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.

Students may have trouble seeing why some equations in column B are not equivalent (particularly the second item, which contains a common error). Encourage these students to choose one pair of equations, solve one equation, and then substitute the solution into the other equation to see what goes wrong.

Invite previously identified students to share their explanations on at least a couple of pairs of equations from each column. If not already clear from students' explanations, emphasize that:

• If two expressions are equal, then performing the same operation—addition (or subtraction) or multiplication (or division)—on both expressions maintains the equality.
• Applying the distributive, commutative, and associative properties (of multiplication or addition) to an expression doesn't change its value, so doing so to one side of a true equation doesn't change the equality.

Explain to students that next they will look at some examples where the moves made to write equivalent equations appear to be acceptable but the resulting equations turn out to be false statements.

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.

Some students may point to a step that is valid and mistakenly identify it as an error. For instance, in the first set of steps, they may object to replacing with , thinking that it should be rearranged to . Push their reasoning with a simpler example. Ask, for instance, if is equivalent to . Remind students that we can think of as and then apply the commutative property of addition to get .

If students hypothesize about two equations being equivalent but are not sure how to check if it's actually the case, suggest that a good way to check is by finding the solution to one equation, then checking whether that value is also a solution to the second equation.

Invite students to share what they thought was the problem with Noah’s work. They are likely to say that Noah seems to have performed allowable moves and did them correctly. Then draw students' attention to the second-to-last step: . Ask students:

• “Earlier, when looking at , we reasoned that there’s no value of that could make this equation true. Is there a value of that could make true?” (Yes, 0.)
• “If there is a value that can make true, what do we know about the original equation? Is it also true?” (Yes, they are all equivalent equations.) "What is its solution?” (0)
• “So the equations are true up through the third step. The last step is where we have a false statement, after Noah shows division by . If is 0, what might be problem with dividing by ?” (If is 0, then we are dividing both sides by 0, which gives an undefined result.)

Explain that dividing by the variable in the equation is not done because if the solution happens to be 0, it could lead us to thinking that there is no solution while in fact there is (the solution is the number 0).

Revisit the lists of acceptable and unacceptable moves compiled in earlier activities. Add ”dividing by the variable” and ”dividing by 0” to the list of unacceptable moves.