SWBAT use combinations of functions to solve a maximization problem.

Function combinations and maximization problems collide to create a challenging and mathematically rich task.

5 minutes

The attached PowerPoint (Intro - Where are the Two Functions Farthest Apart) is the introduction to today's primary task. On each slide students are given two different functions and they are to figure out where the two functions are farthest apart. The first three sets are very straightforward and can be figured out solely by visual inspection; the fourth one will require more thought and some calculations. I go through this series as a class to get them thinking about the constraints of this task (**MP8**).

When the fourth set of functions is shown, I want to have the conversation about why this example is different from the rest. **Heads up:** in my experience many students will think that maximum distance occurs at the vertex of the parabola! If one student announces this to the class, I am confident that another will disagree with him/her. I am prepared to calmly explain that they will investigate this soon!

While the last slide is still displayed, I want students to do two things:

- Write down their
**First Thoughts**about this problem. The First Thoughts can be an idea of how to start the problem, what will be important in the problem, or an approach to solving. It should not be calculations! - Estimate where the x-value that would produce the greatest vertical distance between the two functions.

35 minutes

After students individually complete the two tasks at the end of the Launch, they will randomly be paired up to continue working on the Task - Distance Between Functions. Since we are still earlu in the year, the first thing each pair will do is to introduce themselves and read their First Thoughts to their partner; then they switch roles. Students will then be given the task sheet (Task - Distance Between Functions) and asked the find the maximum vertical distance between the two functions.

Perseverance will be very important with this task! Students may get stuck or may not know how to start, but it is important that they push on and keep working (**MP1**). As a teacher, it is extremely challenging to motivate them to keep working when they hit a wall, but here are some strategies for some roadblocks that you may encounter:

**A student who is completely stuck:** You could ask them how they could find the distance between the two graphs when x = 0.6, for example. This should get them focused on the need to find the y-values and the need for subtraction.

**A student who is confident that the maximum distance occurs when x = 1** (at the vertex of the parabola): Ask them to find the distance between the two functions when x = 0.9. If they find it correctly they will see that this distance is greater than when x = 1.

**A student who uses guess and check to get the correct answer:** Many of my students will use this method and get the correct answer. A few of them will comment on how annoying it is to do that - that's great! Now they are motivated to find a more efficient route. *Many* of my students used guess and check and found the correct answer. Many had their data organized by x-value and the distance between the two functions; once they have all of those values, ask them to find the function that represents that relationship.

**A student who notices that the vertical distance is the same at 0.7 and 0.8: **This is a great observation - ask them to graph these values to think about what type of function this would be. If they realize that is a parabola, ask them to find the equation for it.

**A student who gets the correct answer:** When students subtract the two functions and get the new quadratic functions, they can use an algebraic or graphical method to find out where the distance is maximum. If they use one method, see if they can use the other.

10 minutes

To start the conversation, I plan to pick a student who used guess and check - it was a great jumping off point and the work was easy to follow. Students will probably comment that there has to be an easier way, so immediately the class is thinking about a more efficient process. I am asking students, "How would you find the distance between the two functions, f(x) and g(x) when x = 0.6?"

Next, I ask students to find the distance when x= 0.72. Then when x = 0.4, then 1.1, then 1.3, then 1.8, and so on and so on. Students will probably get so annoyed with me that they will be bursting at the seams to tell me how to find the distance between the two functions at any x value. This might be a good stopping point and have them see if they can finish the problem for homework if they haven't already solved it.