Solving and Graphing Linear Inequalities Algebra I Tony Lucas MATH 410 Spring 2010
Introduction Solving and Graphing Linear Inequalities is a unit addressed in Algebra I. The lesson plans created for this unit will review solving and graphing inequalities in one dimension and introduce the concept of solving and graphing linear inequalities. Students will have prior knowledge of solving and graphing linear equations, which will be used to relate the two concepts and show the similarities and differences between the two. Multiple resources were used to create this unit plan and can be found at the end after the conclusion. The main components of the lesson plan came from the following website, http://www.algebra-class.com/. The teaching methods that I predominantly use in the lesson plans are questioning and collaboration. Most of the lessons involve me asking students questions to help guide us through the problems given in class. After having gone through a few example problems, I assign the students a few in-class problems to work on individually and then collaborate with a partner. For Lesson 4, the entire lesson is structured around collaboration. The prior lesson discussed solving and graphing linear inequalities and in Lesson 4, students work with their partner to investigate linear inequalities with real-world examples. Some lessons, such as graphing linear inequalities, do involve the modeling method in introducing a new concept such as a dotted line for the boundary line. In most of the lessons, I attempt to connect what we are doing to a prior unit, Solving and Graphing Linear Equations, because the Solving and Graphing Linear Inequalities Unit is an extension of it and involves changing the equals sign to inequality signs. I have chosen to primarily teach using questioning and collaboration methods because they allow for students to actively participate in the class and they allow me a method to assess the students. Furthermore, with both methods, but more specifically collaboration, I can assist students better in a smaller and more intimate environment and I can observe and see where
students may need more assistance. Also, questioning allows students to establish their own connections and they can better see why a certain method is used or an operation is formed. I also believe that active students will perform better and develop an appreciation for mathematics. The standards and principles encompassed in the attached lesson plans include the following: National Council of Teachers of Mathematics Principles and Standards: • • • • •
understand relations and functions and select, convert flexibly among, and use various representations for them; understand the meaning of equivalent forms of expressions, equations, inequalities, and relations write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases; use symbolic algebra to represent and explain mathematical relationships; draw reasonable conclusions about a situation being modeled.
North Carolina Standard Course of Study Algebra I: • • •
1.01 Write equivalent forms of algebraic expressions to solve problems; 4.01 Use linear functions or inequalities to model and solve problems; justify results. o a Solve using tables, graphs, and algebraic properties; o b Interpret constants and coefficients in the context of the problem. 4.03 Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results. The lesson plans do not use any technology, except for one question on the quiz at the
end of the unit. The inequalities given are not that complicated and by solving and graphing them by hand will help them prepare for the standardized test at the end of the year. The forms of discourse and writing in the lesson plans include students sharing ideas and writing answer responses to problems. Students will work with each other and share their ideas through group work and answering and asking questions in class. Also, students will be required to write short statements answering questions on the Lesson 4 worksheet and when answering word problems.
The students will be given a variety of problems to solve throughout the unit and are expected to use their prior knowledge and/or new concepts they have just learned to solve the problems. Additionally, problem solving will occur in the unit through students solving word problems using the information that they have already learned. The students will be required to find the information they need in the problem to write inequalities and to solve the problem. An example of problem-solving is in Lesson 4, where the students work together to complete an investigation worksheet on linear inequalities. Students will be assessed in a variety of ways during this unit including: in-class problems, homework, group work, and a quiz. Assessing students through in-class problems will allow me to observe students’ misconceptions and help clarify topics. The in-class problems would be for practice only and not graded. An example of a homework assessment is from Lesson 3 and Lesson 4, in Lesson 3 students were assigned to create two linear inequality problems and an answer key. At the beginning of Lesson 4, students turn in their answer keys, which will be graded for accuracy and they exchange their questions with another person, who completes it for the Do-Now (Warm-Up) part of class. The warm-up would also be collected and graded for effort. Also, in Lesson 4, there will be assessment in the form of group work, in which pairs of students work together to complete a worksheet to turn in for a grade. The worksheet will be graded for accuracy and completeness. Finally, there is an overall quiz on the last day of the unit that covers Lessons 1-3. The quiz will be mainly graded for accuracy. Finally, special needs will be addressed in multiple ways. Students will sit in rows of two desks and the person sitting next to them will be their partner. Their seats will be assigned based off of academic performance and perceived ability in the math course. I will attempt to pair each student with a student that they may complement and learn from, both mathematically and
socially. Also, students who tend to finish assignments faster will have the opportunity to do extra credit work that will challenge them mathematically. The extra credit work is MathStars worksheets for eighth grade. Although the worksheets are meant for middle school students, they do have problems that require students of all levels to think to solve the problems. Also, the student levels in Algebra I may vary from Honors ninth grade students to seniors trying to pass so that they can graduate.
Solvin ng and Grraphing Inequaliities Lesson L 1: Graphing G I Inequalitie es in One Variable V Subject: Algebra I Time Allowed: A 50 minutes Date: R inequualities in onne variable; Graph G inequualities on a number n line;; Goals/Objectives: Review Write ineequality giveen graph NC SCO OS Objectivees: 4.01 Usee linear functtions or ineqqualities to model m and soolve problem ms; justify reesults. Misconcceptions: Ch hanging the inequality siggn when muultiplying by a negative number; n Usinng closed circles when graphing g <, > inequalitiees. Motivatiion: Comparrison in absoolute terms – Joe is olderr than sue; Budgeting/Sa B aving moneyy Organization: 1. Do-Now D 2. Development D t of lesson thhrough explaanation, dem monstration, and a questionning 3. Checking C for Understandding – callingg on studentss individuallly 4. Practice – stu udents will coome to the board b and woork out problems P – Sttudents solvee problems on o board inddividually. 5. Inndependent Practice 6. Strategies a. Pre-A Assessment – to see studeents’ levels of o understannding of ineqqualities. b. Questtioning c. Collab boration 7. Closure C Lesson 1 Notess : 10-155 min Do-Now (Pre-Assesssment, Attachhment A; Prre-Assessmennt Key, Attacchment B) • Distribute D thee Pre-Assesssment, Attachhment A, annd have studeents complette it individuually. • When W the students seem to t be just about done witth the do-now, ask studeents to raise their t hand if they would w like too explain how w they derivved their answer. Call onn different sttudents for each e problem m. • Ask A after each h problem iff anyone hass a question. If a questioon arises, dem monstrate prroblem on th he board. • NOTE: N For Part P II of the Pre-Assessm ment tell studdents that these are the four f basic typpes of inequalities and re-empphasize them m – <, >, <, >. > Lesson : 20-25 miin Write on n Board:
Say: Can anyone tell me what this is? Num C mber Line A how do we And w plot poinnts on a num mber line? With W a closed circle. T That’s right. Now, if we were given the followinng problem: Write on n Board and d Say: Jooe is two yeaars older thaan his sister, Sally. If Saally is five yeears old, how w old is Joe?? Say: W Where would d we plot Joee’s age on thhe number linne? Ask stu udent to com me to board and plot the poin nt.
Now, what if we were given this problem: Write on Board and Say: Joe is at least two years older than his sister, Sally. If Sally is five years old, how old is Joe? Say: Do you notice anything different about this problem? Is at least instead of is So what do we know about Joe’s age? He could be 7 years old or he could be older. On your sheet of paper, try to graph how old Joe is. Keep your graph and at the end of class, we’ll see if you graphed Joe’s age correctly. Today we will be talking about graphing inequalities. On your Pre-Assessment worksheet, we discussed the four basic types of inequalities. Write on Board and Say: a < b, a > b, a < b, and a > b Say: Look at number four on the worksheet. Does anyone know how we can graph the inequality? (Have students (3-4) come up and draw their answers on the board) Let’s think about this, if a is greater than 9, then what can a be? 10, 11, 12, … How can we show that a is all of these numbers? By putting dots at every number,… Close, but what about the other real numbers, like 10.5 or √143 11.9583 … ? We put points between the integers. Yes, but we will have an infinite amount of points. What do we call an infinite collection of points perfectly aligned? A line. Right! So we can draw a line from 9 to the right and shade in the arrow because a can be any of the numbers to the right. Write on Board:
Say: But what about a = 9, can a = 9? No because a is greater than 9. So how can we represent a > 9 if a ≠ 9? Remember that to plot the point a = 9, we use a closed circle. (Wait for students to respond, if no one responds, ask rhetorically: Can we use an open circle?) Write on Board:
Say: What would we do if a > 9? Use a closed circle. Write on Board: Say: What would we do for number five? Place an open circle at -4 and draw a line to the left. Write on Board: And if c < -4? Use a closed circle. Write on Board:
Say: Now if we go back to the problem at the beginning of class, can anyone come up to the board and graph Joe’s age? (Have student come up to board and if the answer is wrong, ask another student to come to the board and assist with the problem). Solution to Problem:
Say: Can anyone tell me the inequality that is graphed on the number line? x > 7 What does x (or letter given) represent? Joe’s age. Right, x (or letter given) is called a variable. Ok, now work on these five problems and if you get stuck, ask your partner for help. Write on Board: (Solutions written under each problem) Graph the following inequalities. 1. a > -3
2. b < 10
Write the inequality for the graph. 3.
4.
5.
x > -1
x<4
-3 < x < 7
Walk around and help students. When the students seem to be just about done with the problems, ask students to raise their hand if they would like to come to the board and write the answers for the problems. Say: Number 5 is called a compound inequality because it combines two inequalities together. The two inequalities combined in number five are x > -3 and x < 7. Since -3 is less than seven, we usually write that on the left side, so we flip the x > -3 inequality to -3 < x, so x is in the middle and we can write the compound inequality -3 < x < 7. For homework, you are to complete the Graphing Inequalities worksheet. (Pass out the Graphing Inequalities Worksheet and let students begin working on it if there is time left.)
Assignment: Graphing Inequalities Worksheet, Attachment C
Attachment A
Attachment B
Attachment C
Solving and Graphing Inequalities Lesson 2: Solving Inequalities in One Variable Subject: Algebra I Time Allowed: 50 minutes Date: Goals/Objectives: Solve inequalities in one variable; Solve inequality word problems NC SCOS Objectives: 1.01 Write equivalent forms of algebraic expressions to solve problems; 4.01 Use linear functions or inequalities to model and solve problems; justify results. Misconceptions: Changing the inequality sign when multiplying by a negative number; Using closed circles when graphing <, > inequalities; Translating words into mathematical symbols – at least, no more than, etc. Motivation: Comparison in absolute terms – Joe is older than sue; Budgeting/Saving money Organization: 1. Do-Now 2. Development of lesson through explanation, demonstration, questioning, and practice 3. Checking for Understanding – calling on students individually, observe students working on practice problems. 4. Practice – students will work on practice problems individually and with a partner 5. Independent Practice – individual work with practice problems 6. Strategies a. Questioning b. Collaboration c. Scaffolding 7. Closure Lesson 2 Notes Do-Now (Graphing Inequalities Worksheet, Attachment A; Graphing Inequalities Worksheet Key, Attachment B) : 10-15 min • Place a transparency copy of the Graphing Inequalities Worksheet on the overhead. • Have students take out their homework and call on students for answers to the homework, making sure each student has had a turn to answer one of the questions. • Using an overhead pen, write the correct answer for each question, using the Graphing Inequalities Worksheet Key if needed. • Ask if there are any questions after going over the homework and if so, address those questions. Lesson Say:
: 25-30 min
Today we are going to solve inequalities. I would like you all to take out a sheet of paper for notes and copy the following problem and try to solve it. (Allow students a few minutes to work on problem) Write on Board: 2y – 5 < 7 Say: (and write on board as students tell you) What do we do first? Add 5 to both sides. And after that? Divide both sides by 2. So y < 6. Write on Board:
2y – 5 < 7 +5 +5 2y – 5 < 12 2 2 2y – 5 < 6 Say: When we solved linear equations, what did we do after we found the solution? Checked the answer. How do you think we should check to see if our statement is true? Plug in a number less than 6. Right. Let’s choose 4. Write on Board: 2y – 5 < 7 2(4) – 5 < 7 8–5<7 3<7 Say: How do we graph the solution? Open circle on 6 with the arrow to the left. Right. So it would look like this Write on Board:
Say: Remember, when you graph the solution, you have an open circle on 6, since y is not equal to 6 and a line to the left indicating all numbers less than 6 are a solution. Ok, now try this problem. Write on Board: 6 > 2(x – 4) Say: (When students appear about finished ask) Who would like to come to the board and show us how they solved the problem and the graph of the solution. Solution to Problem: (Make sure student checks problem, if not remind students that they have to check the solution and for this problem they can choose 7 because it is greater than or equal to. Also, make sure student’s graph has a closed circle and is to the left. If wrong, ask other students to help correct the student’s work.) 6 > 2(x – 4) 6 > 2x – 8 +8 +8 14 > 2x 2 2 7>x Check: 6 > 2(x – 4) 6 > 2(7 – 4) 6 > 2(3) 6>6
Say: Why do we use a closed circle for this graph? Because it is greater than or equal to. Right. Let’s try two harder problems, but this time work with your partner. Remember how we solved linear equations. (Walk around and help students solve problems). Write on Board: 1) 3x + 1 > 4x – 2 2) 3x > (x-2) Say: Let’s start with the first problem. (Use overhead copy of Attachment C and cover up steps. Call on students and ask them for the next step in the problem, uncovering the steps as the students say them. Do the same for problem 2, Attachment D) Say: Now let’s try a word problem. Take five minutes with your partner and try to solve the problem. (Use overhead copy of attachment E and walk around and help students. After five minutes, go through answer with students, writing steps on the overhead copy. Make sure students identify the variable and understand how you arrived at the answer, see Attachment F.) Problem: Keith has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 each week for food, clothes, and movie tickets. • Write an inequality that represents Keith’s situation. • How many weeks can Keith withdraw money from his account? Justify your answer. Say: Ok, now that we have solved the problem, let’s write down the three steps to solving an inequality word problem. Who can tell me what the first step is? The second step? And the third step? (As the students tell you, write the steps on the board.) Write on Board: Step 1: Highlight/Underline the important information in the problem. Step 2: Identify the variable. Step 3: Write the inequality. Say: These steps are the same as solving a linear equation word problem, except in step 3, we change equation to inequality. For homework tonight, you will have a couple of problems similar to the one we just solved. (Pass out worksheet for homework) Assignment: Solving Inequalities: Word Problems, Attachment F
Attachment A
Attachment B
Attachment C
Attachment D
Attachment E
Attachment F Solving Inequalities: Word Problems Name: _____________________________ Date: ___________________ Directions: Solve the following word problems and remember to check your answers. Problem 1: Chris wants to order DVDs over the internet. Each DVD costs $15.99 and shipping for the entire order is $9.99. Chris has no more than $100 to spend. • Write an inequality that represents Chris’ situation.
•
How many DVDs can Chris order without exceeding his $100 limit? Justify your answer mathematically.
Problem 2: Skate Land charges a $50 flat fee for birthday party rental and $5.50 for each person. Joann has no more than $100 to spend on the birthday party. • Write an inequality that represents Joann’s situation.
•
How many DVDs people can Joann invite to her birthday party without exceeding her limit? Justify your answer mathematically.
Solving and Graphing Inequalities Lesson 3: Graphing and Solving Linear Inequalities Subject: Algebra I Time Allowed: 50 minutes Date: Goals/Objectives: Solve linear inequalities; Graph linear inequalities; Build vocabulary NC SCOS Objectives: 1.01 Write equivalent forms of algebraic expressions to solve problems; 4.01 Use linear functions or inequalities to model and solve problems; justify results. Misconceptions: Changing the inequality sign when multiplying by a negative number; When to use a solid/dotted line when graphing linear inequalities; Do not need to shade area on graph. Motivation: Solution space – what are the possibilities of rolling a seven with two dice; How to budget for the cost of buying things. Organization: 1. Do-Now 2. Development of lesson through explanation and demonstration. 3. Checking for Understanding – having students respond to questions. 4. Practice – students will attempt practice problems on the note sheet. 5. Independent Practice – students will create problems and answer keys based off of lesson. 6. Strategies a. Note Sheet b. Scaffolding c. Questioning d. Allow students to create their own problems. 7. Closure Lesson 3 Notes Do-Now (Solving Inequalities: Word Problems worksheet, Attachment A) : 8-10 min • Have students turn in their Solving Inequalities: Word Problems worksheet. Grade according to Solving Inequalities: Word Problems Key, Attachment B. • Before students arrive, write the following on the board: Graph the following equations 1. y = 2x + 2 2. 3y = 2x – 6 3. -4y = 2x - 12 • Ask three students, one for each question, to come to the board and free-hand graph the equations. Their graphs should resemble the following. Ask other students to verify if the graphs are correct.
1.
2.
3. Lesson (Lesson 3: Solving and Graphing Linear Inequalities Notes, Attachment B overhead transparency copy; Lesson 3: Solving and Graphing Linear Inequalities Notes Key, Attachment C) : 25-35 min Pass out Lesson 3: Solving and Graphing Linear Inequalities Notes, Attachment B and place transparency copy on overhead projector. Say (and fill out note sheet as you go along): Number 1: We will use the graphs you drew for our lesson today, so do not put them away. Today we will graph linear inequalities. Graphing linear inequalities is similar to graphing linear equations, however there are some differences, which you will see as we go along. The first inequality we will graph is y < 2x + 2. When graphing linear inequalities, we must first graph the boundary line. To find the boundary line, we will pretend that the linear inequality is a linear equation and graph the line as we did during the warm‐up. However, because the inequality is less than, we will use a dotted line to graph the line. The dotted line indicates that the line does not contain any solutions. (Fill in note sheet). So we will use a number 1 from the warm-up and dot the line on our note sheet. Another difference from graphing linear equations is that we shade one side of the line. To find out where to shade, we pick a test point that is not on the line. For this example, we can choose (0,0). If the test point works, we shade that side of the line, if it fails, we shade on the opposite side of the line. Plug in (0,0) and tell me what side we shade. The side with (0,0). Right. Number 2: Now let’s try number two. Before we graph the boundary line, what should we do first? Solve for y. Right. How do we solve for y? (Perform the steps as the students tell you on the overhead copy.) So this is similar to number two from the warm-up. Will we use a solid or dotted line? Solid. Why? Because it is equal to.
So our boundary line looks exactly like the line we graphed in number two from the warm-up. Now choose a test point and find which side to shade on your own. (After two to three minutes.) Which side do we shade? The side to the left of the line. Number 3: Now try number three on your own and identify three solutions to the inequality. You may use your partner for help. (Allow students 8-12 minutes to work on the problem). Now share your answers with another pair of two and verify that you all have the same answer. (After 3-5 minutes, post answer on overhead.) This is the correct answer. Did we all figure it out? (Take time to answer students’ questions.) Now let’s finish the notes and then you can start on your homework. What is the standard form of a line? y = mx + b So what do you think is the standard form of a linear inequality? y < mx + b, y > mx + b, y < mx + b, y > mx + b. The dotted or solid line that we drew is called the what? Boundary line. To determine which side of the line to shade, we choose a what? Test point. If the test point satisfies the inequality, we shade the side that does or does not contain the test point? And if the test point does not satisfy the inequality, we shade the side that …(wait for students response). For homework tonight, I want you to create two linear inequality problems like we did today in class and create an answer key to those problems. You will turn the answer key in to me at the beginning of class tomorrow and your two problems will be given to someone in the class to solve. Make sure to write them on two separate sheets of paper and write your name on both sheets. (If there is time left, let students start on homework) Assignment: Create 2 linear inequality problems and an answer key.
Attachment A Solving Inequalities: Word Problems Name: _____________________________ Date: ___________________ Directions: Solve the following word problems and remember to check your answers. Problem 1: Chris wants to order DVDs over the internet. Each DVD costs $15.99 and shipping for the entire order is $9.99. Chris has no more than $100 to spend. • Write an inequality that represents Chris’ situation.
•
How many DVDs can Chris order without exceeding his $100 limit? Justify your answer mathematically.
Problem 2: Skate Land charges a $50 flat fee for birthday party rental and $5.50 for each person. Joann has no more than $100 to spend on the birthday party. • Write an inequality that represents Joann’s situation.
•
How many DVDs people can Joann invite to her birthday party without exceeding her limit? Justify your answer mathematically.
Attachment B Solving Inequalities: Word Problems Key Name: _____________________________ Date: ___________________ Directions: Solve the following word problems and remember to check your answers. Problem 1: Chris wants to order DVDs over the internet. Each DVD costs $15.99 and shipping for the entire order is $9.99. Chris has no more than $100 to spend. • Write an inequality that represents Chris’ situation. (2 points: 1 point for attempt, 1 point for correct inequality) •
How many DVDs can Chris order without exceeding his $100 limit? Justify your answer mathematically. (3 points: .5 point for solving inequality for d, .5 point for statement, 1 point for justification, 1 point for correct computations)
Problem 2: Skate Land charges a $50 flat fee for birthday party rental and $5.50 for each person. Joann has no more than $100 to spend on the birthday party. • Write an inequality that represents Joann’s situation. (2 points: 1 point for attempt, 1 point for correct inequality) •
How many DVDs people can Joann invite to her birthday party without exceeding her limit? Justify your answer mathematically. (3 points: .5 point for solving inequality for d, .5 point for statement, 1 point for justification, 1 point for correct computations)
Attachment C – Page 1 Lesson 3: Solving and Graphing Linear Inequalities Notes 1. y < 2x +2
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We use a ____________________ to show that the inequality is less than (<) or greater than (>). This indicates that the line ____________________ contain ____________________ solutions. 2. 3y > 2x – 6
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We use a ____________________ to show that the inequality is less than or equal to (<) or greater than or equal to (>). This indicates that the line ____________________ contain ____________________ solutions.
Attachment C – Page 2 3. -4y < 2x – 12 Identify three solutions.
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The standard form of a linear inequality is: o ____________________, o ____________________, o ____________________, or o ____________________.
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The dotted or solid line is called the ____________________.
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To determine which side of the line to shade, we choose a ____________________.
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If the test point satisfies the inequality, we shade the side that ____________________ the test point. If the test point does not satisfy the inequality, we shade the side that ____________________ the test point.
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Attachment D – Page 1 Lesson 3: Solving and Graphing Linear Inequalities Notes 1. y < 2x +2 Test Point (0,0) y < 2x +2 0 < 2(0) + 2 0 < 2 TRUE, shade side with test point Note: Use dotted line, instead of no line
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We use a dotted line to show that the inequality is less than (<) or greater than (>). This indicates that the line does not contain any solutions. 2. 3y > 2x – 6 Solve for y 3y > 2x – 6 y > 2/3x - 2 Test Point (0,0) 3y > 2x – 6 3(0) < 2(0) – 6 0 < -6 FALSE, shade side without test point
•
We use a solid line to show that the inequality is less than or equal to (<) or greater than or equal to (>). This indicates that the line does contain some solutions.
Attachment D – Page 2 3. -4y < 2x – 12 Identify three solutions. Solve for y -4y < 2x – 12 y > -1/2x + 3 Test Point (0,0) -4y < 2x – 12 -4(0) < 2(0) – 12 -4 < -12 FALSE, shade side without test point Note: Use dotted line, instead of no line
•
The standard form of a linear inequality is: o y > ax + b, o y < ax + b, o y > ax + b, or o y < ax + b.
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The dotted or solid line is called the boundary line.
•
To determine which side of the line to shade, we choose a test point.
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If the test point satisfies the inequality, we shade the side that contains the test point. If the test point does not satisfy the inequality, we shade the side that does not contain the test point.
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Solving and Graphing Inequalities Lesson 4: Graphing and Solving Linear Inequalities Word Problems Subject: Algebra I Time Allowed: 50 minutes Date: Goals/Objectives: Solve linear inequality word problems using information learned earlier. NC SCOS Objectives: 1.01 Write equivalent forms of algebraic expressions to solve problems; 4.01 Use linear functions or inequalities to model and solve problems; justify results; 4.01a Solve using tables, graphs, and algebraic properties; 4.01b Interpret constants and coefficients in the context of the problem; 4.03 Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results. Misconceptions: Changing the inequality sign when multiplying by a negative number; When to use a solid/dotted line when graphing linear inequalities; Do not need to shade area on graph. Motivation: Solution space – what are the possibilities of rolling a seven with two dice; How to budget for the cost of buying things. Organization: 8. Do-Now 9. Development of lesson through student collaboration. 10. Checking for Understanding – students will turn in worksheet. 11. Practice – students will work together to solve two problems. 12. Strategies a. Collaboration b. Questioning 13. Closure Lesson 4 Notes Do-Now : 10-15 min • Have students take out their homework and turn in their answer keys. (Grade for completion) • Have students write their name on their created problems and exchange with the person next to them. • Give students 10 minutes to work on the problems and then collect. (Look at students’ answers and observe misconceptions, grade for effort)
Lesson (Spending with Limits, Attachment A; Spending with Limits, Attachment B used to grade) : 25-35 min Pass out Spending with Limits, Attachment A, one copy for every student. Students will work with the person sitting beside them and they will each turn in a copy. Say: Today we our going to work on linear inequalities word problems. I want you to work with the person sitting beside you to complete the group activity on the worksheet. Be sure to read the directions carefully. Everyone will need to turn in the worksheet at the end of class. (Pass out one red die and one blue die to each pair of students for Part A) Walk around and assist students by asking questions without giving away the answers, like: What if I don’t spend all of my money? What if I borrow a car and only rent the tuxedo? Are there other restrictions on the situation?
When there are about 5 minutes left in class, tell students that what they have not finished will need to be finished for homework. If they are finished, they can go ahead and turn in the worksheet. Remind them that tomorrow there will be a quiz covering the material learned. Assignment: Finish worksheet. Study for quiz.
Attachment A – Page 1
Attachment A – Page 2
Attachment A – Page 3
Attachment A – Page 4
Attachment B – Page 1
Attachment B – Page 2
Attachment B – Page 3
Attachment B – Page 4
Solving and Graphing Inequalities Lesson 4: Graphing and Solving Linear Inequalities Word Problems Subject: Algebra I Time Allowed: 50 minutes Date: Goals: Assess students, Understand misconceptions students may make. Objectives: Assess students’ understanding on material covered. Misconceptions: Changing the inequality sign when multiplying by a negative number; When to use a solid/dotted line when graphing linear inequalities; Do not need to shade area on graph; Using closed circles when graphing <, > inequalities. Motivation: Comparison in absolute terms – Joe is older than sue; Budgeting/Saving money; Solution space – what are the possibilities of rolling a seven with two dice; How to budget for the cost of buying things. Organization: 1. Do-Now 2. Assessment of Students’ Knowledge on Lesson Material 3. Checking for Understanding – Independent Quiz 4. Strategies a. Assessment 5. Closure Lesson 5 Notes Do-Now : 5-10 min • Have students take out their homework and turn in. • Ask students if they have any questions on graphing and solving inequalities before they take the quiz. If they do, answer the questions. Lesson (Inequalities “Big” Quiz, Attachment A; Inequalities “Big” Quiz Key, Attachment B; MathStars Vol.8, No.1, Attachment C; MathStars Vol.8, No.1 Key, Attachment D) : 30-40 min Say: Everybody, clear your desk except for your pencil. You will have until the end of the class to complete the quiz. If you finish early, you can work on the extra-credit Math Stars problems. (Have students turn in quizzes once they are finished.) Assignment: None.
Attaachment A I Inequalit ties “Big g” Quiz Name _________ _ __________________ Date ________________________ Graph the follow wing inequ ualities. (1 pt. each) 1. c > -4 2. 6 > x 3. -33d + 1 > 7 4. -88 < 4m < 16 1 5. 5a – 20 < 20 + a Write an a inequallity for eacch graph. (1 pt. each h) _______________ 6. 7.
_______________ _______________
8. Graph the follow wing linearr inequalitty. (2 pts) 9. 2y 2 + 8x < 4
More Prooblems on Bacck
Graph the following linear inequality. (2 pts) 10. 5x +20 > -10y
Solve the following word problem. (You can use a calculator) 11. Yellow Cab Taxi charges $1.75 flat rate in addition to $0.65 per mile. Katie has no more than $10 to spend on a ride. • Write an inequality that represents Katie’s situation. (Remember to identify your variable.) (1 pt)
• How many miles can Katie travel without exceeding her limit? Justify your answer. (2 pts)
Attachment B Inequalities “Big” Quiz Key Name _________________________ Date _____________________ Graph the following inequalities. (1 pt. each) 1. c > -4 2. 6 > x 3. -3d + 1 > 7 4. -8 < 4m < 16 5. 5a – 20 < 20 + a Write an inequality for each graph. (1 pt. each) ____a > -2____ 6. 7.
____b ≤ 3____ __3 ≤ c < 10__
8. Graph the following linear inequality. (2 pts: 1 pt correct graph, 1 pt attempt) 9. 2y + 8x < 4 NOTE: Dotted line, instead of no line.
More Problems on Back
Graph the following linear inequality. (2 pts: 1 pt correct graph, 1 pt attempt) 10. 5x +20 > -10y
Solve the following word problem. (You can use a calculator) 11. Yellow Cab Taxi charges $1.75 flat rate in addition to $0.65 per mile. Katie has no more than $10 to spend on a ride. • Write an inequality that represents Katie’s situation. (Remember to identify your variable.) (1 pt) Let m = number of miles .65m + 1.75 ≤ 10 • How many miles can Katie travel without exceeding her limit? Justify your answer. (2 pts: 1 pt for answer, 1 pt for justification) .65m + 1.75 ≤ 10 .65m + 1.75 – 1.75 ≤ 10 – 1.75 .65m ≤ 8.25 .65 .65 m ≤ 12.69 Katie can travel 12 miles or less before reaching her limit of $10. Justify: .65m + 1.75 ≤ 10 .65(12) + 1.75 ≤ 10 7.80 + 1.75 ≤ 10 9.55 ≤ 10
Attachment C – Page 1
Attachment C – Page 2
Attachment D – Page 1
Attachment D – Page 2
Conclusion This unit plan covers the main aspects of solving and graphing linear inequalities, while also providing a review of inequalities for students who may have forgotten what inequalities were or who did not learn about inequalities. The aim of the unit was to promote students involvement in the class in hopes that they would better understand the concepts addressed because they had in a sense “discovered” connections to other material. The order of Lesson 3 and Lesson 4 could be changed depending on the skill level of the class as a whole. By making this change, students would build their own connection to graphing linear inequalities based off of their prior knowledge of graphing linear equations. This change promotes the discovery method. Lesson 3 and Lesson 4 are in the order they currently are in because it is usually the typical order that they are taught in classes. Also, in this unit plan there is not a review day for the material because for the most part, the material introduced has been covered in previous math classes and in previous lessons. Also, the material is an extension of solving and graphing linear equations. However, I would recommend that if the makeup of the class consists of students who seem to not have a strong foundation of the concepts taught to include a review day before the quiz. An extra day covering the material may help the students grasp the material better, however that is at the discretion of the instructor and may be limited to time constraints. All in all, the students were assessed in every lesson either through questioning, practice problems, homework, group work, or the quiz. Furthermore, at the end of this unit, students will be ready to progress into the next unit of solving systems of linear inequalities and will be prepared for Algebra II when the concept of absolute value is introduced.
References and Resources •
Improving Student Achievement in Algebra, Grades 7-12. The DuPage1 Mathematics Network. DuPage Regional Office of Education. May 2007 •
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www.dupage.k12.il.us/doc/Linear%20Inequalities%20Investigation.doc
Algebra- Class.com •
http://www.algebra-class.com/solving-inequalities.html
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http://www.algebra-class.com/solving-inequalities-practice.html
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http://www.algebra-class.com/solving-word-problems-in-algebra.html
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http://www.algebra-class.com/solving-word-problems-in-algebra-practice.html
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http://www.algebra-class.com/graphing-inequalities.html
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http://www.algebra-class.com/linear-inequalities.html
Math Stars Newsletter •
http://mathlearnnc.sharpschool.com/UserFiles/Servers/Server_4507209/File/Math %20Stars%20Newsletter/Grade%208%20Math%20Stars.pdf
