SWBAT transform a quadratic into vertex form, and then use this form to find and interpret the graph’s vertex and x-intercepts.

Students use the concepts they have learned about quadratics to find more precise answers to a real world problem.

10 minutes

I begin class by giving students a warm up that covers some of the work we have been doing in previous lessons. It's important to me that students can move fluently between standard and vertex form and I find they need a lot of practice in order to master both tasks. I start class with this warm up and explain that today we'll be taking the vertex form a step further to figure out how we can use vertex form to find the x-intercepts of a quadratic.

45 minutes

Students will be working in homogenous groups today to put together what they've learned about going from standard to vertex form and then solving a quadratic. I like to read through the activity together and let them get to work. They will be doing a written reflection on this work for homework to summarize what they have learned thus far, so I let them work up until the last 5 minutes of class.

Things to watch for:

- Students may struggle with the value of
*a*as -16. Some students may have not yet mastered how to change a standard form quadratic into vertex form when the coefficient of*a*is anything other than 1. I try to have their peers guide them through this process, if possible. - Students may forget to bring the -16 back into the vertex form. I remind them that they can’t just take it out permanently and help them see that they need that -16 to get back to the original quadratic.
- Students may need help and guidance to see that the vertex form will give them both how long it will take the rocket to reach its maximum height and the rocket’s height at that point.
- Students may choose to work with either decimals or fractions.

This is the first time in this unit that students will be solving for x-intercepts. Once students have the quadratic in vertex form, I like to bring the whole class together and put a little sketch on the board. We talk about how the vertex form tells them how long it will take the rocket to reach its maximum point and what that maximum height is. Then I try to elicit from students that setting the equation equal to 0 will tell them how long it takes the rocket to hit the ground if height is a function of time. Once we set the vertex form equal to 0, students may need some guidance about how to solve for x. Because my students have academic interruptions, many of them will need a review about working with square roots.

Depending on how much time I have left in class, I may give students more work on solving quadratics by completing the square, or perhaps give this work to students who are ready for an extension activity.

5 minutes

**Reflection:** Ask students to verbally report out what is most challenging for them about this problem.

**Homework: **Let students know they will be reflecting on their problem solving strategies and steps in tonight’s homework. You may want to give students more than one night for this assignment. Explain to them that this problem ties together many of the main concepts of the unit and it is important that they understand the steps they have taken to solve the unit problem.