At a glance. One lesson of about 60 minutes. Students increase and decrease amounts with a multiplier and by building from 10%, and see why a rise then a fall of the same percentage ends below the start.
Calculator: no calculator. Nothing here needs one. The multipliers are friendly and the check by building from 10% is the learning, not the multiplication.
What students will learn
Know
- Increase by p% is a multiplication by 1 + p/100; decrease by p% by 1 − p/100.
- A rise then a fall of the same percentage ends below the start.
- Successive percentage changes are multiplied, not added.
Understand
- A percentage change is always a percentage of the amount at the start of that step.
- 100% plus a percentage rise is more than 100%; 100% minus a percentage fall is less.
- One multiplier can stand for a whole sequence of changes.
Do
- Increase and decrease an amount by a percentage with a multiplier.
- Check an answer by finding the change and adding or subtracting it.
- Find the effect of two successive changes.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: increase and decrease with a multiplier and by building (questions 1 to 6).
- Criterion B. Investigating patterns: the second figure. Ask students to find the rule for how far below the start the amount ends, as a percentage, and to test it on a new x.
- Criterion C. Communicating: explaining why +50% and −50% do not cancel, using the words original, change and multiplier (question 9).
- Criterion D. Applying mathematics in real-life contexts: fares, sale prices, populations and service charges (the hook and questions 2, 3, 5 and 8).
The one thing to do with the figure
Ask for a prediction before the first slide. “80 goes up by 25%. What is the new price?” Take 100 and 1.25 from the room and let the figure settle it, then try a decrease.
In the second figure, predict first: “up 50% then down 50%. Back where it started?” Most say yes. Then drag x from 5 to 50 and ask what grows as x grows.
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. Ask for the multiplier before the new amount. |
| 10 min | Key idea; the second figure. Students predict 50% up then down before any slider moves. |
| 10 min | Worked example on the bag; students finish the faded service charge in pairs. |
| 5 min | Question 1, the check. If many choose A or D, go back to the first figure and read the multiplier. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 42; 0.07; 115. |
|---|---|
| Worked example | 544 naira. |
| Faded example | 428 rupees. |
| 1. Choosing the multiplier | B, 1.15. A is the change only; C is 1500%; D is a decrease. |
| 2. Increase an amount | 60. 50 × 1.2. |
| 3. Decrease an amount | 480. 640 × 0.75. |
| 4. Increase with a multiplier | 138. 120 × 1.15. |
| 5. Growth of a population | 8480. 8000 × 1.06. |
| 6. Multiplier for a decrease | B, 0.88. A keeps 12%; C is a rise; D multiplies by 12. |
| 7. Up then down | 240. 250 × 1.2 × 0.8 = 250 × 0.96. |
| 8. Two reductions | 360. 500 × 0.8 × 0.9 = 500 × 0.72. |
| 9. Rise then fall | B. A assumes the changes cancel; C gives the size of the fall; D ignores the rise. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used the change alone, the percentage as the multiplier, or the decrease multiplier. | Ask: is the new amount 100% or more? So is the multiplier more than 1? |
| 10 or 160 in questions 2 and 3 | Gave the change only. | Ask what the question asks for: the new amount or the change? |
| 70, 615 or 135 in questions 2 to 4 | Added or subtracted the percentage number. | Show 10% of 50 is 5, not 10, and ask what 20% is. |
| 40 in question 2; 102 in question 4; 7520 in question 5 | Subtracted for an increase. | Ask: did it go up or down? Then draw an arrow. |
| A or C in question 6 | Confused 0.12 with 0.88, or used the increase multiplier. | Write 100% − 12% = 88% and then 0.88. |
| 250 or 300 in question 7 | Assumed the changes cancel, or stopped after the rise. | Use the second figure with x = 20 and a start of 100. |
| 200 in question 7 | Took the second percentage off the original. | Write the amounts in a column: 250, 300, then 20% of 300. |
| 350, 400 or 450 in question 8 | Added 20% and 10%, or applied only one change. | Work a column: 500, 400, 360. Ask which amount the 10% is taken from. |
| A, C or D in question 9 | Assumed the changes cancel, gave the size of the fall, or ignored the rise. | Use the second figure with x = 40 and a start of 100. |
Questions worth asking
- A price rises by 10% then falls by 10%. Higher, lower or the same? Lower: 100 × 1.1 × 0.9 = 99. Listen for “the fall is taken from a larger amount”.
- What single multiplier is the same as × 1.1 × 1.1? 1.21, which is a rise of 21%, not 20%. Ask where the extra 1% comes from.
- Why is the multiplier for 15% off 0.85 and not 0.15? 0.15 keeps only 15%. A fall of 15% leaves 85%.
- By how much must a price fall to undo a rise of 25%? 20%, since 25 out of 125 is 20%. Not 25%.
Differentiation
A way in for anyone stuck
- Give a multiplier card: 5% = 1.05 and 0.95; 10% = 1.1 and 0.9; 20% = 1.2 and 0.8; 25% = 1.25 and 0.75.
- Use the building method (10%, 5%, then add or subtract) before the multiplier, and compare the two answers.
- Write the amounts in a column: start, after the first change, after the second.
Stretch
- Question 8, then: what single percentage change equals a rise of 20% followed by a rise of 10%? (1.2 × 1.1 = 1.32, so 32%.)
- Why does x × x ÷ 100 give the percentage below the start for any rise then fall by x%? Check it for x = 30.
Students learning in English as an additional language
“Increase” and “decrease” both have a direction word. Practise “goes up by” and “falls by” with the same number, and watch “to” against “by”.
| Word | Plain meaning | Example |
|---|---|---|
| increase | to make an amount bigger | a rise of 20% on 50 gives 60 |
| decrease | to make an amount smaller | a fall of 25% on 640 gives 480 |
| multiplier | the number you multiply by to apply a change in one step | 1.25 for a rise of 25% |
| original | the amount before a change | the old fare, 50 rupees |
| reduction | a decrease in a price | a 15% reduction |
| service charge | an extra percentage added to a bill | 7% added to 400 rupees |
Sentence starters for explaining aloud or in writing:
- The multiplier is … because the new amount is …% of the original.
- After the rise the amount is …, and the fall is …% of that, which is …
- These changes do not cancel because …
Two things not to say
“Just add the percentages.” Rises and falls of the same percentage do not cancel, because each is a percentage of a different amount.
“Take the percentage off the original.” Not when there are two changes. The second is taken from the amount after the first.
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.
- MultiplierWhat multiplier increases an amount by 8%? What decreases by 8%?
1.08 and 0.92. - IncreaseIncrease 60 by 10%.
66.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA phone that costs 900 dirhams is reduced by 20%. What does it cost now?
720. 20% of 900 is 180, and 900 − 180 = 720. - Earlier work (Lesson 4 of this unit)40% of a number is 36. What is the number increased by 40%?
The number is 90, and 90 × 1.4 = 126.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA price rises by 25% and then falls by 20%. What is the overall change?
1.25 × 0.8 = 1, so no change at all. - Repeated percentageA bank pays simple interest: 3% of the 1500 rupees saved, the same amount every year. How much interest is paid in 4 years?
3% of 1500 is 45 a year, so 4 × 45 = 180 rupees.
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 10 boxes for the original and shade the extra or the removed boxes.