5 Task Slippery Slope Topic Proportional Relationships, Lines & Linear Equations Common Core State Standards 8EE6 Copyright @ 2016 The Regents of the University of California This work was supported by grant number #DRL-1020393 from the National Science Foundation and grant number 2012-8075 from the William and Flora Hewlett Foundation.

Agenda Set Up Mind Stretch Workout Check Your Pulse Final Lift Whole class, 5 min Individual, 10-15 min Pairs, Individual, Whole class, 10 min Individual, 15 min

How can we use math to check the proportional reasoning of others? Setup

Learning Goals & Expectations The sides of similar triangles are proportional. If two slope triangles are similar, the line(s) formed by their hypotheses will have the same slope How to justify answers and critique others’ reasoning. We are learning … We will be successful when we… Explain how similar triangles are related. Determine the slope from a graph of a line. Provide evidence to support our thinking about another’s work.

High Quality Work Work is accurate & precise Explanations Problem is set up in a way that helps you solve it. Scale: responses use appropriate units. Work has been checked for calculation errors. Explanations Describe what you did and why you did it. Use multiple representations to show your thinking about math. Include a logical argument and evidence to support each answer. It makes sense.

Multiple Representations Use multiple representations to help you think and to show your thinking to others. Representations Create a chart or table Set up an equation or an algebraic rule Sketch a graph Use words to explain and justify

Prior Knowledge The slope (m) of a line is defined as:

Prior Knowledge A right triangle is a triangle where the angle opposite the longest side of the triangle measures 90 degrees (i.e., is formed by perpendicular lines). The longest side of a right triangle is called the hypotenuse. A slope triangle is a right triangle whose hypotenuse connects two points on a line and whose base is parallel to the x axis and whose altitude is parallel to the y axis.

Prior Knowledge Two right triangles are similar, if the length of corresponding sides are proportional. If two right triangles are similar, then the corresponding angles of the two triangles will have the same measurement.

Prior Knowledge Are the corresponding sides of these triangles proportional? Are the two triangles above, similar? 15 5 9 3 4 12

Prior Knowledge Are the corresponding sides of these triangles proportional? Are the two triangles above, similar? Why must the slope of the smaller triangle be the same as the slope of the larger triangle? 15 5 9 3 4 12

The story so far… Maria, Diego, Sam, and Mia are trying to figure out how similar triangles connect to the slope of a line. Can you help them? Is the slope triangle a “right triangle”? How do you know? Where is the hypotenuse of the triangle? How could you find the amount of rise and amount of run if these were not given?

Mind Stretch

Mind Stretch What is the slope of the blue line (line a)? Draw another line with the same slope.

Mind Stretch What is the slope of the blue line (line a)? Draw another line with the same slope. Look at the red line (line b). The slope of line b is steeper than the slope of line a equal to the slope of line a less steep than the slope of line a … how do you know? b

Workout

Workout Maria, Diego and Sam are computing the slope between pairs of points on the line in this drawing. y Maria finds the slope between the points (0,0) and (3,2). Diego finds the slope between the points (3,2) and (6,4). Sam finds the slope between the points (3,2) and (9,6). x

Workout They have each drawn a triangle to help with their calculations. y Which student has drawn which triangle? Write the student’s name inside their triangle. Finish the slope calculation for each student. Explain the relationship between the three triangles by answering the questions on your worksheet. x

Check Your Pulse

Check Your Pulse Compare your answers with a partner. Then discuss… Where do you agree or disagree with your partner’s answer? What part(s) were difficult for you?

Check Your Pulse On your own… self-assess: Circle your level of understanding for the Workout. I have lots of questions. I need help Almost got it, but I need practice Got it. I can explain this to a classmate.

Final Lift

Final Lift point a (-6,-4); point c (-3,-2) change in x = -6 – 3 = -9 Abby draws a triangle (Triangle ABC) like the triangles drawn by Maria, Diego and Sam. She ends up with a different slope. A B C point a (-6,-4); point c (-3,-2) change in x = -6 – 3 = -9 change in y = -4 – -2 = -6 slope = -9 ÷ -6 = -1.5 Is Abby’s work correct? Explain why you think she is correct or incorrect.

Challenge Using a straight edge, draw a line parallel to Line AC. Label this Line DF. Draw a new slope triangle so that the hypotenuse of the new slope triangle is on Line DF. Label this new Triangle DEF. Use Triangle DEF to calculate the slope of Line DF. How does the slope of Line AC compare to the slope of the Line DF? Why?