Learn on PengiReveal Math, AcceleratedUnit 9: Linear Relationships

Lesson 9-2: Compare Proportional Relationships

In this Grade 7 Reveal Math Accelerated lesson from Unit 9, students learn how to compare proportional relationships by analyzing slope across multiple representations, including tables, graphs, and equations. Using real-world contexts like roller coaster lift hills and wheelchair-accessible ramps, students calculate and compare unit rates and slopes to determine which relationship is steeper. The lesson builds skills in interpreting proportional relationships flexibly across different formats and applying slope comparisons to practical situations.

Section 1

Writing the Proportional Relationship Equation y = kx

Property

The variables yy and xx are proportional if

yx=k\frac{y}{x} = k
where kk is a constant. This constant kk is called the constant of proportionality. This relationship can also be expressed as an equation:
y=kxy = kx
This second version says that yy is proportional to xx if yy is a constant multiple of xx. The two equations are two ways to say the same thing.

Examples

  • The cost CC for gallons gg of gas is proportional. If 5 gallons cost 20 dollars, the constant is k=205=4k = \frac{20}{5} = 4. The equation is C=4gC = 4g.
  • The number of words ww you type is proportional to the minutes mm you spend typing. If you type 240 words in 4 minutes, the constant is k=2404=60k = \frac{240}{4} = 60. The equation is w=60mw = 60m.
  • The length in centimeters cc is proportional to the length in inches ii. Since 1 inch is 2.54 cm, the constant of proportionality is k=2.54k = 2.54. The equation is c=2.54ic = 2.54i.

Explanation

Proportional variables have a constant ratio. This means one variable is always a fixed multiple of the other. Think of it like a recipe: doubling the ingredients doubles the serving size. Their graph is a straight line through the origin (0,0).

Section 2

Finding the Constant of Proportionality

Property

In a proportional relationship, the ratio between two quantities, yy and xx, is always constant. This constant value is called the constant of proportionality, represented by the letter kk. It is also known as the unit rate.

k=yxk = \frac{y}{x}

Section 3

Application: Comparing Rates by Comparing Slopes

Property

When multiple proportional relationships are graphed on the same coordinate plane, they can be compared by analyzing their slopes.
The relationship with the larger slope (constant of proportionality) has a steeper line and represents a faster rate of change.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Writing the Proportional Relationship Equation y = kx

Property

The variables yy and xx are proportional if

yx=k\frac{y}{x} = k
where kk is a constant. This constant kk is called the constant of proportionality. This relationship can also be expressed as an equation:
y=kxy = kx
This second version says that yy is proportional to xx if yy is a constant multiple of xx. The two equations are two ways to say the same thing.

Examples

  • The cost CC for gallons gg of gas is proportional. If 5 gallons cost 20 dollars, the constant is k=205=4k = \frac{20}{5} = 4. The equation is C=4gC = 4g.
  • The number of words ww you type is proportional to the minutes mm you spend typing. If you type 240 words in 4 minutes, the constant is k=2404=60k = \frac{240}{4} = 60. The equation is w=60mw = 60m.
  • The length in centimeters cc is proportional to the length in inches ii. Since 1 inch is 2.54 cm, the constant of proportionality is k=2.54k = 2.54. The equation is c=2.54ic = 2.54i.

Explanation

Proportional variables have a constant ratio. This means one variable is always a fixed multiple of the other. Think of it like a recipe: doubling the ingredients doubles the serving size. Their graph is a straight line through the origin (0,0).

Section 2

Finding the Constant of Proportionality

Property

In a proportional relationship, the ratio between two quantities, yy and xx, is always constant. This constant value is called the constant of proportionality, represented by the letter kk. It is also known as the unit rate.

k=yxk = \frac{y}{x}

Section 3

Application: Comparing Rates by Comparing Slopes

Property

When multiple proportional relationships are graphed on the same coordinate plane, they can be compared by analyzing their slopes.
The relationship with the larger slope (constant of proportionality) has a steeper line and represents a faster rate of change.

Examples