# MA.912.AR.2.5

Solve and graph mathematical and real-world problems that are modeled with linear functions. Interpret key features and determine constraints in terms of the context.

### Examples

Algebra 1 Example: Lizzy’s mother uses the function C(p)=450+7.75p, where C(p) represents the total cost of a rental space and p is the number of people attending, to help budget Lizzy’s 16th birthday party. Lizzy’s mom wants to spend no more than \$850 for the party. Graph the function in terms of the context.

### Clarifications

Clarification 1: Key features are limited to domain, range, intercepts and rate of change.

Clarification 2: Instruction includes the use of standard form, slope-intercept form and point-slope form.

Clarification 3: Instruction includes representing the domain, range and constraints with inequality notation, interval notation or set-builder notation.

Clarification 4: Within the Algebra 1 course, notations for domain, range and constraints are limited to inequality and set-builder.

Clarification 5: Within the Mathematics for Data and Financial Literacy course, problem types focus on money and business.

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

• Coordinate Plane
• Domain
• Function Notation
• Range
• Rate of Change
• Slope
• $x$-intercept
• $y$-intercept

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students determined and interpreted the slope and $y$-intercept of a two-variable linear equation in slope-intercept form from a real-world context. In Algebra I, students solve real-world problems that are modeled with linear functions when given equations in all forms, as well as tables and written descriptions, and they determine and interpret the domain, range and other key features. Students will additionally, interpret key features and identify any constraints. In later courses, students will graph and solve problems involving linear programming, systems of equations in three variables and piecewise functions.
• This benchmark is a culmination of MA.912.AR.2. Instruction here should feature a variety of real-world contexts.
• Instruction includes representing domain, range and constraints using words, inequality notation and set-builder notation.
•  Words
•  If the domain is all real numbers, it can be written as “all real numbers” or “any value of $x$, such that $x$ is a real number.”
• Inequality Notation
•  If the domain is all values of $x$ greater than 2, it can be represented as $x$ > 2.
•  Set-Builder Notation
•  If the domain is all values of $x$ less than or equal to zero, it can be represented as {$x$|$x$ ≤ 0} and is read as “all values of $x$ such that $x$ is less than or equal to zero.”
• Instruction includes the use of $x$-$y$ notation and function notation.
• This benchmark presents the first opportunity for students to represent constraints in the domain and range of functions. Students should develop an understanding that linear graphs, without context, have no constraints on their domain and range. When specific contexts are modeled by linear functions, parts of the domain and range may not make sense and need to be removed, creating the need for constraints.
• Instruction includes the understanding that a real-world context can be represented by a linear two-variable equation even though it only has meaning for discrete values.
• For example, if a gym membership cost \$10.00 plus \$6.00 for each class, this can be represented as $y$ = 10 + 6$c$. When represented on the coordinate plane, the relationship is graphed using the points (0,10), (1,16), (2,22), and so on.
• For mastery of this benchmark, students should be given flexibility to represent real-world contexts with discrete values as a line or as a set of points.
• Instruction directs students to graph or interpret a representation of a context that necessitates a constraint. Discuss the meaning of multiple points on the line and announce their meanings in the associated context (MTR.4.1). Allow students to discover that some points do not make sense in context and therefore should not be included in a formal solution (MTR.6.1). Ask students to determine which parts of the line create sensible solutions and guide them to make constraints to represent these sections.
• Instruction includes the use of technology to develop the understanding of constraints.
• Instruction includes the connection to scatter plots and lines of fit (MA.912.DP.2.4) and the connection to systems of equations or inequalities (MA.912.DP.9.6).

### Common Misconceptions or Errors

• Students may express initial confusion with the meaning of $f$($x$) for functions written in function notation.
• Students may assign their constraints to the incorrect variable.
• Students may miss the need for compound inequalities in their constraints. Students may not include zero as part of the domain or range.
• For example, if a constraint for the domain is between 0 and 10, a student may forget to include 0 in some contexts, since they may assume that one cannot have zero people, for instance.

### Strategies to Support Tiered Instruction

• Teacher provides equations in both function notation and $x$-$y$ notation written in slope-intercept form and models graphing both forms using a graphing tool or graphing software (MTR.2.1).
• For example, $f$($x$) = $\frac{\text{2}}{\text{3}}$$x$ + 6 and $y$ = $\frac{\text{2}}{\text{3}}$$x$ + 6, to show that both $f$($x$) and $y$ represent the same outputs of the function.
• Instruction provides opportunities for identifying the domain and range on the $x$- and $y$- axis respectively using a highlighter.
• For example, Tim bought 2 cubic feet of fertilizer and uses a little everyday on his lawn for 6 months, and the amount of fertilizer decreases at a constant rate as shown on the graph. The domain of the function in this context is 0 ≤ $x$ ≤ 6.
• Teacher provides context to visualize and determine if it would make sense for the function to extend to a given area.
• For example, if Garrison bought a house in 2014 and the price increases at a constant rate, he can model this by graphing a linear function where $x$ represents the time since 2014. The domain could include negative values if he wanted to show the estimated price of the house before 2014.
•  For example, if the temperature in Alaska is at 14 degrees Fahrenheit at 6:00 am and drops at a constant rate, this can be modeled by graphing a linear function where $t$ represents the time since 6:00 am. The range could include negative numbers to show the temperature below 0 degrees Fahrenheit.

• The population of St. Johns County, Florida, from the year 2000 through 2010 is shown in the graph below. If the trend continues, what will be the population of St. Johns County in 2025?
• Devon is attending a local festival downtown. He plans to park his car in a parking garage that operates from 7:00 a.m. to 10:00 p.m. and charges \$5 for the first hour and \$2 for each additional hour of parking.
•  Part A. Create a linear graph that represents the relationship between the price and number of hours parked.
•  Part B. What is an appropriate domain and range for the given situation?

### Instructional Items

Instructional Item 1
•  Suppose you fill your truck’s tank with fuel and begin driving down the highway for a road trip. Assume that, as you drive, the number of minutes since you filled the tank and the number of gallons remaining in the tank are related by a linear function. After 40 minutes, you have 28.4 gallons left. An hour after filling up, you have 26.25 gallons left.
•  Part A. Graph this relationship.
•  Part B. Determine how many hours it will take for you to run out of fuel.

*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))
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))
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))
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))
1200388: Mathematics for Data and Financial Literacy Honors (Specifically in versions: 2022 and beyond (current))
1700600: GEAR Up 1 (Specifically in versions: 2020 - 2022, 2022 and beyond (current))
1700610: GEAR Up 2 (Specifically in versions: 2020 - 2022, 2022 and beyond (current))
1700620: GEAR Up 3 (Specifically in versions: 2020 - 2022, 2022 and beyond (current))
1700630: GEAR Up 4 (Specifically in versions: 2020 - 2022, 2022 and beyond (current))
1200384: Mathematics for Data and Financial Literacy (Specifically in versions: 2022 and beyond (current))
7912120: Access Mathematics for Data and Financial Literacy (Specifically in versions: 2022 - 2023, 2023 and beyond (current))
1200710: Mathematics for College Algebra (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.AR.2.AP.5: Given a mathematical and/or real-world problem that is modeled with linear functions, solve the mathematical problem, or select the graph using key features (in terms of context) that represents this model.

## Related Resources

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

## Formative Assessments

Uphill and Downhill:

Students are asked to interpret key features of a graph (intercepts and intervals over which the graph is increasing) in the context of a problem situation.

Type: Formative Assessment

Tech Repairs Graph:

Students are asked to graph an equation in two variables given in context.

Type: Formative Assessment

Lunch Account:

Students are given a linear function in context and asked to interpret its parameters in the context of a problem.

Type: Formative Assessment

Computer Repair:

Students are given a linear function in context and asked to interpret its parameters in the context of a problem.

Type: Formative Assessment

Constraints on Equations:

Students are asked to determine the constraint on a profit equation and to interpret solutions as being viable or not in the context of the problem.

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

What Will I Pay?:

Who doesn't want to save money? In this lesson, students will learn how a better credit score will save them money. They will use a scatter plot to see the relationship between credit scores and car loan interest rates. They will determine a line of fit equation and interpret the slope and y-intercept to make conclusions about interest and credit scores.

Type: Lesson Plan

Why do I have to have a bedtime?:

This predict, observe, explain lesson that allows students to make predictions based on prior knowledge, observations, discussions, and calculations. Students will receive the opportunity to express themselves and their ideas while explaining what they learned. Students will make a prediction, collect data, and construct a scatter plot. Next, students will calculate the correlation coefficient and use it to describe the strength and magnitude of a relationship.

Type: Lesson Plan

Is My Model Working?:

Students will enjoy this project lesson that allows them to choose and collect their own data. They will create a scatter plot and find the line of fit. Next they write interpretations of their slope and y-intercept. Their final challenge is to calculate residuals and conclude whether or not their data is consistent with their linear model.

Type: Lesson Plan

Spaghetti Trend:

This lesson consists of using data to make scatter plots, identify the line of fit, write its equation, and then interpret the slope and the y-intercept in context. Students will also use the line of fit to make predictions.

Type: Lesson Plan

Slippery Slopes:

This lesson will not only reinforce students understanding of slope and y-intercept, but will also ensure the students understand how it can be modeled in a real world situation. The focus of this lesson is to emphasize that slope is a rate of change and the y-intercept the value of y when x is zero. Students will be able to read a problem and create a linear equation based upon what they read. They will then make predictions based upon this information.

Type: Lesson Plan

Whose Line Is It Anyway?:

In this lesson, students will use graphing calculators to explore linear equations in the form y = mx + b. They will observe the graphs of equations with different values of slope and y-intercept. They will draw conclusions about how the value of slope and y-intercept are visible in the appearance of the graph.

Type: Lesson Plan

Testing water for drinking purposes:

The importance of knowing what drinking water contains. How to know what properties are present in different bottled water. Knowing the elements present in water that is advantageous to growth and development of many things in the body. To know what to be alert for in water and to understand the importance of water in general.

Type: Lesson Plan

How Fast Do Objects Fall?:

Students will investigate falling objects with very low air friction.

Type: Lesson Plan

Which Function?:

This activity has students apply their knowledge to distinguish between numerical data that can be modeled in linear or exponential forms. Students will create mathematical models (graph, equation) that represent the data and compare these models in terms of the information they show and their limitations. Students will use the models to compute additional information to predict future outcomes and make conjectures based on these predictions.

Type: Lesson Plan

Don't Blow the Budget!:

Students use systems of equations and inequalities to solve real world budgeting problems involving two variables.

Type: Lesson Plan

My Candles are MELTING!:

In this lesson, students will apply their knowledge to model a real-world linear situation in a variety of ways. They will analyze a situation in which 2 candles burn at different rates. They will create a table of values, determine a linear equation, and graph each to determine if and when the candles will ever be the same height. They will also determine the domain and range of their functions and determine whether there are constraints on their functions.

Type: Lesson Plan

Turning Tires Model Eliciting Activity:

The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of the MEA is applying surface area concepts and algebra through modeling.

Type: Lesson Plan

Exploring Systems with Piggies, Pizzas and Phones:

Students write and solve linear equations from real-life situations.

Type: Lesson Plan

Movie Theater MEA:

In this Model Eliciting Activity, MEA, students create a plan for a movie theater to stay in business. Data is provided for students to determine the best film to show, and then based on that decision, create a model of ideal sales. Students will create equations and graph them to visually represent the relationships.

Type: Lesson Plan

## Original Student Tutorial

Linear Functions: Jobs:

Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Experts

Using Mathematics to Optimize Wing Design:

Nick Moore discusses his research behind optimizing wing design using inspiration from animals and how they swim and fly.

Type: Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

Solving Systems of Equations, Oceans & Climate:

<p>Angela Dial discusses how she solves systems of equations to determine how the composition&nbsp;of ocean floor sediment has changed over 65 million years to help reveal more information&nbsp;regarding&nbsp;climate change.</p>

Type: Perspectives Video: Professional/Enthusiast

Coffee and Crime:

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.

Cash Box:

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or not.

Downhill:

This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

## STEM Lessons - Model Eliciting Activity

Movie Theater MEA:

In this Model Eliciting Activity, MEA, students create a plan for a movie theater to stay in business. Data is provided for students to determine the best film to show, and then based on that decision, create a model of ideal sales. Students will create equations and graph them to visually represent the relationships.

Testing water for drinking purposes:

The importance of knowing what drinking water contains. How to know what properties are present in different bottled water. Knowing the elements present in water that is advantageous to growth and development of many things in the body. To know what to be alert for in water and to understand the importance of water in general.

Turning Tires Model Eliciting Activity:

The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of the MEA is applying surface area concepts and algebra through modeling.

## MFAS Formative Assessments

Computer Repair:

Students are given a linear function in context and asked to interpret its parameters in the context of a problem.

Constraints on Equations:

Students are asked to determine the constraint on a profit equation and to interpret solutions as being viable or not in the context of the problem.

Lunch Account:

Students are given a linear function in context and asked to interpret its parameters in the context of a problem.

Tech Repairs Graph:

Students are asked to graph an equation in two variables given in context.

Uphill and Downhill:

Students are asked to interpret key features of a graph (intercepts and intervals over which the graph is increasing) in the context of a problem situation.

## Original Student Tutorials Mathematics - Grades 9-12

Linear Functions: Jobs:

Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial.

## Student Resources

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

## Original Student Tutorial

Linear Functions: Jobs:

Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

Coffee and Crime:

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.

Cash Box:

The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or not.

Downhill:

This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

## Parent Resources

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

## Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

Coffee and Crime:

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.