Session 1 Teaching Script

Warm-Up (3 minutes)

Teacher (showing Slide 1 of Exponential Functions Slide Deck): "Good morning, everyone! Let’s start with a quick warm-up.

Please look at this table showing x values from –1 to 3 and their corresponding y values for y = 2ˣ. Take 30 seconds to observe how the y-values change as x increases by 1. Think quietly: What pattern do you notice?"

<<Pause 30 seconds>>

Teacher: "Who’d like to share what they noticed?"

• If a student says, “They double,” respond: "Exactly! Each time x goes up by 1, y doubles. Great observation!"
• Solicit 2–3 more responses: "What else did you notice?"

Teacher: "Fantastic. That doubling pattern is the heart of exponential growth. Now let’s define exponential functions formally."

Introduction & Discussion (7 minutes)

Teacher (switch to Slide 2): "An exponential function has the form y = aˣ. Here, ‘a’ is the base (must be positive and not equal to 1), and ‘x’ is the exponent or independent variable. The key is that x appears as a power."

Teacher (show Slide 3): "When the base a > 1, the function grows as x increases. When 0 < a < 1, the function decays. How do you think the graph would look for decay versus growth?"

<<Listen for responses; prompt if needed: “Does the curve rise or fall as x increases?”>>

Teacher (show Slide 4):
"This is the graph of y = 2ˣ. Notice two things:

1. As x → –∞, the curve approaches the x-axis but never touches it. That’s the horizontal asymptote y = 0.
2. As x increases, the curve rises steeply."

Teacher: "Can anyone name a real-world example of exponential growth or decay?"

<<Collect examples such as population growth, compound interest, radioactive decay, viral spread.>>

Teacher: "Excellent! Exponential functions model all of those. Now let’s get hands-on."

Graphing Activity (8 minutes)

Teacher: "Clear your desks for chart paper, graph paper, rulers, and markers. In pairs, choose one base: 2, 3, or ½. Then:

1. Create a table of values for x = –1, 0, 1, 2, 3
2. Plot those points on your graph paper
3. Draw the smooth exponential curve, labeling your axes and scale clearly

You have 8 minutes—go!"

<<Circulate, offering encouragement and guidance. At 6 minutes, give a 2-minute warning.>>

Teacher: "Time’s up! Let’s regroup. What similarities and differences do you notice among the curves?"

<<Briefly discuss asymptote placement, steepness, symmetry.>>

Exponential Game (6 minutes)

Teacher: "It’s time for our Exponential Ball Toss Game! Split into two teams. Each team gets a deck of Graphing Ball Toss Game Cards. Here’s how to play:
• One student tosses the soft ball to a teammate.
• The teammate reads the x-value on the card.
• Your team has 10 seconds to shout out y for f(x) = 2ˣ.
• Correct within time = 1 point. First team to 5 points wins.

Ready? Let’s play!"

<<Run game, track scores, cheer on teams.>>

Teacher: "Great job to both teams—congratulations to our winners!"

Worksheet Practice (4 minutes)

Teacher: "Please take an Exponential Basics Worksheet. For the next 4 minutes, complete:

1. Matching equations to their graphs
2. Definitions: base, exponent, asymptote

I’ll come around if you need help. Begin now."

<<Assist as needed; give 1-minute warning at 3 minutes.>>

Teacher: "Please put your pencils down."

Cool-Down & Exit Ticket (2 minutes)

Teacher: "Finally, grab an exit ticket card and write in one sentence: ‘How does exponential growth differ from linear growth?’ You have 2 minutes."

<<Distribute cards; collect as students leave.>>

Teacher: "Thank you for your hard work today! See you next session, where we’ll build on these ideas."

Exponential Basics Worksheet

1. Matching Functions to Graphs

Below are four graphs labeled A–D (printed on your worksheet). Match each exponential equation to the correct graph by writing the letter in the blank.

1. ___ y = 2^x
   
   
2. ___ y = (1/2)^x
   
   
3. ___ y = 3^x
   
   
4. ___ y = (1/3)^x
   
   

2. Key Definitions

In your own words, define each term below.

a) Base (a)






b) Exponent (x)






c) Horizontal asymptote

3. Table of Values

Complete the table for the function y = 3^x.

4. Graphing

Using graph paper, plot the points from your completed table for y = 3^x. Label your axes and draw a smooth curve through the points.

(Use the back of this page or separate graph paper.)

5. Real-World Application

Describe one real-world situation that could be modeled by an exponential function. Indicate whether it is exponential growth or exponential decay, and explain why.

The Graphing Ball Toss Game Cards have been created with x-values ranging from -2 to 4. Each card displays one of the following x-values: -2, -1, 0, 1, 2, 3, 4. Shuffle and use these cards for the Exponential Game in Session 1.

Session 2 Teaching Script

Warm-Up (3 minutes)

Teacher (showing Slide 1 of Session 2 Slide Deck): "Good morning, everyone! Let’s warm up by comparing exponential growth and decay.

Here you see the table and graph of y = 2ˣ. In the next 30 seconds, please sketch y = (1/2)ˣ on your whiteboard or scratch paper. Think about how the curve behaves differently from the growth curve."

<<Pause 30 seconds>>

Teacher: "Who’d like to share their sketch? How does the decay curve differ visually from the growth curve?"

• If a student says it falls to the right: "Exactly—this curve decreases as x increases. Great observation!"
• Prompt: "What horizontal line do both graphs approach but never touch?"
• Look for “y = 0.”

Teacher: "Excellent. That horizontal asymptote is y = 0 in both cases. Now, let’s talk about solving exponential equations."

Discussion: Solving Exponential Equations (7 minutes)

Teacher (switch to Slide 2 of Session 2 Slide Deck): "To solve an equation of the form aˣ = b, we take a logarithm. In general:

x = logₐ(b)

Or using natural logs:

x = ln(b) / ln(a)

Let’s practice with a few examples."

Teacher (show Slide 3):
"1. 2ˣ = 8. What power of 2 gives 8?"
<<Wait for response: 3>>
"Right—x = 3."

"2. 3ˣ = 27. What is x?"
<<Wait for response: 3>>
"Exactly—x = 3."

"3. (½)ˣ = 4. Which step isolates x?"
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"We get x = ln(4) / ln(½). Does anyone know that value?"
<<Wait; then clarify: –2>>
"Yes—x = –2, because (½)⁻² = 4."

Teacher: "Why is the exponent negative here?"
<<Listen for “base between 0 and 1” or “reciprocal”>>
"Exactly—when 0 < a < 1, negative exponents produce growth in the positive y-value."

Teacher (show Slide 4): "Any questions on using logarithms to solve these exponential equations?"
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Real-World Activity (8 minutes)

Teacher: "Time to apply this to real data. In your groups of three, you’ll work on one scenario: population growth, radioactive decay, or compound interest. I’m handing out the Real-World Exponential Worksheet and chart paper with markers.

Your task:

1. Plot the data points provided.
2. Determine the base a for your model.
3. Write the equation y = aˣ.

You have 8 minutes—let’s begin!"

<<Distribute materials; circulate and support. At 6 minutes, call out a 2-minute warning.>>

Teacher: "Time’s up! Let’s hear from one group: what base did you find and why?"
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Exponential Game (5 minutes)

Teacher: "Next is our Decay vs Growth Card Game! Split into two teams and grab the Decay vs. Growth Card Game.

How to play:

• One student draws a card showing an exponential function (e.g., y = 5ˣ or y = (1/3)ˣ).
• Your team calls out “Growth” or “Decay” and explains how you know (think about whether a > 1 or 0 < a < 1).
• Earn 1 point per correct justification.

First team to 5 points wins—ready? Go!"

<<Run game, track points, and celebrate winners>>

Mini-Project Introduction (5 minutes)

Teacher: "Finally, we’re launching our Exponential Modeling Project. Please open the Exponential Modeling Project Guide.

Project requirements:

1. In groups of 3–4, choose a real phenomenon (e.g., viral spread, investment growth).
2. Gather or imagine data for at least five time intervals.
3. Fit an exponential model y = aˣ to your data.
4. Create a poster explaining your model, the meaning of a, and real-world implications.

You’ll have one week to complete this. For now, form your groups, pick your topic, and use this chart paper to brainstorm data sources and steps. You have 5 minutes—begin now."

<<Circulate; after 4 minutes, prompt groups to finalize their topic.>>

Teacher: "Great choices, everyone. I can’t wait to see your models next class!"

Cool-Down & Exit Ticket (2 minutes)

Teacher: "Before you go, please grab an exit ticket card. In one sentence, explain: ‘How do logarithms help us solve exponential equations?’ You have 2 minutes."

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Teacher: "Thank you for your hard work today. See you next time!"

Real-World Exponential Worksheet

Scenario 1: Bacterial Growth (Tripling Per Hour)

The table below shows the relative size of a bacterial colony that triples each hour (relative amount starts at 1).

1. Complete the missing value in the table for x = 4.
   
   
2. What is the growth factor (base a)?
   
   
3. Write the exponential model in the form: y = aˣ
   
   
   

Scenario 2: Radioactive Decay (Half-Life Per Day)

A radioactive substance loses half of its mass each day. The table below shows its relative mass over four days (starting at 1).

1. Fill in the missing value for x = 4.
   
   
2. What is the decay factor (base a)?
   
   
3. Write the exponential model in the form: y = aˣ
   
   
   

Scenario 3: Compound Interest (10% Annual)

An investment grows by 10% each year. The table below shows the relative account value (starting at 1).

1. Calculate the missing value for x = 4 (round to four decimal places).
   
   
2. What is the growth factor (base a)?
   
   
3. Write the exponential model in the form: y = aˣ
   
   
   

Graphing

Choose one of the three scenarios above.

1. On graph paper, plot the data points from your completed table.
2. Label your axes clearly (x = time, y = relative amount).
3. Draw a smooth curve through the points.
   
   
   
   
   

Reflection

Which scenario (growth or decay) shows the fastest relative change? Explain your reasoning in one or two sentences.

Decay vs Growth Card Game

This game uses a deck of cards where each card shows an exponential function. Teams will draw a card, decide whether the function represents exponential growth or decay, and justify their answer based on the base value.

Cards in the Deck:

• y = 2ˣ
• y = 5ˣ
• y = 1.5ˣ
• y = 10ˣ
• y = (1/2)ˣ
• y = (1/3)ˣ
• y = (0.8)ˣ
• y = (0.25)ˣ

How to Play:

1. Shuffle the cards and place them face down in a stack.
2. Teams take turns drawing the top card.
3. The drawing team announces “Growth” if the base a > 1, or “Decay” if 0 < a < 1, and explains their reasoning (e.g., “Base = 0.8, so it’s decay because each step multiplies by less than 1.”).
4. Correct answers earn 1 point. First team to 5 points wins the game!

Use this game to reinforce recognition of exponential growth vs. decay and to practice interpreting the base value in real time.

Exponential Modeling Project Guide

Project Overview

In this mini‐project, you and your group will select a real‐world phenomenon that follows exponential behavior, collect or imagine data for at least five time intervals, fit an exponential model to your data, and create a visual presentation (poster and brief oral explanation).

Learning Goals:

• Apply exponential functions to real scenarios
• Use data to determine the base a in y = aˣ
• Communicate mathematical reasoning clearly
• Collaborate effectively in small teams

Project Steps

1. Form Your Group (3–4 students).
2. Choose a Phenomenon.
   • Examples: viral spread, investment growth, population change, radioactive decay, bacterial growth.
3. Gather or Create Data.
   • Identify at least five time intervals (e.g., days, hours, years).
   • Record corresponding values (e.g., population size, account balance).
4. Determine the Exponential Model.
   • Calculate the growth/decay factor (base a).
   • Write the equation y = aˣ (adjusting if your initial amount is not 1: y = C·aˣ).
5. Design Your Poster.
   • Include: title, data table, plotted graph, equation, explanation of a, and real‐world interpretation.
6. Prepare a 2–3 minute Presentation.
   • Explain your data source, how you found a, and why your model makes sense in context.

Timeline & Checkpoints

Deliverables

• Poster (one per group) containing:
  • Title and group member names
  • Data table with at least five (x, y) pairs
  • Graph of points and smooth exponential curve
  • Final model equation y = aˣ (or y = C·aˣ)
  • Written explanation of base a and real‐world implications
• Oral Presentation (2–3 minutes)

Assessment Rubric

Expectations & Tips

• Cite Your Data Sources: If you use real data, include the source.
• Check Your Work: Verify calculations and graph scaling.
• Balance Roles: Assign tasks (research, calculation, design, presentation) so everyone participates.
• Practice Your Pitch: Rehearse your oral explanation to stay within 2–3 minutes.
• Ask for Help Early: Consult your teacher if you encounter difficulties.

Good luck—use this opportunity to deepen your understanding of exponential behavior in the world around you!

The separate Session 1 Warm-Up material has been removed since the warm-up is already integrated into the lesson plan, slide deck, and script.