Slope isn't just a number on a graph — it's a rate. €12 per hour. 10 metres per second. 3 kg of hay per day. Whenever a quantity changes at a steady rate, you have a linear relationship and the slope tells you exactly how fast it changes. Lesson 18 is about recognising this in tables and graphs.
Rate of change from a table:
$$\text{rate of change} = \frac{\Delta y}{\Delta x} = \frac{\text{change in output}}{\text{change in input}}$$
If this ratio is the same between every consecutive pair of rows, the relationship is linear (constant rate). If the ratio changes, it is non-linear.
From a graph: On a straight line, pick any two points and compute $\frac{y_2-y_1}{x_2-x_1}$. If the line is straight, the rate of change is constant everywhere.
Worked Example
Worked Example — Is this table linear?
Mia records her savings each week:
| Week ($x$) | Savings in € ($y$) | Change in $y$ |
|---|
| 1 | 62 | — |
| 2 | 74 | $+12$ |
| 3 | 86 | $+12$ |
| 4 | 98 | $+12$ |
$\Delta y = 12$ every time, $\Delta x = 1$ every time → rate of change $= 12/1 = 12$ €/week. Constant → linear.
Warm-Up — Read rate of change from a table (values given)
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Problem 1
Mia buys hay at a fixed price. The table shows total cost vs. kilograms purchased. Find the rate of change.
| kg of hay ($x$) | Total cost €($y$) |
|---|
| 2 | 7 |
| 4 | 14 |
| 6 | 21 |
| 8 | 28 |
Rate of change $= \dfrac{\Delta y}{\Delta x} = \dfrac{14-7}{4-2}$
calculate → interpret units →
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Problem 2
A horse walks along a trail. The table shows distance from the start at different times. Find the rate of change and state whether the pace is constant.
| Time in min ($x$) | Distance in m ($y$) |
|---|
| 0 | 0 |
| 5 | 400 |
| 10 | 800 |
| 15 | 1200 |
Rate of change $= \dfrac{400}{5}$
calculate → units → constant? →
Core Problems — check for linearity, find and interpret rate of change
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Problem 3
Mia runs a riding lesson business. She records weekly income for 4 weeks:
| Week ($x$) | Income € ($y$) |
|---|
| 1 | 120 |
| 2 | 240 |
| 3 | 360 |
| 4 | 480 |
(a) Find the rate of change between each consecutive pair of weeks.
(b) Is this relationship linear?
(c) What does the rate of change mean in context?
compute Δy/Δx each time → check constant → interpret →
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Problem 4
Here are two tables. Decide which is linear and which is non-linear. Explain how you know.
Table A
Table B
calculate Δy/Δx for each → constant? → classify →
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Problem 5
A line on a graph passes through $(0, 5)$ and $(4, 21)$.
(a) Find the rate of change (slope).
(b) Describe what this line looks like: where does it start? Does it rise steeply or gently?
(c) What is the value of $y$ when $x = 7$? (Hint: use the rate of change to extend.)
slope → describe → extend →
From here — write the equation yourself, then solve
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Problem 6
Mia records feed costs for her stable over 5 days:
| Days ($x$) | Total cost € ($y$) |
|---|
| 1 | 8.50 |
| 2 | 17.00 |
| 3 | 25.50 |
| 4 | 34.00 |
| 5 | 42.50 |
(a) Find the rate of change.
(b) Write an equation $y = mx$ for this relationship.
(c) Predict the cost after 12 days.
rate of change → equation → substitute x=12 →
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Problem 7
A graph shows the number of riding lessons Mia gives vs. her total weekly earnings. The line starts at $(0, 0)$ and goes through $(3, 75)$ and $(8, 200)$.
(a) Find the rate of change between the two given points.
(b) Is the relationship linear?
(c) Interpret the rate of change: what does €/lesson tell you about her pricing?
slope formula → check → interpret →
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Problem 8 Challenge
The table below shows the area of a square with side length $x$:
| Side $x$ (cm) | Area $y = x^2$ (cm²) |
|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
(a) Compute the rate of change between each consecutive pair of rows.
(b) Is it constant?
(c) This means area is non-linear. Explain in one sentence why that makes intuitive sense.
Δy/Δx each time → non-constant → explain →
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Problem 9 Open
Design a pricing table for Mia's riding lesson business that has a constant rate of change of exactly €12 per lesson, but starts with a one-time registration fee of €30. How does a starting fee affect the table? Write an equation for your pricing model.
choose x values → compute y = 12x + 30 → write equation → check rate of change →
Show answers
Problem 1
$\Delta y / \Delta x = 7/2 = 3.5$ €/kg · Hay costs €3.50 per kilogram.
Problem 2
Rate of change $= 400/5 = 80$ m/min · Constant throughout — the horse walks at a steady pace of 80 metres per minute.
Problem 3
(a) $\Delta y/\Delta x = 120/1 = 120$ each time · (b) Yes, linear — constant rate of change. (c) Mia earns €120 per week from lessons.
Problem 4
Table A: $\Delta y = 3, 3, 3$ → constant → linear (rate of change $= 3$) · Table B: $\Delta y = 2, 4, 8$ → not constant → non-linear (exponential — doubling each time).
Problem 5
(a) $m = (21-5)/(4-0) = 16/4 = 4$ · (b) Starts at $y=5$ when $x=0$; rises steeply (4 units up per 1 unit across). (c) $y = 5 + 4(7) = 5 + 28 = 33$
Problem 6
(a) Rate of change $= 8.50/1 = 8.50$ €/day · (b) $y = 8.5x$ · (c) $y = 8.5 \times 12 = 102$ euros after 12 days.
Problem 7
(a) $m = (200-75)/(8-3) = 125/5 = 25$ · (b) Yes — rate of change is constant ($25$ €/lesson; check: $75/3 = 25$ ✓). (c) Mia charges €25 per lesson.
Problem 8
(a) $\Delta y$: $3, 5, 7$ — increasing by 2 each time. (b) Not constant → non-linear. (c) As the side grows, the area grows faster because you're adding strips along two sides plus a corner — so the rate accelerates.
Problem 9
Open — equation: $y = 12x + 30$. Example values: $(0,30), (1,42), (2,54), (3,66)$. Rate of change $= \Delta y/\Delta x = 12$ ✓. The registration fee ($30) is the starting value (y-intercept), not part of the per-lesson rate.
Coming up next → Lesson 19: Area and Perimeter
Before pulling everything together with graphing lines ($y = mx + b$), we cover one more German Lehrplan essential: area and perimeter of rectangles, triangles, and composite shapes. You'll use the equation-solving skills from earlier to find missing sides.