Students who pause to think about the structure of a complex equation before taking steps to solve it can find the most efficient solution paths and, sometimes, notice that there is no single solution to be found. The goal of this lesson is to encourage students to make this pause part of their routine and to build their skill at understanding and manipulating the structure of equations through the study of two special types of equations: ones that are always true and ones that are never true.

Students begin the activity sorting a variety of equations into categories based on their number of solutions. The activity ends with students filling in the blank side of an equation to make an equation that is always true and then again to make an equation that is never true.

For the last part of the activity, students may think that any expression that is not equivalent to is a good answer. Ask students:

• “How many solutions do the equations from the Warm-up have? Why?”
• “How many solutions does the equation  have?

Display a list of the equations from the task, leaving enough space to add student ideas next to the equations. The purpose of this discussion is for students to see multiple ways of thinking about and justifying the number of solutions that an equation has.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they categorized each equation. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. 

Next, ask students for different ways to write the other side of the equation for the second problem, and add these to the display. For example, students may have distributed to get while others chose or something with more terms, such as .

End the discussion by asking students for different ways to write the other side of the incomplete equation in the last question. It is important to note, if no students point it out, that all solutions should be equivalent to , where the question mark represents any number other than -10.

In this activity, students are presented with three equations, all with a missing term. They are asked to fill in the missing term to create equations with either no solution or infinitely many solutions, building on the work begun in the previous activity. At the end, students summarize what they have learned about how to tell if an equation is true for all values of or no values of .

The purpose of this discussion is to record the students’ thinking about conditions that must be true for an equation to have no solution or infinite solutions.

Display each equation, leaving a large space for writing. Under each equation, invite students to share what they used to make the equation true for all values of , and record these for all to see. Ask:

• “What did all these answers have in common?” (There is only one possible answer for each equation that will make it be always true.)
• “What strategy did you use to figure out what that answer had to be?” (The solution had to be something that would make the right side equivalent to the left.)

Next, invite students to share what they used to make the equation true for no values of , and record these for all to see. Ask:

• "Why are there different ways to fill in the blank so that the equation is true for no values of ?" (As long as the answer is different from what we chose in Part 1, then the equation cannot be true for any value of .)
• “What was different about Equation C?” (We had to be careful to make sure that the variable coefficient was 3, and we added a constant so that the equation wouldn't have a single solution.)

Ask students to share observations that they made for the last question. If no student mentions it, explain that an equation with no solution can always be rearranged or manipulated to say that two unequal values are equal, such as , which means that the equation is never true.