pAIrmatic Lessons
UNIT 3Slope and proportional relationshipsLesson 2 of 31 Graphing Proportional Relationships2 Slope: Rise Over Run3 Writing Equations in y = mx + b Form

Slope: Rise Over Run

8.EE.6Grade 8Expressions & Equations22 practice questions

Slope tells you how steep a line is. It is the rise divided by the run.

−3−3−2−2−1−111223344556677(−1, 6)(3, −2)

What you'll be able to do

The big idea

Rise over run. Pick two points on a line. The rise is how far y changes. The run is how far x changes. The slope is rise ÷ run.

An example. Take (1, 2) and (4, 8). The rise is 8 − 2 = 6. The run is 4 − 1 = 3. The slope is 6 ÷ 3 = 2.

Keep the order. Subtract in the same order on the top and the bottom. Start with the same point both times.

The sign. A line that goes up to the right has a positive slope. A line that goes down to the right has a negative slope.

Flat and straight up. A flat line has a rise of 0, so its slope is 0. A line that goes straight up has a run of 0. You can't divide by 0, so its slope is undefined.

Plotting. Start at a point. A slope of 2/3 means go right 3, then up 2. Mark the new point. Do it again. Then draw the line.

Why it never changes. Draw a small slope triangle and a big one on the same line. They are similar triangles. So rise ÷ run is the same for both.

Steepness. The size of the slope tells how steep a line is. A slope of −5 is steeper than a slope of 3. Ignore the sign when you compare steepness.

Words you need

Slope
How steep a line is: the rise divided by the run.
Rise
How much y changes from one point to the next.
Run
How much x changes from one point to the next.
Undefined slope
The slope of a line that goes straight up. Its run is 0.
Slope triangle
A right triangle drawn on a line. Its sides show the rise and the run.

Four kinds of slope

The lineIts slope
Goes up to the rightPositive, like 2
Goes down to the rightNegative, like −3
Flat0
Straight upUndefined

Worked examples

Example 1

Find the slope of the line through (−1, 6) and (3, −2).

  1. Rise: −2 − 6 = −8.
  2. Run: 3 − (−1) = 4.
  3. Slope = −8 ÷ 4 = −2. The line goes down to the right.

Answer: −2

−3−3−2−2−1−111223344556677(−1, 6)(3, −2)

Example 2

A line goes through (0, 1) and (4, 4). Draw a slope triangle and find the slope.

  1. Go across from (0, 1) to x = 4. That run is 4.
  2. Go up from there to (4, 4). That rise is 3.
  3. Slope = rise ÷ run = 3/4.

Answer: 3/4

−1−11122334455(0, 1)(4, 4)

Example 3

Graph the line through (1, −2) with slope 2/3.

  1. Plot the point (1, −2).
  2. The slope 2/3 means go right 3, then up 2. That lands on (4, 0).
  3. Do it again to reach (7, 2).
  4. Draw a line through the three points.

Answer: (4, 0), (7, 2)

−3−3−2−2−1−11122334455667788right 3up 2(1, −2)

Common mistakes

Dividing the run by the rise. Slope is rise ÷ run. The change in y goes on top.

Mixing the order. If you start with (6, 5) on top, start with (6, 5) on the bottom too.

Calling a straight-up line's slope 0. Its run is 0, so its slope is undefined. A flat line has slope 0.

Thinking −5 is less steep than 3. Steepness ignores the sign. 5 is bigger than 3, so −5 is steeper.

Practice: 22 questions

12 of 22 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

  1. Find the slope of the line through (−3, 4) and (5, −2).
    −4−4−3−3−2−2−1−1112233445566(−3, 4)(5, −2)
  2. A trail goes downhill. At 2 km along it, a hiker is 1,450 m high. At 6 km, she is 1,150 m high. Find the slope in meters per kilometer. Word problem
  3. Line A goes through (4, −1) and (4, 7). Line B goes through (−2, 5) and (6, 5). Which line has slope 0? Which has an undefined slope?
  4. A ramp rises 1.5 feet over a run of 18 feet. Find its slope as a fraction. Word problem
  5. A plant is 4 cm tall on day 2. It is 13 cm tall on day 5. Ava says the slope is (5 − 2) ÷ (13 − 4) = 1/3. Find her mistake. What is the slope? Word problem
  6. Start at (−2, 1). The slope is −3. Which two whole-number points come next, to the right?

Tier 2 Intermediate One more step to think through

  1. A phone shows 92% charge at 10 minutes. It shows 80% at 25 minutes and 68% at 40 minutes. Find the slope in percent per minute. Word problem
  2. Find the slope of the line through (1/2, 3) and (2, −3).
    −4−4−3−3−2−2−1−111223344(1/2, 3)(2, −3)
  3. A line has slope −3. It goes through (2, 5) and (6, n). Find n.
  4. At hour 6, it is −4°F. At hour 12, which is noon, it is 14°F. Leo writes (14 − (−4)) ÷ (6 − 12) = −3. Find his mistake. What is the slope? Word problem
  5. Two slope triangles sit on the same line. Triangle A has a rise of 6 and a run of 4. Triangle B has a run of 10. Find its rise.
  6. Ana draws a roof line to scale on grid paper. It starts at (0, 1) and has slope 2/5. Which two whole-number points come next, to the right? Word problem

Tier 3 Advanced Several steps - find the hidden structure

  1. Do (−2, 7), (1, 1) and (4, −5) lie on one line? Check the slopes.
    −6−6−4−4−2−222446688(−2, 7)(1, 1)(4, −5)
  2. A road climbs with a slope of 0.06. At the bottom, it is 120 m high. How far across do you go to reach 162 m high? Word problem
  3. A pool holds 1,800 gallons. After 90 minutes of draining, it holds 1,350 gallons. Find the slope in gallons per hour. One hour is 60 minutes. Word problem
  4. Maya has $2 at week 0, $8 at week 3 and $13 at week 5. Kai finds slope 2 from the first two weeks. He says her savings make a line. Find his mistake. Word problem
  5. A drone rises at a steady rate. It is 40 m high at 5 seconds and 76 m high at 14 seconds. How high is it at 20 seconds? Word problem

Tier 4 Challenge Stretch problems, some above grade level

  1. Line A has slope −5. Line B goes through (1, 2) and (3, 8). A ramp rises 4 over a run of 1. Order them from least steep to steepest.
  2. Find the slope of the line through (a, b) and (a + 4, b − 12).
  3. A ski run drops 300 m over 1,200 m across. A second run drops 1/3 as far over 1/5 the distance across. How many times as steep is the second run? Word problem
  4. Roof A rises 2 m over a run of 5 m. Roof B is 1.5 times as steep as Roof A. Its run is 12 m. How far does Roof B rise? Word problem
  5. Two lines are perpendicular when their slopes multiply to −1. Line A goes through (1, 2) and (4, 4). Line B goes through (0, 5) and (2, 2). Are they perpendicular? Above grade level

Answers

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