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Lesson 3More about Constant of Proportionality

Let’s solve more problems involving proportional relationships using tables.

Learning Targets:

  • I can find missing information in a proportional relationship using a table.
  • I can find the constant of proportionality from information given in a table.

3.1 Equal Measures

Use the numbers and units from the list to find as many equivalent measurements as you can. For example, you might write “30 minutes is \frac12 hour.”

You can use the numbers and units more than once.

1

12

0.4

60

50

\frac{1}{2}

40

0.01

3\frac{1}{3}

30

0.3

24

\frac{1}{5}

6

2

centimeter

meter

hour

feet

minute

inch

3.2 Centimeters and Millimeters

There is a proportional relationship between any length measured in centimeters and the same length measured in millimeters.

A ruler where the top half is labeled "centimeters" and the numbers 0 through 5 are indicated on 5 evenly spaced tick marks. The bottom half is labeled "millimeters" and the numbers 0 through 50, in increments of 10, are indicated on 5 evenly spaced tick marks that are placed identical to the top tick marks. There are 9 evenly spaced tick marks between each labeled tick mark.

There are two ways of thinking about this proportional relationship.

  1. If you know the length of something in centimeters, you can calculate its length in millimeters.

    1. Complete the table.
    2. What is the constant of proportionality?
length (cm) length (mm)
9
12.5
50
88.49

  1. If you know the length of something in millimeters, you can calculate its length in centimeters.

    1. Complete the table.
    2. What is the constant of proportionality?
length (mm) length (cm)
70
245
4
699.1
  1. How are these two constants of proportionality related to each other?

  2. Complete each sentence:

    1. To convert from centimeters to millimeters, you can multiply by ________.
    2. To convert from millimeters to centimeters, you can divide by ________ or multiply by ________.

Are you ready for more?

  1. How many square millimeters are there in a square centimeter?
  2. How do you convert square centimeters to square millimeters? How do you convert the other way?

3.3 Pittsburgh to Phoenix

A plane traveling at a constant speed flew over Pittsburgh, Saint Louis, Albuquerque, and Phoenix on its way from New York to San Diego.

Complete the table as you answer the questions. Be prepared to explain your reasoning.

A map of the United States with 5 line segments that represent the distance a plane flew. The first segment is from New York to Pittsburgh, the second segment from Pittsburgh to Saint Louis, the third segment from Saint Louis to Albuquerque, the fourth from Albuquerque to Phoenix, and the fifth from Pheonix to San Diego.
“Map of the path of a plane flying from New York to San Diego” by United States Census Bureau via American Fact Finder. Public Domain.
segment time distance speed
Pittsburgh to Saint Louis 1 hour 550 miles
Saint Louis to Albuquerque 1 hour 42 minutes
Albuquerque to Phoenix 330 miles

  1. What is the distance between Saint Louis and Albuquerque?

  2. How many minutes did it take to fly between Albuquerque and Phoenix?

  3. What is the proportional relationship represented by this table?
  4. Diego says the constant of proportionality is 550. Andre says the constant of proportionality is 9 \frac16 . Do you agree with either of them? Explain your reasoning.

Lesson 3 Summary

When something is traveling at a constant speed, there is a proportional relationship between the time it takes and the distance traveled. The table shows the distance traveled and elapsed time for a bug crawling on a sidewalk.

A 2-column table with 4 rows of data. The first column is labeled "distance traveled, in centimeters" and the second column is labeled "elapsed time, in seconds." Row 1: 3/2, 1; Row 2: 1, 2/3; Row 3: 3, 2; Row 4: 10, 20/3. An arrow in each row points from the value in column 1 to the value in column 2 and "times 2/3" is labeled below the table.

We can multiply any number in the first column by \frac23 to get the corresponding number in the second column. We can say that the elapsed time is proportional to the distance traveled, and the constant of proportionality is \frac23 . This means that the bug’s pace is \frac23 seconds per centimeter.

This table represents the same situation, except the columns are switched.

A 2-column table with 4 rows of data. The first column is labeled "elapsed time, in seconds" and the second column is labeled "distance traveled, in centimeters." Row 1: 1, 3/2; Row 2: 2/3, 1; Row 3:2 , 3; Row 4: 20/3, 10. An arrow in each row points from the value in column 1 to the value in column 2 and "times 3/2" is labeled below the table.

We can multiply any number in the first column by \frac32 to get the corresponding number in the second column. We can say that the distance traveled is proportional to the elapsed time, and the constant of proportionality is \frac32 . This means that the bug’s speed is \frac32 centimeters per second. 

Notice that \frac32 is the reciprocal of \frac23 . When two quantities are in a proportional relationship, there are two constants of proportionality, and they are always reciprocals of each other. When we represent a proportional relationship with a table, we say the quantity in the second column is proportional to the quantity in the first column, and the corresponding constant of proportionality is the number we multiply values in the first column to get the values in the second.

Lesson 3 Practice Problems

  1. Noah is running a portion of a marathon at a constant speed of 6 miles per hour.

    Complete the table to predict how long it would take him to run different distances at that speed, and how far he would run in different time intervals.

    time
    in hours    		 miles traveled at
    6 miles per hour
    1
    \frac12
    1\frac13
    1\frac12
    9
    4\frac12

  2. One kilometer is 1000 meters.  

    1. Complete the tables. What is the interpretation of the constant of proportionality in each case?
      meters kilometers
      1,000 1
      250
      12
      1
      kilometers meters
      1 1,000
      5
      20
      0.3

      The constant of proportionality tells us that:

      The constant of proportionality tells us that:

    2. What is the relationship between the two constants of proportionality?

  3. Jada and Lin are comparing inches and feet. Jada says that the constant of proportionality is 12. Lin says it is \frac{1}{12} . Do you agree with either of them? Explain your reasoning.

  4. The area of the Mojave desert is 25,000 square miles. A scale drawing of the Mojave desert has an area of 10 square inches. What is the scale of the map?

  5. Which of these scales is equivalent to the scale 1 cm to 5 km? Select all that apply.

    1. 3 cm to 15 km

    2. 1 mm to 150 km

    3. 5 cm to 1 km

    4. 5 mm to 2.5 km

    5. 1 mm to 500 m

  6. Which one of these pictures is not like the others? Explain what makes it different using ratios.

    Three ovals labeled L, M and N on a coordinate grid. Each oval has a smaller oval inside. At its widest point, each oval has the following dimesions: Oval L, outside oval width 5 units, outside oval thickness 1 unit, inside oval width 3 units, height 4 units. Oval M, outside oval width 10 units, outside oval thickness, 3, inside oval width 4 units, height 8 units. Oval N, outside oval width 15 units, outside oval thickness 3, inside oval width 9 units, height 12 units.