Lesson 5
Function Representations
These materials, when encountered before Algebra 1, Unit 4, Lesson 5 support success in that lesson.
5.1: Notice and Wonder: Representing Functions (10 minutes)
Warmup
The purpose of this warmup is to elicit the idea that a single function can have many representations, which will be useful when students connect the representations in a later activity. While students may notice and wonder many things about these representations, that they all represent the same function are the important discussion points.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Launch
Display the equation, graph, and table for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a wholeclass discussion.
Student Facing
What do you notice? What do you wonder?
\(f(x) = \frac{2}{3}x  1\)
\(x\)  \(y\) 

1  \(\text{}\frac{5}{3}\) 
0  1 
1  \(\text{}\frac{1}{3}\) 
2  \(\frac{1}{3}\) 
3  1 
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the representations. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the idea that all the representations are connected to the same function does not come up during the conversation, ask students to discuss this idea.
5.2: A Seat at the Tables (15 minutes)
Activity
In this activity, students use equations to find values and complete tables. In the associated Algebra 1 lesson, students examine functions in different representations including equations and tables such as these.
Student Facing
Use the equations to complete the tables.

\(y = 3x  2\)
\(x\) \(y\) 1 3 2 
\(y = 52x\)
\(x\) \(y\) 0 3 5 
\(y = \frac{1}{2}x + 2\)
\(x\) \(y\) 4 3 6 
\(x\) \(y = 2x  10\) 3 7 8
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The purpose of the discussion is to connect representations of the functions. Select students to share their solutions. Ask students:
 “What is a visual way these functions could be represented? How would the table help create that representation?” (A graph is a visual representation of a function. A table can be used to get points on the graph to help see the function.)
 “What might a table look like if the equation did not represent a function?” (There would be two of the same \(x\)value with different \(y\)values.)
5.3: Function Finder (20 minutes)
Activity
In this activity, students use values given in a table to draw and write other representations, then invent a function for their partner to guess the rule. In the associated Algebra 1 lesson, students connect different representations of functions. This activity supports students by strengthening their understanding of the connection between table representations and equations.
Launch
Arrange students in groups of 2.
Student Facing

Use the values in the table to graph a possible function that would have the values in the table.

\(x\) \(y\) 1 3 2 5 3 7 5 11 
\(x\) \(y\) 2 0 0 1 2 2 4 3 
\(x\) \(y\) 2 14 1 12 1 8 2 6

 For each of the tables and graphs, write a linear equation (like \(y = ax + b\)) so that the table can be created from the equation.
 Invent your own linear equation. Then, create a table or graph, including at least 4 points, to trade with your partner. After getting your partner’s table or graph, guess the equation they invented.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The purpose of the discussion is to connect the representations of the functions. Select students to share their solutions including several of the equations that students invented. Ask students:
 “Are there other functions that could be drawn based on the tables other than the equations you wrote?” (Yes, the functions need to include those 4 points, but it could wiggle up and down between those points or curve in different ways. It could even be a graph of just those 4 points.)
 “Is it easier to create a table from an equation or an equation from a table?” (I think it’s easier to create a table from an equation since I can think of a number for \(x\) and substitute it in to get a \(y\)value. Thinking of a rule that might apply to all of the entries in a table is more difficult.)
 “When a person is doing science or figuring out how the world works, do you think they start with an equation and find points that match their equation or start with points like in the table and find an equation to match?” (They usually collect data which could look like the values in the table and then try to understand how to write an equation to describe the values they have.)