At a glance. One lesson of about 60 minutes. Students multiply a ratio to make equal ratios, compare with adding, find a missing number with a scale factor, and check with a cross product.
Calculator: no calculator. The scale factors are whole numbers or halves. The skill is choosing multiplication over addition.
What students will learn
Know
- Equal ratios come from multiplying or dividing both parts by the same number.
- A proportion is two equal ratios, a : b = c : d.
- The scale factor is found from the two numbers that are in the same position.
Understand
- Adding the same amount to both parts changes the shape of a ratio, unless the parts are equal.
- The check a × d = b × c works because both sides are k × a × b.
- A scale factor may be a decimal.
Do
- Find a missing number in a proportion.
- Decide whether two ratios are equal.
- Scale a measurement up or down using a scale factor.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: equal ratios, scale factors and a proportion (questions 1 to 6).
- Criterion B. Investigating patterns: set the amount added to 0, 1, 2, 3 and record whether the ratio is kept. Ask for a rule about what adding does to the cross product.
- Criterion C. Communicating: explaining why multiplying keeps a ratio and adding does not, in words and with the bars (questions 5 and 9).
- Criterion D. Applying mathematics in real-life contexts: paint, pancakes, maps and photographs (the hook and questions 4 to 6).
The one thing to do with the figure
Ask for a prediction before the sliders move. “2 tins of white and 3 of blue. She adds 4 of each to get 6 : 7. Is it the same shade?” Take opinions, then set the figure to add 4 and multiply by 2.
Set the amount added to 0 and ask why the two rows now agree. Then set it to 1 and ask whether a little adding is nearly right. Use the cross products to show it is not.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, with predictions first. Include a change of 0 added. |
| 10 min | Key idea; the scale factor between two numbers in the same position, said aloud. |
| 10 min | Both worked examples; students finish the faded example in pairs. |
| 5 min | Question 1, the check. If many choose A or D, go back to the hollow bars. |
| 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 | 15; 6/7; yes. |
|---|---|
| Worked example 1 | 35. |
| Worked example 2 | Yes, both are 3 : 4. |
| Faded example | 72. |
| 1. Spotting an equal ratio | B, 8 : 14. A adds 2 to both; C changes one part; D adds 8 to both. |
| 2. Missing number, scaling up | 32. 3 × 4 = 12 and 8 × 4 = 32. |
| 3. Missing number, scaling down | 2. 36 ÷ 12 = 3 and 24 ÷ 12 = 2. |
| 4. Proportion in a recipe | 750. 150 × 5 = 750 ml. |
| 5. Scale on a map | 28. 4 × 7 = 28 km. |
| 6. Enlarging a rectangle | 15. 6 × 2.5 = 15 cm. |
| 7. Linking two ratios | 25. a : b : c = 4 : 6 : 5, and 20 is 4 × 5, so c = 5 × 5 = 25. |
| 8. Decimal in a ratio | 7. 3.5 × 2 = 7 and 6 × 2 = 12, so 7 : 12. |
| 9. A change that keeps the ratio | C. A adds to both parts; B and D change the parts by different amounts. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1; 17 in question 2; 21 in question 3 | Added or subtracted instead of scaling. | Ask how many times bigger the known number is, not how much bigger. |
| 96 or 20 in question 2; 1500 or 300 in question 4; 84 or 63 in question 5 | Combined the numbers they could see instead of finding a scale factor. | Write the ratio table with two columns and find the arrow between them. |
| 8 or 12 in question 3 | Divided by the wrong number, or gave the scale factor. | Label each arrow: divide by 12 on top, so divide by 12 underneath. |
| 158 in question 4; 22 in question 5; 18 in question 6 | Added the difference. | Draw the bars with a hollow extra and ask what happens to the ratio. |
| 12 or 48 in question 6 | Rounded the scale factor, or multiplied the two original numbers. | Say 20 ÷ 8 = 2.5 aloud and write it above the arrow. |
| 30 or 50 in question 7 | Did not make the b values match. | Write both ratios in a row with a gap and rewrite until b is the same. |
| 3, 4 or 12 in question 8 | Rounded, or gave the second number. | Ask what makes 3.5 whole. Apply it to both parts. |
| A, B or D in question 9 | Added to both, or changed the parts by different amounts. | Ask for the multiplier of each part: is it the same number? |
Questions worth asking
- How can you tell two ratios are equal without simplifying? Cross products are equal. Listen for “both multiplied by the same number”.
- Why does adding to both parts break a ratio? The bars no longer repeat the original; the added bit is the same length on a short bar and a long one. The one exception is equal parts, such as 5 : 5, which stay equal when the same amount is added to both.
- Can a scale factor be less than 1? Yes, when scaling down. Listen for “it is a fraction or a decimal”.
- Is 0.5 : 1 equal to 1 : 2? Yes, multiplying both by 2. Link to the decimal in question 8.
Differentiation
A way in for anyone stuck
- Give strips of squared paper so students can lay k copies of the original pair side by side.
- Start with scale factors of 2 and 3 only; add halves later.
- Write the two numbers in the same position in the same colour on the page, then draw the arrow.
Stretch
- Question 7, then: find a : b : c in the smallest whole numbers (4 : 6 : 5) and say which ratio you had to change first (a : b, to make the b values match).
- Show that if a : b = c : d then (a + c) : (b + d) is also equal to a : b. Try 2 : 3 and 4 : 6. (6 : 9 = 2 : 3.)
Students learning in English as an additional language
“Scale factor” is the number you multiply by. Say “times” for multiply and “divided by” for divide, and keep “scale” for the whole picture getting bigger or smaller.
| Word | Plain meaning | Example |
|---|---|---|
| proportion | two ratios that are equal | 3 : 5 = 6 : 10 |
| scale factor | the number you multiply or divide both parts by | 6 : 10 is 3 : 5 with scale factor 2 |
| cross product | the first part of one ratio times the second part of the other | 3 × 10 and 5 × 6 |
| equal ratios | ratios that say the same thing in different numbers | 4 : 7 and 8 : 14 |
| enlargement | a copy that is bigger by a scale factor | a photograph printed larger |
Sentence starters for explaining aloud or in writing:
- I found the scale factor by … ÷ … = …
- Both parts are multiplied by …, so the ratio is …
- These ratios are not equal because the cross products are … and …
Two things not to say
“Whatever you do to one side, do to the other.” This covers adding, which breaks a ratio. Say “multiply both by the same number”.
“Cross multiply and divide.” Without the reason, students apply it to sums and differences too.
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.
- Missing number5 : 6 = 25 : ?
30. - Equal or notAre 9 : 12 and 3 : 4 equal?
Yes. Both divide by 3 and 4 × 9 = 36 = 12 × 3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cleaner mixes 3 spoons of powder with 7 litres of water. How much water goes with 12 spoons of powder?
28 litres. 12 ÷ 3 = 4, and 7 × 4 = 28. - Earlier work (Lesson 3)Share 80 in the ratio 3 : 5. Then write the ratio of the first share to the total as three equal ratios.
30 and 50; 30 : 80 = 3 : 8 = 6 : 16.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningAyo says that 4 : 6 and 7 : 9 are equal because both add 3 to each part. Is she right?
No. 4 : 6 is 2 : 3 and 7 : 9 is not: 4 × 9 = 36 but 6 × 7 = 42. - Clearing a decimalWrite 1.5 : 4 using the smallest whole numbers.
3 : 8. Both parts are doubled. - Map scaleA map has the scale 1 : 50000, so 1 cm on the map is 50000 cm on the ground. A path is 4 cm long on the map. How long is it on the ground, in kilometres?
2 km. 4 cm is 4 × 50000 = 200000 cm, and 100000 cm make 1 km, so 2 km.
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 strips of squared paper: three squares and five squares, then copies of each.