At a glance. One lesson of about 60 minutes. It assumes percentage change from Year 7 and geometric sequences from lesson 4. Students write a multiplier, apply it repeatedly, find a rate from a total and set up a calculation. A scientific calculator is used for powers.
Calculator: scientific calculator. Powers such as 1.04 squared are quick on a calculator and tedious by hand. Students estimate first, so the calculator confirms rather than decides.
What students will learn
Know
- A rise of p% is a multiplier of 1 + p/100; a fall is 1 − p/100.
- Repeating a change t times raises the multiplier to the power t.
- A rate from a total over t years is the t-th root of the total multiplier.
Understand
- Each year's percentage is taken from the current amount, which is why compound growth outruns the straight line.
- A rise followed by an equal fall does not return to the start.
- Percentages cannot be added across years.
Do
- Write the multiplier for a rise or fall.
- Compute start × multipliert.
- Find a rate or a number of years from given values.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: multipliers, powers and totals (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students compare the curve with the dashed line for several rates and describe the gap.
- Criterion C. Communicating: a one-line statement of the multiplier and the power before any calculation.
- Criterion D. Applying mathematics in real-life contexts: savings and depreciation (hook and extra questions).
The one thing to do with the figure
Ask for the prediction first. Then set the rate to 5 and move the years slider, asking students to compare the curve with the dashed line. Set the rate to a negative value and ask what the dashed line does that the curve cannot.
Ask: “Why does the curve fall less steeply each year?” A fixed percentage of a smaller amount is a smaller amount, so the curve flattens but never reaches zero.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: eight years at 5%; collect predictions and reasons. |
| 12 min | Figure, key idea, warning card, calculator note. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or C, ask whether the value goes up or down, and how much of it is kept. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 450; 240; 4. |
|---|---|
| Hook | 1.058 is about 1.477, so the extra is about 47.7%, more than 40%. 4000 grows to about 5910 shillings. |
| Worked examples | 46 305; 19 683; 15%. |
| Faded example | 6365.40. |
| 1. Writing a multiplier | B. A is the part lost; C is the multiplier for a rise; D is the percentage unconverted. |
| 2. Compound growth | 8820. 8000 × 1.1025. |
| 3. Depreciation | 3072. 6000 × 0.512. |
| 4. Compound interest total | 2704. 2500 × 1.0816. |
| 5. How many years | 3. 4840 then 5324. |
| 6. Finding the rate | 30. The cube root of 2.197 is 1.3. |
| 7. Two different rates | 16 200. 18 000 × 0.9. |
| 8. Setting up a calculation | D. A uses 0.08; B adds 8 five times; C is an 80% multiplier. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used the part lost, the multiplier for a rise, or the percentage itself. | Ask: if the laptop keeps 65 of every 100 of its value, what multiplies the value? |
| 8800 or 8400 in question 2 | Took 5% of the original twice, or stopped after one year. | Write the value year by year in a table. |
| 2400 or 4800 in question 3 | Subtracted 20% of the original each year, or did one year. | Write ×0.8 three times. |
| 2700 or 2600 in question 4 | Used simple interest, or did one year. | Ask: what is 4% of 2600? |
| 2.5 or 2 in question 5 | Assumed equal yearly interest, or stopped a year early. | Make a table of values for years 1, 2, 3 and compare each with 5000. |
| 119.7 or 39.9 in question 6 | Gave the total rise, or shared it equally over the years. | Ask: what do you multiply by three times to get 2.197? |
| 15 600 or 18 000 in question 7 | Added the percentages, or did one year. | Apply 0.75 and then 0.9 on separate lines. |
| A, B or C in question 8 | Used 0.08, added 8, or used 1.8. | Test each option at one year: which gives 648? |
Questions worth asking
- How can you tell before calculating that compound growth is more than simple growth? Listen for: the second year's percentage is of a larger amount.
- Why is a 20% fall followed by a 20% rise not back to the start? Listen for: the 20% is taken from different amounts.
- How would you find the rate if you only knew the start and the end after 3 years? Listen for: divide, then take the cube root.
- What does a multiplier of 1 mean? No change.
Differentiation
A way in for anyone stuck
- Give a table with columns for year, multiplier and value, filled one row at a time.
- Provide the multipliers for 2%, 5%, 10%, 20% as a card.
- Give the frame: “The multiplier is …. After … years the value is start × …… = …”
Stretch
- Show that doubling by 5% a year takes about 14 years. How long does 10% take? Use a table.
- A price rises by 20% and then falls by 20%. What single percentage change is equivalent?
Students learning in English as an additional language
“Interest” is the money a bank adds to savings. “Compound” means the interest is added to the balance, so next year's interest is calculated on the new balance. “Depreciation” means a fall in value as something ages. “Multiplier” is the number you multiply by to apply a percentage change.
| Word | Plain meaning | Example |
|---|---|---|
| multiplier | the number you multiply by to apply a percentage change | 1.07 for a 7% rise |
| compound interest | interest that is added to the balance and then earns interest itself | 3% a year on a growing balance |
| depreciation | a fall in value as something ages | a van losing 12% a year |
| growth factor | another name for the multiplier when it is more than 1 | 1.2 |
| decay factor | the multiplier when it is less than 1 | 0.7 |
Sentence starters for explaining aloud or in writing:
- The multiplier is … because …
- After … years the value is … × ……
- I estimated … first, and the calculator gave …
Two things not to say
“Multiply by the percentage.” Say instead: change the percentage to a multiplier first.
“Add the percentages for each year.” Say instead: multiply the multipliers.
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.
- WriteWrite the multiplier for a 7% rise and for a 12% fall.
1.07 and 0.88. - FindFind 800 × 1.52.
1800.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA bank in Mumbai pays 6% compound interest a year. How much is 10 000 rupees worth after 2 years?
10 000 × 1.1236 = 11 236. - Earlier work (lesson 4)Find the 5th term of 3, 12, 48, …
768.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs a 20% increase followed by a 20% decrease back at the start? Explain.
No: 1.2 × 0.8 = 0.96, which is a 4% fall. - Working backwardsAfter 2 years at 10% a year a price is 6050. What was it at the start?
6050 ÷ 1.21 = 5000.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
A scientific calculator is needed for the powers; a phone calculator in landscape also works.