Real World, Real Numbers: Ratio and Proportions

# 2. Finding the Unknown in Ratios Suggested 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
• Computer, laptop or tablet
• Coloured counters or tiles

This stage introduces students to the application of proportional reasoning in the construction industry to further demonstrate that ratios and proportions are used in everyday life. This stage focuses on the concept that ratios involve multiplication and division rather than addition and subtraction. It supports students to apply multiplicative thinking to solve ratio problems and to find unknown quantities of a ratio.

Show students the Working with Ratios video clip on ClickView (sign in using your department credentials) to introduce the problem. The example of adhesive mixture as seen in the video clip will be used as a context for this activity.

Present the following problem:

A mixture of adhesive was created using 8 parts of primer and 5 parts of solvent. If 24 mL of primer is used, how much solvent needs to be added?

Ask students to create a visual representation of the ratio. Facilitate students to Think, Pair, Share when approaching the problem and encourage them to note the similarities and differences between their own and their peers’ solutions.

In this example, the sketch of a tape diagram could look like this: In observing the model, students may realise that the need to divide 24 mL by 8, indicating 3 mL for each partition. Therefore 15 mL of solvent needs to be added. Pose other questions to encourage students to find unknown quantities in an adhesive mixture.

If the total amount of adhesive is 26 mL, how much solvent is there?

If there are 6 mL more primer than solvent, how much primer is there?

If a mixture of adhesive is made up of 12 mL of solvent and 20 mL of primer, what could the primer to solvent ratio be?

Enable students by first establishing their understanding of the unit of measurement, if necessary.

Suggest that students draw tape diagrams and support them to mark the parts of the ratio that they already know. Some questions to ask students while they are completing the problem are:

• Can you create a model or draw a tape diagram to show the information in the problem?
• What are the parts?
• How can you mark the information and the unknown on the tape diagram?

Prompt students to calculate what one part will equate to and support them to use counters to model the problem.

The ratios can also be modified so the comparison of the two quantities can be simplified. For example, if the ratio of primer to solvent is 8:4, students may be able to visualise that the amount of solvent is half that of the primer. When calculating how much solvent for 24 mL of primer, students will be able to find half of the amount.

Extend students by including decimal fractions. For example:

• If 28 mL of primer is used, how much solvent is needed?
• If there is 13.5 mL more primer than solvent, how much primer is there?

1. Finding the unknown in part-to-part ratios

Present students with the following scenario, as shown in the video clip in the Get started section:

In a rendering mix, the ratio of ingredients is 6 parts sand, 1 part cement and 1 part lime.

Using this information, can students determine:

• the amount of cement and lime needed if 18 kg of sand is used?
• the amount of sand and lime if 2 kg of cement is used?
• the amount of cement and lime used if there is 10 kg more sand than lime in the mixture? Elicit a class discussion to share students’ strategies in solving the problem.

Enable students by supporting them to use a tape diagram or coloured counters or tiles to help them solve the problem. Ask students to mark the known and unknown parts on their diagrams, or model using concrete materials such as Unifix blocks or counters to represent parts of the ratio.

Extend students by altering the given amount of the ingredients, so that when divided into each part the result is not a whole number. For example, ask, what is the amount of cement and lime needed if 15 kg of sand is used?

2. Finding the unknown in part-whole ratios

Present students with a problem that develops their concept of dividing quantities in ratios. For example:

"An apprentice builder knows that a concrete mix consists of 1 part cement and that the proportion of gravel is more than that of sand.

If the total weight of dry materials needed is 36 kg, what might the ratio of dry materials be? What amount of each dry material do they need? Find as many solutions as possible."

Provide students some time to think about the problem individually and consider the strategies that they are going to employ to solve the problem. Will they draw a diagram? Will they divide the total weight into parts mentally? Encourage students to work collaboratively with a partner and share their ideas with each other. While students are working, observe their approach to the problem. Do students use a tape diagram to help them solve the problem? What strategies do they use? Encourage students to explain their thinking.

Encourage students to annotate their work, explaining how they solved the problem. These student work samples can be used as formative assessment, as explained in the Reflect and consolidate phase.

Enable students by:

• providing them with the ratio of the materials as 1 cement: 5 sand: 6 gravel and asking them to work out how much of each material is needed if the total weight of materials is 36 kg
• asking, "If cement, sand and gravel have equal proportions, what will the ratio be? How can you divide 36 kg into the ratio?"
• encouraging students to use counters to help them find other ratios for the materials, where 1 counter represents 1 kg. What might the ratio of materials in the mixture be, so that it totals 36 kg?
• supporting students to find the volume of water needed in the mixture. Ask them to first find out how much water is needed for 16 kg of dry materials. What about for 24 kg of dry materials? Do they see a pattern?

Extend students by:

• setting further limits on the ratio of dry materials. For example, the concrete mix consists of 1 part cement and there is 20% more gravel than sand. What will be the ratio of dry materials and what is the weight of each material?
• asking students to solve a problem, shown in the example from the video clip: “For every 8 kg of dry materials, 3200 mL of water is needed. How many litres of water will be needed for the concrete mix if 24 kg of dry materials are used? How much water will be needed for 6 kg of dry materials?”

### Areas for further exploration

1. ClickView Working with Ratios follow up activities

ClickView provides access to additional resources and activities for teachers and students as a follow up from the Working with Ratios video clip. Students explore ratios in Mixing Mortar and Concrete and Mixing Paint. Sign into ClickView using your department credentials.

2. Use Thinking Blocks to solve problems

To further consolidate the idea of using tape diagrams to solve problems, invite students to explore proportional situations with tape diagrams using Thinking Blocks.

3. Ratios and Double Number Lines

Double number lines are similar to tape diagrams, but may not show partitions. Introduce students to the use of ratios and double number lines when they are confident with solving problems with tape diagrams with this activity from Khan Academy.

As indicated in the Go deeper section above, formative assessment of student learning could include examining student work samples and observing students’ explanation of their reasoning. To check for student understanding of calculating quantities into ratios, pose questions to allow students to develop their proficiency of reasoning., such as:

Students in 7B were working on a problem:

"Andy decided to split his weekly pay into three purposes - transport, savings and entertainment into the ratio 2:4:6. If he saves \$240 each week, how much money does he spend on transport and entertainment and how much is his weekly pay?"

Below are two students’ responses to the problem.

Student A used tape diagrams as a tool to solve the problem and Student B used a ratio table.

What feedback would you give to each student? Enable students by asking them to identify the misconceptions of the students in the example. Encourage them to explain what went wrong or what was correct about the examples.

Extend students by asking them to list questions they could ask the students in the example that would probe their thinking about the problem.

Clickview, 2020. Working with Ratios. [Online]
Available at: https://www.clickview.com.au/curriculum-libraries/video-details/?id=26444801&cat=3708536&library=secondary
[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].

Khan Academy, n.d. Ratios and double number lines. [Online]