At a glance. One lesson of about 60 minutes. It assumes the index laws and percentage change. Students distinguish a constant ratio from a constant difference, build y = a × bx, and find the first step at which a target is passed by listing powers.
Calculator: no calculator. No calculator: the factors are 2, 3, 5 and small decimals, and the point is to see the power as a step count. Logarithms will turn the listing into one calculation later in the unit.
What students will learn
Know
- y = a × bx: a is the start value and b is the factor for each step.
- Growth has b above 1 and decay has b between 0 and 1.
- A constant ratio between neighbouring terms means exponential; a constant difference means linear.
Understand
- Why the power counts the number of times the factor is used.
- Why the answer to "how many steps?" need not be a whole number.
- Why an exponential model eventually passes any linear model.
Do
- Build y = a × bx from a context.
- Find the first whole step at which a target is passed.
- Find a factor from a start value and a later value.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using powers for repeated multiplication (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say when the power is a whole number and when it falls between two whole numbers.
- Criterion C. Communicating: writing the model and listing powers in a clear table.
- Criterion D. Applying mathematics in real-life contexts: colonies, rumours, cooling and savings (hook and questions 2, 3, 4, 6 and 7).
The one thing to do with the figure
Ask for the prediction first. Then set base 2 and move the target through its four values, and ask students to predict whether the power will be a whole number before they read the tiles.
Ask: “Which base and target give a whole-number power, and what do those targets have in common?” They are powers of the base: 25 for base 5 and 16 or 64 for bases 2 and 4.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the colony in the tray; collect predictions. |
| 12 min | Figure, key idea and warning card; list powers of 2 until a thousand is passed. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or C, ask for the ratio of the second term to the first. |
| 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 | 81; 440; 80. |
|---|---|
| Hook | After 10 hours there are 210 = 1024 cells, and after 9 hours there are 512, so about 10 hours. A hundred or five hundred hours are far too many. |
| Worked examples | 675 rupees; 5; day 5. |
| Faded example | 6 hours (8 × 64 = 512). |
| 1. Telling growth by a factor from growth by an amount | B. A and C add a constant amount; D is the square numbers, which have no constant ratio. |
| 2. Growth by a factor | 384. 12 × 32. |
| 3. Decay by a factor | 10. Three halvings of 80. |
| 4. First step that passes a target | 6. 1458 is short, 4374 passes. |
| 5. Finding the factor | 5. b3 = 125. |
| 6. Which power reaches the target, when a step is longer than one hour | 8. 4 triplings of 2 hours. |
| 7. Comparing growth by a factor and by an amount | 5. 320 against 250. |
| 8. Writing the model from two values | C. A uses the two-step growth as the factor; B adds instead of multiplying; D swaps start and factor. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Chose a list that adds a constant amount, or the square numbers. | Ask for the ratio of each term to the one before, then the difference between each term and the one before. |
| 120 or 300 in question 2 | Multiplied by 2 × 5, or used 52 for 25. | Write out 12, 24, 48, … year by year and count the doublings. |
| 20 or 50 in question 3 | Halved only twice, or subtracted 30 from 80. | Make a table of time and excess: 0, 10, 20, 30 minutes against 80, 40, 20, 10. |
| 5 or 250 in question 4 | Stopped on the last step before the target, or gave the value 3n must reach. | Ask: what is the number of people after 5 hours, and is it more than 1500? |
| 125 or 124 in question 5 | Gave the cube of the factor, or treated the growth as adding a fixed amount. | Ask: how many factors are multiplied together after 3 steps? |
| 4 or 81 in question 6 | Gave the number of triplings, or the value 3n must reach. | Ask: how long does one tripling take, and how many are there? |
| 4 or 320 in question 7 | Stopped one year early, or gave the amount instead of the year. | Fill a table of years 1 to 5 for both plans before comparing. |
| A, B or D in question 8 | Took the two-step growth 16 as the factor, wrote a linear model, or swapped start and factor. | Test each formula at x = 0, 1 and 2: it should give 5, then 20, then 80. |
Questions worth asking
- What is the difference between adding 3 each step and multiplying by 3 each step? Listen for: the second makes the change larger each time because it is a share of a bigger amount.
- Is 25 the same as 2 × 5? Listen for: no, the power counts factors: 2 × 2 × 2 × 2 × 2 is 32.
- Why is the answer not a whole number for a target of 40? Listen for: 40 is between two powers of 2 (32 and 64).
- When does an exponential model overtake a linear model? Listen for: it always does eventually, even if it starts far behind.
Differentiation
A way in for anyone stuck
- Give a table with columns for step, working and value for the first six steps of a doubling colony.
- Provide a list of the powers of 2, 3 and 5 up to 1000 and ask students to use it.
- Give the frame: “The factor is …. After … steps the value is …, which is below/above the target.”
Stretch
- A quantity grows by 20% each step. Write the factor, and show that it takes 4 steps to more than double, by bracketing 1.23 and 1.24.
- Plan A adds 50 each year from 200 and Plan B triples each year from 2. Find the first year that B passes A.
Students learning in English as an additional language
“Factor” here means the number each step is multiplied by. “Exponential” means that the number of steps is in the power. “Decay” means getting smaller by a factor. “Target” is the number we are trying to reach.
| Word | Plain meaning | Example |
|---|---|---|
| exponential growth | increase by the same factor in each step | a colony that triples every hour |
| factor | the number each step multiplies by, written b | b = 1.5 |
| start value | the amount at step 0, written a | a = 200 rupees |
| decay | decrease by a factor between 0 and 1 in each step | a factor of 0.9 |
| power | the number of copies of the base that are multiplied together | the 4 in 74 = 7 × 7 × 7 × 7 |
Sentence starters for explaining aloud or in writing:
- The factor is … because …
- After … steps the value is …, which is below/above the target.
- The power is between … and … because …
Two things not to say
“Exponential means very fast.” Say instead: exponential means multiplying by the same factor, and a factor below 1 makes a quantity shrink.
“The answer is the biggest number in the list.” Say instead: the answer is the position of the first value that reaches the target.
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.
- FindA tree is 2 m tall and its height doubles each year for 4 years. How tall is it?
32 m. - FindWhat power of 2 gives 128?
7.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA phone loses 20% of its value each year and starts at 500 rupees. What is it worth after 2 years?
500 × 0.8 × 0.8 = 320 rupees. - Earlier work (Year 10)Find the common ratio of 6, 18, 54.
3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA population of 50 triples each year. In which year does it first exceed 10 000?
50 × 35 = 12 150, so year 5 (50 × 81 = 4050 is short). - Working backwardsAfter 3 steps a quantity is 108 and the factor is 3. What was the start value?
108 ÷ 27 = 4.
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.
No calculator is needed.