Pre-Algebra
Proportional relationships
Proportional relationships and slope
20 practice questions
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Theory + worked examples
Proportional Relationships and Slope
Texas Pre-Algebra (TEKS) • Standard 8.4(B) • Proportional Relationships
Proportional Relationships and Slope is a topic in Proportional Relationships in the Texas Essential Knowledge and Skills. It is aligned to Standard 8.4(B), which requires students to connect the unit rate of a proportional relationship to the slope of its graph.
In a proportional relationship, the constant of proportionality equals the slope of the graph.
Theory
For a proportional relationship \(y=kx\), the constant of proportionality \(k\) is the slope of the line.
Slope = rise / run = unit rate = \(k\).
Slope equals \(k\).
Slope is the unit rate.
Slope:
\[\text{slope}=\dfrac{\text{rise}}{\text{run}}=k\]
Steeper lines have a larger unit rate.
How to connect slope and unit rate
- Find rise over run from the graph.
- That slope equals \(k\).
- \(k\) is the unit rate of the relationship.
- Compare slopes to compare rates.
Example 1 — Slope from the graph
A proportional line rises \(4\) over a run of \(2\). Find the slope.
Solution
Rise over run.
| \(\dfrac{4}{2}\) | \(=\) | \(2\) |
Example 2 — Slope equals k
For \(y=3x\), what is the slope?
Solution
The coefficient of \(x\).
| \(\text{slope}\) | \(=\) | \(3\) |
Example 3 — Unit rate
A runner covers \(8\) km in \(1\) h, graphed proportionally. Slope?
Solution
The unit rate is the slope.
| \(\text{slope}\) | \(=\) | \(8\text{ km/h}\) |
Example 4 — Compare
Which is steeper: \(y=2x\) or \(y=5x\)?
Solution
Larger \(k\) is steeper.
| \(5\) | \(>\) | \(2\) |
Common pitfalls
Slope equals the unit rate, not just any ratio.
Rise over run, not run over rise.
Bigger \(k\) means a steeper line.
Frequently asked questions
How does slope relate to a proportional relationship?
The slope equals the constant of proportionality.
What is the slope of \(y=3x\)?
\(3\).
Is slope the unit rate?
Yes.
Which line is steeper, larger or smaller k?
Larger \(k\).
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