Algebra Success. LESSON 14: Discovering y = mx + b

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1 T282 Algebra Success [OBJECTIVE] The student will determine the slope and y-intercept of a line by examining the equation for the line written in slope-intercept form. [MATERIALS] Student pages S7 S Transparencies T287, T289, T29, T29, T295 Wall-size four-quadrant grid [ESSENTIAL QUESTIONS]. What is the y-intercept of a line? 2. Where are the slope and y-intercept of a line located in the equation for the line? [GROUPING] Cooperative Pairs, Whole Group, Individual [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Table, Graph, Algebraic Formula, Verbal Description [WARM-UP] (5 minutes IP) S7 (Answers on T286.) Have students turn to S7 in their books to begin the Warm-Up. Students will be using several different methods to find slope. They will be finding slope using the slope formula, the graph of a line, and a table of values. Monitor students to see if any of them need help during the Warm-Up. Give students minutes to complete the problems and then spend 2 minutes reviewing the answers as a class. {Table, Graph, Algebraic Formula} [HOMEWORK]: (5 minutes) Take time to go over the homework from the previous night. [LESSON]: (45 55 minutes M, GP, IP)

2 Algebra Success T28 SOLVE Problem (2 minutes GP) T287, S8 (Answers on T288.) Have students turn to S8 in their books, and place T287 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 to determine the slope and y-intercept of a line by examining the equation of the line. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE} Identify Slope and y-intercept ( minutes M, GP) T287, S8 (Answers on T288.) Use the following modeling activity to model for students how to identify the slope and y-intercept of lines using graphs of the lines. Optional: Graph the points on the wall-size four-quadrant grid. {Table, Graph, Algebraic Formula} MODELING Slope and y-intercept Step : Have students use the equation y = 2 x in Example A to find the y-coordinates for the given x-coordinates in the table. Model how to plug in each x-coordinate into the equation and evaluate to find the corresponding y-coordinate. Complete the table as a class. Step 2: Have students use each ordered pair from the table to plot points on the coordinate grid. Explain that for the first ordered pair, ( - 2, - 4), students should start at the origin (, ) and then move left 2 units and down 4 units; for the second ordered pair, (, - 2), students should start at the origin (, ) and then move left unit and down 2 units; and so on. Step : Model for students how to find the slope of the line using two points on the line. For example, show that to move from the point at (, 2) to the point at (2, 4), students must rise 2 units and run unit. Remind students that slope is equal to rise run, so the slope of the line is 2, or 2. Then work with students to find the slope of the line using the slope formula: (, 2) (2, 4) m = y 2 y = 4 2 x 2 x 2 = 2 = 2 x, y x 2, y 2 Step 4: Remind students that the y-intercept is the point where the line crosses the y-axis. Have students identify the y-intercept of the line in Example A. The line crosses the y-axis at, so the y-intercept of the line is. Repeat the steps above with students to identify the slope and y-intercept of the line described by the equation y = 2 x + in Example B.

3 T284 Algebra Success More with Slope and y-intercept ( minutes IP) T289, S9 (Answers on T29.) Have students complete Example 2 on S9 in cooperative pairs or independently. Make sure students are using the same method as the one modeled in Example. Give your students 8 minutes to complete Example 2 and then use 2 minutes to review the answers. Make sure students have the correct answers on S9, because they will be referring to their answers later in the lesson. {Table, Graph, Algebraic Formula} Slope and y-intercept from Equations (2 minutes M, GP, IP) T288, T29, T29, S8-S (Answers on T292.) minutes M, GP: Have students use the information and answers on S8 and S9 (T288 and T29) to complete the first table on S as a class. {Table, Algebraic Formula, Verbal Description} minutes IP: After completing the first table, give your students minutes to complete the second table independently. If they seem to have trouble, question them about the first table. For example, use questions such as, Where was the slope found on each equation in the first table? (before the x as the coefficient) Where was the y-intercept found on each equation in the first table? (as the constant) {Table, Algebraic Formula, Verbal Description} minutes GP: Complete the paragraph on S with your students. Use T29 to read the paragraph and fill in the blanks. {Table, Algebraic Formula, Verbal Description} minutes GP: Give students 2 minutes to find the slope and y-intercept of the line described by each equation at the bottom of the page. Use minute to review answers. {Algebraic Formula, Verbal Description} SOLVE Problem (8 minutes GP) T29, S (Answers on T294.) Have students turn to S in their books, and place T29 on the overhead. 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. (The function is in the form y = mx + b. The slope is m and the y-intercept is b.) {SOLVE, Algebraic Formula, Verbal Description}

4 Algebra Success T285 If time permits... ( minutes) T295, S2 (Answers on T296.) 8 minutes IP: Have students complete Problems on S2 independently. Ask students to identify the slope and y- intercept first, using the equation y = mx + b and then complete the table and graph the points to make sure their answers are correct. {Table, Graph, Algebraic Formula} 2 minutes: Review the answers to the problems on S2. [Closure]: (5 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. What is the y-intercept of a line? (The point where the line crosses the y-axis.) Where are the slope and y-intercept of a line located in the equation for the line? (When an equation is in slope-intercept form, y = mx + b, the slope is the number in front of x, represented by m, and the y-intercept is the number without a variable, represented by b.) [Homework]: Assign S for homework. (Answers on T297.) [Quiz Answers] T298 T299. B 2. A. A 4. D 5. D 6. B 7. D 8. C 9. B. C The quiz can be used at any time as extra homework or to see how students did on understanding what y = mx + b represents.

5 T286 Algebra Success Here is the key to S7. Warm Up Directions: Find the slope of each of the following.. the line that connects the points (4, - 2) and (, - 5) Slope = (4, - 2) and (, - 5) m = y 2 y = x 2 x 4 = - - = x, y x 2, y 2 2. the line graphed on the coordinate Down 2 units, to the right plane below units Slope = - 2. the line whose table is shown below As the x-coordinates go up by, the y-coordinates go up by 5, slope = 5 x y

6 Algebra Success T287 TRANSPARENCY MASTER Directions: Complete the following SOLVE problem with your teacher. You will only complete the S step. Dan saves money to buy his favorite video game. Dan begins with $5 and saves $2 per week. The function y = 2 x + 5 represents the amount Dan saves, where y represents the amount of money that Dan has saved and x represents the number of weeks that have passed. What is the slope and the y-intercept of the function? S Underline the question. This problem is asking me to find. Directions: For each of the following, Substitute each x-value into the equation and evaluate. This will give you the corresponding y-value. Use the ordered pairs to plot points. Connect the points to form a line. Identify the slope and the y-intercept of the line. Example A. y = 2x slope = y-intercept = B. y = 2x x + slope = y-intercept = x x - 2 y y 2

7 T288 Algebra Success Here is the key to S8. Directions: Complete the following SOLVE problem with your teacher. You will only complete the S step. Dan saves money to buy his favorite video game. Dan begins with $5 and saves $2 per week. The function y = 2 x + 5 represents the amount Dan saves, where y represents the amount of money that Dan has saved and x represents the number of weeks that have passed. What is the slope and the y-intercept of the function? S Underline the question. This problem is asking me to find the slope and y-intercept - of the function. Directions: For each of the following, Substitute each x-value into the equation and evaluate. This will give you the corresponding y-value. Use the ordered pairs to plot points. Connect the points to form a line. Identify the slope and the y-intercept of the line. Example A. y = 2x slope = 2 = 2 y-intercept = B. y = 2x x + slope = 2 = 2 y-intercept = x y x y

8 Algebra Success T289 TRANSPARENCY MASTER Example 2 A. y = x slope = y-intercept = x - y B. y = x + 4 slope = y-intercept = x - y C. y = x slope = y-intercept = x - y

9 T29 Algebra Success Example 2 A. y = x slope = y-intercept = x y Here is the key to S9. B. y = x + 4 slope = y-intercept = 4 x y C. y = x slope = y-intercept = x y -

10 Algebra Success T29 TRANSPARENCY MASTER Directions: Using your answers from S8 and S9, complete the table below. Problem Equation Slope (m) y-intercept (b) Example A Example B Example 2A Example 2B Example 2C Directions: Make some predictions in the table below. Equation Slope (m) y-intercept (b) y = 4x y = 2 x + y = - x 2 y = - 5x + y = mx + b The general equation for a line in slope-intercept form is. The slope of a line is represented with the letter. It is the coefficient of the variable x when the equation for the line is in slope-intercept form. The y-intercept of a line is represented with the letter. It is the constant when the equation for the line is in slope-intercept form. Directions: Find the slope and y-intercept of the line described by each equation below.. y = 4x x 7 2. y = 2 x + 4. y = - 2x

11 T292 Algebra Success Here is the key to S. Directions: Using your answers from S8 and S9, complete the table below. Problem Equation Slope (m) y-intercept (b) Example A y = 2x 2 Example B y = 2x + 2 Example 2A y = Example 2B y = x x + 4 Example 2C y = x Directions: Make some predictions in the table below. 4 Equation Slope (m) y-intercept (b) y = 4x 4 - y = 2 x + 2 y = - x 2 y = 5x y = mx + b m b The general equation for a line in slope-intercept form is y = mx + b. The slope of a line is represented with the letter m. It is the coefficient of the variable x when the equation for the line is in slope-intercept form. The y-intercept of a line is represented with the letter b. It is the constant when the equation for the line is in slope-intercept form. Directions: Find the slope and y-intercept of the line described by each equation below.. y = 4x x 7 slope = 4, y-intercept = y = 2 x + 4 slope = 2, y-intercept = 4. y = - 2x slope = - 2, y-intercept = -

12 Algebra Success T29 TRANSPARENCY MASTER Directions: Complete the following SOLVE problem with your teacher. Dan saves money to buy his favorite video game. Dan begins with $5 and saves $2 per week. The function y = 2 x + 5 represents the amount Dan saves, where y represents the amount of money that Dan has saved and x represents the number of weeks that have passed. What is the slope and the y-intercept of the function? S Underline the question. This problem is asking me to find. O Identify the facts. Eliminate the unnecessary facts. List the necessary facts. L Choose an operation or operations. Write in words what your plan of action will be. V Estimate your answer. Carry out your plan. E Does your answer make sense? (Compare your answer to the question.) Is your answer reasonable? (Compare your answer to the estimate.) Is your answer accurate? (Check your work.) Write your answer in a complete sentence.

13 T294 Algebra Success Here is the key to S. Directions: Complete the following SOLVE problem with your teacher. Dan saves money to buy his favorite video game. Dan begins with $5 and saves $2 per week. The function y = 2 x + 5 represents the amount Dan saves, where y represents the amount of money that Dan has saved and x represents the number of weeks that have passed. What is the slope and the y-intercept of the function? S Underline the question. This problem is asking me to find the slope and y-intercept of the function. O Identify the facts. Eliminate the unnecessary facts. List the necessary facts. Dan begins with $5. He saves $2 per week. The function is y = 2x x + 5. L Choose an operation or operations. Write in words what your plan of action will be. Identify the slope and the y-intercept. V Estimate your answer. the slope and y-intercept will be either 2 or 5 Carry out your plan. y = 2x x + 5 y = mx + b Slope = 2 and y-intercept = 5 E Does your answer make sense? (Compare your answer to the question.) Yes. Is your answer reasonable? (Compare your answer to the estimate.) Yes. Is your answer accurate? (Check your work.) Yes. Write your answer in a complete sentence. The slope is 2 and the y-intercept is 5.

14 Algebra Success T295 TRANSPARENCY MASTER Directions: Identify the slope and the y-intercept from the equation. Then use the table to find points on the line, graph the points, and check your answers for the slope and y-intercept.. y = x slope (m) = y-intercept (b) = x y 2. y = x + 4 slope (m) = y-intercept (b) = x y. y = x slope (m) = y-intercept (b) = x y

15 T296 Algebra Success Here is the key to S2. Directions: Identify the slope and the y-intercept from the equation. Then use the table to find points on the line, graph the points, and check your answers for the slope and y-intercept.. y = x slope (m) = y-intercept (b) = x y 2. y = x + 4 slope (m) = y-intercept (b) = 4 x y 4 5. y = x slope (m) = y-intercept (b) = - x y

16 Algebra Success T297 Here is the key to S. Homework For each of the following, write the slope and y-intercept.. y = 4 x + 2. y = - x Slope = 4, y-intercept = Slope = -, y-intercept =. y = - x + 4. y = 4 x 2 Slope =, y-intercept = Slope = 4, y-intercept = y = 5x x 4 6. y = 2 x + Slope = 5, y-intercept = - 4 Slope = 2, y-intercept = 7. y = x y = - 7 x 5 Slope =, y-intercept = 8 Slope = - 7, y-intercept = y = - x. y = 2 x 6 Slope = -, y-intercept = Slope = 2, y-intercept = - 6

17 T298 Algebra Success Name Quiz Date. What is the slope of the line described by the function y = 4x +? A. 7 B. 4 C. D. 2. What is the y-intercept of the line described by the function y = A. - B. - 2 C. 2 D. - 2x x?. Which equation describes a line that has a slope of? A. y = - x + B. y = x C. y = - 2x x D. y = x 4. Which equation describes a line that has a y-intercept of? A. y = - x x B. y = x C. y = D. y = x + 5. What is the y-intercept of the line described by the function y = 5 x? A. 5 B. C. 5 D. 6. What is the slope of the line described by the function y = - 2 x 5? A. - 5 B. - 2 C. 2 D. 5

18 Algebra Success T Which of the following is true for the function y = - 2x x + 6? A. The graph is a line crossing the y-axis at - 2. B. The graph is a line with a slope equal to 6. C. The graph is a line passing through the point ( - 2, 6). D. The graph is a line crossing the y-axis at (, 6). 8. Which of the following is an equation of a line with a slope of? A. y = x + B. y = x C. y = - x + 2 D. y = 2x What is the slope of the line described by the function y = x + 2? A. 2 B. C. D.. What is the y-intercept of the line described by the function y = 2 x + 4? A. - 4 B. 2 C. 4 D. 6

Mathematics Success Level H

Mathematics Success Level H T473 [OBJECTIVE] The student will graph a line given the slope and y-intercept. [MATERIALS] Student pages S160 S169 Transparencies T484, T486, T488, T490, T492, T494, T496 Wall-size