Lesson 16

Students extend their understanding from the previous lessons to recognize the structure of a linear equation for all possible types of solutions: one solution, no solution, or infinitely many solutions. Students are still using language such as “true for one value of x,” “always true” or “true for any value of x,” and “never true.” Students should be able to articulate that this depends both on the coefficient of the variable as well as the constant term on each side of the equation.

Give students 2–3 minutes of quiet think time followed by a whole-class discussion. 

In order to highlight the structure of these equations, ask students:

• “What do you notice about equations with no solution?” (These equations have equal or equivalent coefficients for the variable, but unequal values for the constants on each side of the equation.)
• “What do you notice about equations that are always true?” (These equations have equivalent expressions on each side of the equation, so the coefficients are equal and the constants are equal or equivalent on each side.)
• “What do you notice about equations that have exactly one solution?” (These equations have different values for the coefficients on each side of the equation and it doesn’t matter what the constant term says.)

Display the equation \(x=12\) for all to see. Ask students how this fits with their explanations. (We can see that there is one solution. Another way to think of this is that the coefficient of \(x\) is 1 on the left side of the equation, and the coefficient of \(x\) is 0 on the right side of the equation. So the coefficients of \(x\) are different, just like the explanation.)

In this activity students solve a variety of equation types; both in form and number of solutions. After solving the 10 equations, groups sort them into categories of their choosing. The goal of this activity is to encourage students to look at the structure of equations before solving and to build fluency solving complex equations. For example, students who notice that equation D, \(3 - 4x + 5 = 2(8 - 2x)\), has the same number of \(x\)s on each side but a different constant know that there are no values of \(x\) that make the equation true. Similarly, equation J has the same number of \(x\)s on each side and the same constants on each side, meaning that all values of \(x\) make the equation true. These up-front observations allow students to avoid spending time working out the steps to re-write the equation into a simpler form where the number of solutions to the equations is easier to see.

Arrange students in groups of 3. Distribute 10 pre-cut slips from the blackline master to each group. After groups have solved and sorted their equations, consider having groups switch to examine another group’s categories. Leave 3–4 minutes for a whole-class discussion.

Select 2–3 groups to share one of their categories’ defining characteristics and which equations they sorted into it. Given the previous activity, the categories “one solution, no solution, all solutions” are likely. Introduce students to the language “infinite number of solutions” if it has not already come up in discussion.

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 \(2x-3,\) the 2 is the coefficient of \(x\), the 3 is a constant, and \(x\) is the variable.

The purpose of this activity is so that students can compare the structure of equations that have no solution, one solution, and infinitely many solutions. This may be particularly useful for students needing more practice identifying which equations will have each of these types of solutions before attempting to solve.

Arrange students in groups of 2. Give students 3–5 minutes of quiet think time followed by 2–3 minutes of partner discussion. Follow with a whole-class discussion.

On the second question, students may think that \(x-3=3-x\) have the same coefficients of \(x\). Recall that \(x\) has a coefficient of 1, while \(\text- x\) has a coefficient of -1.

Review each of the equations and how many solutions it has. If students disagree, ask each to explain their thinking about the equation and work to reach agreement. Once students are satisfied with the solutions, display the following questions for all to see:

• “What do you notice about equations with one solution?”
• “What do you notice about equations with no solutions?”
• “What do you notice about equations with infinitely many solutions?”

Give students brief quiet think time and then ask them to share a response to at least one of the questions with a partner. After partners have shared, invite student to share something they noticed with the class. Record students responses for all to see next to the relevant question.