# MA.912.AR.5.5

Given an expression or equation representing an exponential function, reveal the constant percent rate of change per unit interval using the properties of exponents. Interpret the constant percent rate of change in terms of a real-world context.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Exponential function

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In Algebra I, students generated equivalent algebraic expressions suing the properties of exponents and wrote exponential functions that modeled relationships characterized by having a constant percent of change per unit interval. In Math for College Liberal Arts, students rewrite these exponential functions to reveal the constant percent rate of change per unit interval and interpret its meaning in terms of the given context. In other courses, students will solve and graph mathematical and real-world problems that are modeled with exponential functions, including interpreting key features in terms of the context.
• Students may need support with using the properties of exponents fluently.
• Instruction includes connecting prior knowledge of constant rate of change, including linear functions and making comparisons as noted in MA.912.F.1.6
• When generating equivalent expressions to reveal the constant percent rate of change per unit interval, students should be encouraged to approach from different entry points and discuss how they are different but equivalent strategies (MTR.2.1)
• Instruction includes interpreting percentages of growth/decay from exponential functions and see that ?? can be used to determine a percentage.
• For example, the function $f$($x$) = 500(1.16)$x$ represents 16% growth of an initial value.
• Guide students to discuss the meaning of the number 1.16 as a percent. They should understand it represents 116%. Taking 116% of an initial value increases the magnitude of that value. (Students can test this in a calculator to confirm.) Taking this percentage repetitively leads to exponential growth.
• For example, the function $f$($x$) = 500(0.72)$x$ represents 28% decay of an initial value.
• Guide students to discuss the meaning of the number 0.72 as a percent. They should understand it represents 72%. Taking 72% of an initial value decreases the magnitude of that value. (Students can test this in a calculator to confirm.) Taking this percentage repetitively leads to exponential decay.
• For example, the function $f$($x$) = 500(1)$x$ represents an initial value that neither grows nor decays as $x$ increases.
• Guide students to discuss the meaning of the number 1 when it comes to growth/decay factors. They should understand it represents 100%. Taking 100% of an initial value causes the value to remain the same. (Students can test this in a calculator to confirm.) Taking this percentage repetitively leads to no change in the initial value (explaining the horizontal line that shows when $b$ =1 on the graph).
• Instruction includes real-world problems on compound interest and continuously compounded interest in connection with MA.912.FL.3.4.

### Common Misconceptions or Errors

• Students may not have fully mastered the Laws of Exponents and understand the mathematical connections between the bases and the exponents.
• Students may struggle with representing the growth factor as a percent, rather than a whole number.
• Students may represent the growth rate or decay rate within the formula instead of using the formula $f$($x$) = $a$(1 ± $r$)$x$. To address this misconception, students should record the values being used to showcase the growth or decay to see their mistake.

• Four physicists describe the amount of a radioactive substance in grams, $Q$, left after $t$ years below.
Function 1: $Q$ = 300$e$−0.0577$t$
Function 2: $Q$ = 300($\frac{\text{1}}{\text{2}}$) $\frac{\text{t}}{\text{12}}$
Function 3: $Q$ = 300(0.9439)$t$
Function 4: $Q$ = 252.290(0.9439)$t$−3
• Part A. Compare the four different functions describing the radioactive substance.
• Part B. Determine whether the described functions are equivalent.
• Part C. What is the constant percent rate of change annually of the radioactive substance?
• Part D. Why do you think each of the four physicists described the amount of radioactive substance differently? What information does each tell you?

• A fisherman illegally introduces some fish into a lake, and they quickly propagate. The growth of the population of this new species (within a period of a few years) is modeled by $P$($x$) = 5$b$$x$, where $x$ is the time in weeks following the introduction and b is a positive unknown base.
• Part A. Exactly how many fish did the fisherman release into the lake?
• Part B. Find $b$ if you know the lake contains 33 fish after eight weeks.
• Part C. Instead, now suppose that $P$($x$)= 5$b$$x$ and $b$ = 2. What is the weekly percent growth rate in this case? What does this mean in everyday language?
• Part D. How does the weekly percent growth rate compare in Part B and Part C?

### Instructional Items

Instructional Item 1
• The equation $y$ = 42,500(0.9198)$x$ represents the value of a car $x$ years after its initial purchase. The average rate of depreciation for vehicles is often measured in 5-year intervals. Write an equivalent expression to show the constant percent rate of change over 5 years.

Instructional Item 2
• The function $V$($t$)= 250,000(1.038)$t$ represents the value ($V$) of a home $t$ years after its initial purchase. Interpret the annual percent rate of change in this context.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.AR.5.AP.5: Given an expression or equation representing an exponential function, reveal the constant percent rate of change per unit interval using the properties of exponents.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Exponential Functions - 2:

Students are asked to identify the percent rate of change of a given exponential function.

Type: Formative Assessment

Exponential Functions - 1:

Students are asked to identify the percent rate of change of a given exponential function.

Type: Formative Assessment

## Lesson Plan

You’re Pulling My Leg – or Candy!:

Students will explore how exponential growth and decay equations can model real-world problems. Students will also discover how manipulating the variables in an exponential equation changes the graph. Students will watch a Perspectives Video to see how exponential growth is modeled in the real world.

Type: Lesson Plan

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

## MFAS Formative Assessments

Exponential Functions - 1:

Students are asked to identify the percent rate of change of a given exponential function.

Exponential Functions - 2:

Students are asked to identify the percent rate of change of a given exponential function.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.