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Dividing fractions

Keywords:
Achievement Objectives:

Achievement Objective: NA3-5: Know fractions and percentages in everyday use.
AO elaboration and other teaching resources
Achievement Objective: NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
AO elaboration and other teaching resources

Purpose: 

The purpose of this series of lessons is to develop understanding of the operation of division with fractions.

Specific Learning Outcomes: 
  • Apply the understanding that fractions can be quotients (or the result of division).
  • Model and record fractional division problems which involve a measurement interpretation of division.
  • Understand, show and use the common denominator algorithm for fractional division.
  • Calculate fractional division problems using the common denominator algorithm.
  • Write and solve fractional division problems.
  • Understand, show and use the invert and multiply algorithm for fractional division.
  • Recognise that there are a number of ways to solve fraction division problems.
Description of mathematics: 

In working with division, it is important that students understand and can use the appropriate mathematical language for the component parts of division equations. Just as they have come to understand addend plus addend makes the sum, and factor (multiplier) times factor (multiplicand) results in a product, so too it is helpful if students at this level come to know and use the language of division. That is: the dividend is partitioned by the divisor and this results in the quotient.

a (dividend) ÷ b (divisor) = c (quotient)

Fundamental to a student’s success in dividing fractions is their sound understanding of whole number division and, in particular, their recognition and understanding of the two kinds of division that we ‘do’ in our daily lives.

The first, and perhaps the most common interpretation by students, is sharing or partitive division, which often involves answering the question, ‘How many each?’ or ‘ How many in one (group)?’ The second form of division is the measurement interpretation (sometimes referred to as group division). Here the number in the group, or size of each measure is known. What is unknown, is the number of those groups or measures that can be made from a given amount. This interpretation is often associated with repeated subtraction, as one way to solve this kind of problem is to keep removing the given equal groups (measures) from the whole amount, until nothing is left. Counting each repeated subtraction gives the solution to the question. It is this form of division that is more commonly explored in fractional division. This involves the student in finding how pieces of a given (fractional size) ‘fit within’ another fraction piece. For example, 1 1/2 ÷ 1/4 is interpreted as how many quarters are there in 1 1/2?

When students carry out the operation of division with whole numbers, their expectation is that the quotient will be smaller than the dividend, for example, 20 ÷ 2 = 10, and sometimes smaller than both the dividend and the divisor, for example, 20 ÷ 5 = 4. It is a conceptual shift for students to come to understand that when they are dividing a fraction by a fraction the quotient may be larger than both the dividend and the divisor, for example 1/2 ÷ 1/4 = 2.

There are many occasions when we divide fractions by fractions, however, it is recognising these contexts that may be the challenge. For example, if you pay $14.50 for three quarters of a kilo of prime lamb and you want to know how much that is per kilo, this will involve your dividing a fraction by a fraction. Locating even simple fractional problems in context is particularly important if students are to recognise the nature of the division problem being posed. Having students consider and create their own contexts, identifying situations in their lives when fractional division arises, is also very important.

As with other work in fractions, students benefit greatly from having the opportunity to explore concepts using materials, such as number lines, sets of cubes and regional models, and to draw and use diagrams. It is relatively easy to teach and ask the students to remember the fractional division algorithm of “invert the second fraction and multiply*”. However, remembering this procedure is of little use if the fundamental understanding of fractional division is not first developed. It is also important that students have the opportunity to explore the patterns and relationships evident in the fractional number algorithms, so that they can understand why multiplying by the reciprocal ‘works’.

Students should have opportunities to explore both the common denominator algorithmic approach and the approach that involves multiplying by the reciprocal*. They should also have opportunities to communicate and demonstrate their understanding to others, for in this way they explore further and consolidate their own conceptual understanding.

These ideas are presented in five sessions however, as they include complex concepts that are fundamental to a student’s success with fractions, these sessions can be extended over a longer period of time. Whilst games are introduced and used within sessions to consolidate ideas, they can also be added to the class or group independent activities, or be sent home for family challenges and enjoyment.

Links to the Number Framework

Stages 7 - 8

This unit supports teaching and learning activities in the Student Fractions e-ako 11 and complements the learning activities in Book 7 Teaching Fractions, Decimals and Percentages.

Required Resource Materials: 
Fraction number line
Number line to 100
12 or more clothes pegs
Unifix (or interlocking) cubes
Fraction Strip sets
Fraction strips (Material Master 7-7)
Strips of paper
Scissors
Collections for making sets, for example, coloured counters
Activity: 

Session 1

SLOs:

  • Review, understand and model division with whole numbers.
  • Understand and use the language of division.
  • Recognise and discriminate between partitive and measurement interpretations of division.
  • Recognise and apply repeated subtraction to solve whole number ‘measurement’ division problems.
  • Recognise that fractions can be quotients (or the result of division).

Activity 1

  1. Begin this session by writing this problem on the class/group chart.
    On school sports day there are 180 students. They are put into 12 groups. How many are in each group?
    Ask students to mentally solve the problem then discuss their strategy with a partner.
    Have a student record the problem equation on the chart (180 ÷ 12 = 15) and explain their strategy to the class. (for example: ‘18 ÷ 6 = 3 so 180 ÷ 6 = 30, so 180 ÷ 12 = 15 – halving and doubling the divisor and quotient).
     
  2. Highlight the word each in the written problem.
    Write on the class chart this problem:
    18 bears make 6 groups at the teddy bears’ picnic. How many are in each group?
    Place in front of the students 18 plastic teddies. Have a student model how a younger child might solve the problem.
    Look for the opportunity to highlight that this, like the first problem, is a sharing problem (partitive division) in which we find ‘How many each?’ or “ How many make 1 (team)?”
    Write the words in bold on the chart to refer to later.

Activity 2

  1. Write on the class/group chart and pose this, or a similar question.
    At the junior school sports day there are 86 students. If there are 7 in each group, how many groups are there?
    Have students work in pairs. Tell them that their task is to solve the problem, discuss their strategy and write down how this problem is different from the first problem and suggest (show) how a younger child might solve the problem. Set a short time limit.
     
  2. Have a student record the problem equation on the chart (86 ÷ 7 = 12 and 2r) and explain his strategy to the class. (for example: I just knew 11 x 7 = 77 and I added another 7 to make 84. That left 2).
     
  3. Ask students to share what difference they noticed.
     
  4. Look for the opportunity to highlight that this is a grouping problem (measurement division) in which we find how many equal groups can be made. Write the words in bold on the chart to refer to later.
     
  5. Display a number line to 100 and give a set of (at least) 12 clothes pegs to one student. Have the student model repeated subtraction of groups of 7, placing a peg at each difference. (minuend – subtrahend).
    Tell the students that this kind of division in which we find how many of something can be made from something else, is sometimes called measurement division.  

Activity 3

  1. Have students in pairs write their own summary of the two kinds of division.
    Have them write a simple number problem for each kind of division.
    (NB. This can be challenging. Focus on the language of sharing/each or number of groups). Students pair share and discuss.  
     
  2. Record on the class chart (or in your class dictionary locate) the words: dividend, divisor and quotient.
    Show their meaning like this: 180 (dividend) ÷ 12 (divisor) = 15 (quotient).
    Discuss.
     
  3. Have pairs of students write three division equations of their own and name the parts using the language of division.
     
  4. Now pose the question: Can you show another way of recording 180 ÷ 12 = 15?
    Have students discuss this.
     
  5. Have some students record their suggestions on the group/class chart.
    Look for the opportunity to highlight the fact that a fraction can be a quotient or the result of division.
    Write dividend/divisor = 180/12 = quotient = 15
    Discuss that the fraction 180/12 is both an expression of the division problem and the quotient (or result of the division).
     
  6. Model again with a simple example such as:
    The division problem 1 ÷ 4 can also be written as 1/4. This is both an expression of the division problem and the quotient (or result of the division/answer).
     
  7. Review the learning in this session by rereading together the notes made on the class/group chart.

Session 2

SLOs:

  • Apply the understanding that fractions can be quotients (or the result of division).
  • Understand and model partitive division with a whole number divisor in which the unit does not need to be partitioned.
  • Understand and model a partitive division problem in which the unit must be partitioned.

Activity 1

  1. Review learning from Session 1 by re reading the class chart to review the two kinds of division, the language of division and the fact can be the quotient or result of division.
     
  2. Pose these problems:
    1 pizza shared between 3 people. How much do they each get?
    1 pizza shared between 8 people. How much do they each get?
    2 pizzas shared between 5 people. How much do they each get?
    3 pizzas shared between 4 people. How much do they each get?
    4 pizzas shared between 8 people. How much do they each get?

    Have students discuss in pairs the solutions (fractions) to these whole number problems and quickly write and draw their solutions.
     
  3. Have individual students write on the class chart a division problem as a fraction, helping them to recognise the fraction as both an expression of the division problem and the quotient (solution to the problem).
    For example: 1 shared between 3 people can be written as 1/3. This is also their share each.
    Be sure to use the word ‘quotient’ in your discussions.
     
  4. Make fraction circles available and have students double check by modeling each of the solutions.
    For example, 2 shared between 5 is 2/5
     showing 5 lots of 2/5 makes 10/5 or 2.

Activity 2

  1. Write on the class chart this sharing (partitive) problem:
    Problemdividenddivisorquotient
    I have 3 1/3 metres of fabric.
    I want to make 5 bags for my friends for Christmas.
    How much fabric can I use for each bag?
       

    Read the problem together.
    Discuss another way to write the dividend. For example, 10/3, an improper fraction.
    Write this in numbers and words: 10 thirds.
     
  2. Make strips of paper available to pairs of students.
    Have them solve the problem and discuss.
    Some may approach it this way:

    Showing that each of the five bags will be 2/3 of a metre.
     
  3. You can record discussion this way. Notice the word thirds written after 10.
    Problemdividenddivisorquotient
    I have 3 1/3 metres of fabric.
    I want to make 5 bags for my friends for Christmas.
    How much fabric can I use for each bag?
    3 1/3
    or
    10/3
    510/5 (thirds)
    2 (thirds)
    2/3 of a metre

    NB: Highlight that 10/5 is both the problem stated and the answer (simplified to 2). Because 10 is 10 thirds the answer is 2 thirds.

Activity 3

  1. Make paper strips, fraction pieces and set collections (eg counters) available to the students.
    Distribute Attachment 1, discuss. Together complete the first example. Have students work in pairs to complete the empty sections, including writing their own fraction word problems.
    NB. The unit part in the whole number divisor problems on Attachment 1. does not require further partitioning.
     
  2. Have students pair share their results.

Activity 4

  1. Pose the problem which involves subdivision of the unit:
    The relay course is 1 1/4 kilometres long. Three people are taking part.
    How much of the course do they each run?
    1 3/4 ÷ 3 = ?

    Set a time and have student pairs work to solve the problem, which is 7/4 ÷ 3 = ?
    Record student ideas.
     
  2. Develop with the students the conclusion that some or all of the parts will need to be split or partitioned. Because quarter parts “won’t work” seek suggestions for what will work.
    Agree that if each of the fourth parts was subdivided by three, giving twelfth parts, this might work.
     
  3. Display fraction strip templates or a fraction wall: 1/4 and 3/12 and ask how many twelfths 7 quarters would be. (21/12)

    Ask students to discuss 21/12 ÷ 3 = ?
    Agree that 21/12 ÷ 3 = 7/12.
    Each person will run 7/12 of a kilometre.
     
  4. Explore another of the problems posed on Attachment 2. Work through this together, highlighting the partitioning of the unit by the divisor.

Activity 5

  1. Make paper strips, fraction pieces and set collections (eg. counters) available to the students.
    Distribute Attachment 2, discuss. Have students work in pairs to complete the empty sections.
    NB. The unit parts in the whole number divisor problems on Attachment 2 do require further partitioning.
     
  2. Have students pair share their results.

Activity 6

Conclude this lesson by summarising key ideas developed in this session.

Session 3

SLOs:

  • Understand the measurement interpretation of fractional division (repeated subtraction).
  • Model and record fractional division problems which involve a measurement interpretation of division.
  • Communicate understanding of fractional division problems to others.

Activity 1

Introduce Attachment 3, the Divide and Ride game. Explore the instructions together then have students play in pairs.
This game reinforces concepts developed in Sessions 1 and 2.
Highlight for students that Sessions 1 and 2 have been about sharing division which is also known as partitive division.

Activity 2

  1. Review class notes made in Session 1. In particular highlight and discuss both the difference between sharing (partitive) and group (measurement) division.
     
  2. Tell the students that the focus in this session is on the measurement interpretation of sharing, or trying to find how many groups.
    Pose these problems and ask which is an example of grouping or measurement division.
    You have 3 1/3 small bottles of juice. If you give each friend 2/3 of a bottle of juice, how many friends get a drink? (3 1/3 ÷ 2/3)
    You have 4 2/3 small bottles of juice. You give this to 7 friends. What fraction of a bottle of juice do they each get? (4 2/3 ÷ 7)

    Agree that it is the first example because the task is to find how many equal groups of 2/3 are in 3 1/3.
     
  3. Distribute strips of paper to individual students and have them explore the problem:
    You have 3 1/3 small bottles of juice. If you give each friend 2/3 of a bottle of juice, how many friends get a drink? (3 1/3 ÷ 2/3)
    Demonstrate and discuss their solution.
  4. Provide a fraction number line and pegs, or draw a number line on the class chart.
    Ask for a student volunteer to demonstrate repeated subtraction. (It may be helpful to refer to the whole number example in session 1.)

    This shows that there are 5 repeated subtractions of 2/3, demonstrating in another way that there are 5 x 2/3 in 3 1/3.
     
  5. Distribute to pairs of students the Groups or Shares cards (Attachment 4).
    Have them consider each problem and decide if it is a grouping or sharing problem. Make two piles.
    Have them pair share with another group and see if they agree.
     
  6. Have each student in the group of four select one ‘grouping’ (measurement) problem and together discuss the solution.

Activity 3

  1. Make poster paper available to students who can choose to work in pairs or alone.
    Set a time limit. Have them write their own group (measurement) fraction division word problem using a personal or familiar context, and the problem as an equation. They should show the solution with a diagram.
    Suggest that they might like to use a format similar to a ‘think board’.
    As they finish have them write in their maths diaries their learning so far about division (of fractions).
     
  2. Share some of the results of both tasks.

Session 4

SLOs:

  • Understand, show and use the common denominator algorithm for fractional division.
  • Calculate fractional division problems using the common denominator algorithm.
  • Write and solve fraction division problems.

Activity 1

Begin by further sharing of posters and diary reflections from Session 3, Activity 3, Step 2.

Activity 2

  1. Write a student generated equation (or this one) on the class chart.
    2 1/3 ÷ 1/4 = ? (How many groups of 1/4 are in 2 1/3?)
    Rewrite the problem:
    7/3 ÷ 1/4 = ?
    Ask, Can we easily work with quarters and thirds? (No.)
    What could we use instead?
    Support students with further questioning if necessary to recognise that it would be easier if they were working with a common denominator. Ask for a suitable denominator (twelfths).
     
  2. Make paper or mini whiteboards available to the students, and have them work in pairs to rewrite the equation using the common denominator and solve the problem.
    Have them (attempt to) draw what this looks like.
    Like this:
    28/12 ÷ 3/12 is asking how many 3/12 are there in 28/12
    28/12 ÷ 3/12 = 9 1/12 as 1/12 is 1/3 of 3/12 or 1/4
    The result is that there are 9 1/3 quarters in 2 1/3
  3. Have students share and explain their results, and discuss. Record student findings and ideas on the class/group chart.

Activity 3

Ask for a student summarise what they did have done and record this on the class chart. Highlight the words finding a common denominator.

Activity 4

Pose these problems and provide paper or mini whiteboards.
1 1/2 ÷ 3/5 = ?
3 1/6 ÷ 3/4 = ? 
2 1/4 ÷ 2/3 = ? 
1 2/3 ÷ 5/6 = ? 
Have students work in pairs and choose two to solve. They should:

  • Write a grouping word problem to provide a context for the problem.
  • Calculate the answer.
  • Draw a diagram of their calculation.

Have students pair share results and pose at least two other problems for them to solve within their group. Guide them to choose smaller whole numbers and more common fractions.

Activity 5

Conclude the session by returning to the summary in Activity 3 above.

Session 5

SLOs:

  • Understand, show and use the invert and multiply algorithm for fractional division.
  • Recognise that there are a number of ways to solve fraction division problems.
  • Communicate division of fractions to others.

Activity 1

  1. Have a student read aloud the summary from Session 4.
     
  2. Explain that they are going to explore a little further what is happening with the numbers in fraction division problems.
     
  3. Distribute Attachment 5 to each student. Have them complete this then discuss ideas with a partner.
     
  4. Use an example from one of the students and discuss.

    Recognise that there is a multiplication relationship.
    Ask the students to identify which numbers are being multiplied.
    Guide the discussion to have them notice that the numerator of the dividend is multiplied by the denominator of the divisor. Explain that this important relationship has lead to the algorithm in which the divisor is inverted and the dividend is the multiplied by this.
     
  5. Model inverting the divisor.
    For example, 5 ÷ 1/2 as 5/1 x 2/1 = 10/1 concluding there are 10 halves in 5
    Have students return to Attachment 5, rewriting the equation and inverting the divisor and multiplying for each of the problems.
    Share results.

Activity 2

  1. Write: ‘Invert the divisor and multiply’ is a way to calculate all fraction division problems. True? False?
    Ask students to indicate those who think this is true/false, recording on the class chart the number of students who vote for each.
    Ensure that calculators are available to the students. Have them explore fractional division problems of their choice, identifying any that they think can’t be solved using this algorithm. Encourage students to use smaller numbers as they may be asked to solve their problem in another way (with a diagram, equipment, number line or using a common denominator).
     
  2. Share findings together and explore any ‘apparent’ anomalies. Have students explore any anomalies by using another method.
     
  3. Recognise that this method for calculating fractional division problems is commonly used, is efficient and useful.

Activity 3

  1. Review the notes made on the class/group chart during these sessions.
     
  2. Conclude this session by having students prepare an electronic presentation or a poster of their learning, in which they demonstrate knowledge of and strategies for solving division problems involving fractions.
AttachmentSize
DividingFractionscm1.pdf31.84 KB
DividingFractionscm2.pdf40.8 KB
DividingFractionscm3.pdf435.91 KB
DividingFractionscm4.pdf35.77 KB
DividingFractionscm5.pdf28.7 KB