The Million Dollar Choice Slides THE MILLION DOLLAR CHOICE
Growth Lab: Investigation 01
Additive vs. Multiplicative Patterns
The Ultimate Dilemma
Option A
Receive $1,000,000 cash, right now.
OR
Option B
Receive a magic penny that doubles in value every day for 30 days.
Take a vote: Which one would you choose?
Understanding the Rules
Additive Pattern
Wait... Option A isn't exactly additive, it's a fixed amount. But what if you got $1,000,000 every day? That would be adding a constant amount.
Multiplicative Pattern
Option B is multiplicative. We multiply the previous day's value by 2 to get the next day's value.
The Big Question:
Does multiplication eventually beat addition? And if so, how fast?
Starting the Logbook
On your worksheet, you will calculate the value of the magic penny for 30 days.
Day 1: $0.01
Day 2: $0.02
Day 3: $0.04
Day 4: $0.08
Grab your calculators.
Penny Doubling Worksheet Penny Doubling Log
Growth Lab: Case File 01-A
Name
Date
The Mission
Starting with a single magic penny worth $0.01 on Day 1, calculate its value as it doubles every single day for 30 days. Round to the nearest cent if necessary, but keep the exact value for your next calculation.
Day Value ($) 1 $0.01 2 $0.02 3 4 5 6 7 8 9 10
Day Value ($) 11 12 13 14 15 16 17 18 19 20
Day Value ($) 21 22 23 24 25 26 27 28 29 30
Analysis Findings
1. On which day did the magic penny finally surpass $1,000,000?
2. How much more money would you have after 30 days if you chose the penny instead of the flat $1,000,000?
3. Describe how the value of the penny changed over time. Was it growing at the same speed throughout the 30 days?
Million Dollar Choice Teacher Guide Teacher Facilitation Guide
Investigation 01: The Million Dollar Choice
8th Grade Mathematics
Learning Objectives
Identify the difference between additive growth and multiplicative growth.
Experience the rapid acceleration of exponential patterns.
Construct a data table based on a doubling rule.
Key Concept
Exponential growth starts slowly but eventually outpaces linear growth or large constant values. This is often counterintuitive for middle school students.
Reference Key: Penny Doubling Values
Day 1: $0.01
Day 2: $0.02
Day 3: $0.04
Day 4: $0.08
Day 5: $0.16
Day 6: $0.32
Day 7: $0.64
Day 8: $1.28
Day 9: $2.56
Day 10: $5.12
Day 11: $10.24
Day 12: $20.48
Day 13: $40.96
Day 14: $81.92
Day 15: $163.84
Day 16: $327.68
Day 17: $655.36
Day 18: $1,310.72
Day 19: $2,621.44
Day 20: $5,242.88
Day 21: $10,485.76
Day 22: $20,971.52
Day 23: $41,943.04
Day 24: $83,886.08
Day 25: $167,772.16
Day 26: $335,544.32
Day 27: $671,088.64
Day 28: $1,342,177.28
Day 29: $2,684,354.56
Day 30: $5,368,709.12
Classroom Discussion Prompts
Prompt 1: The Frustration Period
"Around Day 10, how were you feeling about your choice? Who had the most money then?"
Answer: At Day 10, the penny is only worth $5.12. Students who chose the penny may feel they made a bad choice.
Prompt 2: The Multiplier Effect
"Is the amount being added to the penny the same every day? How do you know?"
Answer: No. On Day 1 we add $0.01, but on Day 29 we add over $2.6 million. The growth rate is increasing.
Prompt 3: Real World Connection
"In real life, where have you heard about things 'doubling' or spreading fast like this?"
Examples: Viral videos, bacteria, rumor spreading, compound interest.
Mapping the Growth Slides Mapping the Growth
Growth Lab: Investigation 02
Visualizing Linear vs. Exponential Shapes
Numbers tell a story. Pictures prove it.
Yesterday, we calculated raw data. Today, we map that data onto a coordinate plane.
Think-Pair-Share:
Predict the "shape" of the doubling penny. Will it be a straight line? If not, how will it curve?
Plotting Area
Linear Growth Profile
The Rule: Additive
You add the same amount every time.
The Shape: Straight
The rate of change is constant . It creates a perfect line.
Exponential Growth Profile
The Rule: Multiplicative
You multiply by the same factor every time.
The Shape: The Curve
The rate of change increases . It starts flat and then "explodes" upward.
Growth Curves Activity Growth Visualization Lab
Growth Lab: Case File 02-B
Name
Compare two ways of making money. Plan A: A flat $5,000 every day. Plan B: A penny that doubles every day. Use your data from yesterday to plot Plan B.
Total Amount ($)
$100,000
$75,000
$50,000
$25,000
$0
Days (1 to 20)
Step 1: The Linear Path
Plan A: Add $5,000 per day. Draw a straight line starting at Day 0 ($0) to Day 20 ($100,000).
Plan A (Linear)
Step 2: The Exponential Path
Plan B: Use your doubling penny table. Plot the points for days 1 through 20. Connect them with a smooth line.
Plan B (Exponential)
1. Intersection Point: On which day do the two lines cross?
2. Visual Difference: Why does Plan B look almost flat for the first 15 days?
3. Vocabulary: Which plan has a "Constant Rate of Change"?
Growth Shapes Exit Ticket Exit Ticket: Shape Recognition
Investigation 02 Conclusion
Name
Graph A
Type of Growth:
Linear
Exponential
Graph B
Type of Growth:
Linear
Exponential
The "Why" Question
How do you know the difference between these two graphs just by looking at them? Mention "Rate of Change" in your answer.
Defining Additive Patterns Slides Defining Additive Patterns
Growth Lab: Investigation 03
Identifying Linear Functions in Tables
The DNA of a Line
In a linear relationship, the difference between any two consecutive y-values is always the same.
Scientific Term:
Constant First Difference
Linear Scenarios
Stacking Cups
Each cup adds exactly 1.5 cm to the height of the stack.
Steady Savings
Saving $10 from every paycheck. The balance grows by +$10 every week.
"Same change, every time."
Is it Linear?
To check if a table is linear, calculate the difference between each row's y-value.
Value 2 Value 1 Constant?
Linear Patterns Worksheet Constant Difference Analysis
Growth Lab: Case File 03-B
Scientist
Objective: For each table below, calculate the First Difference (\(y_2 - y_1\)). If the difference is the same for all consecutive rows, the pattern is Linear .
Pattern Alpha
First Differences
Linear? Yes / No
Pattern Beta
First Differences
Linear? Yes / No
Pattern Gamma
First Differences
Linear? Yes / No
Pattern Delta
First Differences
Linear? Yes / No
Scenario Construction
A water tank starts with 50 gallons of water. It is being filled at a rate of 15 gallons per minute. Create a table for the first 4 minutes and prove it is linear.
Time (min) Water (gal) 0 1 2 3
Your Proof:
Linear Patterns Answer Key Answer Key
Investigation 03: Constant Difference Analysis
TEACHER USE ONLY
PATTERN ALPHA
First Differences: +5, +5, +5
Linear? YES
Explanation: The y-value increases by exactly 5 for every 1-unit increase in x.
PATTERN BETA
First Differences: +4, +8, +16
Linear? NO
Explanation: The difference is not constant; it is doubling. This is exponential growth.
PATTERN GAMMA
First Differences: -10, -10, -10
Linear? YES
Explanation: Constant subtraction is still additive growth (linear decrease).
PATTERN DELTA
First Differences: +2, +3, +4
Linear? NO
Explanation: The difference is increasing, but not by a constant factor or amount. (This is quadratic, which we'll learn later!)
Scenario Proof: Water Tank
Time (min) Water (gal) 0 50 1 65 2 80 3 95
Proof Explanation:
"The first difference is constant. Between minute 0 and 1, we add 15 gallons. Between minute 1 and 2, we add 15 gallons. Between minute 2 and 3, we add 15 gallons. Since we add the same amount every minute, the growth is linear."
Defining Multiplicative Patterns Slides Defining Multiplicative Patterns
Growth Lab: Investigation 04
Identifying Exponential Functions in Tables
The DNA of a Curve
In an exponential relationship, the ratio between any two consecutive y-values is always the same.
Scientific Term:
Constant Common Ratio
Exponential Scenarios
Cell Division
A single cell splits into two every hour. The population multiplies by 2 constantly.
Viral Videos
Each viewer shares with 3 friends. The audience multiplies by 3 with every "wave."
"Same multiplier, every time."
Is it Exponential?
To check if a table is exponential, calculate the ratio between each row's y-value.
Value 2 Value 1 Constant?
Exponential Patterns Worksheet Common Ratio Analysis
Growth Lab: Case File 04-B
Scientist
Objective: For each table below, calculate the Common Ratio (\(y_2 \div y_1\)). If the ratio is the same for all consecutive rows, the pattern is Exponential .
Pattern Sigma
Common Ratios
Exponential? Yes / No
Pattern Omega
Common Ratios
Exponential? Yes / No
The Viral Challenge
A new gaming video is uploaded at noon. Every hour, the number of views triples . At 1:00 PM, it has 90 views. Complete the table and identify the common ratio.
Time Total Views 12:00 PM (hour 0) 1:00 PM (hour 1) 90 2:00 PM (hour 2) 3:00 PM (hour 3)
Common Ratio (\(r\)):
Proof Calculation:
Show dividing one row by the previous row.
Exponential Patterns Answer Key Answer Key
Investigation 04: Common Ratio Analysis
TEACHER USE ONLY
PATTERN SIGMA
Common Ratios: 3, 3, 3
Exponential? YES
Explanation: Each y-value is multiplied by 3 to get the next value.
PATTERN OMEGA
Common Ratios: 0.5, 0.5, 0.5
Exponential? YES
Explanation: Multiplying by 0.5 (or dividing by 2) is a constant ratio. This is exponential decay.
Case Study Key: Viral Challenge
Time Total Views 12:00 PM (h 0) 30 1:00 PM (h 1) 90 2:00 PM (h 2) 270 3:00 PM (h 3) 810
Proof Explanation:
"Common Ratio (r) = 3. We divide 90 by 30 to find the base (3). Then we multiply 90 by 3 to get 270, and 270 by 3 to get 810. Since the ratio between all terms is 3, the growth is exponential."
Classification Challenge Slides Classification Challenge
Growth Lab: Investigation 05
Mastering the Growth Code
Detective Toolbox
Tool 1: Additive
Check for Differences
"Does it add (or subtract) the same amount every time?"
Linear
Tool 2: Multiplicative
Check for Ratios
"Does it multiply (or divide) by the same factor every time?"
Exponential
Warning: The "Neither" Zone
Not every pattern in the universe is linear or exponential. Some data is just random, and some follows different rules (like quadratic or cubic).
If it fails both tests:
- No common difference
- No common ratio
Classification: NEITHER
Mission Briefing
1
Examine the Mystery Tables provided in your dossier.
2
Run the Difference Test and the Ratio Test for every table.
3
Classify each as Linear, Exponential, or Neither.
Good luck, Detectives. The clock is ticking.
Classification Challenge Dossier Mystery Table Dossier
Growth Lab: Case File 05-B
Investigator
Mission Brief: Analyze each table. Perform the Difference Test or Ratio Test. Circle the pattern type and provide your "Proof" (the constant value you found).
Table Alpha-01
CLASSIFICATION:
[ ] LINEAR[ ] EXPONENTIAL[ ] NEITHER
Mathematical Proof:
Table Beta-02
CLASSIFICATION:
[ ] LINEAR[ ] EXPONENTIAL[ ] NEITHER
Mathematical Proof:
Table Gamma-03
CLASSIFICATION:
[ ] LINEAR[ ] EXPONENTIAL[ ] NEITHER
Mathematical Proof:
Table Delta-04
CLASSIFICATION:
[ ] LINEAR[ ] EXPONENTIAL[ ] NEITHER
Mathematical Proof:
Table Epsilon-05
CLASSIFICATION:
[ ] LINEAR[ ] EXPONENTIAL[ ] NEITHER
Mathematical Proof:
Growth Lab Final Assessment Growth Lab Final Assessment
Summative Mastery Check
Name
Score
Part 1: Visual Profiles
1. Which of the following graphs represents a constant rate of change?
Choice A
Choice B
2. Describe the shape of an exponential growth graph.
A straight line that always goes through the origin.
A curve that starts shallow and gets steeper over time.
A horizontal line that stays at the same y-value.
Part 2: Analysis Mastery
Analyze the table below. Is it linear or exponential? Justify your answer with a calculation.
Type:
Linear
Exponential
Mathematical Justification:
Part 3: Real-World Scenarios
Scenario A: Bank Account
A savings account adds $50 every month as long as no withdrawals are made.
Linear / Exponential
Scenario B: Social Media
A post gets shared so that the total number of people who have seen it doubles every hour.
Linear / Exponential
Bonus: If you want your money to grow as fast as possible for a long time, which growth pattern would you prefer? Explain why.
