At a glance. One lesson of about 60 minutes. It assumes Lesson 1 and percentage multipliers from Unit 3. Students find a and b from a rate, a half-life or two readings, write y = a times b to the power t, and use it, including at part-periods with a calculator.
Calculator: scientific calculator. A scientific calculator is used for non-integer exponents such as t/4 with t = 7, and for checking powers of decimals; every question is built so the answer can be found by repeated multiplication.
What students will learn
Know
- A rate of r% means b = 1 ± r/100.
- A doubling or half-life time T gives the exponent t/T.
- b = later value ÷ earlier value for two readings one period apart.
Understand
- The base is a multiplier, so a fall is below 1 and a rise is above 1.
- The exponent counts periods, not time units.
- A half-life removes a different amount each time, always half of what is there.
Do
- Write the model from a rate, a doubling time or two values.
- Evaluate it at a whole number of periods.
- Find b from a value at a known x by taking a root.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: base from a rate, the exponent as a count of periods (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to say what percentage change each factor means and to find the factor for a 50% rise.
- Criterion C. Communicating: writing the model with t defined and units stated (the worked examples).
- Criterion D. Applying mathematics in real-life contexts: caffeine in the blood, algae in a pond, machine value, a count each week.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “What will be left after 10 hours?” with a half-life of 4 hours. Most say 15 (halfway between 20 and 10). Slide to show about 14.1.
In the second figure, ask what value after one period means the quantity is unchanged. Then ask what it means when the value is 100.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: coffee at midnight. Collect guesses for the fraction left. |
| 10 min | First figure and key idea: the three ways to find a and b. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A or C, write 100% − 2.5% = 97.5% on the board. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 14; 96; 60. |
|---|---|
| Hook | 16 hours is 3.2 half-lives, and 100 × 0.53.2 is about 10.9 mg, so roughly a tenth is still there at midnight. |
| Worked example | 5 × 24 = 80 g; 500 × 0.82 = 320 g. |
| Faded example | 18 ÷ 6 = 3 half-lives; 360 → 180 → 90 → 45 mg. |
| 1. Base from a percentage fall | B. A is the rate as a decimal; C is a rise; D writes 2.5% as 0.25. |
| 2. A percentage fall repeated | 4860. 6000 × 0.81. |
| 3. A doubling time | 56. 7 × 23. |
| 4. Base from a percentage fall | 1.005. 1 + 0.005. |
| 5. Factor from two readings | 1.3. 325 ÷ 250. |
| 6. A half-life | 15. 120 × 0.53. |
| 7. Base from a point (square root) | 0.6. √0.36. |
| 8. Compare a percentage loss with a fixed loss | 155. 3800 − 3645. |
| 9. Build a model from a tripling time | C. A triples every day; B swaps the 2 and the 3; D multiplies 3 by 2. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Gives the rate as a decimal, a rise factor, or misplaces the decimal point. | Ask: what fraction is kept each hour? |
| 4800, 5400 or 10800 in question 2 | Takes 20% in one go, stops after one week, or multiplies by the exponent. | Write 6000 → ×0.9 → ×0.9 with arrows. |
| 28672, 168 or 28 in question 3 | Uses the hours as exponent, multiplies by the hours, or doubles twice only. | Ask: how many 4-hour periods are in 12 hours? |
| 0.005, 1.05 or 0.995 in question 4 | Gives the rate, misplaces the decimal, or treats the rise as a fall. | Ask: what is 0.5% as a decimal? Is the base above 1? |
| 75, 0.77 or 0.3 in question 5 | Subtracts, divides the wrong way round, or gives the rate. | Say: later over earlier. A rise must give a number above 1. |
| 40, 60 or 24 in question 6 | Divides by the count, halves once, or divides by the hours. | Draw 120, 60, 30, 15 with arrows labelled “5 hours”. |
| 0.36, 1.67 or 0.18 in question 7 | Stops before the root, inverts the ratio, or divides by 1600. | Write 800 × b2 = 288 and solve step by step. |
| 300, 955 or 3795 in question 8 | Takes 30% off in one go, subtracts one 400 only, or uses the 10% lost as the base. | Make a three-row table for each machine. |
| A, B or D in question 9 | Triples every day, swaps 2 and 3, or multiplies 3 by 2. | Test each model at t = 2. |
Questions worth asking
- Why is a 20% fall not b = 0.2? 0.2 is what is removed. What remains is 80%, so the multiplier is 0.8.
- Why is a half-life the same for a big amount and a small one? Each half-life halves whatever is there, so it is a proportion, not an amount.
- How can you tell from two values whether the base is above or below 1? If the later value is larger the ratio is above 1.
- What does t/T count? The number of doubling or halving periods that have passed.
Differentiation
A way in for anyone stuck
- Give a three-column table: period, multiply by, amount, with the first row filled.
- Use a checklist: start value? percentage rate or doubling time? is it a rise or a fall? what is b?
- Keep every question to a whole number of periods before introducing 2.5 half-lives.
Stretch
- A quantity grows by 5% a day. Estimate how many days it takes to double and test it with the power key.
- A substance has a half-life of 4 hours. How much of the start is left after 5.5 hours, to one decimal place of a percentage?
Students learning in English as an additional language
“Half-life” is the time for a quantity to fall to half. “Period” here is one step of time, such as one hour or one week. “Rate” is how much something changes in one period, as a percentage. “Factor” is the number you multiply by.
| Word | Plain meaning | Example |
|---|---|---|
| half-life | the time it takes for a quantity to fall to half | 6 days for a fading dye |
| doubling time | the time it takes for a quantity to double | the days for a patch to cover twice the area |
| growth factor | the number you multiply by for each period | 1.15 for a rise of 15% |
| percentage rate | the percentage change in one period | a rise of 4% a month |
| period | one step of time in the model | one week |
| model | an equation that describes a real situation | y = 400 × 1.1x |
Sentence starters for explaining aloud or in writing:
- The start value is … and the base is … because ….
- The exponent is … because there are … periods in ….
- The quantity is multiplied by … each ….
Two things not to say
“A 20% fall means multiply by 0.2.” It means keep 80%, so multiply by 0.8.
“Half-life means it loses half the original amount every time.” It loses half of what is left.
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.
- BaseA value rises by 15% each year. What is b?
1.15. - DoublingA count of 3 doubles every 2 days. How many after 6 days?
24. Three doublings: 3 × 8.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA phone loses 8% of its battery capacity per year. Find b and the capacity left as a fraction after 2 years.
b = 0.92; 0.922 = 0.8464, about 85%. - Earlier work (Lesson 1)y = 3 × 0.5x. Find y when x = 4.
0.1875.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA quantity multiplies by 1.1 each year. Does it double in fewer than 7 years? Test it.
1.17 = 1.9487, so no; it takes 8 years: 1.18 = 2.1436. - Working backwardsAfter 3 periods a quantity is 135, and it triples each period. What was the start?
5. 135 ÷ 27.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Give each pair a strip of paper to fold in half repeatedly and record the width of the folded strip.