Ratio and Proportion Problem-solving

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1 Getting Started
2 Watch and Notice
3 Try It Together
4 Note Your Method in Your Copy
5 Class Challenge
Pupil practice

1 - Getting Started ~4 mins

Illustration for Getting Started

Key point

Here is a puzzle: 36 cookies are shared between two classes in the ratio 4:5. What is the very first thing you would work out before you can share a single cookie?

Teacher Notes

Take two or three hands-up answers, not open call-outs. You are only fishing for the first step — most pupils will jump to the whole answer, so redirect: what do you have to know before you can share anything?

Listen for someone naming the total number of parts (9) — that is the doorway into the whole lesson. Don't solve the puzzle here; the sharing idea returns straight away in the next step.

👇 Mark your steps as done as you complete them.

+5 XP

2 - Watch and Notice ~9 mins

Three tools solve ratio and proportion problems, and the real skill is picking the right one: sharing when a total is split by a ratio, the unitary method when you find what one is worth then multiply up, and scaling when everything grows by the same factor. Read the question first, decide which kind it is, then choose the tool. We will watch two worked on the ratio bars — look at how each one is cut up before anything is worked out.

Teacher Notes

Sharing first: split 36 in ratio 4:5. Add the parts before sharing, 4 + 5 = 9, so the bar cuts into 9 equal pieces. Ask what is one part worth before revealing one part is 4; shares are 16 and 20.

Unitary next: 5 tickets cost €17.50. Point out the bars are five equal single-ticket units, not a ratio split — the single unit is the pivot. Point at one unit and ask what is one ticket worth before revealing it: divide by five, one ticket is €3.50, then multiply up, eight tickets is €28.

Finish by asking the class to name the method each needs: the cookie hook is sharing, the ticket one is unitary. That contrast is the point.

👇 Mark your steps as done as you complete them.

+10 XP

3 - Try It Together ~10 mins

Let's work this one together in two clear steps, and we will say out loud which method we are using before each step.

Worked example

Step 1: A smoothie recipe for 3 people uses 5 strawberries. We want enough for 9 people. How many strawberries do we need?

Step 2: Now we take those strawberries we just worked out and share them between this smoothie and a second recipe in the ratio 2:3. How many strawberries go to each recipe?

Scale to 15, then share 15 in the ratio 2:3

Teacher Notes

Talk this one through together — pupils take turns at the board and the class agrees or corrects out loud.

This is a mixed problem on purpose: the first step is scaling (× 3 because 9 is three times 3, giving 15 strawberries), and the second step is sharing (the 15 strawberries split 2:3 → 5 parts, one part is 3 strawberries, so the shares are 6 and 9 — both whole). Make it clear the 15 shared in step 2 is exactly the 15 worked out in step 1, nothing new. Set the ratio bars to share the 15 as 2:3 and read the two shares straight from the bars.

Name the method before each step — that naming is the skill being practised.

Watch for the pupil who scales only the strawberries and forgets the other ingredients would scale too; head it off by asking what else in the recipe changes?

+10 XP

4 - Note Your Method in Your Copy ~2 mins

COPYBOOK MOMENT

In your maths copy, for each problem write the method you chose at the top (sharing, the unitary method, or scaling), then your working line by line. Box the final answer.

Worked example

8 identical notebooks cost €12. How much do 5 notebooks cost?

Name the method first, then write the working step by step underneath. Box the final answer when you finish.

Teacher Notes

Walk the room glancing at the method-name each pupil wrote at the top, that is what you are checking, not the arithmetic. This is whole-class copybook practice, not marking. If a pupil has skipped naming the method, prompt them to add it before working.

Board the example with them end to end. Method: the unitary method (find one, then multiply up). One notebook is €12 ÷ 8 = €1.50. Five notebooks is €1.50 × 5 = €7.50. Box €7.50.

Slip to watch: pupils who try to scale 8 straight to 5 and stall on the awkward factor. Point them back to one notebook first, then multiply.

+10 XP

5 - Class Challenge ~11 mins

Key point

Today we crack a five-clue mystery. Each clue is a ratio, proportion or scaling problem, and every answer feeds straight into the next clue. The clues get harder as we go: clue 1 is a straight share, clue 5 needs two methods in one. Solve clue 1 to unlock clue 2.

We crack clue 1 together first so everyone sees how the chain works. Then pupils take turns at the board for clues 2 to 5. Before each clue, name the method out loud (sharing, unitary, or scaling).

Clue 1 (worked together)

€36 is shared between Ana and Ben in the ratio 5:7. How much money does Ana get?

Method: sharing. Add the parts: 5 + 7 = 12 parts. One part is €36 ÷ 12 = €3. Ana's share is 5 × €3 = €15.

Write €15 at the top of the chain. That number unlocks clue 2.

Clue 2

Ana spends her €15 on buns. 5 buns cost €2.50. How many buns does she get with €15?

Clue 3

Those buns would feed 6 people. How many buns are needed to feed 9 people at the same rate?

Clue 4

Share that new total of buns between girls and boys in the ratio 2:3. How many buns do the girls get?

Clue 5 (two methods)

The girls use their buns for a party. A tasting tray of 6 buns served 8 guests. First scale up to find how many times bigger their bun total is than 6, then use that factor to find how many guests they can feed. How many party guests?

Keep the running chain on the board: each answer unlocks the next clue. No numbers come from outside the chain.

Clue 1 (sharing): A €20 prize is shared between two friends in the ratio 3:1. How much does the friend with the larger share get?
Clue 2 (the unitary method): One bun costs 50c. Your budget is your answer from clue 1. How many buns can you buy? (Find how many buns one euro buys, then multiply up.)
Clue 3 (scaling): Your buns from clue 2 are a recipe that feeds 6 people. You now want to feed 9 people. 9 is one and a half times 6, so scale your number of buns up by the same factor. How many buns do you need?
Clue 4 (sharing): Share your buns from clue 3 between the boys' table and the girls' table in the ratio 3:2. How many buns go to the girls' table? (Five parts in all, so find what one part is worth first.)
Clue 5 (scaling then the unitary method): The party is 4 times bigger than the girls' table, so scale the girls' buns from clue 4 up by 4 to find the total for the whole party. Every guest eats exactly 3 buns. How many guests are at the party? (Use the unitary method.)

Answers & strategies (teacher)

Clue 1 (sharing): A €20 prize is shared between two friends in the ratio 3:1. How much does the friend with the larger share get? — €15
Clue 2 (the unitary method): One bun costs 50c. Your budget is your answer from clue 1. How many buns can you buy? (Find how many buns one euro buys, then multiply up.) — 30 buns
Clue 3 (scaling): Your buns from clue 2 are a recipe that feeds 6 people. You now want to feed 9 people. 9 is one and a half times 6, so scale your number of buns up by the same factor. How many buns do you need? — 45 buns
Clue 4 (sharing): Share your buns from clue 3 between the boys' table and the girls' table in the ratio 3:2. How many buns go to the girls' table? (Five parts in all, so find what one part is worth first.) — 18 buns
Clue 5 (scaling then the unitary method): The party is 4 times bigger than the girls' table, so scale the girls' buns from clue 4 up by 4 to find the total for the whole party. Every guest eats exactly 3 buns. How many guests are at the party? (Use the unitary method.) — 72 buns ÷ 3 buns each = 24 party guests

Teacher Notes

Practice round. Model clue 1 fully with the class, then pupils take turns at the board for clues 2 to 5. Keep board work brisk. Before each clue, ask which method it needs.

Clue 1 demonstration (sharing) → €15. Parts: 5 + 7 = 12. One part = 36 ÷ 12 = 3. Ana = 5 × 3 = €15. Start the visible chain on the board now.

Clue 2 (unitary) → 30 buns. One bun = 2.50 ÷ 5 = €0.50. Buns = 15 ÷ 0.50 = 30.

Clue 3 (scaling) → 45 buns. 9 is ×1.5 of 6 (or ÷6 ×9). 30 × 1.5 = 45. Slip to watch: pupils who add 3 people and only add 3 buns.

Clue 4 (sharing) → 18 buns. Parts 2 + 3 = 5. One part = 45 ÷ 5 = 9. Girls = 2 × 9 = 18.

Clue 5 (scaling) → 24 guests. Scale factor: 18 ÷ 6 = 3. Then multiply up the guests: 8 × 3 = 24. Name the method before anyone calculates.

Running chain to keep visible: €15 → 30 buns → 45 buns → 18 buns → 24 guests. Every number comes from the clue before it.

+15 XP

Pupil practice

Module 4 · Ratio and Proportion Measures

Lesson 46 · Ratio and Proportion Problem-solving

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Teacher Class Feed

1 - Getting Started ~4 mins

2 - Watch and Notice ~9 mins

3 - Try It Together ~10 mins

Scale to 15, then share 15 in the ratio 2:3

4 - Note Your Method in Your Copy ~2 mins

5 - Class Challenge ~11 mins

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