Ratio and Proportion: Sharing, Scaling and Best Buys
GCSE MATHS RESIT · EDEXCEL FOUNDATION · 3 OF 7
Ratio and proportion
Sharing fairly, scaling recipes and spotting the best deal. A quarter of the Foundation marks live here.
Paper 1 · Paper 2 · Paper 3
Lesson duration is: 60 minIntroduction
Items in this lesson
GCSE MATHS RESIT · EDEXCEL FOUNDATION · 3 OF 7
Ratio and proportion
Sharing fairly, scaling recipes and spotting the best deal. A quarter of the Foundation marks live here.
Paper 1 · Paper 2 · Paper 3
Lesson 3 of 7, about 60 minutes. Timings: do now 5 min, word web 3 min, sharing in a ratio 15 min, best buys 10 min, recipes and exchange rates 12 min, exam question and exit 15 min.
Ratio and proportion is about 25% of the Foundation marks, and most of it comes down to one move: find the value of one part (or one item) first.
Device-free fallback for the whole lesson: every activity works on mini whiteboards or paper; the notes on each slide say how. Image credit: Daniel Seßler / Unsplash
DO NOW · 4 MINUTES
Four from last lesson
Work out 10% of £45
£4.50
Work out 3/4 of 60
45 (60 ÷ 4 × 3)
Write 0.3 as a percentage
30%
Increase £20 by 10%
£22 (× 1.1)
Students answer on paper as they arrive. Click four times to reveal the answers.
If last lesson's exit poll showed one weak skill, add a fifth question on it.
0.3 as a percentage is a common trip: some write 3%. 0.3 = 3 tenths = 30 hundredths = 30%.
Device-free: questions on the board before the lesson.
Where do you meet ratio outside maths?
Three minutes. Let students type freely; then pick out the useful ones.
Expect: recipes, mixing squash or paint, map scales, betting odds, exchange rates, sharing a bill, gear ratios, screen sizes (16:9), hair dye.
The point: ratio is something they already do. The exam just asks them to write it down.
Device-free: pairs list as many as they can in one minute, then share.
SHARING IN A RATIO
Ali and Bea share £45 in the ratio 2 : 3
Ali
£9
£9
Bea
£9
£9
£9
5 equal parts = £45
1 Add the parts: 2 + 3 = 5
2 One part: £45 ÷ 5 = £9
3 Ali: 2 × £9 = £18
Bea: 3 × £9 = £27
Check: £18 + £27 = £45
Every box is the same size. That is the whole trick of the bar model.
The bar model is drawn to scale: five boxes of equal width, two for Ali and three for Bea.
Four clicks: total parts, one part, both shares, the check.
The big misconception: dividing £45 by 2 and by 3 (giving £22.50 and £15). Ask: "Do those two add up to £45?" (No: £37.50.) The check step catches it every time.
Device-free: draw the bar on the board; students copy it into their books, because they should draw one in the exam too.
Sam and Kai share £60 in the ratio 2 : 3
£60 altogether
Sam · 2 parts
Kai · 3 parts
One part
Sam gets
Kai gets
£36
£20
£12
£40
£24
£30
Three of the amounts are traps. Find one part first.
The bar is to scale: five equal boxes, two teal for Sam and three amber for Kai.
Answers: one part £12 (60 ÷ 5). Sam £24 (2 × 12). Kai £36 (3 × 12).
Traps: £30 (halving £60, ignoring the ratio), £20 (dividing by 3), £40 (two-thirds of £60, treating 2 : 3 as a fraction 2/3).
Device-free: draw the bar on the board, students write the three answers on whiteboards.
BEST BUYS
Which pack is better value?
6 cans for £3.90
£3.90 ÷ 6 = £0.65
65p a can
4 cans for £2.80
£2.80 ÷ 4 = £0.70
70p a can
The 6-pack is better value. Price of one is the fairest comparison.
Three clicks: price per can for each pack, then the verdict.
Exam wording: "Which is the better buy? You must show your working." A correct choice with no working scores nothing. Students must show the two unit prices and then say which is better.
Another route: compare 12 cans of each (£7.80 vs £8.40). Any fair comparison earns the marks.
Device-free: whiteboards.
Which is better value?
500 g for £2.40
750 g for £3.30
They are the same value
Answer: 750 g for £3.30.
Price per 100 g: £2.40 ÷ 5 = 48p, and £3.30 ÷ 7.5 = 44p. The bigger bag is cheaper per 100 g.
Non-calculator route: 250 g of the small bag costs £1.20, so 750 g would cost £3.60. The big bag is £3.30, so it is better.
Common slip: "the cheaper bag is the better buy". Cheaper is not the same as better value.
PROPORTION · RECIPES
Pancakes for 4. You need 6.
Ingredient
For 4
For 2 (÷ 2)
For 6 (× 3)
Flour
120 g
60 g
180 g
Eggs
2
1
3
Milk
300 ml
150 ml
450 ml
Scale down to an easy number first, then up to what you need. Going straight from 4 to 6 is × 1.5, which is harder in your head.
Two clicks: the "for 2" column (halve everything), then the "for 6" column (times 3).
This is the unitary method in disguise: find a small, easy amount first. For awkward numbers, go all the way to 1 person.
Common slip: adding 2 to everything ("6 is 2 more than 4, so add 2"), giving 122 g of flour. Ask: "Would 122 g feed 2 more people?"
Device-free: same table on the board. Image credit: Duncan Kidd / Unsplash
Find one, then scale
1. A recipe for 4 people uses 200 g of pasta. For 1 person you need 50 g, so for 10 people you need 500 g.
2. Squash is 1 part cordial to 4 parts water. In 500 ml of drink there is 100 ml of cordial.
3. 3 notebooks cost £4.50, so 7 notebooks cost £10.50.
Answers: 50, 500, 100, £10.50.
Working: 200 ÷ 4 = 50, × 10 = 500. Squash: 1 + 4 = 5 parts, 500 ÷ 5 = 100 ml. Notebooks: £4.50 ÷ 3 = £1.50 each, × 7 = £10.50.
Distractors match slips: 40 (dividing by 5), 800 (multiplying by 4), 125 (dividing 500 by 4, forgetting the cordial is a part too), £9.50 (a near miss).
Device-free: board and whiteboards.
EXCHANGE RATES
£1 = €1.15
£
€
× 1.15
÷ 1.15
£200 is €230
€69 is £60
Sense check: each pound buys more than one euro, so the euro number is always the bigger one.
The rate is made up for practice; real rates change daily. The exam always gives the rate.
Two clicks: pounds to euros (multiply), then euros to pounds (divide).
The sense check is the most useful habit here: if converting pounds to euros gives a smaller number, it has been done the wrong way round.
Device-free: draw the two boxes and arrows on the board. Image credit: Lukasz Radziejewski / Unsplash
£1 = €1.15. How much is €92 in pounds?
£80
£105.80
£92
£77
Answer: £80 (92 ÷ 1.15).
Slips: £105.80 = multiplied instead of dividing (fails the sense check: euros to pounds should get smaller). £92 = no conversion. £77 = took away 15.
Calculator question; on Paper 1 the numbers would be friendlier.
EXAM-STYLE · PAPER 1 · NO CALCULATOR
Mia and Leo share £84 in the ratio 3 : 4.
How much more money does Leo get than Mia?
(3 marks)
4
minute
00
second
Sketch your bar model here first
4 minutes on the timer. Students work on paper first, then answer on the next slide.
Mark scheme style: M1 for 84 ÷ 7 (= 12). M1 for 3 × 12 and 4 × 12 (36 and 48), or for seeing the difference is one part. A1 £12.
Quick route worth showing afterwards: Leo has exactly one more part than Mia, so the difference is one part = £12. Strong students will spot it; everyone else gets there with the full method.
Common slip: answering £48 (Leo's share) instead of the difference. Underline "how much more" when reading the question.
Device-free: same question on paper; the teacher keeps time.
How much more does Leo get than Mia?
Answer: £12. Accepted forms: 12, £12, 12.00, £12.00.
If many answered 48 or 36, go back to the question and underline "how much more".
Device-free: whiteboards.
How do you feel about ratio now?
I could explain it to someone
Fine with the bar model
Need more practice
Still lost
Exit poll. Anyone choosing "still lost" gets a bar-model question in next lesson's do now and a quick check-in.
Device-free: fist to five as they leave.
BEFORE NEXT LESSON · 10 MINUTES
Three to try at home
1
Share £35 in the ratio 3 : 4.
2
5 pens cost £3.50. How much do 8 pens cost?
3
£1 = $1.30. Change $65 into pounds.
Draw the bar for question 1. Maths Genie has free topic videos at mathsgenie.co.uk
Answers: 1) £15 and £20 (35 ÷ 7 = 5). 2) £5.60 (70p each). 3) £50 (65 ÷ 1.3).
Next lesson: algebra. It looks different, but substitution and solving use the same careful, one-step-at-a-time working.
