The shape of the lesson
By the end of the lesson, learners can
- Ratios in their simplest form: In the lab, choose Simplify and drag the sliders: the blocks regroup until no common factor is left.
- Dividing in a given ratio: Add the parts to find how many equal blocks there are.
- Proportion in context: Recipes, map scales and best-value questions all use the same idea: find the value for ONE unit (one person, one centimetre, one gram), then scale up.
- Same units before you simplify or compare.
- Simplify by dividing every part by the highest common factor.
- Share: add the parts, divide the total by that number, multiply back.
- Difference given: the difference in parts matches the difference in amounts.
- Proportion: find the value for one unit, then multiply; ask whether more of one means more or less of the other.
One picture every idea returns to
Find the value of one part. Divide the amount you know by the number of parts it stands for, then multiply by the number of parts you want.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Equal parts, fair shares.. A ratio compares the sizes of parts. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Ratios in their simplest form | In the lab, choose Simplify and drag the sliders: the blocks regroup until no common factor is left. Project tab 1 of the lab (Ratio, rates and percentages lab, ratio) and let a student drive it while the class predicts. Model the worked example Simplify with mixed units on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Dividing in a given ratio | Add the parts to find how many equal blocks there are. Model the worked example Share an amount on the board, one line at a time. Your turn: Red and green sweets in a jar are in the ratio \(3: 5\). There are 30 green sweets. How many sweets are there altogether? | Pairs |
| 21 to 30 min | Teach and check: Proportion in context | Recipes, map scales and best-value questions all use the same idea: find the value for ONE unit (one person, one centimetre, one gram), then scale up. Model the worked example Adapt a recipe on the board, one line at a time. | Cooperative Concept Mapping |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 12 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Proportion in context, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Cooperative Concept Mapping
Teams receive the key terms on cards and arrange them on a large sheet, drawing and labelling the links. Compare maps between teams before showing the model map.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Simplifying \(1.5\ \text{kg}: 600\ \text{g}\) as \(1.5: 600 = 1: 400\) is wrong: the parts were in different units. | Convert first, then simplify. |
| To share \(\$360\) in the ratio \(2: 3: 4\), do not divide by 2, 3 or 4. | Divide by the TOTAL number of parts, 9. |
| Map scales hide a units trap: \(1\ \text{km} = 100\,000\ \text{cm}\), not 1000 cm. | Write the conversion down before you divide. |
Ratio and proportion: worksheet
6 questions, 12 marks. Written in the style of the papers; no past paper question is reproduced.
Write each ratio in its simplest form. (a) \(45 : 60\) (b) \(2.4\ \text{m} : 80\ \text{cm}\)
Write \(5 : 8\) in the form \(1 : n\).
Share \(\$630\) in the ratio \(2 : 3 : 4\).
A map has scale \(1 : 200\,000\). Two towns are 7.5 km apart. How far apart are they on the map, in centimetres?
Pack A contains 400 g of cereal and costs \(\$2.60\). Pack B contains 650 g and costs \(\$4.10\). Which pack is better value? Show your working.
Green paint is made by mixing blue and yellow paint in the ratio \(2 : 7\). Tom has 1.5 litres of blue paint and 5 litres of yellow paint.
Work out the greatest amount of green paint he can make.
Answer key and marking notes
- (a) HCF is 15: \(3 : 4\)
- (b) \(240\ \text{cm} : 80\ \text{cm} = 3 : 1\)
Divide both parts by 5: \(1 : 1.6\)
- \(2 + 3 + 4 = 9\) parts; one part \(= 630 \div 9 = \$70\).
- Shares: \(\$140,\ \$210,\ \$280\).
- \(7.5\ \text{km} = 750\,000\ \text{cm}\).
- Map distance \(= 750\,000 \div 200\,000 = 3.75\) cm.
- Pack A: \(2.60 \div 4 = \$0.65\) per 100 g.
- Pack B: \(4.10 \div 6.5 = \$0.631\) per 100 g (3 s.f.).
- Pack B is better value.
- Using all the blue: 1 part \(= 1.5 \div 2 = 0.75\) l, needing \(7 \times 0.75 = 5.25\) l of yellow. He only has 5 l, so yellow runs out first.
- Using all the yellow: 1 part \(= 5 \div 7\) l.
- Green paint \(= 9 \times \dfrac{5}{7} = 6.43\) litres (3 s.f.).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Write \(2.4\ \text{m} : 80\ \text{cm}\) in its simplest form.
- A\(3 : 1\)
- B\(3 : 100\)
- C\(30 : 1\)
- D\(1 : 3\)
\(\$560\) is shared between Mia and Joe in the ratio \(3 : 5\). How much does Mia receive?
- A\(\$210\)
- B\(\$350\)
- C\(\$186.67\)
- D\(\$336\)
A map has a scale of \(1 : 50\,000\). Two towns are 3.8 cm apart on the map. What is the real distance between them?
- A1.9 km
- B19 km
- C190 km
- D0.19 km
12 identical machines can complete an order in 20 days. How many days would 15 of these machines take, working at the same rate?
- A16 days
- B25 days
- C17 days
- D23 days
Blue and yellow paint are mixed in the ratio \(7 : 4\). The mixture contains 18 litres more blue paint than yellow paint. How many litres of paint are there altogether?
- A66 litres
- B42 litres
- C198 litres
- D29 litres
\(x : y = 2 : 3\) and \(y : z = 4 : 5\). Find \(x : y : z\) in its simplest form.
- A\(8 : 12 : 15\)
- B\(2 : 3 : 5\)
- C\(8 : 12 : 20\)
- D\(6 : 12 : 15\)
Quiz key
Change metres to centimetres: \(2.4\ \text{m} = 240\ \text{cm}\). Then \(240 : 80 = 3 : 1\) (divide both by 80). The tempting \(3 : 100\) comes from simplifying \(2.4 : 80\) without converting. \(30 : 1\) uses 1 m = 1000 cm, and \(1 : 3\) writes the parts the wrong way round.
There are \(3 + 5 = 8\) parts, so one part is \(560 \div 8 = \$70\). Mia gets \(3 \times 70 = \$210\). \(\$350\) is Joe's share. \(\$186.67\) comes from dividing by 3 instead of by the total number of parts, and \(\$336\) from working out \(\dfrac{3}{5}\) of 560.
Real distance \(= 3.8 \times 50\,000 = 190\,000\) cm. Since \(1\ \text{km} = 100\,000\ \text{cm}\), this is \(190\,000 \div 100\,000 = 1.9\) km. The tempting 190 km comes from dividing by 1000, as if 1 km were 1000 cm. 19 km and 0.19 km are slips in the power of ten.
This is inverse proportion. The work is \(12 \times 20 = 240\) machine-days, so 15 machines take \(240 \div 15 = 16\) days. The tempting 25 days treats it as direct proportion (\(20 \times \dfrac{15}{12}\)), which would mean more machines take longer. 17 and 23 come from subtracting or adding the 3 extra machines.
The difference is \(7 - 4 = 3\) parts, so 3 parts \(= 18\) litres and one part \(= 6\) litres. The total is \(11 \times 6 = 66\) litres. 42 litres is only the blue paint. 198 comes from treating 18 litres as ONE part (\(18 \times 11\)), and 29 from adding 18 and 11.
Make \(y\) equal to 12 in both: \(x : y = 2 : 3 = 8 : 12\) and \(y : z = 4 : 5 = 12 : 15\). So \(x : y : z = 8 : 12 : 15\). The tempting \(2 : 3 : 5\) just joins the ratios, but the 3 and the 4 stand for the same \(y\), so they must be matched first. \(8 : 12 : 20\) scales \(z\) by 4 instead of 3, and \(6 : 12 : 15\) scales \(x\) by 3 instead of 4.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.