~ 2.7 ~

Comparing Relationships with Tables

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Learning Targets

I can decide if a relationship represented by a table could be proportional and when it is definitely not proportional.

Notes

Here are the prices for some smoothies at two different smoothie shops:

Smoothie Shop A

Smoothie Shop B

For Smoothie Shop A, smoothies cost $0.75 per ounce no matter which size we buy. There could be a proportional relationship between smoothie size and the price of the smoothie. An equation representing this relationship is

p=0.75s

where s represents size in ounces and p represents price in dollars. (The relationship could still not be proportional, if there were a different size on the menu that did not have the same price per ounce.)

For Smoothie Shop B, the cost per ounce is different for each size. Here the relationship between smoothie size and price is definitely not proportional.

In general, two quantities in a proportional relationship will always have the same quotient. When we see some values for two related quantities in a table and we get the same quotient when we divide them, that means they might be in a proportional relationship—but if we can't see all of the possible pairs, we can't be completely sure. However, if we know the relationship can be represented by an equation is of the form y=kx, then we are sure it is proportional.

Identify proportional relationshipsPractice telling whether or not the relationship between two quantities is proportional by looking at a table of the relationship.

Activities

7.1 Adjusting a Recipe

A lemonade recipe calls for the juice of 5 lemons, 2 cups of water, and 2 tablespoons of honey.

Invent four new versions of this lemonade recipe:

One that would make more lemonade but taste the same as the original recipe.
One that would make less lemonade but taste the same as the original recipe.
One that would have a stronger lemon taste than the original recipe.
One that would have a weaker lemon taste than the original recipe.

7.2 Visiting the State Park

Entrance to a state park costs $6 per vehicle, plus $2 per person in the vehicle.

How much would it cost for a car with 2 people to enter the park? 4 people? 10 people? Record your answers in the table.
For each row in the table, if each person in the vehicle splits the entrance cost equally, how much will each person pay?
How might you determine the entrance cost for a bus with 50 people?
Is the relationship between the number of people and the total entrance cost a proportional relationship? Explain how you know.

Add to Your Notes

Notice that the relationship is NOT proportional.

The cost per person is different for different number of people in a vehicle, i.e. the quotients of the entries in each row are not equal for all rows of the table.
The ratio of people in the vehicle to total entrance cost are not equivalent ratios. You can't just multiply the entries in one row by the same constant to get the entries in another row.
Each number of people and corresponding total entrance cost is not characterized by the same unit rate. You can't multiply the entries in the first column by the same number (constant of proportionality) to get the numbers in the second column.

7.3 Running Laps

Han and Clare were running laps around the track. The coach recorded their times at the end of laps 2, 4, 6, and 8.

Han's run:

Clare's Run:

Is Han running at a constant pace? Is Clare? How do you know?
Write an equation for the relationship between distance and time for anyone who is running at a constant pace.

Add to Your Notes

In this lesson, we learned some ways to tell whether a table could represent a proportional relationship. Revisit one or more of the activities in the lesson, highlighting the following points:

If the quotient is the same for each row in the table, the table couldrepresent a proportional relationship.
It can be helpful to compute and write down this quotient for each row.
The quotient is the constant of proportionality for the relationship (if the relationship is proportional).
If all the quotients are not the same, the table definitely does not represent a proportional relationship.
The relationship between the two quantities in a proportional relationship can be expressed using an equation of the form y=kx.

Summary

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