**This project will be completed over the course of a week.**

### Pre-Project

I utilize a flipped classroom. Before the rollout of the project, I assign the students video notes for the theory of optimization. I like this format because it lessens the “sticker shock” of optimization, reinforces concepts learned about derivatives and function analysis, and introduces the students to the vocabulary. Additionally, I would encourage students to turn on closed captioning or slow down the playback speed to aid with the processing of the material. I recommend that students take notes on notebook paper. If you are not using a flipped classroom, you could do some of the examples that are in the video as part of your direct instruction. Of particular importance is Example 1:

I start with this example because this example has a low barrier to entry, can be solved without Calculus, and Calculus is introduced as a more efficient way of solving this number problem.

### Day 1: Introduction to Optimization Week

#### Bellwork [10 minutes]

For the bellwork on the Day 1 PPT I like giving an area maximization problem, based on a perimeter because students do not need any Calculus to solve it and they can engage in the question of optimization. For example:

(2) “Good fences make good neighbors” – You have been given 200 feet of fence to make an enclosure for your dog. Your goal is to create a rectangle that has a perimeter of 200 ft WHILE maximizing the area of fence. Choose a length and a width that maximize the area. Why is your proposal better than other possible proposals?

After 10 minutes of individual work time, you might want to have students talk with their table partners about how they solved the area maximization problem. Students should come up with the 10×10 square for the solution. When going over the methods of this problem, you want to embrace non-Calculus solutions. One question I love for motivating optimization based thinking is: *why is your solution better than other possible solutions?*

#### The Box Problem 1.0 [30-40 minutes]

To get students familiar with a basic Calculus optimization problem, assign individual students the optimization box problem. Students can still solve this problem without Calculus. There is a rubric so that you can assess students on where they are at with optimization. I instruct my students to call me over when they are done so that I can assess their solution with my rubric. NOTE: Non-Calculus solutions can still score reasonably well.

- Advanced students will be able to come up with a Calculus-based equation based on length, width, and height in one variable with minimal to no help. They will know to take the derivative and set it equal to 0 and will intuitively use the quadratic formula.
- Mid level students are comfortable with taking derivatives and can factor or use Quadratic formula with prompting too. These students will need a little help getting started as they transition from V=lwh to an equation with 1 variable. Once they are able to do that, they are able to take derivatives relatively independently and use quadratic formula solving.
- Low level students will need lots of guidance on this. Encourage these students to draw out specific examples. Ask questions like what would the length, width, and height be if 1 cm was cut out? If 2 cm was cut out? These lower level students who are able to construct those specific cases are able to see that some solutions work better than others and are able to get a concept about how cutting more height reduces the area of the base.

#### Start Group Work [10-15 minutes]

Assign students groups of 3-4. On Day 1 groups should look through the packet and choose a problem that they want to focus on. They should work through the questions on the proposal so that they have the materials and a game plan for next class.

### Day 2: Crafting The Optimal Solution

#### Check In [10 minutes]

Students should fill in this Google Form so that you can assess how your students and groups are processing the expectations of the projects. Be on the lookout for any questions. After the 10 minutes is over, debrief and go over questions that students have.

#### Work Time/Presentation Rehearsals [50 minutes]

I post something like this up on the screen while students are working:

As groups are working, you want to circulate and check in with each group. I love discussing with students about the idiosyncrasies of each problem (i.e. how you have to subtract 3x for the length of the pizza box, or how x represents the number $5 rent increases). Groups can choose from:

- Creating posters.
- Using graphs/tables to sell their solutions – Non equation based methods such as a table of values or graphs are actually user friendly in the real world if the audience of the presentation is someone who doesn’t know Calculus.
- Usage of technology: Some of my more astute groups recognize the need for a graphing Calculator like Desmos. My innovative groups even went to the level of graphing the original function and its derivative and comparing the graphs of those.

Ask probing questions like: How are you going to frame this problem to someone who doesn’t understand Calculus? Why did you choose the positive number rather than the negative number of your solution? Is a decimal number practical? What if you need a whole number, which whole number would you choose?

### Day 3: Presentation Day

#### Rehearse Presentations [10 minutes]

Students should start in their group and rehearse their presentations. They should iron out last minute details.

#### Presentations [ 30 minutes]

I shoot for 5 minutes per group/presentation to add an element of time efficiency:

- I use the Peer Feedback Google Form to ensure that students are engaged in watching others’ presentations and give students an opportunity to offer feedback to each other.
- I always make sure that I ask each group a question and allow other peers to ask questions to each group after they present their solution.