Teacher Candidate: Grade Level/Subject Unit Title Lesson Title Duration Lesson Outcomes Chiara Shah 9 th /Algebra I Unit 4: Solving and Graphing Inequalities 6.1 Solving One-Step Linear Inequalities 45 Minutes Students will be able to: 1) Make a conjecture about what happens when multiplying or dividing inequalities by a negative number (NJCCCS 4.1.A.1, 4.1.A.2, 4.1.A.3, 4.1.B.1, 4.3.B.1; NCTM 2.2, 9.1, 9.2; NJPTS 7.iii4, 7.iii.5) 2) Graph inequalities, as evidenced by their successful completion of problems 23-29 odd on page 337 of their textbook. (NJCCCS 4.1.A.1, 4.1.A.2, 4.1.B.1, 4.B.3.1; NCTM 5.1, 10.4; NJPTS 7.ii.4, 7.iii.5) 3) Solve inequalities, as evidenced by their successful completion of problems 15, 17, and 19 on page 337 of their textbook. (NJCCCS 4.1.B.1, 4.3.B.1; NCTM 9.2; NJPTS 7.iii.4, 7.iii5) Standards NJCCCS STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS) ALL STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS. A. Number Sense 1. Extend understanding of the number system to all real numbers. 2. Compare and order rational and irrational numbers. 3. Develop conjectures and informal proofs of properties of number systems and sets of numbers. B. Numerical Operations 1. Extend understanding and use of operations to real numbers and algebraic procedures. STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES. B. Functions and Relationships 1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs. 1
Standards NCTM Standards NJPST Modifications & Accommodations Materials and Use of Instructional Technology Standard 2: Knowledge of reasoning and Proof 2.2 Make and investigate mathematical conjectures Standard 5: Knowledge of Mathematical Representation 5.1 Use representations to model and interpret physical, social, and mathematical phenomena Standard 9: Knowledge of Number and Operations 9.1 Analyze and explain the mathematics that underlies the procedures used for operations involving integers, rational, real, and complex numbers 9.2 Use properties involving number and operations, mental computation, and computational estimation Standard 10: Knowledge of Different Perspectives on Algebra 10.1 Analyze patterns, relations, and functions of one and two variables 10.4 Use mathematical models to represent and understand quantitative relationships 7. Standard Seven: Special Needs. Teachers shall adapt and modify instruction to accommodate the special learning needs of all students. iii. Teachers engage in activities to: (4) Meet the needs of all learners by using a wide range of teaching techniques to accommodate and modify strategies, services and resources, including technology; and (5) Make appropriate provisions, in terms of time and circumstances for work, task assigned, communication and response modes, for individual students who have particular learning differences or needs. 1) Calculators will be provided to those students who need them. (NJPTS 7.iii.4, 7.iii.5) Item Description 1) Textbook 1) 2) Calculators 2) 2
P R O C E D U R E S Attendance, Announcements, Homework Today is the start of a new chapter, chapter 6, Solving and Graphing Inequalities Anticipatory Set This activity targets learner outcome #1: Make a conjecture about what happens when multiplying or dividing inequalities by a negative number (NJCCCS 4.1.A.1, 4.1.A.2, 4.1.A.3, 4.1.B.1, 4.3.B.1; NCTM 2.2, 9.1, 9.2; NJPTS ) DO NOW Investigate what happens to an inequality when it is multiplied or divided by positive and negative numbers. Write the following table on the board. Multiply by 2 Multiply by -2 Divide by 2 Divide by -2 5 > 2 4 < 8 3x < 7 Have students complete it as soon as they arrive in class. Ask students for the answers for each cell. (Do not switch direction of symbols yet, unless class appears to already know this rule or unless they notice something is wrong.) 5 > 2 4 < 8 3x < 7 Multiply by 2 10 > 4 4 < 8 6x < 14 Multiply by -2 10 > 4 4 < 8 6x < 14 Divide by 2 5 2 > 1 1 < 2 Divide by -2 3x 2 < 7 2 5 2 > 1 1 < 2 3x 2 < 7 2 Go through each cell in the table above and ask: Is this true? Is this true? How about this one? (Note: Tell the students, we are not sure if the ones with the variables are true because it depends on what number we plug in for x. Right now, because we haven t solved for 3
x, x can be any value we wish. BUT, regardless, to make this inequality match the original, the symbol needs to be switched.) A: The inequalities in the cells circled above are not true. Q: How can we make them all true? A: Change the direction of the inequality sign. Q: Can anyone spot a pattern? A: We multiplied or divided by a negative number. Q: Do you think that every time you multiply or divide by a negative number that you ll need to reverse the inequality symbol? A: Yes. Q: Can we make this a rule? A: Yes. Write rule on the board. RULE: When you multiply or divide an inequality by a negative number you need to change the direction of the inequality symbol. Let s look at the third column of the table. We couldn t tell earlier if the inequality was true or false because of the variable. But now that we know this rule, we need to change the direction of the inequality symbol. Do it now. 5 > 2 4 < 8 3x < 7 Multiply by 2 10 > 4 4 < 8 6x < 14 Multiply by -2 10 < 4 4 > 8 6x > 14 Divide by 2 5 2 > 1 1 < 2 Divide by -2 3x 2 < 7 2 5 2 < 1 1 > 2 3x 2 > 7 2 4
Practice problems: Students should work independently on the following problems. Page 333, problems 8-17. This is a Section/Chapter pre-work page that investigates the symbol reversal issue. NOTE: These problems assess if learner outcome #1 was met. Lesson PART I: Graphing Inequalities The following activities targets learner outcome #2: Graph inequalities, as evidenced by their successful completion of problems 23-29 odd on page 337 of their textbook. (NJCCCS 4.1.A.1, 4.1.A.2, 4.1.B.1, 4.B.3.1; NCTM 5.1, 10.4) While students are working on the above problems independently, put a new table on the board. Statement Inequality Graph All real numbers less than two All real numbers greater than negative two All real numbers less than or equal to one All real numbers greater than or equal to zero Work with students to fill in the table. Explain how to graph using open ( less than or greater than ) and closed ( less than or equal to or greater than or equal to ) dots. 5
Statement Inequality Graph All real numbers less than two x < 2 On board All real numbers greater than negative two x > 2 On board All real numbers less than or equal to one x 1 On board All real numbers greater than or equal to zero x 0 On board Graphing: Independent Practice problems: page 337, 23-45 odd. These problems assess if learner outcome #2 was met. 6
PART II: Solving Inequalities Learner Outcome #3: Solve inequalities, as evidenced by their successful completion of problems 15, 17, and 19 on page 337 of their textbook. (NJCCCS 4.1.B.1, 4.3.B.1; NCTM 9.2) Solving in equalities is the same as solving equalities. We need to isolate the variable. Q: What does isolate mean? A: Get the variable by itself. The only thing that s different is that we need to make sure we change direction of the symbol. Q: When do we change direction of the symbol? A: When multiplying or dividing by a negative number. Q: How do we solve this? On board: x 3 = 5 A: Add three to both sides. Q: So how do you think we solve this? On board: x 3 < 5 A: Add three to both sides. On board: x 3 < 5 +3 +3 x < 8 Right. Let s try a few more. I m going to right six problems on the board. I want you to work on them at your desk. I ll walk around to help if you need it. 7
On board: 1. x + 6 10 subtract six from both sides 2. x 5 3 add 5 to both sides 3. 2 > n 4 add four to both sides 4. a 3 12 multiply both sides by 3 5. 5m > 10 divide both sides by -5 don t forget to switch signs 6. 1 2 x 3 multiply both sides by -2 don t forget to switch signs Solving Inequalities. Independent Practice. Page 337, Problems 15,17, 19. These problems assess if learner outcome #3 was met. Closure Question for Students Answer 1) When do we need to change 1) When we multiply or divide by a negative number direction of an inequality symbol? 2) How to we solve inequalities? 2) The same way as equalities, except need to switch the inequality symbol if we multiply or divide by a negative number 3) When do we use an open dot and 3) Open dot is used for less than and greater than when do we use a closed dot when and a closed dot is used for greater than or equal to graphing inequalities? and less than or equal to 8
Homework Page 337, 22-44 even. Reflection What Went Well (Strengths of lesson) Students in both periods noticed immediately that something was wrong when they started multiplying the inequalities by negative numbers. This is constructivism at its greatest! As soon as a student raised his hand, I asked, Does anyone else notice something strange going on? I let a few more students nod or raise their hands before asking, What do you think is happening? WHY is this happening? What Didn t Go Well (Weaknesses of lesson) Because this is not my class and I have not taught them the way I intend to teach this when it s my own class, they didn t get the answer to this question, or they didn t really understand it. The answer is that negative means opposite, or a flip through the origin on the number line. Because of the negative everything in the problem gets flipped across the number line. Therefore, what was smaller is now larger and what was larger is now smaller. Instead, I told them that Negative means opposite, right? So we need to do the opposite. The opposite of greater than is less than. The opposite of less than is greater than. Also, I said we need to do this because it s obvious from the numbers on the board that the sign needs to be switched. Even still, some students didn t understand why you need to change direction of inequality symbol when there are variables in the inequality. It took a few more examples of multiplying regular, non-variable inequalities, by negative numbers for them to be comfortable with the idea that it s got to happen with all inequalities that are multiplied/divided by negatives. Notes to Self/Notes for Next Time When its my own class this changing directions of inequality symbol would be a GREAT writing assignment. Don t tell them why it happens in class. Assign them to complete a reflection asking Why do you think we need to change the inequality symbol? What is happening when you multiply or divide by a negative? Actually, this little chart and activity would be a great independent problem for students to do after they finish the chapter 5 test. Then, Icould assign the reflection writing assignment as homework the night of the test, and they can come in today, on the first lesson of the chapter, prepared to answer WHY? and begin solving inequalities. I could probably combine sections 6.1 (solving one-step inequalities) with section 6.2 (solving multistep 9
inequalities) into one day s instruction. Professional Development Recommendations By changing this lesson to include a reflection writing assignment, I would be targeting the following NJPTS: 1. Standard One: Subject Matter Knowledge. Teachers shall understand the central concepts, tools of inquiry, structures of the discipline, especially as they relate to the New Jersey Core Curriculum Content Standards (CCCS), and design developmentally appropriate learning experiences making the subject matter accessible and meaningful to all students. iii. Teachers engage in activities to: (1) Promote the development of critical and creative thinking, problem solving and decision making skills by engaging students in formulating and testing hypotheses according to the methods of inquiry and standards of evidence within the discipline; AND 4. Standard Four: Instructional Planning and Strategies. Teachers shall understand instructional planning, design long and short term plans based upon knowledge of subject matter, students, community, and curriculum goals, and shall employ a variety of developmentally appropriate strategies in order to promote critical thinking, problem solving and the performance skills of all learners. ii. Teachers value and are committed to the development of students critical thinking, independent problem solving and performance capabilities. AND 1. Standard One: Subject Matter Knowledge. Teachers shall understand the central concepts, tools of inquiry, structures of the discipline, especially as they relate to the New Jersey Core Curriculum Content Standards (CCCS), and design developmentally appropriate learning experiences making the subject matter accessible and meaningful to all students. i. Teachers know and understand: (3) That literacy skills and processes are applicable in all content areas and help students to develop the knowledge, skills and dispositions that enable them to construct meaning and make sense of the world through reading, writing, listening, speaking and viewing; and 10