# Multiplying fractions

Purpose

The purpose of this series of lessons is to develop understanding of the multiplication of fractions.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-5: Know fractions and percentages in everyday use.
NA4-2: Understand addition and subtraction of fractions, decimals, and integers.
Specific Learning Outcomes
• Record in words, the actions and results of finding a fraction of a fraction.
• Record and respond to written multiplication equations.
• Use arrays to model and solve multiplication equations that involve subdividing the unit.
• Notice, explain and generalise what is happening to the numbers in a multiplication algorithm.
• Pose and solve their own fraction multiplication problems.
• Understand and use number properties when multiplying fractions.
• Explore and demonstrate the reciprocal relationship between multiplication and division.
Description of Mathematics

This series of lessons builds upon students’ understanding and use of equivalent fractions as they solve fraction addition and subtraction problems. As in earlier units of work, an emphasis is given to having students model operations with fractions, using a range of materials and record using words and symbols. Having a sound knowledge of basic multiplication and division facts is fundamental to the students’ success in working with and understanding equivalent fractions, and, in this unit, multiplying fractions.

There are three key understandings that underpin multiplication of fractions. The first is that multiplying two fractions involves finding the fraction of another fraction. For example, 1/2 x 1/4 is interpreted as 1/2 of 1/4. The second is that when two fractions less than one are multiplied, the product is always less than either factor. In multiplying whole numbers, students expect a product that is larger than either factor. Multiplying fractions requires a conceptual shift for the students who must clearly understand they are finding a part of a part. Thirdly, by understanding the commutative property, students can make problems simpler by changing the order of the factors.

In multiplying fractions students come to recognise that the language of ‘times’ and ‘of’ are interchangeable. The use an array model to visualise and solve problems involving finding a fraction of a fraction, by breaking an area into parts, horizontally and vertically, scaffolds the move from a whole number understanding to a fractional one.

Beginning with problems that involve working with unit parts without subdivision (eg. 1/3 of 3/8), establishes the conceptual understanding of the multiplication operation with fractions. Once this is clearly understood, working with unit parts that involve subdivision (eg. 3/4 of 2/3), focuses the students on recording and calculations as they explore the relationships between the numbers.

Using realistic contexts for finding fractions of fractions is important. Having students respond to these, and create context of their own, will help them recognise the practical application of fraction multiplication.

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 the 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.

Required Resource Materials

Fraction Strip sets

Fraction strips (Material Master 7-7)

Coloured fraction overlays

Activity

Session 1

SLOs:

• Review finding a fraction of a whole number.
• Recognise that a fraction of a fraction results in a smaller part.
• Record in words the actions and results of finding a fraction of a fraction.
• Solve problems involving finding a fraction of a fraction, using a regional model.

Activity 1

1. Begin this session by posing these three problems. Have the students discuss in pairs the solutions to each:
There are 33 children in Nina’s class. She was asked how many would be in her team if it included 1/3 of the class. She said 12. Is she right?
15 students in Jo’s class were sitting on the mat. This was 3/5 of the class. How many students are in Jo’s class?
18 students in Roly’s class are on the mat. 2/5 are yet to come into the classroom. How many are outside?
Have the students pair share their solutions.

2. On the class chart write:
Finding fractions of whole numbers.
Nina’s class: 1/3 of 33 is:
Jo’s class: 3/5 of something is 15 so 5/5 is:
Roly’s class: 3/5 of something is 18 so 2/5 is:
Have individual students come and record their results on the class chart and explain their solutions on behalf of their group.
Remind the students that they have been finding fractions of whole numbers.

Activity 2

1. Ask: What if you find a fraction of a fraction. Will the result be larger or smaller than both fractions?
Discuss. Record ideas on the class chart.

2. Provide pairs of students with strips of paper and coloured pens.
Pose the problem: Can you use the materials to show one half of one half.
Have students share the results.

Pose another problem: Use the materials to show three quarters of one half.
Have students share what they did and discuss the results.

3. Distribute Fraction Strips (Material Master 7-7) to the students to supplement the paper strips. Give students time to become familiar with the fraction strips reading the unit fractions shown and talking about the subdivisions that they see.
Pose the question: What is one third of one half?
Have students find one half on the fraction wall and look for the subdivision of one half into thirds, identifying that one sixth is one third of one half.
Distribute and discuss Attachment 1. Explain that they can refer to the fraction wall or use the paper strips and scissors to complete it. Highlight the importance of their writing realistic stories for each.

4. Have students pair share and check their results.
Review the focus of the learning so far: We have been finding fractions of whole numbers and finding fractions of fractions of an area.

Activity 3

1. Conclude the session by having a number of students read aloud (in random order) word problems they have written. The other students work in pairs to image paper strips or the fraction wall, and respond.

2. Pose aloud word problems that build on these examples using repeated addition to reach solutions:
One third of one half is one sixth, so what is two thirds of one half? (2/6 or 1/3)
One fifth of one half is one tenth, so what is three fifths of one half? (3/10)
One third of one third is one ninth, so what is two thirds of one third? (2/9)
One quarter of one half is one eighth, so what is three quarters of one half? (3/8).

3. Conclude that a fraction of a fraction results is a smaller part.

Session 2

SLOs:

• Understand that the language ‘of ‘ and ‘times’ is interchangeable.
• Record and respond to written multiplication equations.
• Use arrays to model multiplication equations.
• Use arrays to model and solve multiplication equations that involve subdividing the unit.

Activity 1

Refer to problem 1 from session 1.
There are 33 children in Nina’s class. She was asked how many would be in her team if it was 1/3 of the class. She said 12. Is she right?
Finding fractions of whole numbers.
Nina’s class: 1/3 of 33 is:
Pose: Nina recorded the equation for this problem as 33 ÷ 1/3 = 11. Is she correct? Why? Why not? (33 ÷ 1/3 = 99 because there are 99 thirds in 33 wholes)
Write 1/3 of 33 = 11 and 1/3 x 33 = 11.
Discuss.

Activity 2

Write on the class chart
1/4 x 2 = ?
2/3 x 12 = ?
Have students work in pairs to solve the problems, and pair share their results including any pictures or diagrams used.

Activity 3

Explore fractions of fractions:
Distribute think board sheets (Attachment 2) to the students. Have each student complete a think board for each of these two problems.
2/3 x 9/10 = ?
1/2 of 4/9 = ?
Encourage students to use diagrams like those used for Attachment 1.
They should look like this:
2/3 x 9/10 = 6/10

1/2 of 4/9 = 2/9

Have them discuss their think boards with a partner.

Activity 4

Have students locate their copy of Attachment 1 (Session 1, Activity 2, Step 3). Have students record written equations for each of the examples.
Have them discuss anything they notice about the numbers in these equations (in which they are multiplying unit fractions).
Ask, is what you noticed true the equations 2/3 x 9/10 and 1/2 x 4/9? (When we are multiplying whole numbers the product is larger than the factors. When we multiply fractions the product is smaller than both factors).

Activity 5

1. Distribute a set of Fraction Overlays to each pair of students.
Give them time to explore the equipment, then ask pairs of students to demonstrate anything they have found out with the equipment.

2. Together have them model several examples using the equipment and record the diagram and equations on the class/group chart. For example:

NB: Students recognise in these examples that the purple fraction is the result of multiplying the two factors.

3. Pose these problems for the students to solve using the fraction overlays, recording each equation with factors and products as they do so.
3/5 x 3/4 = ?
2/3 x 2/5 = ?
1/5 x 1/4 = ?
5/6 x 1/2 = ?

4. Conclude the session by summarising learning in this session on the class/group chart. For example:
When we are multiplying whole numbers the product is larger than both factors.
When we multiply fractions the product is smaller than both factors because we are finding a fraction of a fraction.

Session 3

SLOs:

• Notice, explain and generalise what is happening to the numbers in a multiplication algorithm.
• Pose and solve their own fraction multiplication problems.
• Work with and show understanding of multiplication involving mixed numerals, that is, fractions greater than one, for example, 1/2 of 2 2/3.
• Recognise that mixed numeral fraction multiplication problems can be solved by changing both to improper fractions or by applying the distributive property.

Activity 1

1. Begin this session by referring to the examples used in Session 2 and the summary from Session 2, Activity 5, Step 4.
If it has not already become clear, highlight what they notice happening with the numbers in the equations.
When we multiply fractions the product is smaller than both factors.
The product is the result the numerators being multiplied together and the denominators being multiplied together.

2. Make Fraction Overlays available to the students.
Each pair should complete (at least one) A3 think board poster of one fraction multiplication problem, which includes a diagram of fraction overlays in the equipment section. These will be used for class display.

Activity 2

Students play the game Multiplifraction (Attachment 3)
How to play
Play with a partner.
The winner is the person who collects the most sets of 3 cards.
There are 15 sets in total.

1. The dealer shuffles the cards and deals 7 to each player. The remainder are placed face down in a pile in front of the two players.

2. Players check their hands for any complete sets of three: a picture card, an expression and a product (a single fraction). Complete sets are placed face up in front of the player.

3. The dealer begins by asking their partner for a card they are seeking to complete a set for which they have at least one member card in their hand. The request must state what the picture card would look like, or state the equation or state the product.
If the partner does not have this card, they say: “Multiplifraction” and the dealer takes a card from the pile.

4. The other player then makes their request.

5. The game continues until all cards are used.

Activity 3

1. Pose the problem: “Manu has 2 1/2 spare cans of paint left over from painting his house. He uses 3/4 of this to paint his shed. How many cans is this?”
Have the students explore the problem in pairs then bring their solutions to a group/class discussion.

2. Ask selected students demonstrate their solutions on the class chart.
Have students recognise two ways they can approach the problem:
3/4 x 2 + 3/4 x 1/2 = 1 1/2 + 3/8 = 1 7/8 (distributive property) or
3/4 x 5/2 = 15/8 = 1 7/8 (changing the mixed numeral into an improper fraction)
On the class chart draw what this would look like:

3. Have students work in pairs to explore each of these problems and show both ways of reaching the solutions.
“Maryanne 3 1/3 metres of fabric. She uses 2/3 this to make her dress for the school formal. How many metres did she use?”
“The floor area of Owen’s bedroom is 8 1/4 square metres. 3/4 of this is taken up with furniture. How many square metres of spare floor space does he have?”

4. Have the students pair share both solutions for each of these problems:
2/3 x 3 1/3
3/4 x 8 1/4

Activity 4

1. Brainstorm some fraction contexts together then have each student write a word problem in which both factors are mixed numerals. (For example: 2 1/2 x 3 1/6). Guide them to use whole numbers less that 10.
As they each finish their own word problem have them work on their solutions.

2. Explore at least one word problem created by a student.
Highlight once more the two ways they can approach the problem:
• changing the mixed numeral into an improper fraction.
• finding the sum of four partial products
For example: 2 1/2 x 3 1/6
• changing the mixed numeral into an improper fraction: 5/2 x 19/6 = 95/12 = 7 11/12 or
• finding the sum of four partial products
2 x 3= 6
2 x 1/6 = 2/6
1/2 x 3 = 1 1/2
1/2 x 1/6 = 1/12
Show with a diagram like this:

Finding the sum of the partial products: 6 + 2/6 + 1 1/2 + 1/12 = 6 + 4/12 + 1
6/12 + 1/12 = 7 11/12

3. Take two other student created problems. Record these on the class chart.
Have students work in pairs to solve the one of their choice.

4. Have students pair share their solutions.

5. Have students solve the other problem on their own then record in their maths diaries their learning from this session.

Session 4

SLOs:

• Understand and use the commutative property when multiplying fractions.
• Understand and explain the distributive property when multiplying with fractions.

Activity 1

1. Have the class/ group sitting in pairs on the mat. Begin by saying to one student so that all students can hear: “Please distribute scrap paper to each pair of students.” Write the word distribute on the class chart:
Ask the students to provide definitions of distribute and record their suggestions.
(If you are regularly adding to a class maths dictionary, you might like to record their ideas here. For example: ‘to spread’, ‘to share out’, ‘to give out’.)
Display this diagram from Session 3, Activity 4, Step 2, and write beneath it, ‘distributive property.’
Explain that this is the mathematical term for the process of expanding and solving the problem in this way.

Ask why this might be called the ‘distributive property’ and record the students' suggestions.

2. Have students consider suggest another fraction problem which could be solved by using the distributive property. Record suggestions.

Activity 2

1. On the class chart record this equation and the questions:
1/4 x 1/2 = 1/2 x 1/4 True? False?
Require the students to work in pairs to decide on their answer (true or false) and, using the paper distributed in Activity 1, Step 1, to prove how they know in two different ways (eg. using an area model and an algorithm).

2. Point out that one side of the classroom is the “true” side and the other is the “false” side. Have students indicate their thinking by moving to that side of the room.

3. Ask a selection of students to justify their position, and show their ‘proof’.

4. Consolidate this understanding by having 8 students stand in a group in the middle of the room.
Ask another student to show, by “using’ the group of 8 students what 1/4 of 1/2 equals. (1/8 or one student).
Reassemble the group of 8 students. Have another student “use” the group to demonstrate 1/2 of 1/4 (1/8 or one student).

Activity 3

1. Query whether this works when whole numbers are involved.
Pose and write the problem 5 x 2/3 = 2/3 x 5 True? False?
Repeat Activity 2, Steps 1-3 above.

2. Have 5 groups of 3 students stand in a group in the middle of the room.
Ask another student to show, by “using’ the five groups of three students model 2/3 + 2/3 + 2/3 + 2/3 + 2/3:
Ask 1/3 of the students in each group to sit down, leaving 2 students + 2 students + 2 students + 2 students = 2 students = 10. These students are then put back into groups of three making 3 1/3 groups. Reassemble the 5 groups of 3 students and combine them into one group of 15. Have another student “use” the group to demonstrate 2/3 of 5 groups (15 students) is 10 students. Have them regroup to make 3 1/3 groups.

Activity 4

1. Write on the class chart: Commutative property. Explain that this is a common mathematical term for what has been happening in 2 and 3 above. Ask students to suggest what commutative might mean and record their suggestions. (‘to change place’, ‘to travel’) and confirm that it means that the order of the numbers can be changed without changing the result).

2. Write on the class chart: Changing the order doesn’t change the result when we:
• add fractions. True? False? (true)
• subtract fractions. True? False? (false)
• multiply fractions. True? False? (true)
• divide fractions. True? False? (false) (encourage prediction)
Have students discuss these in pairs, then, as a class, reach an agreement.

Activity 5

Conclude the session by having the students suggest summary statements for today’s learning and record these on the class chart.

Session 5

SLOs:

• Communicate their understanding of the multiplication of fractions to others.
• Explore and demonstrate the reciprocal relationship between multiplication and division.

Activity 1

1. Begin this session by reviewing conclusions from session 4.
Ask one student to: “Please distribute A3 poster paper to each pair of students.” Set a time limit and have students work in pairs to create a poster, which includes a story context, and which explains either the distributive or commutative property as it applies to multiplication of fractions.
(Alternatively, have students prepare an electronic presentation to explain either of these properties to parents).

2. Fast finishers play Multiplifraction or fraction games from session 3.

Activity 2

Ask students to form pairs. Distribute cards from Attachment 4 to each pair. Have the pairs sort the cards into True and False piles. Pairs should then pair share, taking turns to read a card and talk about and justify their decision, using examples.

Activity 3

1. Conclude this session by reviewing class notes from each session.

2. Pose this problem: 1/2 x 1/4 > 1/2 ÷ 1/4. True? False?
Have students make their own decision independently.
Select several students to explain their thinking.
Conclude the session by indicating that they will be exploring this further in another session.

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