Time To Hit The Slopes. Exploring Slopes with Similar Triangles

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1 Time To Hit The Slopes Eploring Slopes with Similar Triangles Learning Goals In this lesson, ou will: Use an equation to complete a table of values. Graph an equation using a table of values. Use transformations to create similar triangles. Use similar triangles to show the slope is the same between an two distinct points on a non-vertical line in a coordinate plane. 015 Carnegie Learning Where might ou see each of the signs shown? What do ou know about the triangles on the signs? For the signs that include numbers, what do ou think those numbers represent? How do triangles and steepness, or slope, relate to each other? In this lesson, ou will eplore how ou can use our knowledge of triangles to connect algebraic and geometric concepts. 9.5 Eploring Slopes with Similar Triangles 1

2 Problem 1 Mr. Green s Challenge Mr. Green posted the following question on his math blog: Is the slope of a line the same between an two points on the line? He told his students if the were able to provide a convincing answer to his question using similar triangles to support their reasoning he would bake them a pie on Pi Da. Mr. Green encourages teamwork, so Cooper and Rlee decide to accept the challenge together. Cooper suggests the begin b drawing a non-vertical line 5 on the coordinate plane, as shown. = 1. Consider the non-vertical line that was drawn. a. State at least one reason Cooper suggested drawing the line through the origin. b. State at least one reason Cooper suggested the equation Carnegie Learning Chapter 9 Similarit

3 Rlee uses the equation to set up a table of values.. Complete the table of values and plot the values on the coordinate plane. = Net, Cooper uses a point on the line to create a right triangle. (, ) 015 Carnegie Learning 3. Use Cooper s method to form the remaining right triangles b connecting each point on the line to its horizontall aligned point on the -ais. How man total right triangles were formed? 9.5 Eploring Slopes with Similar Triangles 3

4 . Cooper All the right triangles formed are similar to each other. Use transformations to prove Cooper s claim. Can ou prove Cooper's claim using different reasoning? 5. Identif the coordinates of a fifth point on the graph of the line 5. a. If a horizontal line is drawn from the location of the fifth point to the -ais, does the point form another similar right triangle? b. How could ou prove that this new triangle is similar to the other triangles? c. How can more right triangles similar to the triangles formed b Cooper and Rlee be created? 015 Carnegie Learning Chapter 9 Similarit

5 Cooper sees the connection between transformations and similar triangles, but he does not understand how these similar triangles can be used to show the slope is the same between an two distinct points on the non-vertical line, 5.. Help Rlee eplain the connection to Cooper. a. Use the slope formula to determine the slope between an two points on the line. b. If a dilation or rotation is used to create various right triangles on the graph, is this enough information to determine the right triangles formed are similar? c. In general, what is the relationship between the ratios associated with the corresponding sides of similar triangles? It might be helpful to look at our graphs. d. How can ou use a graph to determine the slope of an two points on a non-vertical line? 015 Carnegie Learning e. How are the quotients of the lengths of the legs of an two of these right triangles related? 9.5 Eploring Slopes with Similar Triangles 5

6 Given the same line and same points, Rlee uses a different method to create similar right triangles. = (, ) (, ) (, ) (, ) 7. Eplain wh Rlee s right triangles are similar.. How does Rlee s graph support the claim that the slope of a line is the same between an two points on the line? 015 Carnegie Learning Chapter 9 Similarit

7 9. Sketch two additional sets of similar right triangles using the graph of the line 5 and the points identified in Rlee s table of values. = = After reviewing Cooper and Rlee s work, Mr. Green had one final question. 10. Is the conclusion true for all equations written in the form 5 m? 015 Carnegie Learning 9.5 Eploring Slopes with Similar Triangles 7

8 Problem A Different Point of View Reggie and Maureen also decided to accept Mr. Green s challenge and work together. Reggie drew the non-vertical line 5 5 on the graph as shown. = 5 Net, Maureen uses Reggie s equation to set up a table of values. 1. Complete the table of values and plot the values on the coordinate plane. = Carnegie Learning Chapter 9 Similarit

9 . Use the graph of the line shown to form right triangles using an method. 3. Use an method to justif that these triangles are similar. 015 Carnegie Learning 9.5 Eploring Slopes with Similar Triangles 9

10 . Identif the coordinates of a sith point on the graph of the line 5 5. a. If a horizontal line is drawn from the location of the sith point to the -ais, use transformations to justif wh this new triangle is similar to the triangles above. b. How man similar triangles can be formed on the graph of the line 5 5? How do ou know? 5. Use the slope formula to determine the slope between an two points on the line.. Sketch two additional sets of similar right triangles using the graph of the line 5 5 and the points identified in Maureen s table of values. 015 Carnegie Learning 10 Chapter 9 Similarit

11 After reviewing Reggie and Maureen s work, Mr. Green had one final question. 7. Is the conclusion true for all equations written in the form 5 m 1 b?. In Problem 1, ou concluded that similar triangles could be used to show the slope of a line is the same between an two points on the line 5 m. Likewise, in Problem, ou concluded that similar triangles could be used to show the slope of a line is the same between an two points on the line 5 m 1 b. Use our knowledge of transformations and similarit to eplain how ou could use our conclusion from Problem 1 to justif the conclusion in Problem. 015 Carnegie Learning 9.5 Eploring Slopes with Similar Triangles 11

12 Problem 3 Strategies! Use the given equation to complete the table of values and graph. Then use the points on the graph to sketch similar triangles that ma be used to show the slope of a line is the same between an two points on the line Carnegie Learning 1 Chapter 9 Similarit

13 Carnegie Learning Be prepared to share our solutions and methods. 9.5 Eploring Slopes with Similar Triangles 13

14 1 Chapter 9 Similarit 015 Carnegie Learning