Cookies, Curriculum, and AB Calculus
Well, the 2014 AP Calculus AB Free-Response exam questions have been released, and lively conversation among calculus teachers is guaranteed!
One of this year’s questions stands out for me. It is fairly difficult, but I feel confident my students scored well on it. Question 2c asked students to setup an equation involving integral expressions for locating a vertical line that divides the area of a region in half. Here is an image of the problem:
When I saw my students on Friday after the exam, several students said the free response questions had a “cookie problem”. They were referring to my Cookies and Pi lesson, in which I had students plan three vertical cuts to divide a Black & White Cookie into four equal pieces, equal with respect to the amount of cookie and the amount of chocolate and vanilla frosting on each piece:
This lesson includes scaffolds and differentiation for lowering the floor to ensure viable entry points for all students, and extensions for students ready to delve more deeply into variants of this problem.
My lesson connects nicely with this AP problem because while students immediately recognize that the first vertical cut should be through the center of the cookie, they then need to plan their next vertical cut. Before giving the radius of the cookie, I develop students’ intuition with predictions for where the cuts should be made, supported by this GeoGebra applet for why the middle pieces need to be narrower and taller than the outer pieces.
Later in the Cookies and Pi lesson, given a 5 cm radius for the cookie, students set up several different equations with integral expressions and make sense of other students’ setups that approach the problem in different ways. Each equation can identify the location of the correct vertical cut.
Here are some of the equations students might generate:
corresponding to the area of white frosting in one piece of cookie
The following lesson, Accumulate This!, continues our work from the Cookies and Pi lesson by investigating variable limits of integration from a graphical perspective. All in all, the rigor and depth of these lessons gives me confidence that my students scored well on this question. It is a good feeling to have at this point in the year.Check out more from Jason here!