Ratio and ProportionRuth Bull – 2017

1.30pm to 3:30pm

Aims of the 3 sessions:

Session 1: Differences between Ratio and Proportion, links to

fractions, decimals and percentages and identifying areas of

multiplicative reasoning in the primary curriculum. Considering

some models and images. Gap task

Session 2: Singapore bar modelling and use of a table to support

ratio and proportion problems. Equivalent ratios and fractions.

Ideas for assessing understanding. Gap task planning a lesson.

Session 3: Looking at scaling. Conversion graphs. Ratio and

proportion in shape. Adapting recipes and developing reasoning

further in R and P. What are the pitfalls? How can we support

further?

Aims of session 1

What is Ratio? What is proportion? What is the difference? How

are they linked to fractions, decimals and percentages?

Where is it in the primary national curriculum?

To consider some models and images that may support

understanding

Ratio and proportion are part of

multiplicative reasoning.

Think about your own class situation. Where do children use

multiplicative reasoning?

Think about your own situation as an adult – where do you use

multiplicative reasoning?

What is Ratio? What is Proportion?

Ratio Proportion

2 : 3

2 yellows “for every” 3

blues

Yellow circles are two-

thirds (2/3) of the blue

ones.

There are one and half

times blue circles (3/2)

than yellow ones.

There are 5 circles in

total.

2/5 are yellow circles

2 “in” 5 are yellow

3/5 are blue circles

3 “out of” 5 are blue

If we had 10 circles, then

4/10 would be yellow.

2 4

5 10=

3 6

5 10=

There’s lots of differences between the classes in your school –

… Some have more boys than girls

… Less United fans than Rovers fans

… More cat-lovers than dog-lovers

… Less pizza-munchers than chicken-dippers

Form 6A has 30 pupils:

10 Red Rovers fans

20 Blue United fans

Form 6B has 18 pupils:

9 Red Rovers fans

9 Blue United fans

You’re a Rovers fan.

Which class would you rather be in?

Discuss and take a vote.

Did You Pick 6A?

6A: 10 + 20 = 30 6B: 9 + 9 = 18

–

Rani, a mad Rovers fan, picked 6A:

“10 is bigger than 9,

so there’s more Rovers fans

in 6A than 6B.

Yey!”

Did You Pick 6B?

6A: 10 + 20 = 30 6B: 9 + 9 = 18

- But other red Rovers fans picked 6B:

MARTIN SAYS

“I’m not going to 6A - I’d

be outnumbered 2 United

to every 1 Rovers fan!”BECKY SAYS:

“Only one third

of 6A Rovers

fans. It’s one

half in 6B.

HANNAH SAYS:

“6A only has 10 out

of 30 Rovers fans.

That’s 1 out of 3.”

Who is right?

Discuss and take a vote!

2 to 1

is a ratiothird and halfare fractions

1 out of 3 is a

proportion

All answers are correct! But - 3 different sorts of answer.

Which do you prefer? Why?

What’s the Difference - Summing Up: .

“1 to every 2” is a Ratio“1 Rovers fan to every 2 United fans”“The ratio of Rovers to United fans is 1 to 2”“The ratio of United to Rovers fans is 2 to 1”

“1 out of 3” is a Proportion

“1 out of every 3 fans is a Rovers fan”

“The proportion of Rover fans is 1 out of 3”

“One third” is a Fraction

“One third of all fans are Rovers fans

“1/3 of all fans are Rovers fans”

Ratios compare PART WITH

PART

Proportions compare PART WITH WHOLE

Fractions compare PART

WITH WHOLE

using shorthand such as 1/3

3 different ways to

say the same thing

3 different ways to

compare numbers

So comparing totals isn’t the only way to

look at these numbers…

Form 6A has 30 pupils:

10 Rovers fans

20 United fans

Form 6B has 18 pupils:

9 Rovers fans

9 United fans

…You can compare part with part using RATIO

…You can compare part with whole using

PROPORTIONS or FRACTIONS

Eg: 1/3 are Rovers fansEg: 1 out of 3 are Rovers fans

Eg: Rovers : United = 1 to every 2 = 1:2

Ratio and Proportion

Ratio and Proportion: Different Words!

Ratio: “to every” or “for every”1 Rovers fan to every 2 United fans

Proportion: “out of” or “in”1 out of 3 football fans is a Rovers fan

1 in 3 football fans is a Rovers fan

Card Sort

Ratio

Four for every ten

Ten to every four

4:10

2:5

3 United fans to every 2

Rover fans

3:1 drivers speed to

those who keep the limit

Proportion

Two out of five

Four out of ten

9 girls out of 10 use soap

3 boys out of 10 use

deodorant

3 in 4 pizza munchers

love olives

Three out of four drivers

speed at some time

Fraction

Two fifths

Four tenths

4/10

2/5

Half of all girls in 6D

love History

A third of all RTAs

involve drinking

Three quarters of drivers

speed

Defining terms

As a proportion

1 in every 8

As a fraction

1/8 (4/32 simplified)

As a decimal

0.125

As a percentage

12.5%

As a ratio

1 : 7 4 : 28 (32 in total)

There are 32 teachers on a course, and 4 of them are

vegetarian.

Where are ratio and proportion in the

NC?

Notes and guidance (non-statutory) - Year 2 Statistics

Pupils record, interpret, collate, organise and compare information (for

example, using many-to-one correspondence in pictograms with

simple ratios 2, 5, 10).

Where are ratio and proportion in the

NC?

Upper Key Stage 2 – Year 5 and year 6

The principal focus of mathematics teaching in upper key stage 2 is to

ensure that pupils extend their understanding of the number system and

place value to include larger integers. This should develop the

connections that pupils make between multiplication and division with

fractions, decimals, percentages and ratio.

Year 6 – Ratio and Proportion NC

Statutory requirements

Pupils should be taught to:

▪ solve problems involving the relative sizes of two quantities where missing values can be

found by using integer multiplication and division facts

▪ solve problems involving the calculation of percentages [for example, of measures, and

such as 15% of 360] and the use of percentages for comparison

▪ solve problems involving similar shapes where the scale factor is known or can be found

▪ solve problems involving unequal sharing and grouping using knowledge of fractions and

multiples.

Notes and guidance (non-statutory)

Pupils recognise proportionality in contexts when the relations between quantities are in the

same ratio (for example, similar shapes and recipes).

Pupils link percentages or 360° to calculating angles of pie charts.

Pupils should consolidate their understanding of ratio when comparing quantities, sizes and

scale drawings by solving a variety of problems. They might use the notation a:b to record

their work.

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.3

5

Year 6 – Ratio and Proportion NC

Similar shapes – What do we mean?

The rectangles pqrs and PQRS are similar. What is the length of PS?

Similar shapes – What do we mean?

Congruent shapes – Identical shape and

size. Which shapes are congruent to each other?

Video clip

More on ratio

If there are 35 people in the room, of which 15 are women and 20

are men, we can write this in three ways:

In words “15 to 20”

Ratio notation 15:20

Fractional notation 15/20

This can be simplified to ¾. This means that you can also express

the ratio of women to men as: ¾ 3:4 or “3 to 4”

A rational number is a number that can be written as a ratio.

That means it can be written as a fraction, in which both the

numerator (the number on top) and the denominator (the

number on the bottom) are whole numbers. The number 8 is a

rational number because it can be written as the fraction 8/1.

The order matters

There are 3 ducks and 2 geese

in a certain park. Express the

ratio of ducks to geese in all

three formats

Express the ratio of

geese to ducks in all

three formats.

3:2

3

2

3 to 2

2:3

2

3

2 to 3

Percentages

A percentage is a proportion out of 100

15% means 15

100

If on a car journey of 200 miles a total of 140 miles is single carriageway,

we could say that 70% of the journey is single carriageway, meaning 70

miles in each 100 miles. (Direct proportion)

Examples:

45 % of a class are girls. What proportion of the class are

boys?

Exploration of proportionality on a

multiplication grid

For follow up work see www.ncetm.org.uk/resources/10334

Identify proportional relationships between numbers, columns or rows of a multiplication grid. For example the second column is twice as big as the first column.

What is the relationship between column two and column three?

What is the relationship between column three and column two?

What is the relationship between row ome and row three?

Exploring Equivalent ratios with

Cuisenaire rods

Choose two rods and then find two others which are in the same proportion to each other. These represent equivalent ratios

What is the value of the other rods when:

• the blue rod is 1?

• the brown rod is 4?

• the red rod is 1/5th?

Take any colour rod, give it a value and express the others proportionally.

KS2 SAT Problem

In a class, 18 of the children are girls.

A quarter of the children are boys

Altogether how many children are there in the class?

Be prepared to justify your method

Algebraic approach

Let t = total, b = boys and g = girls

So t = b + g

Also, ¼ t = b so t = 4b

We know g = 18

Now we have two unknowns and two equations:

Substituting: 4b = b + 18 which means 3b = 18 and b= 6

This means t = 6 + 18 = 24

Bar model approach

?total

boys A quarter are boys

18girls

The total is made from boys and girls.

So girls must represent the rest which is ¾

18÷ 3 = 6

So each ¼ must represent 6

Other approaches

boys

Girls

18

18 ÷ 3 = 6

Carroll diagram – looking for depth

Give me a question that has the answer

3 : 7

40 %

2 : 1

2/3

Boys

Glasses

No

Glasses

Girls

1

42

3

Aims of session 1

What is Ratio? What is proportion? What is the difference? How

are they linked to fractions, decimals and percentages?

Where is it in the primary national curriculum?

To consider some models and images that may support

understanding