Unit 2, Lesson 3
More about Constant of Proportionality
Grade 7
3.2
Activity
There is a proportional relationship between any length measured in centimeters and the same length measured in millimeters.
A ruler. Centimeters on top. Millimeters on bottom. Centimeters, 0 to 5, by 1's. Millimeters, 0 to 50, by 10's. 1 centimeter = 10 millimeters, 2 centimeters = 20 millimeters, 3 centimeters = 30 millimeters, 4 centimeters = 40 millimeters, 5 centimeters = 50 millimeters.
There are two ways of thinking about this proportional relationship.
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If the length of something in centimeters is known, its length in millimeters can be calculated.
- Complete the table.
- What is the constant of proportionality?
length (cm) length (mm) 9 12.5 50 88.49 -
If the length of something in millimeters is known, its length in centimeters can be calculated.
- Complete the table.
- What is the constant of proportionality?
length (mm) length (cm) 70 245 4 699.1 - How are these two constants of proportionality related to each other?
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Complete each sentence:
- To convert from centimeters to millimeters, the value in centimeters is multiplied by ________.
- To convert from millimeters to centimeters, the value in millimeters is divided by ________, or multiplied by ________.
3.3
Activity
On its way from New York to San Diego, a plane flew over Pittsburgh, Saint Louis, Albuquerque, and Phoenix traveling at a constant speed.
Complete the table as you answer the questions. Be prepared to explain your reasoning.
A United States map with 5 segments representing the distance a plane flew. The segments are New York to Pittsburgh, Pittsburgh to Saint Louis, Saint Louis to Albuquerque, Albuquerque to Phoenix and Phoenix to San Diego.
| 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 |
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What is the distance between Saint Louis and Albuquerque?
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How many minutes did it take to fly between Albuquerque and Phoenix?
- What is the proportional relationship represented by this table?
- Diego says the constant of proportionality is 550. Andre says the constant of proportionality is . Do you agree with either of them? Explain your reasoning.
Student Lesson 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.
We can multiply any number in the first column by 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 . This means that the bug’s pace is seconds per centimeter.
Table with 2 columns and 4 rows of data. The columns are: distance traveled (cm) and elapsed time (sec). The table has the ordered pairs (the fraction 3 over 2 comma 1), (1 comma the fraction 2 over 3), (3 comma 2) and (10 comma the fraction 20 over 3). Each pair has an arrow pointing from the value in the first column to the value in the second column. The arrow represents multiplying the first value by the fraction 2 over 3 to calculate the second value.
This table represents the same situation, except the columns are switched.
We can multiply any number in the first column by 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 . This means that the bug’s speed is centimeters per second.
Table with 2 columns and 4 rows of data. The columns are: elapsed time (sec) and distance traveled (cm). The table has the ordered pairs (1 comma the fraction 3 over 2), (the fraction 2 over 3 comma 1), (2 comma 3) and (the fraction 20 over 3 comma 10). Each pair has an arrow pointing from the value in the first column to the value in the second column. Times the fraction three over 2 is below the table.
Notice that is the reciprocal of . 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 by to get the values in the second.
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