Unit 8: Ratio and Rate

Unit 8: Ratio and rate

Planning guidance

Unit objectives

Within this unit, students will learn to:

Interpret a:b and a:b:c, where a, b and c are whole numbers
Compare two or more quantities by ratio
Relate ratios to fractions
Write equivalent ratios and find the missing term in a pair of equivalent ratios
Express ratios involving rational numbers in their simplest form
Find the ratio of two or three given quantities
Find one quantity given the other quantity and their ratio
Express one quantity as a fraction of another given their ratio
Express one quantity as a fraction of another given the two quantities
Find the whole/one part when a whole is divided into parts in a given ratio
Calculate average rate
Solve up to 2-step word problems involving ratio

Students’ understanding of multiplication is often based on the repeated addition of integers, which is limiting when looking at topics such as ratio and proportion.The challenge is therefore to provide students with an understanding of multiplicative reasoning that is not repeated addition. Using activities that draw on measure and similarity where the scale is continuous can encourage a shift away from the use of repeated addition and build a deeper understanding of proportional reasoning. Examples with ratios involving doubling and halving should be used sparingly, since this relationship is often conceptually different for students than multiplication by other numbers. Cuisenaire rods should be used to not only demonstrate the idea of ratio – connecting to multiples and factors, but also to link ratio to previous work on fractions. Ideally these would be the rods with no markings on them so that the students did not assign particular values to them.

This unit should take three weeks. Whilst there are three sections of work in this unit, we leave it to teachers to determine how much time they wish to spend on each section. Time should be spent on the key ideas and concepts in the first two sections.

Suggested Structure

The actual number of lessons spent on each section will depend on the individual class and the number of lessons available to teachers. We suggest the following based on a total of 9, 12 or 15 lessons over three weeks:

Section / 9 lessons / 12 lessons / 15 lessons
Interpreting and defining ratio / 3 / 4 / 5
Using ratio / 4 / 5 / 6
Rate (speed/distance/time) / 2 / 3 / 4

Interpreting and defining ratio

Objectives

Interpret a:b and a:b:c, where a, b and c are whole numbers
Compare two or more quantities by ratio
Relate ratios to fractions
Use bar models to represent ratios and fractions
Write equivalent ratios and find the missing term in a pair of equivalent ratios
Express ratios involving rational numbers in their simplest form
Find the ratio of two or three given quantities
Find one quantity given the other quantity and their ratio
Express one quantity as a fraction of another given their ratio
Express one quantity as a fraction of another given the two quantities
Apply ratios to similarity questions

During the section students must have the opportunity to explore rations and their properties using concrete manipulatives if at all possible. Beads, Cuisenaire rods, cubes and counters can all be used to support understanding. It is vital that students are able to appreciate the multiplicative rather than additive nature of ratios and their links to fractions. Many of the tasks provided are designed to generate discussion and could be used a number of ways to develop students understanding such as through talk tasks, group work or whole class discussions.

Strong links should be made between ratios and fractions and different representations should be discussion – such as when the ratio 1:2 might correspond to a fraction of and when it might co0rrespond to . In addition, bar models are used throughout to reinforce concepts and to support students as needed.

If needed, time from the final section of this unit (working with rates) can be used here as it is vital that students have a good grasp on the concepts underpinning this unit.

For further guidance, including the Department Tasks for this section, click Interpreting and defining ratio underPlanning resources on the main page for this unit.

Using ratio

Objectives:

Divide a quantity in a given ratio
Find the whole/one part when a whole is divided into parts in a given ratio
Make links between the ratios and proportion
Solve problems involving ratios and proportion (such as scaling recipes)
Compare ratios (e.g. which squash is stronger/which paint is darker)
Apply ratios to similar shapes
Solve up to 2-step word problems involving ratio

Having explored the meaning and key properties of ratio students can now apply their knowledge to a wide variety of problems. As always, they should be encouraged to develop their problem solving strategies by sharing their ideas and methods with each other. Many of the problems provided (for example scaling recipes) can be completed in a number of ways and students should be given the chance to discuss the merits of different methods.

Throughout this section bar models are used when demonstrating sharing into given ratios and finding parts/wholes when given parts and links can be made between these models and when working with fractions.

For further guidance, including the Department Tasks for this section, click Using ratiounder Planning resources on the main page for this unit.

Rate (speed, distance, time)

Objectives:

Understand the concept of rate in a variety of contexts including
Exchange rates
Speed/distance/time
Calculate average rates
Solve complex, multi-step problems involving rates

The amount of time spent on this section will depend on whether students feel confident enough with the earlier parts of this unit which should be prioritised.

During the final section, students will explore the concept of rate in a variety of contexts and making links between rate, ratio and proportion. The primary context is speed, distance and time and students are given the opportunity to explore connections between the three quantities through investigative tasks before looking at the formal definition. It is important that students to develop an intuitive understanding of rates so the focus is not on applying rules and formula but thinking carefully through increasingly complex problems.

For further guidance, including the Department Tasks for this section, click Rate (speed, distance, time) under Planning resources on the main page for this unit.