The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.
Use the equation , where represents degrees Fahrenheit and represents degrees Celsius, to complete the table.
temperature
temperature
20
4
175
Use the equation , where represents the length in centimeters and represents the length in inches, to complete the table.
length (in)
length (cm)
10
8
Are these proportional relationships? Explain why or why not.
5.3
Activity
Here are some cubes with different side lengths. Complete each table. Be prepared to explain your reasoning.
How long is the total edge length of each cube?
side
length
total
edge length
3
5
What is the surface area of each cube?
side
length
surface
area
3
5
What is the volume of each cube?
side
length
volume
3
5
Which of these relationships is proportional? Explain how you know.
Write equations for the total edge length , total surface area , and volume of a cube with side length .
5.4
Activity
Here are six different equations.
Predict which of these equations represent a proportional relationship.
Complete each table using the equation that represents the relationship.
Six identical tables, each with 3 columns and 4 rows of data. All have first rows: x, y, the fraction y over x. All have the same x values: 2, 3, 4 and 5. All have a different equation above it: y = x + 4, y = 4x, y = the fraction 4 over x, y = the fraction x over 4, y = 4^x and y = x^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?
Student Lesson Summary
If two quantities are in a proportional relationship, then their quotient is always the same. This table represents different values of and , two quantities that are in a proportional relationship.
20
100
5
3
15
5
11
55
5
1
5
5
Notice that the quotient of and is always 5. To write this as an equation, we could say . If this is true, then . (This doesn’t work if , but it works otherwise.)
If quantity is proportional to quantity , we will always see that has a constant value. This value is the constant of proportionality, which we often refer to as . We can represent this relationship with the equation (as long as is not 0) or .
Note that if an equation cannot be written in this form, then it does not represent a proportional relationship.