Algebra Success. LESSON 16: Graphing Lines in Standard Form. [OBJECTIVE] The student will graph lines described by equations in standard form.

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1 T328 [OBJECTIVE] The student will graph lines described by equations in standard form. [MATERIALS] Student pages S125 S133 Transparencies T336, T338, T340, T342, T344 Wall-size four-quadrant grid [ESSENTIAL QUESTIONS] 1. Which form of an equation for a line is more useful when graphing the line, slope-intercept form or standard form? 2. Can the equations for all lines be written in standard form? [GROUPING] Cooperative Pairs, Whole Group, Individual [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Graph, Algebraic Formula, Verbal Description [WARM-UP] (5 minutes IP) S125 (Answers on T335.) Have students turn to S125 in their books to begin the Warm-Up. For each problem, students will be isolating the given variable. Monitor students to see if any of them need help during the Warm-Up. Give students 3 minutes to complete the problems and then spend 2 minutes reviewing the answers as a class. {Algebraic Formula} [HOMEWORK]: (5 minutes) Take time to go over the homework from the previous night. [LESSON]: (45 55 minutes M, GP, IP)

2 T329 SOLVE Problem (2 minutes GP) T336, S126 (Answers on T337.) Have students turn to S126 in their books, and place T336 on the overhead. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to convert any equation for a line into slope-intercept form. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE} Slope and y-intercept (8 minutes M, GP) T336, S126 (Answers on T337.) After learning how to graph a line in slope-intercept form, many students think it is possible to always use the coefficient of x as the slope and the constant as the y-intercept. Use the following modeling activity to help students understand that they must isolate the y-variable and write the equation in slope-intercept form before identifying the slope as the coefficient of x and the y-intercept as the constant. Many students will make mistakes during this activity. Making mistakes will help students understand why they cannot find the slope or y-intercept without putting the line in slope-intercept form first. {Graph, Algebraic Formula, Verbal Description}

3 T330 MODELING Use an Equation not in Slope-Intercept Form Step 1: Discuss Problem 1 with students. If necessary, remind students that the two things they used in the last lesson to graph a line were the slope and y-intercept. Step 2: Discuss Problem 2 with students. Students may likely say that the coefficient of x, 2, is the slope. Some students may have other answers, but write down 2 to use as the example. Remind students that in the last lesson they learned that, in the equation for a line, the slope of the line was the coefficient, or the number in front of the x. Step 3: Discuss Problem 3 with students. Students may likely say that the constant, 3, is the y-intercept. Some students may have other answers, but write down 3 to use as the example. Remind students that in the last lesson they learned that, in the equation for a line, the y-intercept was the constant. Step 4: For Problem 4, have students use a slope of 2 and a y-intercept of 3 to graph the line described by the equation 2x x y = 3. Remind students about the steps they used in the last lesson to graph a line. If necessary, have students use the graphic organizer on S119 from Lesson 15 to graph this line. Optional: Graph on the wall-size four-quadrant grid. The steps are listed below: Identify if the line is diagonal, horizontal, or vertical. (diagonal) Identify the slope. (m = 2) Identify the y-intercept. (b = 3) Make a point on the y-intercept. (0, 3) Use the slope, rise, to find other points. run Connect the points.

4 T331 Step 5: For Problem 5, have students choose two points on the line they graphed, such as (0, 3) and (1, 5). Students can use any two points on the graphed line. Then, for Problem 6, have students plug the ordered pairs into the original equation and solve. Point out that the values used do not make the equations true: 2x x y = 3 2 x y = 3 original equation 2(0) 3 = 3 2(1) 5 = 3 plugged in ordered pairs 0 3 = = Explain to students that if they just choose the coefficient of x as the slope and the constant as the y-intercept, these will not always be the correct slope and y-intercept when the equation is not written in slope-intercept form. Isolate the y-variable (20 minutes M, GP) T338, T340, S127, S128 (Answers on T339, T341.) 10 minutes M, GP: Use the following modeling activity to model for students how to convert an equation for a line into slope-intercept form in order to graph the line. {Graph, Algebraic Formula, Verbal Description} MODELING Graph Using Standard Form Step 1: Discuss Problem 7 with students. Explain that the points chosen did not work because the equation for the line was not in slope-intercept form when students chose the coefficient of x as the slope and the constant as the y-intercept. Step 2: For Problem 8, first remind students that in the last lesson they referred to y = mx + b as the slope-intercept form of an equation, and if an equation for a line is in slope-intercept form, then students know the slope is the coefficient of x and the constant is the y-intercept. Discuss with students that they must isolate the y-variable in an equation to put the equation in slope-intercept form.

5 T332 Step 3: For Problem 9, work with students to rewrite the equation 2x x slope-intercept form, as shown below: 2x x y = 3 2x x 2 x Subtract 2 x from both sides. - y - = - 2x Divide by - 1 to isolate the y-variable. 1 y = 2x x 3 y = 3 in Some students will want to add the 2x x first because there is a minus sign between the 2x x and the y. Explain that students should subtract the 2x, because the sign of the 2x x is in front of the 2, not behind. Remind students that 2x x y is the same as 2 x + - y. The negative belongs to the y, not the 2x. Point out that when students subtract 2x x from 3, they cannot combine the numbers. They can only combine a 2x x with another like term such as 3x. 3 2x x has to stay as 3 2x x or - 2x x + 3. Many students will not divide all three terms by - 1. This will result in an incorrect answer. Other students may not even see the need to divide by - 1. Discuss with students that they must divide each term by - 1. Remind - 1 y and that students must isolate the students that - y is the same as y-variable completely by dividing all terms by - 1 to keep the equation balanced. Step 4: For Problems 10 11, have students identify the slope and y-intercept in the slope-intercept form of the equation 2x x y = 3 ( y = 2 x 3; slope = 2; y intercept = - 3). Step 5: For Problem 12, have students use a slope of 2 and a y-intercept of - 3 to graph the line described by the equation 2x x y = 3, or y = 2 x 3. Remind students about the steps they used in the last lesson to graph a line. If necessary, have students use the graphic organizer on S119 from Lesson 15 to graph this line. Optional: Graph the line on the wall-size four-quadrant grid. The steps are listed below: Identify if the line is diagonal, horizontal, or vertical. (diagonal) Identify the slope. (m = 2) Identify the y-intercept. (b = - 3) Make a point on the y-intercept. (0, - 3) Use the slope, rise, to find other points. run Connect the points.

6 T333 Step 6: For Problem 13, have students choose two points on the line they graphed, such as (0, - 3) and (1, - 1). Students can use any two points on the graphed line. Then, for Problem 14, have students plug the ordered pairs into the original equation (in standard form) and solve. 2x x y = 3 2 x y = 3 original equation 2(0) ( - 3) = 3 2(1) ( - 1) = 3 plugged in ordered pairs = = 3 3 = 3 3 = 3 Point out to students that each set of ordered pairs makes the equation true. Step 7: Discuss Problem 15 with students. Explain that the equation for a line is written in standard form when it is written as ax + by = c. Remind students that when an equation is written in standard form, students cannot use the coefficient of x as the slope or the constant as the y-intercept. 10 minutes M, GP: Have students turn to S128 in their books, and place T340 on the overhead. Repeat Steps 3 6 above to model how to convert each equation in standard form into slopeintercept form, identify the slope and y-intercept, graph the line, and check the points used. Convert from Standard Form (10 minutes GP, IP) T342, S129 (Answers on T343.) 8 minutes IP: Have students work in cooperative pairs to complete Problems 3 and 4 on S129 (T342). Monitor students closely as they complete the problems and make sure they do not have any questions. {Graph, Algebraic Formula} 2 minutes GP: Use 2 minutes to review the answers as a class. SOLVE Problem (5 minutes GP) T344, S130 (Answers on T345.) Remind students that the SOLVE problem is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Students can isolate the y-variable to put the equation in slope-intercept form.) {SOLVE, Algebraic Formula, Verbal Description}

7 T334 If time permits... (10 minutes IP) S131 (Answers on T346.) 7 minutes IP: Have students complete Problems 1 3 on S131 independently. {Graph, Algebraic Formula} (3 minutes) Review the answers as a class. [Closure]: (5 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. Which form of an equation for a line is more useful when graphing the line, slope-intercept form or standard form? (This is an opinion question, but a good answer would be that slope-intercept form is more useful, because you know the slope and y-intercept.) Can the equations for all lines be written in standard form? (The equations for all diagonal lines can be written in the form ax + by = c. The equations for vertical lines are still written as x = a number and the equations for horizontal lines are still written as y = a number.) [Homework]: Assign S132 and S133 for homework. (Answers on T347, T348.)

Mathematics Success Grade 8

Mathematics Success Grade 8 T429 [OBJECTIVE] The student will solve systems of equations by graphing. [PREREQUISITE SKILLS] solving equations [MATERIALS] Student pages S207 S220 Rulers [ESSENTIAL QUESTIONS]