Lesson 4: Solving and Graphing Linear Equations


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1 Lesson 4: Solving and Graphing Linear Equations Selected Content Standards Benchmarks Addressed: A2M Modeling and developing methods for solving equations and inequalities (e.g., using charts, graphs, manipulatives, and/or standard algebraic procedures) A2H Recognizing the relationship between operations involving real numbers and operations involving algebraic expressions. A4H Solving algebraic equations and inequalities using a variety of techniques with the appropriate tools (e.g., handheld manipulatives, graphing calculator, symbolic manipulator, or pencil and paper) GLEs Addressed: Grade Solve and graph solutions of multistep linear equations and inequalities (A2M) Grade Graph and interpret linear inequalities in one or two variables and systems of linear inequalities (A2H) (A4H) 16. Interpret and solve systems of linear equations using graphing, substitution, elimination, with and without technology, and matrices using technology (A4H) Lesson Focus This lesson is intended to help students see the connections between algebraic equations and linear graphs. It should include all of the following: Graphing algebraic equations with two variables using points on the line Writing linear equations from problem situations Determining scale for x and y axis Interpreting linear graphs and relating them to problem situations GEE 21 Connection The skills that will be addressed in this lesson include the following: Solve and graph linear equations Solve and graph real world problems (addition, subtraction, multiplication, division) involving linear equations and systems of linear equations Solve and graph multistep equations Translating Content Standards into Instruction A. The first thing we want students to comprehend is the connection between the algebraic equation and its linear graph. Start the lesson by writing the simple equation x + y = 1 from Teacher Blackline #1. Discuss how the solutions would look for this equation. Could x = 4? If x = 4, what would y 36
2 have to be to make the equation true? That is one ordered pair (4,6) that would be a solution. Are there any more solutions? What about (6,4)? Is that a different solution? Could x =? What would y have to be? Could x be a fraction? a negative number? a decimal? Get the students to come up with several examples. In the discussion with the class, be sure to ask the students to decide what the value of x depends on. Help the students arrive at the conclusion that the value of y depends upon the value of x. Now ask the students to graph all of the solutions to the equation, x + y = 1. Do they notice any pattern? Are there any other solutions that have not been listed? Where would those solutions go on the graph? Put a few more solutions on the graph. Could we possibly list all the solutions to the equation, x + y = 1? How could we show all solutions on the graph? If we connect all the points in a straight line and put arrows on both ends, then we can represent all possible solutions to the equation. Name another solution to the equation and ask the students if it falls on the line. It is important for the students to know that all points on the line are solutions to the equation, and all solutions to the equation lie on the line. Calculator Note: The teacher should lead the students through graphing this line on their graphing calculators. Since all linear equations graphed on the calculator must be in the form of y =, the teacher should lead the students through solving the equation, x + y = 1, for y. The students should enter the equation into the calculator, graph the equation, and use the trace function to determine the values of the points on the line. B) The students should work to make connections between a problem situation, the equation that represents the situation, and its graph using example B on Teacher Blackline # 1: In order to play golf at the Scottsville City Golf Course, a person must first join the club for a onetime membership fee of $125 and then pay a $12 green fee each time he plays a round of golf. Ask the students to decide the total cost of joining the club but not play a round of golf. Ask the students to determine the total cost of playing one round of golf. What would be the total cost of 6 rounds of golf? Discuss with the students what quantities are changing (varying). Since the total cost and the number of rounds of golf are changing, those two quantities would be our variables. Discuss with the students what quantity should be the x variable and what quantity should be the y variable. Remember y depends upon x. After the students understand that the total cost depends upon the number of rounds of golf played, then we can assign variables. x = the number of rounds of golf y = the total cost of playing golf Ask the students to come up with an equation to represent the situation. (y = x) Discuss with the students the best way to graph the situation. Should they use the values they have already come up with or should they use the 37
3 equation? How could they use the equation to come up with some ordered pairs to graph? Before the students begin to graph the equation, they will need to decide upon the scale for the xaxis and the scale for the yaxis. Discuss whether or not they need to be the same scale. Make sure the students understand that the scale does not need to be 1 unit in length. The y quantities start at 125 and increase by 12. The x quantities start at and increase by 1. After the scales for the xaxis and the yaxis are decided upon, make sure that the students label the xaxis with the proper title (number of rounds of golf) and the yaxis with the proper title (total cost per round). Discuss with the students what part of the coordinate system will they need. All four quadrants? The negative sides of the axis? Why aren t these needed in this situation? After the students have decided on the scales and have properly labeled the x and y axes, they should decide how to graph the line. Two points are needed, but three are best because the third point serves as a double check of the line. After the points are graphed, they should be connected in a line. The teacher should lead the students through a discussion about the connections between the situation, the graph, and the equation. Some questions that should be raised are Does the line go through the origin? Why not? Where does the line cross the yaxis? What is the meaning of that point on the graph in the actual situation? How is that point related to the equation? Does the line cross the xaxis? Why not? What is the slope of the line? How can you tell from the graph that a round of golf (without the membership fee) is $12? Should the points on the graph really be connected? Does that accurately represent the situation? Calculator Note: Get the students to graph the equation on their graphing calculators. Show the students how to adjust their window, if needed. The calculator will graph the negative values for this equation, but we will ignore those points. Show the students how to access the table on the graphing calculator that corresponds to the graph. C. Next we want to make sure that the students can analyze a graph and describe it using an equation or a real situation. Have the students work with a partner and analyze the graph on Teacher Blackline #2. Have the partners discuss the graph and write down as much information as they can gather from the graph. The partners should assist each other in raising questions about the graph. After giving the partners 38
4 some time to analyze, the teacher should lead a class discussion on what information was discovered. The teacher should be sure that information gathered is supported by what physically appears on the graph. Do not allow the students to make statements without saying why. After a complete analysis of the graph, the teacher should discuss with the students how to write the equation that goes with the graph. The simplest way for the students to do this might be to identify the yintercept and the slope of the line. With the slope and yintercept identified, the students can use y = mx + b to write an equation. Sources of Evidence about Student Learning A. As students work through the examples, the teacher should monitor their discussions as they work between the three different representations of a problem situation, the word problem, the equation, and the graph of that situation. B. Have students do the Student Worksheet provided with the lesson. 39
5 GEE 21 Connection Sample items similar to what students might see on the GEE 21 test include: 1. This graph was made to compare the costs of renting copy machines from Ames Business Products and from Beck s Office Supply. What information is given by the point of intersection of the two lines? 7 Total Copies per Month Beck Ames Number of Copies Made a. The number of copies for which the fixed permonth charge is equal to the cost of copies. b. The price per copy for renting a copies from both companies. c. The fixed permonth charge for renting a copier from both companies. d. The number of copies for which the total cost is the same for both companies. Louisiana GEE 21 Sample Questions, January, 21 4
6 2) Which equation could describe the line of best fit for the graph below? U. S. Population POPULATION (in millions) YEAR a) y = 5x b) y = 5x c) y = 1 x Massachusetts Grade 1 MCAS ReTest Study Questions, 1999 Attributes of Student Work at the GotIt Level A. When students are graphing equations, they should be able to determine placement of x and yaxis, determine the scale for each axis, and accurately graph two or more points from the equation. B. Students should be able to interpret a problem situation and accurately represent it with a linear equation in two variables. Students should be able to identify which variable represents the x quantity and which variable represents the y quantity by determining what variable depends upon the other. C. Students should be able to use their own, as well as given, linear graphs to answer questions about problem situations. Students should be able to interpret the meaning of the x and yintercepts, as well as the slope as it applies to the problem situation. D. Students should be able to compare two linear graphs and determine the meaning of the point of intersection. 41
7 Lesson 4: Solving and Graphing Linear Equations Teacher Blackline #1 A. x + y = 1 B. In order to play golf at the Scottsville City Golf Course, a person must first join the club for a onetime membership fee of $125 and then pay a $12 green fee each time he plays a round of golf. 42
8 Lesson 4: Solving and Graphing Linear Equations Teacher Blackline #2 Loan for Used Car Remaining Debt in Dollars Weeks Since Loan 43
9 Lesson 4: Solving and Graphing Linear Equations Student Worksheet 1) Your parents want to throw you a party for your graduation. They have decided to hire a band for $5 and figure about $15 per person for food and drink. Write an equation to describe the situation. Then graph the equation on the grid provided. Be sure to label all parts of the graph. Use the graph and the equation to determine the cost of inviting 5 people. 2) Use the graph below to answer the following questions Sally's Pie Shop Daily Profit Profit in Dollars Number of Pies Sold a) How many pies must Sally sell in one day to break even? b) If Sally has made a profit of $15, how many pies has she sold? c) How much profit does Sally make on each pie? d) What does the yintercept of 3 mean? e) If Sally sells 5 pies in one day, what will be the profit? 44
10 Lesson 4: Solving and Graphing Linear Equations Student Worksheet 3) You are in charge of purchasing the signs for the school s annual garage sale. One local company Quick Signs charges a $25 set up fee plus $4 per sign. Another company Signs of the Times charges a $3 set up fee plus $3 a sign. Write an equation for the cost of purchasing signs from Quick Signs and write an equation for the cost of purchasing signs from Signs of the Times. Graph each equation on the grid below. Label the lines and answer the following questions. a. Name the coordinates of the point of intersection. What is the meaning of that point of intersection? b. When was Signs of the Times cheaper than Quick Signs? c. When was Quick Signs cheaper than Signs of the Times? d. Explain what company you would choose and why. 45
11 Lesson 4: Solving and Graphing Linear Equations ANSWERS Teacher Blackline #1 A. x + y = Y X In order to play golf at the Scottsville City Golf Course, a person must first join the club for a onetime membership fee of $125 and then pay a $12 green fee each time he plays a round of golf. y = x total cost number of rounds of golf 46
12 Teacher Blackline #2 Y = 2x + 3 Loan for Used Car Remaining Debt in Dollars Weeks Since Loan Things to be discussed from the graph: 1) The original loan was for $3. 2) The loan will take 15 weeks to be totally paid off 3) The scale on the xaxis is one. 4) The scale on the yaxis is 2. 5) The payment is $2 per week. 47
13 Student Worksheet 1) y = x, where y is the cost of the party and x is the number of people in attendance. Cost in $ Number of People The cost for 5 people is $125. 2) Use the graph below to answer the following questions a) Sally must sell 6 pies to break even. b) For a profit of $15 Sally would have sold 9 pies. c) Sally makes $5 per pie. d) It costs Sally $3 just to open her shop. She starts out $3 in the hole. e) If Sally sells 5 pies in on day, her profit would be $22. Sally's Pie Shop Daily Profit Profit in Dollars Number of Pies Sold 48
14 3) Quick Signs y = x, where y is the total cost of the signs and x is the number of signs Signs of the Times y = 3 + 3x, where y is the total cost of the signs and x is the number of signs 1. The point of intersection is (5,45). This is the point where the two companies charge the same for the same number of signs. 2. Signs of the Times is cheaper than Quick Signs for more than 5 signs. 3. Quick Signs is cheaper than Signs of the Times for less than 5 signs. 4. Explanations may vary. If you need more than 5 signs, you would choose Signs of the Times. If you need less than 5 signs, you would choose Quick Signs total cost Quick Signs Signs of the Times number of sign GEE 21 Connection 1. d 2. a 49
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