Exploring Slope. High Ratio Mountain Lesson 11-1 Linear Equations and Slope
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1 Eploring Slope High Ratio Mountain Lesson 11-1 Learning Targets: Understand the concept of slope as the ratio points on a line. between any two Graph proportional relationships; interpret the slope and the y-intercept (0, 0) of the graph. Use similar right triangles to develop an understanding of slope. SUGGESTED LEARNING STRATEGIES: Create Representations, Marking The Tet, Discussion Groups, Sharing and Responding, Interactive Word Wall Misty Flipp worked odd jobs all summer long and saved her money to buy passes to the ski lift at the High Ratio Mountain Ski Resort. In August, Misty researched lift ticket prices and found several options. Since she worked so hard to earn this money, Misty carefully investigated each of her options. Activity 11 High Ratio Mountain Ski Resort Student Lift Ticket prices Daily Lift Ticket $0 10-Day Package $80 upon purchase and $20 per day (up to 10 days) Unlimited Season Pass $90 1. Suppose Misty purchases a daily lift ticket each time she goes skiing. Complete the table below to determine the total cost for lift tickets. Number of Days Total Cost of Lift Tickets 2. According to the table, what is the relationship between the cost of the lift tickets and the number of days? Activity 11 Eploring Slope 1
2 ACTIVITY 11 Lesson Let d represent the number of days for which Misty bought lift tickets and C represent Misty s total cost. Write an equation that can be used to determine the total cost of lift tickets if Misty skis for d days. 4. Model with mathematics. Plot the data from the table on the graph below. The data points appear to be linear. What do you think this means? y Total Cost of Lift Tickets Days MATH TIP Vertical change is the number of spaces moved up or down on a graph. Up movement is represented by a positive number. Down is a negative number. Horizontal change is the number of spaces moved right or left on a graph. Movement to the right is indicated by a positive number. Movement to the left is indicated by a negative number.. Label the leftmost point on the graph point A. Label the net 6 points, from left to right, points B, C, D, E, F, and G. 6. Reason quantitatively. According to the graph, what happens to the total cost of lift tickets as the number of days increases? Justify your answer. 7. Describe the movement, on the graph, from one point to another. A to B: Vertical Change Horizontal Change B to C: Vertical Change Horizontal Change C to D: Vertical Change Horizontal Change D to E: Vertical Change Horizontal Change E to F: Vertical Change Horizontal Change F to G: Vertical Change Horizontal Change 14 SpringBoard Mathematics Course /PreAlgebra, Unit 2 Equations
3 Lesson 11-1 Activity a. The movements you traced in Item 7 can be written as ratios. Write ratios in the form vertical change to describe the movement from: horizontal change A to B: B to C: C to D: D to E: b. Vertical change can also be described as the. Similarly, the horizontal change is often referred to as the. Therefore, the ratio vertical change can also be written as horizontal change. Determine the and from A to C in Item 4. Write the ratio as. Reading and Writing Math When writing a ratio, you can also represent the relationship by separating each quantity with a colon. For eample, the ratio 1:4 is read one to four. Continue to use the data from Item 4. Determine the and for each movement described below. Then write the ratio. c. From B to E: d. From A to E: e. From B to A: f. From E to B: Activity 11 Eploring Slope 1
4 ACTIVITY 11 Lesson Describe the similarities and differences in the ratios written in Item 8. How are the ratios related? 10. Make sense of problems. What are the units of the ratios created in Item 8? Eplain how the ratios and units relate to Misty s situation. MATH TIP In similar triangles, corresponding angles are congruent and corresponding sides are in proportion. 11. How do the ratios relate to the equation you wrote in Item? 12. The ratio between any two points on a line is constant. Use the diagram below and what you know about similar triangles to ratios are equivalent for the movements eplain why the described. 10 Z From W to V: 6 From W to Z: W V = 6 = SpringBoard Mathematics Course /PreAlgebra, Unit 2 Equations
5 Lesson 11-1 ACTIVITY 11 The slope of a line is determined by the ratio between any two points that lie on the line. The slope is the constant rate of change of a line. It is also sometimes called the average rate of change. All linear relationships have a constant rate of change. The slope of a line is what determines how steep or flat the line is. The y-intercept of a line is the point at which the line crosses the y-ais, (0, y). MATH TERMS Slope is the ratio of vertical change to horizontal change, or. 1. Draw a line through the points you graphed in Item 4. Use the graph to determine the slope and y-intercept of the line. How do the slope and y-intercept of this line relate to the equation you wrote in Item? READING MATH 14. Complete the table to show the data points you graphed in Item 4. Use the table to indicate the ratio and to determine the slope of the line. Number of Days Total Cost of Lift Tickets : : : slope: The slope of a line,, is also epressed symbolically as y. is the Greek letter delta, and in mathematics it means change in. Activity 11 Eploring Slope 17
6 ACTIVITY 11 Lesson 11-1 Check Your Understanding 1. Find the slope and the y-intercept for each of the following. Remember to use the ratio. a. y b. 4 y CONNECT TO SPORTS Longboards are larger than the more trick-oriented skateboards. Longboards are heavier and sturdier than skateboards. Some people even use them instead of bicycles. c. 4 y d. y e. Look back at the figure for Item 12. Would a point P that is 9 units up from point W and 1 units to the right be on the line that contains points W, V, and Z? Use similar triangles to eplain your answer. 16. John is longboarding at a constant rate down the road. If 2 minutes after he leaves his house he is 1,000 feet away and at minutes he is 2,00 feet from his house, what would his average rate of change be? 18 SpringBoard Mathematics Course /PreAlgebra, Unit 2 Equations
7 Lesson 11-1 Activity 11 LESSON 11-1 PRACTICE The Tran family is driving across the country. They drive 400 miles each day. Use the table below to answer Items Day Total Miles Driven Complete the table. 18. Draw a graph for the data in the table. Be sure to title the graph and label the aes. Draw a line through the points. 19. Write an equation that can be used to determine the total miles, M, driven over d days. 20. Find the slope and the y-intercept of the line you created, using the graph you drew or the equation you wrote. Eplain what each represents for the Tran family s situation. The graph below shows the money a student earns as she tutors. Use the graph to answer Items y Money Earned Tutoring $0 $00 Money Earned $20 $200 $10 $100 $ Weeks Tutoring What is the slope of the line? 22. What is the y-intercept of the line? 2. Write an equation that can be used to determine how much money, D, the student has earned after w weeks. 24. Attend to precision. Calculate how much money the student will have earned after 2 weeks. Activity 11 Eploring Slope 19
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