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. As students use the equations to build the tables they look for and make use of structure (MP7).

The purpose of the discussion is to connect representations of the functions. Select students to share their solutions. Here are some questions for discussion:

• “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 -value with different -values.)

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. Students look for and express regularity in repeated reasoning when they use the patterns in the table and graph to write an equation (MP8). 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.

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. Here are some questions for discussion:

• “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 and substitute it in to get a -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 that could look like the values in the table and then try to understand how to write an equation to describe the values they have.)