The goal of this discussion is to describe what makes a pattern linear, exponential, or neither.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to why Andre’s pattern is not linear or exponential. 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. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, here are some possible discussion questions:

• “How did you recognize linear patterns?” (Linear patterns have a constant rate of change so you add (or subtract) the same amount each time the step increases by 1 to find the next number.)
• “How do you recognize exponential patterns?” (Exponential patterns have a constant growth factor so you multiply by the same number each time to find the next number in the pattern, or, when you divide sequential numbers in the pattern, you always get the same quotient.)

The purpose of this discussion is for students to share strategies they used to complete the last few rows of each table and to clarify how they knew the relationship was linear, exponential, or neither. If time permits, highlight connections between the visual patterns and how the total number of dots is increasing in the tables.

Display a completed table for both patterns, and refer to it when discussing the following questions:

• “How did you extend the pattern to figure out how many dots were in step ?” (I drew the figures for each step.)
• “How could you predict the number of dots in step 10? Or step 20?” (I could use the expression for each pattern.)
• “What advice would you give classmates to help them recognize a linear relationship? An exponential relationship?” (If the same amount is added each time, it is linear. If the same amount is multiplied each time, it is exponential.)
• “Did anyone see the pattern in question 1 growing in a different way?”