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Meet Slope

Author:Jakob Swanson, Tim Brzezinski

SOME SIMPLE INSTRUCTIONS

1. Throughout the class, I would like you to work through this activity independently. 2. There are 5 questions that I will collect answers from these questions that are labeled with a ****. You will write these answers on a separate sheet of paper with your name on it and I will collect it. These will be at different points in the class and not all at once. Some of your responses will be celebrated by me writing it on the board but it will remain anonymous. 3. If there are any questions that you would rather write or draw on a separate sheet of paper, that is fine. I will come around to check your work.

Learning Objectives (Today we will...)

Comprehend the term “slope” to mean the quotient of the vertical distance and the horizontal distance between any two points on a line. Draw a line on a coordinate grid given its slope and describe (orally) observations about lines with the same slope. Justify (orally) that all “slope triangles” on one line are similar by using transformations or Angle-Angle Similarity.

Math Objective

We are working towards... 8.EE.B.6 [Using] similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx  for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.

A Quick Review

Which is the best representation of a quotient?

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

What do we call lines that do not intersect?

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

How do you know?

****Out of the 3 rollercoasters that I drew below which of them is the steepest? How do you know?

Image

How do we determine slope?

Play around with the points on the line in the grid below. Notice how the "slope" changes as you move the points around.

****How does this program know what the slope of the ramp is?

What does it mean when the slope of the ramp =0?

What does it mean for the slope to be UNDEFINED!

What would the slope of this ride be and why?

Is the slope a quotient?

Select all that apply
  • A
  • B
Check my answer (3)

****Explain your previous answer.

Lines with the same slope

In the activity below there are two lines. Using the bar in the top left, you can adjust the slope of the lines.

****As you play around with the slope of the two lines, what happens when you set the slopes equal to each other?

Another Quick Review

How can I show two triangles are similar? Select all that apply.

Select all that apply
  • A
  • B
  • C
Check my answer (3)

Slope Triangles

In the ramp example above, there was a right triangle created to connect the two points. This triangle is called a slope triangle because it has a right angle and the hypotenuse (the longest side, across from the right angle) is along the line connecting two points. In the activity below, the orange triangle is also a slope triangle for the same reasons.
Check the boxes to the right of the graph to reveal the other triangles.

Is Triangle 1 a slope triangle?

Select all that apply
  • A
  • B
Check my answer (3)

Is Triangle 2 a slope triangle?

Select all that apply
  • A
  • B
Check my answer (3)

Why is Triangle 3 a slope triangle?

What is similar between Triangle 1,2 and 3? Select all that apply.

Select all that apply
  • A
  • B
  • C
Check my answer (3)

I claim that all 4 triangles in this activity are similar.

****How would you defend this argument?

Extended Questions

Rewrite the following in your own words. Comprehend the term “slope” to mean the quotient of the vertical distance and the horizontal distance between any two points on a line.

Your skateboarding down a hill, you're going pretty fast and you estimate the slope of the hill to be around -5. Your friend is skating right next to you and claims his hill has a slope is -6. Is this possible if you are on the same hill? Justify your answer using the idea of parallel lines.

Using what you know about slope, which rollercoaster from the drawings was the steepest?

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