000? Why?
4. Repeat with fractions that are close to 0 but still greater than zero.
Can a fraction have 5 as a numerator but still be close to zero? Give an example.
5. As the students nominate fractions encourage them to give explanations that focus on the relative size of the fractional parts.
6. 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.
7. 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 shaded area 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 given 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. Look for creative methods like estimating that the white area combines to 1/4 so the shaded area must be 3/4
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 (numerator and denominator) 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 pairs of 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 locate the fractions 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. When placing the fractions on the number line, students should justify their choice with logical arguments. For example, three tenths is closer to one half than zero because 3/10 is 2/10 away from 5/10 and 3/10 away from zero.
3. Ask each pair of students to join with another pair to see if they agree with one another’s order and placement of fractions.