At a glance. One lesson of about 60 minutes. Students write ratios in a stated order, read part to part against part to whole on a counter figure, and put amounts in one unit before comparing.
Calculator: no calculator. Every number is a small whole number or converts by 100. The skill is reading the comparison, not computing.
What students will learn
Know
- A ratio a : b means a of the first for every b of the second.
- The order of the numbers follows the order of the words.
- Both amounts must be in the same unit.
Understand
- A part-to-part ratio says nothing about the size of the whole; a part-to-whole ratio needs the parts added first.
- To find a share of the whole, add the parts first.
- A ratio is not a difference: 2 : 9 does not mean 2 more.
Do
- Write a ratio of two or three amounts in the order asked.
- Move between part to part and part to whole.
- Convert to a common unit before writing a ratio.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: ratio notation, order and part to whole (questions 1 to 6).
- Criterion B. Investigating patterns: set the first group to 3 and move the second from 1 to 9. Ask what the part-to-whole read-out does and what the ratio does, then write a general statement.
- Criterion C. Communicating: explaining in a sentence why 4 : 6 means 4 out of 10 and not 4 out of 6, using the words part and whole (questions 3, 7 and 9).
- Criterion D. Applying mathematics in real-life contexts: bead bags, fruit stalls, club membership and ribbon lengths (the hook and questions 1, 2, 5 and 6).
The one thing to do with the figure
Ask for a prediction before the sliders move. “3 red beads and 5 blue beads. What is blue to red? How many beads are red, out of how many?” Take 3 : 5, 5 : 3 and 3/5 from the room, then build it.
Set both groups equal and ask what the read-outs say. Then set the first group to 1 and the second to 8 and ask which read-out is the one that is not 1 out of 8. Name the difference on the board: part to part, part to whole.
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 pair of equal groups. |
| 10 min | Key idea; swap the order and read both ratios aloud, using the words “for every”. |
| 10 min | Worked example on the tarts; students finish the faded pens example in pairs. |
| 5 min | Question 1, the check. If many choose C or D, go back to the figure's third read-out. |
| 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 | 3/4; 1/4; 6. |
|---|---|
| Worked example | 12 : 20 and 20 : 32. |
| Faded example | 6 : 9. |
| 1. Writing a ratio in the order asked | B, 4 : 7. A is the order reversed; C and D are part to whole. |
| 2. Total number of parts | 12. 5 + 7 = 12. |
| 3. Reading a ratio in words | A. B turns a part-to-part ratio into part-to-whole without adding; C reverses the order; D reads the ratio as a difference. |
| 4. Part to whole | 13. 7 + 2 + 4 = 13. |
| 5. Same units first | 300. 3 × 100 = 300, so the ratio is 300 : 40. |
| 6. Part to the rest | 16. 9 + 7 = 16, so walk to not walk is 14 : 16. |
| 7. Ratio to percentage | 75. 3 out of 4 = 3/4 = 75%. |
| 8. Three-part ratio | 5. 2 + 3 = 5, so atlases to the rest is 7 : 5. |
| 9. Part to part against part to whole | B. A and D ignore the order; C uses part to part as if it were part to whole. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; 7 : 4 written for limes to mangoes | Wrote the numbers in the order they appear in the sentence, not the order the question asks. | Underline the first thing named in the question and say it aloud: limes first, so limes go first. |
| C or D in question 1; 7 in question 2 | Used part to whole, or counted only one group of the whole. | Ask: how many are there altogether? Write the whole next to the ratio before choosing. |
| 2 or 35 in question 2 | Subtracted the parts, or multiplied them, instead of adding to get the whole. | Ask what the whole club is made of, then add the two groups. |
| B in question 3 | Read 2 : 9 as 2 out of 9. | Add the parts. Draw a bar of 11 equal boxes and shade 2. |
| 6 or 9 or 11 in question 4 | Left out a colour when finding the whole. | Count each colour on the board, then add: every one of the sweets, red included. |
| 3, 30 or 3000 in question 5 | Did not convert, or multiplied by 10. | Write 1 m = 100 cm on the line before the working. |
| 30, 7 or 9 in question 6 | Answered a different question: the whole class, or one of the two other groups. | Write the question in two groups: walk, do not walk. Say which group each pupil is in. |
| 25, 3, 33 or 33.3 in question 7 | Gave red as a percentage, gave the parts of blue only, or compared the two parts only. | Draw 4 boxes, shade 3, and ask what fraction of the boxes is shaded. |
| 12, 3 or 2 in question 8 | Counted every book, or only one of the two kinds that are not atlases. | Cross out the atlases and add what is left. |
Questions worth asking
- Does 3 : 5 mean 3/5 of the beads are red? No. 3/5 compares red with blue only if blue were the whole. Listen for “3 out of 8” and the word whole.
- If I double both groups, does the ratio change? The numbers change but the comparison does not: 6 : 10 says the same as 3 : 5. Do not push the rule yet; it is lesson 2.
- Can a ratio have a zero in it? Yes, 0 : 5 is possible but unusual. Listen for “none of the first for every 5 of the second”.
- Which is bigger, 4 : 7 or 7 : 4? They are not numbers to be ordered: they are different comparisons. Listen for the order of the words.
Differentiation
A way in for anyone stuck
- Give each student two colours of counters so they can build every ratio before writing it.
- Start with two groups only and keep the whole in view by writing the total beside the ratio.
- Draw a bar of 7 equal boxes under the counters with 2 shaded: the bar shows the part and the whole together, and it is the picture lesson 3 uses.
- Use a sentence frame for every ratio: “for every … of the first there are … of the second”.
Stretch
- Question 8, then: write all three possible part-to-whole fractions for the shelf in the ratio 2 : 3 : 7. (2/12, 3/12 and 7/12.)
- A bag has red and blue beads and red to blue is 2 : 3. Could the bag hold 17 beads? Why not? (Red and blue come in groups of 5 beads, so the total must be a multiple of 5.)
Students learning in English as an additional language
“Ratio” comes from the Latin for “reckoning”. Say the sentence “for every … there are …” each time a colon appears, and read the colon as “to”.
| Word | Plain meaning | Example |
|---|---|---|
| ratio | a comparison of two amounts, written with a colon | red to blue is 3 : 5 |
| part | one of the amounts being compared | the 3 in 3 : 5 |
| whole | all the parts added together | 3 + 5 = 8 beads in all |
| order | which amount is named first | limes to mangoes is not mangoes to limes |
| same unit | both amounts measured in the same way | 300 cm and 40 cm, not 3 m and 40 cm |
Sentence starters for explaining aloud or in writing:
- For every … there are …, so the ratio of … to … is …
- The whole is … + … = …, so the first part is … out of …
- I put both amounts in … first because …
Two things not to say
“A ratio is just a fraction.” A ratio of part to part is not a fraction of the whole until the parts are added.
“The bigger number goes first.” The order is the order of the words in the question.
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.
- OrderA basket has 8 plums and 5 figs. Write the ratio of figs to plums.
5 : 8. - Part to wholeRice to beans is 3 : 4 in a bowl. How many parts is the whole bowl?
7.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cricket squad has 9 batters, 5 bowlers and 1 wicketkeeper. Write the ratio of bowlers to players who are not bowlers.
5 : 10. - Earlier work (Year 7, Unit 5, lesson 1)Write 10/15 in simplest form, then say how that fraction is not the same as the ratio 10 : 15.
2/3. The ratio 10 : 15 compares the two parts; 10/15 would mean 10 out of 15 in the whole.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningThe ratio of cats to dogs at a shelter is 3 : 4. Can there be 20 animals? Explain.
No. Cats and dogs come in groups of 3 + 4 = 7, and 20 is not a multiple of 7. - UnitsWrite the ratio of 2 hours to 30 minutes with both in minutes.
120 : 30.
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 two colours of counters, beads or squares of paper on a desk.