Modeling Growth: Mastering Exponential Equations

PreAlgebra Grades High School 4:02 Video
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Lesson Description

This lesson explores the exponential growth model and its applications in solving real-world problems. Students will learn to write and solve exponential equations using systems of equations.

Video Resource

Exponential Growth Model Example

Mario's Math Tutoring

Duration: 4:02
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Key Concepts

  • Exponential Growth Formula
  • Writing Exponential Equations from Data
  • Solving Systems of Equations
  • Modeling Real-World Phenomena

Learning Objectives

  • Students will be able to define and apply the exponential growth formula.
  • Students will be able to translate word problems into exponential equations.
  • Students will be able to solve systems of equations to determine unknown parameters in exponential growth models.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general form of exponential functions and their characteristics. Briefly discuss real-world scenarios where exponential growth occurs (e.g., population growth, compound interest).
  • Exponential Growth Formula (5 mins)
    Introduce the exponential growth formula: y = a * b^x, where 'y' is the final amount, 'a' is the initial amount, 'b' is the growth factor, and 'x' is the time. Explain each variable and its role in the model. Refer to the video at 0:10.
  • Word Problem Example (10 mins)
    Present a word problem involving exponential growth (refer to the video at 0:24). Guide students through the process of identifying the given information and translating it into mathematical terms. Emphasize the importance of correctly identifying the initial amount and the growth factor.
  • Writing Equations and Solving (15 mins)
    Demonstrate how to create a system of two equations from two data points given in the word problem (refer to the video at 0:43). Explain different methods for solving the system (e.g., substitution, elimination). Walk through the steps of solving for the unknown variables (usually 'a' and 'b'). Provide several example problems and show how the data are represented as coordinates (1:00).
  • Practice and Application (10 mins)
    Provide students with additional word problems to solve independently or in small groups. Encourage them to discuss their approaches and solutions with each other. Emphasize the importance of checking their answers and interpreting the results in the context of the problem.

Interactive Exercises

  • Desmos Graphing Activity
    Students will use Desmos to graph exponential growth functions and explore how changing the parameters 'a' and 'b' affects the shape of the graph. They will also graph multiple exponential equations on the same axes to solve systems of equations graphically.
  • Group Problem Solving
    Divide students into small groups and assign each group a complex exponential growth word problem. Groups will work together to translate the problem into equations, solve the system, and present their solution to the class.

Discussion Questions

  • What are some real-world examples of exponential growth?
  • How does the growth factor 'b' affect the rate of growth? What does it mean if b < 1?
  • What are the limitations of using exponential growth models to predict future outcomes?

Skills Developed

  • Mathematical Modeling
  • Problem-Solving
  • Analytical Thinking
  • Critical Thinking

Multiple Choice Questions

Question 1:

The general form of an exponential growth function is given by:

Correct Answer: y = a * b^x

Question 2:

In the exponential growth formula y = a * b^x, 'a' represents the:

Correct Answer: Initial amount

Question 3:

If b > 1 in the exponential growth formula, the function represents:

Correct Answer: Exponential growth

Question 4:

To solve for the parameters in an exponential growth model using two data points, you typically need to:

Correct Answer: Solve a system of equations

Question 5:

An exponential growth model is most appropriate for which of the following scenarios?

Correct Answer: A savings account earning compound interest

Question 6:

What is the primary difference between exponential growth and exponential decay?

Correct Answer: Exponential growth has a rate greater than 1, decay has a rate less than 1.

Question 7:

Which method is most suitable for solving a system of exponential equations?

Correct Answer: Graphical Analysis using Desmos

Question 8:

In an exponential growth model, what does the variable 'x' typically represent?

Correct Answer: Time or number of periods

Question 9:

When solving for the growth factor in an exponential growth model, which mathematical operation is most commonly used?

Correct Answer: Taking Roots

Question 10:

What is a key assumption made when using an exponential growth model?

Correct Answer: Resources are unlimited

Fill in the Blank Questions

Question 1:

The exponential growth formula is y = a * b^x, where 'b' is called the _______ _______.

Correct Answer: growth factor

Question 2:

When using two data points to determine the exponential growth equation, you create a _______ of equations.

Correct Answer: system

Question 3:

In the context of exponential growth, 'a' always represents the _______ value.

Correct Answer: initial

Question 4:

If the growth factor 'b' is 1.05, the growth rate is _______ %.

Correct Answer: 5

Question 5:

To solve an exponential growth equation, you need to isolate the _______ variable.

Correct Answer: unknown

Question 6:

Exponential growth is often observed in _______ populations where resources are abundant.

Correct Answer: biological

Question 7:

The process of finding the values of 'a' and 'b' in the exponential growth equation is called _______ the model.

Correct Answer: fitting

Question 8:

A population that doubles every hour exhibits __________ growth.

Correct Answer: exponential

Question 9:

The limitation to the exponential growth model is that in real life, growth is never truly __________ due to limited resources.

Correct Answer: unrestricted

Question 10:

Graphs of exponential functions never intersect the __________ axis, unless there is an added vertical shift.

Correct Answer: x

Educational Standards

PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.3.1:Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.2:Build functions to model and validate relationships among functions.

Teaching Materials

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