Hands-on Calculus ActivitiesThe University of Arizona

Mathematics Instruction Colloquium

Liana Dawson

November 6, 2007

Introduction (and disclaimer) Examples of various hands-on and nontraditional

calculus activities

Most from a Project NExT Workshop attended during Summer 2007 Presented by Julie Barnes, Western Carolina University

(anything with a * is from this workshop)

Disclaimer: I have not used any of these activities, nor do I claim they are all appropriate for students at the U of A

Calculus I - Differentiation

Functions on the floor* (also for precalculus)

Optimization of a cereal box Lynette Boos, Trinity College

Matching Notecards Kristin Camenga, Houghton College

Filling a Vase Robert Kowalczyk & Adam Hausknecht, UMass Dartmouth,

(http://www2.umassd.edu/temath/TEMATH2/Examples/ExamplesContent.html)Sarah Mason, Davidson College

Functions on the Floor First and Second Derivatives

Materials: Adding machine tape, tape, rope Place several axis systems on the floor using adding

machine tape Provide students with a collection of conditions on

derivatives and second derivatives Give students jump ropes (other heavy rope) to create

graphs with given conditions

Similar activities can be used for precalculus (give increasing/decreasing, concavity), parametric equations

Students can also draw axes on paper and use string to create graphs

Optimization of a cereal box Materials: cereal boxes (or cardboard

rectangles), cereal, scissors, tape Students must create a box with the

maximum volume by using one side of the cereal box and cutting squares out of the corners

Students can compare volumes by filling their boxes with cereal

Matching Notecards Differentiation/Integration functions

Materials: Notecards Create two sets of notecards – one with the

original function and one with the derivative Pass out the cards and have students find their

“match”

Activity can be expanded by including graphs of the functions and the derivatives, the limit definition of derivatives, or chains of functions where students need to line up according differentiation

Filling the Vase Inflection Points

Materials: Oddly shaped vase, measuring cup, water

Pour in water 50 mL at a time to simulate a constant rate

Have students graph the depth as a function of time

Discuss concavity and points of inflection

Calculus I - Integration

Gum and Riemann Sums*

Cookie Calorie Counter*

Gum and Riemann Sums

Materials: sticks of gum, handouts

Pass out a sketch of a simple graph

Have students slide around the sticks of gum so that the top of the gum is hitting the function where the top of the rectangles should be for left, right, and midpoint sums.

Cookie Calorie Counter Riemann Sums

Materials: Strangely shaped cookie, graph paper Hand out cookies and graph paper Have students trace their cookie on the graph paper. Tell students the approximate number of calories per

square unit Students must approximate the number of calories per

cookie

Calculus II

Party Favors*

Play-doh*

Bundt Cake*

Who sucks the most Stu Schwartz, www.mastermathmentor.com

Play-Doh Materials: Play-doh, dental floss Students use play-doh to model solids of

revolution Example: Let A be the region bounded by exp(x),

y=0, x=0, x=1 Model the solid obtained by rotating A around the x-axis Model the solid obtained by rotating A around the y-axis Model the solid obtained by rotating A around y=-2

Use floss to cut the objects to see cross-sections

Bundt Cake Materials: bundt cake and pan,

plastic wrap, calculator, graph paper, measuring cups, rice

Give each student a slice of the cake wrapped in plastic wrap

Have the students trace the slice on their paper Give the students the diameter of the cake, and

have them use their calculator (quartic regression) to find an equation that approximates the curve

Have students approximate the volume of the cake Students can fill the pan with rice to check their

approximation

Who sucks the most Work Materials: paper cups, water, straws, rulers Each student receives a cup of water and a straw. Time the students on how long it takes them to drink

the water through the straw (the straws should be kept perpendicular and very close to the bottom of the cup)

Students then must calculate the volume of water in the cup and the amount of work needed to empty the cup through the straw

Students then calculate their “sucking power” and the horsepower of their mouth.

Calculus III More Play-doh

Kathy Ivey, Western Carolina University

Level curves of heads Hope McIlwain, Mercer University

3-D axes*

Topographical maps*

Running hills Cayley Rice, Albion College

More Play-Doh

Level Curves Materials: Play-doh Sketch the level curves

for a play-doh “mountain”

Have students create a 3-d model of the mountain

Level Curves of heads

Materials: Adding machine tape, tape Tape several lines on the floor. Ask for volunteers who are 5’10”, 5’8”, 5’6”,

and 5’4” (or whatever heights that work for your class). Have these students stand on their respective lines

Students can visual the surface by looking at the tops of the volunteers’ heads.

Topographical maps

Level Curves Materials: Photos, topographical maps Have students match photographs with

topographical maps

Topographical Maps

3-D axes

Materials: Yarn, tape Create 3-d axes using the yarn and tape Have students plot points, curves, or more

complex examples

3-D axes

Example: Plot z = x^2+y^2 by plotting points and connecting them with string. Make a “mesh” out of

yarn

Running Hills

Gradients Materials: Hills, yarn Students are each told to pick a spot on the hill Students walk along the level curve from their spot a

few steps, and lay down a piece of yarn along their path.

Students then must find the steepest direction and walk along it.

Students should “discover” that the gradient is perpendicular to the level curves

Mathematical Visualization Toolkit Java applet developed by the Department of

Applied Mathematics, University of Colorado, Boulder

http://amath.colorado.edu/java/

Basic plotting tools Applications: tangent slider, Riemann sums,

solids of revolution

Discussion

What classes these activities might be appropriate?

Other activities?