Suggested Learning Intentions

- To evaluate different rates and prove that one is better than the other

Sample Success Criteria

- I can identify ratios as quotients and rates
- I can find the best deal
- I can compare rates
- I can use a range of manipulatives to model and justify my thinking

- Coloured counters or tiles

This stage focuses on comparing simple rates in ‘best value’ situations. Students build on their knowledge of ratio comparisons to decide on the quality of rates. Students will have the opportunity to explore and compare rates in real world context, and to justify their reasons for choosing one rate over the other.

Begin by brainstorming various ratios that involve 2 variables. For example:

- in speed - km per hour
- in prices - $ per kg
- in population density - number of people per square kilometre.

Explain to students that when ratios involve 2 measurements with different units, they are known as rates. Remind students that in the previous lesson, they have already worked with rates. For example, when making calculations of water to dry materials in concrete mix, in the previous stage *Finding The Unknown in Ratios*.

Present the following problem, adapted from the ‘Pod Tunes or New Tunes’ task in the Fractions and Decimals Online Interview:

Sometimes people buy music cards to download songs from the internet. Here are 2 such cards - Pod Tunes and New Tunes.

With Pod Tunes, you get 16 songs for $24. With New Tunes, you get 12 songs for $20.

Which music card provides better value?

Ask students to describe the ratios for each of the two music cards. Discuss what they think ‘better value’ means. Encourage students to use their learning from Comparing Ratios to think about this problem.

Facilitate students to use the Think, Pair, Share routine to first attempt this problem individually then sharing their solutions with their peers.

The activities in this stage have been adapted from the Rates and Ratio tasks published by Sullivan, P. (2018) and are used with permission.

Before beginning this activity, it may be helpful to explain to students that according to data from Australia State of Environment in 2011, transport accounts for the second largest source of carbon emissions in Australia. All new conventional light vehicles sold in Australia are now required to display a Fuel Consumption Label on the windscreen.

Present these problems to the students:

Yusif drove 400km and his car used 36 L of petrol. Lan drove 300km and her car used 24 L. Who has a more fuel efficient car? Solve this in 2 different ways. Use manipulatives to explain your reasoning.

A PHEV (plug-in hybrid electric vehicle) claims it uses 1.7 L per 100km. How much petrol will Yusif and Lan save if they travelled the same distance? What is the fraction and percentage savings?

**Enable** students by prompting and supporting them to use tools that may help them visualise the problem, such as drawing a tape diagram or a double number line. Guide students to unpack the meaning of ‘fuel efficient’ and remind them the less petrol the car uses, the more fuel efficient it is.

**Extend** students by modifying the values to include decimal numbers. For example, when Yusif drove 560 km, his car used 47.6L of petrol.

**Areas for further exploration**

**1. Best Value**

Ratios can be used to determine the best buy. The study of rates supplements the concept of comparing value and unit price in Money and Financial Mathematics. The Mathematics Curriculum Companion offers various resources that help students investigate and calculate ‘best buys’.

The following activity from the Rates and Ratio tasks published by Sullivan, P. (2018) can also be used:

A box of 12 bottles of drink costs $8.00. It is also possible to buy single bottles for 75c each.

If I want to buy 12 bottles of this drink, which is a better deal?

**2. Paper size**

In Australia, paper comes in different sizes from A0 to A5. Students can explore the six paper sizes and describe how they are related to each other. Teachers can facilitate a discussion of how the paper sizes can be described as a ratio.

Opportunities for formative assessment of student learning include work samples from the tasks, with annotations that provide evidence of students’ reasoning.

To consolidate students' understanding of simple rates, the following questions can be posed:

The manufacturer's specifications suggest that my car has a fuel consumption of 8 L/100 km. When I measured my fuel consumption carefully, I noticed that my car used 36 L of fuel to travel 400km. Is this more efficient that the manufacturer's claim?

Work this out in 2 different ways, explaining your reasoning.

**Enable** students by first asking them to calculate the actual amount of fuel used when I travelled 100 km. Encourage students to use a diagram or materials to support their problem solving.

**Extend** students by presenting the problem:

The manufacturer's specifications suggest that my car has a fuel consumption of 8 L/100 km. When I measured my fuel consumption carefully, I noticed that my car used 31.1 L of fuel to travel 371 km. Should I be happy or not?

Encourage students to explain their reasoning and demonstrate using a variety of manipulatives.

Commonwealth of Australia, 2011. *Emission sources. *[Online]

Available at: https://soe.environment.gov.au/theme/climate/topic/emission-sources#:~:text=The%20energy%20sector%20(comprising%20stationary,LULUCF)%20(Figure%203.10).

[Accessed 15 March 2022].

Harvard Graduate School of Education, 2015. *Project Zero: Think, Pair, Share. *[Online]

Available at: http://pz.harvard.edu/sites/default/files/Think%20Pair%20Share.pdf

[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. *Investigate and calculate ‘best buys’. *[Online]

Available at: https://fuse.education.vic.gov.au/MCC/CurriculumItem?code=VCMNA250

[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. *Mathematics assessment. *[Online]

Available at: https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/mathsassess.aspx

[Accessed 15 March 2022].

The Department of Infrastructure, Transport, Regional Development and Communications, n.d. *Fuel consumption label. *[Online]

Available at: https://www.infrastructure.gov.au/vehicles/environment/fuel_consumption_label.aspx

[Accessed 15 March 2022].

Other stages

1. Building Understanding of Ratios

EXPLORESuggested Learning Intentions

- To recognise the relationships between quantities or measures in ratios

Sample Success Criteria

- I can model a ratio using diagrams, objects and other manipulatives
- I can describe ratios as a comparison between quantities
- I can increase or decrease quantities while keeping the ratio constant
- I can identify equivalent ratios

2. Finding the Unknown in Ratios

EXPLORESuggested Learning Intentions

- To build fluency in applying our understanding of ratios to solve problems

Sample Success Criteria

- I can calculate the quantity for each part of a ratio
- I can calculate the unknown value of a ratio
- I can demonstrate my thinking and justify my solutions using a range of manipulatives

3. Comparing Ratios

EXPLORESuggested Learning Intentions

- To apply our knowledge of equivalence to compare ratios

Sample Success Criteria

- I can simplify and compare ratios
- I can identify if two ratios have the same value
- I can describe the effects of adding parts of a ratio
- I can model and demonstrate my thinking using a range of manipulatives