Fraction Benchmarks

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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 a 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.

Opportunities for Adaptation and Differentiation

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 and area models (e.g. cubes, counters, etc.), 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, and explicitly modelling the use of 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 (mahi-tahi) 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
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, 6/10, 10/8, 11/12, 1/10, 3/8, 2/5, 9/10.
    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 6/10 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 11/12 close to one? Is it more or less than one? How much less?
  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, "9/10 is 9 parts and the parts are tenths. If we had one more tenth it would be 10/10 or 1 so 9/10 is very close to 1".
  5. Use Fraction Strips to physically model each fraction, if needed. Locate the fractions on a number line using the one unit as the space between 0 and 1.
  6. 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:
    3/10, 5/6, 5/9, 4/9, 18/20, 13/20, 2/8, 9/12, 1/5
    Once more encourage the students to explain their decision for each fraction.
  7. Add 1/4 and 6/8 to the list of fractions. Ask the students which group each fraction fits into. Use Fraction Strips to support students to understand why these fractions are exactly in between the benchmarks. Locate the 'between' fractions on the number line you created previously.
  8. Challenge the students to work in pairs, and develop a story that demonstrates how different fractions are close to 0, close to 1/2 and close to 1. Students could use the fractions provided in the earlier questions, or come up with a new list of fractions to use (or for another pair of students to use). This opportunity to investigate fraction benchmarks, in a relevant and meaningful context, will help to reinforce students’ understandings and can be used as formative assessment. During this task, take the opportunity to work with smaller groups of students and rectify any misunderstandings. At the conclusion of the session, pairs of students could be challenged to solve the questions created by other pairs of students.

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.
    Do you notice a pattern as the fraction gets closer to one? (5/6, 7/8, 9/10, 11/12)
    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.
    Which of these fractions is closest to zero, 1/3, 1/4, 1/5, or 1/6? Explain why?
    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 student from 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. Consider pairing more knowledgeable students with less knowledgeable students to encourage tuakana-teina.
  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.
      A rectangle with a fraction of it shaded red.
    Ask for volunteers to record their estimate on the board. As they record estimates, ask each student to share their reasoning. Listen, without judgement to the estimate and then discuss why any given estimate might be a good one. Encourage students to share their justifications and ask questions of each other. There is 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 the following shapes and number lines.
    Three shapes, each with a fraction of them shaded red.
    Three number lines from zero to one, each with a question mark on them indicating a fraction.
  3. Ask the students to work in pairs. Direct one student from each 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.
  4. Broaden the selection of shapes that students find fractions of. Include symmetric polygons like hexagons and octagons, as well as circles and other ellipses (ovals).

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.
    2/5 or 2/8
    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 2/8 is smaller than 2/5 because eighths are smaller than fifths. Some students may know that 2/8 = 1/4. Use Fraction Strips to check students predictions.
  2. Write the following two fractions on the board and ask the students to tell you which is larger.
    4/5 or 4/6.
    Encourage explanations that show that the students understand that both these fractions have the same count (numerator) but fifths are larger than sixths. Therefore 4/5 is greater than 4/6. Write 4/5 > 4/6. Use Fraction Strips to check students' predictions.
  3. Repeat with 5/8 and 7/8. In this case the size of parts is the same (the denominator) but the number of parts (numerator) is different. Record 5/8 < 7/8 or 7/8 > 5/8. Check with Fraction Strips, if needed.
  4. 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.
  5. Give the students pairs of fractions and ask them to make decisions about which fraction is larger. Support students with fraction strips if necessary but encourage prediction before using the materials.
    7/10 or 6/10
    6/8 or 6/12
    3/8 or 4/10 (more difficult)
    9/8 or 4/3 etc
  6. Provide students with this open challenge.
    One fraction is greater than the other.
    The two fractions have different numerators and different denominators.
    What might the fractions be?

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". Suitable fractions are available on Copymaster 1. 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. Support students with Fraction Strips but only if necessary.
    A number line with locations of the fraction benchmarks zero, one half, one, and two marked on it.
  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.
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Level Three