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National Curriculum Resource Tool

Materials to support teachers and schools in embedding the National Curriculum

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KS3 - Ratio and Proportion and Rates of Change

New Curriculum

Pupils should be taught to:

  • change freely between related standard units (for example time, length, area, volume/capacity, mass)
  • use scale factors, scale diagrams and maps
  • express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1
  • use ratio notation, including reduction to simplest form
  • divide a given quantity into two parts in a given part:part or part:whole ratio; express the division of a quantity into two parts as a ratio
  • understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
  • relate the language of ratios and the associated calculations to arithmetic of fractions and to linear functions
  • solve problems involving percentage change, including: percentage increase, decrease and the original value problems and simple interest in financial mathematics
  • solve problems involving direct and inverse proportion, including graphical and algebraic representations
  • use compound units such as speed, unit pricing and density to solve problems

Non-Statutory Guidance

Links and Resources

Making connections to other topics within Key Stage 3

Number

Pupils will need to be conversant with a whole range of number topics to support their learning and development within this area. In particular they will need to be secure in multiplication and division operations with fractions, percentages and decimals. They will also need to understand and use inverse operation.

Algebra

Some aspects of this area will require pupils to use and interpret mathematical relationships both algebraically and graphically. Connections should be made when teaching direct and inverse proportion.

Geometry

Connections between this area of study should be made when working with similar triangles and other similar shapes as the concept of proportionality is the basis of similarity.

Making connections to this topic in Year 6

In year 6 pupils will have used ratio and proportion in a range of contexts:

  • Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (e.g. similar shapes, recipes).
  • Pupils link percentages of 360° to calculating angles of pie charts.
  • Pupils 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 e.g. ‘for every egg you need three spoonfuls of flour’, ‘35 of the class are boys’. These problems are the foundation for later formal approaches to ratio and proportion.

Cross-curricular and real life connections

  • Science: science can provide the context for many basic ratio problems such as NRICH’s Oh Harry!
  • Geography: statistics on populations in different parts of Canada at different periods, given as percentages and also allowing calculations of average rates of change
  • Art: see proportion wheel for creating art cards of different sizes by reducing pictures in different proportions
  • Ratio and Proportional Reasoning A. Watson, K. Jones, D. Pratt, 2013 The authors use research about students’ understanding of ratio and proportional reasoning to shed light on difficulties which seriously hinder success across mathematics. They examine the strengths and limitations of various teaching approaches and make suggestions about how teachers might work with colleagues on teaching ratio and proportional reasoning.

  • The struggle to achieve multiplicative reasoning 11-14, Margaret Brown, Dietmar Küchemann and Jeremy Hodgen, 2010. Multiplicative reasoning is a key competence for many areas of employment and everyday life, and for further mathematical study. It is however a complex conceptual field. The ICCAMS (Improving Competence and Confidence in Algebra and Multiplicative Structures) project, with multiplicative reasoning as one of its two focus themes, has in Phase 1 conducted a broadly representative survey of attainment which suggests that standards in this area have not risen since the 1970s and that relatively few students are achieving competence in relevant areas. Student difficulties are illustrated by evidence from group interviews in Phase 2 of the project.

  • Ratio or Proportion?, Nrich Team, Discussion of the similarities and differences between ‘ratio’ and ‘proportion’ and how the word ‘proportion’ is used in two different ways.

  • Misconceptions with the Key Objectives, has a section (from page 60) identifying common misconceptions in ratio and proportion and suggesting effective questions to tackle them.

Ratio (mathematical moments)

This is a collaborative task to support pupils to develop concepts about equivalent ratios. (You need to be logged in to the National STEM Centre website to access this document)

Understanding ratio (Learning mathematics in context):

This is a collaborative task on combining quantities in a given ratio. Pupils will learn to: identify the ratio of two quantities and use the notation. Understanding of the meaning of ratio and equivalent ratios will be developed. Pupils combine quantities and divide quantities in a given ratio.

Proportion (mathematical moments)

This activity is intended to give pupils the opportunity to explore the proportional relationships between numbers using a relatively familiar resource (a multiplication grid) and make links with other areas of mathematics. (You need to be logged in to the National STEM Centre website to access this document)

Toad in the whole:

This activity uses proportion and the understanding of the multiplicative relationships between quantities to solve a problem that superficially appears a simple question of adding amounts of money.

Mixing Lemonade:

This activity uses ratio and proportional reasoning to determine the relative strength of mixing cordial with water. There are a range of ways of determining the answer using fractions, percentages or pie charts to illustrate the answer.

Calculating scales for scale drawing (Learning Mathematics in Context):

This activity develops pupils understanding of scales written in the form 1:x. It develops the ability to carry out calculations involving scales using a variety of units.

Developing Proportional Reasoning (Improving Learning in Mathematics N6):

Pupils are given four direct proportion problems to solve, taken from different areas of the mathematics curriculum. They then compare their methods for solving these with methods produced by other pupils.

Using Percentages to Increase Quantities (Improving Learning in Mathematics N7):

Pupils make links between percentages, decimals and fractions, represent percentage increase and decrease as multiplication and recognise the inverse relationship between increases and decreases using proportional reasoning.

Four on the road:

This activity involves using graphical representations to solve speed, time and distance problems.

Examples of what children should be able to do, in relation to each (boxed) Programme of Study statement

Examples of what children should be able to do, to show a broad grasp of the content of this topic

Example A

Simplify the ratios expressed below in different units.

  • 2m : 50cm
  • 450g : 5kg

Example B

2 parts of blue paint mixed with 3 parts of yellow paint makes green. A boy has 50ml of blue paint and 100ml of yellow.

What is the maximum amount of green he can make?

Example C

The ratios of Lycra to other materials in two stretch fabrics are 2 : 25 and 3 : 40.

Which fabric has the greater proportion of Lycra?

Example D

The cost of 4 kg of apples is £3.36

The total cost of 3 kg of apples and 2.5 kg of pears is £4.12

Work out the cost of 1 kg of pears.

Example E

A Large tub of popcorn costs £3.80 and holds 200 g.

A Regular tub of popcorn costs £3.50 and holds 175 g.

Mike says that the 200 g Large tub is the better value for money.

Lucy says that the 175 g Regular tub is the better value for money.

Who is correct?

Explain the reasons for your answer. You must show all your working.

Example F

This is a recipe for fruit squash for six people.

Scroll

Tracy made fruit squash for ten people.

How many litres of lemonade did she use?

John used 2 litres of orange juice for the same recipe.

How many people was this enough for?

Example G

This map has a scale of 1cm to 6km.

Map

The road from Ridlington to Carborough measured on

the map is 5.2cm long.

What is the length of the road in kilometres

Example H

The length of a coach is 15 metres.

Joe makes a model of the coach.

He uses a scale of 1:24

Work out the length, in centimetres, of the model coach.

coach used under the Attribution-ShareAlike 2.0 Generic CC BY-SA 2.0 licence
Image adapted from photograph by
Glen Wallace some rights reserved

Example I

A truck and van left from Paris in diametrically opposite directions. The truck travelled for 4

hours at 76 mph. The vehicles were 564 miles apart. Find the van's average speed.

Example J

diagrams

Shapes ABCD and EFGH are mathematically similar.

  • Calculate the length of BC.
  • Calculate the length of EF.

Example K

Graph

The graph above shows the relationship between gallons and litres.

Does the graph illustrate direct proportion? How do you know?

  • Change 13 gallons to litres.
  • Change 500 litres to gallons.

Explain how you worked out your answers.