Slope & Unit Rate — Drills
Slope & Unit Rate has 8 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Unit rate — A rate with denominator 1 — how much for exactly one.
- Proportional — Two quantities with a constant ratio: yx = k always.
- Slope — How much y changes for each 1 that x increases.
- Rise — The vertical change, y2 - y1.
- Run — The horizontal change, x2 - x1.
From two points
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the slope of the line through (-6, 0) and (-1, 7).
Worked answer
7/5
Rise: 7 - (0) = 7
Run: -1 - (-6) = 5
m = 75
Problem 2
Find the slope of the line through (-2, -2) and (-1, 1).
Worked answer
3
Rise: 1 - (-2) = 3
Run: -1 - (-2) = 1
m = 3
Special cases
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
What is the slope of the line x = 7?
Worked answer
undefined
Every point has the same x, so the run is 0.
Dividing by zero is not allowed, so the slope is undefined.
Problem 2
What is the slope of the line y = -8?
Worked answer
0
Every point has the same y, so the rise is 0.
m = 0
Unit rate
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A machine covers 230 litres in 10 minutes. Find the rate in litres per minute.
Worked answer
23
230 ÷ 10 = 23
23 litres every minute.
Problem 2
A machine covers 248 pages in 8 minutes. Find the rate in pages per minute.
Worked answer
31
248 ÷ 8 = 31
31 pages every minute.
Table
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A table shows x: 4, 8, 12 and y: 28, 56, 84. Find the unit rate yx.
Worked answer
7
28 ÷ 4 = 7
56 ÷ 8 = 7 ✓
The unit rate is 7.
Problem 2
A table shows x: 4, 8, 12 and y: 40, 80, 120. Find the unit rate yx.
Worked answer
10
40 ÷ 4 = 10
80 ÷ 8 = 10 ✓
The unit rate is 10.
Proportional
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Is the relationship y = 3x proportional? Answer yes or no.
Worked answer
yes
At x = 0, y = 0.
It passes through the origin, so it is proportional.
Problem 2
Is the relationship y = 5x proportional? Answer yes or no.
Worked answer
yes
At x = 0, y = 0.
It passes through the origin, so it is proportional.
Comparing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Line A passes through (0, 0) and (5, 40). Line B is y = 6.5x. Which has the larger slope, A or B?
Worked answer
A
Line A: m = 405 = 8
Line B: m = 6.5
8 > 6.5, so A.
Problem 2
Line A passes through (0, 0) and (5, 10). Line B is y = 5x. Which has the larger slope, A or B?
Worked answer
B
Line A: m = 105 = 2
Line B: m = 5
5 > 2, so B.
From a graph
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A line rises 4 units for every 5 units to the right. Find the slope.
Worked answer
4/5
Rise = 4, run = 5
m = 45
Problem 2
A line falls 6 units for every 4 units to the right. Find the slope.
Worked answer
-3/2
Rise = -6, run = 4
m = -32
Finding the missing value
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A line through (-3, 2) and (3, k) has slope 7. Find k.
Worked answer
44
k - (2)3 - (-3) = 7
k - (2) = 7 × 6 = 42
k = 44
Problem 2
A line through (3, 8) and (8, k) has slope 2. Find k.
Worked answer
18
k - (8)8 - (3) = 2
k - (8) = 2 × 5 = 10
k = 18