Cogito
Grade 8 Math · Chapter 3 · Lesson 1
Why Slope Is the Same Everywhere
The similar triangles you just built.
10 problems · about 19 minutes · 8.EE.B.6
What this lesson teaches
The child explains why the slope between any two points on a line is constant, using similar triangles.
- Slope is rise over run.
- Slope triangles on one line are similar.
- Similar triangles share one ratio, so the slope cannot vary.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Angle-Angle Similarity: Angles 25° and 95°. What is the third?
Answer 60
Why 60°.
Review — Dilations on the Coordinate Plane: (4, −3) dilated by 5. What is the new y?
Answer -15
Why (20, −15).
Warm Up
Straightforward practice. Get the method working first.
4 problemsRise 12, run 4. Slope?
Answer 3
Why 3.
Why is slope the same everywhere on a line?
Answer The slope triangles are similar, so their sides share one ratio.
Why Similarity forces the ratio to match.
Rise 10, run 5. Slope?
Answer 2
Why 10 ÷ 5.
Rise 3, run 6. Slope, as a decimal?
Answer 0.5
Why 3 ÷ 6.
Build It Up
The same ideas with more to keep track of.
2 problemsFalls 9 over a run of 3. Slope?
Answer -3
Why Negative.
What is the slope of a horizontal line?
Answer 0
Why No rise at all.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Slope: A line rises 6 for a run of 2. What is its slope?
Answer 3
Why 3.
Going Down: A line falls 8 over a run of 4. What is its slope?
Answer -2
Why −2.