5.3 Writing Expontential Equations

Last updated about 1 year ago

26 questions

Note from the author:

OBJECTIVES & STANDARDS
Math Objectives
Recognize and define the parts of an exponential function
Write exponential functions to model a given situation
Evaluate and solve exponential functions
Common Core Math Standards
Link to all CCSS Math
CCSS.PRACTICE.MP4
CCSS.PRACTICE.MP7
CCSS.HSF.LE.A.1.C
CCSS.HSM
Personal Finance Objectives
Analyze return on an investment given starting investment, rate of return and time periods
National Standards for Personal Financial Education
Investing
2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
DISTRIBUTION & PLANNING
Distribute to students
Student Activity Packet
Application Problems

Instructions

OBJECTIVES & STANDARDS
Math Objectives
Recognize and define the parts of an exponential function
Write exponential functions to model a given situation
Evaluate and solve exponential functions
Common Core Math Standards
Link to all CCSS Math
CCSS.PRACTICE.MP4
CCSS.PRACTICE.MP7
CCSS.HSF.LE.A.1.C
CCSS.HSM
Personal Finance Objectives
Analyze return on an investment given starting investment, rate of return and time periods
National Standards for Personal Financial Education
Investing
2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
DISTRIBUTION & PLANNING
Distribute to students
Student Activity Packet
Application Problems

Intro/Warm-Up

Learn It

Practice It

Application Problems

5.3 Writing Expontential Equations

Last updated about 1 year ago

26 questions

Note from the author:

OBJECTIVES & STANDARDS
Math Objectives
Recognize and define the parts of an exponential function
Write exponential functions to model a given situation
Evaluate and solve exponential functions
Common Core Math Standards
Link to all CCSS Math
CCSS.PRACTICE.MP4
CCSS.PRACTICE.MP7
CCSS.HSF.LE.A.1.C
CCSS.HSM
Personal Finance Objectives
Analyze return on an investment given starting investment, rate of return and time periods
National Standards for Personal Financial Education
Investing
2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
DISTRIBUTION & PLANNING
Distribute to students
Student Activity Packet
Application Problems

Instructions

OBJECTIVES & STANDARDS
Math Objectives
Recognize and define the parts of an exponential function
Write exponential functions to model a given situation
Evaluate and solve exponential functions
Common Core Math Standards
Link to all CCSS Math
CCSS.PRACTICE.MP4
CCSS.PRACTICE.MP7
CCSS.HSF.LE.A.1.C
CCSS.HSM
Personal Finance Objectives
Analyze return on an investment given starting investment, rate of return and time periods
National Standards for Personal Financial Education
Investing
2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
DISTRIBUTION & PLANNING
Distribute to students
Student Activity Packet
Application Problems

Intro/Warm-Up

DISCUSSION PROMPTS: Repeated Doubling
Answer each question using any appropriate math notation.

If you were asked to start with the number 5 and double it three times, how would you write this problem using math notation?_______ 

If you were asked to start with the number 5 and double it nine times, how would you write this problem using math notation?_______ 

If you were asked to start with the number 5 and double it one hundred times, how would you write this problem using math notation?_______ 

Learn It

An exponential function is a function in the form y = ab^x where
a is the starting value of the function
b is the number that is repeatedly multiplied, commonly referred to as the growth or decay factor. Assuming positive numbers, then the following is true:
If b > 1, then the function will grow.  This is also referred to as exponential growth
If b is between 0 and 1 (fractions and decimals) then the function will shrink.  This is also referred to as exponential decay
If b = 1, then the function will not grow or decay because multiplying by 1 does not change the starting value.
x is the number of times that you will be multiplying by b


*Remember your order of operations!!!! 
PEMDAS

Given the exponential function

State the starting value:_______ 
State the growth/decay factor:_______ 
Is the function growing or decaying?_______ 
Describe in words what this function is doing. Refer to the language in the intro activity._______ 
What is the result if x = 3?_______ 

Practice It

Use what you have learned about the parts of exponential functions to write an equation for each of the following situations and answer the questions.

Everytime Pinocchio lies, his nose doubles in size.  His nose is 1.5 inches long before he has told any lies.
a. Write an equation that represents this situation where x is the number of lies and y is the size of Pinochhio’s nose after x lies._______ 
b. Use your equation to calculate how long Pinocchio’s nose will be after 6 lies._______ inches

You make a $10,000 investment that has historically doubled every 7 years.
Write an equation that represents this situation where x is the number of years and y is the investment value after x years._______ 
Use your equation to calculate the value of your investments after 15 years.$_______ 

As with other types of equations, you can solve exponential functions for any variable. In this section, we will solve for the exact initial value but only estimate the exponent since solving for exponents is beyond what we have learned so far.

Sari makes an investment with the goal of having $20,000 after 30 years. If the growth factor is 1.12, how much money will she need to invest to reach her goal?_______ 

Charlotte has $12,000 to invest and would like this investment to grow to $18,000. If the growth factor is 1.09, how many years will it take to reach her goal? Use guess and check to find an approximate answer_______ years

Application Problems

You are given $10,000 that will triple every year for the next 5 years.

Write an equation that a represents this situation where y represents the value of the investment after 5 years._______ 

Your equation has three numbers: starting value, growth factor, and the exponent. You are allowed to double one of these numbers. Which one do you think will result in the highest outcome? Explain your reasoning.

Level 2: 
The world record for folding a piece of paper in half over and over again is 12 folds by Britney Gallivan in 2018.
Every time she folded her paper in half, the thickness doubled!  Let’s pretend that we can beat Britney’s record and fold a piece of paper as many times as we want.

Start by writing an equation that represents starting with a 0.1 mm thick piece of paper that is folded in half x number of times._______ 

Based on your equation, approximately how thick was the piece of paper Britney folded?_______ 

Level 3: 
We can represent the motion of a tennis ball bouncing using math.  Here’s a graph that lets you see different bounces from different heights.  You can move the sliders to adjust the height of the ball and how high each additional bounce will be.
In this video, the initial height of the ball is 36 inches and each bounce is represented in the table below.

Let’s create a decay model to represent the ball’s height after a certain number of bounces.

Calculate the decay factor for each bounce by dividing the second height by the first height, then the third by the second, until you reach the bottom of the table. Make a list of those decay factors in the space below to three decimal places.

5.3 Writing Expontential Equations

5.3 Writing Expontential Equations

Your equation has three numbers: starting value, growth factor, and the exponent. You are allowed to double one of these numbers. Which one do you think will result in the highest outcome? Explain your reasoning.

Calculate the decay factor for each bounce by dividing the second height by the first height, then the third by the second, until you reach the bottom of the table. Make a list of those decay factors in the space below to three decimal places.

Your equation has three numbers: starting value, growth factor, and the exponent. You are allowed to double one of these numbers. Which one do you think will result in the highest outcome? Explain your reasoning.

Which double resulted in the highest return on your investment?

Think about the change that resulted in the highest return on your investment. What does that represent in the real world context of the original situation?

After performing this activity, if you had to give advice to someone making an investment that results in exponential growth, what would you tell them?

Complete the table using your equation. As the numbers get larger, you may need to convert millimeters to centimeters, meters, or kilometers. Be sure to include units with your numbers!

Choose any three rows of your table and find an object that exists in the world that is approximately the same size. For example, if your measurement was 7 mm, you could say that an iPhone 13 is about 7mm thick. Write your answers in the space provided below.

Imagine that there was an investment that performed similarly by starting small and doubling after a certain amount of time has passed. What is the best advice that you would give to someone that was holding this investment?

Calculate the decay factor for each bounce by dividing the second height by the first height, then the third by the second, until you reach the bottom of the table. Make a list of those decay factors in the space below to three decimal places.

Test your function by substituting a value on your table for x and seeing if it is close to the actual height measured in the video. Does your equation appear to be a good representation of the ball’s height?

Use your equation to make a prediction about the height of the ball after 7 bounces.

Which double resulted in the highest return on your investment?

Think about the change that resulted in the highest return on your investment. What does that represent in the real world context of the original situation?

After performing this activity, if you had to give advice to someone making an investment that results in exponential growth, what would you tell them?

Complete the table using your equation. As the numbers get larger, you may need to convert millimeters to centimeters, meters, or kilometers. Be sure to include units with your numbers!

Choose any three rows of your table and find an object that exists in the world that is approximately the same size. For example, if your measurement was 7 mm, you could say that an iPhone 13 is about 7mm thick. Write your answers in the space provided below.

Imagine that there was an investment that performed similarly by starting small and doubling after a certain amount of time has passed. What is the best advice that you would give to someone that was holding this investment?

Test your function by substituting a value on your table for x and seeing if it is close to the actual height measured in the video. Does your equation appear to be a good representation of the ball’s height?

Use your equation to make a prediction about the height of the ball after 7 bounces.