Grade 7

Intermediate

7 min

Lesson 6 of 12

The Constant of Proportionality

Not Started

Define and calculate the constant of proportionality (k) and represent it in equations.

Have you ever wondered how a map can represent hundreds of miles on a tiny piece of paper, or how a chef scales a recipe for two people up to two hundred without ruining the taste?

Learning Objectives

1.Identify the constant of proportionality (k) in tables, graphs, and word problems
2.Write linear equations in the form $y = kx$ to model real-world relationships
3.Interpret the constant of proportionality as a 'unit rate' in specific contexts

The 'Unit Rate' in Disguise

In a proportional relationship, two quantities grow or shrink at the same rate. We call this special rate the constant of proportionality, represented by the letter

k

. Think of

k

as the 'magic multiplier' that turns your input (

x

) into your output (

y

). If you know

k

, you can predict any value in the relationship. To find

k

, we use the simple ratio:

k = \frac{y}{x}

This value

k

is also known as the unit rate, telling us how much of

y

we get for exactly one unit of

x

A grocery store sells 3 apples for $1.50. Let's find the constant of proportionality ($ k $). 1. Identify your variables:$ x $is the number of apples,$ y $is the total cost. 2. Use the formula:$ k = \frac{y}{x} = \frac{1.50}{3} $. 3. Calculate:$ k = 0.50 $. This means the cost is$ 0.50 per apple.

Quick Check

If a car travels 150 miles in 3 hours at a constant speed, what is the constant of proportionality (k)?

Divide the distance (y) by the time (x).

Answer

k = 50

The Power of the Equation

Once you find

k

, you can write an equation that describes the entire relationship. Every proportional relationship can be written as:

y = kx

On a coordinate plane, this relationship always forms a straight line that passes through the origin

(0, 0)

. If the line doesn't pass through

(0, 0)

, it isn't proportional! On a graph, you can find

k

by picking any point

(x, y)

on the line (except the origin) and calculating the ratio

\frac{y}{x}

Imagine a graph showing a car's travel. The line passes through the point $(4, 240)$ , where $x$ is hours and $y$ is miles. 1. Find $k$ : $k = \frac{240}{4} = 60$ . 2. Write the equation: $y = 60x$ . 3. Interpret: The car travels at a constant speed of 60 miles per hour.

Quick Check

If the constant of proportionality is k = 2.5, what is the equation for the relationship?

Answer

y = 2.5x

Advanced Application: Scaling Up

The constant of proportionality allows us to solve complex problems by creating a mathematical model. Whether you are calculating the flow of water from a leaking faucet or the interest earned on a savings account, $k$ remains the same regardless of how large $x$ or $y$ become. This is why it is called a constant. In word problems, look for phrases like 'per,' 'for every,' or 'at a rate of' to help identify $k$ quickly.

A faucet leaks $1.5$ gallons of water every $10$ minutes. How much water is wasted in one hour ( $60$ minutes)? 1. Find $k$ : $k = \frac{1.5}{10} = 0.15$ gallons per minute. 2. Set up the equation: $y = 0.15x$ . 3. Substitute $x = 60$ : $y = 0.15(60)$ . 4. Solve: $y = 9$ gallons.

Key Takeaways

1The constant of proportionality $k$ is the ratio of $y$ to $x$ ( $k = \frac{y}{x}$ ).
2Proportional relationships are always written in the form $y = kx$ .
3On a graph, a proportional relationship is a straight line passing through $(0, 0)$ .
4The value of $k$ represents the unit rate (the amount of $y$ for one unit of $x$ ).

Quick Check

Multiple Choice

A table shows that 5 pounds of flour cost $10. What is the constant of proportionality ($ k$)?

Multiple Choice

Which equation represents a proportional relationship with a constant of 0.75?

True/False

A graph of a proportional relationship must always pass through the point (0, 0).

What's Next?

Review Tomorrow

In 24 hours, try to explain to a friend why the equation $y = 2x + 1$ is NOT a proportional relationship.

Practice Activity

Find a nutrition label on a snack. Calculate the constant of proportionality for calories per gram ( $k = \frac{calories}{grams}$ ).

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The Constant of Proportionality

Learning Objectives

The 'Unit Rate' in Disguise

The Power of the Equation

Advanced Application: Scaling Up

Key Takeaways

Quick Check

What's Next?

The Constant of Proportionality

Learning Objectives

The 'Unit Rate' in Disguise

The Power of the Equation

Advanced Application: Scaling Up

Key Takeaways

Quick Check

What's Next?