8.5.E 8.5.F 8.5.H

Proportional vs. Not Proportional 8th Grade

Texas Essential Knowledge and Skills (TEKS): 8.5.E
solve problems involving direct variation;
Texas Essential Knowledge and Skills (TEKS): 8.5.F
distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0;
Texas Essential Knowledge and Skills (TEKS): 8.5.H
identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems; and

The relationship between two variables, say x and y, is proportional
if there exists a constant, say k, such that y equals kx.

In a proportional relationship, the constant is
often referred to as the constant of proportionality.

Consider the known relationship between inches and feet. There are 12 inches in 1 foot.
If we want to know the number of inches, y, in a given number of feet, x, we can use the equation y = 12x.
This equation represents a proportional relationship, and 12 is the constant of proportionality.
In some situations, this constant is referred to as the unit rate.
In this situation, the unit rate can be interpreted as 12 inches per foot.

Sometimes values are presented in a table.
How can we determine if a table represents a proportional relationship?
We have to determine if there exists a constant such that each output value is the product of the constant and the corresponding input value.

Example:

Recognize that each y-value is 10 times the corresponding x-value.
So, there is a constant, in this case 10, such that each y-value is the product of the constant and the corresponding x-value.
Therefore, the table represents a proportional relationship.

If a relationship exists between two variables such that their ratio is a
non-zero constant, the relationship is referred to as a direct variation.

The constant is often referred to as the
constant of variation, or constant of proportionality.

If two variables, say x and y, are in direct variation, then y = kx for some constant k.
In this relationship, we say y varies directly with x or y varies directly as x.

Example:

Since y varies directly with x, the relationship can be represented by an equation of the form y = kx, where k is some constant.
The constant, k, is referred to as the constant of proportionality.
Use the given values to solve for k.

y	=	kx
16	=	k(8)
16 ÷ 8	=	k(8) ÷ 8
2	=	k

Example:

Since y varies directly with x, the relationship can be represented by an equation of the form y = kx, where k is some constant.
The constant, k, is referred to as the constant of variation.
Use the given values to solve for x.

y	=	kx
14	=	2x
14 ÷ 2	=	2x ÷ 2
7	=	x

Proportional vs. Not Proportional

Texas Essential Knowledge and Skills (TEKS): 8.5.E

solve problems involving direct variation;

Texas Essential Knowledge and Skills (TEKS): 8.5.F

distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0;

Texas Essential Knowledge and Skills (TEKS): 8.5.H

identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems; and

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