Which would you rather have: $1,000,000 today, or a penny doubled every day for a month?
Learning Targets
I can compare and contrast exponential growth and decay.
I can determine values of simple exponential functions in context.
π§βπ« This lesson reintroduces exponential functions from a previous course through geometric sequences. Students explore exponential decay (passport photo shrinking by 80%) and exponential growth (algae doubling). They reason about constant growth factors, the relationship between the factor and growth/decay, and why exponential growth is deceptively slow at first.
Standards: HSF-LE.A.1.c, HSF-LE.A.2
Prep Checklist
Rulers
Warm-Up β± 5 min
Bank Accounts
π§βπ« This warm-up reminds students about arithmetic and geometric sequences. Read the prompt aloud. Give 2 minutes of individual think time, then 2 minutes of partner share. Listen for: equal differences (arithmetic) vs. equal quotients (geometric). Select one example of each to share during synthesis.
A bank account has a balance of $120 on January 1. Describe a situation in which the account balance for each month (February 1, March 1, β¦) forms each type of sequence. Write the first three terms of each sequence.
Work on your own, then share with a partner
Describe a situation that forms an arithmetic sequence. Write the first 3 terms.
Describe a situation that forms a geometric sequence. Write the first 3 terms.
Hint: arithmetic β Greek for "the art of counting" geometric β geo + metric, Greek for "Earth measurement" (measuring a field: area = length Γ width)
β± 2 min individual think time before partner work
Warm-Up Synthesis β± 3 min
Arithmetic vs. Geometric β What Is the Difference?
π§βπ« Invite students to share examples. Highlight: arithmetic sequences have equal differences between terms; geometric sequences have equal quotients. Select one of each so that the arithmetic initially grows faster. Ask: "Will the arithmetic sequence always stay ahead?" Do not resolve β plant the seed for the lesson.
Arithmetic
\(120,\quad\)
\(120 +\) \(\quad \ldots\)
Each month I add to the previous balance.
Geometric
\(120,\quad\)
\(120 \times\)\(\quad \ldots\)
Each month I multiply the previous balance by .
The purpose of this discussion is to surface the core difference between arithmetic and geometric growth patterns.
“What is the same about your two sequences?”Both start at $120 and increase each month.
“What is different about how each sequence grows?”The arithmetic sequence adds the same amount each time. The geometric sequence multiplies by the same factor each time.
“Do you think the arithmetic sequence will always have a larger balance than the geometric sequence?”Students may disagree β that is fine. The goal is to plant the question, not resolve it.
click to advance discussion βΆ
Try Saying
“I notice that the arithmetic sequence ___ while the geometric sequence ___.”
Try Saying
“I think the ___ sequence will eventually be larger because ___.”
Sample Responsesclick to reveal βΆ
Arithmetic: Each month the owner deposits $20. No interest. $120, $140, $160, β¦
Geometric: Each month the account earns 10% interest. No other deposits. $120, $132, $145.20, β¦
Key idea: Successive terms in an arithmetic sequence have equal differences. Successive terms in a geometric sequence have equal quotients (ratios).
Activity 1 Β· Launch β± 3 min
Shrinking a Passport Photo
π§βπ« Tell students they will watch a short video. Give 1 minute of quiet think time. Ask: "What did you notice? What do you wonder?" Invite 2β3 students to share, then explain the task: Elena needs to shrink a photo to meet passport requirements. Provide rulers if requested. Do not over-explain the 80% scaling β let students figure out how to use it.
Elena needs a passport photo. The measurement from her chin to the top of her head must be between 25 mm and 35 mm. Her current photo measures 150 mm for that distance. She has a photocopier that reduces the height to 80% of its previous value each time.
Your task
Find the chin-to-head measurement after scaling by 80% a total of 3 times. Then find it after 6 times.
How many times must the image be scaled for the measurement to be less than 35 mm?
How many times must it be scaled for the measurement to be less than 25 mm?
β± 1 min individual think time before partner work
Activity 1 Β· Work Time β± 10 min
Shrinking a Passport Photo
π§βπ« Monitor for strategies in order of abstraction: (1) listing out successive values, (2) repeated multiplication expression like \(150 \cdot 0.8 \cdot 0.8 \cdot 0.8\), (3) exponent notation like \(150 \cdot 0.8^n\), (4) a function \(f(n) = 150 \cdot 0.8^n\). Select students with different strategies to share in synthesis. Common error: students who add 80% instead of multiplying by 0.8 β watch for answers like 120, 90, 60. Another error: students who subtract 20% of the original each time (getting 150, 120, 90, 60) instead of 20% of the current value.
If Students Are Stuck
“Tell me about a time when you or someone else had to change the size of a photo.”
“Sketch and label the sides of a photo that is half the size, or 50%, of the original. What do you notice?”
Amplify Β· Activity Builder Β· Type-in
Setup (do once):
1. Add a Text component β include all student-facing text from the slide.
2. Add a Math Response component β name it answer. Example: 76.8
3. Add a Note component below it
4. Click </> on the Note β paste CL below
5. Straight quotes only β curly quotes break CL
Problem 1: After 3 scalings
content:
when answer.submitted and numericValue(answer.latex) = 76.8
"Correct! 76.8 mm"
when answer.submitted and numericValue(answer.latex) = 77
"Correct! About 77 mm (76.8 mm exactly)"
when answer.submitted and numericValue(answer.latex) = 90
"Almost! That is 150 minus 20% of 150 three times. But each copy is 80% of the PREVIOUS copy, not 80% of the original. Try: 150 times 0.8, then times 0.8 again."
when answer.submitted and numericValue(answer.latex) = 120
"Not quite - 120 is 80% of 150 (one scaling). What happens after two more scalings?"
when answer.submitted
"Multiply 150 by 0.8, then multiply that result by 0.8 again, and again. What do you get?"
otherwise ""
Problem 2: After 6 scalings
content:
when answer.submitted and numericValue(answer.latex) >= 39.3
and numericValue(answer.latex) <= 39.4
"Correct! About 39.3 mm"
when answer.submitted and numericValue(answer.latex) = 40
"Correct! Approximately 40 mm (39.3 mm exactly)"
when answer.submitted and numericValue(answer.latex) = 30
"Good start - but 30 is 150 minus 20% of 150 six times. Each copy is 80% of the PREVIOUS copy. Start from your answer to part 1 and multiply by 0.8 three more times."
when answer.submitted
"Start from 76.8 mm (after 3 scalings) and multiply by 0.8 three more times. What do you get?"
otherwise ""
Problem 3: Scalings for less than 35 mm
content:
when answer.submitted and numericValue(answer.latex) = 7
"Correct! After 7 scalings the measurement is about 31.5 mm, which is less than 35."
when answer.submitted and numericValue(answer.latex) = 6
"You're close - after 6 scalings the measurement is about 39.3 mm. That is still more than 35. Try one more."
when answer.submitted and numericValue(answer.latex) = 8
"Almost! 7 scalings is enough. After 7 the measurement is about 31.5 mm. Check: 150 times 0.8 to the 7th."
when answer.submitted
"Keep multiplying by 0.8 until you get below 35 mm. How many times does that take?"
otherwise ""
Problem 4: Scalings for less than 25 mm
content:
when answer.submitted and numericValue(answer.latex) = 9
"Correct! After 9 scalings the measurement is about 20.1 mm, which is less than 25."
when answer.submitted and numericValue(answer.latex) = 8
"You're close - after 8 scalings the measurement is about 25.2 mm. That is still more than 25. Try one more."
when answer.submitted and numericValue(answer.latex) = 10
"Almost! 9 scalings is enough. After 9 the measurement is about 20.1 mm."
when answer.submitted
"Continue from 35 mm and keep multiplying by 0.8. How many total scalings until you get below 25?"
otherwise ""
Work with your partner
Find the chin-to-head measurement after the image has been scaled by 80%:
(a) 3 times (b) 6 times Explain or show your reasoning.
How many times must the image be scaled by 80% for the measurement to be less than 35 mm?
How many times must the image be scaled by 80% for the measurement to be less than 25 mm?
Try Saying
“First I ___, then I ___.”
Try Saying
“I multiplied by 0.8 because ___.”
β Challenge
(Challenge 1) If Elena instead uses a copier that reduces the height to 75% each time, how many copies does she need for the measurement to be between 25 mm and 35 mm? Is there more than one valid number of copies?
(Challenge 2) Write an expression using an exponent that gives the chin-to-head measurement after \(n\) scalings at 80%. Use your expression to find the measurement after 12 scalings.
Activity 1 Β· Synthesis β± 5 min
Strategies for Finding the Shrinking Values
π§βπ« Sequence student sharing from most concrete to most abstract: (1) listing values, (2) repeated multiplication, (3) exponent notation, (4) function notation. Record each on the board. Key question: "Which method is most efficient for jumping ahead?" Reinforce that repeated multiplication can be expressed efficiently with an exponent: \(150 \cdot 0.8^n\).
The purpose of this discussion is to connect different strategies and show how repeated multiplication leads naturally to exponent notation.
“How is the 80% value used in each of the methods?”In listing, we multiply each value by 0.8. In repeated multiplication, 0.8 appears as a factor multiple times. In exponent notation, 0.8 is the base raised to a power.
“Which method is most efficient for finding the next value in the sequence?”Listing β just multiply the current value by 0.8.
“Which method is most efficient for jumping ahead several positions in the sequence?”Exponent notation β compute \(150 \cdot 0.8^n\) directly without finding every value in between.
Reinforce: repeated multiplication can be expressed efficiently with an exponent. \(150 \cdot 0.8 \cdot 0.8 \cdot 0.8 = 150 \cdot 0.8^3\).
click to advance discussion βΆ
Try Saying
“I think the ___ method is most efficient because ___.”
Try Saying
“I notice that 0.8 appears in every method as ___.”
Sample Responsesclick to reveal βΆ
(a) After 3 scalings: \(150 \cdot 0.8^3 = 150 \cdot 0.512 = 76.8\) mm (about 77 mm)
(b) After 6 scalings: \(150 \cdot 0.8^6 \approx 39.3\) mm (about 40 mm)
Less than 35 mm: 7 times β \(150 \cdot 0.8^7 \approx 31.5\) mm
Less than 25 mm: 9 times β \(150 \cdot 0.8^9 \approx 20.1\) mm
Math As A Second Language
The Exponential Function
Math Β· We Say Β· Meaning
β Math (given)
\(f(x) = a \cdot b^x\)
β² We Say
“f of x equals a times b to the x”
β Meaning
An exponential function has a constant starting value a multiplied by a growth factor b raised to the input x. Example: \(f(x) = 10000 \cdot 2^x\) models a population doubling each day.
π§βπ« This is the general form. Connect both activities: passport photo β \(f(n) = 150 \cdot 0.8^n\) (a = 150, b = 0.8, decay). Pond β if we let x = days since May 12, then coverage = \(\frac{1}{4096} \cdot 2^x\) (a = 1/4096, b = 2, growth). Read the We Say card aloud. Emphasize "b to the x" β students should practice saying this.
Math As A Second Language
Exponential Decay β Growth Factor Less Than 1
Math Β· We Say Β· Meaning
β Math (given)
\(0 < b < 1\)
β² We Say
“b is between 0 and 1”
β Meaning
When the growth factor b is between 0 and 1, the exponential function models decay β the output decreases as x increases. Example: b = 0.8 means 80% remains each step.
π§βπ« Connect directly to the passport photo: the growth factor was 0.8, which is between 0 and 1. Each copy was 80% of the previous β the photo shrank. This is exponential decay. Read the We Say card aloud together.
Activity 2 Β· Launch β± 3 min
Pond in a Park
π§βπ« Read the problem stem aloud. Draw a simple rectangle on the board to represent the pond. Ask: "How might the algae look on Day 1? Day 2?" Let students suggest β the point is to build intuition about doubling. Then ask students to proceed with the task.
On May 12, a fast-growing species of algae is accidentally introduced into a pond in a park. The area of the pond that the algae covers doubles each day. If not controlled, the algae will cover the entire surface of the pond on May 24.
Your task
On which day is the pond halfway covered?
On May 18, Clare visits the park. A caretaker says the pond will be completely covered in less than a week. Clare looks at the pond and thinks the caretaker must be mistaken. Why might she find the claim hard to believe?
β± 1 min individual think time before partner work
Activity 2 Β· Work Time β± 10 min
Pond in a Park
π§βπ« Monitor for students who use tables, work backward from May 24, or reason about fractions. Common misconception: students think halfway covered = halfway through the time (May 18), which reflects linear not exponential thinking. If students think May 18 is the answer, ask: "If the pond is half covered on May 18, what happens on May 19?" For Clare's question, look for students who calculate the actual coverage on May 18.
If Students Are Stuck
“Can you explain how you knew when the pond would be half covered?”
“If the pond is half covered on May 18th, what will the pond look like on May 19th?”
Amplify Β· Activity Builder Β· Type-in
Setup (do once):
1. Add a Text component β include all student-facing text.
2. Add a Math Response component β name it answer. Example: May 23
content:
when answer.submitted and answer.latex = "May 23"
"Correct! If the area doubles each day and the pond is full on May 24, it was half full the day before."
when answer.submitted and answer.latex = "May23"
"Correct! If the area doubles each day and the pond is full on May 24, it was half full the day before."
when answer.submitted and answer.latex = "23"
"Correct! May 23. If the area doubles each day, the pond was half full just one day before it was completely full."
when answer.submitted and answer.latex = "May 18"
"I see what happened - you found the date halfway between May 12 and May 24. But the algae does not grow at a constant rate. If the area DOUBLES each day and the pond is full on May 24, when was it half full?"
when answer.submitted and answer.latex = "May18"
"I see what happened - you found the date halfway between May 12 and May 24. But the algae doubles. If the pond is full on May 24, when was it half full?"
when answer.submitted
"Think backward: the pond is 100% covered on May 24. It doubles each day, so one day before it was full, how much was covered?"
otherwise ""
Work with your partner
On which day is the pond halfway covered?
On May 18, Clare visits the park. A caretaker says the pond will be completely covered in less than a week. Clare looks at the pond and thinks the caretaker must be mistaken. Why might she find the claim hard to believe?
Try Saying
“I think the pond is halfway covered on ___ because ___.”
Try Saying
“Clare might not believe the caretaker because ___.”
β Challenge
(Challenge 1) What fraction of the pond was covered on May 12, the day the algae was introduced? Express your answer as a fraction and as a percentage.
(Challenge 2) If 50% of the pond is covered at the start of May 23, approximately how much is covered halfway through May 23? Explain your reasoning.
Activity 2 Β· Synthesis β± 5 min
Why Is Exponential Growth So Surprising?
π§βπ« Invite students to share responses. Display organizing strategies (tables, working backward). After discussing Clare's reasoning, use Stronger and Clearer Each Time: students write a first draft explaining why Clare is skeptical, share with 2β3 partners who give feedback, then revise. If time allows, ask: "If 50% of the pond is covered at the start of May 23, how would you figure out how much is covered halfway through May 23?" Accept strategies β graphing, tables, writing an equation. Do not resolve yet.
The purpose of this discussion is to surface why exponential growth is deceptive β it starts slowly and then accelerates dramatically.
“On which day is the pond halfway covered, and how do you know?”May 23. If the area doubles each day and the pond is fully covered on May 24, it was half covered the day before.
“Why might Clare find the caretaker's claim hard to believe?”By May 18, only about \(\frac{1}{64}\) of the pond is covered β roughly 1.5%. Looking at the pond, she would see almost no algae and would not believe it could fill the entire pond in 6 days.
“What fraction of the pond was covered on May 12?”\(\frac{1}{2^{12}} = \frac{1}{4096}\), or about 0.02%.
Key takeaway: exponential growth is deceptive. The quantity stays small for a long time, then grows very rapidly near the end.
click to advance discussion βΆ
Try Saying
“Clare might not believe the caretaker because on May 18 only ___ of the pond is covered, which ___.”
Try Saying
“I notice that exponential growth is surprising because ___.”
Math As A Second Language
Exponential Growth β Growth Factor Greater Than 1
Math Β· We Say Β· Meaning
β Math (given)
\(b > 1\)
β² We Say
“b is greater than 1”
β Meaning
When the growth factor b is greater than 1, the exponential function models growth β the output increases as x increases.
π§βπ« Connect to the pond problem: the growth factor was 2 (doubling), which is greater than 1. Each day the area was multiplied by 2. This is exponential growth. Read the We Say card aloud together. Ask: "How does this compare to the passport photo, where b = 0.8?"
Lesson Synthesis β± 5 min
Comparing Growth and Decay
π§βπ« Guide discussion to surface both similarities (geometric sequences, constant growth factor, exponential expressions) and differences (shrinking vs. growing, factor less than 1 vs. greater than 1, bounded vs. unbounded). Foreshadow: students will explore many more exponential situations in this unit and use exponential functions to solve problems.
Today we explored two situations β a passport photo shrinking and algae growing. Both involved exponential change.
“How are the shrinking passport photo and growing algae situations similar?”In each case, there is a geometric sequence. Both can be represented by exponential expressions or equations. Both involve multiplying by a constant factor.
“How are the shrinking passport photo and growing algae situations different?”
The photo gets smaller while the algae grows. The photo has a growth factor less than 1 (0.8). The algae has a growth factor greater than 1 (2).
The pond is eventually full of algae β there is a natural upper limit. The passport photo can continue to shrink (approaching 0 but never reaching it).
The photo changes in discrete steps (one copy at a time, forming a geometric sequence). The algae changes continuously β the area does not suddenly double at one instant.
Preview: in this unit, you will explore many more situations involving exponential change and use exponential functions to model and solve problems.
click to advance discussion βΆ
Try Saying
“The passport photo and the algae are similar because both ___.”
Try Saying
“The passport photo and the algae are different because ___.”
By the end of this lesson, you can:
Compare and contrast (orally) exponential growth and decay.
Determine values of simple exponential functions in context.
Calculate values that are changing exponentially.
Reference Β· How ToCool-Down β± 5 min
π§βπ« Leave this slide on screen during the cool-down. The reference table on the left is a scaffold β students who do not need it can ignore it. What to look for: (1) Does the student recognize that Han is wrong about 2 rounds = one-third? (2) Can the student correctly compute \(450 \cdot (1/3)^4\)? Common error on Problem 1: students agree with Han because \(450 \div 3 = 150\) and stop β they do not apply the second round of cutting. Common error on Problem 2: students compute \(450 \div 12\) (dividing by 3 four times means dividing by 12) instead of \(450 \cdot (1/3)^4\).
Amplify Β· Activity Builder Β· Cool-Down
Setup (do once):
1. Add a Text component β paste all student-facing text.
2. Add a Math Response component β name it answer. Example: 50
content:
when answer.submitted and answer.latex = "50"
"Correct! After two rounds, each rectangle is 50 square inches - one ninth of the original, not one third."
when answer.submitted and answer.latex = "150"
"Almost! 150 is the area after ONE round of cutting. Han cut into thirds TWICE. What is one third of 150?"
when answer.submitted
"After round 1, each piece is 450 divided by 3 = 150. After round 2, each piece is 150 divided by 3 = ? Try that calculation."
otherwise ""
Problem 2: Area after 4 rounds
content:
when answer.submitted and answer.latex = "50/9"
"Correct! 50/9 square inches, or about 5.6 square inches."
when answer.submitted and answer.latex = "\\frac{50}{9}"
"Correct! 50/9 square inches, or about 5.6 square inches."
when answer.submitted and answer.latex = "5.56"
"Correct! About 5.56 square inches (exactly 50/9)."
when answer.submitted and answer.latex = "5.6"
"Correct! About 5.6 square inches (exactly 50/9)."
when answer.submitted and numericValue(answer.latex) >= 5.5
and numericValue(answer.latex) <= 5.6
"Correct! About 5.6 square inches (exactly 50/9)."
when answer.submitted and numericValue(answer.latex) = 37.5
"Not quite - it looks like you divided 150 by 4, which assumes linear reduction. Each round divides by 3. Start from 50 (after 2 rounds) and divide by 3 twice more."
when answer.submitted
"After 2 rounds the area is 50. For rounds 3 and 4, divide by 3 each time: 50 divided by 3, then that result divided by 3. What do you get?"
otherwise ""
Cutting into Thirds
Reference Β· How To
Step
What to do
Example: 450 inΒ² cut into thirds
1βΆ
click to reveal
β
2βΆ
click to reveal
β
3βΆ
click to reveal
β
β
click to reveal
β
Cool-Down
Han has a thin rectangular sheet of paper. Its area is 450 square inches. He cuts it into thirds, then cuts one of the resulting rectangles into thirds again.
On your own
Han believes that after two rounds of cutting, the rectangles will be 150 square inches, or one-third of the original area. Do you agree? Explain your reasoning.
What is the area of the rectangles, in square inches, after 4 rounds of cutting in the same way? Explain or show your reasoning.
Lesson Summary
Exponential Change
Key Idea
Sometimes quantities change by the same factor at regular intervals. This relationship can be modeled by an exponential function.
\[f(x) = a \cdot b^x\]
Example: A bacteria population is 10,000 on the day of measurement and doubles each day.
Day 1: 20,000 Β· Day 2: 40,000 Β· Day 3: 80,000
If \(x\) is the number of days since measurement, then the population on day \(x\) is \(f(x) = 10{,}000 \cdot 2^x\).
The population each day also forms a geometric sequence because each term is found by multiplying the previous term by 2.
