 # Standard #: MAFS.912.A-CED.1.3 (Archived Standard)

This document was generated on CPALMS - www.cpalms.org

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

### General Information

Subject Area: Mathematics
Domain-Subdomain: Algebra: Creating Equations
Cluster: Create equations that describe numbers or relationships. (Algebra 1 - Major Cluster) (Algebra 2 - Supporting Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes

### Test Item Specifications

N/A

Assessment Limits :
In items that require the student to write an equation as a constraint, the equation may be a linear function.

In items that require the student to write a system of equations to represent a constraint, the system is limited to two variables.

In items that require the student to write a system of inequalities to represent a constraint, the system is limited to two variables

Calculator :

Neutral

Clarification :
Students will write constraints for a real-world context using equations, inequalities, a system of equations, or a system of inequalities.

Students will interpret the solution of a real-world context as viable or not viable.

Stimulus Attributes :
Items must be set in a real-world context.

Items may use function notation.

Response Attributes :
Items may require the student to choose an appropriate level of accuracy.

Items may require the student to choose and interpret the scale in a graph.

Items may require the student to choose and interpret units. Items may require the student to apply the basic modeling cycle.

### Sample Test Items (1)

 Test Item # Question Difficulty Type Sample Item 1 The production cost, C, in thousands of dollars, for a toy company to manufacture a ball is given by the model C(x)=75+21x-0.72x², where x is the number of balls produced in one day, in thousands. The company wants to keep its production cost at or below \$125,000. The graph shown models the situation. What is a reasonable constraint for the model? N/A MC: Multiple Choice

#### Related Courses

 Course Number1111 Course Title222 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)) 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)) 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)) 1200500: Advanced Algebra with Financial Applications (Specifically in versions: 2014 - 2015 (course terminated)) 1200410: Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated)) 1200700: Mathematics for College Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated)) 1304300: Music Technology and Sound Engineering 1 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current)) 1304310: Music Technology and Sound Engineering 2 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current)) 7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 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)) 1200335: Algebra 2 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2019 (course terminated)) 1200375: Algebra 1-A for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current)) 7912100: Fundamental Algebraic Skills (Specifically in versions: 2013 - 2015, 2015 - 2017 (course terminated)) 1207300: Liberal Arts Mathematics 1 (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated)) 7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current)) 7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current)) 1200387: Mathematics for Data and Financial Literacy (Specifically in versions: 2016 and beyond (current))

#### Formative Assessments

 Name Description The New School Students are asked to represent constraints using inequalities given in a problem context. Sugar and Protein Students are asked to model a problem involving constraints using inequalities. 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.

#### Original Student Tutorials

 Name Description Solving Systems of Linear Equations Part 6: Writing Systems from Context Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial. This part 6 in a 7-part series. Click below to explore the other tutorials in the series. Solving Inequalities and Graphing Solutions Part 2 Learn how to solve and graph compound inequalities and determine if solutions are viable in part 2 of this interactive tutorial series. Click HERE to open Part 1. Solving Inequalities and Graphing Solutions: Part 1 Learn how to solve and graph one variable inequalities, including compound inequalities, in part 1 of this interactive tutorial series. Click HERE to open Part 2.

#### Perspectives Video: Expert

 Name Description Problem Solving with Project Constraints 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!

#### Perspectives Video: Professional/Enthusiasts

 Name Description Solving Systems of Equations, Oceans & Climate Angela Dial discusses how she solves systems of equations to determine how the composition of ocean floor sediment has changed over 65 million years to help reveal more information regarding climate change. Hurricane Dennis & Failed Math Models What happens when math models go wrong in forecasting hurricanes? Download the CPALMS Perspectives video student note taking guide. Gear Heads and Gear Ratios Have a need for speed? Get out your spreadsheet! Race car drivers use algebraic formulas and spreadsheets to optimize car performance.

 Name Description 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 mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not. Writing Constraints The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question (i.e., "constraining" said values). In particular, it is worth differentiating the role of constraint equations from more functional equations, e.g., formulas to convert from degrees Celsius to degree Fahrenheit. The task has students interpret the context and choose variables to represent the quantities, which are governed by the constraint equation and the fact that they are non-negative (allowing us to restrict the graphs to points in the first quadrant only). The four parts are independent and can be used as separate tasks. Bernardo and Sylvia Play a Game This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game. Dimes and Quarters Students are given a word problem that can be solved by using a pair of linear equations. This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. This is an important step in the process of solving systems. Growing Coffee This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b). MIT BLOSSOMS - The Broken Stick Experiment: Triangles, Random Numbers and Probability This learning video is designed to develop critical thinking in students by encouraging them to work from basic principals to solve a puzzling mathematics problem that contains uncertainty. One class session of approximately 55 minutes is necessary for lesson completion. First-year simple algebra is all that is required for the lesson, and any high school student in a college-preparatory math class should be able to participate in this exercise. Materials for in-class activities include: a yard stick, a meter stick or a straight branch of a tree; a saw or equivalent to cut the stick; and a blackboard or equivalent. In this video lesson, during in-class sessions between video segments, students will learn among other things: 1) how to generate random numbers; 2) how to deal with probability; and 3) how to construct and draw portions of the X-Y plane that satisfy linear inequalities.

#### Unit/Lesson Sequence

Name Description
Sample Algebra 1 Curriculum Plan Using CMAP

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

### Using this CMAP

To view an introduction on the CMAP tool, please .

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All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

#### Video/Audio/Animation

 Name Description Basic Linear Function This video demonstrates writing a function that represents a real-life scenario.

#### Original Student Tutorials

 Name Description Solving Systems of Linear Equations Part 6: Writing Systems from Context: Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial. This part 6 in a 7-part series. Click below to explore the other tutorials in the series. Solving Inequalities and Graphing Solutions Part 2: Learn how to solve and graph compound inequalities and determine if solutions are viable in part 2 of this interactive tutorial series. Click HERE to open Part 1. Solving Inequalities and Graphing Solutions: Part 1: Learn how to solve and graph one variable inequalities, including compound inequalities, in part 1 of this interactive tutorial series. Click HERE to open Part 2.

#### Perspectives Video: Expert

 Name Description Problem Solving with Project Constraints: 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!

 Name Description 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 mathematical practice standard MP4 (Model with mathematics), and crucial as the system has an integer solution for both situations, that is, whether or not we include the dollar on the floor in the cash box or not. Writing Constraints: The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question (i.e., "constraining" said values). In particular, it is worth differentiating the role of constraint equations from more functional equations, e.g., formulas to convert from degrees Celsius to degree Fahrenheit. The task has students interpret the context and choose variables to represent the quantities, which are governed by the constraint equation and the fact that they are non-negative (allowing us to restrict the graphs to points in the first quadrant only). The four parts are independent and can be used as separate tasks. Bernardo and Sylvia Play a Game: This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game. Dimes and Quarters: Students are given a word problem that can be solved by using a pair of linear equations. This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. This is an important step in the process of solving systems. Growing Coffee: This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b).

#### Video/Audio/Animation

 Name Description Basic Linear Function: This video demonstrates writing a function that represents a real-life scenario.

#### Perspectives Video: Expert

 Name Description Problem Solving with Project Constraints: 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!