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
Grade: 912
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


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Related Resources

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.

Lesson Plans

Name Description
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Students with investigate the amount of space that could be saved by flattening cardboard boxes. The analysis includes linear graphs and regression analysis along with discussions of slope and a direct variation phenomenon.

Solving Linear Equations in Two Variables

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Preserving Our Marine Ecosystems

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Alternative Fuel Systems

The Alternative Fuel Systems MEA provides students with an engineering problem in which they must develop a procedure to decide the appropriate course for an automobile manufacturer to take given a set of constraints. The main focus of the MEA is to apply the concepts of work and energy to a business model.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Don't Blow the Budget!

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

Efficient Storage

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Feasible or Non-Feasible? - That is the Question (Graphing Systems of Linear Inequalities)

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Exploring Systems with Piggies, Pizzas and Phones

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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
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Download the CPALMS Perspectives video student note taking guide.

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Problem-Solving Tasks

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

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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.

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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).

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Unit/Lesson Sequence

Name Description
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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 .

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

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Video/Audio/Animation

Name Description
Basic Linear Function

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Student Resources

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!

Problem-Solving Tasks

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.



Parent Resources

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!

Problem-Solving Tasks

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).



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