Proportional Relationships and Slope

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Concept Review

The Hidden Language of Slopes: Reading Stories in Lines

Imagine you're tracking your phone's battery as it drains throughout the day. If you graphed this, would the line be steep or gentle? Would it go up or down? That line tells a story — and the slope is the narrator.

When two quantities have a proportional relationship, they grow or shrink together at a constant rate. Think of it like a recipe: if 2 cups of flour make 12 cookies, then 4 cups make 24 cookies, and 6 cups make 36 cookies. The ratio stays the same — that's proportional.

From Recipe to Graph

Let's graph our cookie recipe. Put cups of flour on the x-axis and number of cookies on the y-axis:

•(2, 12) — 2 cups make 12 cookies
•(4, 24) — 4 cups make 24 cookies
•(6, 36) — 6 cups make 36 cookies

Connect these points and you get a perfectly straight line that passes through the origin (0,0). This is the signature of a proportional relationship — it's always a straight line through zero.

🔍 The Unit Rate Secret

Here's the amazing part: the slope of this line is exactly the same as the unit rate!

Slope = (24 - 12) ÷ (4 - 2) = 12 ÷ 2 = 6 cookies per cup

Unit rate = 12 cookies ÷ 2 cups = 6 cookies per cup

They're identical! The slope tells you the rate at which y changes for every 1 unit of x.

Reading the Slope Story

Every proportional relationship graph whispers its story through slope:

↗Steep upward: Fast growth (like a rocket launch)
↗Gentle upward: Steady growth (like walking up a hill)
↘Downward: Decrease (like that phone battery draining)

🔑 Key Takeaway

Just like your phone's battery graph tells the story of how fast it's draining, every proportional relationship's slope reveals its hidden rate. Master reading slopes, and you can decode the stories that numbers are trying to tell you — from speed to cost per item to growth over time.

Sample questions

1. A proportional relationship is represented by the equation y = 3.5x. What is the slope of its graph?

○ 1

○ 0

○ Cannot be determined

✓ 3.5

Answer: 3.5 — In y = mx, m is both the constant of proportionality and the slope.

2. Which graph represents a proportional relationship with a unit rate of 2?

✓ A line passing through (0,0) and (1,2)

○ A line passing through (0,0) and (2,1)

○ A line passing through (0,1) and (1,3)

○ A line passing through (0,0) and (2,4)

Answer: A line passing through (0,0) and (1,2) — Unit rate = slope = rise/run = 2/1 = 2. Also must pass through origin for proportional.

3. A car travels at a constant speed of 60 miles per hour. Which equation represents this proportional relationship where d is distance and t is time?

○ d = t + 60

✓ d = 60t

○ t = 60d

○ d = 60/t

Answer: d = 60t — Distance = rate × time, so d = 60t. This is proportional with slope 60.

Skills in this topic

Graph proportional relationships, interpreting the unit rate as the slope of the graph
Compare two different proportional relationships represented in different ways
Understand that the slope (m) is constant between any two points on a line
Use similar triangles to explain why the slope is the same between any two distinct points on a non-vertical line
Derive the equation y = mx for a line through the origin

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