 # Real-World Functions

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Examples, videos, and solutions to help Grade 8 students examine and recognize real-world functions with discrete rates and continuous rates.

### New York State Common Core Math Grade 8, Module 5, Lesson 4

Lesson 4 Student Outcomes

• Students examine and recognize real-world functions, such as the cost of a book, as discrete rates.
• Students examine and recognize real-world functions, such as the temperature of a pot of cooling soup, as continuous rates.

Lesson 4 Student Summary

Not all functions are linear. In fact, not all functions can be described using numbers.
Linear functions can have discrete rates and continuous rates.
A rate that can have only integer inputs may be used in a function so that it makes sense, and it is then called a discrete rate. For example, when planning for a field trip, it only makes sense to plan for a whole number of students and a whole number of buses, not fractional values of either.
Continuous rates are those where any interval, including fractional values, can be used for an input. For example, determining the distance a person walks for a given time interval. The input, which is time in this case, can be in minutes or fractions of minutes.

Lesson 4 Classwork

Discussion
In Table A, the context was purchasing bags of candy. In Table B, it was the distance traveled by a moving object. Examine the tables. What are the differences between these two situations?

Example 1
If 4 copies of the same book cost \$256, what is the unit rate for the book?

Example 2
Water flows from a faucet at a constant rate. That is, the volume of water that flows out of the faucet is the same over any given time interval. If 7 gallons of water flow from the faucet every 2 minutes, determine the rule that describes the volume function of the faucet.

Example 3
You have just been served freshly made soup that is so hot that it cannot be eaten. You measure the temperature of the soup, and it is 210°F. Since 212°F is boiling, there is no way it can safely be eaten yet. One minute after receiving the soup the temperature has dropped to 203°F. If you assume that the rate at which the soup cools is linear, write a rule that would describe the rate of cooling of the soup.

Example 4
Consider the following function: There is a function G so that the function assigns to each input, the number of a particular player, an output, their height. For example, the function G assigns to the input, 1 an output of 5’ 11”.

Exercises 1–3
1. A linear function has the table of values below related to the number of buses needed for a fieldtrip.
a. Write the linear function that represents the number of buses needed, y, for x number of students.
b. Describe the limitations of x and y.
c. Is the rate discrete or continuous?
d. The entire 8th grade student body of 321 students is going on a fieldtrip. What number of buses does our function assign to 321 students? Explain.
e. Some 7th grade students are going on their own field trip to a different destination, but just 180 are attending. What number does the function assign to 180? How many buses will be needed for the trip?
f. What number does the function assign to 50? Explain what this means and what your answer means.

2. A linear function has the table of values below related to the cost of movie tickets.
a. Write the linear function that represents the total cost, y, for x tickets purchased.
b. Is the rate discrete or continuous? Explain.
c. What number does the function assign to 4? What does the question and your answer mean?

3. A function produces the following table of values.
a. Can this function be described by a rule using numbers? Explain.
b. Describe the assignment of the function.
c. State an input and the assignment the function would give to its output.

Closing
• We know that not all functions are linear and, moreover, not all functions can be described by numbers.
• We know that linear functions can have discrete rates and continuous rates.
• We know that discrete rates are those where only integer inputs can be used in the function for the inputs to make sense. An example of this would be purchasing books compared to books.
• We know that continuous rates are those where we can use any interval, including fractional values, as an input. An example of this would be determining the distance traveled after minutes of walking.

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