8 th Grade Mathematics

Transcription

1 8 th Grade Mathematics Unit #2 : Understanding and Representing Linear Relationships Pacing: 40 Days Unit Overview In this unit, students will build upon informal knowledge of one-variable linear equations, to solve equations, which include cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation. Students will use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Students will solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students will solve systems of equations using graphing, substitution and elimination. In addition, students will be able to write and solve systems of equations from word problems. Students will use their understandings of linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems. Students should engage in mathematical tasks that provide an opportunity to connect this content to the Mathematical Practices. Prerequisite Skills Vocabulary Mathematical Practices Computation with whole numbers and decimals, including application of order of operations Operations with fractions Plotting points in four quadrant coordinate plane Understanding of independent and dependent variables Characteristics of a proportional relationships Operations with integers Truncate Equations Rational Coefficient Multiplicative Inverse Equivalence Properties of Equality Two-Step Equation Null Set Identity Constant of Proportionality Constant of Variation Constant Rate of Change Direct Variation Linear Relationships Point-Slope Form Rise Run Slope Slope-Intercept Form Standard Form Substitution Systems of Equations x-intercept y-intercept Ordered Pair Intersecting Lines Parallel Lines MP.1: Make sense of problems and persevere in solving them MP.2: Reason abstractly and quantitatively MP.3: Construct viable arguments and critique the reasoning of others MP.4: Model with mathematics MP.5: Use appropriate tools strategically MP.6: Attend to precision MP.7: Look for and make use of structure MP.8: Look for and express regularity in repeated reasoning

2 Common Core State Standards Progression of Skills Major Standards (70%) Supporting Standards (20%) 8.SP.2 Know that straight lines model relationships. 8.SP.3 Use the equation of a linear model to solve problems. 8.EE.5: Graph Proportional Relationships and Interpret the Unit Rate as the Slope of the Graph 8.EE.6 Use similar triangles to explain why the slope m is the same. 8.EE.7: Solve Linear Equations in One Variable 8.EE.8: Analyze and Solve Pairs of Simultaneous Linear Equations According to the PARCC Model Content Framework, Students build on previous work with proportional relationships, unit rates, and graphing to connect these ideas and understand that the points (x, y) on a non-vertical line are the solutions of the equation y = mx + b, where m is the slope of the line as well as the unit rate of a proportional relationship (in the case b = 0). 8.EE.7 Students have been working informally with one-variable linear equations since as early as kindergarten. This important line of development culminates in grade 8 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational numbers. 7 th Grade 8 th Grade 9 th Grade 7.EE.2: Equivalent Expressions Understand 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. For example, a a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05." 7.EE.4: Solve Real-life Mathematical Problems Using Expressions and Equations 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. N/A 8.EE.7a: Linear Equations and Number of Solutions 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 a = b results (where a and b are different numbers). 8.EE.7b: Solve Multi-Step Linear Equations Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.EE.8c: Solve Contextual Problems with Systems of Linear Equations 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. HSA.CED.A.1: Create equations that describe numbers or relationships Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. HSA.REI.C.5: Solve systems of equations Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. HSA.REI.C.7: Solve systems of equations Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x 2 + y 2 = 3. 2 Page

3 Big Ideas What does the graph of a proportional relationship look like? How does the graph compare to the slope of the related line in a proportional relationship? How can the use of tables, graphs, and equations represent linear relationships and how can each be analyzed to predict rates of change? How are changes in linear relationships affected and shown in a table, graph, or equation? Linear equations may have one solution, an infinite amount of solutions, or no solution and it is important to know the constraints for each type of solution set. There are several methods for solving systems of equations such as numerically, graphically, or algebraically using substitution or elimination Know/Understand Linear equations with one variable may have one solution, infinitely many solutions or no solutions dependent on what the equation looks like when simplified where it is x = a or a = b Solving linear equations in one variable with rational number coefficients may necessitate expanding the expression using the distributive property and collecting like terms Linear equations may be solved using inverse operations or by graphing the equation and finding the value of x based on the value of y Linear models can be represented with a linear equation that represents the fit line of a bivariate measurement data on a scatter plot Slope and intercept of the equation of a linear model provides information about the two quantities reflected in bivariate measurement data Equation of a linear model can be used to solve problems related to bivariate measurement data Two-way table organizes and summarizes data between two different variables using the columns and rows to display the association between the two variables To make a two way table, data on each category must be collected Bivariate data can be presented in a two way table. Patterns of associate in bivariate data can be identified and quantified using a two way table. Students understand that relative frequencies for rows or columns of the two way table can be used to describe possible associations between the two variables Students Will Be Able To Expressing the solution to a problem involving very small or very large quantities in scientific notation with the appropriate size measurement Explaining the appropriate size measurement and value of a quantity expressed in scientific notation and generated by technology when performing operations between very large or very small numbers Identifying whether a linear equation has one solution, infinitely many solutions, or no solutions by simplifying the equation to find x = a or a = b. Providing examples of a linear equation in one variable with one solution, infinitely many solutions, or no solutions and explain why. Students use the distributive property to expand expressions in a linear equation and collect like terms across the linear equation Solving one variable linear equations using inverse operations, applying the distributive property or collecting like terms or by graphing the linear equations on a coordinate plane or representing it with pictures Writing the equation for a linear model to reflect bivariate measurement data on a scatter plot Interpreting what the slope and intercept of the equation of a linear model means in the context of the bivariate measurement data. Using the equation of a linear model to solve problems about bivariate measurement data Constructing a two-way table summarizing data on two variables or bivariate data Interpreting a two-way table to understand possible patterns of association between the two variables Calculating the relative frequencies for rows or columns of the two-way table Describing possible patterns of association between two variables using the calculated relative frequencies 3 Page

4 Lesson Learning Objective SWBAT 1 Use properties of equality and inverse operations to solve one-step equations with rational coefficients Unit Sequence Sample Assessment Item from Exit Ticket In the eighth grade, 417 students voted for the new mascot to be a bearcat. This was! of! the total number of students in the eighth grade. How many students are in the eighth grade? Instructional Resources Chapter 2 Lesson Use properties of equality and inverse operations to solve two-step equations with rational coefficients 1) What is the solution to this equation? ¾ y -1/2 = 5/8 1. y = 9/20 2. y = 4/5 Chapter 2 Lesson y = 27/32 4. y = 3/2 4 Page

5 3 Represent real world scenarios by writing one and twostep equations with one variable; solve for the unknown Examples for mini lesson or guided practice: Represent each scenario by writing an equation, then solve for the unknown: 1) Miriam read a book with an unknown number of pages. The first week she read five less than!! of the pages. The second week she read 171 pages and finished the book. Write an equation that represents the total number of pages in the book. 2) Subtract seven more than twice a number from the square of one-third of the number to get zero. Write and solve an equation for the following scenario: Bruce bought two books. One book costs four dollars more than three times the other. Together, the two books cost him $72. Find the cost of each book. Chapter 2 Lesson 3 _files/collected_learning/middl e_school/algebra/solving_equa tions.pdf 4 Flex Day (Instruction Based on Data) Recommended Resources: Chapter 2 Problem-Solving Investigation (Pages ) Chapter 2 Mid-Chapter Check (Page 140) Chapter 2 Inquiry Lab (Pages ) 5 Solve equations with variables on both sides of the equal sign 6 Angel transformed the following equation from 6x + 4 x = 2(x + 1) to 10 = 2(x + 1). He then stated that the solution to the equation is x = 4. Is he correct? Explain. Chapter 2 Lesson 4 essons/ essons/152 essons/151 5 Page

6 7 Solve multi-step equations Solve real world problems by identifying the solution or range of solutions to a linear equation in one variable 8 Analyze and represent contextual situations with equations, identify whether there is one, none, or many solutions, and then solve to prove conjectures about the solutions. Sample Problem for Mini- Lesson or Guided Practice: Two linear equations are given below. Equation 1: 5x = 8x 3(x + 2) Equation 2: 4x = 2(x+5) 1. How many solutions does equation 1 have? Show or explain how you found your answer. 2. How many solutions does equation 2 have? Show or explain how you found your answer. 1) Which statement regarding the number of solutions for the linear equation shown below is true? 4(3X+8) 9 = 2(6X 8) + 39 a. There are infinitely many solutions. b. There are exactly two solutions. c. There is exactly one solution. d. There is no solution. 2) Give an example of a linear equation that has infinitely many solutions. Explain how you know your equation has infinitely many solutions. Chapter 2 Lesson 5 om/activity/article/tictac-equations/ 9 Flex Day (Instruction Based on Data) Recommended Resources: 10 Explore proportional relationships related to average speed and constant speed in order to write a linear equation in two variables. Chapter 2 Review (Pages ) The table represents Rebeca s travel time. Based on the table, what is her constant speed? Explain your reasoning. x (hours) y (miles) Engage NY Module 4 Lesson 10 Resource for Remediation: Chapter 3, Lesson 1 6 Page

7 11 Evaluate numerical data to determine the rate of change. Plot points on a coordinate plane to represent rate of change problems. 12 Use a table to find solutions to a given linear equation and plot the solutions on a coordinate plane Vicky reads at a constant rate. She can read 5 pages in 9 minutes. We want to know how many pages, p, Vicky can read after t minutes. Write a linear equation in two variables that represents the number of pages Vicky reads in any given time interval. Find three solutions for the linear equation 4x 3y = 1, and plot the solutions as points on a coordinate plane. Engage NY Module 4 Lesson 11 Resource for Remediation: Chapter 3, Lesson 1 Engage NY Module 4 Lesson 12 x Linear equation: 4x 3y =1 y 13 Given a set of coordinate points or a line graphed on a coordinate plane, calculate the slope of the line. Sketch a line that has a negative slope. Explain how you know your slope is negative. Engage NY Module 4 Lesson 15 Resource for Remediation: Chapter 3, Lesson 2 14 Derive the formula for a direct variation. Write equations in y=mx form to represent mathematical and real world situations. What is the relationship among the unit rate, slope, and constant rate of change of a proportional linear relationship? Chapter 3, Lesson 3 7 Page

8 15 Derive the equation for a non-proportional linear relationship by using the slope formula. Write equations in slope-intercept form to represent mathematical and real world situations. The entrance fee to the carnival is $8 and each ride cost $2. The total cost y for a trip to the carnival for x number of rides can be represented by the equation y = 2x + 8. Interpret the slope and y-intercept. Chapter 3, Lesson 4 16 Use x- and y-intercepts to graph equations on a coordinate plane Monty has a total of $290 in ten dollar and five dollar bills. This can be represented by the function 10x + 5y = 290. Interpret the x- and y-intercepts. 17 Based on the information you are given, determine which form you must use to write a linear equation 18 Prove that any point on the graph of y = mx + b is on a line l and that any point on a line l is a point on the graph of y = mx + b. The equation of a line is y = -2x + 7. Write an equation in point-slope form for the same line. Explain the steps that you used. Graph the equation y =! x 10 using! the y-intercept and slope. Chapter 3, Lesson 5 Chapter 3, Lesson 6 Engage NY Module 4 Lesson 19 Graph the equation 5x 4y = 40 using intercepts. What can you conclude about the equations y =! x 10 and 5x 4y =! 40? 19 Compare two different proportional relationships represented by graphs, equations, and tables to determine which has a greater rate of change. Water flows out of Pipe A at a constant rate. Pipe A can fill 3 buckets of the same size in 14 minutes. Write a linear equation that represents the situation. The figure below represents the rate at which Pipe B can fill the same sized buckets. Write a linear equation that represents the situation for Pipe B. Engage NY Module 4 Lesson 22 8 Page

9 20 Deduce that: 1) two equations in the form of ax + by = c and a'x + b'y = c' graph as the same line when a'/a = b'/b = c'/c and at least one of a or b is nonzero. 2) the graph of a linear equation ax + by = c, where a, b, and c are constants and at least one of a or b is nonzero, is the line defined by the equation ax + by = c. Which pipe fills buckets faster? Explain. Given the equation 3x y =11, write another equation that will have the same graph. Explain why. 21 Flex Days (Instruction Based on Data) Recommended Resources: Solving Linear Equations with One Variable Baseball Jerseys 22 Engage NY Lesson 21 We Predict Chapter 3 Inquiry Lab (Pages ) Mid-Chapter Check (Page 220) 23 Write equations with various solution types and provide step-by-step solutions with justifications See Performance Task Engage NY Module 4 Lesson 23 Performance Task: Writing for a Math Website 24 Identify solutions of systems of linear equations in 2 variables * Note: Students must understand that simultaneous linear equations may be graphed on a coordinate plane using ordered pairs and/or linear equations and may be identified using a line in order to graph systems of linear equations on a coordinate pair using ordered pairs and/or the equations. Can you determine whether a system of equations has a solution by looking at the graph of the equations? Explain. Chapter 3 Lesson 7 Pgs essons/64 9 Page

10 25 Solve systems of linear equations using tables and graphs Graph a system of two linear equations in the grid below so that the system has a single solution of ( 2, 3). Chapter 3 Lesson 7 Pgs Solve systems of equations algebraically using substitution *Note: At this point, students should reach the conclusion that solutions to a system of linear equations may be found algebraically, graphically, or by reasoning about the equations. 28 Solve systems of two linear equations algebraically, graphically, and by reasoning about their possible intersection. Leroy has five times as many points as Jay. Jay has 20 fewer points than Leanne. Write a system of equations and find each player s points? The local swim center is making a special offer. They usually charge $7 per day to swim at the pool. This month swimmers can pay an enrollment fee of $30 and then the daily pass will only be $4 per day. (a) Suppose you do not take the special offer. Write an equation that represents the amount of money you would spend based on how many days you go to the pool if the passes were bought at full price. MyMath Chapter 3 Lesson 8 ns/ htm Chapter 3 Inquiry Lab (Pages ) essons/1017 (b) Write a second equation that represents the amount of money you would spend if you decided to take the special offer. 10 Page

11 29 Apply the method elimination in order to solve systems of equations 30 Explain the relationship between the number of solutions a system of linear equations has and the types of lines that represent these systems. *Note: Students will deduce that the systems of linear equations that have one solution or point of intersection are intersecting lines, those with infinite number of solutions are equivalent lines, and those with no solutions are parallel lines. 31 Write real world problems into systems of linear equations and solve using elimination and substitution methods What is the first step to solving this system of equation algebraically? Justify your reasoning. 4x + 5y = 13 3x + 3y = 9 Consider the equation 5x 2y = 3. If possible, write a second linear equation to create a system of equations that has: (a) Exactly 1 solution. (b) Exactly 2 solutions. (c) No solutions. (d) Infinitely many solutions. For each of these cases, how many such equations are possible? Small boxes contain DVDs, and large boxes contain one gaming machine. Three boxes of gaming machines and a box of DVDs weigh 48 pounds. Three boxes of gaming machines and five boxes of DVDs weigh 72 pounds. How much does each box weigh? Write and solve a system of equations. Engage NY Module 4 Lesson 28 my.org/math/algebra/sy stems-of-eq-andineq/fast-systems-ofequations/v/solvingsystems-of-equationsby-elimination What are the Coefficients? Classifying Solutions to Systems of Equations ns/ ns/1019 Engage NY Module 4 Lesson Page

12 32 Flex Day (Instruction Based on Data) Recommended Resources: Engage NY Lesson 30 Chapter 3 Review and Reflect (Pages ) 33 Make sense of real world problems and persevere in solving them *Teaching Tip: Provide a variety of structures for students to work within this week (i.e. Socratic discussions, homogenuous groups and hetereogenuous groups, etc; if the majority of the class is struggling, provide a gradual release over the course of the week so they are working more independently by the end of the week) Cara s Candles DVD Club Field Day Stacking Cups Free Throw Percentages How Much Did They Cost? Fencing Planning a Party Real World Problem Solving *Teaching Tip: Build in structures this week to maximize opportunities for students to construct mathematical arguments and critique the arguments of their peers (MP 3) /03/g8_03_05.pdf 39 Performance Task: Stained Glass Window *Note: create an opportunity for students to discuss and debate their strategies and solution at the end of the lesson *This does not need to be treated as an assessment students may work independently, collaboratively, and/or receive some scaffolding in small groups as needed 40 MCLASS Beacon End of Unit Assessment Appendix B *This assessment will be administered online 12 Page

13 Standard Appendix A: Unpacked Standards Guide Source: Public Schools of North Carolina NCDPI Collaborative Workspace Unpacking What do these standards mean a child will know and be able to do? 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distancetime equation to determine which of two moving objects has greater speed. 8.EE.5 Students build on their work with unit rates from 6 th grade and proportional relationships in 7 th grade to compare graphs, tables and equations of proportional relationships. Students identify the unit rate (or slope) in graphs, tables and equations to compare two proportional relationships represented in different ways. Example 1: Compare the scenarios to determine which represents a greater speed. Explain your choice including a written description of each scenario. Be sure to include the unit rates in your explanation. Scenario 1: Scenario 2: y = 55x x is time in hours y is distance in miles Solution: Scenario 1 has the greater speed since the unit rate is 60 miles per hour. The graph shows this rate since 60 is the distance traveled in one hour. Scenario 2 has a unit rate of 55 miles per hour shown as the coefficient in the equation. Given an equation of a proportional relationship, students draw a graph of the relationship. Students recognize that the unit rate is the coefficient of x and that this value is also the slope of the line. 13 Page

14 8.EE.7 Solve linear equations in one variable. 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 a = b results (where a and b are different numbers). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.EE.7 Students solve one-variable equations including those with the variables being on both sides of the equals sign. Students recognize that the solution to the equation is the value(s) of the variable, which make a true equality when substituted back into the equation. Equations shall include rational numbers, distributive property and combining like terms. Example 1: Equations have one solution when the variables do not cancel out. For example, 10x 23 = 29 3x can be solved to x = 4. This means that when the value of x is 4, both sides will be equal. If each side of the equation were treated as a linear equation and graphed, the solution of the equation represents the coordinates of the point where the two lines would intersect. In this example, the ordered pair would be (4, 17) = = = 17 Example 2: Equations having no solution have variables that will cancel out and constants that are not equal. This means that there is not a value that can be substituted for x that will make the sides equal. -x + 7 6x = 19 7x Combine like terms -7x + 7 = 19 7x Add 7x to each side 7 19 This solution means that no matter what value is substituted for x the final result will never be equal to each other. If each side of the equation were treated as a linear equation and graphed, the lines would be parallel. Example 3: An equation with infinitely many solutions occurs when both sides of the equation are the same. Any value of x will produce a valid equation. For example the following equation, when simplified will give the same values on both sides. 1 3 (36a 6) = (4 24a) a + 3 = 3 18a 14 Page

15 If each side of the equation were treated as a linear equation and graphed, the graph would be the same line. 8.EE.8 Analyze and solve 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 because 3x + 2y cannot simultaneously be 5 and 6. a. 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 15 Page Example 4: Two more than a certain number is 15 less than twice the number. Find the number. Solution: n + 2 = 2n = n 8.EE.8 Systems of linear equations can also have one solution, infinitely many solutions or no solutions. Students will discover these cases as they graph systems of linear equations and solve them algebraically. Students graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations. Students recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions. By making connections between algebraic and graphical solutions and the context of the system of linear equations, students are able to make sense of their solutions. Students need opportunities to work with equations and context that include whole number and/or decimals/fractions. Students define variables and create a system of linear equations in two variables Example 1: 1. Plant A and Plant B are on different watering schedules. This affects their rate of growth. Compare the growth of the two plants to determine when their heights will be the same. Solution: Let W = number of weeks Let H = height of the plant after W weeks Plant A Plant B W H W H 0 4 (0, 4) 0 2 (0, 2) 1 6 (1, 6) 1 6 (1, 6) 2 8 (2, 8) 2 10 (2, 10) 3 10 (3, 10) 3 14 (3, 14)

16 through the second pair. 2. Based on the coordinates from the table, graph lines to represent each plant. Solution: Plant Plant A 3. Write an equation that represents the growth rate of Plant A and Plant B. Solution: Plant A H = 2W + 4 Plant B H = 4W At which week will the plants have the same height? Solution: 2W + 4 = 4W + 2 Set height of Plant A equal to height of Plant B 2W 2W + 4 = 4W 2W+ 2 Solve for W 4 = 2W = 2W = 2W = W After one week, the height of Plant A and Plant B are both 6 inches. Check: 2(1) + 4 = 4(1) = = 6 Given two equations in slope-intercept form (Example 1) or one equation in standard form and one equation 16 Page

17 in slope-intercept form, students use substitution to solve the system. Example 2: Solve: Victor is half as old as Maria. The sum of their ages is 54. How old is Victor? Solution: Let v = Victor s age v + m = 54 Let m = Maria s age v = ½ m ½ m + m = 54 Substitute ½m for v in the first equation 1½ m = 54 m = 36 If Maria is 36, then substitute 36 into v + m = 54 to find Victor s age of 18. Note: Students are not expected to change linear equations written in standard form to slope-intercept form or solve systems using elimination. For many real world contexts, equations may be written in standard form. Students are not expected to change the standard form to slope-intercept form. However, students may generate ordered pairs recognizing that the values of the ordered pairs would be solutions for the equation. For example, in the equation above, students could make a list of the possible ages of Victor and Maria that would add to 54. The graph of these ordered pairs would be a line with all the possible ages for Victor and Maria. Victor Maria Page