Optimization Week

• Model a Calculus optimization problem by carefully defining variables, motivating an equation that is based on the variables defined, and ultimately creating an equation in terms of one variable. 
• Frame the optimization problem in non-Calculus terms  to cater to an audience that may not know Calculus.
• Perform differentiation to optimize the equation that the student or group of students have set forth. 
• Justify their solution and explain why the solution that they found is better than all other possible solutions.

This project is a week of collaboration and deep engagement in optimization problems. I structure this week by starting to define the concepts of optimization through modeling optimization problems, getting students to engage in a Geometry level problem that could be solved without Calculus, and then build in the tasks of mathematical modeling and using the concept of a derivative to maximize or minimize.

After the basic instruction on optimization, I move towards having the students engage in a single optimization problem in a group of 3-4. I structure the group component similarly to the expectations in the working world. I have selected approximately 4 problems that are very much rooted in multiple applications of optimization that might be encountered in the working world. These problems include:

• How to cut cardboard to create a pizza box that maximizes the volume. 
• The price to set an airline ticket to maximize the revenue of an airline.
• How many floors to build an apartment building to minimize the cost per floor.

The project culminates in a group presentation to the whole class, where the students have to sell the class on their solution and why their solution is better than all other possible solutions. By the nature of group work, groups will have to learn to work together and delegate their roles. I utilize a peer grading/feedback form so that (1) students can offer each other constructive feedback and (2) presentations can be assessed to whether their classmates understand their presentations.

In this year’s iteration of Optimization Week, I have added technology requirements to further enhance this project. Such additions include: 

• Video Notes Over Optimization
• Peer Feedback Google Form (Make sure to create your own copy to share with your students).
• Several of my students created Google Slides to showcase their optimization projects. In their group, these students came up with the idea of going to Desmos (an online graphing Calculator) and using the graphical perspective to enhance their arguments.

This project features a variety of different ways to assess students. The rubric is structured in a way to assess students on not only the Calculus standard of Optimization, but also the clarity of mathematical communication.

I utilize the rubric data from the individual Optimization problem to gauge where students are at in terms of Optimization thinking before they start on their group work and to know how much assistance they will need in completing the project:

The group presentations are assessed around 3 essential questions that I assess and the peers of the classroom assess, these questions are effectively. The peer friendly Google Form question below shows the 3 essential tenets of Optimization and how the projects are assessed:

After the Google Forms are filled out, I send compile Google Form data and send each group an email that is similar to the following email for high quality feedback:

• Optimization Week Video Note Taking: This is about a 30 minute video going over the basic concept of optimization and developing Optimization from the non-Calculus level to the Calculus Level.
• Optimization Week Intro PPT: This PPT has an overview of optimization.
• Optimization Week Project Document: This has the individual optimization problem, the project proposal page, and the group optimization problems for the group members to choose from. Additionally, there is a rubric on the back 2 pages that goes through the expectations of how the group presentations are assessed.
• Day 2 Check In Google Form: This helps solidify the proposal for the group part of the project and checks in with students about how they are processing the expectations of the project.
• Day 2 Work Time Expectations PPT
• Peer Feedback Google Form (Make sure you create your own copy to share with your students).
• Exemplar Pizza Box PPT: This group did an exceptional job of incorporating technology and hitting on all the points of the rubric. This group was able to create a high quality optimization argument with both an analytical and graphical perspective.

Physical Materials to Have in Classroom

• Poster-sized paper
• Straightedges
• Students should have a graphing calculator or a device with internet to use Desmos.

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.

From the Indiana Academic Standards – Mathematics: Calculus

C.AD.9: Solve optimization real-world problems with and without technology.

C.AD.7: Use first and second derivatives to help sketch graphs modeling real-world and other mathematical problems with and without technology. Compare the corresponding characteristics of the graphs of f, f’, and f”.

PS.1: Make sense of problems and persevere and solve them: Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway, rather than simply jumping into a solution attempt. They consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” and “Is my answer reasonable?” They understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students understand how mathematical ideas interconnect and build on one another to produce a coherent whole.