Models

3D Printing

Many of the Manipulative Calculus models were produced using 3D printing. In the early phases of the project, we used Harvard University's Active Learning Labs to 3D print several copies of the models as we tested out different lessons and designs. When it came time to scale production to accommodate all Harvard undergraduates in Math 21a, Ultimaker sponsored the project by 3D printing many of the models for us. We used two Ultimaker 3 printers, which have dual-extrusion printer heads, allowing for dissolvable support material and more complex designs. We finished most of the models with blackboard paint so that students could write and erase in chalk on the models. All models were designed entirely in OpenSCAD.

Thermoforming

Inspired by the Raising Calculus to the Surface project, our function graph models were made using thermoformed PETG plastic. We designed and 3D printed a 6x6" mold and used the Active Learning Labs' thermoformer to produce 50 copies of the model. The resulting surface is transparent, dry-erasable, and durable.

We were careful to select a function for the graph that would

1. have features that would embody concepts learned in the class (e.g. critical points),

2. have an explicit formula that is simple enough to be used in calculations, and

3. lack symmetries that would allow students to avoid engaging with the concepts directly.

We decided on a degree-3 function of two variables, and designed the mold in OpenSCAD.

Laser-Cutting

The flat models (plane, washer, and disks) used for our volumes of revolution lesson were produced by laser-cutting 1/8 inch acrylic in the Instructional Physics Lab. The cylindrical shells were produced by machining cast acrylic (note that these could also be 3D printed). The models were sized to fit together precisely for the purposes of the lesson. The cut files, specs, and source code are available here.

Low-Tech Props

To illustrate certain objects in multivariable calculus, we need only very simple models. For coordinate axes, we assembled kits with K'Nex pieces. For lines and planes we use a kit with K'Nex pieces and small chipboard sheets.