Analyzing and Solving Pairs of Simultaneous Linear Equations

Transcription

1 Analyzing and Solving Pairs of Simultaneous Linear Equations Mathematics Grade 8 In this unit, students manipulate and interpret equations in one variable, then progress to simultaneous equations. They look at the structure of equations (SMP.7) to predict whether there is one solution, no solution, or infinite solutions. Students see the relationships between equations and between algebraic and graphical representations. They extend prior knowledge of the properties of equality, representing quantities as variables, and writing simple equations to construct and solve systems of two linear equations that model real-world and mathematical situations (SMP.4). Students will be able to analyze the context of equations and problems in order to determine which method for solving simultaneous equations is most efficient (SMP.3). Before starting this unit, students need to complete the unit on Proportions, Lines, and Equations (8.EE.5, 8.EE.6) These Model Curriculum Units are designed to exemplify the expectations outlined in the MA Curriculum Frameworks for English Language Arts/Literacy and Mathematics incorporating the Common Core State Standards, as well as all other MA Curriculum Frameworks. These units include lesson plans, Curriculum Embedded Performance Assessments, and resources. In using these units, it is important to consider the variability of learners in your class and make adaptations as necessary. Draft 8/ 2013 Page 1 of 137

2 Table of Contents Unit Plan Lesson Lesson Lesson Lesson Lesson 5.86 Lesson 6.98 Appendix: Optional Pre-Lesson. 107 Unit Resources CEPA Draft 8/ 2013 Page 2 of 137

3 Stage 1 Desired Results ESTABLISHED GOALS 8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x=a, a=a, or the given results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.EE.8 Analyze and solve linear equations and pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 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 Transfer Students will be able to independently use their learning to Apply mathematical knowledge to analyze and model mathematical relationships in the context of a situation in order to make decisions, draw conclusions, and solve problems. Meaning UNDERSTANDINGS Students will understand that U1 Linear equations in one variable are used to model real-world situations. They may result in one solution, infinitely many solutions, or no solution. U2 Simultaneous equations are used to model real-world situations. Solutions may result in one common solution, an infinite number of solutions or no common solution. Students will know 1. The properties of operations and the properties of equality can be used to write and solve equivalent equations with rational number coefficients. 2. Simultaneous linear equations can be solved using multiple methods (tables, graphing, algebraic methods). 3. The solution of simultaneous linear equations solved algebraically corresponds to the point(s) of intersection of their graphs. 4. Academic Vocabulary: model, coefficient, ESSENTIAL QUESTIONS 1. How can equations be used to represent real-world and mathematical situations? 2. Given a particular problem situation, how do we determine which method of solving simultaneous equations will be the most useful? Acquisition Students will be able to 1. Interpret a word problem, define the variable(s), and write the equation(s). 2. Use the properties of operations (associative, distributive, commutative, and inverse) to transform equations. 3. Use the properties of equality to transform equations (see 2011 MA Curriculum Framework for Mathematics, p. 185) 4. Solve problems involving linear equations in one variable. 5. Analyze solutions to draw conclusions in the context of the problem. 6. Analyze when a linear equation in one variable has one Draft 8/ 2013 Page 3 of 137

4 because 3x + 2y cannot simultaneously be 5 and 6. c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. SMP.3 Construct viable arguments and critique the reasoning of others. SMP.4 Model with mathematics. SMP.7 Look for and make use of structure. elimination; graphical; linear; point of intersection; simultaneous; slopeintercept form; substitution; system; one, infinite, no solution; identity; infinite solution (x=a); infinite solutions (a=a); or no solution (a = b). 7. Produce examples of linear equations that represent one solution, infinite solutions, and no solutions. 8. Solve problems involving simultaneous equations using multiple methods (tabular, graphical, and algebraic). 9. Write systems of equations to represent mathematical and real-world problems. 10. Estimate solutions to simple cases by inspection of equations and/or graphs. 11. Prove that the solution of simultaneous equations solved algebraically is the point(s) of intersection of the lines graphed in a plane. 12. Justify the accuracy of solutions. Stage 2 - Evidence Evaluative Criteria Assessment Evidence See CEPA Rubric. CURRICULUM EMBEDDED PERFOMANCE ASSESSMENT (PERFORMANCE TASKS) PT Powering Up Patriot School Students act as technology consultants, using simultaneous equations to develop a recommendation for the purchase of laptops and tablets for a school. They prepare a report (including a cost analysis and an itemized budget) for the principal, superintendent, and members of the school committee. OTHER EVIDENCE: OE Formative Assessments: Lesson 1 Writing Equations to Represent Situations Lesson 3 Ticket to Leave Number of Solutions One, None, or Infinitely Many Lesson 4 Creating Systems of Equations Lesson 5 Substitution Cards Stage 3 Learning Plan Draft 8/ 2013 Page 4 of 137

5 Summary of Key Learning Events and Instruction 1. Solving equations in one variable. 2. Categorizing linear equations as having one solution, no solutions, or infinite solutions. 3. Solving linear equations through graphing. 4. Solving simultaneous equations by substituting for one of the variables. 5. Solving simultaneous equations by eliminating one of the variables. 6. Connecting learning through a multi-step problem that requires knowledge of linear equations and systems of linear equations. 7. Curriculum Embedded Performance Assessment (CEPA). Appendix: Optional Pre-lesson: Modeling real-world and mathematical situations with equations in one variable Draft 8/ 2013 Page 5 of 137

6 Brief Overview of Lesson: Lesson 1: Solving Equations in One Variable Students extend their understanding of writing algebraic equations by reasoning out verbal statements. (Example: If you double a number and add 8, the result is 20 can be written as 2n + 8 = 20). Attention to the nuances of language as well as to the relationships of unknowns and values is essential. After reviewing the properties of operations and the properties of equality, students transform and solve equations, developing a formal process for solving equations in one variable. Students check their solutions by calculator or by substituting the solution back into the original equation. They connect equations to real-world situations in order to consider whether solutions make sense in the context of the story problem. As you plan, consider the variability of learners in your class and make adaptations as necessary. Prior Knowledge Required: 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. 6.EE.3 Apply the properties of operations to generate equivalent expressions. 6.EE.6 Use variables to represent numbers and write expressions when solving real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers. 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7. EE.2Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7. EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Estimated Time (minutes): 2 Days/ 50 minute lessons Draft 8/ 2013 Page 6 of 137

7 Resources for Lesson: Classwork: Problem Set Verbal Expressions Equations Homework Classwork: Solving Equations Homework: Working Across the Equal Sign Concrete models: Algebra tiles (or template for algebra tile cut-outs) Virtual manipulative algebra tiles are available at: Draft 8/ 2013 Page 7 of 137

8 Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations Content Area/Course: Grade 8 Mathematics Lesson 1: Solving Equations in One Variable Time (minutes): Two 50-minute lessons By the end of this lesson students will know and be able to: Write linear equations in one variable to model real-life and mathematical situations. Interpret a word problem, identify the variable, and write the equation. Use the properties of operations and the properties of equality to transform equations. Solve problems involving linear equations in one variable. Analyze solutions to draw conclusions in the context of the problem. Determine the accuracy of solutions. Essential Question(s) addressed in this lesson: How can equations be used to represent real-world and mathematical situations? Standard(s)/Unit Goal(s) to be addressed in this lesson: 8.EE.7 Solve linear equations in one variable. 8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. SMP.3 Construct viable arguments and critique the reasoning of others SMP.4 Model with mathematics SMP.7 Look for and make use of structure Targeted Academic Language: model Teacher Content and Pedagogy In Grades 6 and 7, students have used the properties of operations and properties of equality to generate equivalent expressions and to solve linear equations in one variable. They are able to add, subtract, factor, and expand linear expressions with rational coefficients. However, during the transition to the new standards, some students may need additional instruction and practice in these concepts. Draft 8/ 2013 Page 8 of 137

9 In this lesson, students move to a mathematical translation of a more abstract verbal expression that does not have a story problem context. They collect language/ terminology to parallel parts of an equation. If students can write word problems that are modeled by given equations, they are demonstrating proficiency in moving between word problem contexts and algebraic expressions and equations. Prior knowledge needed for this lesson: Write, read, and evaluate expressions in which letters stand for numbers. Write expressions that record operations with numbers and with letters standing for numbers Use variables to represent quantities and write expressions when solving real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Solve real-world/ mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Construct simple equations and inequalities to solve problems by reasoning about quantities. Anticipated Student Preconceptions/Misconceptions: Students may consider n = 6 as a solution. They need to multiply or divide both sides of the equation by -1 to get n isolated. When using the distributive property, students are often confused about the signs of the multiplied addends. Lesson Sequence: Introduce the unit: You may recall that you have already studied how to read, write, and evaluate expressions and equations. You also used equations to solve real world math problems. In order to assess students background knowledge, ask students: What is the difference between and expression and an equation? Solicit student responses. Remind them they also studied Properties of Operations and ask the students what they remember about the properties. Today we are beginning a new unit in which you will continue to analyze and solve linear equations. You will also learn how to solve pairs of linear equations. Today s lesson will focus on writing and solving linear equations to model real-life and mathematical situations (SMP.4 Model with Mathematics) Opening: Entry ticket: Write a word problem that can be modeled by this equation: Draft 8/ 2013 Page 9 of 137

10 x + (x 4) = 40 Students may need to look back at previous problems. For students who find this very challenging, give a general context: Someone has x of something, and someone else has 4 less of the same thing. Together, they have 40. Share several problems. If possible, choose some problems that do not fit the equation. Ask the class to modify the word problem to reflect the equation. Pose the following problem: If a number is doubled, and 8 is added, the result is 20. Students are used to reading from left to right, and in this example, they will be able to start on the left. o If a number is doubled (multiply 2 x n) o and 8 is added (2n + 8) o the result is 20 (2n + 8 = 20) Sometimes a verbal expression needs to be interpreted in a different order: The result is 27 when you subtract 12 from a number. n 12 = 27 or 27 = n 12 Double a number which results in three plus five times the number. 2x = 3 + 5x Draft 8/ 2013 Page 10 of 137

11 Create a wall chart where students can add language that indicates mathematical relationships or operations. Students should have a blank copy in their folders or notebooks, and as the unit progresses, they can add to this page. Sample Mathematical Language Chart Multiplication Division twice as many x2 half as many 2 tripling a number x3 O O O O O O O O Addition the sum of..add together Subtraction 6 fewer than -6 O O O O O O O O O O O O O O O O O O Students practice translating verbal expressions into equations in one variable by completing a short problem set. Classwork: Problem Set Students solve their equations. They then pick one or two problems to discuss with a partner, explaining their solution strategies. As students listen to the solutions of their peers, they decide whether the solutions make sense and ask questions to better understand and evaluate the work (SMP.3 Construct viable arguments and critique the reasoning of others). Draft 8/ 2013 Page 11 of 137

12 The teacher may link some of these strategies to Properties of Operations. Note: Students may memorize the names of these properties, but unless they have a strong conceptual understanding, they will not be able to apply them when problem-solving. Although this is not the focus of this lesson, whenever possible, the properties should be referenced in the equation-solving process. x + 3 = 17 Use the subtraction property of equality to subtract 3 from both sides of the equation. x = 17 3 x = 14 Homework: Students translate six verbal expressions into equations; they then write three additional word problems to match given equations. Verbal Expressions Equations Homework (***See Optional Enrichment) DAY 2 Explicitly connect the lesson to the work from Day 1. Yesterday we worked on writing a word problem that could be modeled by a specific equation. You also used the Properties of Operations to solve equations. Today you will work with writing equations to model problem solving situations. Pose the following problem: In an election, Candidate A received half the votes of Candidate B. Candidate C received 1000 votes less than Candidate B. All three candidates combined got 3,000 votes. How many votes did each candidate get? Step 1: Students write the equation. B + (B/2) + (B-1000) = 3000 Step 2: Students simplify. B + B/2 + B 1000 = B + B/2 = (2B + B/2) = 2(4000) 4B + B = B = 8000 Draft 8/ 2013 Page 12 of 137

13 B = 1600 Candidate A: 800 votes Candidate B: 1600 votes Candidate C: 600 votes Step 3: Check Does the solution work in the original equation? Analyze the equation-solving process (SMP.3 Construct viable arguments and critique the reasoning of others). o What steps did you take? Why? Remind students that in order to solve a linear equation, the variable must be isolated on one side of the equation (it does not matter which side). Like terms must be combined, and simplified if possible. Students apply what they have learned about solving equations by completing a problem set of four problems (SMP.4 Model with mathematics). Classwork: Solving Equations They are required to: o Write an equation to match a story problem. o Use properties of equality and properties of operations to isolate the variable and solve the problem (SMP.7 Look for and make use of structure) o Write down the steps that they took to solve the problem. Teacher Note: Working across the equal sign: Students have had ample opportunity to manipulate expressions, but they may not be used to working across the equal sign. Take time to look at this problem: 12n = 144 Most students will easily be able to solve this by relying on a known math fact 12 x 12. However, they may not fully understand how the equation can be manipulated to arrive at the solution. Students need to justify the solution as coming from the step before. Ask students: What happened to the 12n? How does n = 12 relate to 12n = 144? For many students, models (such as a balance scale) can help them to visualize concepts of equality. Draft 8/ 2013 Page 13 of 137

14 Some balance beam problems can be found here: Students generate a reference chart of solution strategies based on these and prior problems. SITUATION STRATEGY EXAMPLE 7n + 5n = 144 Like terms on the same side of the equation Combine like terms 12n = 144 Like terms on opposite sides of the equation Fractional coefficients Collect all like terms on one side (doesn t matter which side) Multiply by reciprocal Negative variable Multiply/divide by -1 Distributive property Like terms and constants on the same side of the equation Multi-step equation Multiply each addend in the sum Use the commutative and associative properties to rearrange/combine like terms Work backward to isolate the variable, but simplify first 2n + 4 = 3n 7 11 = n 2/5 n = 20 5 (2/5n) = 5(20) -n = 10 n = -10 7(a + 6) = 112 7a + 42 = 112 6a 8 3a + 7 = 11 (6a 3a) + (-8 + 7) = 11 3a + -1 = 11 3a + 4 = -7a a + 4 = 7a +2 10a = -2 a = -1/5 Optional Enrichment Sheet: Working Across the Equal Sign: Students solve problems that demonstrate understanding of the properties of equality and manipulation of equations. Draft 8/ 2013 Page 14 of 137

15 Closing: On the chart, put a green sticker on the strategy that is easiest for you. Put a red sticker on the strategy that you find the most difficult. Summarize the day s lesson by asking students to reflect upon what was learned, and sharing either with the class or a partner. Preview outcomes for the next lesson: Students will be able to identify if a problem has no solution, one solution, or an infinite number of solutions. Draft 8/ 2013 Page 15 of 137

16 Lesson 1 Classwork: Problem Set: Verbal Expressions to Equations Name: Class: Date: A. Write the following verbal expressions as equations: 1. The sum of a number n and 12 is The quotient of 15 divided by a number x is equal to The product of 5 and a number n is times a number is greater than or equal to A distance of 55 meters that is 20 meters short than x meters. 6. A bill of $96 for n baseball caps, each costing d dollars. 7. A weight of 115 pounds that is 40 pounds heavier than p pounds. 8. Doubling d dollars is equal to $228. B. Solve your equations: This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 16 of 137

17 Lesson 1 Classwork: Problem Set: Verbal Expressions to Equations ANSWER KEY A. Write the following verbal expressions as equations: 1. The sum of a number n and 12 is 16. n + 12 = 16, n = 4 2. The quotient of 15 divided by a number x is equal to 3. 15/x = 3, x = 5 3. The product of 5 and a number n is 25. 5n = 25, n = times a number is greater than or equal to x 50, x 5 5. A distance of 55 meters that is 20 meters short than x meters. 55 = x 20, x = A bill of $96 for n baseball caps, each costing d dollars. nd = 96 or 96/n = d 7. A weight of 115 pounds that is 40 pounds heavier than p pounds. p + 40 = Doubling d dollars is equal to $228. 2d = 228, d = $114 B. Solve your equations. See above. This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 17 of 137

18 Lesson 1 Homework: Verbal Expressions to Equations Name: Class: Date: Write and solve equations for the following verbal expressions: 1. A number decreased by 75 is equal to A number minus one-half is equal to three-fourths. 3. The sum of 32 and 88 and a number is equal to A number increased by 22 is Twice a number is less than One and one-fourth times a number is 15. Write a story to go with each equation. The first one is done for you. 1. 2(25) + n = 90 Sandy raked leaves for two days and made $25 each day. How much more does she need to make to earn $90? = y s + s = p = 104 This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 18 of 137

19 Lesson 1 Homework: Verbal Expressions to Equations ANSWER KEY Write and solve equations for the following verbal expressions: 1. A number decreased by 75 is equal to -25. n 75 = - 25 (Solution: 50) 2. A number minus one-half is equal to three-fourths. n = 3 4 (Solution: 5 4 or 11 4 ) 3. The sum of 32 and 88 and a number is equal to n = 156 (Solution: 36) 4. A number increased by 22 is -93. n + 22 = - 93 (Solution: -115) 5. Twice a number is less than 56. 2n < 56 (Solution: n < 28) 6. One and one-fourth times a number is n = 15 or 5 n = 15 (Solution: 12) 4 4 Write a story to go with each equation. The first one is done for you. 1. 2(25) + n = 90 Sandy raked leaves for two days and made $25 each day. How much more does she need to make to earn $90? = y 2 3. s + s = p = 104 This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 19 of 137

20 Lesson 1 Classwork Problem Set: Solving Equations Name: Class: Date: 1. We saved $50 to buy an anniversary gift for our parents. You saved y dollars. I saved $ How much money did you save? Equation: Solution Steps What I did in this step 2. A car gets 28 miles per gallon. If the car was driven for 250 miles, how many gallons of gas were used? Equation: Solution Steps What I did in this step This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 20 of 137

21 Lesson 1 Classwork Problem Set: Solving Equations 3. Mrs. O Neill saved some money last summer. She spent half of it on some new clothes for her children. She took the remainder, and spent one-third of that on a new tire for her car. She had $200 left. What was the amount of money she saved last summer? Equation: Solution Steps What I did in this step 4. John owns three times as many video games as Mitch. Mitch owns twice as many video games as Sam. If the three boys own 72 video games altogether, how many video games does John own? Equation: Solution Steps What I did in this step This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 21 of 137

22 Lesson 1 Homework: Working Across the Equal Sign Name: Class: Date: Solve each equation. Put an X to show which Property (or Properties) of Equality you used to solve the equation. Equation Solution Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality 1. p 3 = = r 3 3. k 2 = q = 6 5. m + 2 = = -3u 7. 4 = v 3 8. j 4 = 4 9. s + 5 = = v g 1 = = 2 + 2d = 8 2k This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 22 of 137

23 = 2h + 2 Equation Solution Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality = 8x = -m h = w = = d r + 12 = 20 This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 23 of 137

24 Lesson 1 Homework: Working Across the Equal Sign ANSWER KEY Solve each equation. Put an X to show which Property (or Properties) of Equality you used to solve the equation. Equation Solution Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality 1. p = 1 p = 3 X = r 3 r = 9 X 3. k = 3 k = 6 X q = 6 q = 3 X 5. m + 2 = 6 m = 4 X 6. 3 = -3u u = -1 X 7. 4 = v 3 v = 12 X 8. j = 4 j = 16 X 4 9. s + 5 = 17 s = 12 X = v-2 v = 10 X 11. 2g 1 = = 2 + 2d = 8 2k g = 5 X X d = 3 X X k = 2 X X This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 24 of 137

25 Lesson 2 Homework: Working Across the Equal Sign ANSWER KEY = 2h + 2 h = 4 X X Equation Solution Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality = 8x = -m h = 5 x = 1 X X m = 6 X X h = 4 X X 18. w + 8 = 10 w = 6 X X = d 3 1 d = 6 X X 20. 4r + 12 = 20 r = 2 X X This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 25 of 137

26 Brief Overview of Lesson: Lesson 2: How Many Solutions? Students will categorize linear equations in one variable into one of the following three categories: 1. One solution: the equation is sometimes true; it is true for one specific value of x. 2. No solution: the equation can never be true; there are no values of x that will make the equation true. 3. Infinite solutions: the equation is always true; no matter what value is assigned to x, the equation will be true. As you plan, consider the variability of learners in your class and make adaptations as necessary. Prior Knowledge Required: 8.EE.7 Solve linear equations in one variable. 8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Estimated Time (minutes): Two 50 minute lessons Resources for Lesson Formative Assessment: Shopping Scenarios Directions for Always, Sometimes, Never Activity Solution for Categorizing Equations as Always True, Sometimes True, Never True Ticket to Leave Homework: Working with Linear Equations Chart paper, sticky notes, internet access and projector with speakers Sorting and Classifying Equations Overview video. (9:50) Sorting and Classifying Equations: Class Discussion video. (8:24) Draft 8/ 2013 Page 26 of 137

27 Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations Content Area/Course: Grade 8 Mathematics Lesson # 2: How Many Solutions? Time (minutes): Two 50 minute lessons By the end of this lesson students will know and be able to: Identify the different forms of solutions and connect the form to the number of solutions (one solution: x=a; infinite solution: a=a; no solution: a = b) Produce examples of linear equations that represent one solution, infinite solutions, and no solutions. Estimate solutions to simple cases by inspection of equations (and/or graphs). Essential Question(s) addressed in this lesson: How can equations be used to represent real-world and mathematical situations? Targeted Academic Language: solution, identity, infinite Standard(s)/Unit Goal(s) to be addressed in this lesson: 8.EE.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x=a, a=a, or the given results (where a and b are different numbers). SMP.3 Construct viable arguments and critique the reasoning of others. SMP.7 Look for and make use of structure. Teacher Content and Pedagogy After working in pairs or small groups to make decisions regarding assigned equations, students will be asked to critique the work of others (SMP.3 Construct viable arguments and critique the reasoning of others). Collaboration and opportunities to share work and explanations are an important part of this lesson. Students should try to justify their decisions mathematically rather than I tried some other numbers and they didn t work. Draft 8/ 2013 Page 27 of 137

28 Use this Math Talk protocol during problem-solving activities: Think about the problem what strategies could you use? How do the problems and strategies compare to work you have done previously? Share your ideas with a partner. Work on your own to rethink the problem and arrive at an answer. Share your solution with a partner. Share with entire class. Notes on One, Infinite, or No Solutions Linear equations in one variable can have one solution, infinite solutions, or no solutions. When the equation has one solution, the variable has one value that makes the equation true, and the equation is sometimes true. Example: 10 3x = 4. The only value for x that makes this equation true is 2. When the equation has infinite solutions, the equation is true for all real numbers, and the equation is always true. Example : 3x + 6 = 3 (x + 2). When the distributive property is applied correctly, the expressions on each side of the equal sign are equivalent; therefore, the value for the two sides of the equation will be the same regardless of which value is chosen for x. Example: 2x + 3x + 9 = 5x + 9 When like terms are combined, the expressions on each side of the equal sign will be equivalent. When an equation has no solution, there is no value that could possibly make the equation true. The equation is never true. Example: 2x + 5 = 2 (x + 5) Draft 8/ 2013 Page 28 of 137

29 The two expressions are not equivalent. The equation simplifies to 5 = 10.Because all of these equations are linear, there will only be ONE solution when the equation is sometimes true. It is important for students to explore this. As equations become more complex (exponents), equations that are sometimes true will sometimes have more than one solution. Key mathematical terms can be reinforced during this discussion: solution identity infinite Notes for Always, Sometimes, or Never True Activity The use of different color markers on the poster (a different color for each member of the group) will allow the teacher to monitor who is completing the work. As students are creating their posters in groups, the teacher observes and encourages student thinking via effective questioning: How do you know for sure? What are you thinking? Sam put the equation in this column. Nick, do you think it should go there? Why/why not? (SMP 3) Can you convince me that it would be true for EVERY number? (SMP 3) How could you change this equation, so it would belong in a different column? Can you create a new equation for each column? What do you notice about the equations that are in this column? Anticipated Student Preconceptions/Misconceptions Errors with application of properties. Example: assuming that subtraction is commutative (5 x = x - 5) Incorrect use of the equal sign. Example: student writes 5 x = 6 = x = 1. Finding one solution and missing other solutions; not continuing to find other values or not finding a value that does not work. Belief that an equation is not true because the expression on each side of the equal signs looks different. Failure to fully simplify expressions. Lesson Sequence Draft 8/ 2013 Page 29 of 137

30 DAY 1 Opening: Explicitly connect the lesson to the previous lesson. Remind students that equations can be solved in a variety of ways. Reference the chart that the class made the day before. Activator: Present students with the three shopping scenarios in the Shopping Scenarios handout. Students work with a partner to interpret the equations provided to determine how many boxes of cookies each person bought. Do a quick share-out following this introductory activity. What do students notice? (The Shopping Scenarios problems have one solution, no solution, or infinite solutions. Students may be confused by equations that are not one-solution equations.) Students begin to look at the structure of equations to draw conclusions about the number of solutions. (SMP.7) Display the following equation: 3x + 2 = 11 Ask: Can you think of a value for x that makes this equation FALSE? Display the equation again: 3x + 2 = 11 Ask: Can you think of a value for x that makes this equation TRUE? When a value is found, challenge students to try to find other values for x that will make the equation true. Ask: Would we describe this equation as always true, never true, or sometimes true? When is it true? (x = 3) Are there any other values for x that make it true? How do you know? Repeat the same process with the following equations: Draft 8/ 2013 Page 30 of 137

31 x + 7 = 7 + x (always true for all values of x ) 5 x = 2 x + 20 (sometimes true; x = -5) x 3 = x + 2 (never true) Students are encouraged to look at the structure of the equations in order to make generalizations about the solutions (SMP.7). Include the correct mathematical label : o sometimes true = one solution o never true = no solution o always true = infinite solutions Post each equation with its solution. Encourage multiple students to contribute. Elicit a variety of answers, encouraging students to think outside the box. Discuss the reasons for their choices. Record exemplary responses. Post these, so students can reference them at a later date. IF necessary, have students support their answers with calculations. Always, Sometimes, or Never True small group activity: o o o o Watch video: Sorting and Classifying Equations Overview video (9:50) at (Enter the title in the search field.) Explain to students that they will be completing a similar activity. Divide students into groups of three. Distribute the set of equations, chart paper, markers (a different color for each student in the group), and directions. Review the three possible outcomes. Draft 8/ 2013 Page 31 of 137

32 ALWAYS TRUE or INFINITE SOLUTIONS NEVER TRUE or NO SOLUTION SOMETIMES TRUE or ONE SOLUTION means The equation is true for any value of x. means There are no values of x that make the equation true. means There is at least one value of x that makes the equation true. To prove that an equation is SOMETIMES TRUE, you need two examples: one value for x that makes the equation true, and one that makes it false. Review directions: Directions for Always, Sometimes, Never True. Students spend the remainder of the class period completing the poster. Lesson Closing- have students reflect upon the lesson and share out either with a partner or in a whole class discussion. DAY 2 Introduce the lesson and explicitly connect it to the previous lesson. Share posters Conduct a gallery walk, so students can critique each others work. Display charts in different areas of the room (hang on walls or lay on tables/desks). Before beginning, distribute sticky notes to each student. Provide the following directions: If you disagree with where an equation has been placed, write three things on a sticky note: 1) Why you disagree. 2) Which column the equation belongs in. Draft 8/ 2013 Page 32 of 137

33 3) Why you think the equation belongs there. Place the sticky note on the chart. This task requires students to critique the reasoning of others (SMP.3). If you see a comment that has already been written and you agree, put a check mark on the sticky note. If you see an explanation that you particularly like or something you didn t think of, leave a note telling the group what you liked about it. (SMP.3) Students walk silently around the room and review the work of others. Set a limit regarding the number of students looking at each poster. Or, students can be divided equally among the charts and given a time limit before moving on to the next poster. It is not necessary to have all students critique all posters; two are sufficient. Closing/Summarizer o o o o o o o o Conduct a whole class discussion. Highlight exemplary explanations. Give me an equation that is (ask for one of each type) Why did you put that equation in that column? Can anyone add to this explanation? What did you learn by looking at other students work? Which equations were the most difficult to categorize? Why? Is there an easy way to tell which column an equation belongs in just by looking at it? Watch Sorting and Classifying Equations: Class Discussion (8:24) Ticket to Leave: Students use their understanding of the structure of equations to classify equations as always, sometimes, or never true, and create an example of each type of equation on their own. (SMP.7) Draft 8/ 2013 Page 33 of 137

34 Homework: Working with Linear Equations Formative Assessment: Three Shopping Scenarios check student understanding of the meaning of the terms and variables in linear equations. Preview outcomes for the next lesson: Students will be able to solve simultaneous linear equations through the use of graphing. Draft 8/ 2013 Page 34 of 137

35 Lesson 2 Shopping Scenarios Formative Assessment: What Does the Equation Tell Me? Name: Class: Date: Read each shopping scenario. Review the equations that have been written to represent them. What can you say about the number of packages of cookies each person bought? Description Equation What can you say about the number of packages of cookies each person bought? A. B. Noelle went shopping and spent $19. She bought one container of ice cream and some packages of cookies. Last week, on Monday, Anna bought 3 packages of cookies and spent $5 more on other items On Tuesday, she bought the exact same thing. This week, Anna bought 6 packages of cookies. She also spent $10 more on other items. She said she spent the same amount of money last week as this week. 5c + 4 = 19 2(3c+ 5) = 6c + 10 C. Last week, Jared bought 8 packages of cookies and spent $8 more on some cheese spread. This week, on Monday, Tuesday, Wednesday, and Thursday, Jared bought two packages of cookies each day. 8c + 8 = 4(2c) This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 35 of 137

36 He said he spent the same amount of money last week as this week. This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 36 of 137

37 Name: Class: Date: Lesson 2: Directions for Always, Sometimes, or Never True Group Activity 1. Each person in your group selects a marker of a different color. This is the only color that you may write with. 2. Divide your chart paper into three equal columns. 3. Label the columns: Always True Infinite Solutions Sometimes True One Solution Never True No Solution 4. Select an equation. Try out different values for x. Discuss with your group. Record the equation in the column where it belongs. 5. In writing, next to each equation, explain why you chose to place the equation where you did. If you think the equation is sometimes true, give values of x for which it is true and for which it is false. If you think the equation is always true or never true, explain how you can be sure this is the case. Guidelines: 1. Take turns recording the equation and writing the explanation. 2. Be sure you agree with your partners. If you are not sure, challenge the explanation. 3. Describe your thinking in your own words. This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 37 of 137

38 Lesson 2: Equations for Always, Sometimes, Never Activity Name: Class: Date: Decide whether each equation is always true, sometimes true, or never true. Record each equation in your ALWAYS TRUE, SOMETIMES TRUE, NEVER TRUE chart. Remember to record your reasoning and some examples next to each equation on the chart x = x x = x x + 5 = x x 5 = 2x 5. x 2 = x 6. 2(x + 1) = 2x x = x 8. 7x + 14 = 7(x + 2) x = 5 2x+4 2 = x x 5 = 5(x + 1) 12. 4x = x x = x + x 5 3 x + 8 = 1 (3x + 16) 4 4 This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 38 of 137

39 Lesson 2: Equations for Always, Sometimes, Never Activity ANSWER KEY Name: Class: Date: ANSWER KEY Always True Infinite Solutions 3 + x = x + 3 Sometimes True One Solution 2 x = x 2 (Solution: x = 2) Never True No Solution x + 5 = x 3 7x + 14 = 7(x + 2) 3x 5 = 2x (Solution: x = 5) 2(x + 1) = 2x + 1 2x+4 2 = x + 2 x 2 = x (Solution: x = 0) 5x 5 = 5(x + 1) 6x = x 1 x x = x + x (Solution: x = 0) 3 x + 8 = 1 (3x + 16) x = 5 (Solution: x = 1) 4x = 4 (Solution: x = 1) This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 39 of 137

40 Lesson 2: Ticket to Leave Name: Class: Date: For which values are the following equations true? Equation For what values of x is the equation true? Number of Solutions (one, none, infinite) Ex. 6x + 3 =15 Only true when x = 2. one none infinite x = 15 one none infinite 2. x - 3 = 3 - x one none infinite 3. 3 (x + 4) = 3x + 4 one none infinite 4. x = 6 one none infinite (x + 3) = 2x + 6 one none infinite Create three examples of your own: 6. One solution 7. None 8. Infinite solutions This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 40 of 137

41 Lesson 2: Ticket to Leave ANSWER KEY Name: Class: Date: For which values are the following equations true? Equation For what values of x is the equation true? Number of Solutions (one, none, infinite) Ex. 6x + 3 =15 Only true when x = 2. one none infinite x = 15 Only true when x = -3. one none infinite 2. x - 3 = 3 - x Only true when x = 0. one none infinite 3. 3 (x + 4) = 3x + 4 Never true one none infinite 4. x = 6 Only true when x = 12. one none infinite (x + 3) = 2x + 6 True for all real numbers. one none infinite Create three examples of your own: 6. One solution 7. None 8. Infinite solutions This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 41 of 137

42 Lesson 2 Homework: Working with Linear Equations Name: Class: Date: x x x x y y y y A B C D 1. Which of these tables of values satisfy the equation y = 2x + 3? 2. Explain how you know. 3. Complete the tables below for the lines y = 2x + 3 and x = 1-2y. Then use the values to graph the lines. y = 2x + 3 x = 1-2y x -2 0 x 0 5 y 5 y 0 This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 42 of 137

43 4. Do the equations y = 2x + 3 and x = 1-2y have one common solution, no common solutions, or infinitely many common solutions? Explain how you know. over Lesson 2 Homework: Working with Linear Equations 5. Draw a straight line on the graph that has no common solutions with the line y = 2x + 3. What is the equation of your new line? Explain your answer. This work is licensed by the MA Department of Elementary & Secondary Education under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit Draft 8/ 2013 Page 43 of 137