Lesson 3

The purpose of this warm-up is for students to use repeated reasoning to write an algebraic expression to represent a rule of a function (MP8). The whole-class discussion should focus on the algebraic expression in the final row, however the numbers in the table give students an opportunity to also practice calculating the square of numbers written in fraction and decimal form.

Arrange students in groups of 2. Give students 1–2 minutes of quiet work time and then time to share their algebraic expression with their partner. Follow with a whole-class discussion.

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 \(s^2\) or \(A\) there. Tell students we can write the equation \(A = s^2\) 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, \(s\) is the independent variable and \(A\) the dependent variable, and we say “\(A\) depends on \(s\).”

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.

The goal of this discussion is for students to describe the connections they see between the different entries for the 4 descriptions. 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:

• “How did you know that this diagram matched with this description?” (We remembered the formula for the circumference of a circle, so we knew description A went with diagram D.)
• “Where in the equation do you see the rule that is in the diagram?” (The equation is the dependent variable set equal to the rule describing what happens to the independent variable in the diagram.)
• “Explain why you chose those quantities for your independent and dependent variables." (We know the independent variable is the input and the dependent variable is the output, so we matched them up with the input and output shown in the diagram.)

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 vs. the other, which can depend on the types of questions one wants to answer.

Identify students who efficiently rewrite the original equation in the third problem and the last problem to share during the discussion.

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.

Some students may be unsure how to write rules for the number of dimes as a function of the number of quarters and vice versa. Prompt them to use the provided equation and what they know about keeping equations equal to create the new equations.

Select previously identified students to share their rules for dimes as a function of the number of quarters and quarters as a function of the number of dimes, including the steps they used to rewrite the original equation.

Tell students that if we write an equation like \(d = 125 - 2.5q\), this shows that \(d\) is a function of \(q\) because it is clear what the output (value for \(d\)) should be for a given input (value for \(q\)).

Display the diagrams for all to see:

When we have an equation like \(0.1d + 0.25q = 12.5\), we can choose either \(d\) or \(q\) to be the independent variable. That means we are viewing one as depending on the other. If we know the number of quarters and want to answer a question about the number of dimes, it is helpful to write \(d\) as a function of \(q\). If we know the number of dimes and want to answer a question about the number of quarters, it is helpful to write \(q\) as a function of \(d\). 

Ensure students understand that we can’t always do this type of rearranging with equations and have it make sense because sometimes only one variable is a function of the other, and sometimes neither is a function of the other. For example, students saw earlier that while squaring values is a function, the reverse—that is, identifying what value was squared—is not. We will continue to explore when these different things happen in future lessons.