Use the third problem on the Warm-Up to start today’s investigation. The big question is: How can you use the function rule to find the vertex of the parabola efficiently? To put this in context, ask students: "Why is it even helpful to find the vertex of a parabola?" My students have done some simple optimization problems in this unit already, and they will do some more challenging ones in upcoming lessons. The vertex helps us find the optimal value.
A sub-question of this larger question is: How do you find the vertex of the parabola when the function rule is given in different forms? Students may or may not realize the advantage of writing a function in vertex form, and they may or may not understand that they can use the process of completing the square to do this.
The big answer to the question is that there are two key ways to find the vertex of a parabola:
(1) You can find both the x-intercepts and then find the midpoint of those intercepts in order to find the x-coordinate of the vertex. Note that this only works when there are two x-intercepts.
(2) You can use the method of completing the square to rewrite the function in vertex form to identify the vertex.
The goal is for students to figure these two methods out on their own, which is why the functions provided are written in many different forms. At some point during the lesson, you can write examples of these two methods on the board and then struggling students can refer to these examples during the work time. Note that it is important not to do this in the form of a direct instruction lesson because as soon as you explicitly teach a skill or method to the entire class, students no longer have to think and make sense of what they are doing (MP1).
Again, this is a good opportunity for students to check their answers using graphing technology if they want to (MP5). Students who easily tackle the first problems can work on the more challenging problems. These problems are more challenging because a is not a factor of b or c so students will need to use rational numbers when finding the x-intercepts and vertex.
The Exit Ticket is written to get students thinking about the deeper questions behind the processes they were working on today. It is easy to get caught up in the algorithms and calculations, which is why it is essential to set aside time at the end of class to bring students’ attention to the ideas behind their calculations (MP2). These exit ticket questions also focus on the parabolas that might not fit students’ algorithms perfectly.
It is also set up to make students think more deeply and abstractly about their methods. Every parabola has a vertex, but not every parabola has two x-intercepts. Ask students to justify each of their answers using examples and diagrams (MP3). Ask them to justify their answers even if they are wrong answers—the purpose is for students to think through their reasoning and try to support their ideas with examples.
It is often difficult to get students to engage in discussions like this one because they are used to thinking that math means “doing problems,” not writing and thinking. I explicitly address this by telling students that in this class math means thinking about what you are doing and being able to explain it using different kinds of reasoning. Obviously one explanation does not change students’ minds, but putting time into this each day, and occasionally formally assessing the quality of students’ answers (and providing them with clear formative feedback) will help make this message more clear.