Calculus: Language of Wizards

1. Calculus: Language of Wizards Bob Robey Los Alamos National Laboratory

2. Introduce Calculus Terms Early • Introduce terms as early as 3rd-4th grade • Include in sequence of math operations of increasing complexity (see following slides) • Hands-on exercises to get concepts as early as 5th and 6th grade • Goal should be to read and write calculus terms, not perform intricate algebraic manipulations

3. Repetition Forward • Everyday Math has been successful at raising scores by repetition of previous material • Works well with improving scores of lagging students (but bores the stronger students) • For stronger students, extend the repetition forward! • Introducing the terms and concepts early reduces the fear and increases the success rate • I can do this! I’ve heard it before

4. Adding ConceptAddition through Integration • + -- Addition • x – Multiplication • Area of a rectangle • ∑ -- Summation of discrete items • Area under stairs • ∫ -- Summing complex shapes • Area of a skateboard ramp

5. Division ConceptDividing through Derivatives • Division -- y/x • Ratio, Percentages • Slope -- ∆x/ ∆t or (x2-x1)/(t2-t1) • Average speed (velocity) from Santa Fe to Socorro • Derivative -- ∂x/∂t • Speed (velocity) at one point, e.g. the center of the Big I in Albuquerque (may be different than the average)

6. Exercise 1 • Car or Bicycle example -- we are most comfortable at an early age with the concept of speed, velocity, distance and acceleration which makes this an excellent first exercise • Start by calculating distance traveled for a constant speed

7. Exercise 1 (cont) • Given a plot of the velocity during a trip, how far did they go? • Show several ways to solve: • Summing rectangles • Calculate average and multiply by time • Cut into one hour strips and lay end to end. Measure total

8. Exercise 1 (cont) • Repeat exercise for the plot below • See what kind of solution they come up with to calculate the area under the last part. Almost any solution will be an advanced integration technique.

9. Exercise 1 (cont) • Ride a bicycle across a playground or track varying speed and measuring it using a bicycle speedometer at a few points. Graph. Calculate the distance traveled. • Accelerate a bicycle over a short known distance. Measure time and velocity at the end. Graph and calculate acceleration assuming a constant acceleration over the distance. • On a highway trip, record the speed every 15 minutes. Graph. Calculate the distance traveled. • Collect GPS data, graph distance versus time. Now calculate and graph velocity and acceleration.

10. Exercise 2 • You are building a skate park. You need the volume of the concrete required • Assume a 1 ft depth so all you have to do is calculate the area • Calculate the concrete for a 10 ft flat entry that is 1 ft thick • Calculate the concrete for 3 steps • Calculate the concrete for a half-pipe cross-section

11. Exercise 3 (hard!) • River gauging stations are critical to manage our water supplies. Visit a nearby gauging station and determine the flow rate for different heights. • First measure the area of the streambed by measuring the depth at 1 ft intervals across the river • Measure the velocity at different points • Estimate the cubic feet per minute for a given height • Maybe do this as a class exercise or have a water engineer demonstrate on a field trip