The two ideas are connected, but the words are not interchangeable
Slope
The graph number:
$$m=\frac{\Delta y}{\Delta x}$$
It tells how the line rises or falls when you move right.
Rate of change
The story meaning of that number:
$$\frac{\text{output units}}{\text{input units}}$$
Examples: metres per second, euros per lesson, litres per day.
On a graph of a real situation, the slope's value gives the rate of change. The axes give it meaning and units.
1 · CalculateFind $\Delta y/\Delta x$.
2 · Read axesWrite “$y$-units per $x$-unit”.
3 · Percent gateConvert only if the question asks for a gradient and rise/run use compatible units.
The percent gate: For a physical gradient with compatible length units,
$$\text{percentage gradient}=m\times100\%$$
A slope of $4$ is therefore a $400\%$ gradient. But a rate of $4$ €/hour stays 4 €/hour. It does not become $400\%$.
One number, three correct descriptions
Plain coordinate graph: $m=4$, so the slope is $4$.
Distance-time graph: $m=4$, so the rate is $4$ metres per second.
Equal-unit trail gradient: $m=4$, so the gradient is $400\%$.
Six quick checks
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Problem 1 · Name both
A graph has no real-world labels and passes through $(1,3)$ and $(3,11)$. Find its slope. Can you give a meaningful real-world rate of change?
$m=\dfrac{11-3}{3-1}$
calculate slope → check whether the axes provide units →
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Problem 2 · Add the units
A horse drinks $12$ litres of water over $3$ days at a constant rate. On a graph, $x$ is days and $y$ is litres consumed. Find the slope and state the rate of change.
$m=\dfrac{12\text{ L}}{3\text{ days}}$
number → y-units per x-unit → percent or not? →
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Problem 3 · Pass the percent gate Gradient
A trail rises $8$ m over a horizontal run of $20$ m. Find (a) the slope and (b) the percentage gradient.
rise/run → decimal slope → multiply by 100% →
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Problem 4 · Interpret the graph
A distance-time graph passes through $(2,10)$ and $(6,30)$, where $x$ is time in hours and $y$ is distance in kilometres. Find (a) the slope and (b) the rate of change with units. Should the answer be written as a percentage?
subtract consistently → calculate → read axes → decide about percent →
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Problem 5 · Percentage points Careful
A phone battery falls from $100\%$ to $84\%$ in $2$ hours. The graph's $y$-axis is battery percentage and its $x$-axis is hours. Find the slope and describe the rate precisely.
change in displayed percentage ÷ time → include sign → say “percentage points per hour” →
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Problem 6 · Error detective Explain
A student says: “Whenever the slope is $4$, the rate of change is $400\%$.” Correct the sentence. Give one example where $400\%$ is right and one where it is wrong.
state the rule → equal-unit gradient example → different-unit rate example →
Show answers
Problem 1
$m=(11-3)/(3-1)=8/2=4$. The slope is $4$. Without labelled axes or a situation, there are no meaningful real-world units to name.
Problem 2
Slope $m=12/3=4$. Rate of change $=4$ litres per day. This is not a percentage gradient because the axes measure different quantities.
Problem 3
(a) $m=8/20=0.4$. (b) Percentage gradient $=0.4\times100\%=40\%$. Both measurements are lengths, so the conversion is meaningful.
Problem 4
$m=(30-10)/(6-2)=20/4=5$. The rate of change is $5$ km/h. It should not be written as $500\%$ because kilometres and hours are different kinds of units.
Problem 5
$m=(84-100)/(2-0)=-16/2=-8$. The battery decreases by 8 percentage points per hour. It is not a percentage gradient and we do not multiply $-8$ by $100$.
Problem 6
Correction: a slope of $4$ becomes $400\%$ only when reporting an equal-unit gradient. A rise of $4$ m over $1$ m across is a $400\%$ gradient. Earnings increasing by €4 each hour have a rate of €4/hour, not $400\%$.
Tutor cue: After every calculation, ask only: “What does the $y$-axis measure, and per what $x$-unit?” Ask about percent last, never first.
Ready to continue with Lesson 20
Keep the three-part answer habit: number, units, interpretation.