Exponential Showdown Teacher Guide
Illustrative Mathematics Algebra 1 • Unit 5 (Exponential Functions)
Lesson 5: Comparing Linear and Exponential Growth
HQIM Lesson Guide 50–60 Minutes
Standards
HSF-LE.A.1, HSF-LE.A.2, HSF-LE.A.3
SMP: SMP4 (Modeling), SMP7, SMP8
Math Language Routines
MLR7: Compare & Connect
MLR8: Discussion Supports
Learning Targets
• Explain why exponential growth outpaces linear growth.
• Distinguish equal differences vs. equal factors.
Lesson Narrative
In previous lessons, students investigated quantities that change exponentially by repeated multiplication. Here, students directly contrast linear functions (constant rate of change, \(f(x) = mx + b\)) with exponential functions (constant growth factor, \(g(x) = a \cdot b^x\)). Students use tables, equations, and graphs to generalize that exponential growth will eventually exceed linear growth, no matter the linear slope.
Access for Multilingual Learners (MLR8)
Provide sentence frames during paired synthesis: "Plan A is changing by adding _____ each day, while Plan B is changing by multiplying by _____ each day."
Access for Students with Disabilities
Representation: Color-code differences in green for addition (\(+100\)) and factors in purple for multiplication (\(\times 2\)) across table rows to highlight structural differences.
5.1 Warm-up: Notice and Wonder — Two Pay Plans 5–10 min
Launch: Display Slide 2 without showing calculations. 1 minute quiet think time. Prompt: "What do you notice? What do you wonder?"
Activity Synthesis: Record observations on board. Highlight student intuitions regarding Plan A ($100/day) looking superior initially compared to Plan B ($1 doubling daily).
5.2 Activity: Tracking the Coins (Tables to Equations) 15 min
Launch: Pairs work to complete the table for Day 1 through Day 15 on Handout. Emphasize writing expressions for Day \(n\).
Activity Synthesis (MLR7 Compare & Connect): Select 2 student papers: one showing repeated addition (\(100 + 100 + \dots\)) and one showing exponent notation (\(2^{n-1}\)). Ask: "Where do we see the constant rate in Plan A? Where do we see the growth factor in Plan B?"
Illustrative Mathematics • Algebra 1 Unit 5 Lesson 5 Page 1 of 2
HQIM Instructional Facilitation
Activity 5.3, Synthesis & Formative Assessment Rubric
Lesson 5 (Day 1)
5.3 Activity: The Long-Term Race & "Are You Ready for More?" 15 min
Student Task: Students examine graph trajectories and identify the crossover window (Days 11–12).
"Are You Ready for More?" Guidance (IM Extension)
Prompt: If Plan C pays \(\$10,000\) per day, on which day does Plan B (\(2^{n-1}\)) surpass Plan C (\(10,000n\))?
Solution: At Day 20: \(C(20)=\$200,000\) vs \(B(20)=\$524,288\). (Crossover occurs between Day 18 and 19).
Lesson Synthesis (10 min)
Display synthesis anchor chart. Ask students:
- "How can you tell from a table whether data is growing linearly or exponentially?" (Look for equal differences vs. equal ratios).
- "Why will an exponential function with base \(b > 1\) always eventually exceed a linear function?" (Repeated multiplication grows faster than repeated addition).
Cool-Down 5.4: Outgrowing Linear — Scoring & Next Steps
5 min
| Student Response | What it Means | Next-Day Instructional Pivot |
|---|
| Identifies P as linear (adding 25) and Q as exponential (multiplying by 1.5). | Strong conceptual mastery of additive vs. multiplicative models. | Advance to Lesson 6 (Exponential Decay models). |
| Confuses \(1.5^t\) with \(1.5t\) (claims Q is linear with slope 1.5). | Misinterprets exponent as coefficient. | Warm-up with a 5-minute Number Talk comparing \(3 \cdot x\) vs \(3^x\). |
Handout Solutions
5.2 Table: Day 1: ($100, $1), Day 5: ($500, $16), Day 10: ($1,000, $512), Day 11: ($1,100, $1,024), Day 12: ($1,200, $2,048), Day 15: ($1,500, $16,384).
Rules: \(A(n) = 100n\); \(B(n) = 2^{n-1}\) or \(0.5(2^n)\).
5.3 Analysis: Linear successive differences = \(m\); Exponential quotients = \(b\).
5.4 Cool-Down: \(P(t)\) is linear (\(+25\)); \(Q(t)\) is exponential (\(\times 1.5\)). At \(t=2\), \(P(2)=100 > Q(2)=22.5\). At \(t=10\), \(Q(10) \approx 576.65 > P(10)=300\).
Illustrative Mathematics • Algebra 1 Unit 5 Lesson 5 Page 2 of 2
Exponential Showdown Slides
Illustrative Mathematics • Algebra 1
Unit 5 • Lesson 5
Comparing Linear and
Exponential Growth
How do additive and multiplicative patterns behave over short and long periods?
Standard: HSF-LE.A.1 Routine: Compare & Connect (MLR7)
IM HQIM Certified
Warm-Up 5.1
Notice and Wonder: Two Pay Plans
1 Min Quiet Think
Plan A: Steady Pay
You receive $100 every day for 15 days.
Day 1: $100 • Day 2: $200 • Day 3: $300
Plan B: Doubling Penny
You receive $1.00 on Day 1, and your pay doubles each day for 15 days.
Day 1: $1 • Day 2: $2 • Day 3: $4
• What do you notice? • What do you wonder? • MLR8: "I notice that Plan A starts with..."
Activity 5.2
Tracking the Coins: Table Analysis
Handout Section 5.2
Early Days (1–5)
Linear Dominates
Day 5: Plan A = $500
Day 5: Plan B = $16
Plan A feels like the clear winner!
The Turning Point
Days 10 & 11
Day 10: A = $1,000 vs B = $512
Day 11: A = $1,100 vs B = $1,024
Gap rapidly shrinking.
Late Game (Day 15)
Exponential Surge
Day 15: Plan A = $1,500
Day 15: Plan B = $16,384
> 10× greater than Plan A!
Compare & Connect (MLR7): Plan A: \(A(n) = 100n\) (Additive) | Plan B: \(B(n) = 1 \cdot 2^{n-1}\) (Multiplicative)
Activity 5.3
The Long-Term Race: Graphical Trajectories
Comparing Slopes
Linear Model f(x) = mx + b
- Increases by equal differences over equal intervals.
- Rate of change (slope) remains constant.
- Graph is a straight line.
Exponential Model g(x) = a • bx
- Increases by equal factors over equal intervals.
- Rate of change accelerates rapidly as \(x\) grows.
- Graph curves upward steeper and steeper.
Key Theorem: As \(x\) increases without bound, any exponential function with \(b > 1\) will always outgrow any linear function.
Exponential Showdown Student Handout
Illustrative Mathematics • Algebra 1
Lesson 5: Comparing Linear and Exponential Growth
Name: ______________________
Date: _________
Period: ____
5.1 Warm-up: Notice and Wonder — Two Pay Plans
5 mins
Plan A: Receive $100 each day for 15 days. | Plan B: Receive $1 on Day 1, and the pay doubles each day for 15 days.
What do you notice?
What do you wonder?
5.2 Activity: Tracking the Coins
15 mins
1. Complete the table to find the daily payout for each plan on each day.
| Day \((n)\) | Plan A: Steady Pay (\$) | Plan B: Doubling Penny (\$) |
|---|
| 1 | $100 | $1 |
| 2 | $200 | $2 |
| 3 | $300 | $4 |
| 4 | | |
| 5 | | |
| 10 | | |
| 11 | | |
| 12 | | |
| 15 | | |
| Day \(n\) (Equation) | \(A(n) =\) | \(B(n) =\) |
2. On which day does Plan B first pay more than Plan A? Describe the mathematical pattern causing this shift.
3. How does the rate of change for Plan A compare to the rate of change for Plan B as \(n\) increases?
Illustrative Mathematics • Algebra 1 Unit 5 Lesson 5 Page 1 of 2
5.3 The Long-Term Race • Are You Ready for More? • Cool-Down
Unit 5 Lesson 5
5.3 Activity: The Long-Term Race
Fill in the blanks to generalize the difference between linear and exponential relationships:
Linear Functions (\(f(x) = mx + b\))
Grow by equal ________________ (addition) over equal intervals. The rate of change is ________________.
Exponential Functions (\(g(x) = a \cdot b^x\))
Grow by equal ________________ (multiplication) over equal intervals. The rate of change ________________.
Are You Ready for More? (Extension)
Challenge
Suppose Plan C pays $10,000 every day (\(C(n) = 10,000n\)). Would Plan B (doubling penny, \(B(n) = 2^{n-1}\)) still eventually surpass Plan C? If so, estimate or calculate on which day Plan B wins.
