Lesson 8
Comparing Relationships with Equations
Let’s develop methods for deciding if a relationship is proportional.
8.1: Notice and Wonder: Patterns with Rectangles
![Three rectangles on a coordinate grid. The rectangles are 3 units long by 1 unit wide, 6 units long by 2 units wide and 9 units long by 3 units wide.](https://cms-im.s3.amazonaws.com/T2FEP3gMY7HKGzdUzazV4w85?response-content-disposition=inline%3B%20filename%3D%227-7.2.D2.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.D2.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T192639Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=550bfde14d697273207ea219ec0b1738ee499d5ea47b0d7b0759e0278e7d913e)
Do you see a pattern? What predictions can you make about future rectangles in the set if your pattern continues?
8.2: More Conversions
The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.
- Use the equation \(F =\frac95 C + 32\), where \(F\) represents degrees Fahrenheit and \(C\) represents degrees Celsius, to complete the table.
temperature \((^\circ\text{C})\) temperature \((^\circ\text{F})\) 20 4 175 - Use the equation \(c = 2.54n\), 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\(\frac12\) - Are these proportional relationships? Explain why or why not.
8.3: Total Edge Length, Surface Area, and Volume
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](https://cms-im.s3.amazonaws.com/fQVT2tupXgcxR8hbrWYpHVDT?response-content-disposition=inline%3B%20filename%3D%227-7.2.D2.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.D2.Image.02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T192639Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=506ac574c15c22a2935b50cd64a67eab619605d6e4365bd0e0523bd852f3358e)
- How long is the total edge length of each cube?
side
lengthtotal
edge length3 5 \(9\frac12\) \(s\) - What is the surface area of each cube?
side
lengthsurface
area3 5 \(9\frac12\) \(s\) - What is the volume of each cube?
side
lengthvolume 3 5 \(9\frac12\) \(s\) - Which of these relationships is proportional? Explain how you know.
-
Write equations for the total edge length \(E\), total surface area \(A\), and volume \(V\) of a cube with side length \(s\).
- A rectangular solid has a square base with side length \(\ell\), height 8, and volume \(V\). Is the relationship between \(\ell\) and \(V\) a proportional relationship?
- A different rectangular solid has length \(\ell\), width 10, height 5, and volume \(V\). Is the relationship between \(\ell\) and \(V\) a proportional relationship?
- Why is the relationship between the side length and the volume proportional in one situation and not the other?
8.4: All Kinds of Equations
Here are six different equations.
\(y = 4 + x\)
\(y = \frac{x}{4}\)
\(y = 4x\)
\(y = 4^{x}\)
\(y = \frac{4}{x}\)
\(y = x^{4}\)
![Six identical tables, each with 3 columns and 4 rows of data.](https://cms-im.s3.amazonaws.com/AP3RMQCnp1t1QGJXUfxwhxF7?response-content-disposition=inline%3B%20filename%3D%227-7.2.D2.Image.03.1.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.D2.Image.03.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T192639Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=883a4724b6f3711dfb7f0880337a29fbc252c5232bde04779e5b2d32b7a65a58)
- Predict which of these equations represent a proportional relationship.
- Complete each table using the equation that represents the relationship.
- Do these results change your answer to the first question? Explain your reasoning.
- What do the equations of the proportional relationships have in common?
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 \(b=5a\). (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 \(y=kx\).
Note that if an equation cannot be written in this form, then it does not represent a proportional relationship.