Discussion Question 5
Marty
one day of a calculus class.
The plan should include the following:
(1) A short statement of prerequisites for this class. What do you
expect the students to be able to do before this class?
(2) A one-hour long plan for a lesson.
The lesson should, as we discussed in class, be the first time that
students see or hear the word "derivative". Moreover, for this
discussion post, you are not allowed to use "slope of tangent lines"
in order to introduce derivatives. There are many other things you
can try instead, including first-order approximations, differentials
and implicit differentiation, purely algebraic derivatives,
qualitative derivatives, (real) real-world problems in which
derivatives are used, optimization, etc... This discussion is meant
to challenge your preconceptions of what calculus is!
Also, in your lesson plan, describe the time spent on simple lecture,
small group activities, large-group interaction, and other modalities.
If your lesson plan includes examples (of functions, problems, etc..),
describe these examples in detail.
Feel free to respond to other people's lessons, and offer refinements
as well.
Jacob West
Prerequisites:
Basic arithmetic.
Familiarity with the idea of a function as something that takes inputs
and produces an output (all numbers),
and possibly (though not necessarily) also with the graph of a
function.
Lesson Plan:
1. Motivation and Discussion.
Let's start with some physical motivation. In classical physics, or
rather, in the world around us, we often talk about "objects", usually
the stuff of matter like this desk, chair, you, me, spacecraft, cannon
balls, you get the idea. Stuff. Things. We often think of these
things as possessing certain inherent qualities, like position. If a
thing exists, then it has to exist somewhere. So where is it? Well,
how about we use GPS coordinates? Many cars on the road today have
GPS devices with this kind of position information. So we attach a
number (or set of numbers) to the idea of position. If you want to
know where something is, then I can give you the numbers on my GPS
device and you'll have all the information you need to find it. Of
course, when your car moves, then it's position has changed! In fact,
this happens whenever anything moves! You're not surprised? OK, so
what do you call that? When something changes position? We just
called it moving a moment ago, but what does that mean? We have this
idea 'moving' that we're all familiar with, but we'd like to have a
good way to describe it, like we had with position. With position, we
gave it a number that precisely defined everything about it, but one
immediate definition of moving is "not standing still", that is, no
position, so no number. If your car is moving down the road, how fast
is it going? Check your speedometer, there's a number there! What
does that number represent? In the US, it's usually "miles-per-hour",
which means that if you can keep your speedometer pointed to that
number, then that's how many miles you travel in one hours time. This
is your current "speed", which is a number, but now rather than
position it involves both distance (miles) *and* time (hour), which
are themselves numbers, but we can turn them into a single number by
dividing. Of course, I'm lying a little. Distance is a number, but
it's not *really* a number, it's really two numbers! We usually think
of distance as a final position (a number) minus an original position
(another number). Of course, the same goes for time, an hour only
makes sense with respect to some initial time: an hour from *now*,
perhaps. So now we've got speed, which is a number, but it's really
an equation like this: (end position - start position) / (end time -
start time)! This kind of equation represents an example of a more
general idea: how much does one thing change with respect to another
thing. In other words, how much did my position change when time
changed by an hour? How many times did your heart beat in 30 sec.
time? For every kilometer that the Earth travels around the Sun, how
far does the Moon travel around the Earth? How much do gas prices
change with respect to distance away from Los Angeles? What examples
can you come up with?
2. Definition Attempt.
Whenever we have a number that tells us how much a certain thing
changes with respect to how much another thing changes, we'll call it
a "derivative".
3. Time to draw a wavy line on the chalk board. Let's see if we can
talk about the derivative of this line. Let's call the first "thing"
in our definition above the height of some point on this line from the
bottom of the chalk board, and the second "thing" will be the distance
of some point on the line from the left edge of the chalk board. We'd
like to figure out how well our definition captures the idea of
something changing with respect to something else. At this point,
I'll just describe how this discussion might go. We'd start by
picking two points on the line with the same height, but some distance
apart (with hopefully a large amount of variation in between), and
discuss how well our definition describes how much this line changed
over that distance (i.e. poorly). This might be done in groups
working at different regions of the board. Discuss how we might
improve our definition to better match what we're trying to describe
about this line, in particular, motivate the idea of accuracy and how
this is improved by looking at smaller regions of the curve. How
small a region should we consider? Motivate and discuss the idea of a
limit. Discuss how we might reformulate our definition using this
idea. Demonstrate this with secant lines between points. Could also
potentially discuss linear approximations here.
4. Definition.
A derivative is a number defined at some point x, by the limit of
(something at x - something near x) / (x - near x) as (near x) goes to
x. At this point, I'd probably write the limit more formally as well.
5. Any Additional Discussion and Questions.
Discuss how this formal definition captures our intuitive
understanding, and how broadly it applies.
Jen Mogel
Students should know about limits and average rate of change at this
point.
OK, I think the idea of rate of change of a car's position is a good
idea, even using a GPS, which is very modern and can help give a
conceptual understanding to the idea of derivative. However, there
are some things I would change. First, I would look at examples
beyond finding the rate of change of a distance, since, as Marty
pointed out, most students are unfamiliar on how to compute a distance
using a GPS. I would actually stare with rate of change of other
things, like amount of water in a tank, how fast is a car's speed
changing over time or over distance, how fast a heart is beating,
etc. Actually the heart beating one might be fun to try for a small
class by collecting data from a student. Anyway! I let's say I take
the water in a funny sized tank for example. I would present a table
of values for the students, like the volume, which could be read on
the side of the tank at any given time, and height, also read on the
side. I would have values for these variables for times like 10:00,
10:30, 10:45, ..., 10:49, 10:49:30, 10:49:45, 10:49:55, 10:49:57, and
for other values after 10:50, really close to the 10:50 time. From
these, we, as a class would compute the average rates of change over
all these times of either the height or the volume. We will notice
that as our time interval gets smaller, the ave. rate of change
stabilizes and we'll estimate the instantaneous rate of change of the
water volume and height at 10:50. We could even do the example with
finding the rate of change of the volume over the height of water, but
I think doing an example over time first might be easier to handle for
this first time.
After this example I would give the definition of a derivative as the
instantaneous rate of change of a certain quantity like distance,
speed, volume, height, over another quantity, like time, distance, or
height. I would equate the derivative to a general rate of change,
like speed is the derivative of distance traveled, and acceleration is
the derivative of speed. (not heroin) I would then give them another
table of speeds of a car at a given distance from a point, than ask my
students to find the derivative of the speed with respect of distance
from a starting point, let's say, 50 miles.
It is after this, I would move on to graphs. I would do the same
thing with a graph, construct a table of values to use average rate of
speed to estimate the derivative of the graph at a certain x value.
Then we would take this and use it to derive the limit of the
difference quotient method of finding the derivative.
All right, why this way? First it moves away from the formula notion
of a derivative and into the real world notion of instantaneous rate
of change. It gives more of a motivation of the traditional
derivative definition.
Well, Clayton has finished mopping the kitchen floor so it's time for
my dinner. See you in class tomorrow.