At a glance. One lesson of about 60 minutes. Students find percentage increase and decrease from two amounts, work back to an original from a new amount, and use a scientific calculator once, with an estimate.
Calculator: scientific calculator. A scientific calculator is used for an awkward division such as 250 ÷ 1350. The estimate and the choice of the original are done by hand, because a calculator divides whatever it is given.
What students will learn
Know
- Percentage change = change ÷ original × 100.
- The original is the amount you start from, the before.
- To find the original, divide the new amount by the multiplier.
Understand
- The same gap is a different percentage when the original is different.
- A rise and a fall between the same two amounts have different percentages.
- Taking a percentage off the new amount does not undo the change.
Do
- Find a percentage increase or decrease from two amounts, to a given accuracy.
- Find the original amount from a new amount and a percentage change.
- Compare changes of different amounts using percentages.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: percentage change and the original from the new amount (questions 1 to 6).
- Criterion B. Investigating patterns: the first figure. Ask students to find pairs of amounts with a change of 20 and different percentages, and to state the rule linking the original to the size of the percentage.
- Criterion C. Communicating: explaining why a rise of 25% and a fall back are not both 25%, using original and new (question 9).
- Criterion D. Applying mathematics in real-life contexts: transactions, fares, discounts and visitor numbers (the hook and questions 2 to 5 and 7).
The one thing to do with the figure
Ask for a prediction before the first slide. “80 to 100 is a rise of 25%. Is 100 back to 80 a fall of 25%?” Take 25% from the room and then let the figure show 20%.
In the second figure, ask students to predict where “taking the percentage off” will land before they move the slider. Then show that dividing by the multiplier lands exactly on the original every time.
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 predict the wrong route before the figure shows it. |
| 10 min | Worked examples; students finish the faded phone price in pairs. |
| 5 min | Question 1, the check. If many choose A or C, go back to the first figure and name the original. |
| 15 min | Your turn, Levels 1 to 3. Question 7 is the calculator question. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 1.3; 25%; 13.3. |
|---|---|
| Worked example | 20% increase; 450 rupees. |
| Faded example | 25% decrease. |
| 1. Which calculation | B. A uses the new amount as the divisor; C gives the new price as a percentage of the old; D is the change. |
| 2. Percentage increase | 25. 15 ÷ 60 × 100. |
| 3. Percentage decrease | 20. 100 ÷ 500 × 100. |
| 4. Percentage increase | 15. 6 ÷ 40 × 100. |
| 5. Original after a reduction | 480. 360 ÷ 0.75. |
| 6. Comparing percentage changes | B. A compares the changes without the original; C misreads 20 to 30; D compares the new amounts. |
| 7. Percentage decrease on a calculator | 18.5. 250 ÷ 1350 × 100 = 18.5185... |
| 8. Original after an increase | 300. 345 ÷ 1.15. |
| 9. Up then back down | B. A uses the same gap; C swaps the originals; D reads the change as a percentage. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Divided by the new amount, gave the new as a percentage of the old, or stopped at the change. | Name the before and the after. The before is the divisor. |
| 15 in question 2; 100 in question 3; 6 in question 4; 250 in question 7 | Gave the change only. | Ask: is 15 a percentage or an amount of the original? |
| 0.25 in question 2 | Stopped at change divided by original. | Ask what 0.25 means as a number of hundredths. |
| 125 in question 2; 80 in question 3; 115 in question 4; 81.5 in question 7 | Gave the new amount as a percentage of the original. | Subtract the result from 100%, or find the change first. |
| 25 in question 3; 22.7 in question 7 | Divided by the new amount. | Mark the original on the figure and divide by it. |
| 450 or 385 in question 5 | Added the percentage to the new amount, or added the percentage number. | Use the second figure: a fall leaves 100% minus the percentage, so divide by that multiplier. |
| 270 in question 5 | Took another percentage off the new amount. | Ask: did the price go up or down to reach the new amount, and so is the original larger or smaller? |
| A, C or D in question 6 | Compared the changes and ignored the originals. | Work each as change over original. |
| 293.25 or 396.75 in question 8 | Applied the percentage to the new amount. | Draw the bar: the new amount is 115% of the original. |
| 330 in question 8 | Took 15 off the new amount, treating the percentage as rupees. | Ask: is 15 a number of rupees or a number of hundredths? |
| A, C or D in question 9 | Used the same gap, or swapped the originals. | Label before and after on both changes. |
Questions worth asking
- A price goes from 50 to 60 and later from 60 to 50. Are the percentages the same? No: +20% and about −16.7%. Listen for “the original is different”.
- Why can a percentage decrease never be more than 100%? A fall of 100% leaves nothing. A percentage increase has no upper limit.
- Why divide by the multiplier and not take off the percentage? Because the percentage was taken from the original, which is smaller than the new amount.
- If a price doubles, what is the percentage increase? 100%. The new amount is 200% of the original.
- A pass rate rises from 40% to 50%. Is that a 10% increase? It is a rise of 10 percentage points, but the percentage increase is 10 ÷ 40 = 25%. The original, 40, is the divisor.
Differentiation
A way in for anyone stuck
- Give a three-row template: original, new, change. Fill it in before any division.
- Use amounts where the original divides 100 cleanly: 20, 25, 40, 50, 80, 200.
- Use the multiplier from lesson 5 for the working-back questions, and check with a forward calculation.
Stretch
- Question 7, then: by how much must the number of visitors rise from 1100 to return to 1350? (250 ÷ 1100 = 22.7%.) Why is it not 18.5%?
- A shop raises a price by 20% and then lowers it by 20%. What percentage of the starting price does it end at, and what single percentage change is that? (96%, a fall of 4%.)
Students learning in English as an additional language
“From … to …” and “by” do different jobs. Practise “rises from 80 to 100” and “rises by 25%” as pairs.
| Word | Plain meaning | Example |
|---|---|---|
| percentage change | the change as a percentage of the original | 80 to 100 is a 25% increase |
| original | the amount you start from | 80 in 80 to 100 |
| new amount | the amount you end at | 100 in 80 to 100 |
| increase, decrease | a change up, a change down | from 400 to 300 is a decrease |
| multiplier | the number that applies a percentage change in one step | 0.8 for a fall of 20% |
| discount | a reduction in price | 20% discount on a jacket |
Sentence starters for explaining aloud or in writing:
- The change is …, the original is …, so the percentage change is …
- The new amount is …% of the original, so the original is … ÷ …
- The percentages are different because the original is …
Two things not to say
“Always divide by the bigger number.” Divide by the original, which is the smaller number for a rise and the larger for a fall.
“Just add the percentage on to get back.” Adding or taking away the percentage of the new amount does not return to the original.
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.
- Percentage changeA price goes from 50 to 65. Find the percentage increase.
30%. - DecreaseA price goes from 80 to 60. Find the percentage decrease.
25%.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA bakery sold 250 loaves on Monday and 200 on Tuesday. Find the percentage decrease.
20%. The change is 50, and 50 ÷ 250 = 0.2. - Earlier work (Lesson 5 of this unit)A jacket costing 450 is cut by 20% and the sale price is then raised by 20%. Find the final price and the overall percentage change.
432, which is 18 below 450: a decrease of 4%.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA price falls by 20%. By what percentage must it rise to return to the original?
25%: 100 falls to 80, and 20 ÷ 80 = 25%. - ReverseAfter a 12% decrease a price is 440. What was it before?
440 ÷ 0.88 = 500.
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? Write the original on one strip of paper and the new amount on another and lay them side by side.