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Calculus Optimization

William Thompson, Diana Gedye, Will Thompson, David Thomas | Published: June 14th, 2024 by K20 Center

  • Grade Level Grade Level 12th
  • Subject Subject Mathematics
  • Course Course AP Calculus
  • Time Frame Time Frame 3-4 class period(s)
  • Duration More 180 minutes


In this lesson, students will explore the calculus concepts of optimization and learn to increase their ability to identify, to model, and to set up functions to solve calculus word problems, focusing on the conceptual understanding over calculations. This is done through a mix of activities and by playing through Mission Prime, a digital game-based learning (DGBL) module. The DGBL module gives students the opportunity to explore and apply the concepts within an interactive world that provides them with three-dimensional visualizations of the word problems they are attempting to solve.

Essential Question(s)

Why do we need optimization? What is it used for?



Students use the Two-Minute Paper strategy to begin thinking about optimization and to refresh what they learned in the previous lesson.


Students play the first two missions of Mission Prime to explore the concepts of calculus optimization.


Students use the Example and Non-Examples strategy to learn how they would use optimization in their daily lives.


Students extend their understanding of the concepts by playing the third mission of the Mission Prime DGBL module.


Students apply the concepts they've learned to a physical optimization problem to show their understanding of the concepts.


  • Student devices with internet access

  • K20 Game Portal accounts or iPad apps of Mission Prime for each student

  • Whiteboard

  • Writing utensils

  • Paper

  • Graph paper

  • Scissors

  • Tape

  • Box Optimization Worksheet (attached)

  • Mission Prime Solutions (attached)


At the start of class, hold up a can of soup and ask students what they know about optimization and how that relates to the can you are holding.

Have each student take out a piece of paper and, using the Two-Minute Paper strategy, have them write everything they know or think they know related to the concept and how it might apply. Once students have finished writing, have some students share out what they have written.


Once students have finished sharing from their papers, introduce them to the Mission Prime DGBL module. Click here to learn more about the game. Play through the game at least once before teaching it so that you have a general understanding of the story and the characters that students will encounter as they play.

Prepare students to play the game on their computers or tablets and have them play through the first two missions, which should take roughly 30-45 minutes. You do not need to give them further instructions here. The game will introduce them to its mechanics, concepts, and story. At this point, take time to walk around the room, helping students who are confused or stuck and observing their progress. A handout with the proper order of operations for each mission is in the Attachments section should you need it.


Having played some of the game and seen how optimization is applied within its virtual world, use the Example and Non-Examples strategy to help students think about how they would use optimization in their daily lives.

Form students into groups or continue using the groups they played the game in. Have them come up with four examples and four non-examples with explanations. Then, have each group share out one of each with the class and discuss them.

If students' examples don't already show it, you might remind them that optimization does not only pertain to physical items. It can also be used to maximize the crop yield of a field or the return on investment of a business venture. Optimization means finding the maximum or minimum values of a quantity or finding when these maximums or minimums occur.


Now, have students to go back and play mission three. This should take around another 30-45 minutes to complete. As mentioned previously, it is not required that players complete mission four, but if you have students who complete mission three very quickly, you can have them continue on to the final mission. The final mission will take another 30-45 minutes to complete.


Now that students have played the game and applied their optimization knowledge in a virtual world, it's time to have them translate that to the physical world. Have students form groups of four or five. If they were in groups to play the game, you can continue using those same groups.

Give each group some graph paper and the attached Box Optimization Worksheet. Give students 15-20 minutes to discuss the problem, create their paper boxes, and complete the worksheets. Then, have groups share out the volumes they ended up with. Record these numbers on the board.

Find whose box had the highest volume and have them share the processes they used to determine the size of the squares they were going to cut out of the graph paper. Make sure to clear up any misconceptions that students might have related to the concepts here as well.