Share a total
Share £40 in the ratio 2 : 3
2 + 3 = 5 boxes, so 1 box = £40 ÷ 5 = £8
Amir: 2 × £8 = £16. Beth: 3 × £8 = £24
Ratio and sharing Total, one share or the difference: how does one bar answer them all?
Foundation and Higher · AQA · 45 minutes
Students meet one bar model and three clues: a total, one share and a difference. Each time they find one box first. Simplifying, sharing and ratio to fraction follow. Slides to present, a vote that shows who's stuck, practice and homework.
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Every point on this line (but 0, 0) has x : y = 2 : k. Move k: the whole-number points marked on it are equal ratios. Which is the simplest, and why is it the first one after (0, 0)?
Whole-number points: (0, 0), (2, 3), (4, 6), (6, 9), (8, 12)
The idea in one minute
A ratio such as 2 : 3 says how many equal boxes each person gets: 2 boxes and 3 boxes. Draw them as bars, and every ratio question asks the same thing first: what is one box worth? The clue may be the total (all 5 boxes), one person's share (their boxes) or the difference (the boxes one has more). Once one box is known, every share is a multiplication.
The lesson goes picture first. Students see one bar and three clues before anyone explains, and find that each clue gives the same box. Then they simplify ratios (equal ratios lie on one line through the origin), share totals, work back from one share or a difference, and turn a ratio into fractions. The core is for Foundation and Higher; the stretch is a ratio that changes, for Higher.
By the end, students can
- write a ratio in its simplest form, with the units the same
- share an amount in a ratio of two or three parts
- find the shares when one share or the difference is given
- turn a ratio into the fractions of the whole it gives
- draw a bar model and say which boxes a clue covers
Key words
- ratio
- simplest form
- share
- part
- bar model
- difference
Before this lesson
Students should know their times tables, find a highest common factor and say what a fraction of an amount is. Ratio is taught here, from bar models.
Taught in Fractions: the four operations.
The lesson, step by step
60 minutes in total. Each step says what changes for this length.
5 min 1 · Hook · students think
One bar, three clues
Students find one box from a total, a share and a difference before anyone explains.
Say“Amir and Beth share money in the ratio 2 : 3. Here are three clues. For each one on its own, work out what one box is worth. Don't show anyone yet.”
Ask- How many boxes does the £40 cover?
- For the third clue: which boxes does Beth have that Amir doesn't?
Watch forDividing £40 by 2 or by 3: ask how many boxes the whole £40 is.
For the teacher
Give two minutes of silent work on the slide, then gather answers clue by clue. The total covers all 5 boxes; Beth's share covers her 3 boxes; the difference covers the one box Beth has more.
Don't name the methods yet. The question is the point: why did three different clues give the same box?
Gather each clue's box with Ask the class. That all three give the same box is the point of the lesson.
Shorter: Do clue A together; leave B and C for the explanation (−2 min).
12 min 2 · Explain · you explain
Simplest form, then sharing with bars
Simplify a ratio; see equal ratios on one line; then share with a bar: find one box first.
Say“Every ratio question starts the same way: which boxes does the number I'm given cover? Then find one box.”
Ask- 12 : 18 and 2 : 3: how do you know they are the same ratio?
- Amir has 2 of the boxes. Is his fraction 2/3 or 2/5? Why?
Watch forDividing the total by the larger number of the ratio: count all the boxes.
The difference given
Ratio 2 : 3, and Beth gets £8 more than Amir
3 − 2 = 1 box, so 1 box = £8
Total: 5 × £8 = £40
For the teacher
First slide: simplest form. Divide both numbers by the same number until only 1 divides both. Make the units the same first: 50p : £2 is 50 : 200, which is 1 : 4.
Second slide: y = kx/2 holds every pair with x : y = 2 : k. Its whole-number points are the equal ratios, and the first one is the simplest form. Equal ratios multiply; adding the same to both (2 : 3 to 4 : 5) leaves the line.
Third and fourth slides: the bar. Amir 2 boxes, Beth 3, every box the same. A total covers all the boxes; a share covers that person's boxes; a difference covers the boxes one has more. Find one box, then multiply.
Fifth slide: Amir has 2 of the 5 boxes, so 2/5 of the money, not 2/3. (2/3 is something else: Amir's share as a fraction of Beth's.)
Vote before moving k on the line slide (Ask the class): which whole-number point will come first when k is 6?
Shorter: Skip the line slide; one worked example (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one. Then you do its twin, which is almost the same. Draw the bar each time.”
Ask- Total, one share or the difference? Which boxes does it cover?
Watch forTreating the difference as the total: shade the boxes it covers.
Share a total
Share £35 in the ratio 3 : 4
7 boxes = £35, so 1 box = £5
£15 and £20
Then theirs
Share £54 in the ratio 4 : 5
The difference given
Ratio 2 : 5; the second gets 18 more. The total?
5 − 2 = 3 boxes = 18, so 1 box = 6
Total: 7 × 6 = 42
Then theirs
Ratio 3 : 7; the second gets 20 more. The total?
One share given
Squash : water = 1 : 4, with 150 ml of squash
1 box = 150 ml, so 5 boxes = 750 ml of drink
Then theirs
Cement : sand = 1 : 3, with 12 kg of sand. The total?
For the teacher
Work the first example aloud, drawing the bar and writing each line. Then students solve its twin while you watch who draws the bar first.
Do the same for each pair. The second pair catches students who divide the difference by a ratio number; the third, those who treat the sand as the whole.
Open the answers slide once most have finished their twins.
Shorter: Do the first two pairs only (−3 min).
25 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: The stretch, then: why can't you add 6 to both sides of a ratio?
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Draw the bar: which boxes does the £12 cover?
Watch forUsing the £12 as if it were the total, or as one box.
Draw the bar for each one, then find one box first. The stretch is optional.
The question that shows who's stuck
- Amir and Beth share money in the ratio 3 : 5. Beth gets £12 more than Amir. How much does Amir get?
In your Classroom, as they answer (an example) - A8
- B6
- C11
- D5
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- In its simplest form, 12 : 18 = ___ : ___
- Share £48 in the ratio 3 : 5. The larger share is £___
- A 35 cm ribbon is cut into two pieces in the ratio 2 : 5, the shorter piece on the left. The line measures the ribbon in cm from its left end. Drag the marker to where the cut goes.They drag a marker on a number line from 0 to 35.
- Ella and Finn share a pizza in the ratio 2 : 3. The pizza is cut into 10 equal slices. Shade Ella's share.They shade parts of a circle cut into 10 equal parts.
- Don't answer these yet. For each one, is the number you are given the total, one share or the difference?They sort 6 items into The total and One share and The difference.
- Red and blue counters are in the ratio 2 : 7. What fraction of the counters are red?
- Paint is mixed blue : yellow = 2 : 3. A painter uses 12 litres of yellow. She makes ___ litres of paint in all.
- 63 sweets are shared in the ratio 2 : 3 : 4. The largest share has ___ more sweets than the smallest.
- Stretch: in a club, boys : girls = 3 : 4. Then 6 more boys join, and now there are as many boys as girls. How many girls are in the club?
Support · For question 7: draw blue 2 boxes and yellow 3 boxes. The 12 litres of yellow cover 3 boxes, so find one box first, then count every box.
For the teacher
Everyone chooses before anyone explains. Whatever they chose, ask them to draw the bar and shade the boxes the £12 covers. Then ask whose share the question wants.
Questions 1–2 build fluency; 3–4 change the look (a ribbon to cut, a pizza to shade); 5–6 ask for reasons (sort the clues, a fraction); 7–8 are exam-style. The stretch is a ratio that changes.
Students show their working on paper or the board; the quiz keeps their final answers.
If the class splits, draw the 3 : 5 bar together and shade the boxes the £12 covers before they practise.
Shorter: After the vote, questions 1–5 only (−6 min).
Longer: The stretch, then: why can't you add 6 to both sides of a ratio? (+10 min).
10 min 5 · Exit · students show
Which boxes?
One difference problem and its bar show who finds one box from the right boxes.
For this length: Write your own question with a total, one share and a difference that all give the same box
Say“Draw the bar first. Then answer, and say which boxes the £30 covers.”
Watch forDividing £30 by 4 or by 5 instead of by the 3 boxes Cara has more.
Draw the bar first. One or two sentences for the second question.
- Cara and Dan share some money in the ratio 4 : 1. Cara gets £30 more than Dan. They share £___ in all.
- Describe your bar for the first question: how many boxes each person has, and which boxes the £30 covers.
For the teacher
Cara and Dan share money in the ratio 4 : 1, and Cara gets £30 more. Students draw the bar, find the total and say which boxes the £30 covers.
Then answer the exit questions.
The bars show who finds one box from the right boxes: start the next lesson with two of them. Set the homework.
Longer: Write your own question with a total, one share and a difference that all give the same box (+5 min).
One to one: ask the student to draw the bar and shade the boxes the clue covers before any number is written, and to check every answer against all three facts: the ratio, the total and the difference.
Homework: bars and boxes
Draw a bar model for questions 2 to 6. Say whether you are given the total, one share or the difference, and find one box first. Check each answer against every fact in the question.
- Write 24 : 36 in its simplest form.
- Share £56 in the ratio 3 : 4.
- At a rescue centre, cats : dogs = 5 : 3. There are 15 dogs. How many cats are there?
- A drink is orange juice : lemonade = 1 : 4. It has 900 ml more lemonade than orange juice. How much drink is there?
- On a trip, adults : children = 3 : 7. What fraction of the people are children? The number of adults is what fraction of the number of children?
- Money is shared in the ratio 2 : 3 : 5. The middle share is £45. How much money is shared?
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching ratio and sharing: common questions
How do you teach sharing in a ratio with a bar model?
Draw a bar for each person, split into equal boxes: 2 boxes and 3 boxes for 2 : 3. Then ask which boxes the number in the question covers: all of them (a total), one person's (a share) or the extra boxes one person has (a difference). Divide to find one box, then multiply for each share. The same picture works for every kind of question.
Why do students get ratio difference questions wrong?
Most have learnt one method, add the parts and divide, and use it whatever number they are given. A difference covers only the boxes one person has more, so it must be divided by the difference in parts. Asking students to say first whether they have a total, one share or a difference, before any working, catches most of these errors.
Is 2 : 3 the same as the fraction 2/3?
No. In the ratio 2 : 3 the first person has 2 of 5 equal boxes, so 2/5 of the whole, and the second has 3/5. The fraction 2/3 does appear, but as a comparison of the two shares: the first share is 2/3 of the second.
Is this ratio lesson suitable for Foundation GCSE?
Yes. Simplifying, sharing a total, one share or a difference given, and ratio to fraction are on both tiers. The three-part ratio questions suit both; the stretch, a ratio that changes when some are added, is Higher.
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Curriculum reference: AQA GCSE Mathematics 8300: R4 (ratio notation and simplest form), R5 (dividing a quantity in a ratio, and ratio in contexts), R6 (a relationship as a ratio or a fraction) and R8 (ratios as fractions and as linear functions). The changing-ratio stretch is Higher.
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