The shape of the lesson
By the end of the lesson, learners can
- Percentages of amounts: In the lab, choose Percentage of and drag the percentage slider along the double number line: the top shows the percentage, the bottom shows the amount.
- Increase, decrease and change: A multiplier does an increase or decrease in one step.
- Reverse and repeated percentages: (Extended) When you know the amount AFTER a change, the new amount is a percentage of the original.
- Percentage of: multiply by the decimal: 18% \(\to 0.18\).
- As a percentage: \(\dfrac{\text{part}}{\text{whole}} \times 100\), same units.
- Increase/decrease: multiply by \(1 \pm \dfrac{r}{100}\).
- Percentage change: change ÷ ORIGINAL \(\times 100\).
- Reverse (Extended): divide the new amount by the multiplier.
One picture every idea returns to
Use a multiplier: an increase of \(r\%\) multiplies by \(1 + \dfrac{r}{100}\), a decrease of \(r\%\) by \(1 - \dfrac{r}{100}\). To go back to the original (reverse percentage), DIVIDE by the multiplier.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Out of a hundred.. Per cent means out of 100. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Percentages of amounts | In the lab, choose Percentage of and drag the percentage slider along the double number line: the top shows the percentage, the bottom shows the amount. Project tab 3 of the lab (Ratio, rates and percentages lab, percentages) and let a student drive it while the class predicts. Model the worked example As a percentage on the board, one line at a time. Your turn: Write 45 as a percentage of 36. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Increase, decrease and change | A multiplier does an increase or decrease in one step. Model the worked example Increase and decrease on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Reverse and repeated percentages | (Extended) When you know the amount AFTER a change, the new amount is a percentage of the original. Model the worked example Reverse percentage (Extended) on the board, one line at a time. | Mix-Pair-Share |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 14 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Reverse and repeated percentages, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Mix-Pair-Share
Students mix around the room until you call pair; they pair with the nearest classmate, discuss the prompt, then mix again. Three prompts of rising difficulty, three different partners.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Divide by the WHOLE, not by the bigger number out of habit. 'What percentage of 36 is 45?' has 36 on the bottom, giving 125%, not 80%. | Return to Percentages of amounts: ask a pair to state the correct version and show it on the lab before moving on. |
| Percentage change uses the ORIGINAL value on the bottom. | From \(\$64\) to \(\$52\) is \(\dfrac{12}{64}\), not \(\dfrac{12}{52}\). |
| For a reverse percentage, do NOT take 20% of the sale price and add it on: \(56 + 0.2 \times 56 = 67.20\), not 70. | The 20% was of the ORIGINAL price, so divide by 0.8. |
Percentages: change, reverse percentages: worksheet
6 questions, 14 marks. Written in the style of the papers; no past paper question is reproduced.
Work out (a) 18% of 350 (b) 135% of 80.
Express 42 seconds as a percentage of 2 minutes.
A bus fare of \(\$85\) increases by 12%. Calculate the new fare.
Ella bought a car for \(\$12\,500\) and sold it for \(\$9800\). Calculate her percentage loss.
(Extended) In a sale all prices are reduced by 30%. A jacket costs \(\$63\) in the sale. Calculate its price before the sale.
A laptop is priced at \(\$750\).
Option A: pay cash and receive an 8% discount.
Option B: pay a 15% deposit, then 10 monthly payments of \(\$66\).
(a) How much more does Option B cost than Option A?
(b) Write the total cost of Option B as a percentage of \(\$750\).
Answer key and marking notes
- (a) \(0.18 \times 350 = 63\)
- (b) \(1.35 \times 80 = 108\)
- 2 minutes \(= 120\) seconds.
- \(\dfrac{42}{120} \times 100 = 35\%\)
- Multiplier 1.12.
- \(85 \times 1.12 = \$95.20\)
- Loss \(= 12\,500 - 9800 = \$2700\).
- \(\dfrac{2700}{12\,500} \times 100 = 21.6\%\)
- The sale price is 70% of the original: multiplier 0.7.
- \(63 \div 0.7 = \$90\)
- Option A: \(750 \times 0.92 = \$690\).
- Option B: deposit \(0.15 \times 750 = \$112.50\); payments \(10 \times 66 = \$660\); total \(\$772.50\).
- (a) \(772.50 - 690 = \$82.50\)
- (b) \(\dfrac{772.50}{750} \times 100 = 103\%\)
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Express 54 cm as a percentage of 2.4 m.
- A22.5%
- B2.25%
- C2250%
- D444%
Decrease \(\$260\) by 35%.
- A\(\$169\)
- B\(\$91\)
- C\(\$351\)
- D\(\$225\)
The price of a game rises from \(\$75\) to \(\$96\). Find the percentage increase.
- A28%
- B21.9%
- C21%
- D128%
A laptop costs \(\$1200\). Buying on credit, Tom pays a 15% deposit and then 24 monthly payments of \(\$46.50\). How much more than \(\$1200\) does he pay, as a percentage of \(\$1200\)?
- A8%
- B7.41%
- C108%
- D93%
(Extended) After a 12% pay rise, Nina's salary is \(\$30\,240\). What was her salary before the rise?
- A\(\$27\,000\)
- B\(\$26\,611.20\)
- C\(\$34\,363.64\)
- D\(\$25\,200\)
(Extended) The price of a bike is increased by 25%. The new price is then reduced by 20%. What is the overall effect on the original price?
- ANo overall change
- BA 5% increase
- CA 5% decrease
- DA 45% increase
Quiz key
\(2.4\ \text{m} = 240\ \text{cm}\), so \(\dfrac{54}{240} \times 100 = 22.5\%\). The tempting 2.25% uses 2400 cm (1 m = 1000 cm). 2250% divides 54 by 2.4 without converting, and 444% divides the whole by the part.
The multiplier is \(1 - 0.35 = 0.65\), so \(260 \times 0.65 = \$169\). The tempting \(\$91\) is the decrease itself (35% of 260), not the new amount. \(\$351\) is an increase of 35%, and \(\$225\) subtracts 35 dollars instead of 35%.
Change \(= 96 - 75 = 21\). Percentage increase \(= \dfrac{21}{75} \times 100 = 28\%\). The tempting 21.9% divides by the NEW price, 96. 21% gives the change in dollars as if it were a percentage, and 128% is the new price as a percentage of the old.
Deposit \(= 0.15 \times 1200 = \$180\). Payments \(= 24 \times 46.50 = \$1116\). Total \(= \$1296\), which is \(\$96\) more. As a percentage of the cash price: \(\dfrac{96}{1200} \times 100 = 8\%\). The tempting 7.41% divides by the credit total 1296 instead of the original 1200. 108% is the total as a percentage of the price, and 93% compares only the monthly payments with 1200.
New salary \(=\) old \(\times 1.12\), so old \(= 30\,240 \div 1.12 = \$27\,000\). The tempting \(\$26\,611.20\) takes 12% off the NEW salary (\(30\,240 \times 0.88\)), but the 12% was of the old salary. \(\$34\,363.64\) divides by 0.88, and \(\$25\,200\) divides by 1.2.
The multipliers are 1.25 and 0.80, and \(1.25 \times 0.80 = 1\). So the final price equals the original price: no overall change. The tempting 5% increase comes from \(25 - 20 = 5\), but the 20% is taken from the larger, increased price, so it removes more money. 45% adds the two percentages.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.