Unit 14: Systems of Equations and Inequalities. Learning Objectives 14.2

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1 Unit 14 Table of Contents Unit 14: Systems of Equations and Inequalities Learning Objectives 14.2 Instructor Notes The Mathematics of Systems of Equations and Inequalities Teaching Tips: Challenges and Approaches Additional Resources Instructor Overview Tutor Simulation: Getting the Mixture Right Instructor Overview Puzzle: Apples and Oranges Instructor Overview Project: Nutty Economics Common Core Standards Some rights reserved. See our complete Terms of Use. Monterey Institute for Technology and Education (MITE) 2012 To see these and all other available Instructor Resources, visit the NROC Network. 14.1

2 Unit 14 Learning Objectives Unit 14: Systems of Equations and Inequalities Lesson 1: Graphing Systems of Equations and Inequalities Topic 1: Graphing Systems of Linear Equations Learning Objectives Solve a system of linear equations by graphing. Determine whether a system of linear equations is consistent or inconsistent. Determine whether a system of linear equations is dependent or independent. Determine whether an ordered pair is a solution of a system of equations. Solve application problems by graphing a system of equations. Topic 2: Graphing Systems of Inequalities Learning Objectives Solve a system of linear inequalities by graphing. Determine whether an ordered pair is a solution of a system of inequalities. Solve application problems by graphing a system of inequalities. Lesson 2: Algebraic Methods to Solve Systems of Equations Topic 1: The Substitution Method Learning Objectives Solve a system of equations using the substitution method. Recognize systems of equations that have no solution or an infinite number of solutions. Solve application problems using the substitution method. Topic 2: The Elimination Method Learning Objectives Solve a system of equations when no multiplication is necessary to eliminate a variable. Solve a system of equations when multiplication is necessary to eliminate a variable. Recognize systems that have no solution or an infinite number of solutions. Solve application problems using the elimination method. 14.2

3 Lesson 3: Systems of Equations in Three or More Variables Topic 1: Solving Systems of Three Variables Learning Objectives Solve a system of equations when no multiplication is necessary to eliminate a variable. Solve a system of equations when multiplication is necessary to eliminate a variable. Solve application problems that require the use of this method. Recognize systems that have no solution or an infinite number of solutions. 14.3

4 Unit 14 Instructor Notes Unit 14: Systems of Equations and Inequalities Instructor Notes The Mathematics of Systems of Equations and Inequalities This unit extends the skills learned from plotting and analyzing individual equations and inequalities into working with linear systems. Students will learn three techniques for solving these systems graphing, substitution, and elimination. They ll also see how to use these methods to analyze application problems. Teaching Tips: Challenges and Approaches Students are very familiar with finding a single number or a range of numbers when they solve mathematical problems. In this unit, they'll be asked for solutions that are ordered pairs (x, y) instead. This will be a conceptual hurdle for many of them. In order to help students build an intuitive feel for the concept of a linear system, we start the unit by using graphing to explore systems. Just a glance at the graph of a system reveals if the lines cross or cover one another. We then move on to the substitution method, which many students will also be able to understand and use without much difficulty. The idea is fairly easy to grasp, and once the substitution is performed, all they have to do is follow the familiar steps for solving equations and inequalities. But students are likely to struggle with elimination (or addition method) as a means for solving linear systems. We suggest teaching this technique last, after graphing (easiest to understand), and substitution (most familiar mathematics). Graphing Systems Students learned how to graph single equations on the coordinate plane in the previous unit, so they'll have little trouble adding a second line. The sight of two (or more) equations on a graph conveys the idea of a linear system more clearly than words or symbols. It also helps students understand the three possible scenarios for the solutions of linear systems: 14.4

5 [From Lesson 1, Topic 1, Presentation] The vocabulary that describes these different scenarios uses familiar words consistent/inconsistent and dependent/independent in unfamiliar ways. This can be confusing, so you may need to spend extra time discussing what these terms mean mathematically. Once students are comfortable solving systems of equations with graphing, it is effective to move on to graphing systems of linear inequalities rather than introducing other methods for working with equations. This will strengthen their grasp of the systems concept. When graphing inequalities, use colors to help students see where the shaded areas overlap: 14.5

6 [From Lesson 1, Topic 2, Topic Text] Students tend to forget to draw dashed boundary lines when the inequality is less than or greater than. You can help them realize the importance of this distinction by having them find points in all four of the colored regions and on the two boundary lines of graphs like the one above, and checking those coordinates in the inequalities. Substitution Most students can easily look at the graph of a system and see if the lines cross or cover one another. It's then an easy step for them to find the solution if the problem is nice the ordered pair lies right on an intersection on the grid. But what happens if the solution isn t easy to read from the graph? This is the motivation behind knowing other ways to solve systems of equations. Solving systems of equations by substitution is probably the easiest algebraic method for students to understand. The idea is fairly easy to grasp, and once the actual substitution is performed, all they have to do is follow the familiar steps for solving equations and inequalities. Without a graph to look at, students may need to be reminded that sometimes lines don't cross and systems have no solution. Consider this example: 14.6

7 From Lesson 2, Topic 1, Topic Text] Many students get to -8 = 4 and think the solution is (-8, 4). They'll catch this mistake on their own and learn from it if you require that all ordered pair solutions be plugged into either equation to see if they work. You can also remind them of the possibility of an infinite number of solutions by asking them to solve the system above if the second equation is 10x 2y = -8 instead of 10x 2y = 4. Make sure that your students understand that when solving a system of equations algebraically instead of graphically, there are still three possibilities: one solution, no solution, and an infinite number of solutions. Elimination Students are very likely to struggle with elimination as a method for solving linear systems, but it's important they understand this method as it will be used in later mathematics, such as matrices, vectors, linear programming, and so forth. Students like the elimination method when adding two equations together automatically eliminates one of the variables. Work a number of these simple problems so students get comfortable with the method. Then show your students a system where neither of the variables cancels out. Here's an example: 14.7

8 From Lesson 2, Topic 2, Topic Text] Ask your students how to change one of the equations so that they could simply add the two equations together and one of the variables would be eliminated. Hopefully someone will suggest that if the y term in the second equation were -4y, the equations could be added and the y s would be eliminated. Try to get them to say that the second equation needs to be multiplied by -4 in order to eliminate the y terms. Once they've gotten the idea of rewriting one of the equations, ask how they would eliminate the x variable instead of the y. Understanding this will be crucial for solving harder systems of equations in the future. When solving by elimination there are two specific areas students will likely find difficult: choosing factors and keeping track of the changing equations. Choosing the appropriate factors to multiply the equations takes both intuition and practice. It is a good idea to present students with a number of systems with two equations and ask them how they would solve it using elimination. In the beginning it isn t necessary for them to even solve the system. They just need the practice of finding what to multiply the equations by. The other stumbling block, keeping track of what happens to equations as they are manipulated, can be reduced by encouraging students to label their work as they go. The following layout may be helpful: 14.8

9 Solve by elimination 2x3y 8 3x4y 46 Operations Equations Labels multiply equation 1 by -3 multiply equation 2 by 2 add new equations 2x3y 8 3x4y 46 6x 9y 24 6x8y 92 6x 9y 24 6x8y 92 equation 1 equation 2 new equation 1 new equation 2 simplify substitute value of y into original equation 1 solve for x write x and y as coordinates 17y 68 y 4 2x 3(4) 8 x 10 xy, 10,4 Solution to the System One common mistake that students make is that they multiply the variable terms by a factor but forget to multiply the constant term after the equals sign by that same factor. In order to catch this type of mistake, encourage your students to check their solution in both of the original equations. Three Equation Systems Students studying intermediate algebra need to know how to solve systems with three equations and three variables. The technique is the same as solving a system with two equations, except that it is more involved and students are more apt to make mistakes. Labeling the equations during this process is even more important here than it was with two equation systems. 14.9

10 Students may try to eliminate the x-variable in one pair of equations and the z-variable in another pair. Make sure they understand that they must eliminate the same variable in both pairs of equations. Be aware of the fact that one system of three equations with three unknowns may take a long time for a student to work through. Because of the complexity of these problems, we suggest giving problems that have only integer values for each of the variables in the solution. Then when students get a fraction for one of the variables, they'll know something isn t correct. Usually it is easier to go back and start the problem from the beginning rather than looking back to try and find the error (sometimes the actual error was one of copying the problem down incorrectly). Students know that they should check their answer and many will. Unfortunately, they will only check it in one of the three equations. Explain to them it is possible to have their solution work in one of the equations but not in the other two. They should be checking their solution in all three given equations. Keep in Mind Although students are quite familiar with algebraic statements by this point in the course, understanding and solving systems of multiple equations and inequalities can still be difficult. One thing that is useful to students is to take a simple system such as x + y = 5 x y = 3 and solve this using all three methods of graphing, substitution, and elimination. It should be noted that more than likely the students will know that the solution is (4, 1) without using any of these methods! This is when you explain that it is necessary to know these methods so that they can be used when the solution isn t so readily apparent. Most of the material in this unit has been geared to both beginning and intermediate students. More difficult examples and problems included for intermediate students could be used to challenge the beginning algebra student. However, the topics of graphing systems of inequalities and solving systems of equations in three variables are not appropriate for beginners. Additional Resources In all mathematics, the best way to really learn new skills and ideas is repetition. Problem solving is woven into every aspect of this course each topic includes warm-up, practice, and review problems for students to solve on their own. The presentations, worked examples, and topic texts demonstrate how to tackle even more problems. But practice makes perfect, and some students will benefit from additional work

11 Solving systems by graphing can be found at At the prompt students can type in an equation and its graph will appear. If they input multiple equations, both the graph and the solution to the system of equations are given. This also will work for inequalities and systems of inequalities, but the resulting graphs do not necessarily display the familiar x- and y- axes and dotted lines. Practice solving two equations with two unknowns using the substitution and elimination methods at (get additional problems on this site by clicking on change or more on this topic ). More practice solving application problems (mixture and distance) can be found at and Students can enter the coefficients of x, y, and z at in order to solve systems of equations with three unknowns. Summary This unit teaches students how to solve systems of linear equations using graphing, substitution, and elimination. Students will also learn how to represent systems of inequalities on the coordinate plane. Finally, they ll practice using these techniques to solve application problems. To help students understand the idea of systems, we suggest beginning with graphing. To help them become skilled at substitution and elimination, show them how to label and organize problems and make sure they get plenty of practice

12 Unit 14: Systems of Equations and Inequalities Purpose Instructor Overview Tutor Simulation: Getting the Mixture Right Unit 14 Tutor Simulation This simulation allows students to demonstrate their ability to write and solve systems of equations in order to understand real world problems. Students will be asked to apply what they have learned to solve a problem involving: Writing and solving equations Writing systems of equations Solving systems of equations Problem Students are presented with the following problem: A merchant and a scientist are both conducting experiments. The merchant is blending two different types of teas and wants to come up with a combination that is priced to sell. The scientist is mixing different acid solutions to come up with a desired result. You will work through each of the problems to determine the right mixtures to get the correct combinations that the merchant and scientist need. Recommendations Tutor simulations are designed to give students a chance to assess their understanding of unit material in a personal, risk-free situation. Before directing students to the simulation, Make sure they have completed all other unit material. Explain the mechanics of tutor simulations. o Students will be given a problem and then guided through its solution by a video tutor; o After each answer is chosen, students should wait for tutor feedback before continuing; o After the simulation is completed, students will be given an assessment of their efforts. If areas of concern are found, the students should review unit materials or seek help from their instructor. Emphasize that this is an exploration, not an exam

13 Unit 14: Systems of Equations and Inequalities Instructor Overview Puzzle: Apples and Oranges Unit 14 Puzzle Objectives Apples and Oranges is a puzzle that represents systems of equations visually, by grouping objects in various combinations. To solve the puzzles, students must translate the images into two or more equations, and then use substitution or elimination to find the value of each variable. Students will need to understand how to write and analyze systems of equations to succeed. Figure 1. Apples and Oranges uses fruits and other objects to test a student's grasp of techniques for solving systems of equations. Description This puzzle has three levels of difficulty. In each level, multiple objects are arranged in various combinations. Players must write an equation that represents each grouping, and then use the resulting system of equations to solve for each variable. When the player clicks on the right answer, the algebraic equations underlying the problem and the solution are shown

14 At the easy level there are three kinds of objects (apples, oranges, and cherries) grouped into two arrangements. The medium level also presents a two equation system, but the three objects (opals, emeralds, and rubies) are grouped in a more complicated way. At the hard level, there are four objects (rockets, spaceships, ray guns, and asteroids) arranged into a system of three equations. Players score points for solving each puzzle correctly. The ten puzzles at each level are scripted, but they are sufficiently difficult to solve visually that there is opportunity for replay. Apples and Oranges is suitable for both individual play and group learning in a classroom setting

15 Unit 14: Systems of Equations and Inequalities Student Instructions Instructor Overview Project: Nutty Economics Unit 14 Project Introduction Linear systems of equations and inequalities are used widely in the fields of business and economics to maximize profit, minimize cost, and balance the production and consumption of goods. In fact, there are entire careers in Production and Operations Management that are built around this skill. Task In this project you will play the role of consultant to a retailer of nuts. You will apply your knowledge of systems of linear equations and inequalities to balance the production of two types of goods in such a way that the profit will be maximized. Instructions Work with at least one other person to complete the following exercises. Solve each problem in order and save your work as you progress as you will create a professional presentation in the end. First Problem Defining the Quantities: The Golden Nut specialty store has obtained 60 lbs. of cashews and 50 lbs. of premium almonds at a good price from its supplier. Since the almonds are more expensive and more of a delicacy than the cashews, it is better to sell the cashews mixed with almonds in order to get a better price for them. The storeowner decides to sell two products: Hearty Nut Mix and Premium Blend. The company hires you to determine how much of each product they should mix together to maximize their profit. o o If each pound of Hearty Nut Mix contains 4 ounces of almonds and 12 ounces of cashews, write an expression that represents the number of pounds of each nut in x pounds of the mix. [Hint: What fraction of the pound are cashews, and what fraction are almonds?] If each pound of Premium Blend contains 8 ounces of almonds and 8 ounces of cashews, write an expression that represents the number of pounds of each nut in y pounds of the mix

16 o Developmental Math An Open Curriculum Complete the following table and use it to write an expression that represents the total number of pounds of cashews used for both mixes together. Then, write an expression for the total number of almonds used for both mixes together. Type of Mix No. of Pounds of Mix No. of Pounds of Cashews No. of Pounds of Almonds Hearty Nut Mix x Premium Blend y o Finally, your market research shows that since the Premium Blend is much more popular, you can afford to charge quite a bit more for it, obtaining a profit of $0.45 per pound for it while you can only earn a profit of $0.30 per pound for the Hearty Nut Mix. Write an expression for the amount of profit that you would make from selling x pounds of Hearty Nut Mix and y pounds of Premium Blend. Second Problem Constraints: It might seem like selling all Premium Blend, since it obtains a better profit, might give the most profit. But this is not necessarily so. Explain why. [Hint: Remember that we are not going to sell cashews or almonds by themselves at this point.] o o You have a limited amount of each type of nut. This places a constraint on the problem that we must account for. Write an inequality that expresses the fact that the total number of cashews cannot be greater than 60 lbs. and the total number of almonds cannot be greater than 50 lbs. [Hint: Use the table from the first problem above.] Write down two other inequalities that represent constraints on the variables x and y. [Hint: What types of values for x and y make sense considering that they represent pounds? These will be simple inequalities.] Third Problem Possibilities: The next step in the process is to graph the inequalities and determine the points that satisfy all of them at once, because these are the only values of x and y that we can realistically use with our supply of nuts. Sketch all four inequalities on the same graph and shade in the region that represents those points that satisfy all of the inequalities. Fourth Problem Maximizing Profit: We are now ready to determine the maximum profit

17 o o Developmental Math An Open Curriculum The points in the region you graphed represent all possible x and y values that we would be allowed to use to maximize our profit. This is a lot of points to check just to find the maximum profit! Fortunately, we can eliminate many of them. It turns out that both the maximum and minimum profit will occur on the edges of this region (in fact, on the extreme points of the edge the corners). Therefore, compute the x and y coordinates of each of the corners of your region. This will require you to solve some systems of equations. Since we know the maximum (and minimum) profit must occur at one of these corners, plug the x and y values for each corner into the expression for profit from the first problem, and record your answers in the table below. Then, identify the maximum profit and tell how many pounds of each mix should be made. [Note: The table below may contain more rows than you need to complete the problem.] Expression for Profit: Corner (x,y) Profit o By completing the following chart, determine how many nuts are left unmixed

18 Mix Pounds Made Pounds of Cashews in the Mix Pounds of Almonds in the Mix Hearty Nut Mix Premium Blend Total Pounds of Cashews Used= Total Pounds of Almonds Used= Conclusions Compare your expressions, computations, and the graph with those from another group. Work to make sure that your explanation is clear and concise. Prepare a final report that clearly identifies and explains the quantities involved in the problem, the expression for profit, the constraints, the region of possible x and y values, the possible combination of mixes that could result in maximum profit, and the maximum profit and the number of pounds of each mix that should be made. Finally, present your solution to the class. Instructor Notes Assignment Procedures Problem 1 Since three quarters of the mix are cashews and one quarter is almonds, then there are 3 x pounds of cashews and 1 x 4 pounds of almonds in x pounds of the mix

19 Since each type of nut comprises half of the mix, then there are 1 y 2 pounds of cashews and 1 y 2 pounds of almonds in y pounds of the mix. The answers are in the table below. Accordingly, then, the total number of pounds of cashews used is 3 1 x+ y 4 2, and the total number of pounds of almonds used is 1 1 x+ y 4 2. Type of Mix No. of Pounds of Mix No. of Pounds of Cashews No. of Pounds of Almonds Hearty Nut Mix x 3 x 4 1 x 4 Premium Blend y 1 y 2 1 y 2 The profit would be 0.30x+0.45y. Problem 2 If we did sell all Premium Blend, we would have an excess of cashews (10 lbs.) that we didn t sell in a mix. The problem states that we obtain a better price for them when they are mixed with almonds. We could possibly increase our profit by making some Hearty Nut Mix in order to sell more of our cashews in the mixes. This is a subtle point that requires some good intuition on the part of the student. In the end, whether this would actually increase profit depends largely on how different the profit margin is between the two mixes. If the profit on the Premium Blend is significantly higher than that for the Hearty Nut Mix, then it may pay to sell all Premium Blend and leave some cashews unmixed.] Using the results in the table above, we find that 3 x+ 1 y and 1 x+ 1 y Since neither x nor y can be negative, we obtain x 0 and y 0. Problem 3 The region is shown in the figure below. In the field of Linear Programming, this region would be called the feasible region, since it contains all allowable x and y values

20 Problem 4 The coordinates are (x,y)=(0,0), (80,0), (0,100), and (20,90). The answers are in the table below. The maximum profit attainable is $46.50, and we should make 20 pounds of the Hearty Nut Mix and 90 pounds of the Premium Blend. As an optional exercise that ties in with the first question under the second problem, you may wish to have students re-examine the total profit if the profit on the Premium Blend is above $0.60 per pound

21 Expression for Profit: 0.30x+0.45y Corner (x,y) Profit (0,0) $0.00 (80,0) $24.00 (0,100) $45.00 (20,90) $46.50 As shown in the table below, we efficiently use all of each type of nut. Mix Pounds Made Pounds of Cashews in the Mix Pounds of Almonds in the Mix Hearty Nut Mix 20 Premium Blend pounds pounds 2 Total Pounds of Cashews Used= pounds pounds 2 Total Pounds of Almonds Used= 60 pounds 50 pounds 14.21

22 Recommendations Developmental Math An Open Curriculum have students work in teams to encourage brainstorming and cooperative learning. assign a specific timeline for completion of the project that includes milestone dates. provide students feedback as they complete each milestone. ensure that each member of student groups has a specific job. Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The students instructions list several websites that provide information on numbering systems, game design, and graphics. The following are other examples of free Internet resources that can be used to support this project: An Open Source Course Management System (CMS), also known as a Learning Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular among educators around the world as a tool for creating online dynamic websites for their students. or Allows you to create a secure online Wiki workspace in about 60 seconds. Encourage classroom participation with interactive Wiki pages that students can view and edit from any computer. Share class resources and completed student work. Allows students to collaborate in real-time from any computer. Google Docs provides free access and storage for word processing, spreadsheets, presentations, and surveys. This is ideal for group projects. The leading open-source office software suite for word processing, spreadsheets, presentations, graphics, databases and more. It can read and write files from other common office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded and used completely free of charge for any purpose

23 Rubric If you had your students complete the optional exercise suggested in the fourth problem, you may wish to add to the following rubric a requirement that students explain, intuitively, why it makes sense that selling all Premium Blend and leaving some cashews unmixed would be advantageous in that case. Score Content Presentation/Communication The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution. The solution completely addresses all mathematical components presented in the task. The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results. Mathematically relevant observations and/or connections are made. The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution. The solution addresses all of the mathematical components presented in the task. The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem. Most parts of the project are correct with only minor mathematical errors. The solution is not complete indicating that parts of the problem are not understood. The solution addresses some, but not all of the mathematical components presented in the task. The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning. Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures. There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made. Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem. There is precise and appropriate use of mathematical terminology and notation. Your project is professional looking with graphics and effective use of color. There is a clear explanation. There is appropriate use of accurate mathematical representation. There is effective use of mathematical terminology and notation. Your project is neat with graphics and effective use of color. Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics. There is some use of appropriate mathematical representation. There is some use of mathematical terminology and notation appropriate for the problem. Your project contains low quality graphics and colors that do not add interest to the project

24 1 Developmental Math An Open Curriculum There is no solution, or the solution has no relationship to the task. No evidence of a strategy, procedure, or mathematical reasoning and/or uses a strategy that does not help solve the problem. The solution addresses none of the mathematical components presented in the task. There were so many errors in mathematical procedures that the problem could not be solved. There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem. There is no use or inappropriate use of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.). There is no use, or mostly inappropriate use, of mathematical terminology and notation. Your project is missing graphics and uses little to no color

25 Unit 14 Correlation to Common Core Standards Unit 14: Systems of Equations and Inequalities Learning Objectives Common Core Standards Unit 14, Lesson 1, Topic 1: Graphing Systems of Linear Equations Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.8.EE. Expressions and Equations CATEGORY / CLUSTER Analyze and solve linear equations and pairs of simultaneous linear equations. STANDARD 8.EE.8. Analyze and solve pairs of simultaneous linear equations. EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) Grade: Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-CED. Creating Equations STANDARD Create equations that describe numbers or relationships

26 EXPECTATION A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities STANDARD Solve systems of equations. EXPECTATION A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities STANDARD EXPECTATION A- REI.11. Represent and solve equations and inequalities graphically. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Unit 14, Lesson 1, Topic 2: Graphing Systems of Inequalities Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.8.EE. Expressions and Equations CATEGORY / CLUSTER Analyze and solve linear equations and pairs of simultaneous linear equations. STANDARD 8.EE.8. Analyze and solve pairs of simultaneous linear equations. EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) Grade: Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra 14.26

27 CATEGORY / CLUSTER A-CED. Creating Equations STANDARD Create equations that describe numbers or relationships. EXPECTATION A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities STANDARD EXPECTATION A- REI.12. Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Unit 14, Lesson 2, Topic 1: The Substitution Method Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.8.EE. Expressions and Equations CATEGORY / CLUSTER Analyze and solve linear equations and pairs of simultaneous linear equations. STANDARD 8.EE.8. Analyze and solve pairs of simultaneous linear equations. EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) Grade: Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-CED. Creating Equations 14.27

28 STANDARD Developmental Math An Open Curriculum Create equations that describe numbers or relationships. EXPECTATION A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities STANDARD Solve systems of equations. EXPECTATION A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Unit 14, Lesson 2, Topic 2: The Elimination Method Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.8.EE. Expressions and Equations CATEGORY / CLUSTER Analyze and solve linear equations and pairs of simultaneous linear equations. STANDARD 8.EE.8. Analyze and solve pairs of simultaneous linear equations. EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) Grade: Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-CED. Creating Equations STANDARD Create equations that describe numbers or relationships

29 EXPECTATION A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities STANDARD Solve systems of equations. EXPECTATION A-REI.5. 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. EXPECTATION A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Unit 14, Lesson 3, Topic 1: Solving Systems of Three Variables Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.8.EE. Expressions and Equations CATEGORY / CLUSTER Analyze and solve linear equations and pairs of simultaneous linear equations. STANDARD 8.EE.8. Analyze and solve pairs of simultaneous linear equations. EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) EXPECTATION 8.EE.8(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. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03, 4.01) Grade: Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra 14.29

30 CATEGORY / CLUSTER A-CED. Creating Equations STANDARD Create equations that describe numbers or relationships. EXPECTATION A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities STANDARD Solve systems of equations. EXPECTATION A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables