I wrote Quiz Graphing Polynomials to test my students' ability to create a sketch of a polynomial without a graphing calculator and interpret features of the graph of a polynomial [MP2]. Students work independently to complete this quiz and they are allowed up to an hour to complete it. Graphing quizzes often take students longer that other quizzes, so allow a longer time than I usually do for quizzes.
In this hands-on activity, students will compete to make a box that holds the most Skittles. Students will select one partner to work with and each pair will be given an 8 cm X 13 cm index card, scissors and tape, and a cup of skittles. The goal is to make an open-topped box by cutting the same size square out of each corner and then folding up and taping the sides of the box. I ask students to cut out squares that are integer number of cm in length. They understand the the goal of the task is to maximize the Volume of a Box.
The context for this activity is quite common. Problems based on this concept appears in many textbooks and in several grade levels under the CCSS. Nonetheless, I find that at least half of my students cannot visualize the problem effectively without actually making the box. Also, my students tend to enjoy competitions, so trying to make the box that will hold the most Skittles is fun for them [MP4].
As students work to make the largest box possible, I circulate and make notes to myself on which students are modeling the volume mathematically. Some students will likely see that the volume of the box can be represented by V=x(8-2x)(13-2x). However, they may not be able recall the procedure for finding the local maximum using the graphing calculator.
When students have completed their boxes, I ask for the number of Skittles from each group and declare the winner. If there are two boxes that hold a very similar number of Skittles, I will ask the two teams to count the Skittles publicly so we can be sure that the right team gets the prize (usually a pencil & eraser set).
I summarize the activity by asking students to share how they went about solving the problem. I start with a pair of students who approached the task by cutting out the a random square without calculating the volume. After they explain their method [MP3], I ask for a show of hands of other groups that used the same method. I then ask a group that calculated the volume of a few different size boxes before making any cuts in the card. Finally, I show my students how to solve the problem by modeling the volume with the polynomial V=x(8-2x)(13-2x) and using the graphing calculator to fund the dimensions of the box that maximize volume.
As an Exit Ticket, I send students a quick poll that asks them to determine the dimensions of the largest open-topped box that can be made with a standard index card that measures 3" by 5." I use a quick poll on the for this exit ticket so that we all have a good idea of how well the class understood how to use a polynomial function to model the volume of a box with a polynomial [MP4].