Learn on PengiReveal Math, Course 3Module 4: Linear Relationships and Slope

Lesson 4-1: Proportional Relationships and Slope

In this Grade 8 lesson from Reveal Math, Course 3 (Module 4), students learn to graph and compare proportional relationships using tables, equations, and coordinate graphs while identifying the constant rate of change and slope. The lesson connects unit rate to slope by showing that in any proportional linear relationship, the slope equals the unit rate, calculated as the change in one quantity divided by the change in another between any two points on the line. Students practice interpreting slope in real-world contexts, such as weekly savings and unit conversions, building foundational skills in linear relationships ahead of broader work with linear equations.

Section 1

Identifying Proportional Relationships

Property

Two quantities are in a proportional relationship if the ratio between them is constant. This can be verified in two main ways:

  1. Using a table: Test for equivalent ratios. For any pair of corresponding quantities (x,y)(x, y), the ratio yx\frac{y}{x} must be the same for all non-zero pairs.
  2. Using a graph: The graph of the relationship must be a straight line that passes through the origin (0,0)(0, 0).

Examples

  • A table shows hours worked and earnings. If 2 hours earns 30 dollars, 3 hours earns 45 dollars, and 5 hours earns 75 dollars, the relationship is proportional because the rate is always 15 dollars per hour.
  • A recipe calls for 2 cups of flour for every 1 cup of sugar. A graph of flour vs. sugar would be a straight line through (0,0)(0,0) and (1,2)(1,2), showing it's proportional.
  • A cell phone plan costs 10 dollars per month plus 1 dollar per gigabyte. A graph of the cost would be a line starting at (0,10)(0,10), not the origin, so it is not proportional.

Explanation

Think of it like a recipe. If you double the flour, you must double the sugar. A proportional relationship means two quantities scale up or down together at a steady rate. The graph is a straight line starting from zero because zero input means zero output.

Section 2

Slope as rate of change

Property

The slope of a line gives us the rate of change of one variable with respect to another.

Formula for slope:

m=ΔyΔx=y2−y1x2−x1,x1≠x2m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}, x_1 \neq x_2

Examples

Find the slope between (−1,4)(-1, 4) and (3,−2)(3, -2): m=−2−43−(−1)=−64=−32m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}.

Section 3

Interpreting Slope as a Rate

Property

  1. The slope of a line measures the rate of change of yy with respect to xx.
  2. The units of Δy\Delta y and Δx\Delta x can help us interpret the slope as a rate.

Examples

  • If a graph plots distance (in miles) vs. time (in hours), a slope of 65 means the speed is 65 miles per hour.
  • A graph shows a phone's battery percentage vs. time in hours. A slope of −10-10 means the battery is draining at a rate of 10 percent per hour.
  • A company's cost to produce widgets is graphed with cost (in dollars) on the y-axis and number of widgets on the x-axis. A slope of 1.5 means each additional widget costs 1.50 dollars to produce.

Explanation

The slope's number gets its real-world meaning from the units on the axes. By combining the y-axis unit 'per' the x-axis unit, you can explain exactly what the rate of change signifies, like 'cost per ticket' or 'feet per second'.

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Module 4: Linear Relationships and Slope

  1. Lesson 1Current

    Lesson 4-1: Proportional Relationships and Slope

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  2. Lesson 2

    Lesson 4-2: Slope of a Line

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  3. Lesson 3

    Lesson 4-3: Similar Triangles and Slope

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  4. Lesson 4

    Lesson 4-4: Direct Variation

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  5. Lesson 5

    Lesson 4-5: Slope-Intercept Form

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  6. Lesson 6

    Lesson 4-6: Graph Linear Equations

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Lesson overview

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Section 1

Identifying Proportional Relationships

Property

Two quantities are in a proportional relationship if the ratio between them is constant. This can be verified in two main ways:

  1. Using a table: Test for equivalent ratios. For any pair of corresponding quantities (x,y)(x, y), the ratio yx\frac{y}{x} must be the same for all non-zero pairs.
  2. Using a graph: The graph of the relationship must be a straight line that passes through the origin (0,0)(0, 0).

Examples

  • A table shows hours worked and earnings. If 2 hours earns 30 dollars, 3 hours earns 45 dollars, and 5 hours earns 75 dollars, the relationship is proportional because the rate is always 15 dollars per hour.
  • A recipe calls for 2 cups of flour for every 1 cup of sugar. A graph of flour vs. sugar would be a straight line through (0,0)(0,0) and (1,2)(1,2), showing it's proportional.
  • A cell phone plan costs 10 dollars per month plus 1 dollar per gigabyte. A graph of the cost would be a line starting at (0,10)(0,10), not the origin, so it is not proportional.

Explanation

Think of it like a recipe. If you double the flour, you must double the sugar. A proportional relationship means two quantities scale up or down together at a steady rate. The graph is a straight line starting from zero because zero input means zero output.

Section 2

Slope as rate of change

Property

The slope of a line gives us the rate of change of one variable with respect to another.

Formula for slope:

m=ΔyΔx=y2−y1x2−x1,x1≠x2m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}, x_1 \neq x_2

Examples

Find the slope between (−1,4)(-1, 4) and (3,−2)(3, -2): m=−2−43−(−1)=−64=−32m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}.

Section 3

Interpreting Slope as a Rate

Property

  1. The slope of a line measures the rate of change of yy with respect to xx.
  2. The units of Δy\Delta y and Δx\Delta x can help us interpret the slope as a rate.

Examples

  • If a graph plots distance (in miles) vs. time (in hours), a slope of 65 means the speed is 65 miles per hour.
  • A graph shows a phone's battery percentage vs. time in hours. A slope of −10-10 means the battery is draining at a rate of 10 percent per hour.
  • A company's cost to produce widgets is graphed with cost (in dollars) on the y-axis and number of widgets on the x-axis. A slope of 1.5 means each additional widget costs 1.50 dollars to produce.

Explanation

The slope's number gets its real-world meaning from the units on the axes. By combining the y-axis unit 'per' the x-axis unit, you can explain exactly what the rate of change signifies, like 'cost per ticket' or 'feet per second'.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Module 4: Linear Relationships and Slope

  1. Lesson 1Current

    Lesson 4-1: Proportional Relationships and Slope

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  2. Lesson 2

    Lesson 4-2: Slope of a Line

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  3. Lesson 3

    Lesson 4-3: Similar Triangles and Slope

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  4. Lesson 4

    Lesson 4-4: Direct Variation

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  5. Lesson 5

    Lesson 4-5: Slope-Intercept Form

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  6. Lesson 6

    Lesson 4-6: Graph Linear Equations

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