📐 Grade 8 · Lesson 6 of 15
Rate Race: Constant Rates Are Just Slope
Speed, price-per-item, litres-per-minute — every steady rate is the slope of a straight line through the origin.
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The big idea: A constant rate — kilometres per hour, dollars per kilogram, pages per minute — describes something that changes by the same amount every step. Plot it and you get a straight line, and the rate is exactly its slope. The faster the rate, the steeper the line. This turns 'who is faster?' into 'whose line is steeper?'
A rate is a story about steady change
Imagine a runner moving at a steady 6 kilometres per hour. After 1 hour she has covered 6 km; after 2 hours, 12 km; after 3 hours, 18 km. Every hour adds the same 6 km. When something grows by the same amount each step, plotting it produces a perfectly straight line.
That “same amount each step” is the rate — and, as you will see, it is exactly the line's slope.
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Unit rate = slope
A constant rate (the amount of change per one unit) is the slope of the line that graphs it. Because 0 of the input gives 0 of the output, the line passes through the origin (0, 0).
\[ y = m\,x \]
A proportional relationship: m is the unit rate (e.g. km per hour), x is the input (hours), y is the output (km). This is y = mx + b with b = 0.
Steeper line, faster rate
Two runners start at the same place. The faster one pulls ahead more with every passing hour, so her distance line climbs more steeply. Comparing rates becomes comparing steepness: the steeper line wins the race.
Use the tool below to feel this. The Slope (m) slider is the rate — crank it up and the line steepens, showing a faster runner covering more distance per hour.
🎮 Interactive: the rate is the slope LIVE
Read m as a rate — say kilometres per hour. Slide m up and the line steepens: more distance for each hour that passes. For a pure rate, keep the Intercept (b) at 0 so the line starts at the origin (a runner who begins at the start line, distance 0 at time 0).
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Finding a unit rate from any two points
On a rate graph, pick any point and divide its y by its x. A line through (2, 12) gives 12 ÷ 2 = 6 per unit. That is just rise over run measured from the origin — slope again, wearing the word “rate.”
📝 Worked example: A printer produces 90 pages in 3 minutes at a steady rate. Write the relationship and find its unit rate.
- Unit rate = pages ÷ minutes = 90 ÷ 3 = 30 pages per minute.
- That rate is the slope, and the printer starts at 0 pages, so b = 0.
- Write it as y = mx with m = 30.
✓ Pages y after x minutes: <strong>y = 30x</strong>. The line's slope, 30, is the printing rate.
📝 Worked example: Store A sells 4 apples for $3; Store B sells 6 apples for $5. Which store is cheaper per apple?
- Store A unit price = 3 ÷ 4 = $0.75 per apple.
- Store B unit price = 5 ÷ 6 ≈ $0.83 per apple.
- The cheaper store has the smaller cost-per-apple — the gentler slope on a cost-vs-apples graph.
✓ Store A is cheaper at $0.75 per apple versus about $0.83 at Store B.
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Rate only equals slope when change is constant
The rate-as-slope idea works for steady rates that graph as a straight line through the origin. If a runner speeds up and slows down, the graph curves and there is no single slope — different parts have different steepness.
Check your understanding
1. A car travels at a constant 80 km/h. On a distance-vs-time graph, the number 80 is the line's…
A constant speed is the rate of change of distance over time — the slope of the line.
2. A proportional relationship graphs as a line that always passes through which point?
With y = mx, putting x = 0 gives y = 0, so the line goes through the origin (0, 0).
3. Line P has slope 5 and line Q has slope 2, both through the origin. Which represents the faster rate?
A steeper line (larger slope) means more output per unit of input — a faster rate. Slope 5 beats slope 2.
4. A hose fills a tank and the graph passes through (4, 20) litres. What is the fill rate?
Unit rate = y ÷ x = 20 ÷ 4 = 5 litres per minute — the slope from the origin to that point.
5. Which relationship is a constant rate you can describe with a single slope?
Being paid a fixed $15 per hour adds the same amount every hour — a constant rate that graphs as a straight line, slope 15. The others speed up, slow down, or curve.
✅ Key takeaways
- A constant rate is the slope of the line that graphs it.
- Proportional relationships have the form y = mx and pass through the origin (0, 0).
- Steeper line = faster rate; comparing rates is comparing steepness.
- Read a unit rate by dividing y by x at any point (rise over run from the origin).
- This only holds for steady change — a curved graph has no single slope.