Key Terms Sentence Frames Scaffolded Questions

Similar Triangles I with Marcus because . What relationship is the graph table demonstrating? The graph?

Slope The triangles I drew helped me see .

Proportions The equation I generated to compare is .

How are the table and the graph similar? Different?

Equation What is slope? How can you determine the slope of this relationship?

Task Description Standards

In this task, students are asked to 8.EE.B.6

Mathematical Understandings Extending Understandings

Upon successful completion of these tasks students will demonstrate the ability to:

Sketch 3 or more similar triangles associated with the graph:

● Creates the rise/run ratio for each triangle, and demonstrates the rise/runs to be equivalent.

● Calculates the slope using the endpoints of the hypotenuses of each similar triangle and uses the ratio of the difference of the y-values to the difference of the x-values to verify the slope of each

● Calculates the slope of one similar triangle and notes that since the triangles are similar, all slopes must be equal.

Employ prior knowledge of proportions and slope to develop a conceptual understanding of why the slope of a line remains constant.

Give an equation in the form y = mx + b or y = mx to represent a line graphed on a coordinate plane.

To move students towards deeper understanding, they should be given opportunities to:

Use multiple representations to demonstrate and explain why any two points on a non-vertical line generate the same slope by using similar slope triangles.

Graph the equation of a line, identify the slope, and create a visual representation that shows that the slope is the same between any two points on a line using similar triangles formed by the vertical and horizontal lines from a point on a non-vertical line.

Supporting Strategies

If students are struggling to access this grade level task, additional supports and strategies could be employed as students are engaging with the task.

Additional ResourcesFinding the Slope of a Line on a Coordinate Grid Proving Slope is Constant Using SimilarityWrite Linear Equations from TablesCompare Proportional Relationships by Analyzing Tables, Graphs and Equations

Marcus' Lemonade Stand

The graph and the data below give two different ways Marcus shows the cost of running a lemonade stand.

a. Marcus thinks the slope is 0.75. Use 3 or more similar triangles in the graph above to justify why you agree or disagree with Marcus.

b. Write an equation for the cost of running the lemonade stand in terms of cups of lemonade sold.

Marcus' Lemonade Stand

Question: Marcus thinks the slope is 0.75. Use 3 or more similar triangles in the graph above to justify why you agree or disagree with Marcus.

Standard(s) 8.EE.B.6

Assessing Questions

This student correctly sketches three triangles associated with the graph and creates the rise/run ratio for each triangle.They state that Marcus is correct and use the slope found when calculating the rise/run ratios and showing that 3/4 is equivalent to 0.75.

How did you generate your triangles?

This student shows two similar triangle relationships each with a rise/run of ¾. The student also calculates the endpoints of the hypotenuses of each similar triangle, and uses the ratio of the difference of the y-values to the difference of the x-values to verify the slope of each triangle. The student agrees with Marcus noting that a slope of0.75 is equivalent to ¾ .

How did you generate the values for your triangle?

Student Work Part A

Question: Write an equation for the cost of running the lemonade stand in terms of cups of lemonade sold.

Standard(s) 8.EE.B.6

Assessing Questions

This student writes an appropriate equation to model the relationship between the cost and cups of lemonade.

How did you determine your equation?

What does the ¾ represent?

What does the 4 represent?

This student writes an equation in y=mx+b form and describes the meaning of the slope.The student describes how they calculated the slope and the meaning of the y-intercept. The student also substitutes values into the equation to demonstrate.

Student Work Part B