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Fill in the table of input-output pairs for the given rule. Write an algebraic expression for the rule in the box in the diagram.
| input | output |
|---|---|
| 8 | |
| 2.2 | |
The purpose of this discussion is for students to connect a rule to a table of input-output pairs and an algebraic expression. Students are also reintroduced to the terms “independent variables” and “dependent variables” in the context of the inputs and outputs of functions.
Select students to share how they found each of the outputs. After each response, ask the class if they agree or disagree. Record and display responses for all to see. If both responses are not mentioned by students for the last row, tell students that we can either put or there. Tell students we can write the equation to represent the rule of this function.
End the discussion by telling students that while we’ve used the terms “input” and “output” so far to talk about specific values, when a letter is used to represent any possible input we call it the independent variable, and the letter used to represent all the possible outputs is the dependent variable. Students may recall these terms from earlier grades. In this case, is the independent variable and the dependent variable, and we say “ depends on .”
The purpose of this activity is for students to make connections between different representations of functions and start transitioning from input-output diagrams to other representations of functions. Students match input-output diagrams to descriptions and come up with equations for each of those matches. Students then calculate an output given a specific input and determine the independent and dependent variables.
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time and time to share their responses with their partner and come to agreement on their answers. Follow with whole-class discussion.
Record your answers to these questions in the table provided.
| description | a | b | c | d |
|---|---|---|---|---|
| diagram | ||||
| equation | ||||
| input = 5 output = ? |
||||
| independent variable |
||||
| dependent variable |
The goal of this discussion is for students to describe the connections they see between the different entries for the four descriptions. This discussion should also highlight the naming of independent and dependent variables for the four functions using the letters shown in the input-output diagrams.
Display the table for all to see, and select different groups to share the answers for a column in the table. As groups share their answers, ask:
The purpose of this activity is for students to work with a function where either variable could be the independent variable. Knowing the total value for an unknown number of dimes and quarters, students are first asked to consider if the number of dimes could be a function of the number of quarters and then asked if the reverse is also true. Since this isn't always the case when students are working with functions, the discussion should touch on reasons for choosing one variable versus the other, which can depend on the types of questions one wants to answer.
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by partner discussion for students to compare their answers and resolve any differences. Follow with a whole-class discussion.
Select students who efficiently rewrite the original equation in the third problem and the last problem to share during the discussion.
Jada had some dimes and quarters that had a total value of \$12.50. The relationship between the number of dimes, , and the number of quarters, , can be expressed by the equation .
If students are not sure how to determine the number of dimes if there are 4 quarters, ask:
The goal of this discussion is to clarify the confusion that can happen with some relationships—which variable is independent and which variable is dependent?
Begin the discussion with a quick show of hands from anyone that responded yes to the third question and the final question. Invite 2–3 students who responded yes to share their responses. Highlight students’ use of the original equation to calculate the value of the other variable.
Tell students that when we have an equation like , we can choose either or to be the independent variable. That means we are viewing one as depending on the other.
Display the diagrams for all to see, which show what the expressions would be if we rewrote the original equation in the ways described:
Ensure students understand that this type of rearranging with equations doesn’t always make sense because sometimes only one variable is a function of the other, and sometimes neither is a function of the other. The Lesson Synthesis highlights an example from earlier where swapping the independent and dependent variables does not work. We will continue to explore when these different things happen in future lessons.
We can sometimes represent functions with equations. For example, the area, , of a circle is a function of the radius, , and we can express this with this equation:
We can also draw a diagram to represent this function:
In this case, we think of the radius, , as the input and the area of the circle, , as the output. For example, if the input is a radius of 10 cm, then the output is an area of cm2, or about 314 cm2. Because this is a function, we can find the area, , for any given radius, .
Since is the input, we say that it is the independent variable, and since is the output, we say that it is the dependent variable.
We sometimes get to choose which variable is the independent variable in the equation. For example, if we know that
then we can think of as a function of and write
or we can think of as a function of and write