# MA.912.F.1.8

Determine whether a linear, quadratic or exponential function best models a given real-world situation.

### Clarifications

Clarification 1: Instruction includes recognizing that linear functions model situations in which a quantity changes by a constant amount per unit interval; that quadratic functions model situations in which a quantity increases to a maximum, then begins to decrease or a quantity decreases to a minimum, then begins to increase; and that exponential functions model situations in which a quantity grows or decays by a constant percent per unit interval.

Clarification 2: Within this benchmark, the expectation is to identify the type of function from a written description or table.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Functions
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Exponential Function
• Linear Function

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students determined whether a linear relationship is also a proportional relationship. In Algebra I, students determine whether a linear, quadratic or exponential function best models a situation. In later grades, students will fit linear, quadratic, and exponential functions to statistical data.
• Instruction should include identifying function types from tables and from written descriptions.
• When examining written descriptions, guide students to see that linear functions model situations in which a quantity changes by a constant amount per unit interval; that quadratic functions model situations in which a quantity increases to a maximum, then begins to decrease or a quantity decreases to a minimum, then begins to increase; and that exponential functions model situations in which a quantity grows or decays by a constant percent per unit interval.
• When considering tables, instruction guides students to understand that linear relationships have a common difference per unit interval (or a constant rate of change), quadratic relationships produce a common second difference, and exponential relationships produce a constant percent rate of change per unit interval (MTR.5.1). It is important to note that using the common second difference method is not an expectation for mastery in Algebra I.
• Considering tables like the one below, lead students to discover that there is a common difference of 0.4 between successive $y$-values. Plotting these points using graphing software will verify that they are collinear.

• Considering tables like the one below, lead students to discover that there is a common second difference of -8 between successive $y$-values. Plotting these points using graphing software will verify that they form a parabola.

• Considering tables like the one below, lead students to discover that there is no common difference or second difference. In this case, there is a common ratio of $\frac{\text{3}}{\text{1}}$ between successive $y$-values. Plotting these points using graphing software will verify that they form an exponential graph.

• Students should note that the search for common differences and ratios only works when the $x$-values are equidistant from each other. Lead them to check for this when presented with tables of values to consider.
• It is important to note that other function types could produce these relationships, making the connection to classifying different function types in MA.912.F.1.1.

### Common Misconceptions or Errors

• Students may interpret any relationship that increases/decreases at a non-constant rate as being an exponential relationship. Have these students verify an exponential relationship by looking for common ratios. If they are interpreting a written description, direct them to make a sample table of values from the context to examine.
• Some students may miscalculate first and second differences that deal with negative values, especially if they perform them mentally. In these cases, have students quickly write out the subtraction expression [i.e., −14 – (−2)] so they can see that they are subtracting a negative value and should convert it to adding a positive value.
• It is often helpful to have these students draw a blank number line with a mark for 0 to use for their calculations. Students who solve −14 + 2 to equal −16 could place their pencil tip to the left of 0 on the number line in a position that could represent −14. Ask them which direction they would move to represent adding 2. When students see movement to the right, toward zero, they should understand that the magnitude of the negative number decreases, resulting in −12 rather than −16.

### Strategies to Support Tiered Instruction

• Instruction includes verifying an exponential relationship by looking for common ratios. When interpreting a written description, make a sample table of values from the context to examine the type of function.
• Teacher co-creates a graphic organizer to compare exponential and quadratic functions.
• For example, a Venn Diagram can be used with the common middle section including the non-constant rate of change.

• Teacher provides opportunities to write out subtraction sentences next to each line of the table when determining first and second differences.

• A scientist is monitoring cell division and notes that a single cell divides into 4 cells within one hour. During the next hour, each of these cells divides into 4 cells. This process continues at the same rate every hour.
• Part A. What type of function could be used to represent this situation?
• Part B. Justify your reasoning.

### Instructional Items

Instructional Item 1
• Sarah is spending the summer at her grandmother’s house. The table below shows the amount of money in her bank account at the end of each week. What type of function could be used to model the total amount of money in Sarah’s bank account as a function of time?

*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.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200370: Algebra 1-A (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (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))
7912080: Access Algebra 1A (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200375: Algebra 1-A for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1210305: Mathematics for College Statistics (Specifically in versions: 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.F.1.AP.8: Select whether a linear or quadratic function best models a given real-world situation.

## Related Resources

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

## Formative Assessments

Students are given a linear and an exponential function, one represented verbally and the other by a table. Then students are asked to compare the rate of change in each in the context of the problem.

Type: Formative Assessment

Prove Linear:

Students are asked to prove that a linear function grows by equal differences over equal intervals.

Type: Formative Assessment

Prove Exponential:

Students are asked to prove that an exponential function grows by equal factors over equal intervals.

Type: Formative Assessment

Linear or Exponential?:

Students are given four verbal descriptions of functions and asked to identify each as either linear or exponential and to justify their choices.

Type: Formative Assessment

Fit a Function:

Students are given a set of data and are asked to use technology to create a scatter plot and write a function that fits the data set.

Type: Formative Assessment

## Lesson Plans

How Hot Is It?:

This lesson allows the students to connect the science of cricket chirps to mathematics. In this lesson, students will collect real data using the CD "Myths and Science of Cricket Chirps" (or use supplied data), display the data in a graph, and then find and use the mathematical model that fits their data.

Type: Lesson Plan

Span the Distance Glider - Correlation Coefficient:

This lesson will provide students with an opportunity to collect and analyze bivariate data and use technology to create scatter plots, lines of best fit, and determine the correlation strength of the data being compared. Students will have a hands on inquire based lesson that allows them to create gliders to analyze data. This lesson is an application of skills acquired in a bivariate unit of study.

Type: Lesson Plan

Appreciation for Car Depreciation:

Students will use information from the internet or a car dealership's advertisement to identify a car and determine the future value of the car using different depreciation rates over different intervals of time. Students will graph their data to show exponential decay and compare to a linear decrease on the same graph.

Type: Lesson Plan

## Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Water Usage Rates:

A seafaring teacher filters all the good information you need to understand water purification rates for distance traveling.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

Algae Blooms:

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

What functions do two graph points determine?:

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

US Population 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

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.

Rumors:

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

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.

Linear or exponential?:

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

This task asks students to analyze data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context. This variant of the task is not scaffolded; for a more scaffolded version, see Basketball Bounces, Assessment Variation 1.

Students are asked to select the best model for a given context and use the model to make predictions. This task assesses students’ modeling skills. Students are tasked to distinguish between situations that can be modeled with linear and exponential functions and recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

## MFAS Formative Assessments

Fit a Function:

Students are given a set of data and are asked to use technology to create a scatter plot and write a function that fits the data set.

Students are given a linear and an exponential function, one represented verbally and the other by a table. Then students are asked to compare the rate of change in each in the context of the problem.

Linear or Exponential?:

Students are given four verbal descriptions of functions and asked to identify each as either linear or exponential and to justify their choices.

Prove Exponential:

Students are asked to prove that an exponential function grows by equal factors over equal intervals.

Prove Linear:

Students are asked to prove that a linear function grows by equal differences over equal intervals.

## Student Resources

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

Algae Blooms:

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

What functions do two graph points determine?:

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

US Population 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

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.

Rumors:

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

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.

Linear or exponential?:

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

## Parent Resources

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

Algae Blooms:

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

What functions do two graph points determine?:

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

US Population 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

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.

Rumors:

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