At a glance. One lesson of about 60 minutes. Students work back from a part and its percentage to the whole through 1%, and find what percentage one amount is of another, including above 100%.
Calculator: no calculator. Nothing here needs one. The numbers divide by 5, 10 and 25, and the route through 10% or 5% is the idea.
What students will learn
Know
- The whole is found from a part by dividing by the percentage and multiplying by 100.
- A part as a percentage of a whole is part ÷ whole × 100.
- The whole is bigger than the part when the percentage is less than 100%.
Understand
- Working back undoes finding a percentage of an amount.
- The question words “of” and “is” show which amount is the whole and which is the part.
- A percentage can be more than 100% when the part is bigger than the whole.
Do
- Find the whole from a part and its percentage.
- Find what percentage one amount is of another.
- Check an answer by working forward, and by asking whether the whole should be bigger than the part.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: working back to the whole and finding a percentage of a whole (questions 1 to 6).
- Criterion B. Investigating patterns: the first figure. Ask students to find the percentage at which the whole equals the part, and to explain why 1% always comes first.
- Criterion C. Communicating: explaining why 18 × 40 ÷ 100 does not find the whole, in words (question 9).
- Criterion D. Applying mathematics in real-life contexts: mangoes, books, school rolls and town populations (the hook and questions 3, 4, 7 and 8).
The one thing to do with the figure
Ask for a prediction before the first slide. “The part is 64 and it is 40% of the whole. Is the whole more or less than 100?” Take “more, because 40% is less than half” from the room, then press Show the whole to settle it.
Then set the percentage to 100 and ask what the part is. In the second figure, drag the part past the whole and ask students to say what the bar tells them about the percentage.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure. Students make the part larger than the whole. |
| 10 min | Worked example on the tank; students finish the faded school bus example in pairs. |
| 5 min | Question 1, the check. If many choose A or D, go back to the first figure and ask which number is the part. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 36 and 108; 45%; 400. |
|---|---|
| Worked example | 90 litres. |
| Faded example | 140 students. |
| 1. Whole from a percentage | C, 150. A works forward (20% of 30); B adds; D multiplies the numbers. |
| 2. Whole from 10% | 140. 14 × 10. |
| 3. Whole from 50% | 170. 85 × 2. |
| 4. Whole from 25% | 240. 60 × 4. |
| 5. Part as a percentage | 25. 18/72 = 1/4. |
| 6. Choosing the calculation | B. A inverts the division; C finds 15% of 60; D adds. |
| 7. Whole from a percentage above 100 | 5000. 6500 ÷ 130 = 50, then × 100. |
| 8. Whole, then what is left | 72. 120 − 48, or 60% of 120. |
| 9. Working back, not forward | B. A accepts an answer smaller than the part; C adds; D divides by 100 only. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or D in question 1 | Worked forward (20% of 30) or multiplied the numbers. | Ask: is the answer bigger or smaller than 30? Then draw the bar with 30 under the 20%. |
| 1.4 or 1400 in question 2 | Divided by 10, or multiplied by 100 instead of 10. | Draw ten blocks of 10%. The 14 is one block. |
| 42.5 or 85 in question 3 | Halved when they should have doubled, or gave the part. | Ask: if half the book is 85 pages, is the book more or fewer than 85? |
| 15 in question 4 | Found 25% of 60. | Underline “that is 60 students” and ask which amount is the 25%. |
| 54 or 4 in question 5 | Subtracted, or divided the whole by the part. | Ask what fraction 18 is of 72 before writing a percentage. |
| A, C or D in question 6 | Inverted the division, or found a percentage of the amount. | Put the part on top and the whole underneath, then scale. |
| 8450 or 6370 in question 7 | Worked forward or subtracted 130. | Ask: did the town grow or shrink? So is the old size bigger or smaller? |
| 120 or 48 in question 8 | Stopped at the whole, or gave the pages already read. | Underline the question: still to read. |
| 28.8 in question 8 | Worked forward: took 60% of the pages read. | Ask: is the book bigger or smaller than the 48 pages already read? |
Questions worth asking
- If 30% of a number is 45, is the number bigger or smaller than 45? Bigger, because 30% is less than the whole. Listen for the bar picture.
- Why do we go through 1%? Because 1% is the smallest benchmark, from which any percentage can be built. 10% or 5% is quicker when they divide.
- Can a percentage be 200%? Yes: the part is twice the whole. A is 200% of B means A = 2B.
- Why do we multiply by 100 at the end? Because 1% is one hundredth of the whole, so 100% is 100 of them.
Differentiation
A way in for anyone stuck
- Draw a bar of ten boxes and write the part inside the right number of boxes. The whole is all ten boxes.
- Start with 10%, 25% and 50%, where the route is a division by 10, 4 and 2.
- Give a sentence frame: “The part is … and it is …%. So 1% is …”
Stretch
- Question 7, then: after a 20% fall a population is 4800. What was it before? (4800 is 80%, 1% is 60, so 6000.)
- Show that a is b% of c means c is (a ÷ b × 100). Make up your own numbers for it. Is it still true when b is over 100?
Students learning in English as an additional language
“Of” and “is” carry the meaning. Practise “30% of the whole is 45” as “the part is 45 and it is 30%” before the questions.
| Word | Plain meaning | Example |
|---|---|---|
| whole | the full amount; 100% | the whole book, 120 pages |
| part | a share of the whole | 48 pages out of 120 |
| work back | undo a calculation to find the start | from 30% being 45 back to 100% |
| 1% | one hundredth of the whole | 1% of 120 is 1.2 |
| inverse | the operation that undoes another | dividing undoes multiplying |
| check | test an answer by working in the other direction | 40% of 160 is 64 |
Sentence starters for explaining aloud or in writing:
- The part is … and it is …%, so 1% is …
- The whole should be bigger than the part because …
- … is …% of … because … ÷ … = …
Two things not to say
“Cross-multiply.” That is a method for a later unit, and it hides the reason: 1% is one hundredth of the whole.
“Just divide by the percentage.” Dividing 45 by 30 gives 1.5, which is 1% of the whole, not the whole. Say what each number means.
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.
- Whole from 10%10% of a number is 23. What is the number?
230. - What percentageWhat percentage is 12 of 48?
25%.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA family in Dubai pays 270 dirhams a week in rent. That is 45% of its weekly income. What is the weekly income?
600 dirhams. 1% is 6, and 6 × 100 = 600. - Earlier work (Lesson 3 of this unit)Find 35% of 600.
210. 10% is 60: 3 × 60 + 30 = 210.
3Challenge
Reasoning, generalising and working backwards.
- Working backwards, two stepsAfter spending 40% of her money, Amara has 90 rupees left. How much did she start with?
90 is 60% of the start, 10% is 15, so 150 rupees. - EstimatingIs 41 out of 160 closer to 20%, 25% or 30%?
25%: 25% of 160 is 40.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw a bar of ten boxes on the board and label the part in boxes.