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Volume + Calculus

It’s #throwbackthursday again–time for another oldie, but goodie! Back in March 2009, my AP Calculus students completed a Pi Day project in which they found the volume of an object using the disc method of finding volume by integration. It is still one of my favorite math projects, and it only took about a week of class time! See below for a synopsis of the project phases and tips for how to facilitate this project with your Calculus students (and ideas for adapting the project for lower grades).

Project Timeline (One/two 90-minute class periods per phase)

  • Preliminary Phase | Object Selection | Students must choose a receptacle that has a circular cross section. They are not told specifically about the cross section, but just that they must be able to use the disc method to find its volume. Additionally, the object’s shape must be complex enough to require a piecewise, non-linear function to model its profile (i.e. no cylinders, partial spheres/cones, etc.) Object must be approved by teacher before student can proceed to next phase.
  • Phase One | Trace & Plot | Students are given graph paper to trace the profile of the object, using the x or y-axis as the center line. Then, they must identify a minimum of 50 ordered pairs to model the object’s shape and record them on a printed t-chart or spreadsheet.
  • Phase Two | Piecewise Modeling | Using the list of ordered pairs recorded in Phase 1, students use the table and regression functions on their graphing calculators to find a piecewise function that models the profile of their shape. The function should be continuous and differentiable for all values in its range (or domain depending on its axis of symmetry).
  • Phase Three* | Find Volume | Using the appropriate formula based on the axis of symmetry, students are now ready to find the volume of their object. My students were required to show their work for this phase, but they were also allowed to check their work with their graphing calculators.  This is particularly important to prepare them for the AP exam, on which they will be required to find integrals by hand and using their calculators. *This did not take a full class period, but it could be extended to a full class period if you allow students check each others’ work.
  • Phase Four | Poster + Presentation | Students created and presented a poster that summarized their work.  Students were required to include a photo or drawing (photo preferred) of their object, a printed graph of their piecewise model (most students used Geogebra for this, but Desmos would be appropriate too), their volume calculations, and possible sources of error for their calculations. The Error part is important so that students think through issues such as the thickness of their object or mistakes in their modeling methods/ estimation.

Tips

  • Keep suitable objects on hand to assign to students who do not bring their own or bring inappropriate objects–with no circular cross section or too simple in shape.  Be sure to add this to your rubric and deduct points if the student is unable to bring an appropriate shape to class in a timely manner or select an appropriate shape from your set.
  • Remind students that while 50 points is the minimum that they have to find, there is no maximum. Also, encourage students to identify non-integer ordered pairs. The more points they identify, the more accurate their regression models will be.
  • Inspect piecewise function graphs before Phase 3. Require students whose graphs do not closely resemble their objects’ profiles to identify more ordered pairs.
  • If you require that students calculate volume in square centimeters, they can check their work by measuring its actual volume in milliliters. My students discovered this on their own, and most of them added this information to their final posters.  Even I was surprised at how accurate their calculations were!

2009; Kay Butler, P.F. Taylor Academy

Adaptions

  • Pre-Algebra/Algebra 1/Geometry | Phase One Only | Students could simply trace and identify points for any object (circular cross section not important) Then, using Geogebra or Desmos, they could plot the points and adjust for any errors. If you print out the plotted points, students could “connect the dots” to see their shape come to life! For Geometry students, students could choose a real-life object that conforms to a specified shape or choose from a list of shapes provided by you.
  • Algebra 1/Algebra 2 | Phases One & Two | Students who are familiar with linear or quadratic functions can choose an object whose shape or profile can be modeled with a single function. This could be introduced not only for real objects but also for photos of objects/shapes, as shown to the right.
  • Algebra 2/ Pre-Calculus | Phases One, Two & Four | Advanced students may be required to find and object or photo that can only be modeled with a piecewise or other specific advanced functions (i.e. exponential, logarithmic, trigonometric, cubic, etc.). Students at this level could also present their findings with a poster or slide show.

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