The purpose of this activity is to recall the connection between equations and graphs. This will be useful when students create an equation to represent a situation and graph it in an associated Algebra 1 lesson.

In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). As students revise their writing, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6).

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “For each point, which graph or graphs is it on? How can you tell by using the equation?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”
• “How do you know ? What else do you know is true?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

After Stronger and Clearer Each Time, highlight the connection between an equation and the graph. Here are sample questions to promote class discussion:

• "How can you figure out which graphs include a specific ordered pair?" (I search for the ordered pair on the coordinate plane, and if it appears on the line, then I know that it is a solution for the respective equation.)
• "How do the coordinates for ordered pairs on a given line relate to the line's equation?" (The coordinates are solutions to the equation of the line.)
• "Could you find which graphs include specific ordered pairs without using the graph?" (Yes, you can use the equation to decide if an ordered pair lies on the respective line. If substituting the coordinates into an equation makes the equation false, then we know that particular ordered pair does not lie on the line.)

In this partner activity, students take turns to practice applying their skills at connecting multiple representations of the same linear relationship.

As students analyze different representations, they practice reasoning quantitatively and abstractly (MP2). In making connections across representations, they practice looking for and making use of structure (MP7).

The practice will pay off in the associated Algebra 1 lesson when they create graphs and equations to describe the relationship between two quantities from a situation.

Once all groups have completed the Card Sort, discuss:

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

The goal of this discussion is to help students understand that rows in tables and points on lines represent solutions to the related linear equation.

Display the graph, table, and equation for Mai’s running speeds. 

Possible questions for discussion:

• "If  is 10, what does equal? How do you know?" (0.4 because .)
• "What does the point mean in the story?" (Mai can run 0.4 miles in 10 minutes.)
• "How can you see from the graph what is if is 10?" (I can find the ordered pair on the graph and see that when  is 10,  is 0.4.)
• "How can you see from the table what is if is 10?" (I can see that 10 minutes is paired with 0.4 miles.)
• "How can you use the equation to find out what is if is 10?" (I can substitute 10 for  in the equation  and solve for .)
• "When would you choose to use a graph to tell you an  value if you know the value? When would you use a table? An equation?" (I would use a graph if the - and -values were easy to see. I would use a table if the -value was listed in the table. I would use an equation if the -value would not be clear from the table or graph.)
• "If is 3.6, what does equal? How do you know?" (90 because )
• "What does the point mean in the story?" (Mai can run 3.6 miles in 90 minutes.)
• "How can you see from the graph what is if is 3.6?" (I can find the ordered pair on the graph and see that when  is 3.6,  is 90.)
• "How can you see from the table what is if is 3.6?" (I can see that 3.6 miles is paired with 90 minutes.)
• "How can you use the equation to find out what is if is 3.6?" (I can substitute 3.6 for  in the equation  and solve for .)