Suggested 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

This stage consolidates students’ understanding of part-to-part ratios and supports students to apply their understanding of ratios as multiplicative comparisons to investigate the effect of adding parts to a ratio. It gives students the opportunity to make connections between ratios and fractions when comparing two ratios.

Begin by watching the ClickView video, Understanding Ratios and Proportions (sign into ClickView using your department credentials). Discuss the example of part-to-part ratios in the video (ratios of blue and white paint) and part-whole ratios (the proportion of yellow lolly in the packet).

Present the following scenario:

“When purchasing paint, a customer selects from colour palette cards and the paint shop assistant mixes the colour. The assistant follows a formula for each colour and mixes the colours according to a ratio so that they get the colour exactly right.

Today you are the paint shop assistant. A customer has asked you to mix blue and white paint to make 5 different tints of pale blue. They want to know the ratios you use so they can have more of the same colour made later if needed."

Students record the amount of blue and white paint they used for each tint on the 'Creating a paint palette' student sheet (see Materials and texts section above). Students can use a teaspoon to measure and add paint, with one teaspoon of paint representing ‘one part’.

As students are mixing the paint, prompt them to consider the relationship between quantities as a ratio and to think about how they can use these ratios to determine the ‘blueness’ of the paint tint, that is, if the tint is lighter or darker. Examples of prompting questions to pose include:

- What do you notice about the colour tint in each mixture?
- How can you express the relations between the two colours as a ratio?
- What is happening to the amount of blue paint and the amount of white paint as the tint of colour changes?
- What ratio of blue and white paint should give the same tint as…? How can you increase or decrease the amount of paint without altering the colour tint?
- What do you notice about the ratio as the tint of colour varies?
- What questions can you pose?

Students record and explain their thinking in their maths learning journal and share their ideas with the class. Observe students as they are answering the questions and justifying their thinking. Can they explain which component of the mixture should have a higher amount to create a darker tint? Can they form a connection between the tint of colour and the ratio?

**Enable** students by asking them to systematically increase the amount of white paint while keeping the portion of blue paint the same and describing what is happening.

**Extend** students by challenging them to create the same tint of colour if the amount of blue paint is halved or if a fractional quantity of blue paint is used, such as ¾ teaspoon.

Present students with the ratio of paint of two different tins:

Ask students to compare the two tins of paint and determine if one tin has a darker tint of blue than the other or if they both have the same tints.

Observe students’ approach as they solve the problem, as there are several ways in which the comparison can be made. Examples of strategies that students may use include:

- Figuring out how much white paint is needed to mix one part of blue paint (finding the unit rate or unitising).
- Comparing ratios as fractions - blue paint compared to white paint (⅔ compared with ¾).
- Comparing blue paint as a fraction of the total (two-fifths compared with three-sevenths).
- Converting to percentages.
- Using multiples of one or both tins until either the amounts of white paint or the amounts of blue paint are equal.

Encourage students to solve the problem in different ways. Allow students to reflect on each approach and determine which strategy they found preferable and why. Observe and record the approach students use to solve the problem for formative assessment purposes, as described in the Reflect and consolidate phase.

Select and invite students to share their ideas with the class, prompting them to explain their reasoning. Promote class discussion with the Number Talk approach. It is important that students are given an opportunity to share their strategy regardless of whether the solution is correct or not. It may be useful to assure students that while they are watching their peers’ solutions, they are welcome to change their minds. This encourages students to take learning risks and promotes collaborative learning.

Encourage other students to add on, respectfully challenge, question and critique their peers’ approaches by asking questions such as:

- Do you agree or disagree with this method?
- Why do you think so?
- What would you do differently?
- How can you improve on this strategy?
- Can you build on what ...is saying?

**Enable** students by adjusting the amount of each component of the mixture so that comparing between ratios is made simpler. The following are some examples of how each amount can be modified:

- If Tin B consists of 4 parts of blue paint and 6 parts of white paint, the comparison will be simpler as both ingredients are twice as much as Tin A and therefore the mixtures have equal concentrations.
- Altering the ratio of blue and white paint in Tin B to 4 parts of blue to 5 parts of white. When comparing part-to-whole fractions, students can double the amount of blue paint in Tin A and compare the amount of white to determine which tin is lighter blue.
- Tin A is made up of 2 parts of blue and 5 parts of white and Tin B is made up of 5 parts of blue and 10 parts of white paint. In this example, the process of unitising is simplified (2.5 parts of white paint for every one part of blue in Tin A compared with 2 parts of white paint in Tin B).

**Extend **students by adapting fractional values to the components of the mixture.

**Areas for further exploration**

- An alternative to mixing colour tints is comparing and modifying cooking recipes. Nana’s Chocolate Milk consists of a series of videos that presents students with a scenario and challenges students to work out the amount of ingredients needed for a recipe.
- The Mixing Paint and Mixing More Paint activities on Nrich Maths challenges students to mix pots of paint to create the target shade of pink paint.

This stage explores the concept of simplifying, comparing between and finding equivalent ratios.

Opportunities for formative assessment of student learning in this stage of the sequence include capturing how students' reason to solve the problem. As students are exploring each problem, encourage them to write down their explanation for their reasoning or to ask them to describe two different ways they could arrive at their solutions.

Check for student understanding of the concept by having them consider the following concept cartoon, then ask them if they agree or disagree with the statements made by Harry and Mariam. Why or why not?

ClickView, 2009. *Understanding ratio and proportion. *[Online]

Available at: https://clickv.ie/w/gtYl

[Accessed 15 March 2022].

Meyer, D., n.d. *nana's chocolate milk. *[Online]

Available at: http://threeacts.mrmeyer.com/nana/

[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), 2019. *Number Talks Lesson Demonstration. *[Online]

Available at: https://vimeo.com/303392767/baa9419718

[Accessed 15 March 2022].

University of Cambridge, n.d. *NRICH: Mixing More Paints. *[Online]

Available at: https://nrich.maths.org/4794

[Accessed 15 March 2022].

University of Cambridge, n.d. *NRICH: Mixing Paints. *[Online]

Available at: https://nrich.maths.org/4793

[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

4. Finding the Better Deal

EXPLORESuggested 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