At a glance. One lesson of about 60 minutes. It assumes the idea of a multiplier from Unit 3 and linear functions from Year 9. Students read a, b and the kind of change from y = a times b to the power x, evaluate the model, find b from a table and a from a point, and compare an exponential with a line.
Calculator: scientific calculator. A scientific calculator is needed for powers such as 0.8 to the power 3; the order of operations (power before the multiplication) is the main point of the calculator note.
What students will learn
Know
- y = a × bx has y-intercept a and factor b for each step.
- b above 1 is growth; b between 0 and 1 is decay.
- A constant ratio means exponential; a constant difference means linear.
Understand
- A curve that starts below a line can still overtake it.
- The graph of an exponential decay approaches 0 and never reaches it.
- The exponent is a count of multiplications, not a multiplier.
Do
- Evaluate y = a × bx for a given x.
- Find b from a table by dividing consecutive values.
- Find a by substituting a point.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the form of an exponential, evaluating it, a and b (questions 1 to 7).
- Criterion B. Investigating patterns: the second figure. Ask students to find how the first day of overtaking changes with the spread, and whether it ever fails to happen.
- Criterion C. Communicating: writing the model from a description and stating the base in words (the worked examples).
- Criterion D. Applying mathematics in real-life contexts: a culture of cells, signal loss in a cable, a pond, a machine losing value.
The one thing to do with the figure
Make students commit before they slide. In the second figure, ask “On which day will the lilies first be bigger?” for the spread 7 before pressing anything.
In the first figure, ask what the value will be at x = 0 for any base. Then change the base and check that it does not move.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: lilies and reeds. Collect predictions about whether and when. |
| 10 min | First figure and key idea: a is the value at 0, b is the factor. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose B or D, write 30 = 1 on the board and ask why. |
| 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 8. |
The answers
| Warm-up | 14; 3; 32. |
|---|---|
| Hook | On day 6 the lilies cover 64 against the reeds' 73, and on day 7 they cover 128 against 85. So the lilies first lead on day 7. |
| Worked example | y = 800 × 1.5x; 1.53 = 3.375; 2700 cells. Signal: decay, y = 40. |
| Faded example | 0.83 = 0.512; 250 × 0.512 = 128 thousand units. |
| 1. The value of a at x = 0 | C. A multiplies by x; B gives the value at x = 1; D gives the bases. |
| 2. Evaluate an exponential | 108. 4 × 27. |
| 3. Evaluate a decay model | 3. Five halvings. |
| 4. Find the base from a table | 1.5. 9 ÷ 6. |
| 5. Identify decay | A. B has base 4, so it grows; C is a falling line; D is a rising line. |
| 6. Find a from a point | 5. 80 ÷ 16. |
| 7. Compare a line and an exponential | 594. 640 − 46. |
| 8. Choose the model for a table | D. A doubles instead of halving; B is a line; C starts at 40. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Multiplies by x, gives the value at x = 1, or gives the bases. | Ask: what is any number to the power 0? Then what is left? |
| 1728, 36 or 12 in question 2 | Raises 4 × 3 to the power 3, uses two factors of 3, or stops after one factor. | Write 4 × 3 × 3 × 3 as three separate multiplications by 3. |
| 240, 48 or 1.5 in question 3 | Multiplies the base by the exponent, halves once, or halves six times. | Draw the chain 96, 48, 24 and count the arrows. |
| 3, 6 or 4.5 in question 4 | Gives a first difference, the start value, or a later difference. | Ask: if the pattern is multiplying, should you divide or subtract? |
| B, C or D in question 5 | Picks a growth base, a line that falls, or a line that rises. | Ask: what happens to y when x goes up by 1? |
| 20, 40 or 1280 in question 6 | Divides by the base once, divides by the exponent, or multiplies by the power. | Write a × 16 = 80 before dividing. |
| 74, 274 or 604 in question 7 | Uses 10 × 2 × 6, doubles 5 times, or leaves the start out of the line. | Make a six-row table for both plans. |
| A, B or C in question 8 | Doubles instead of halving, uses a line, or starts at the wrong value. | Test each equation at x = 0, then at x = 1. |
Questions worth asking
- Why is the value at x = 0 always a? Any number to the power 0 is 1, so the factor b has not been applied yet.
- Why can the lilies be behind and then ahead? The line adds the same amount each day, the lilies add more each day because they multiply the larger area.
- Can y = a × bx ever be 0? No, for a above 0 and b above 0: it gets close and never arrives.
- How is an exponential different from a straight line in a table? Equal ratios of consecutive y-values, not equal differences.
Differentiation
A way in for anyone stuck
- Give the table of 2 × 1.2 to the power x for x = 0 to 5 with the ratio column blank.
- Use a checklist: what is a? what is b? is b above or below 1? what is the power?
- Allow the calculator's repeated-multiplication mode (multiply the last answer by b each time) before introducing the power key.
Stretch
- Find b in y = 3 × bx if the curve passes through (2, 48) and b is above 0.
- For which spreads of the reeds do the lilies first lead on day 5? Use the second figure's idea of testing days.
Students learning in English as an additional language
“Growth” means getting bigger and “decay” means getting smaller. “Base” is the number that is multiplied again and again. “Factor” here means the number you multiply by. “Exponent” is the small raised number that counts the multiplications.
| Word | Plain meaning | Example |
|---|---|---|
| exponential function | a function where y is multiplied by the same factor each time x goes up by 1 | y = a × bx |
| base | the number that is raised to a power; here the factor for each step | the 1.3 in 7 × 1.3x |
| growth | values getting bigger because the base is above 1 | a population that multiplies each year |
| decay | values getting smaller because the base is between 0 and 1 | how far a bowl of soup is above room temperature as it cools |
| y-intercept | the value of y where the graph crosses the y-axis, which is a | the start of a model |
| exponent | the power, the count of multiplications | the 2 in 52 |
Sentence starters for explaining aloud or in writing:
- The start value is … and the factor for each step is ….
- This is growth / decay because the base is ….
- The ratio of consecutive values is …, so the model is ….
Two things not to say
“Exponential just means fast.” It means a constant ratio; a decay can be slow.
“The exponent tells you what to multiply by.” It counts how many times the base is used.
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.
- Evaluatey = 2 × 5x. Find y when x = 3.
250. - ClassifyIs y = 90 × 0.8x growth or decay?
Decay, because 0.8 is less than 1.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA town's number of recycling bins is y = 400 × 1.1x after x years. Find y at x = 2.
484. - Earlier work (Unit 3)A sequence has first term 7 and common ratio 3. Find the fourth term.
189. 7 × 27.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan y = a × bx be negative if a and b are both above 0? Explain.
No: a positive base to any power is positive, and a positive times a positive is positive. - Working backwardsy = a × 2x passes through (3, 56). Find a.
7. 56 ÷ 8.
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 squared paper and ask them to fold it in half repeatedly, counting the layers.