Lesson 11 | Linear Functions and Applications | 11th Grade Mathematics

Objective

Write systems of linear inequalities from a contextual situation.

Common Core Standards

Core Standards

The core standards covered in this lesson

A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Creating Equations

A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A.REI.D.12 — Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Reasoning with Equations and Inequalities

A.REI.D.12 — Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Foundational Standards

The foundational standards covered in this lesson

7.EE.B.4.B

Expressions and Equations

7.EE.B.4.B — Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Define variables and write linear inequalities that model a context.
Identify possible solutions to that context.
Identify an objective function that helps to define when a solution is "better" or "worse."
Identify solutions that are "better" or "worse" based on an objective function.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

A clothing manufacturer has to use at least 1,000 yards of cotton to make room for new inventory to make shirts and pajamas. A shirt requires 1 yard of fabric, and a pair of pajamas requires 3 yards of fabric. It takes 2 hours to make a shirt and 3 hours to make a pair of pajamas, and there are 1600 hours available to make all of the clothing. Let $$x$$ represent the number of shirts and $$y$$ represent the number of pajamas.

What is the smallest number of pajamas the manufacturer can make and meet all of these conditions?

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 1 > Topic C > Lesson 24 — Exploratory Challenge

Algebra I > Module 1 > Topic C > Lesson 24 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Phil and Cath make and sell boomerangs for a school event. The money they raise will go to charity. They plan to make them in two sizes: small and large.

Phil will carve them from wood.

The small boomerang takes 2 hours to carve, and the large one takes 3 hours to carve.
Phil has a total of 24 hours available for carving.

Cath will decorate them. She only has time to decorate 10 boomerangs of either size.

Phil and Cath want to make as much money for charity as they can.

The small boomerang will make $8 for charity.
The large boomerang will make $10 for charity.

How many small and large boomerangs should they make?
How much money will they then make?

Guiding Questions

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References

MARS Formative Assessment Lessons for High School Maximizing Profits: Selling Boomerangs

Maximizing Profits: Selling Boomerangs from the Classroom Challenges by the MARS Shell Center team at the University of Nottingham is made available by the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed July 29, 2017, 10:08 a.m..

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Below is student work for Anchor Problem #2. Describe improvements that this student could make to his work to fully answer the question.

References

MARS Formative Assessment Lessons for High School Maximizing Profits: Selling Boomerangs — Sample Response: Jeremiah

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems that extend Anchor Problem #1 to include writing an objective function and identifying some “better” and “worse” amounts of shirts and pajamas based on the objective function.

Linear Systems: The Busing Problem
Illustrative Mathematics Bernardo and Sylvia Play a Game
JMAP A.CED.A.3: Modeling Systems of Linear Inequalities — Modeling Systems of Linear Inequalities
Mathematics Vision Project: Secondary Mathematics One Module 5: Systems of Equations and Inequalities — all lessons in modules have some application of writing linear systems from contextual problems

Lesson 10

Lesson 12

Lesson Map

Topic A: Features of Linear Functions

Identify features of linear functions from equations, verbal descriptions, tables, and graphs.

A.SSE.A.1.A F.IF.B.4 F.IF.C.9

Write linear functions that represent contextual situations.

F.IF.A.2 F.IF.B.4 F.IF.B.5

Solve for a desired quantity in a linear function.

A.CED.A.1 A.CED.A.4

Find the inverse of a contextual situation graphically and describe the meaning of the function and its inverse.

F.BF.B.4.A F.BF.B.4.C

Find the inverse of contextual and non-contextual situations algebraically.

F.BF.B.4.A F.BF.B.4.B F.BF.B.4.C

Topic B: Systems of Functions and Constraints

Write a system of functions for contextual situations and solve algebraically. Describe the solutions in context of the problem.

A.CED.A.2 A.REI.C.6 F.IF.B.5

Describe the number of solutions of a system of equations. Verify algebraically.

A.REI.C.6 A.REI.D.11

Solve a system of three equations in three variables.

A.REI.C.6

Identify the solution to a system of an absolute value equation and a linear function algebraically and graphically.

A.REI.D.11

Graph and identify solutions to systems of linear inequalities.

A.CED.A.3 A.REI.D.12 F.IF.B.5

Write systems of linear inequalities from a contextual situation.

A.CED.A.3 A.REI.D.12

Topic C: Piecewise Functions

Write and evaluate piecewise functions from graphs. Graph piecewise functions from algebraic representations.

A.CED.A.2 F.IF.C.7.B

Write piecewise functions from contextual situations.

A.CED.A.2 A.CED.A.3 F.IF.B.4 F.IF.C.7.B

Linear Functions and Applications

Objective

Common Core Standards

Core Standards

Creating Equations

Reasoning with Equations and Inequalities

Foundational Standards

Expressions and Equations

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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