Real World, Real Numbers: Ratio and Proportions

# 1. Building Understanding of Ratios Suggested 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

This activity introduces students to the different types of ratios and how they can be applied in everyday situations. Students build an understanding of the various tools that can be used to represent ratios and that can assist them in solving ratios problems, such as ratio tables and tape diagrams.

Introduce the concept of ratios as multiplicative comparisons between two quantities by presenting the following scenario:

Yesterday, I overheard part of my Dad’s conversation with his friend. I heard Dad say, “The ratio is 10 to 4,” but I didn’t hear the rest of the conversation. What could Dad be talking about? Facilitate a discussion about what ratios are and the situations in which ratios can be used. Some examples of ratios include:

• In data: a sporting team won 10 games and lost 4 games
• In craft: 10 balls of wool to make 4 beanies
• In finance: each day, he spends \$10 on transport and \$4 on food
• In probability: the ratio of red balls to gold balls in a lucky dip is 10 to 4.

Discuss how ratios can be written and what the order of the ratio means in each situation. After that, present the following problem, adapted from the Bar Model Method: In Secondary Mathematics, Lesson 1 from reSolve: Maths by Inquiry. Tim has 15 marbles. 6 of them are yellow and the rest are blue.

What is the ratio of:

• The number of yellow marbles to the number of blue marbles?
• The number of yellow marbles to the total number of marbles?

Prompt discussion by asking students the following questions:

• What is a ratio and how can we represent it?
• What quantities are being compared in the problem and how are these quantities related to each other?
• What unknown quantity do you need to find?
• Can each of these ratios be described as a fraction? Note that the ratio of yellow to blue marbles is a part-to-part ratio and is not described as a fraction, as fractions are not part-to-part. The ratio of yellow marbles to the total number of marbles is a part-whole ratio. This can be adapted to say that six-fifteenths (or simplified as two-fifths) of the marbles are yellow.

Encourage students to suggest how they can represent the information visually, by modelling using coloured counters or tiles or a diagram and have them explain their suggestions.

Enable students to use counters or draw diagrams to create a visual representation of the problem and to find a solution. Introduce and model how to create a tape diagram as a tool to solve proportional problems. For example, introduce students to ratios as quotients by presenting the following scenario:

Tim bought the marbles from a shop. They are sold in bags of four for \$2. • How can you represent this information visually?
• How can you describe the ratio of the marbles to the cost?
• Using this information, what questions can you ask about the price of marbles? Students may ask questions such as, “How much does 20 marbles cost?”  Enable students by prompting:

• How many quantities are being compared in the ratio? Is there a missing quantity you should find first before you compare? Why?
• How do we show the number of yellow marbles, the number of blue marbles and the total?

• Can ratios be simplified like fractions? How can you prove your thinking?

This stage aims to develop students’ understanding that ratios involve relationships between quantities and these quantities vary together. It allows students to think of ratios in two ways - as composed units and as multiplicative comparisons. Students explore how to keep a ratio constant when increasing or decreasing the quantities in a ratio.

1. Ratios as composed unit

Pose the following problem:

Sean is making orange drinks for everyone at camp, by mixing orange cordial with water in a jug. The ratio of orange cordial to water is 4:10.

How much cordial and how much water will be needed to make 2 jugs? How much for 5 jugs? What about half a jug?

Students use Think, Pair, Share to discuss the problem and how they might solve it. Encourage them to note the similarities and differences between their own and their peers’ solutions. Provide coloured counters or tiles for students to model their thinking.

Composed units means thinking of the ratio as one unit and scaling the values up or down. For example, if one jug consists of 4 parts of cordial to 10 parts of water, they may think of this as a unit and think about other multiples that would also be true. In this case, 2 jugs will need 8 parts cordial and 20 parts water; 3 jugs, 12 parts cordial to 30 parts water, and so on. This is iterating (repeating). The composed unit can also be partitioned (breaking into equal sized parts). In this example, half a jug will contain 2 parts cordial and 5 parts water.

Discuss how students can use a ratio table to organise their thinking, such as:

 Number of jugs ¹/₂ 1 2 3 Cups of cordial 2 4 8 12 Cups of water 5 10 20 30

Ask students to describe the relationship. Encourage students to describe the relationship between the ratios by asking:

• What do they notice about the ratio of orange cordial to water in each jug? How are they similar and how are they different?
• What do you notice about the relationships between the amount of cordial and the amount of water as the number of jugs increases or decreases?

Guide students to see that even though the ratios are different, they have the same value. That means that they are equivalent ratios.

Students may notice covariation in the ratios. That means that the two different quantities in a ratio vary together. When the amount of cordial increases, so does the amount of water. And as the amount of water changes, so does the amount of cordial.

Further explore the concept of covariation and equivalent ratios by preparing cards of different numbers of objects, such as the ‘Are they the same?’ cards (see Materials and texts section above) and asking students to find the cards with matching ratios of objects.

2. Ratios as multiplicative comparisons

Explore the multiplicative relationships between two quantities by using the example of camp groups to pose the following problems:

The best tasting refreshment after a long hike while on camp is a drink that has the concentration of 4 cups of orange cordial to 10 cups of water.

Amy added 12 cups of orange cordial into a large keg. How much water should she add to maintain the same taste?

If Amy only has one cup of cordial, how much water should she add to make a delectable drink? Allow some time for students to think about this problem, then engage students in a discussion about how it can be solved. Strategically select students to demonstrate their thinking, prompting them to explain their reasons for their working.

Enable students by encouraging them to use counters or tiles to make a model of the ratio. For example, they may line up orange and blue counters to represent the ratio of orange cordial to water. Prompt students to consider how many blue counters will there be for each orange tile.

Extend students by asking them to think about this: If 4L of orange cordial is mixed with 10L of water, how much orange cordial will be needed for 1L of water? How much water would be needed for 40L of orange cordial?

3. Applying proportional reasoning to solve ratio problems

Using the scenario of making orange cordial drinks, further explore ratios as multiplicative thinking and develop students’ proportional reasoning by posing the following problem:

Sean wants to make orange cordial drinks for the campers with the concentration of 10 cups of water to 4 cups of orange cordial. How many cups of orange cordial will be needed for 15 cups of water? Anticipated student solutions include:

• Recognising a pattern: Students may notice that if 10 cups of water require 4 cups of cordial, then 20 cups of water (another 10 more cups) require 8 cups of cordial (another 4 more cups). If only 5 more cups of water are added, that is half the difference, resulting in 6 cups of cordial.
• Using a ratio table: Students may simplify 10:4 as 5:2. If 15 cups of water are added, that is 3 times more than the ratio, so the number of cups of orange cordial will be 6. • Using tape diagrams to visualise unitising: Students may think of finding the unit rate, or what one part equates to if 10 parts equates to 15 cups. By calculating 15 divided by 10, they could work out that for each part will be 1.5 cups. Seeing that there are 4 parts of cordial, students could multiply 4 by 1.5 to result in 6 cups.  Check for student misconceptions such as using additive thinking rather than multiplicative thinking. For example, students may think that 5 more cups of water were added, and they may proceed to add 5 more cups of cordial, giving the response as 9 cups.

As students are working, monitor their responses and ask students to analyse their solution by proving that the ratio is kept constant when the amount of orange cordial and water vary. Prompt students to make their learning visible using counters, tiles, or diagrams, and guide them to clarify their thinking. Can they see the relationship between the numbers in the new ratio formed?

Enable students by modelling how to use coloured counters to represent the ratio. Ask them to show with counters how much juice and water will be needed for half a batch of orange cordial drink. How many cups of water is in half a batch? Then ask students to show how many cups of orange cordial will be in 15 cups of water.

Extend students by encouraging them to consider a more challenging ratio:

If the orange drink is mixed using 4 cups of water and 1.5 cups of orange cordial, how many cups of orange cordial will be needed for 10 cups of water?

### Areas for further exploration

1. Ratios in the Classroom

Students conduct surveys to investigate ratios in the classroom (links to a downloadable document). For example, the ratio of students who walk to those who use a form of transport.

2. The Bar Model Method

The Bar Model Method unit from the Resolve teaching resources consists of eight lessons using the bar model, a tool used to solve problems in many topics in Mathematics, including ratios. Six lessons show how the bar model can be used as a tool to assist in solving problems involving fractions, ratios, and percentages. Two lessons can be used as part of introductory algebra.

This stage focuses on developing students’ concept of ratios and how quantities can vary while the ratios are kept constant.

Opportunities to track students’ learning progress through formative assessment include student self-reflection in a learning journal or using exit tickets to capture students’ self-evaluation of their learning against the success criteria.

Prompts for exit tickets (see Materials and texts section for template) could include:

• What helped me understand was…
• What I want to know next is…
• I didn’t know … before, now I know... To assess students’ ability to think of ratios multiplicatively instead of additively, present a problem that allows students to explore the distinction between multiplicative thinking and additive thinking.

This activity is inspired by the Adding Parts problem within the Paint With Numbers learning sequence from reSolve: Maths By Inquiry.

Carly is making an orange drink for a party. The suggested recipe is 2 parts of orange cordial to 3 parts of water. Carly added 2 L of cordial and 3 L of water into a large keg, making 5 L altogether.

However, she realised that she needed to make 7 L of orange drink. She decided to add another 1 L of cordial and another 1 L of water into her 5 L of orange drink to make 7 L.

Will this new batch of drink have the same concentration as the original 5 L batch? Why or why not? Ask students to share their thinking and justify their reasoning.

Enable students by encouraging them to use coloured counters or tiles as a visual representation. Prompt students to think about how much water will be needed for 1 L of cordial while maintaining the ratio. Model how to use counters to work this out. When students have worked out that for 1 L of cordial, 1.5 L of water is needed, then ask them to consider if Carly’s actions will result in the same concentration of drink.

Extend students by asking students to work out how much orange cordial and water should Carly have added to make 7 L while maintaining the ratio. To make an extra two litres of drink in the same concentration, Carly should have added 800 mL of orange cordial and 1.2 L of water.

Australian Academy of Science, 2020. Paint with Numbers. [Online]
Available at: https://www.resolve.edu.au/paint-numbers
[Accessed 15 March 2022].

Australian Academy of Science, 2022. Bar Model Method: In Secondary Mathematics. [Online]
Available at: https://www.resolve.edu.au/bar-model-secondary-maths?lesson=1650
[Accessed 13 April 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].

NSW Government, n.d. Exit tickets. [Online]
Available at: https://app.education.nsw.gov.au/digital-learning-selector/LearningActivity/Card/543
[Accessed 15 March 2022].

State Government of Victoria, (Department of Education and Training), n.d. Ratio in the Classroom. [Online]
Available at: http://www.education.vic.gov.au/Documents/school/teachers/teachingresources/discipline/maths/ttratioclassroom.doc
[Accessed 15 March 2022].