Associate Lead Teacher – Maths

Manor Croft Academy, Dewsbury

darwink@manorcroft.org.uk

@Arithmaticks

Session Outline

• The Big Idea
• Why is this important?
• Where the obsession began – Speed & Compound Measures
• ‘Conventional’ Direct Proportion Problems
• ‘Covert’ Direct Proportion Problems

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Proportionality and some related concepts appear everywhere in the curriculum but are often treated implicitly.

Anne Watson, Keith Jones and Dave Pratt (2013)

Key Ideas in Teaching Mathematics: Research-based guidance for ages 9-19, page 12

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The Big Idea

Any two numbers can be connected by a multiplicative relationship with a single multiplier.

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This includes:

• Use of ‘proportion’ as a verb, rather than a noun
• A shift from additive relationships to multiplicative relationships�e.g. ‘lots of’ and scaling
• Maintaining proportionality through multiplication and division
• Eventually the use of a single (fractional) multiplier
• Recognising when to use multiplicative relationships across the curriculum

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The Big Idea

Student strategies were categorised by Karplus et al. (1883, p. 221) as:

• “Between” : comparing variables of the same unit
• “Within”: comparing the parts of each individual ratio
• “Other”: intuitive methods, such as rated addition

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• E. g. ‘If 2 units are £1, how much for 10 units?’,

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Between:

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2 units : £1

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10 units: £5

Within:

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2 units : £1

10 units: £5

×5

×5

The National Curriculum

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas.”

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Year 6:

• Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes).
• Pupils solve problems involving unequal quantities, for example, ‘for every egg you need three spoonfuls of flour’, ‘ 3/5 of the class are boys’. These problems are the foundation for later formal approaches to ratio and proportion.
• They use commutativity and inverse relations to develop multiplicative reasoning (for example, 4 × 5 = 20 and 20 ÷ 5 = 4).

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas.”

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Year 6:

• Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes).
• Pupils solve problems involving unequal quantities, for example, ‘for every egg you need three spoonfuls of flour’, ‘ 3/5 of the class are boys’. These problems are the foundation for later formal approaches to ratio and proportion.
• They use commutativity and inverse relations to develop multiplicative reasoning (for example, 4 × 5 = 20 and 20 ÷ 5 = 4).

The National Curriculum

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas.”

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KS3:

• Extend and formalise their knowledge of ratio and proportion in working with measures and geometry, and in formulating proportional relations algebraically
• Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
• Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas.”

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KS3:

• Extend and formalise their knowledge of ratio and proportion in working with measures and geometry, and in formulating proportional relations algebraically
• Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
• Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning

The National Curriculum

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas.”

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KS4:

• Extend and formalise their knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically and graphically
• Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning
• (+ statements on algebraic direct & inverse proportion, and rates of change)

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas.”

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KS4:

• Extend and formalise their knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically and graphically
• Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning
• (+ statements on algebraic direct & inverse proportion, and rates of change)

The Curriculum & Exams

However, many other topic areas implicitly assess this reasoning, e.g. fractions, unit conversions, compound measures, which fit in different ‘topic area’ descriptors in NC.

Edexcel

AQA

Issues with proportionality

• Viewing relationships as additive and considering only differences

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• Difficulty working with non-integer multipliers

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• “Multiplication always makes it bigger/Division always makes it smaller”

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• Recognising where to use multiplicative thinking

Issues with proportionality

• Student approaches to multiplicative questions varied according to the problem type and its context.
• e.g. Students used rated addition for problems stating facts such as “1 pizza can be shared between 3 people. How many pizzas for 12 people?”
• BUT students applied common factors and multiples to problems involving �identifying the best value for 3-, 6- and 12-month subscription deals.

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Watson et al (2013), Lamon (1993, 1994) and Karplus et al. (1983)

meaning accumulates through use, over time, in many different mathematical, real-life and �scientific contexts

Anne Watson, Keith Jones and Dave Pratt (2013)

Key Ideas in Teaching Mathematics: Research-based guidance for ages 9-19, page 42

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Where it all started…

John drives 60 miles in 40 minutes.

What is his average speed in mph?

What would your students do?

What problems do they come up against?

Where it all started…

D

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T

M

D

V

F

P

A

Speed - Attempt 1 - Bar Models

John drives 60 miles in 40 minutes.

What is his average speed in mph?

30 miles

30 miles

30 miles

Speed - Attempt 2 – Double Number Lines

John drives 60 miles in 40 minutes.

What is his average speed in mph?

Speed – Attempt 2 – Ratio Tables

John drives 60 miles in 40 minutes.

What is his average speed in mph?

60 miles

40 mins

90 miles

60 mins

30 miles

20 mins

I hour

Compound Measures - Density

A piece of wood has a mass of 7g and a volume of 10cm³�What is the density in g/cm³? �

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7g

10cm³

0.7g

1cm³

Compound Measures - Density

A piece of material has a mass of 42g and a density of 6g/cm³�What is the volume of the material?

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6g

1cm³

42g

7cm³

Compound Measures - Pressure

A block is placed on a table. �It has pressure 20N/cm² and is being placed with force of 50N. �What is the area in contact with the table?

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20N

1cm²

50N

2.5cm²

10N

0.5cm²

Within/Between?

So far our thinking has been between, quite intuitively…�But we can exploit the within relationships too!

20N

1cm²

50N

2.5cm²

6g

1cm³

42g

7cm³

The BIG IDEA - Direct Proportion

NCETM. Via Jo Morgan @ Resourceaholic

https://drive.google.com/drive/folders/1i_LSylx0dR7avIywR5WBab2-DfnEi7Og

The BIG IDEA - Direct Proportion

The BIG IDEA - Direct Proportion

NCETM. Via Jo Morgan @ Resourceaholic

https://drive.google.com/drive/folders/1i_LSylx0dR7avIywR5WBab2-DfnEi7Og

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The BIG IDEA - Direct Proportion

NCETM. Via Jo Morgan @ Resourceaholic

https://drive.google.com/drive/folders/1i_LSylx0dR7avIywR5WBab2-DfnEi7Og

15

The BIG IDEA - Direct Proportion

NCETM. Via Jo Morgan @ Resourceaholic

https://drive.google.com/drive/folders/1i_LSylx0dR7avIywR5WBab2-DfnEi7Og

2

inappropriate use of whole number thinking

S. J. Norton (2005)

The construction of proportional reasoning.�Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. 4, p. 17-24.

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NCETM. Via Jo Morgan @ Resourceaholic

https://drive.google.com/drive/folders/1i_LSylx0dR7avIywR5WBab2-DfnEi7Og

1

1

Your Turn!

NCETM. Via Jo Morgan @ Resourceaholic

https://drive.google.com/drive/folders/1i_LSylx0dR7avIywR5WBab2-DfnEi7Og

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Don Steward’s ‘Boxes’

https://donsteward.blogspot.com/2012/03/boxes.html

https://donsteward.blogspot.com/2019/02/boxes-resources.html

MathsBot to the rescue!

‘Conventional’ Direct Proportion

The cost of 6 cups of coffee is £9

a) Work out the cost of 12 cups.

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b) Work out the cost of 10 of cups.

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6

£9

12

£18

6

£9

10

£15

‘Conventional’ Direct Proportion

12

3

20

5

36

9

B

4B

‘Conventional’ Direct Proportion

2

8

7.5

30

100

400

F

‘Conventional’ Direct Proportion

50

25

18

9

200

100

B²

2B²

a) A=2B²

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5

3

10

B

Ratio - Anne & Bob

8

£45

£75

8

£320

£200

8

£72

£192

2

£180

£300

Percentages

Find 30% of £620

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Increase £340 by 15%

100%

£620

30%

£186

10%

£62

10%

£34

115%

£391

5%

£17

Reverse Percentages

A watch costs £180, inclusive of VAT at 20%.�What is the cost of the watch before VAT?

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120%

£180

100%

£150

20%

£30

Frations

Find ¾ of £96

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¾ of a number is 42. What is the number?

4

£96

3

£72

1

£24

3

42

4

56

1

14

Pie Charts

70^o

110^o

40^o

140^o

Stratified Sampling

18

12

9

11

Similarity

4

2

6

Similarity

160cm²

4840cm²

96cm³

4

121

2

11

8

1331

15672cm³

Trigonometry

https://donsteward.blogspot.com/2012/03/boxes.html

‘Covert’ Direct Proportion

https://mathshko.com/2018/05/01/proportional- reasoning/

Don’t just take my word for it…

Always working algebraically is not necessarily the most efficient.

Students had a better grasp of the topics and why it works, rather than blindly following a formula that they don't understand

The transition between hours and then having to work with minutes was much smoother as they were able to use the same method.

Students have a better understanding of the origins of the units for compound measures.

Engagement and success improved.

Students are able to attempt ‘higher tier’ proportion questions with understanding, where algebra held them back before

I understood more clearly, never mind the students!

Tell your physics teachers - this is loads easier

I like the tables - they help me keep all the boxes linked with “times-ing”

I used to make mistakes with the formula, but get more right now.

Developing proportional reasoning is a �medium-term project

Anne Watson, Keith Jones and Dave Pratt (2013)

Key Ideas in Teaching Mathematics: Research-based guidance for ages 9-19, page 66

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Associate Lead Teacher – Maths

Manor Croft Academy, Dewsbury

darwink@manorcroft.org.uk

@Arithmaticks