Web viewStudents will learn what a system of linear equations is and they will learn how to solve them using graphing on paper and graphing with calculators

McDougall-Littell Algebra I:

Solving systems of linear equations and inequalities

Sabrina Allen

Kyla Basnett

Courtney O’Donovan

Anna Fricano

Table of Contents

Unit Introduction 3

Unit Calendar 4

Day 1 10

Day 2: Sabrina’s full lesson 11

Day 3: Kyla’s full lesson 16

Days 4-5 24

Days 6-7: Courtney’s full lesson 26

Days 8-10 34

Day 11: Anna’s full lesson 37

Days 12-14 44

Assessment Items 47

Unit Introduction

This unit provides an opportunity to study the methods and strategies for

solving systems of linear equations and inequalities. The overarching goal of the

unit is for students to develop strategies for solving systems and an understanding

of why it is helpful to have multiple strategies from which to choose. In order to

achieve this goal, students should develop an understanding of how systems of

equations and inequalities arise; in other words, students should understand that

systems are a way to represent particular constraints on a pair of unknowns.

Depending on how these unknowns and constraints arise, different strategies may

be useful for solving them.

Before beginning this unit, students should have experience solving and

graphing linear equations and inequalities. In particular, students should be able to:

graph linear equations; solve linear equations in one variable algebraically;

distribute factors over a polynomial (distributive property); and graph linear

inequalities.

Students in this classroom have diverse needs. Since 50% of our students

struggle with motivation, we work to provide as much opportunity as possible for

students to stay engaged in class work. As often as possible, implement problem-

based methods in order for students to stay accountable for their learning. In

addition, we allow students the opportunity to earn back points on quizzes and tests

by making corrections to them. We made this decision with the goal of keeping

students motivated to learn concepts even after they have been assessed on the

concepts.

Unit Calendar

Date Brief Description of Content and LessonTechnology, Special Activities,

Manipulatives, Problem-Based, Instructional Strategies?

Mon 5/5 Unit Hook (mini lesson): Students will solve a fun problem using information they already know. Once they are done, I will reveal to them that they will learn multiple ways to solve these types of problems that take less time and are more efficient.

The instructional strategy here is problem-based learning, with one slight difference. The students will not actually learn the new method of solving the problem today, but rather throughout the rest of the unit. I have written a problem that is hopefully interesting to a lot of students, in order to engage them and motivate them to find the answer.

Tue 5/6 7.1 Solve by graphing (full lesson): Students will learn what a system of linear equations is and they will learn how to solve them using graphing on paper and graphing with calculators. They will solve multiple problems, including the problem they solved during the unit hook. This will teach them a new way to solve a problem that would otherwise require a lot more time.

This lesson will require the usage of some technology. The students will learn how to use the graphing calculators to look at the graphs of the lines in their problems. They have already learned how to graph with calculators, so this process will just be a review.

Wed 5/7 7.2 Solve by substitution (full lesson): Students apply their new knowledge of solving linear systems by graphing to a scenario that has two equations involving money. Here students will see that solving by graphing works but can be difficult when the numbers aren't integers. The lesson then introduces the method of substitution to the students through a three step procedure. Students then apply this new knowledge to solve other linear systems by substitution. Students will be assigned a journal entry as homework discussing the new material they have learned thus far.

The instructional strategy here is similar to problem-based learning. The students are given a problem that they are able to solve with their previous knowledge, but the choice of problem will transition well into the method of substitution because of the decimals in the equations. Throughout this lesson, students are working in groups to solve word problems by finding two equations, two unknowns, and solving the linear system using substitution.

Thur 5/8 7.2 cont./7.1-7.2 Wrap-up (mini-lesson): Students spend time reviewing their journal entries from the day before. The class then transitions into other word problems where solving by substitution would be the most effective method. For the rest of the period, students will be assessed on their understanding on linear systems and how to solve them graphically and algebraically by substitution.

The instructional strategy for this lesson is to have the lesson be student lead. Students have much opportunity to go to the board and explain their understanding to the journal entry and word problem. As a class, we discuss the method to solving word problems of this nature using substitution. For the second half of the period, students are being assessed individually by being required to work on a problem set for review.

Fri 5/9 Assessment - Quiz over sections 7.1 and 7.2 (mini-lesson): Students will not learn new material today, but rather they will take a quiz over the two sections they’ve learned so far. The purpose of this assessment day is for me to be able to see where the students are at in their understanding of the material. This class is shortened, so the short quiz should take up most of the hour.

For today's quiz, the two students who have a low reading level will be able to go to a different room to take the quiz with extra time. The rest of the students will take it in the classroom.

Mon 5/12 7.3 Solve by adding/subtraction (full lesson): Students will receive their quizzes from the previous week. Time will be spent to go over the answers to the quizzes and address any questions students may have. After going over the quiz, students will have an activity to complete involving algebra tiles. They will be investigating what it means to eliminate variables in a system of equations to solve the system. This will go up to the end of the period. They will turn in their work before they leave.

At the beginning of this lesson, students will be going over their quizzes. Students will be held accountable for correcting their mistakes on their quizzes so that they may learn from them. Students will be using manipulatives, specifically algebra tiles, to begin to learn how to solve systems of equations by adding and subtracting equations. The learning strategy used is investigation. Students will see themselves how they can solve systems of equations by eliminating variables.

Tue 5/13 7.3 cont. (full lesson): The class will begin by having students discuss the activity with the algebra tiles that they did the previous day. They will

The teaching strategy is to have students learn through examples.

then learn how to algebraically solve systems of equations using elimination by addition and subtraction. They will be taking notes on this and working out examples on their own. Students will use their knowledge on solving systems of equations using substitution. They will then have homework that they will be allowed to begin if time remains.

They will be taking notes and applying the new material to solve systems using this new method. There is an emphasis on student collaboration as students will work in partners to work through examples.

Wed 5/14 7.4 Solve by multiplying first (mini-lesson): Students will apply their knowledge of solving systems of linear equations using elimination by addition and subtraction to solve them using elimination by multiplying first. They will work together to solve word problems from which they create a system of equations and must solve using elimination by multiplying first.

Students will again learn through examples by applying their new knowledge to solve systems by elimination. This lesson will be centered around problem solving as students will not only have to practice solving systems by elimination but also create equations from word problems which they learned the prior week.

Thur 5/15 7.5 Special types of linear systems (mini-lesson): Students will learn about the 'special' types of linear systems of equations, i.e. systems with no solutions and systems with infinitely many solutions. Students will work in pairs on an investigation in which they are expected to use all of the methods they have learned so far (solving by graphing, solving by substitution, solving by adding, solving by multiplying first). Through their work, students will discover that there are some occasions when systems do not have solutions and some occasions when systems have infinitely many solutions. The homework tonight will be to study for the test the following day.

The instructional strategy is to let students work on a problem in order discover systems of equations with no solutions and systems of equations with infinitely many solutions. The purpose for this is two-fold. First, by encountering these situations on their own, students will be more likely to realize that they are special, or different from the systems of equations that they have studied thus far. Second, letting students work on a problem provides a good opportunity to review all of the solution techniques that they have learned. This will be helpful for the quiz they will take the following day.

Fri 5/16 7.5 cont./Quiz over 7.3-7.5 (mini-lesson): The lesson today is split into two parts. First, students will have time to ask questions about their homework, and the teacher will give a quick review of the topics covered

The strategy today is to give students some time to ask questions and clear up confusion before they take the

in 7.5. In particular, the teacher should highlight the point that there are exactly three possibilities for the solutions to systems of equations: the system can have exactly one solution, no solutions, or infinitely many solutions. The first half of the lesson should take approximately 20-25 minutes. Students will have the rest of the period to work on their quiz, which covers sections 7.3-7.5.

quiz. The choice to have students take a quiz that covers section 7.5 the day after they learn that section is intentional, since 7.5 does not cover any new procedures. Rather, section 7.5 just highlights certain special cases, for which students can apply the same procedures as any other system of equations.

Tue 5/20 7.6 Solving systems of linear inequalities by graphing (full lesson): Today will begin with a review of the quiz that students took on Friday. Students will get quizzes back and we will go over answers as a class. Then, students will investigate the process of finding the solution to a system of linear inequalities by graphing. The lesson builds off two pieces of prior knowledge: solving systems of equations by graphing, and graphing linear inequalities. After a brief warm-up in which students are asked to consider how they might find a single point that satisfies two different inequalities, they will work in pairs to complete a worksheet that asks them to make use of a Geogebra applet for graphing pairs of inequalities. The lesson will take the full class period, as well as the first 25 minutes of Day 12.

This lesson makes extensive use of a Geogebra applet that is available for public use on the web. The lesson lends itself particularly well to the use of technology for two reasons. First, the applet makes use of different colors of shading to make it clear to students where there are areas of intersection. Second, using the applet allows students to focus on the concept of the lesson - that solutions of systems of inequalities come from the intersection of two half-planes - rather than getting caught up in tedious details of drawing and shading graphs by hand. The primary instructional strategy is for students to develop the procedures for graphing systems of inequalities on their own, rather than having it presented in a lecture format. This seems fitting, because students already have experience with graphing inequalities, so the concepts in the lesson should be fairly accessible.

Wed 5/21 7.6 cont./Review for Chapter 7 exam (mini lesson): Today's lesson is split The purpose of today is to make sure

into two parts. The first half of the class is outlined in the lesson plan for 7.6. Briefly, students will have approximately 10 minutes to finish their worksheet from the previous day, and we will spend 15 minutes as a class going over the concepts from the lesson. The second half of the lesson will provide a brief review of concepts for the test over Chapter 7. First, students will have a chance to ask questions about the homework from the night before. Then, the teacher will remind students of the different topics covered in each of the sections, which students will be held accountable for on the chapter test. If time remains, students may ask general questions about the content of the chapter.

that students are prepared for their exam the following day. In line with that purpose, the lesson is designed so that most of the time is spent on the concepts from section 7.6. This seems important since students have just learned this material and have not had an opportunity to take a quiz or other form of summative assessment to test their understanding. Since they did a homework assignment the previous night, they will have a chance to ask questions and clear up misconceptions about the content from the homework assignment. Finally, the last part of the review gives students a chance to remember what are all of the concepts and procedures that they need to remember for the chapter test.

Thur 5/22 Chapter 7 exam (mini lesson): Students will be given a chapter exam to assess their understanding on solving linear systems graphically, by substitution, and by elimination. They will be held accountable to know special types of linear systems and systems of linear inequalities. The students will be allowed the whole period to work on the chapter exam.

The instructional strategy is to hold students accountable and assess their understanding by giving a chapter exam. Students will get the opportunity to see the answers to the problems the next day in class.

Fri 5/23 Go over Chapter 7 Exam (mini-lesson): The students will receive their graded exams. As a class, we will go through the entire exam, and we will walk through the procedures and answers to each questions. Students are required to pay attention and take notes. Those that take notes on the chapter exam by writing the correct methods to solving the problems will be able to turn their notes in for extra participation points.

Students are allowed to see their exams and relate their method to approaching the problems to the correct method and answers. Since students aren't able to correct their exam for extra points back, we will allow them to turn in notes they take during this class period for some participation points. This will

motivate students to pay attention, which will allow them to see what they did wrong (if anything). This is a crucial element in learning.

Day #: 15/5

Lesson Title: Unit Hook – Systems of Equations and Inequalities

Goal: The goal of this lesson is to get the students interested in the topic of the unit through doing a fun problem that will be easier to solve through using the methods they will learn the next few weeks.

Objectives: Students will be able to use tables and plug and chug methods to solve the problem given to them.

Lesson Summary (one paragraph maximum)

I will present the following problem to the students: During his first season with the Chicago Bears, Julius Peppers (a defensive player) had 8 sacks. Brian Urlacher (another defensive player) has had 41 sacks for the Bears since he joined the team in 2000, making his average about 5 sacks per season. Assuming that Urlacher continues to maintain this average, and Peppers averages 8 sacks per season, when will their total number of sacks be the same?The students will be separated into groups to solve this problem using whatever method they know how to use. I will preface the rest of the unit saying that this chapter teaches us different methods to solving problems such as this one, that probably take less time than you spent solving this one. Throughout this chapter, we will look back at this problem and solve it using all the methods we will learn.

HW There is no homework today.

Name: Sabrina AllenDay #: 2Date: 5/12/2011Grade Level: 8Course: 8th grade mathTime Allotted: 50 minutesNumber of Students: 25

Solving Linear Systems by Graphing

I. Goal(s): To what a system of linear equations is and to know how to solve them through graphing.

II. Objective(s):1. Students will be able to explain what a system of linear equations is and what

it means to be a solution of a system of linear equations.2. Students will be able to graph a pair of linear equations to determine their

point of intersection (aka the solution).3. Given a story problem, students will be able to solve it by writing the

equations that represent the data given, graph them, and find the solution.4. Students will be able to extend the idea of a system of linear equations to a

system that has more than two equations.

III. Materials and Resources:-Graphing calculators (one for each student)-Smart board to project my graphing calculator for the students to see-Pencil and paper (graph paper if it’s available) for the students to graph with

IV. Motivation1. I will start by revisiting the problem we did on the unit hook day: During his

first season with the Chicago Bears, Julius Peppers (a defensive player) had 8 sacks. Brian Urlacher (another defensive player) has had 41 sacks for the Bears since he joined the team in 2000, making his average about 5 sacks per season. Assuming that Urlacher continues to maintain this average, and Peppers averages 8 sacks per season, when will their total number of sacks be the same?

2. “Today we are going to learn how to solve problems like this using graphs. First let’s come up with equations that represent each player.”

a. Note: I am not using the term “system of linear equations” yet because I will define that at the beginning of the lesson procedure.

b. I will help the students derive the following two equations: y=8+8x (Peppers) and y=41+5x (Urlacher).

3. “What do you think will happen to the graphs when their total number of sacks is the same?” We will have a short mini discussion on what people think might happen.

4. I will then get out the graphing calculator on the smart board and show the students what they look like.

a. The two graphs intersect at one point, which is where the number of sacks is equal. The point is (11, 96), so in 11 more seasons, they will both have a total of 96 sacks.

V. Lesson Procedure1. I will start by defining a new vocabulary word: A system of linear

equations, or simply a linear system, consists of two or more linear equations in the same variables.

a. “We have already learned about linear equations (how to graph them, how to find them given a graph). Now we simply have a set of linear equations; two or more linear equations that we are going to use together.”

b. For example, y=8+8x & y=41+5x is a system of linear equations. 2. When we have a system of linear equations, we usually want to solve them,

or find the point at which the graphs intersect. a. A solution of a system of linear equations in two variables is an

ordered pair that satisfies each equation in the system. 3. This definition may be a little bit confusing at first, so I will take a few

minutes to break down the definition.a. What does it mean to be in two variables?b. What is an example of a linear equation not in two variables?c. What does it mean for the ordered pair to satisfy both equations?

i. The point (11, 96) is a solution to both the equation for Pepper’s sacks and Urlacher’s sacks.

Transition: As we saw earlier, graphing the equations in the set and finding where they intersect will give us the solution to a system of linear equations. Let’s take a look at some examples.

4. Example: y-2x=3 & 2x+2y=12a. First put the equations into y-equals form to get y=2x+3 & y=-x+6b. Next graph the two equations (I will graph the two equations on the

xy-plane white board for the students to see).y=2x+3: The y-intercept is 3 and the slope if 2y=-x+6: The y-intercept is 6 and the slope is -1 (insert image here)

c. As you can see from the graphs, the two intersect at (1,5), thus x=1 and y=5 is the solution to this system of linear equations.

d. To check the answer, simply plug in (1,5) to our equations: 5=2(1)+3 & 5=-1+6

5. Now I will give the students a system for them to solve on their own by graphing the equations on their own paper. After going over this problem we will go into using the graphing calculators.

a. x+2y=-6 & y-3=1/4xb. Once the students have given me their answer, we will check it on the

graphing calculators.6. “Let’s check our answer on the graphing calculators and review how to graph

lines on the calculators. Everyone get out their calculators, go to they y= menu, and clear any graphs that may already be in there.”

a. Type in the two equations and click graph. Once the graphs have appeared and you like the window setting (zoom standard is fine) click 2nd and then trace. Scroll down to the intersect choice and hit enter. You will then have to enter a close guess for the first and second curves. Then guess where the intersection is, and the calculator will tell you the point of intersection.

b. The point of intersection, and therefore the solution of the system of linear equations is (-8,1).

c. I will spend enough time here to make sure everyone knows how to properly use the graphing calculator to find the intersection point.

Transition: We have just learned what a system of linear equations is and how to solve them using graphing, both by hand and with a calculator. This will help us do problems like the one about Urlacher and Peppers.

VI. Closure1. I will have the students do one more problem where they have to come up

with the equations and then find the intersection of the two graphs to make sure they understand the material. This will also give them practice reading and coming up with equations to represent what the words say.

2. Kirigami is the Japanese art of making paper designs by folding and cutting paper. A student sells small and large greeting cards decorated with kirigami at a craft fair. The small cards cost $3 per card, and the large cards cost $5 per card. The student collects $95 for selling a total of 25 cards. How many of each type of card did the student sell?

a. Equation 1: x+y=25 y=25-xb. Equation 2: 3x+5y=95 y=19-

(3/5)xc. The graphs of these two equations

shows that the solution to the system of equations is (15,10). This answer can also be verified by plugging this point into both equations.

VII. Extension1. What if we have 3 equations?

a. Does the following system of linear equations have a solution?y=2-2xy=x-1y=4x-4

b. If it does, what will the graphs look like? The graph has all three lines intersecting at one point, (1,0).

c. How many lines can you have intersecting through one point? Is there a maximum? Is there a maximum number of equations you can have in a system of equations such that the system as a solution?

d. I will discuss these questions with the students if time allows.

VIII. AssessmentThroughout this lesson I will watch the students as they work on the problems on their own as a way of assessing their understanding. I will also pay close attention to who is discussing and answering questions, and I will try and make sure everyone is participating so that I know they understand what we are learning. My main form of assessment will be the homework assignment I will assign them to do that night, as well as a quiz over the material. The homework assignment will be problems 7, 11, 15, 16, 17, 18, 25, 26, 30 from the section (page 431-432). The quiz will be on Friday and it will cover information from section 7.1 and 7.2.

IX. StandardsNCTM Process Standards:

NCTM Problem Solving: “Apply and adapt a variety of appropriate strategies to solve problems.” This lesson teaches the first of several ways to solve systems of linear equations.

NCTM Representation: “Create and use representations to organize, record, and communicate mathematical ideas.” The students will create graphs to help them understand the solutions to the systems of linear equations.

Common Core Standards: A-CED.2. Create equations in two or more variables to represent

relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED.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.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Homework Assignment: numbers 7, 11, 15, 16, 17, 18, 25, 26 & 30 from the section (page 431-432)

7. Multiple Choice Which ordered pair is a solution of the linear system x+y=-2 and 7x-4y+8?

a. (-2, 0) b) (0, -2) c) (3, 3) d) (3, 6)

11. Error Analysis Describe and correct the error in solving the linear system below.

Equation 1: x-3y=6Equation 2: 2x-3y=3

Graph-and-Check Method Solve the linear system by graphing. Check your solution.15. x-y=2 & x+y=-816. x+2y=1 & 2x+y=-417. 3x+y=15 & y=-1518. 2x-3y=-` & 5x+2y=2625. (1/5)x-(2/5)y=(-8/5) & (-3/4)x+y=326. -1.6x-3.2y=-24 & 2.6x+2.6y=26

30. Challenge The three lines below form a triangle. Find the coordinates of the vertices of the triangle.

Line 1: -3x+2y=1 Line 2: 2x+y=11 Line 3: x+4y=9

Name: Kyla BasnettDay #: 3Date: May 7, 2011Grade Level: 8Course: AlgebraTime Allotted: 50 MinutesNumber of Students: 25

I. Goal(s): To develop an understanding of how to solve linear systems algebraically by

using substitution. II. Objectives:

The student will distinguish which equation to use to solve a variable for, i.e. the equation which has one of its variables with coefficient 1 or -1.

The student will appropriately substitute the expression they solved a variable for into the other equation, and solve for the variable.

The student will accurately substitute the value for this variable into the first equation.

The student will check the values they got for the variables by substituting them into the equations and comparing them to the graph of the linear system.

The students will use this new procedure to solve word problems involving two equations with two unknowns.

III. Materials and Resources: Each student will need a pencil, paper, and graphing calculator. Each student will need the textbook, motivational problem, and note taking

guide that are all attached at the end. The classroom will need a Smartboard or a way to project a calculator.

IV. Motivation:1. Before the lesson begins, have students do number 22 on page 431 in their

textbooks as a warm-up. This question is designed to take about 5 minutes for students to do individually. This will help them recall the information they learned on solving linear systems by graphing so that they can apply this knowledge to today’s lesson. Walk around the room while students are working on the warm-up; have them draw their own graphs to solve the system, but allow them to check their answers on their graphing calculators. Call on two students to present and explain their answers to the class. Restate their answers by showing the graph to the linear system on the Smartboard.

2. Break the students up into five groups of five. Allow students to work with their normal groups unless they don’t have normal groups. If students haven’t been working in groups lately, then break the class up to where there is a variety of work ethic and intelligence in each group. Distribute this question to each group and have the question on the Smartboard in front of the class. Distribute only one

question to each group so that they will more likely work together. Read the problem to the class. “We have some information about the first basketball game of the year, but we are trying to determine how many students attended and how many adults attended so that we can accurately prepare for the next game. The game brought in a total of $637.50. We know that admission for students is $2.00 and admission for adults is $4.50. We also know that there were a total of 225 people at the basketball game. Using this information, determine how many students were at the game and how many adults were at the game. Work in your groups and use what you know about linear systems to solve this problem.” The idea here is to teach with a problem, so walk around the room and make sure that students have the approach to create two equations from this information. Students should name two variables to represent students and adults, say x=students and y=adults. The two equations students should have are 2.00x + 4.50y = 637.50 and x + y = 225. Give students about 10 minutes to work on this problem. If groups are having trouble with coming up with equations, after 5 minutes of work, have a group that got the two equations to go to the Smartboard to write the two equations they got. Then proceed with the group work to solve the two equations. Students will probably try and graph these two equations to solve for x and y since this is what they learned previously. However, students will soon figure out that this may not be a sufficient method since there are decimals in the equation. Note: The correct answer to this problem is 150 students and 75 adults.

V. Lesson Procedure:3. After allowing the groups to work about 10 minutes on the problem, ask the

groups to share what they came up with for a solution. Allow about 10 minutes for different groups to explain what they received as an answer and how they received the answer. Many students may have received an answer by estimating where the two lines intersect on the graph. At this time, don’t go over the correct answer, just allow students to share their different approaches and techniques for solving this problem.

Transition: “We could solve these two equations in the same way by graphing them like we did yesterday. However, did anyone find it problematic since there are decimals in one of the equations? What if it cost $2.32 for students and $3.76 for adults? How can we find the solution to these equations when graphing is trivial in cases like this?”Give students wait time to ponder these questions.

4. Introduce to the students the new procedure of solving linear systems by substitution. Distribute the note taking guide to each student for them to copy the notes on this new procedure. The students should follow the notes that are displayed on the Smartboard. “You previously learned how to solve linear systems by graphing, but now you will learn a new method to solve linear systems

and that is by substitution. Solving linear systems by substitution is beneficial for solving for tubing costs like we will see in Example 32.”

5. Students should copy this procedure in their note taking guide:Step 1: Solve one of the equations for one of its variables. When possible, solve for a variable that has coefficient 1 or -1.Step 2: Substitute the expression from Step 1 into the other equation and solve for the other variable. Step 3: Substitute the value from Step 2 into the revised equation from Step 1 and solve.

6. Begin with a simple example for students to go through this three step procedure. As a class, solve the linear system y = 3x +2 (equations 1) and x + 2y = 11 (equation 2). This is included in the note taking guide. Call on three individual students to complete one of the three steps in front of the class.Step 1: Solve for y. Equation 1 is already solved for y.Step 2: Substitute 3x + 2 for y in Equation 2 and solve for x.

x + 2y = 11 Write Equation 2.x + 2(3x+2) = 11 Substitutes 3x + 2 for y.7x + 4 =11 Simplify.7x=7 Subtract 4 from each side.x=1 Divide each side by 7.

Step 3: Substitute 1 in for x in the original Equation 1 to find the value of y.y = 3x + 2 = 3(1) + 2 = 3 + 2 =5The Solution is (1,5).

Check: Substitute 1 for x and 5 for y in each of the original equations.y = 3x + 2x + 2y = 115 = 3(1) + 21 + 2(5) = 11

Therefore, our solutions are correct.7. “When we solve for linear systems using substitution, we can use a graph to

check for reasonableness of our solution. For example, let’s graph these two lines.” Graph the two equations on the Smartboard. Students will see that the intersection of the two lines is the point (1, 5) which is the solution we got when using substitution.

8. Move to the other example on the note taking guide. Have students work on this problem individually. Walk around the room and make sure students are accurately following the three step procedure.

9. Briefly go through this problem on the Smartboard after students have had adequate time to finish the problem.The equations are x – 2y = -6 and x = 2y – 6. The solution to this is (-2, 2). Make sure that students are substituting this value back into the original problem to make sure it makes the expressions true.

VI. Closure:10. “At the beginning of class, I gave you all a problem where we needed to find the

number of students and adults who were at the basketball game. Many of you

probably found this difficult to solve by trying to graph the linear equations, but now we have a more efficient method for solving problems like this. For tonight’s homework, I want you to go back and resolve this problem from the beginning of class in your journals. After resolving the problem, I want you to right a journal entry about your approach to the problem at the beginning of class and your approach to the problem at the end of class. Explain the method that would be the easiest to use in this situation, and also distinguish the two methods we have learned to use on solving linear systems. Do both methods produce the same solution? How can you determine which method to use when given a problem?”

11. Pass out the slip of paper that has these directions for the journal article printed on for each student to take home. Tell them we will discuss the journal article at the beginning of next class.

Lesson ContinuedDate: May 8, 2011Grade Level: 8Course: AlgebraTime Allotted: 25 MinutesNumber of Students: 25

VII. Motivation:12. Begin class by having students open their journals to the journal entry they

finished the night before. Walk around the room to check for completion of the journal entries (journal entries should be a half a page to a page). Call on one student to go to the front of the room to explain his or her journal entry to the class. Allow a class discussion to arise by calling on a different student to revoice what the previous student explained. Review how solving these equations by graphing work in this situation, but how solving by substitution may be a better or simpler option for this particular problem since money and decimals are involved. This should take about 10 minutes to discuss the journal entry.

VIII. Lesson Procedure Continued:13. Allow students to apply this information from the journal entry to work in

their groups to solve the tubing problem that is attached. Allow time for students to discuss the problem in their groups, but as a class, go over the two equations that are needed to solve this problem before proceeding to solve the problem. They are: Let x denote a regular tube and y denote a cooler tube.

15x + 7.50y =360 and x + y = 26 are the two equations. Students should solve the second equation for x or y and substitute it in for the corresponding variable in equation one. The answer is 22 regular tubes and 4 cooler tubes.

14. Have a student go to the board and explain what his or her group did to solve this problem. Analyze the problem as a class by being clear on how the equations were found, and go through the three step procedure in determining the number of each tube that was rented.

The problem and discussion should take about 15 minutes.VIII. Closure:

“Now we are familiar with two methods to solving a system of linear equations. Not only are we curios about the point of intersection of two lines, but we also apply systems of linear equations to real life scenarios as in the basketball and tubing problems. We can now solve these systems algebraically by substitution and represent the solutions visually by showing the intersection of the two lines. We will now transition into working individually to solve systems by graphing or by substitution. Since we have two methods of solving, it is good to use the other method to check if your solution is correct.”

IX. Extension:

Give students a couple more guided practice problems so they can practice distinguishing which variable to solve for and substitute for. Allow students to work in their groups to find the solutions. Walk around the room to keep students working on the problems, and after adequate time, share the answers with the students for them to check their work. Be sure to ask any questions students may have about this new procedure. Here are three problems to give the students as an extension. (in Textbook)1. y = 2x +5 2. x – y = 3 3. 3x + y =7

3x + y = 10 x + 2y = -6 -2x + 4y = 0Solutions: 1. (1, 7) 2. (0, -3) 3. (2,1)

X. Assessment: Assessment takes place throughout this entire lesson. It begins at the beginning of class when students are asked to recall information from the previous lesson of solving linear systems by graphing. Students are then assessed on their ability to use this fairly new knowledge to the problem about the attendance of the first basketball game. Students are kept accountable by presenting their conclusions to the class. Also, after students learn how to solve linear systems by substitution, they are assessed by having to solve a few problems on their own and in groups using this new method. As a homework assignment, students are asked to write a normal journal entry on how to solve the problem from the beginning of class. Students should realize that solving linear systems by graphing and solving linear systems using substitution are both accurate ways to solve linear systems. However, when considering equations that don’t necessarily have integer coefficients, solving by substitution is the best way to go about solving the system. Students are to write an explanation similar to this and rework the problem using substitution.

XI. Standard(s): Analyze and solve linear equations and pairs of simultaneous linear

equations. 8. EE. 8. “Solve systems of linear equations algebraically. Solve

real world and mathematical problems leading to two linear equations in two variables.”

In this lesson, students solve systems algebraically using substitution, and they relate this to real-world problems with the basketball game attendance and tubing problems.

NCTM Problem Solving: “Apply and adapt a variety of appropriate strategies to solve problems.” Students will use this new knowledge of solving linear systems using substitution to various word problems.

NCTM Connections: “Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.” Students will connect their previous knowledge of solving linear systems by graphing to that of substitution.

NCTM Representation: “Select, apply, and translate among mathematical representations to solve problems.” Students will recognize when it is more

sufficient to solve a linear systems by substitution rather than solving by graphing.

NCTM Communication: “Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.” Students will work in groups to discover how to solve problems where substitution is a good method. They will communicate their findings to the class, and write about them for the teacher.

Motivational Problem:We have some information about the first basketball game of the year, but we are trying to determine how many students attended and how many adults attended so that we can accurately prepare for the next game. The game brought in a total of $637.50. We know that admission for students is $2.00 and admission for adults is $4.50. We also know that there were a total of 225 people at the basketball game. Using this information, determine how many students were at the game and how many adults were at the game.

Note Taking Guide!Method for solving linear systems using substitution:Step 1:Step 2:Step 3:

Practice!Solve the systemEquation 1: y = 3x +2 Equation 2: x + 2y = 11 Step 1:Step 2:Step 3:

Solution:Check:

One Your Own!Solve: Equation 1: x – 2y = -6

Equation 2: x = 2y – 6.

In Groups!#32 in Book: TUBING COSTS A group of friends takes a day-long tubing trip down ariver. The company that offers the tubing trip charges $15 to rent a tube

for a person to use and $7.50 to rent a "cooler" tube, which is used to carryfood and water in a cooler. The friends spend $360 to rent a total of26 tubes. How many of each type of tube do they rent?

Day #: 45/8

Lesson Title: Solving Systems of Linear Equations Graphically and Algebraically by Substitution.

Goal: To assess student understanding of solving linear equations by graphing or by substitution.

Objectives: The student will distinguish when to solve a system graphically or by substitution.

The student will represent solutions to a system of linear equations in two ways.

The student will review and get to ask questions before the sections 7.1-7.2 quiz.


The first portion of this lesson will be a continuation of the lesson from the day before. After the lesson from the day before is completed, students will be given a problem set for assessment. This problem set is designed for students to do individually during the remaining of the class period. The problem set will include problems similar to those in the previous lesson plans where students will have to use both techniques that they know thus far for solving the systems. Students are allowed to receive help from those around them, and they will be encouraged to ask any questions before the quiz they will have the next day. Explain to the students that the quiz will be similar to the problem set they are working on. Students will be expected to solve linear equations by graphing and by substitution for the next day’s quiz.

HW Students should get the problem set done in class, but if not they must finish it for homework. Also, students are to study for the quiz tomorrow.

Day #: 55/9

Lesson Title: Assessment day

Goal: The goal of this lesson is assess how well the students understand the material up to this point.

Objectives: Students will demonstrate understanding of the material they have learned.


Once class starts, I will pass out a short quiz. The two students who have a low reading level will be able to take the quiz in a different room and will be allowed extra time if needed. The other students will take it in the classroom. It should take students about 10-15 minutes to complete. Once the students complete the quiz, they will hand it to me, I will mark the ones they have wrong, and I will give it back for them to redo. I will do this a maximum of two times for each student, and then they will hand in their final quiz to be graded.We have decided to do this for quizzes do to the fact that a lot of students in the classroom lack motivation. They don’t care enough to look at their mistakes once they get their quiz back with a final grade. By allowing them to redo the problems helps encourage them to look over their mistakes and attempt to fix and understand them.Class is only 25 minutes, so this should take all period.


Lesson Plan on Solving Systems of Equations by Elimination

Courtney O’DonovanClass: Algebra 1Day #: 6-7Grade: 8thNumber of Students: 25Date: May 12-13, 2011

Goal: Students will learn how to solve systems of equations using elimination by adding and subtracting.

Objectives: Students will solve systems of equations using elimination by addition and subtraction. Students will collaborate with each other as they work through problems using algebra tiles and examples.

Standards:8.EE b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

NCTM Standards:The NCTM Standard being addressed in this lesson is problem solving and representation. Students will be completing an activity in which they will use algebra squares to represent linear equations. This way, students can begin to see how to use elimination to solve systems of equations. They can see how the variables are eliminated by either adding or subtracting the equations. Problem solving will be emphasized so that students know they can use whatever method they would like to solve the systems. The NCTM Standard of representation is also being addressed. Students are using algebra squares to represent the equations and their solutions to solve the system of linear equations.

Materials:ElmoPaper14 bags of Algebra Tiles 25 Algebra Tile Activity worksheets25 worksheets for homework

Warm-Up:Solve this system using substitution and check your answers using the graph to find the point of intersection:2y +x = - 4 y – x = -5

I will allow students to work on this Warm-Up for about 3 minutes. During their working time, I will walk around the classroom and observe their work and answer questions they may have. We will then go over the problem as a class. I will ask a student to come up to the board and show the class how he or she solved the problem. I will then ask if anyone has any questions about using the substitution method for solving systems of equations.

● If there are still questions and concerns, I will address them. I will then have the following problem to work out as a class and discuss the concept again.

5x - y = 27 - 2x + y = 3

If there are not further questions, I will continue with the lesson. .

(15 min) I will then proceed to pass out their graded quizzes. I will go over the answers to the quiz. I will answer any questions that they have.

Following this, will be the motivation for Section 7.3.

Transition: “For the past couple of days, we have been solving systems of linear equations using graphing and substitution. We will now learn a new method to solve equations called elimination.”

Motivation:I will begin by asking students, “What does it mean to eliminate something?” I expect students to respond by saying to get rid of it or some variation of this. If no one has an answer, I will ask them what they think it means.Then I will say, “We can apply this idea of elimination to solve systems of equations by getting rid of or eliminating variables.”I will then proceed to have students do the following activity using algebra tiles.

Transition: “We are now going to do an activity about solving systems of equations using algebra tiles. While we do this think about what it means to eliminate something and how that relates to the activity.”

Algebra Tiles Activity:1. I will have students break into groups of two (there will be one group of three) and I will

hand out to each group a bag filled with algebra tiles. (They will have seen and worked with algebra tiles before so I will not need to explain how to use the tiles but each bag will contain a key to identify each type of tile.)

2. I will then explain how we are going to use the tiles to solve linear equations. I will have the following system of equations to use as an example and show them how to work it. I will use the Elmo to project the tiles on to the board so students can work through the example with me.

“In your groups, use the algebra tiles to represent these two equations. Follow along with me as I go. If you have any questions as we go through them, please raise your hand and I will call on you.”Ex. x + y = 5 x - y = 3I will address any questions they have about the activity and then have them begin to

work in their groups working out a couple of problems. 3. I will then give each group of students a piece of paper with two systems of equations. It

will state: Solve the following linear systems using the algebra tiles. Explain how you solve the systems?1. 3x - y = 5

x + y =3

2. x + 4y = 10 x + 2y = 6

4. The students will work on the problems for about 15 minutes. I will walk around the class checking in with students’ progress and answering any questions they might have. Then we will come back as a class and discuss problems. I will ask the groups to share their results on the board and explain to the class what they did.

Closure:I will tell students that tomorrow we will review how to formally solve systems of linear equations using elimination by addition and subtraction. I will ask them to hand in their worksheets for class credit.

Warm-Up:(5-10 min) Discuss the activity that we did the previous day.

Transition: “Now please take out a piece of paper and a pencil to take notes with and we will begin to discuss what it means to solve systems of equations by elimination by adding and subtracting.”

Procedure (25 min):1. Using the Elmo, I will write notes on a piece of loose leaf paper, and the class will take notes on what I write down and what we discuss as a class involving the method of elimination. I will begin by asking the students what was the definition of elimination that we discussed yesterday. We will write down the definition. 2. We will then what it means to eliminate by adding and subtracting as it pertains to systems of equations. I will stress that we are getting rid of a variable and that is where elimination comes into play. This may confuse them at first so I am prepared to address any questions they may have.3. Then I will explain when we eliminate by addition and when we eliminate by subtraction.

We add the two equations if the coefficients of one variable are opposite of each other(i.e. -6x and 6x, the coefficients are -6 and 6 respectively so they are opposite of each

other so we add the equations because then -6x+6x =0 and we eliminate the x variable)

Ex. -6x + 3y = 4 6x + 2y = 6We add these equations because the coefficients attached to the x are -6 and 6. Since -6 and 6 are opposite integers. We subtract two equations if the coefficients of one variable are the same (i.e. 6x and 6x

both have coefficients of 6, so we subtract the equations because 6-6=0 and that eliminates the x-variable)

Ex. 6x + 3y = 4 6x + 2y = 6We subtract these equations because looking at the x-variable the coefficients attached are 6 and 6. Since these are the same integers in order to eliminate

4. Then we will discuss the general steps for solving systems of linear equations by elimination.Step 1: Add or subtract the equations to eliminate one variable.Add equations if the coefficients of one variable are oppositesSubtract equations if the coefficients of one variable are the same.Step 2: Solve the resulting equation for the other variable.Step 3: Substitute in either original equation to find the value of the eliminated variable.

5. I will then give an example to model these step. I will have students return to the system of equations in the Warm-Up. I will ask the students about what we can eliminate from these two equations. I expect them to say the y-variables. If they do not, I will ask by looking at the two equations, what variable, x or y, can we get rid of. Again, I will look for the students to say the y-variables. Then I will discuss that this is a method of solving equations called elimination by addition and subtraction. By adding these two equations, we are eliminating the y-variables. 5x - y = 27- 2x + y = 3

6. Then I will tell the students to add these two equations together by putting the equations in column form. This means that they need to line up the variables under each other and the constants on the right side of the equal sign. I will then show them how to add the equations together and then to solve for x.

3 7. Now I will explain to them that they will need to use their knowledge of substitution to finish the problem. I will say, “Now that we have solved for x, we need to find the y-value. We do this by substituting our x-value into either of the equations. Which equation should we use?” After the class chooses the equation they want to substitute x in for, I will then discuss how this is just like the substitutions they were doing before and ask someone to come to the board and finish the problem. 4. I will ask them to compare the answers that they got for solving the system by substitution and solving the system by elimination. They will see that they got the same answers and so both methods work.5. I will then have one more problem for us to solve as a class. It will be an elimination problem by subtraction. This will be the system:

I will first ask the class which variable we should eliminate. I anticipate they will say x because the x-values have the same coefficient. If they do not come up with this, I will tell them which one and explain why we choose x. I will then ask if the coefficients on the x-values are opposites. Then I will show them they are not because the coefficient on both x-variables are 1 and if we used addition we would still have the x-variable and we would not have eliminated any variable. Since the signs on the coefficients are the same, we subtract the equations.8. In order to subtract the equation, I will show them that they must do the following:

I will explain that they are not only subtracting the x-values but they must subtract all the terms in the bottom equation from all the terms in the top equation. To make this clearer for students, I will show them that they can distribute the negative sign to all the terms in the bottom equation to get:

So now they just subtract and get the following answer: y = 19. (10 min) I will then tell them that they will use substitution again to find the x-value now that they have the y-value. I will ask them to choose from the original equations which one they would like to use. I will emphasize that they need to choose from the original equations in order to get the correct solution. So as a class we will work out the rest of the problem. 10. Now that we have solved a couple of systems as a class, I will have the students work on the some in partners. This way they can begin to use the method on their own. After giving them about 5 minutes to work, during which I will be walking around observing student work and answering questions, I will have a pair come to the board and show their work and explain what

they did to the class. I will then ask if everyone came up with the same answer and if the are any questions, I will address them then. Closing: (5 min) I will close by reviewing this new method of solving systems by elimination. I will ask them what are the steps involved by solving systems of equations. I will also discuss how there is multiple ways of solving these systems of equations and the way the choose to solve the them is up to them. Students will turn in their work on the the examples for class participation for the day.

Extension:If time allows, I will allow students to begin their homework for that night which will be a worksheet with the systems of equations that they will need to solve by elimination by adding and subtracting. There will also be a part in which students can choose which method, of the methods they have learned thus far, to solve them.

Assessment:I will be assessing students throughout the lesson. I will be walking around when they are working on the algebra tiles to begin to see what the student’s are coming up with. The worksheets that they turn in will be checked for class participation but I will go through the worksheets and see if students understand the concept. I will also be observing and assessing their work on the examples in class they turned in. The final assessment will be a homework sheet that students will complete and turn in for a grade. This will be able to help me in determining where students stand on solving systems of equations by elimination. It will help me decide whether or not to take more time to go over this concept or to see if they understand enough to move on to solving systems of equations by multiplying.

Name__________________________ Date____________________

Fun with Algebra TilesDirections: Solve the following systems of equations using the algebra tiles. Explain what how you solved the system.

a. 3x - y = 5 x + y =3

b. x + 4y = 10 x + 2y = 6

Name____________________________ Date___________

Solving Systems of Equations by Elimination

Solve the systems of equations using elimination. 1. x+ 2y = 13 2. 3x - y = 30

-x +y = 5 - 3x + 7y = 6

2. 5x+ 6y = 50 4. 4x - 9y = - 21- x + 6y = 26 4x + 3y = -9

Solve the systems of equations using any method you have learned so far. 5. x + 2y = 1 6. 6x + 12y = - 6

- 2x+y =- 4 2x + 5y = 0

Solve the following word problem: Two small jugs and one large jug can hold 8 cups of Kool-aid. One large jug minus one small jug holds 2 cups of Kool-aid. How many cups of Kool-aid can each jug hold?

Day #: 8 5/14

Lesson Title:Solving Systems of Equations using Elimination by Multiplying first

Goal: The goal of the lesson is to have students solve systems of linear equations using elimination by multiplication.

Objectives: ● Students will identify that they cannot solve a system of equations by adding or subtracting.

● Students will solve the systems using elimination by multiplying first and then adding or subtracting the equations.

● Student will work together to solve problems. .● Students will set up systems of equations from word

problems and apply the method of elimination to solve them.Lesson Summary (one paragraph maximum)

The lesson will begin with a warm-up. Students will be given three minutes to complete a system of equations using elimination by addition or subtraction. After the warm-up, students will be presented with a system that they cannot solved by simply adding or subtracting the equations. We will discuss the reasons why they cannot solve it. Students will learn through examples that they will need to multiple either one or both equations by a factor in order to be able to make the coefficients of one of the variables opposite of each other. From here they will be able to add or subtract the equations. They will work in groups to solve word problems where they will need to set up a system of equations and solve by elimination.

HW The homework will come from the book. Students will be assigned to do page 455 #9-17 odd, 37, 38.

Day #: 95/15

Lesson Title:Special types of linear systems

Goal: Students will recognize and understand the meaning of systems of linear equations in which there are either no solutions or infinitely many solutions.

Objectives: Students will recall and apply previously learned techniques for solving systems of equations.

Given a systems of linear equations, students will identify whether the system has one solution, zero solutions, or infinitely many solutions.

Students will be able to construct two systems of equations, with zero solutions and infinitely many solutions, respectively.


Students will work in groups to apply the procedures they have learned (solving equations by graphing, substitution, adding and subtracting) to solve systems of equations. The teacher will provide students with a worksheet, which should contain 9 systems of equations—3 each of one solution, no solution, infinitely many solutions. The teacher will not tell students ahead of time that they will discover new types of systems. Students will be instructed on the worksheet to create a graph and solve algebraically—either by substitution or adding/subtracting—for each system. Then they will sort their systems based on the graphs they created. Finally, students will create a system of equations that will create two parallel lines, and a system of equations that will only give one line. If students finish before the end of the period, allow volunteers to share and explain the graphs they have created.

HW The homework for today is to study for tomorrow’s quiz over sections 7.3-7.5.

Day #: 105/16

Lesson Title:Special types of linear systems (cont) / Quiz over 7.3-7.5

Goal: To recognize the number of solutions to systems of equations and to apply strategies for solving by adding/subtracting on a quiz.

Objectives: Students will recognize when a system of linear equations has one solution, no solutions, or infinitely many solutions.

Students will be able to construct linear equations with either no solution or infinitely many solutions.

Students will display their understanding of solving systems by adding/subtracting on a quiz.


This lesson is divided into two parts. In the first half of the lesson (20-25 minutes), the teacher will discuss with the class the three options for solutions to linear systems: systems can have either zero solutions, one solution, or infinitely many solutions. Students should have their worksheets from yesterday, and the teacher will refer to examples from the worksheet for each type of system. Students should have time to ask questions over the material. In the second half of the lesson, students will take a quiz covering sections 7.3-7.5. The topic of the quiz is solving systems of equations by elimination; at least one of the systems on the quiz should be a system with no solution. Consistent with our prior quiz, students will have an opportunity to have their quizzes checked and re-worked before they turn them in for a final grade (see Day 5). Allow at least 25 minutes for students to work on the quiz.


Linear EquationsName: Anna FricanoDay #: 11Date: Tue, May 20Course: Algebra ITime Allotted: 75 minutes (over two days)Number of students: 25

I. Goal: To create and use graphical representations of linear inequalities in order to solve

systems of linear inequalities in two variables.II. Objectives: Students will relate the ideas of graphing linear inequalities and solving systems of

equations by graphing to solve systems of inequalities by graphing. Given a graph of two inequalities, students will be able to interpret the graph,

describing the shading and the boundary lines of the graph. Students will be able to compare graphs of inequalities, addressing issues of which

region is shaded, and whether boundary lines are included or excluded. Students will be able to create graphs to represent systems of two linear

inequalities.III. Materials and Resources For each student, a piece of graph paper (for the Motivation), and a copy of the

worksheet, “Systems of Linear Inequalities”. For each pair of students, a computer with Internet access.I. Wrap-up Previous Week (10 minutes) Go over the quiz that students took last Friday. The quiz covered sections 7.3

through 7.5. Since students had an opportunity to revise their quiz before turning it in, I expect

that there will be few questions about the quiz. However, answer any questions that arise at this time.

Remind students of Friday’s test, and that the quiz would be a good resource for studying for the test.

II. Motivation (20 minutes)Give students the following problem as a warm-up:

“Think about the following two inequalities:

y ³2 x+38 x+2 y<14

1. Graph each of these inequalities.2. Write a sentence to describe how to find the points that satisfy one of these

inequalities.3. Find at least one ordered pair that satisfies both of the inequalities. How did you

find it?4. (If you have time) Can you find more ordered pairs?”

Give the class 5-10 minutes to work on the warm-up. I expect it will take students some time, since they learned how to graph linear inequalities at the end of Chapter 6, and we are now at the end of Chapter 7. As students are working, maintain a presence among them, even if they do not ask for help. I expect that since many students struggle to stay motivated, they will need a push if they don’t immediately recall how to graph linear inequalities. Remind students that they graphed linear inequalities in section 6.7, and they may refer to their notes if they need. Once students have had time to work, go over the solutions. Prepare a coordinate grid on the Smartboard (ahead of time), and allow two students to graph each of the linear inequalities on the Smartboard. Allow several students to share their solutions to parts 3 and 4. I expect that most students will come up with a point by a guess and check method. While students are working, pay close attention to whether anyone develops a more systematic way of finding points that satisfy both inequalities. If so, ask this student to share his/her strategy. To transition into today’s lesson: “At the end of Chapter 6 you all learned about graphing linear inequalities. Over the past couple of weeks in Chapter 7, we have been learning about solving systems of equations. Today, we’re going to put those two ideas together. We’re going to solve systems of inequalities”.

III. Lesson Procedure (30 minutes)1. Split students into pairs (one group of 3 is okay, if necessary). Decide on pairs

prior to the class period.2. Give students the following instructions, “Today you are going to work with your

partner to think about how to solve systems of inequalities graphically. Instead of graphing things by hand, we’ll be using a Geogebra applet on the web. Once I put you into pairs, I’m going to pass out a worksheet to each student. Although you are working together on the computer, I expect every student to write on his or her own paper, and I expect everyone to turn in the worksheet at the end of class. Does anyone have any questions?” Answer questions before passing out the worksheet.

3. Pass out a copy of the worksheet “Systems of Linear Inequalities” to each student.

4. Give students the rest of the class period (20 minutes) and the start of 5/21 (10 minutes) to work on the worksheet.

5. As students work, walk around the room and observe their work. I expect that students will struggle initially with questions 1 and 2 on the worksheet, because they will have difficulty explaining in words the procedures that they apply to solve systems of equations and to graph inequalities. If students do not remember how to do these things, refer them to the work that they have done in their notes: “When did we graph systems of linear equations? Can you refer back in your notes? Try to describe the process that you think through in your head.” Help students to recall this information so that they do not lose interest and motivation to move forward with the activity.

6. Make sure that all groups are able to find the appropriate URL, and make sure that all groups figure out how to use the applet. Since students have experience using the computers in our classroom, I expect that problems with connecting to the web and using the applet will be minimal.

7. Pay particular attention to students’ written answers on the second page, on questions 6-8. As students work, ask questions to encourage students to use the graphical representation in order to make sense of the solutions of the systems of inequalities: “How did you find the solutions in 6b?” “Is that similar to what you said in question 7?” “Describe to me the differences between the three systems in question 8. Show me how the graph represents the differences in the solution sets of each of the systems.”

At the end of the first day of the lesson (5/20), give a quick wrap-up of the lesson. Give students time to log out of computers and pack up their papers. Tell students, “Tomorrow we will spend the first half of class finishing this activity. When you get to class tomorrow, find your partner and log-on to a computer. We will begin class by spending about 10 more minutes working on the computers. Then we will discuss our findings.” Collect students’ worksheets to read through them before the second day of the lesson. Return them at the start of the second day. Students have a homework assignment tonight, so make sure they have this assignment written down before leaving.

IV. Closure (15 minutes)The closure of the activity should provide an overview of how to represent the solution set to a system of linear inequalities by graphing. First, go over the solution to question 9 on the worksheet. I expect students to develop the following steps: 1) graph the first inequality, shading the area that represents the solution set, 2) graph the second inequality, again shading the area that represents the solution, 3) the solution to the system of inequalities is the intersection of these two shaded areas. Allow students to present the steps they develop. If the class does not come up with this, or an equivalent, procedure, then guide the class through this process. However, I expect that by the second day students will be comfortable with this procedure. After going over the solution to question 9, go back to question 8c. Allow students to graph each of these systems (have the Geogebra applet projected at the front of the room) and describe the differences between them. Through this discussion, make explicit that dotted lines mean that the line is not included in the solution set, and solid lines mean that the line is included in the solution set. By the end of the lesson, students should be comfortable with 1) the process for graphing systems of inequalities and 2) knowing the meaning of and appropriately using dotted and solid lines.

V. ExtensionIf students finish the worksheet early, pose the following problem:“Consider the following system of inequalities:

{ y>7 x−8 ¿ ¿¿¿Describe the solution set to this system of inequalities. Justify your answer algebraically, and/or with a picture”.

VI. AssessmentAssessment in this lesson comes in a few different forms, each of them serving primarily a formative function. The first assessment occurs during the Motivation/Warm-up. As students work on the problem, I can observe the

strategies and ideas that students develop; moreover, as students present their solutions at the board, I can assess whether the class has a general understanding of what their classmates say. Next, I can assess students’ understandings as they work on the computers with their partners. Since students are all expected to do their own writing, I should have evidence of what each student is doing during the period. Based on what I observe of their answers on the worksheet, I can probe students to see what is their level of understanding. I plan to collect students’ worksheets at the end of the lesson. Although I will not assign a grade to the worksheets, I will read students’ answers to the question to gauge their understanding of the concept. This will be particularly important after the first day of the lesson. Since the second day of the lesson is also the day before the chapter test, it will be important to make sure that students have an understanding of the concept before the day of the test. Finally, I will give students a short homework assignment at the end of the first day of the lesson, in order to give them practice in graphing systems of linear inequalities before the chapter test.

VII. HomeworkStudents will do a homework assignment from section 6.5 in the book. The assignment is as follows:Section 7.6, pages 469-471# 6-8, 13, 23, 38Students may use the Geogebra applet on their homework, but they need to have something to turn in. In particular, they either need to print out the graphs they create with Geogebra, or they need to copy the graphs onto paper.

VIII. Common Core Math StandardsA-REI.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.

This standard really encompasses the work that students do in lesson 7.6. Students first have a review of graphing inequalities, and then they describe the solution set of a system of inequalities as the intersection of two half-planes. The lesson also gives treatment to the distinction between including or excluding the boundary lines.

IX. NCTM Process StandardsRepresentations—At the core of this lesson is students’ developing ability to relate symbolic and graphical representations of systems of inequalities. Students have the chance to create graphical representations based on symbolic representations, and they use these representations to communicate mathematical ideas, in particular to describe how to find the solution set to a system of inequalities.

Communication—Integrated into this lesson, which is primarily done through the use of technology, are several opportunities for students to communicate mathematical ideas, both verbally—with their partners and to the teacher, as well as in writing—through their work on the worksheet and on the homework assignment.

Students will use the mathematical notation and representations they create in order to effectively communicate mathematical ideas.

Name Date Period

1. In Chapter 6, we learned about graphing linear equations. Describe how to find solution sets to systems of linear equations by graphing.

2. Earlier in Chapter 7, we learned about graphing linear inequalities. Describe how to graph linear inequalities.

3. Today, we want to combine those two ideas, to find solutions of systems of inequalities graphically. Open a web browser and go to the following URL.

http://www.geogebra.org/en/upload/files/english/dtravis/sys_of_line_inequalities.html

4. First, get some practice with the applet. Use the slider in quadrant 2 to graph the inequalityy>2 x+3

. How are the solutions to this inequality represented on the graph you created? Describe the line, shading, color, etc.

5. Now use the slider in quadrant 1 to graph the inequality y≤−x+7

. How are the solutions to this inequality represented on the graph you just created?

6. Consider the inequalities that you graphed in numbers 4 and 5.a. Name 5 points that satisfy both inequalities.

b. How can you use the graphs you just made to find points that satisfy both y>2 x+3

and y≤−x+7

?

Systems of Linear Inequalities

7. Let’s try another example. Describe the solution set to the following system (use the sliders to graph the inequalities):

{ y>x ¿ ¿¿¿

8. The boundary of the set of solutions is made up of the line segments surrounding the solution area.

a. How can we tell, by looking at the graph, whether the boundary lines are included?

b. How can we tell by looking at the algebraic inequality whether the boundary lines are included?

c. Describe the difference between the solution sets of the following 3 systems.

{ y<x ¿ ¿¿¿

{ y≤x ¿¿¿¿

{ y≤x ¿¿¿¿9. Based on the work you did in the previous exercises, describe the steps for finding the

solution set to a system of inequalities graphically.

10. Using the applet, play with the sliders until construct a system of inequalities with no solution. How do you know that there are no solutions? Write down your two inequalities.

Day #: 125/21

Lesson Title:Chapter 7 Review

Goal: To review strategies for solving systems of equations and inequalities and understand the concepts to be assessed on the upcoming test.

Objectives: Students will recall the different strategies for solving systems of linear equations and inequalities.

Students will relate different strategies for, and representations of, solutions of systems of equations.

Students will apply the strategies that they have learned in situations similar to those they will encounter on the exam.


The Chapter 7 review makes up 25 minutes of today’s period (for the first 25 minutes, see lesson plan for 5/20). The first priority for today is for students to have an opportunity to ask questions about the homework from the previous night. I expect that students will have some trouble with # 38, since the in-class activity did not require them to construct linear systems. After students have had time to ask questions, the teacher should highlight the main topics covered in the chapter, so that students have an outline by which to study. In particular, point out the different strategies that they have covered: graphing, substitution, and elimination. If time remains at the end of the period, students will have time to ask questions over the chapter.

HW The homework tonight is to study for the Chapter 7 test.

Day #: 135/22

Lesson Title: Chapter Exam over Chapter 7

Goal: To assess student knowledge of the material covered in all of chapter 7.

Objectives: The student will distinguish which method is appropriate to solving the given linear system.

The student will display his or her understanding to chapter 7 and be graded on that understanding.

The student will be expected to know how to work previous homework assignments and quizzes without using notes.


Students will get the entire period to work on the chapter exam. Make sure students are all seated before passing out the exam. Since students have known about the exam ahead of time, there will not be time for them to review directly before the exam. Students are to have only pencils and a calculator on their desks. The two students with a low reading level are able to take their exam in another room with an aid. These students will be allowed extra time if they need to take it. Have the rest of students begin the exam together, and give students a ten-minute warning before the bell rings. After receiving all the exams and before dismissing the class, tell the students they will receive their exams back tomorrow and be given an opportunity to receive extra participation points.

HW There will be no homework since the students will be given all class period to finish the exam. They will get the opportunity to see their exams with the correct answers the next day.

Day #: 145/23

Lesson Title: Chapter 7 Exam Solutions

Goal: To allow students to ask any questions about the chapter 7 Exam. To address common misconceptions that occurred on the chapter

7 Exam.

Objectives: The student will be able to compare his or her solution to the correct solution.

The student will gain a better understanding of the material when recognizing why he or she missed a problem.

The student will see different methods to solving one problem. The student will be given an opportunity to boost their grade by

participating in today’s class.Lesson Summary (one paragraph maximum)

At the start of class, the students will receive their graded exams back. Students will be given a couple of minutes to look over their exams, so they can be aware of what problems they did incorrectly (if any). As a class, we will go over the entire exam, and students are expected to take notes with the correct solutions to the problems. Each problem will be worked on the board for students to copy in their notes. Students will have opportunities to go to the board and explain how he or she solved the problem. During this time, students will be allowed to ask any questions they have about the material on the exam. For a closure, tell the students that they will be able to turn in their notes or the correct solutions to the problems for an extra participation points that will essentially help their exam grades.

HW The students will be allowed to turn in their notes from today’s class in order to potentially receive extra participation points. There will be no new homework assigned.

Sample Assessment Items# Learning Objective Assessment Item1 Students will create graphical

representations of linear equations and inequalities.

You are going to do some yard work to earn money to pay for a summer road trip. You need to earn at least $500 total. You can make $10 for every garden that you weed and $15 for every lawn that you mow. Also, you know that you can mow at least 10 lawns (your parents and a couple of neighbors have already agreed to hire you). Set up and graph a system of inequalities to represent this situation. Name a lawn-garden combination that will earn you enough money for your road trip.

2 Students will use algebraic methods (i.e. substitution and adding/subtracting) to solve systems of equations.

"Solve the following systems of equations algebraically:

2x + 5y = 125y = 4x + 6

5x - 4y = 27- 2x + y = 3

12x - 7y = - 2- 8x + 11y = 14"

3 Given a system of equations, students will solve and represent the solution in multiple ways.

Solve this system algebraically and represent your solution graphically.y-3x=-2y-6=4x

4 Students will construct a system of equations to represent a word problem.

Steve has twice as much money as Sarah. If Sarah gives Steve $20, they will have the same amount. How much money do Steve and Sarah have individually?

5 Students will explain how the graphical representation of a system of equations indicates the number of solutions to a system.

Given that a system of equations that has no solutions, what will the graphs of the two lines look like? What about a system that has infinitely many solutions? Can a system of linear equations have exactly 2 solutions?

Web viewStudents will learn what a system of linear equations is and they will learn how to solve them using graphing on paper and graphing with calculators

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