Students sort different equations during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Monitor for the different ways in which groups choose to categorize the equations, but especially for categories that distinguish between the number of solutions. 

Students should look for ways to sort the equations without solving them. In particular, they should pay attention to the coefficients of the variable term as well as any constant terms.

Select 2–3 groups to share one of their categories’ defining characteristics and which equations they sorted into that category. Given the Warm-up, categories based on the number of values that make the equations true labeled “one value,” “no values,” or “any values” are likely. Introduce students to the language “infinite number of solutions” if it has not already come up in discussion.

Ask students if anyone noticed a way to categorize the cards without completely solving them. If not, ask students to look for what the equations have in common when the equations got to the point where they looked like .

During the discussion, it is likely that students will want to refer to specific parts of an expression. Encourage students to use the words “coefficient” and “variable.” Define the word constant term as the term in an expression that doesn't change, or the term that does not have a variable part. For example, in the expression the 2 is the coefficient of , the 3 is a constant, and is the variable.

The purpose of this activity is to give students an opportunity to compare the structure of equations that have no solution, one solution, and infinitely many solutions. This may be particularly useful for students who need more practice identifying which equations will have each of these types of solutions before they attempt to solve the equations.

Briefly review each of the equations and how many solutions it has. If students disagree, ask them to explain their thinking about the equation and work to reach agreement.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to what they notice about equations with 1, 0, or infinitely many solutions. 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 their partner clarify and strengthen their 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 that 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, make clear for students how the form of the expressions on each side of the equation can help make it clear to identify the number of solutions. In particular, it is helpful to rewrite the original equation into an equivalent equation of the form . For example, it is not as easy to see how many solutions there are for the equation . Rewriting the equation using the distributive property and combining like terms on each side, we can get and can more easily see that there are no solutions to this equation.

Students should notice that, when the original equation is rewritten into an equivalent equation of the form ,

• There is 1 solution if the coefficients and are not equal.
• There is no solution when the coefficients of are equal () and the constant terms are not equal ().
• There are infinitely many solutions when the coefficients of are equal () and the constant terms are also equal ().