Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 38181
Feasible or Non-Feasible? - That is the Question
(Graphing Systems of Linear Inequalities)
In this lesson, students learn how to use the graph of a system of linear inequalities to determine the feasible region. Students practice solving word
problems to find the optimal solution that maximizes profits. Students will use the free application, GeoGebra (see download link under Suggested
Technology) to help them create different graphs and to determine the feasible or non-feasible solutions.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera,
Computer for Presenter, Computers for Students, LCD
Projector, Adobe Acrobat Reader, Microsoft Office, Java
Plugin, GeoGebra Free Software (Download the Free
GeoGebra Software)
Instructional Time: 3 Hour(s) 30 Minute(s)
Freely Available: Yes
Keywords: graphing, linear inequalities in the plane, feasible, maximize, region, system of linear inequalities, linear
programming
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Extension_Answer_Key.docx
Formative_Answer_Key.pdf
High_School_Flipbook.pdf
Homework.pdf
Homework_Answer_Key.docx
KWL_Chart.docx
Rubric_-_Graphing_Systems_of_Inequalities.pdf
Station_2_Answer_1.docx
Station_2_Answer_2.docx
Summative_Assessment.pdf
Summative_Assessment_-_Systems_of_Inequalities.pdf
Worksheet_-_Slope_Intercept_Form_Graphing_-_Practice_with_Geogebra.pdf
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LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Given a word problem, students will be able to construct constraints using systems of inequalities and interpret solutions as feasible or non-feasible options in a
page 1 of 7 modeling context by using GeoGebra and plain graph paper.
Prior Knowledge: What prior knowledge should students have for this lesson?
1. Students should know that a linear function is a function that can be graphically represented in the Cartesian coordinate plane by a straight line. A linear function is
a first degree polynomial of the form, f(x) = m x + b, where m and b are constants and x is a real variable. The constant m is called slope and b is called yintercept.
2. Students should be able to graph equations in slope-intercept form.
3. Students should be able to identify the four inequality symbols.
4. Students should be able to graph by hand linear inequalities and systems of linear inequalities in the plane.
5. Students should be fluent in translating word problems into algebraic expressions and equations.
6. Students know how to make a K-W-L chart:
K stands for what you already KNOW about the topic;
W stands for WHAT YOU WANT TO LEARN about the topic;
L stands for WHAT YOU LEARNED after the topic was taught.
The student will fill in the the first two columns prior to the lesson. The third column will be filled in after the lesson is taught.
KWL_Chart
Guiding Questions: What are the guiding questions for this lesson?
What is a feasible solution?
A feasible solution is one that satisfies all constraints.
Solutions that violate one or more constraints are called "infeasible."
What is an optimal solution?
The optimal solution is a combination x and y that maximizes profit (or minimizes cost)
The optimal solution MUST be a feasible solution.
If the region of feasible solutions is bounded, then the optimal solution lies at a vertex.
Teaching Phase: How will the teacher present the concept or skill to students?
Demonstrate how to graph systems of inequalities
1. Graph each inequality separately.
2. Lightly shade the solution region for each.
3. Find where all shaded regions overlap and mark it more darkly.
4. Label it "FS" for feasible solution.
Examples The objective function Cost = f(x,y) defines the cost of making (or selling) x units of one item and y units of another item.
The objective function Profit = g(x,y) defines the profit achieved by making (or selling) x units of one item and y units of another item.
Explain how to find an optimal solution
1. Identify all vertices and write down their coordinates.
2. One-by-one, substitute the coordinates (x,y) of each vertex into the objective function.
3. The coordinate pair that optimizes the objective function (by maximizing profit or minimizing cost) is an optimal solution.
4. Mark the optimal solution on the graph by circling it, drawing an arrow, putting a star next to it, or write a short comment.
Teacher concludes by defining terminology and showing how to use GeoGebra to plot linear inequalities.
Define Important Terminology
1. A feasible solution is one that makes ALL equations / inequalities true.
2. An infeasible solution is one that makes ONE (or more) of the equations / inequalities false.
3. The objective function describes the outcome for a particular combination of x and y.
4. x and y are called "decision variables" because the decision-maker must decide on the best values to use for x and for y.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Students will arrange desks in a 2x3 pattern to create four different stations (24 desks total). The class will then divide into groups of 5 or 6 per group, to create four
groups. Each group will cycle through the four stations. All groups will begin at Station 1.
To grade the station work, the teacher can use the following rubric:
Rubric - Graphing Systems of Inequalities
Station 1
Students answer the following questions:
What is a system of equations? (A system of equations is when you have two or more variables that can be represented with a set of two or more equations)
How many solutions can a system of equations have? (One solution, No solution, Infinitely Many Solutions)
What is the difference between a system of equations and a system of inequalities? (The solutions of a system of linear equation gives a point of intersection in a
line or plane. The solution of a system of linear inequalities give areas of intersection where any specific pair of points or lines are solutions)
Name the inequality symbols: (less than, less than or equal to, greater than, greater than or equal to) (, >=)
What are boundary lines?(solid and dotted lines)
When is the boundary line solid and when is it dotted? (Solid line when using greater than or great than or equal to symbol and dotted line when using the less
than or less than or equal to symbol)
What do these tell you about the points on the new line?(It tells you whether or not the points on the line are solutions to to the inequality)
What does feasible region means? (The feasible region contains all the points that satisfy all the constraints)
What is a constraint? (The constraint is the inequality)
Station 2
page 2 of 7 Graph each system of constraints using Geogebra. Name all the vertices. Then find the value of x and y that maximize the objective function:
Example 1)
x<5
y≤4
x≥0, y≥0
Maximum for P= 3x + 2y
Station 2 Answer 1
Example 2)
x + y ≥ 6
x ≤ 8
y ≤ 5
Maximum for P = x+ 3y
Station 2 Answer 2
<p?Station 3
Students go to www.geogebratube.org/student/m17.
Students will create at least three systems of inequalities using the GeoGebra applet and describe what the feasible region would be in terms of shape and vertices.
Station 4
Students will answer three word problems using systems of inequalities and GeoGebra.
1. 1. Sarah is selling bracelets and earrings to make money for summer vacation. The bracelets are selling at a cost of $2 and earrings are selling at a cost of $3.
Sarah wants to make at least $500. Write an inequality to represent the income from the jewelry sold. Sarah knows that she will sell more than 50 bracelets.
Write an inequality to represent this situation. Graph the two inequalities and shade the intersection. Identify a solution. How many bracelets and earrings
would Sarah sell?
Answer:
let x = number of bracelets sold
let y = number of earring sold
2x + 3y >=500
x>50
Possible Solution: (200, 200) Sarah could sell for example, 200 bracelets and 200 pairs of earrings.
1. 2. Jason is buying wings and hot dogs for a party. One package of wings cost $7. Hot dogs cost $4 per pound. He must spend less than $40. Write an inequality
to represent the cost of Jason's food for the party. Jason knows that the will be buying at least 5 pounds of hot dogs. Write an inequality to represent this
situation. Graph both inequalities and shade the intersection. Identify two solutions and justify your answers.
Answer:
let x = number of wing packages
let y = the number of pounds of hot dogs
page 3 of 7 7x + 4y <40
y>=5
Solution:(1,6) He could buy 1 package of wings and 6 pounds or hot dogs or he could buy (1,5) 1 package of wings and 5 pounds of hot dogs.
1. 3. The boys and girls soccer clubs are trying to raise money for new uniforms. The boys' soccer club is selling candy bars for $2 apiece and the girls' soccer
club is selling candles for $4. They must raise more than $800. Write an inequality to express the income from the two fundraisers. The girls expect to sell at
least 100 candles. Write an inequality to represent this situation. Graph both inequalities on a grid and shade the intersection. Give two possible solutions to this
system. Justify your answer.
Answer:
let x = number of candy bards
let y = number of candies
2x + 4y >800
y >= 100
>Solution: (200, 200) The boys could sell 200 candy bars and the girls could sell 200 candies or (400, 300) the boys could sell 400 candy bars and the girls
could sell 300 candies.These are just two feasible solutions.
page 4 of 7 Station 5
Create a word problem that requires writing a system of inequalities that has a minimum of two constraints. Describe the objective function. Graph the problem using
GeoGebra and give the optimal solution.
Example: You can work a total of no more than 41 hours each week at your two jobs. Housecleaning pays $5 an hour and your sales job pays $8 an hour. You need to
earn at least $254 each week to pay your bills. Write a system of inequalities that show the various number of hours you can work at each job.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Homework Assignment
Homework: Homework Questions
Answer Key: Homework Answer Key
Problem #1
Mary babysits for $4 per hour. She also works as a tutor for $7 per hour. She is only allowed to work 13 hours per week. She wants to make at least $65 per week.
Write and graph a system of inequalities to represent this situation.
Problem #2
Graph the system of constraints. Name all the vertices. Then find the values of x and y that maximizes the objective function: N = 100x + 40y
x + y ≤ 8
2x + y ≤ 10
x≥ 0
y ≥ 0
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
After students have cycled through four all stations, teacher will bring the class back together. Teacher will ask all students to take out their K-W-L charts and fill in the
L column with what they learned.
Summative Assessment
Students will be given a summative assessment in the form of a written test
Test: Summative_Assessment_-_Systems_of_Inequalities
Answer Key: Summative_Assessment - Answer Key
Teacher will return graded assessment with comments for class review.
Formative Assessment
The teacher will review with students how to graph linear equations and inequalities. The teacher can show equations using GeoGebra, the document camera, or
other technology that is available.
Students will use GeoGebra to practice graphing equations and inequalities
Worksheet: Formative Assessment - Worksheet
Answer Key: Formative Assessment - Answer Key
Students and teacher will go over worksheet together. Teacher will ask follow-up questions such as:
1. What is rise/run?
2. Where is the starting point on the graph when you start to graph your function?
page 5 of 7 3. What is the difference between a positive slope and a negative slope?
4. Why is it necessary to know how to graph in slope-intercept form?
5. What is the difference between an equation and an inequality?
The answers to these questions will reveal whether there are needs for remediation in plotting equations and inequalities. It will also show which students need
additional support in using GeoGebra software.
6. What are the features of the graph of a linear inequality?
A linear inequality involves a linear expression in two (or more) variables that uses any of the relational symbols such as <, >, ≤ or ≥.
A linear inequality divides a plane into two parts:
If the boundary line is solid, then the linear inequality must be either ≥ or ≤. This means that all points on the line are included in the solution because they satisfy
the inequality and make it a true statement.
If the boundary line is dotted, then the linear inequality must be either > or < and the points on the line do not satisfy the inequality and therefore are not included
in the solution.
If the inequality is < or ≤, then you would shade below the line. If the inequality is > or ≥, then you would shade above the line. Only when the inequality is in
slope-intercept form can we use these rules for shading.
How do you know whether an inequality will have a solid or dotted line for its boundary?
How can you determine which side of the line to shade to indicate the solution to the inequality?
Does a ">" sign always mean to shade above the line?
Does a "<" sign always mean to shade below the line?
Feedback to Students
Stress the importance of carefully labeling graphs and including explanatory notes when plotting equations and inequalities using GeoGebra.
Explain to students that, by selecting an Object's Properties in GeoGebra, it is possible to change an object's labeling to show name, value, and captions. Similarly,
attributes such as color, style, and text appearance can be changed.
Misconceptions to be on the alert for include:
1. Believing that equations of linear equations and inequalities exist only in math books, without seeing the usefulness of them for modeling real-world phenomena.
2. Assuming that the labels and scales and a graph are not important, and that the reader automatically knows what they are.
3. Interchanging rise and run when computing the slope of a line.
4. Not knowing when to include the "or equal to" bar when translating the graph of an inequality.
5. Shading the wrong side of an inequality.
6. Not understanding what variables represent.
7. Forgetting to check whether or not the boundary line should be dotted rather than solid.
Source: KATM High School Flipbook
While teaching this lesson, the teacher should be aware of possible misconceptions and address each one in an appropriate way.
For example, an easy way to address Misconception #5 is to use the Origin (0,0) as a reference point. Substitute (0,0) into the inequality to determine whether it is
true or false. If true, the side of the inequality that contains the Origin should be shaded. If false, the side of the inequality that does NOT contain the Origin should be
shaded.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
1. Students can pair up with another student during individual practice. Also, during the rotation of stations, students can also pair up with another student in their
group. Students can have extended time at each station if needed. Students will be given extra time for homework assignments such as submitting 3/4 of the
assignment on next class day and 1/4 on the day after. If applicable, student will have reduced assignments to ensure that work is understood and handed in on
time.
2. Students could use a computer on campus, either in a computer lab or the media center, or come to the teacher"s classroom before or after school or during
lunchtime.
3. Students who are members of DECA or in a marketing class, can use their teachers as an additional resource for real-world problems in business.
Extensions:
Students can visit local businesses such as retail stores to discuss how each business optimizes their profits using systems of inequalities. Students can also research
corporate business routes, such as UPS, Fed Ex or smaller businesses such as lawn services, pool services or newspaper delivery routes to determine their uses of
system of inequalities and how they find optimal routes.
Also, students can work on these two additional linear programming problems:
1. A vertex of a feasible region does not always have whole-number coordinates. Sometimes you may need to round coordinates to find the solution. Using the
objective function and the following constraints, find the whole-number values of x and y that minimize C. Then find C for those values of x and y. **Hint** You
might want to change the scale on your graph.
{x + 2y ≥50
{2x + y ≥60
{x ≥ 0, y ≥ 0 Minimize C = 6x + 9y
2. Suppose you want to buy some tapes and CDs. You can afford as many as 10 tapes or 7 CDs. You want at least 4 CDs and at least 10 hours of recorded music.
Each tape holds about 45 minutes of music and each CD holds about an hour. Write a system of inequalities to model the problem. Let x represent the number
of tapes purchased and let y represent the number of CDs purchased. Graph your system of inequalities.
Extension Answer Key
page 6 of 7 Suggested Technology: Document Camera, Computer for Presenter, Computers for Students, LCD Projector, Adobe Acrobat Reader, Microsoft Office, Java Plugin,
GeoGebra Free Software
Special Materials Needed:
Students need to be familiar with the basic functions of GeoGebra. This can be done prior to the lesson and/or students can view tutorials on www.geogebra.org.
Students should be encouraged to download GeoGebra on their home computer by following the following steps:
1. Go to www.geogebra.org.
2. Click download.
3. Click webstart.
Geogebra will be downloaded as an icon on their desktop.
Student supply list:
1. pencils
2. calculators
3. rulers (for straight lines)
4. colored pencils (for shading)
SOURCE AND ACCESS INFORMATION
Contributed by: Wendy Moskowitz
Name of Author/Source: Wendy Moskowitz
District/Organization of Contributor(s): Broward
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.A-CED.1.3:
Description
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. ★
page 7 of 7