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TEKS A.6 CF Minding your m’s and b’s TAKS Objective 3 The student will demonstrate an understanding of linear functions. TEKS Math Concepts (A.6) Linear functions. The student understands the meaning of slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to: (A) develop the concept of slope as a rate of change and determine slopes from graphs, tables, and algebraic representations; (B) interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs; (C) investigate, describe, and predict the effects of changes in m and b on the graph of y mx b ; (D) graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept; (E) determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic expressions; (F) interpret and predict the effects of changing slope and y- intercept in applied situations; and (G) relate direct variation to linear functions and solve problems involving proportional change. Overview Students can graph linear equations written in the form y mx b . Students can examine patterns in the graphs to determine the effects of changing the values of m and b in the equation y mx b. TAKS Objective 3 page 1 TEKS A.6C&F Instructional Strategies Discovery & Cooperative Learning, Direct Teaching Students will work in partners to graph linear functions and discover patterns that illustrate the effects of increasing/decreasing the slope, m, and the y-intercept, b, have on the graph of y mx b . Students will extend these patterns they discover to reason what would happen in real- world situations when a rate of change occurs or the initial starting conditions are changed. Lesson Objectives 1. Students will identify the effects of increasing/decreasing the slope, m, has on the graph of a linear equation, y mx b. 2. Students will identify the effects of increasing/decreasing the y-intercept, b, has on the graph of a linear equation, y mx b. 3. Given a real-world scenario, students will describe what changing m or b means in the context of the problem. For Teacher’s Eyes Only This lesson is built on the previous lesson (TEKS A.6 AB How does your slope rate?) of developing an understanding of slope as the rate of changes that is represented as a ratio of y to x. This lesson emphasizes the slope intercept form of y=mx+b where m represents the ratio of y to x. Changing Slope Changing the value of the slope of a line has the following effects: m 0 line rises from left to right (increasing); as m gets larger the graph of the line gets steeper m 0 line falls from left to right (decreasing); as m gets smaller the graph of the line gets flatter (less steep) m 0 line has no rise horizontal line m = undefined ; line has no run vertical line (recall this is NOT a function) TAKS Objective 3 page 2 TEKS A.6C&F The y-intercept was introduced in the previous lesson (TEKS A.6 AB How does your slope rate?) with Job #2. The ‘What Have I Earned? #2’ is meant to build on the meaning of what is a y-intercept. Changing y-intercepts As b increases, the graph of the line translates up As b decreases, the graph of the line translates down Misconceptions Misconception In the formula, y mx b , b represents the x-intercept. Mathematics Concept b represents the y-intercept on the graph of the function and can be found on the y-axis. Rebuild Concept Provide students with practice locating and identifying the y-intercept of a linear function. Misconception A line with a negative slope, for example a slope of -4, is less steep than a line with a positive slope, for example a slope of 4. Mathematics Concept The steepness of a line is determined by the m . So a line with slope of -4 has the same steepness as a line with slope of 4 since 4 4 4 . The positive and negative sign only identify whether the line is rising (increasing) or falling (decreasing). TAKS Objective 3 page 3 TEKS A.6C&F Rebuild Concept Provide students with practice identifying the slopes of various lines and the characteristics of the line that result from the value of the slope. Student Prior Knowledge Students have prior experience working with the concept of slope as a rate of change as well as identifying the slope of a line given the graph of the line (TEKS A.6.A). Students also have prior experience working with making conjectures and predictions about functional relationships (TEKS A.1.E). Materials Activity Sheets Map Colors Graphing Calculators Rulers TAKS Objective 3 page 4 TEKS A.6C&F 5 E’s ENGAGE The learner is introduced to a new experience and must draw from prior experiences to make sense of the engage activity. 1. Show students Display Sheet #1 Story Lines. Have students work in pairs to read and answer the prompt. 2. Once students have had a chance to answer the prompt, have students share their answers with the whole class. EXPLORE During the explore activity, the student becomes directly involved with a particular phenomena by manipulation of materials that are used to discover the phenomena. 1. Students will need to work in pairs. Distribute a copy of Activity Sheet #1 Charting Slopes and y-intercepts to each student. 2. One student in the partner pair will complete Sheet A while the other student completes Sheet B. 3. Once they are finished, both students will compare and discuss their data and then answer the questions (#4-#6) that follow. 4. Once all students have had time to complete the activity, analyze the activity using the following questions: a. As the slope increases, what happened to the graph of the linear equation? b. As the slope decreases, what happened to the graph of the linear equation? c. As the y-intercept increases, what happens to the graph of the linear equation? d. As the y-intercept decreases, what happens to the graph of the linear equation? TAKS Objective 3 page 5 TEKS A.6C&F EXPLAIN The student communicates in verbal and written form about the information derived from the learning experience. In discussing the results of Activity Sheet #1 Charting Slopes and y-intercepts with the students, ask the students to verbalize what effect changing the slope had on the graph of y mx b . As the slope, m, in y mx b is changed the “steepness” of the graph changes. This idea of steepness can be related to how quickly the data is increasing or decreasing. Point out that both negative and positive slope can characterized the same if we talk about the change in the m . As m increases, the graph of the line becomes steeper. Absolute value bars are used to cover both cases when m is a positive and a negative value (when the line is rising and falling). The sign of the slope simply tells us whether the line is rising or falling (whether we are increasing or decreasing). As the y-intercept, b, in y mx b is changed the position of the line is changed. As b increases the graph of the line is translated up on the y-axis. As b decreases the graph of the line is translated down on the y-axis. In terms of real-world situations, changes in slope indicate a rate of change in the original situation has increased or decreased. A change in the y-intercept indicates that an initial starting condition has changed. Tie these concepts back to Display Sheet #1 Story Lines. Ask students to elaborate more about their scenarios in terms of changing slopes and y- intercepts and what they would mean in the context of the scenario. ELABORATE During the elaboration phase, students expand their knowledge by making connections about what they have learned and applying this new knowledge to real world situations. 1. Distribute Activity Sheet #4 Savings, Slope, and Intercept to each student. 2. Have students work in pairs following the directions on the activity sheet to answer the questions. 3. After students have had an opportunity to complete the questions, debrief the activity. TAKS Objective 3 page 6 TEKS A.6C&F EVALUATE Evaluation throughout the learning experience is an ongoing process and has a diagnostic function. 1. Distribute Activity Sheet #5 Changing Slopes and Intercepts to each student. 2. Students will work individually to complete this activity. TAKS Objective 3 page 7 TEKS A.6C&F TAKS Objective 3 page 8 TEKS A.6C&F A B C Pick any 2 lines from the graph and write a scenario to model the relations represented by each linear graph. Be sure to identify the 2 lines you use in creating your scenario. TAKS Objective 3 page 9 TEKS A.6C&F Charting Slopes and y-intercepts You will work in partners for this activity. Each person will have their own chart and a graphing calculator. 1. Record the slope and y-intercept of each equation in your chart using a map color. You will need a different map color for each equation. 2. Enter each equation into your calculator and graph it. Using the calculator (graph or table of values) plot the y-intercept and draw the graph of your equation using the same map color that you used to record the slope and the y-intercept. Note: all of your lines will be graphed on the same coordinate grid. 3. Compare your chart with your partner’s chart. Do you notice any patterns? You and your partner should answer the following questions together. 4. What happens to the graph of your equation as your slope (m) values in your equation increase? decrease? 5. What happens to the graph of your equation as your y-intercept (b) values in your equation increase? decrease? 6. What effect does the sign (positive/negative) of the slope have on the graph of your equations? TAKS Objective 3 page 10 TEKS A.6C&F Charting Slopes and y-intercepts Record Sheet A – Partner I Part I Equation Slope y-intercept y x 1 y x 2 1 y x 2 1 y x 4 1 y x 4 y 2x y 2x y 4x y 8x y 8x Part II Equation Slope y-intercept y x y x 2 y x 5 y x 10 y x 3 y x 6 y x 10 TAKS Objective 3 page 11 TEKS A.6C&F Charting Slopes and y-intercepts Record Sheet B – Partner II Part I Equation Slope y-intercept y x y x 2 y x 3 2 y x 3 1 y x 6 1 y x 6 y 3x y 3x y 5x y 5x Part II Equation Slope y-intercept y x y x 2 y x 5 y x 8 y x 3 y x 6 y x 8 TAKS Objective 3 page 12 TEKS A.6C&F Savings, Slope, and Intercept You decide to open a savings account to start saving money for college. After your first week at your new job, you take $50 of your paycheck and open a savings account. You continue to deposit $50 at the end of each week into your savings account. 1. Draw a graph to represent this situation. Label the horizontal axis as “weeks” and the vertical axis as “dollars.” 2. What is the slope of the line that you have drawn? What does it represent? 3. What is the y-intercept of your line? What does it represent? 4. If you increased the amount of money that you save each week, how would your graph change? What part of the graph does this represent? 5. If you decrease the amount of money that you save each week, how would your graph change? What part of the graph does this represent? 6. Suppose you already had a savings account before you got your job. This savings account already had $200 in it. How would this change your graph? What part of the graph does this represent? TAKS Objective 3 page 13 TEKS A.6C&F Now suppose instead of putting money into your bank account, you are withdrawing money. You start with $500 in your account. You withdraw $10 each day with your debit card for lunch. 7. Draw a graph to represent this situation. Label the horizontal axis as “weeks” and the vertical axis as “dollars.” 8. What is the slope of the line that you have drawn? What does it represent? 9. What is the y-intercept of your line? What does it represent? 10. If you increased the amount of money that you withdraw each week, how would your graph change? What part of the graph does this represent? 11. If you decrease the amount of money that you withdraw each week, how would your graph change? What part of the graph does this represent? 12. Suppose you started with $200 in the savings account instead of $500. How would this change your graph? What part of the graph does this represent? TAKS Objective 3 page 14 TEKS A.6C&F Changing Slopes and Intercepts 1. How does changing the value of m affect the graph of y mx b ? 2. How does changing the value of b affect the graph of y mx b ? On the left you are given the graph of a linear equation. Some changes are made to the slope and the y-intercept. Redraw the graph using the information that you are given. 3. Slope is increased by 3 y-intercept stays the same Equation___________________ Equation___________________ TAKS Objective 3 page 15 TEKS A.6C&F 4. Multiply the slope by -1 y-intercept stays the same Equation___________________ Equation___________________ 5. Slope stays the same y-intercept increases by 4 Equation___________________ Equation___________________ 6. Slope is doubled y-intercept increases by 2 Equation___________________ Equation___________________ TAKS Objective 3 page 16 TEKS A.6C&F

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