Lesson 5Comparing Relationships with Equations

Learning Goal

Let’s develop methods for deciding if a relationship is proportional.

Learning Targets

I can decide if a relationship represented by an equation is proportional or not.

Warm Up: Notice and Wonder: Patterns with Rectangles

Three rectangles on a coordinate grid. The dimensions are as follows: Top rectangle, length 3 units; width 1 unit. Middle rectangle, length 6 units; width 2 units. Bottom rectangle, length 9 units, width 3 units.

Do you see a pattern? What predictions can you make about future rectangles in the set if your pattern continues?

Activity 1: More Conversions

The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.

Use the equation $F = \frac{9}{5} C + 32$ , where $F$ represents degrees Fahrenheit and $C$ represents degrees Celsius, to complete the table.
temperature $(^{\circ} C)$
temperature $(^{\circ} F)$
$20$
$4$
$175$
Use the equation $c = 2.54 n$ , where $c$ represents the length in centimeters and $n$ represents the length in inches, to complete the table.
length (in)
length (cm)
$10$
$8$
$3 \frac{1}{2}$
Are these proportional relationships? Explain why or why not.

temperature $(^{\circ} C)$	temperature $(^{\circ} F)$
$20$
$4$
$175$

length (in)	length (cm)
$10$
$8$
$3 \frac{1}{2}$

Activity 2: Total Edge Length, Surface Area, and Volume

Problem 1

Here are some cubes with different side lengths. Complete each table. Be prepared to explain your reasoning.

Three cubes of different sizes: first cube has side length 3, second cube side length 5, and thrid cube has side length 9 and 1/2

How long is the total edge length of each cube?
side
length
total
edge length
$3$
$5$
$9 \frac{1}{2}$
$s$
What is the surface area of each cube?
side
length
surface
area
$3$
$5$
$9 \frac{1}{2}$
$s$
What is the volume of each cube?
side
length
volume
$3$
$5$
$9 \frac{1}{2}$
$s$

side length	total edge length
$3$
$5$
$9 \frac{1}{2}$
$s$

side length	surface area
$3$
$5$
$9 \frac{1}{2}$
$s$

side length	volume
$3$
$5$
$9 \frac{1}{2}$
$s$

Problem 2

Which of these relationships is proportional? Explain how you know.

Problem 3

Write equations for the total edge length $E$ , total surface area $A$ , and volume $V$ of a cube with side length $s$ .

Are you ready for more?

Problem 1

A rectangular solid has a square base with side length $ℓ$ , height 8, and volume $V$ . Is the relationship between $ℓ$ and $V$ a proportional relationship?

Problem 2

A different rectangular solid has length $ℓ$ , width 10, height 5, and volume $V$ . Is the relationship between $ℓ$ and $V$ a proportional relationship?

Problem 3

Why is the relationship between the side length and the volume proportional in one situation and not the other?

Activity 3: All Kinds of Equations

Here are six different equations.

$y = 4 + x$	$y = \frac{x}{4}$	$y = 4 x$
$y = 4^{x}$	$y = \frac{4}{x}$	$y = x^{4}$

Predict which of these equations represent a proportional relationship.
Complete each table using the equation that represents the relationship.
$Six identical three column tables with 4 rows of data: The first column is labeled "x", the second column is labeled "y", and the third column is labeled "the fraction y over x". Row 1: x, 2. Row 2: x, 3. Row 3: x, 4. Row 4: x, 5. Each table has an equation above it, as follows: Table 1, Equation 1: y equals 4 + x; Table 2, Equation 2: y equals 4x; Table 3, Equation 3: y equals the fraction 4 over x; Table 4, Equation 4: y equals x the fraction x over 4; Table 5, Equation 5: y equals 4 to power x; Table 6, Equation 6: y equals x to the power 4;$
Do these results change your answer to the first question? Explain your reasoning.
What do the equations of the proportional relationships have in common?

Lesson Summary

If two quantities are in a proportional relationship, then their quotient is always the same. This table represents different values of $a$ and $b$ , two quantities that are in a proportional relationship.

$a$	$b$	$\frac{b}{a}$
$20$	$100$	$5$
$3$	$15$	$5$
$11$	$55$	$5$
$1$	$5$	$5$

Notice that the quotient of $b$ and $a$ is always 5. To write this as an equation, we could say $\frac{b}{a} = 5$ . If this is true, then $b = 5 a$ . (This doesn’t work if $a = 0$ , but it works otherwise.)

If quantity $y$ is proportional to quantity $x$ , we will always see this pattern: $\frac{y}{x}$ will always have the same value. This value is the constant of proportionality, which we often refer to as $k$ . We can represent this relationship with the equation $\frac{y}{x} = k$ (as long as $x$ is not 0) or $y = k x$ .

Note that if an equation cannot be written in this form, then it does not represent a proportional relationship.