prop & ratios stage 4

Level/Framework stage/Group

stage 4

Planning notes
need to learn how fractions work
Focus Achievement Objectives
Resource logo
Level Two
Number and Algebra
Secondary activities
These exercises and activities are for students to use independently of the teacher to practice number properties
  • Shading a fraction of an object that has been partitioned into equal pieces
  • Shading a fraction of a set of objects
  • Completing a shape (or set of objects) if only part of it is given
Resource logo
Level Two
Number and Algebra
Secondary activities
These exercises and activities are for students to use independently of the teacher to practice number properties.
  • Divide items into equal pieces
  • Share by dealing pieces
  • Use the symbols for the mixed numbers and improper fractions
Resource logo
Level Three
Number and Algebra
Secondary activities
These exercises and activities are for students to use independently of the teacher to practice and develop their number properties.

add and subtract fractions with the same denominator
convert fractions (tenths) to decimals
find fractions of a quantity

Resource logo
Level Three
Number and Algebra
Secondary activities
These exercises and activities are for students to use independently of the teacher to practice number properties.

Using multiplication facts to find a simple fraction of an amount

Resource logo
Level Three
Number and Algebra
Units of Work
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.
  • State which of two fractions is larger.
  • Explain why a fraction is close to 0, 1/2 or 1.
Source URL: https://nzmaths.co.nz/user/387/planning-space/prop-ratios-stage-4

Fair Shares

Purpose

These exercises and activities are for students to use independently of the teacher to practice number properties

Achievement Objectives
NA2-5: Know simple fractions in everyday use.
Specific Learning Outcomes
  • Shading a fraction of an object that has been partitioned into equal pieces
  • Shading a fraction of a set of objects
  • Completing a shape (or set of objects) if only part of it is given
Description of Mathematics

Fractions ratios and proportions, stages 3 to 4, counting from one by imaging to advanced counting

Required Resource Materials
Homework exercises with answers (PDF or Word)
Activity

Prior knowledge. Students should be able to:

  • Identify shapes that have been cut correctly in to fractional pieces (where all are an equal size), and those that have not
  • Explain the functions of the two different numbers in a fraction

Background

These exercises are for students with little fractional knowledge and focus on the ‘parts of a whole’ construct of fractions.

When marking, it is important to discuss with students that it does not matter which pieces are shaded, as long as the correct number are shaded. Likewise for exercise 3, there are a range of different shapes that could be the whole figure. Note that several shapes have been left unshaded in the answers for exercise 1, so students come to you for this discussion.

Attachments

Fraction wafers

Purpose

These exercises and activities are for students to use independently of the teacher to practice number properties.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
Specific Learning Outcomes
  • Divide items into equal pieces
  • Share by dealing pieces
  • Use the symbols for the mixed numbers and improper fractions
Description of Mathematics

Proportions and Ratios, EA (Stage 5)

Required Resource Materials
  • Practice exercises with answers (PDF or Word)
  • Homework with answers (PDF or Word)
Activity

Prior knowledge 

  • The activity ‘fraction circles’ (Book 7, pages 9 and 10) as well as fair shares (Book 7, pages 2 to 4) should have already been taught
  • Identify that in a fraction, all the pieces are the same size
  • Recognise symbols for 1/2, 1/4 etc
  • The link between the number of equal pieces an item is cut into, and the name of the fraction

Background

There are a several significant ideas that arise in this activity, and need to be addressed in the teaching that precedes these activities.

  1. It cannot be assumed that students have the skill to accurately divide a shape into equal pieces – if there are no marks or guides to work from. In the process of halving, students pass through a developmental process. A child’s understanding of one half develops through as number of stages before it can be represented precisely. As a continuous quantity, involving tasks like cutting lengths of string, these stages are:
    • One half as a multiple sequence of subdivisions, (where preschoolers simply do not know when to stop cutting…);
    • One half as a single subdivision where there is gross inequality in each part (the whole is simply cut into 2 parts);
    • One half as a single subdivision with remainder (here some attention to equality is made with the first cut, then ‘a remainder’ is usually cut off the large piece and discarded;
    • One half as a single subdivision into 2 equal parts that uses all of the material (here eye movements and/or a finger movements are used to check the relative size of the pieces before cutting, and the size of the pieces is checked after cutting).
      Cutting into 4, 3 and 5 pieces all require different strategies for success
  1. The concept of the fraction as 5 items each cut into 2 pieces is being created here. This can lead to a confusion between proper fractions and mixed numbers (3/4 and 3 3/4) as the numerator is seen as the 3 wholes…
    Working with cutting shapes and dividing written lines should be included with the written problems.

Comments on the Exercises

Exercise 1
Asks students to draw pictures to help solve problems that involve a number of people sharing equally a set of items. For example, 4 people share 3 bags of popcorn between them.  How much of a bag will each person get. In this questions there are more people than items so the answers are less than 1.

Exercise 2
Asks students to draw pictures to help solve problems that involve a number of people sharing equally a set of items. But in these questions the answers are greater than 1.

 

Decimal Fractions of a Set

Purpose

These exercises and activities are for students to use independently of the teacher to practice and develop their number properties.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Specific Learning Outcomes

add and subtract fractions with the same denominator
convert fractions (tenths) to decimals
find fractions of a quantity

Description of Mathematics

Proportions and Ratios, AA (Stage 6)

 

Required Resource Materials
Practice exercises with answers (PDF or Word)
Activity

Prior knowledge.

  • Explain why 10/10 is another name for one whole
  • Identify mixed numbers and explain how they relate to improper fractions
  • Find fractions of a quantity

Background

In this activity students make the connection between tenths written as a fraction and written as a decimal.  Students add together tenths, and find tenths of a set for example 7/10 of 20.

Comments on the Exercises

Exercise 1: Adding tenths
Asks students to solve word problems that involve adding tenths. For example, 7/10 + 1/10. With the exception of the last 2 problems, in this exercise the answers are less than one, and as the questions require students to convert their answers to decimals, simplification of the fractions is not required, but could be discussed as another way of ‘tidying up’ the numbers.

Exercise 2: Adding tenths
Asks students to solve word problems that involve adding tenths.  For example, 7/10 + 5/10. It also includes adding mixed fraction.  In this exercise, all of the answers are greater than one. Question 7 and beyond represent a slight change in the pattern. Here each rooster is eating more than one worm, so the answers can be sorted by ‘adding like to like’, that is adding the whole numbers and the fractional parts separately.

Exercise 3: Adding and subtracting tenths. Finding “tenths of …”
Asks students to combine the skills of adding fractions to one whole and finding the fraction of an amount. In the last problem, Tim and Ted are equally sharing nine tenths, meaning they get four and a half tenths each. Discussing how to write such fractions should be covered either before the exercise is sat, or afterwards as a debrief, where the students are asked to explain how they met this challenge. Going back to materials (decimats are a good resource here) to display (4/2)/10 and sorting out why it equals 9/20 is useful.

Exercise 4: Finding tenths of coins/working with money.
Asks students to not only work out fractions of an amount, but also work out how much money that have with this number of coins

 

Birthday cakes

Purpose

These exercises and activities are for students to use independently of the teacher to practice number properties.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Specific Learning Outcomes

Using multiplication facts to find a simple fraction of an amount

Description of Mathematics

Proportions and Ratios, AA (Stage 6)

Required Resource Materials

Practice exercises with answers (PDF or Word)

Homework with answers (PDF or Word)

Activity

Prior knowledge.

  • Find a simple fraction of an amount using addition facts

Background

Remember that students who have not reached stage 6 in the "basic facts" domain have not learned all of their basic multiplication facts, so a multiplicative strategy based purely on the numbers may not work for all students. Keeping to simpler numbers has been attempted in the activities for this reason.

Comments on the Exercises

Birthday Cakes 1
Asks students to use multiplication basic facts to find fractions of a set.  For example, how many candles on each piece of cake if there are 6 pieces and 30 candles. Asks students to find unit fractions, for example 1/3 of 24 =.

Birthday Cakes 2
Asks students to practice activities parallel to Birthday Cakes 1.

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 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.
Attachments

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