### Clarifications

*Clarification 1:*Within the Algebra 1 course, exponential functions are limited to the forms , where

*b*is a whole number greater than 1 or a unit fraction, or , where .

*Clarification 2:* Within the Algebra 1 course, tables are limited to having successive nonnegative integer inputs so that the function may be determined by finding ratios between successive outputs.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Algebraic Reasoning

**Standard:**Write, solve and graph exponential and logarithmic equations and functions in one and two variables.

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Exponential

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In middle grades, students solved problems involving percentages, including percent increases and decreases and write equations that represent proportional relationships. In Algebra I, students write exponential functions that model relationships characterized by having a constant percent of change per unit interval. In later courses, students will further develop their understanding of this feature of exponential functions.

- Provide opportunities to reference MA.912.AR.1.1 as students identify and interpret parts of an exponential equation or expression as growth or decay and connect them to key features of the graph.
- Problems include cases where the initial value is not given.
- Instruction includes guidance on how to determine the initial value or the percent rate of
change of an exponential function when it is not provided.
- For example, if the initial value of (0,3) is given, students can now write the
function as $f$($x$) = 3$b$
^{$x$ }. Guide students to choose a point on the curve that has integer coordinates such as (2, 12). Lead them to substitute the point into their function to find $b$. Students should recognize that exponential functions are restricted to positive values of $b$, leading to the function $f$($x$) = 3(2)^{$x$ }.

- For example, if the initial value of (0,3) is given, students can now write the
function as $f$($x$) = 3$b$
- Instruction includes interpreting percentages of growth/decay from exponential functions
expressed in the form $f$($x$) = $a$$b$
^{$x$}and see that $b$ can be used to determine a percentage.- For example, the function $f$($x$) = 500(0.72)
^{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).

- For example, the function $f$($x$) = 500(0.72)

### Common Misconceptions or Errors

- Students may not understand that exponential function values will eventually get larger than those of any other polynomial functions because they do not fully understand the impact of exponents on a value.

### Strategies to Support Tiered Instruction

- Teacher provides students with a graphic displaying key terms within an exponential function.

- Instruction includes comparing quadratic (polynomial) to exponential functions using graphs or tables with whole-number inputs to show how exponential functions will quickly exceed quadratic (polynomial) functions.

### Instructional Tasks

*Instructional Task 1 (MTR.4.1, MTR.5.1, MTR.7.1)*

- Karl and Simone were working on separate biology experiments. Each student documented
their cell population counts over time in the chart below.

- Part A. Do the number of cells in Simone’s experiment increase at a constant percentage rate of change? If so, what is the percentage rate? If not, describe what is happening to the number of cells. Does this change represent growth or decay? Justify your answer.
- Part B. Write exponential functions to represent the relationship between the quantities for each student’s experiment. In which experiment are the number of cells changing more rapidly? Justify your answer.
- Part C. Graph these functions and determine their key features.

*Instructional Task 2 (MTR.2.1,*

*MTR.7.1*)- The population of J-Town in 2019 was estimated to be 76,500 people with an annual rate of
increase of 2.4%.
- Part A. Write an equation to model future growth.
- Part B. What is the growth factor for J-Town?
- Part C. Use the equation to estimate the population in 2072 to the nearest hundred people.

### Instructional Items

*Instructional Item 1*

- Write an exponential function that represents the graph below.

Instructional Item 2

- A forester has determined that the number of fir trees in a forest is decreasing by 3% per year. In 2010, there were 13,000 fir trees in the forest. Write an equation that represents the number of fir trees, $N$, in terms of $t$, the number of years since 2010.

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

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorial

## Problem-Solving Tasks

## MFAS Formative Assessments

Students are asked to explain the relationship between the set of solutions and the graph of an exponential equation.

Students are asked to write and solve an equation that models an exponential relationship between two variables.

Students are asked to write function rules for sequences given tables of values.

Students are asked to write an exponential function from a written description of an exponential relationship.

Students are asked to write an exponential function represented by a table of values.

Students are asked to write an exponential function given its graph.

## Original Student Tutorials Mathematics - Grades 9-12

Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial.

## Student Resources

## Original Student Tutorial

Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial.

Type: Original Student Tutorial

## Problem-Solving Tasks

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

Type: Problem-Solving Task

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

Type: Problem-Solving Task

This problem solving tasks asks students to find the values of points on a graph.

Type: Problem-Solving Task

This problem solving task asks students to graph a function and find the values of points on a graph.

Type: Problem-Solving Task

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Type: Problem-Solving Task

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

Type: Problem-Solving Task

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

Type: Problem-Solving Task

This problem solving tasks asks students to find the values of points on a graph.

Type: Problem-Solving Task

This problem solving task asks students to graph a function and find the values of points on a graph.

Type: Problem-Solving Task

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Type: Problem-Solving Task

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

Type: Problem-Solving Task