At a glance. One lesson of about 60 minutes. Students find the value of one, scale to the amount wanted, compare best value by the price of one, and test whether a relationship is proportional.
Calculator: no calculator. The numbers are chosen to divide cleanly or to give halves and tenths. The skill is picking the method, not computing.
What students will learn
Know
- The value of one is the amount divided by the number of items.
- The amount wanted is the value of one times the number wanted.
- Best value compares the price of one.
Understand
- A proportional relationship has the same value of one throughout.
- A fixed start charge or a constant gap is not proportional.
- A scale factor and the value of one give the same answer.
Do
- Solve a proportion problem by the value of one or a scale factor.
- Compare two packs by the price of one.
- Test a relationship by asking whether doubling one doubles the other.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: proportion, value of one, scale factor (questions 1 to 6).
- Criterion B. Investigating patterns: change the known pack with the number wanted fixed and describe what stays the same. Ask which values of the number wanted give a whole-number scale factor.
- Criterion C. Communicating: writing each step of a proportion problem with its units and a sense check (questions 5, 7 and 8).
- Criterion D. Applying mathematics in real-life contexts: market stalls, recipes, fuel, packing and paint (the hook and questions 2 to 8).
The one thing to do with the figure
Ask for a prediction before the sliders move. “3 mangoes cost 18 rupees. How much for 5?” Ask how they got it before saying whether it is right. Collect scale factor and value of one methods side by side.
In the figure, set the known pack to 5 and the number wanted to 7, so that the scale factor is awkward. Ask why the value of one still gives a clean answer. Then set the number wanted to 1 and ask what the figure has just told them.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, with predictions and methods first. |
| 10 min | Key idea; both methods on the tailor example. |
| 10 min | Best value example; students finish the faded cyclist example in pairs. |
| 5 min | Question 1, the check. If many choose A, C or D, go back to the figure's arrows. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 28; 24; 40. |
|---|---|
| Worked example 1 | 15 m. |
| Worked example 2 | Pack B. |
| Faded example | 12 km. |
| 1. Cost from the cost of a pack | B, 56 rupees. A multiplies items by items; C adds; D multiplies cost by items. |
| 2. The cost of one | 4. 24 ÷ 6 = 4. |
| 3. A rate and the amount wanted | 36. 12 ÷ 3 = 4, and 4 × 9 = 36. |
| 4. Scaling a recipe | 21. 14 × 1.5 = 21 eggs. |
| 5. Best value | 0.5. 36 ÷ 8 = 4.5 and 25 ÷ 5 = 5, so 5 − 4.5 = 0.5. |
| 6. Fuel and distance | 84. 12 × 7 = 84 km. |
| 7. Two quantities changing | 180. 15 × 4 × 3 = 180. |
| 8. A mix with a limit | 32. 12 : 20 = 3 : 5, so 12 + 20 = 32. |
| 9. What is proportional | B. A has a fixed start charge; C adds a constant; D starts from a non-zero amount. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1; 144 or 18 in question 2 | Multiplied, added or subtracted the numbers they could see. | Ask what one item costs first; write it before any other working. |
| 18 in question 3; 18 in question 4; 62 in question 6 | Added the extra amount. | Ask whether doubling the number doubles the answer. |
| 1.75 in question 4; 4, 12 in questions 3 and 6 | Stopped at the value of one. | Underline the amount wanted and finish with it. |
| 0.25 in question 2 | Divided the wrong way: notebooks per shilling instead of shillings per notebook. | Say the unit: shillings per notebook, so shillings ÷ notebooks. |
| 108 in question 3; 168 in question 4; 420 in question 6 | Multiplied the two given numbers. | Label each number with its unit and ask what units the answer has. |
| 11, 3 or 4.5 in question 5 | Compared the totals or the numbers of pencils, or gave the wrong shop's price. | Write the price for one of each shop side by side. |
| 60, 270 or 360 in question 7 | Changed only one of the two quantities. | Build it in two steps: one worker for one hour, then 4 workers, then 3 hours. |
| 37, 40 or 20 in question 8 | Used all of both colours, or answered with the white only. | Ask: which colour runs out first? Write the matching amount of the other. |
| A, C or D in question 9 | Accepted a relationship with a start amount or a constant gap. | Test with a table: does doubling one quantity double the other? |
Questions worth asking
- When is the scale factor quicker than the value of one? When it is a whole number or a simple fraction. Listen for “it is exactly 3 times”.
- How do you know the answer makes sense? More items cost more, and it is close to an estimate. Listen for a reason in context.
- Is a bigger pack always better value? Not always; compare the price of one. Link to the what-if.
- Is the speed of a runner proportional to time? The distance is, if the speed is steady. Listen for “the speed stays the same”.
Differentiation
A way in for anyone stuck
- Give a two-column ratio table template with a row for the known amounts and a row for one.
- Keep to whole-number values of one for the first questions.
- Ask students to write the unit beside every number before they start.
Stretch
- Question 8, then: if she had 18 litres of blue and the same 25 litres of white, which colour would run out first and how much paint would she make? (White: 25 ÷ 5 = 5 lots of 5, which needs 15 litres of blue; 15 + 25 = 40 litres.)
- A pack of 6 costs 21 pesos and a pack of 10 costs 33 pesos. Which pack has the lower price per item, and by how much? (3.5 against 3.3, so the larger pack, by 0.2 pesos an item.)
- 6 workers take 4 hours to pack a load. How long do 3 workers take? Is the time proportional to the number of workers? (8 hours. Halving the workers doubles the time, so the time is not in direct proportion to the workers: doubling one does not double the other.)
Students learning in English as an additional language
“Per” means “for each”. Say “shillings for each notebook” and write it as shillings per notebook, so that students link the division to the word.
| Word | Plain meaning | Example |
|---|---|---|
| proportion | two quantities where one is always the same multiple of the other (also: two equal ratios, Lesson 4) | cost and number of items |
| value of one | the amount for a single item | 9 pesos for one item |
| scale factor | the number you multiply both quantities by | 12 items is 3 times 4 items |
| rate | how much of one thing for each one of another | 4 pages a minute |
| best value | the lowest price for one | 6.5 taka a pen against 7 |
Sentence starters for explaining aloud or in writing:
- One … costs … ÷ … = …
- So … cost … × … = …
- This is not proportional because when … doubles, … does not.
Two things not to say
“Just cross multiply.” It hides the meaning of the value of one and it is applied to relationships that are not proportional.
“The bigger pack is always better value.” Compare the price for one.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Value of one7 bottles of water cost 49 naira. What does one bottle cost?
7. - ScalingA car travels 90 km in 2 hours at a steady speed. How far in 5 hours?
225 km.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA gardener plants 24 seedlings in 8 minutes. At the same rate, how many in 20 minutes?
60. - Earlier work (Lesson 3)Share 63 rupees in the ratio 2 : 7. The smaller share buys 5 items. What does one item cost?
2.8. The shares are 14 and 49, and 14 ÷ 5 = 2.8.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningEsi says that 10 oranges cost 50 pesos, so 20 oranges cost 100. Her brother says that 20 cost 90 because there is a discount. Who is using proportion?
Esi is. With a discount the price per orange changes, so the cost is not proportional. - Working backA recipe uses 9 eggs for 6 people. Maya has 12 eggs. For how many people can she cook?
8 people. One person needs 1.5 eggs and 12 ÷ 1.5 = 8.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Use a two-row table on paper: amount wanted on the top row and cost on the bottom.