SWBAT determine if an equation has one solution, no solutions or an infinite number of solutions.

The structure of an equation gives insight into the number of solutions the equation has.

15 minutes

This opening activity (true_and_false_equations2_independent.pdf) lets students get right to work on solving some complex linear equations. I ask students to work on these questions individually. I choose to post the answers when students begin working. I explain to students that having the answers posted challenges them to be precise in their work (MP6). If they do not get the same solution as the posted answer, they should go back through their work and try to identify the error. Having the answers posted also takes the emphasis off of students getting "an answer" and puts the emphasis on the process of solving an equation.

20 minutes

I display the first slide from True_and_False_Equations 2 and I have students work individually to categorize the three equations as "sometimes true", "always true", and "never true." I expect that most of my students will do this by solving the equations. After about 3-4 minutes of letting students think, I will ask them pair up with a partner (think-pair-share) to discuss and justify their categorizations (MP3). While students are working and discussing, make note of which students are categorizing correctly.

Next, I will put up the correct categorizations (on the next slide) and ask students to compare their work with the answers on the slide. I will call on students who have categorized the equations correctly to explain their reasoning to the class. I will then allow several students to add on to each explanation where applicable. Finally, I will ask students to consider how to match the terms "infinite solutions" "one solution" or "no solutions" with each equation. I want them to write down their ideas first. Then, I will ask students to share their explanations with their partner and critique each other's justification (MP3). I plan to allow a few students to share their ideas and guide the thinking of the class towards understanding the following:

- Equations that are always true will have an infinite number of solutions.
- Equations that are never true will have no solutions
- Equations that are sometimes true will have (in this case) one solution. In general, these equations will have a finite number of solutions.

5 minutes

This ticket out (true_false_equations2_close) will allow students to show their thinking around the concept of number of solutions. In each question students can fill in the blank or make up an equation to show the appropriate number of solutions.