intro_activities


 * Potential Calculus Intro Activities**

Given a sheet of paper/posterboard, figure out the size of the square pieces you need to cut out of the corners to construct (after folding) the largest possible shoe box bottom.
 * Materials: several pieces of paper/posterboard; ruler/meter stick; scissors

The game of pool is all about planning and careful consideration of where to hit the pool balls on the table, bouncing off rails as you need. Given a pool table (drawn on a large sheet of paper/posterboard) with the cueball in position "A" and the 8-ball at position "B," which "bounce spot" do you aim at on the side rail (call it position "C") that will result in the shortest total distance for the cue ball to travel?
 * Materials: large piece of paper/posterboard; ruler/meter stick; string (a few meters); cellotape
 * Follow-up Question: How could you "prove" what you find (e.g., using algebra, using geometry, using physical principles, etc.)?

Find the surface area (the largest flat surface) of the following cut-out shapes.
 * Examples might range from regular n-gons (e.g., triangles, rectangles) to highly irregular ones (e.g., cookie-cutter shapes, non-conic-section curves). Possible methods might include using equations, subdividing into and counting of known shapes (area units), converting uniform material area into weight
 * Materials: several objects, ranging from cardboard cut-outs to metal sheets, all of which have a relatively large, recognizable 2-dimensional polygon for a surface; ruler/meter stick; string (a few meters); cellotape; pre-marked grid/graph paper; scissors; large (big enough to cover the objects) pieces of paper/posterboard; large pieces of cardboard; kitchen scale (that can weigh small quantities)
 * Follow-up Question: What other methods could you use to find the area, and what are the limitations of each method?

Predict the location at which you could place a mirror in order to see around a corner.
 * Materials: Small mirror; meter stick(s); string (several meters); scissors; masking tape (one roll);
 * Follow-up Question:

Use the string to create "graphs" on the paper that correspond to each of the described situations. The entire group needs to agree on the particular graph, so feel free to move the string around during your discussions before you tape it to the page.
 * Materials: Descriptions of particular situations (printed in words), varying from everyday language (e.g., words like "speed" and "distance") to particular mathematics terms (e.g., things like "y-axis," "concave up," "increasing slope")
 * Variation: Use a rope and a room-sized xy-axis to have students "act out" shapes of graphs from a top-down perspective

Draw a graph depicting the distance (vertical axis) versus time (horizontal axis) relationship described in the following situation: On Friday Xelli and her husband Raffe left in their car for their scenic weekend vacation hideaway, 100 miles from their home. When they had driven about a quarter of the way, they realized they forgot one of their suitcases. Disappointed, but determined not to spoil their weekend adventure, they drove back home, picked up the suitcase, and got on the road again. This time, they drove faster. Halfway to their destination however, they became unsure of the directions. They stopped for a bite at a local restaurant, where they got directions, finally making the last leg of their journey to their weekend vacation spot. Let t=0 denote the time Xelli and Raffe first leave their home, and let the vertical axis on your graph denote distance (in miles) they are away their home at any given time.
 * Materials: graph paper (several pieces)
 * Follow-up Question: In what ways might your graph of this information be different from others' graphs, yet still equally correct?
 * Follow-up Question: How could you use the graph you created to estimate the speed that Xelli and Raffe were traveling at any given moment?

Given a series of graphs (distance and other scalars versus time), do the following (depends on how the activity is set up):
 * Describe the situation depicted in words (i.e., create a narrative that interprets the graph)
 * Arrange the graphs in a particular order (e.g., from smallest to largest distance traveled, from smallest to largest slope)
 * Find the graph that corresponds to a particular situation (e.g., dramatic decrease in the stock market, rocket launching, car braking quickly at a stop light, cake temperature as it cools, runner's speed during a marathon, height of a human from birth to adulthood)

Given a topographic map (2-dimensional map with contour lines), describe what you can know about the terrain depicted.
 * Variation: a map without elevation markings - what can we say (and not say) about the terrain?
 * Variation: choose a route or location for a particular task (e.g., constructing a road, building a roller coaster, blazing a hiking trail, scouting a location for a scenic lookout, determining the boundaries for a population of high-altitude animals, locating likely locations for rock outcroppings)

Given the following series of shapes, describe what patterns you see and make a prediction about other figures that would fit your patterns.
 * Materials: a set of circumscribed n-gons with increasing n-values
 * [[image:http://www.coolmath.com/images/ctri.gif width="150" height="150" caption="triangle in circle"]][[image:http://www.coolmath.com/images/csqr.gif width="150" height="150" caption="square in circle"]][[image:http://www.coolmath.com/images/chex.gif width="150" height="150" caption="hexagon in circle"]][[image:http://www.coolmath.com/images/cdec.gif width="150" height="150" caption="decagon in circle"]]