Slope & Unit Rate — Print

Slope is a rate of change wearing a geometric coat: how much y moves for every one step in x.

What the practice questions cover

These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

A wheelchair ramp runs from the pavement up to a door. How steep is it?

"It rises two metres" is not an answer — over what distance? A two-metre rise over one metre is a wall; over a hundred metres it is nearly flat.

the same ramp, measured at two places across 3 up 2 across 3 up 2 near the bottom or near the top, it climbs 2 for every 3 across

So measure both. Near the bottom: across 3, up 2. Near the top: across 3, up 2 again.

Anywhere you measure, the ramp climbs 2 for every 3 across. That unchanging ratio is what makes it a ramp and not a hill.

A surface whose ratio changed partway would bend — and a bent line is not a straight line.

Slope is that ratio: rise over run. It is the same number as a unit rate, which is why "75 miles per hour" and "slope 75" are the same idea in different clothes.

Slope as rise over run, the same anywhere on the line slope is a rate: how far up for every step across run 3 rise 2 run 3 rise 2 rise run = 2/3 both triangles give 2/3 that is what makes it a straight line going down as you move right makes the rise negative

One idea, two names

A line climbing 3 units for every 4 across has slope 34. A car covering 150 miles every 2 hours has a unit rate of 75 mph.

They are the same calculation: change in the output divided by change in the input.

Once you see that, every "find the unit rate" question is a slope question and every slope question is about a rate.

Slope formula

m = y2 - y1x2 - x1. Rise on top, run on the bottom. Subtract in the same order in both places.

Reading slope from a graph

Find two points that sit exactly on lattice crossings. Count up or down to get the rise, then across to get the run.

Up is positive rise, down is negative. Right is positive run.

Pick points far apart — a long triangle is far more accurate to read than a short one.

Count, do not guess

Put your pencil on the first point. Count the grid squares up, out loud. Then count across. Write the fraction. Only then simplify.

Positive, negative, zero, undefined

Uphill left to right: positive. Downhill: negative.

A horizontal line has rise 0, so m = 0. Flat, and going nowhere vertically.

A vertical line has run 0, and dividing by zero is not allowed, so its slope is undefined. Not zero — undefined.

Zero and undefined are opposites

Horizontal: slope 0, equation y = 5. Vertical: slope undefined, equation x = 5. Learn them as a pair or you will swap them.

Proportional relationships

If a line passes through the origin, then y = mx and the relationship is proportional.

In a proportional table, yx is the same for every row. That common value is the unit rate.

If the line misses the origin, the relationship is still linear, but not proportional — there is a starting amount.

Comparing two rates

The steeper line has the larger unit rate. To compare a graph with a table, work out both unit rates and compare the numbers.

Worked examples

Slope from two points

Find the slope of the line through (2, 3) and (8, 12).

  1. Rise: 12 - 3 = 9 Change in y, second minus first.
  2. Run: 8 - 2 = 6 Change in x, in the same order.
  3. m = 96 = 32 Always simplify the fraction.

m = 32

A negative slope

Find the slope through (-1, 7) and (3, -5).

  1. Rise: -5 - 7 = -12 The line drops as x increases.
  2. Run: 3 - (-1) = 4 Subtracting a negative adds.
  3. m = -124 = -3 Down 3 for every 1 across, which fits a falling line.

m = -3

Unit rate from a table

A table shows 2 hours for $26, 5 hours for $65, 8 hours for $104. Find the hourly rate, and say whether it is proportional.

  1. 26 ÷ 2 = 13 Dollars per hour for the first row.
  2. 65 ÷ 5 = 13 and 104 ÷ 8 = 13 Same ratio every row.
  3. So the rate is $13 per hour and the relationship is proportional. A constant ratio means the graph passes through the origin.

$13 per hour, proportional

Words to know

Slope
Rise divided by run. The letter m.
Rise
The vertical change, y2 - y1.
Run
The horizontal change, x2 - x1.
Unit rate
The amount per one — miles per hour, dollars per pound. The same number as the slope.

Mistakes to watch for

Putting the run on top. Slope is rise over run — vertical over horizontal — because it answers "how much does y change for each step in x." Upside down, a steep line comes out shallow.

Subtracting in opposite orders. If you do y2 - y1 on top you must do x2 - x1 underneath. Swap one and the sign flips, turning an uphill line into a downhill one.

Calling a vertical line slope zero. Its run is zero, and dividing by zero is not allowed, so the slope is undefined. It is the horizontal line that has slope 0 — learn the pair together or you will keep swapping them.

Calling every straight-line relationship proportional. Proportional needs the line to pass through the origin, so that yx is the same in every row. A $20 joining fee breaks that, even though the graph is perfectly straight.

The m in y = mx + b has no agreed origin. The popular story that it stands for French monter, "to climb," appears to be invented — Descartes wrote in French and never used it.

One historian traced the earliest m to an 1844 British textbook, where the author simply used the letters m and b without comment.

In other countries the letters differ: Germany uses y = mx + n, Sweden y = kx + m. The mathematics is identical; only the habit differs.

Problem 1

Find the slope through (1, 2) and (5, 10).

Worked answer

2

Rise: 10 - 2 = 8

Run: 5 - 1 = 4

m = 84 = 2

Problem 2

Find the slope through (0, 3) and (4, 9).

Worked answer

3/2

Rise: 9 - 3 = 6

Run: 4 - 0 = 4

m = 64 = 32

Problem 3

Find the slope through (2, 8) and (6, 2).

Worked answer

-3/2

Rise: 2 - 8 = -6

Run: 6 - 2 = 4

m = -64 = -32

Problem 4

What is the slope of the line y = -4?

Worked answer

0

Every point has y = -4

Rise is 0

m = 0

Problem 5

A car travels 210 miles in 3 hours. Find the speed in miles per hour.

Worked answer

70

210 ÷ 3 = 70

70 mph

Problem 6

What is the slope of the line x = 7?

Worked answer

undefined

Run is 0

Dividing by zero is not allowed

The slope is undefined

Problem 7

Find the slope through (-3, 5) and (1, -7).

Worked answer

-3

Rise: -7 - 5 = -12

Run: 1 - (-3) = 4

m = -3

Problem 8

A printer prints 1,260 pages in 18 minutes. Find the rate in pages per minute.

Worked answer

70

1260 ÷ 18 = 70

70 pages per minute

Problem 9

A table shows x: 3, 6, 9 and y: 12, 24, 36. Find the unit rate yx.

Worked answer

4

12 ÷ 3 = 4

24 ÷ 6 = 4 ✓

The unit rate is 4

Problem 10

Which relationship is proportional?

  1. y = 6x
  2. y = 6x + 2
  3. y = 6
  4. y = x2
Worked answer

y = 6x

y = 6x gives y = 0 when x = 0

The others do not, or are not linear

Problem 11

Line A passes through (0,0) and (4, 10). Line B is y = 2x. Which has the larger slope?

Worked answer

A

Line A: m = 104 = 2.5

Line B: m = 2

2.5 > 2, so A

Problem 12

A line passes through (2, 5) and falls 3 units for every 4 units right. Find the slope.

Worked answer

-3/4

Rise = -3, run = 4

m = -34

Problem 13

A line through (3, k) and (7, 14) has slope 2. Find k.

Worked answer

6

14 - k7 - 3 = 2

14 - k = 8

k = 6

Problem 14

Runner A covers 9 km in 45 minutes; runner B covers 14 km in 80 minutes. Who is faster, in km per minute?

Worked answer

A

A: 9 ÷ 45 = 0.2 km/min

B: 14 ÷ 80 = 0.175 km/min

A is faster

Problem 15

A line through (-2, 1) and (c, 13) has slope 4. Find c.

Worked answer

1

13 - 1c + 2 = 4

12 = 4(c + 2)

3 = c + 2, so c = 1

Problem 16

A table shows x: 2, 5, 8 and y: 7, 17.5, 28. Is it proportional? Answer yes or no.

Worked answer

yes

7 ÷ 2 = 3.5

17.5 ÷ 5 = 3.5

28 ÷ 8 = 3.5

Constant, so yes — and y = 3.5x

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