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Lesson A: �Factors and multiples

Objectives

  • Find factors and multiples of numbers less than 100
  • Find common factors and common multiples of two numbers
  • Solve problems involving HCF and LCM
  • Understand how to use representations to provide insight �into solving problems

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What do these words mean?

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Explain the two words to the person next to you,�or write their meanings in your notebook.

Factors

Multiples

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Chocolate bar factory

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How many pieces are in this chocolate bar?

A chocolate bar containing 24 pieces of chocolate �is described as a 24-bar.

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24-bars

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YOUR TURN

6 pieces by 4 pieces

What other possible 24-bars are there?

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Possible 24-bars

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LET’S

LOOK

6 pieces by 4 pieces

8 pieces by 3 pieces

12 pieces by 2 pieces

24 pieces�by 1 piece

The bars produced by the factory are at least 2 pieces wide

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Multiplying

6

6

4

6

4

24-bar

6 × 4 = 4 × 6

We can multiply the two numbers in any order.

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Factors and multiples

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KEY IDEA

3

8

24-bar

3 is a factor of 24

8 is a factor of 24

24 is a multiple of 3

24 is a multiple of 8

24 is a common multiple of 3 and 8.

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Factors of 24

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KEY IDEA

24

1

24-bar

All numbers �have a factor of 1 and the number itself.

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Packing chocolate bars into trays

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Cardboard tray

The trays are described by the number of pieces of chocolate that fit the base layer.

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Describing trays

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We describe this as a 96-tray.

96 pieces of chocolate �fit on the base layer.

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Why are these 96-trays?

11

The chocolate factory makes these two 96-trays.

24

4

6

16

Explain why these are 96-trays.

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Packing 96-trays

12

4

24

Which 24-bars are being packed in this 96-tray?

6 by 4

Can any other �24-bars be packed in this 96-tray?

4

24

12 by 2

12 by 2

12 by 2

12 by 2

6 by 4

6 by 4

6 by 4

Handout�available

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A different 96-tray

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YOUR TURN

Which 24-bars can be packed in this 96-tray?

6

16

6

16

8 by 3

8 by 3

8 by 3

8 by 3

6

16

4 by 6

4 by 6

4 by 6

4 by 6

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The ‘16 by 6’ tray

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Can ‘12 by 2’ bars be packed into this 96-tray?

6

16

12

DISCUSS

2

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Packing ‘24 by 4’ 96-trays

15

24

4

12 by 2

12 by 2

12 by 2

24

4

6 by 4

6 by 4

6 by 4

12 by 2

6 by 4

These bars fit in the ‘24 by 4’ 96-trays because:

24 is a �common multiple �of 12 and 6

4 is a �common multiple of 2 and 4

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Packing ‘16 by 6’ 96-trays

16

6

16

8 by 3

8 by 3

8 by 3

16

6

4 by 6

4 by 6

4 by 6

8 by 3

4 by 6

These 24-bars fit in the ‘16 by 6’ 96-trays because:

16 is a �common multiple �of 8 and 4

6 is a �common multiple� of 3 and 6

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Other possible 96-trays?

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The chocolate factory makes these two 96-trays:

24

4

6

16

What other possible 96-trays are there?

DISCUSS

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All possible 96-trays

18

24

4

6

16

2

48

3

32

12

8

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Working in pairs

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YOUR TURN

  • Work in pairs.
  • Take turns.
  • Explain your reasoning to your partner.
  • Agree on the answers before writing them down.

Handout�available

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Task A

Find a 96-tray that can be used to pack all three types of 24-bars:

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20

4

6

2

12

3

8

‘12 by 2’ bars are the only bars that fit

‘8 by 3’ bars are the only bars that fit

‘12 by 2’ and ‘6 by 4’ bars both fit

‘8 by 3’ and ‘6 by 4’ bars both fit

‘12 by 2’, ‘8 by 3’ and ‘6 by 4’ bars all fit

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REVIEW

Possible 96-trays:

  • ‘48 by 2’ tray
  • ‘32 by 3’ tray
  • ‘24 by 4’ tray
  • ‘16 by 6’ tray
  • ‘12 by 8’ tray

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Task B

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REVIEW

All possible types of 30-bars?

10

3

30

15

2

1

6

5

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Task C

24-bars:

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REVIEW

4

6

2

12

3

8

30-bars:

10

3

15

2

6

5

The chocolate factory wants a tray that can be used to pack 24-bars and 30-bars.

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Highest Common Factor / Lowest Common Multiple

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DISCUSS

  • How can working out the lowest common multiple help in packing chocolate bars?
  • How can working out the highest common factor help in packing chocolate bars?

24-bar

30-bar

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Highest common factor

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  • Factors of 24

1, 2, 3, 4, 6, 8, 12, 24

  • Factors of 30

1, 2, 3, 5, 6, 10, 15, 30

6 is the highest common factor of 24 and 30.

How can working out the highest common factor help in packing chocolate bars?

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Lowest common multiple

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  • Multiples of 24

24, 48, 72, 96, 120, 144, 168, 192, …

  • Multiples of 30

30, 60, 90, 120, 150, 180, 210, 240, …

120 is the lowest common multiple of 24 and 30.

How can working out the lowest common multiple help in packing chocolate bars?

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Packing 24-bars and 30-bars

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24-bars or 30-bars

120

6

Packing 24-bars:

30 ‘4 by 6’ bars

30 ‘8 by 3’ bars

30 ‘12 by 2’ bars

Packing 30-bars:

24 ‘5 by 6’ bars

24 ‘10 by 3’ bars

24 ‘15 by 2’ bars

120

6

6

6

6

6

6

120

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Packing 24-bars and 30-bars

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What other dimensions could the tray have?

24-bars or 30-bars

120

6

Is there a limit?

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Practice question

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REVIEW

Find the highest common factor (HCF) of 72 and 90.

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Q24 from May 2019, 1MA1/1F

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Handout�available

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Lesson review: �Factors and multiples

Objectives

  • Find factors and multiples of numbers less than 100
  • Find common factors and common multiples of two numbers
  • Solve problems using HCF and LCM
  • Understand how to use representations to provide insight into �solving problems

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Suggested further steps/areas to work on

  • Find all the factor pairs, HCF and LCM using prime factorisation

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Lesson A: �Credits

Photo acknowledgements

Shutterstock.com: Vincent Giordano Photo

Text acknowledgements

Pearson Education Ltd: Pearson Edexcel GCSE (9 – 1) In Mathematics (1MA1) Foundation (Non-Calculator) Paper 1F

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