This Math Talk focuses on solutions to equations. It encourages students to think about solutions to equations and to rely on properties of operations to mentally solve problems. It also prompts students to recall that dividing a number by 0 leads to an undefined result, preparing them for the work later in the lesson. (In that activity, students will consider why dividing by a variable is not considered an acceptable move when writing equivalent equations or solving equations.)

To determine if 0 is a solution to the equations, students could substitute 0 into the expressions and evaluate them. For some equations, however, the answer can be efficiently found by making use of structure (MP7). In explaining their strategies, students need to be precise in their word choice and use of language (MP6).

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”

Previously, students saw that certain moves can be made to an equation to create an equivalent equation. In this activity, they deepen that understanding by explaining why, if a given equation is true for a certain value of the variable, performing one of those moves leads to a second equation that is also true for the same variable value. This requires more than simply stating what the moves are, and offers students opportunities to construct logical arguments (MP3).

Clarify the distinction with an example. Ask students: "Consider these equations: and . The first equation is a true statement for a certain value of . Can you explain why the second equation must also be true for the same value of ?"

Explain that an answer such as “Adding 5 to each side of the first equation gives the second equation” is a description of the move rather than an explanation for why the second equation must be true.

An explanation may sound something like: "We know that and 16 are equal when has a particular value. If we add the same number, like 5, to both  and to 16, the results are still equal for the same variable value. This means the statement must be true for the same value of ."

In this partner activity, students take turns giving explanations, and listening to and critiquing a partner's explanations (MP3). As students discuss their thinking, listen for explanations that are particularly clear so that they can be shared with the class later.

So far, students have seen only one-variable equations that have a solution. For these equations, performing acceptable moves always led to equivalent equations that have the same solution. In this activity, students encounter an example in which the given equation has no solutions and performing the familiar moves leads to an untrue statement.

Prior to this point, students have added, subtracted, multiplied, and divided a number on both sides of an equation. They have also added a variable expression to (or subtracted a variable expression from) an equation. They recognize these moves as allowable for solving equations. Here, students also come across an equation that is divided by a variable expression and make sense of why it leads to a false statement.