Stage 4 -number – ratio and rates
Determine a ratio; write a ratio in its simplest form; recognise the difference between ratios and rates; calculate rates, using speed, distance and time
Strategy
Students can:
- determine a ratio
- write a ratio in its simplest form
- recognise the difference between ratios and rates
- calculate rates, using speed, distance and time
Activities to support the strategy
Activity 1
A ratio is a relationship between two or more quantities, measured in the same unit.
1. The teacher dilutes a quantity of concentrated fruit juice or cordial. Discuss how much water is required compared to the amount of concentrated liquid. Refer to the instructions, e.g.
Discuss what this means. Ask:
- How much concentrate would you need for every 4 glasses of water?
- How much concentrate would you need for every 8 glasses of water?
- How much water would you need for every 3 parts of concentrate?
In mathematics this comparison of concentrate to water can be shown as:
Students brainstorm and discuss other examples of when ratios are used in everyday life.
2. The teacher models how to determine the ratio using the following everyday example.
John earns $11 pocket money each week. Mark, his brother, earns $9. Determine the ratio of:
Emphasise to students the importance of the order in a ratio, using the above solutions as examples.
Have students determine other ratios, e.g. using the class roll data, students can calculate:
- the ratio of girls to boys
- the ratio of boys to girls
- the ratio of girls to the total number of students
- the fraction of boys to the total number of students.
3. Students use a line divided into equal parts to determine ratios
- the ratio of AB to AD
- the ratio of BC to CD
- the ratio of AB to CD
- AB : AE
- DE : BE
Students work in pairs to determine other ratios using a line.
4. Show students how to find equivalent ratios by multiplying or dividing. Emphasise that to make an equivalent ratio, both terms must be multiplied or divided by the same number.
Work through this example.
Have students follow the same process to simplify these ratios. Give students examples in both the ratio form and the fraction form.
5. Additional information on ratios can be found at Math mate - ratios
Activity 2
A rate is a quantity (measured in units) compared with another quantity (measured in different units).
Some examples of rates include:
- speed (measured in kilometres per hour) comparing distance travelled (kilometres) and time taken (hours)
- electricity usage (measured in kilowatts per hour) compares the amount of energy (kilowatts) and the time taken (hours)
- exchange rate (measured in yen per $A) compares the amount of Japanese yen you can exchange for each Australian dollar.
Introduce the rate of speed as one of the most common rates in everyday life. Display the formula for calculating speed.
1. The teacher models how to calculate speed using examples where the distance and time are given.
Examples should include problems which require:
- substitution into the formula and no conversion of units - e.g. a runner completes a 14 kilometre fun run in 2 hours. What is his average speed in km/h?
- substitution into the formula and conversion of units
2. The teacher introduces other problems in which the speed and either distance or time are given.
Show students that an easy way to remember the three formulas for calculating speed, distance or time is to use this triangle.
3. The teacher models how to calculate speed, involving conversion of units (change from m/s to km/s), e.g. A runner completes a 400 metre race in 50 seconds. What is his average speed in km/h?
References
Australian curriculum
ACMNA173: Recognise and solve problems involving simple ratios.
NSW syllabus
MA4-7NA: Operates with ratios and rates, and explores their graphical representation.
Student resources
Distance, speed and time
.