Equivalent and simplest ratios

Classroom resources

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Students will be able to

• I can find an equivalent ratio by multiplying or dividing both terms by the same number.
• I can write a ratio in simplest form.
• I can scale a recipe up using equivalent ratios.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 8 min

Before class

Do the Run board yourself first. Students need equivalent fractions (the same multiplying and dividing) and highest common factors.

The one idea: equivalent ratios are copies of the pattern: both terms multiplied (or divided) by the same number. The common mistake is additive thinking (Ana: 2 → 6 is +4, so 3 → 7). This lesson is the hinge for sharing in a ratio: students who cannot scale and simplify will struggle next.

Link to equivalent fractions: 2 : 3 behaves like 2/3 for scaling, but a ratio is not a fraction (part to part).

Running the lesson

1. Predict: students who chose 4 : 5 added 2 to both terms.
2. Run: ask how many blocks of each colour with 3 copies, then how the numbers are related to 2 and 3.
3. Investigate: ask students to draw Ana's recipe as blocks.
4. Modify: for the impostor, ask what the multiplier would have to be.
5. Make: collect the multipliers used: all give the same ratio.

The hard part (on purpose)

Multiplying both terms, not adding the same amount to both. Ask 'how many copies is that?' before any arithmetic.

When students get stuck

Answers to the checks

1. which is the same as 2 : 3?: Which ratio is equivalent to $2 : 3$?
   4 : 6. Both terms are doubled. 4 : 5 and 3 : 4 each add the same number to both terms, which changes the ratio.
2. copies of a ratio: Make 3 copies of 2 : 3. What is the new ratio?
   
   Three copies give 6 : 9. Each term is multiplied by 3.
3. Fill in the blank: Fill in the blank: $2 : 3 = 10 : ?$
   15
   2 → 10 is × 5, so 3 → 3 × 5 = 15.
4. Find the mistake: Ana scales a recipe. Click the line where her working goes wrong.
   line 4
   Line 4: adding 4 to the milk too is wrong. Flour went × 3 (2 to 6), so milk goes × 3: 3 × 3 = 9 cups.
5. Simplify: Simplify $12 : 18$. It is $2 : ?$
   3
   The highest common factor of 12 and 18 is 6. 12 ÷ 6 = 2 and 18 ÷ 6 = 3.
6. Simplest form: The board shows 4 : 6. Change it to the simplest ratio equal to 4 : 6.
   
   4 : 6, divide both by 2, gives 2 : 3.
7. Divide by ten: Simplify $20 : 30$. It is $2 : ?$
   3
   Divide both terms by 10: 2 : 3.
8. Scale the recipe: A recipe uses 2 cups of flour for every 3 cups of milk. How many cups of milk go with 10 cups of flour?
   15
   × 5: 2 : 3 = 10 : 15.
9. Spot the impostor: Which of these is NOT equivalent to $6 : 8$?
   9 : 11
   6 : 8 → 9 : 11 adds 3 to both terms. 6 → 9 is × 1.5, but 8 × 1.5 = 12, not 11.
10. Divide by five: Simplify $15 : 25$. It is $3 : ?$
    5
    Divide both by 5: 3 : 5.
11. Multiply: Fill in the blank: $5 : 2 = ? : 8$
    20
    2 → 8 is × 4, so 5 × 4 = 20.
12. Pink paint: Pink paint is 3 parts red to 1 part white. How many parts of white go with 12 parts of red?
    4
    3 → 12 is × 4, so white is 1 × 4 = 4.
13. Review: part to whole: Review: the ratio of blue to yellow is $3 : 4$. Blue to the whole is $3 : ?$
    7
    3 + 4 = 7, so blue to the whole is 3 : 7.
14. big copies of 3 : 5: Make a ratio equal to 3 : 5 with both terms bigger than 20.
    
    7 copies give 21 : 35, 8 copies give 24 : 40.

Solutions

One possible solution for each task. Other correct programs are fine.

scale and simplify › Simplest form

Open task: any program that does what the step asks.

scale and simplify › Divide by ten

Open task: any program that does what the step asks.

scale and simplify › Scale the recipe

Open task: any program that does what the step asks.

scale and simplify › Spot the impostor

Open task: any program that does what the step asks.

big copies of 3 : 5

Open task: any program that does what the step asks.

Stretch and support

Stretch: A ratio is 2 : 3. Find all ratios equivalent to it with both terms less than 30, and say how you know you found all of them.

Support: Keep the copy blocks on the board and count the groups.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Equivalent and simplest ratios".

The idea: Equivalent ratios and simplification: I can Students find equivalent ratios by scaling up and down, and express a ratio in simplest form.

What understanding looks like:
- Correctly simplifies a ratio to lowest terms.
- Generates equivalent ratios by multiplying or dividing both parts by the same value.
- Does not confuse simplifying a ratio with subtracting the same value from both parts.

Mistakes students often make here:
- A student adds the same number to both terms (Ana's slip).
- A student simplifies one term only.
- A student cannot find the multiplier.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Ratio as a relationship between quantities), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Equivalent and simplest ratios".

Exit ticket

Simplify 24 : 36, and find the missing term: 3 : 5 = 12 : ?

24 : 36 = 2 : 3. 3 → 12 is × 4, so 5 × 4 = 20.

What each step is for

• which is the same as 2 : 3?: Committing shows whether they multiply both terms (4 : 6) or add the same amount to both (4 : 5).
• copies of a ratio: Copies of the same pattern of blocks keep the ratio; the number of copies is the multiplier.
• multiply or divide both: Equivalent ratios multiply both terms by the same factor; Ana adds. Simplifying divides both terms by the same factor.
• scale and simplify: Simplifying on the board, simplifying by hand, scaling a recipe and spotting an impostor use the same rule from several angles.
• equivalent ratios: Simplify, scale up, and scale by a factor found from one term; the review keeps part to whole fresh.
• big copies of 3 : 5: Finding copies of 3 : 5 with both terms over 20 shows the multiplier is what keeps the ratio, and many copies work.

Where it leads

Next: Sharing in a ratio, which uses equivalent ratios to split a total.

Where it leads: rates and unit rates, scale drawings, similar shapes, and proportion in Years 8 and 9.