At a glance. One lesson of about 60 minutes. Students build a bar of equal parts, find the value of one part, share a total in a ratio, and work back from one known share or a difference.
Calculator: no calculator. Every total divides by the number of parts without a remainder. The skill is choosing the steps, not computing.
What students will learn
Know
- Add the ratio numbers to find the number of parts.
- One part is the amount divided by the number of parts.
- A share is its ratio number times one part.
Understand
- The shares always add back to the amount.
- A ratio such as 3 : 2 does not mean the shares are 3 and 2.
- A known share or a difference gives the value of one part.
Do
- Share an amount in a ratio of two or three parts.
- Find the total from one share or from a difference.
- Check that the shares add to the amount.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding one part and the shares (questions 1 to 6).
- Criterion B. Investigating patterns: change only the value of one part and describe what happens to the total and both shares. Then change only the ratio and describe it.
- Criterion C. Communicating: writing each step of a share with its reason, as in the worked example (questions 5, 7 and 8).
- Criterion D. Applying mathematics in real-life contexts: market takings, rent, a garden, mixed nuts and a prize (the hook and questions 4 to 7).
The one thing to do with the figure
Ask for a prediction before the sliders move. “Tunde and Bisi share the takings in the ratio 3 : 2. What fraction does Tunde get? How much each if it is 100 naira?” Take 3/5, 60 and 40 and also the wrong 3/2 and 3 and 2.
Then set the ratio and the value of one part, and ask for the amount shared before the read-out: 7 parts of 5. Reverse it: “the amount shared is 35 and the ratio is 2 : 5; what is one part?”
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, with predictions first. |
| 10 min | Key idea; practise saying the three steps aloud with a total of your own. |
| 10 min | Both worked examples; students finish the faded stickers example in pairs. |
| 5 min | Question 1, the check. If many choose A or D, go back to the figure's bar of equal parts. |
| 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 | x = 9; 3 : 5; 14. |
|---|---|
| Worked example 1 | 40 and 32 naira. |
| Worked example 2 | 63. |
| Faded example | 21 and 35 stickers. |
| 1. The larger share | C, 90. A divides by one ratio number; B is the smaller share; D reads 2 : 3 as 2/3. |
| 2. Value of one part | 12. 108 ÷ 9 = 12. |
| 3. Sharing in a ratio | 120. 150 ÷ 5 = 30, and 4 × 30 = 120. |
| 4. A share in a ratio | 480. 840 ÷ 7 = 120, and 4 × 120 = 480. |
| 5. Three-part share | 18. 60 ÷ 10 = 6, and 3 × 6 = 18. |
| 6. Sharing with units | 1800. 2400 ÷ 4 = 600, and 3 × 600 = 1800 g. |
| 7. A share, then a change | 58. 120 ÷ 12 = 10, so 70 and 50; 50 + 8 = 58. |
| 8. Total from a difference | 108. One part is 12, so 9 × 12 = 108. |
| 9. Checking a share | B. A multiplies by the number of parts; C reverses the order; D divides by each ratio number. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 (60 is the smaller share); 27 or 21.6 in question 2; 37.5 in question 3; 280 or 210 in question 4; 20 in question 5 | Divided the total by one ratio number (800 in question 6 is 2400 ÷ 3), or read the ratio as a fraction. | Ask how many equal parts the bar has in all. Draw the bar and count. |
| 9 in question 2 | Gave the number of parts, not the value of one. | Write the unit on the answer line: pencils per part. |
| 30 or 146 in question 3; 360 in question 4 | Gave the smaller share, or subtracted a ratio number. | Colour the larger share on the bar and say which colour is asked for. |
| 12 or 30 in question 5 | Gave a different kind of flower's share. | Underline the kind of flower asked for and the matching ratio number. |
| 600 or 7.2 in question 6 | Used the other share, or multiplied the total by a ratio number. | Write 2.4 kg = 2400 g first, then the parts and the division. |
| 50, 42 or 62 in question 7 | Gave the shares before the move, or moved the 8 pesos the wrong way. | Draw two boxes for the shares and an arrow for the move. |
| 12, 60 or 48 in question 8 | Used the difference as the total, or gave one colour. | Ask how many parts the difference is, then how many parts there are in all. |
| A, C or D in question 9 | Multiplied by the number of parts, reversed the order, or divided by each number. | Check that the shares add to 135 and that the smaller number of the ratio goes with the smaller share. |
Questions worth asking
- Why add the ratio numbers first? They count the equal parts in the whole. Listen for “so I know how many equal parts there are”.
- How can you check a share without redoing it? Add the shares and compare with the total, and check that the bigger ratio number has the bigger share.
- What if the total is not a multiple of the number of parts? One part is a decimal or fraction and the shares may not be whole numbers; link to the what-if.
- In 40 shared 1 : 3 and 40 shared 2 : 3, which first share is bigger? The first shares are 10 (one part of 10) and 16 (two parts of 8). In 2 : 3 each part is smaller, but the first share has two of them.
Differentiation
A way in for anyone stuck
- Give squared paper and let students draw the bar of equal parts for every question before any arithmetic.
- Start with ratios whose parts add to 10 or 5, so the division is friendly.
- Give a three-line frame to write under every share: parts, one part, shares.
Stretch
- Question 7, then: after the move, what is the new ratio in simplest form? (The shares become 70 − 8 = 62 and 50 + 8 = 58, so the ratio is 62 : 58 = 31 : 29.)
- Three people share 360 in the ratio 1 : 2 : 3. The first gives half of their share to the third. What are the three shares now? (60, 120, 180 becomes 30, 120, 210.)
Students learning in English as an additional language
“Share” and “in the ratio” can both be confusing. Say “divide into equal parts, then give so many parts to each”, and point to the bar as you say it.
| Word | Plain meaning | Example |
|---|---|---|
| share | the part of a total that one person gets | Tunde's share of the takings |
| total | the whole amount before it is shared | 100 naira |
| parts | the equal pieces the total is cut into | 3 + 2 = 5 parts |
| one part | the value of each equal piece | 100 ÷ 5 = 20 |
| difference | how much more one share is than another | 5 parts and 4 parts differ by 1 part |
Sentence starters for explaining aloud or in writing:
- There are … + … = … parts in all, so one part is …
- The shares are … × … = … and … × … = …, which add to …
- The difference is … parts, and that is worth …
Two things not to say
“Divide by the numbers in the ratio.” That gives the shares in the mistake the warning card shows.
“Just use a ratio table.” Show the bar first; a table is a record of the bar, not a replacement for it.
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.
- SharingShare 40 in the ratio 3 : 5. Find each share.
15 and 25. - One partA total of 54 is shared in the ratio 2 : 7. How much is one part?
6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo cousins pay for a gift of 90 rupees in the ratio 2 : 3. How much does the one who pays more give?
54. 90 ÷ 5 = 18, and 3 × 18 = 54. - Earlier work (Lesson 2)Share 90 mm of rope in the ratio 12 : 18, after writing the ratio in simplest form.
12 : 18 = 2 : 3, so 36 mm and 54 mm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningTwo shares are in the ratio 5 : 3 and the larger is 15 more than the smaller. What is the total?
60. The difference is 2 parts = 15, so one part is 7.5 and the total is 8 × 7.5 = 60. - Working backA total is shared 3 : 4 and the larger share is 28. What is the total?
49. One part is 7, and 7 parts make 49.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All three 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 strip of paper folded into equal parts.