Students will graph parametric equations of lines.

Materials:

Graphing calculator, or graphing utility, with simultaneous, parametric graphing capabilities

. The question arises, "What's** t** ?" The table
of values helps students see that** t **is any number. More
problems are discussed and assigned. Given information is changed in order
to vary the practice and solidify the concepts. For example, the problem
reads, "Write the parametric equations for the line** y=2x-3**
" or, "Find the parametric equations for the line through (-4,-3)
and (5,2)". Students also reverse the process and write an equation
in slope-intercept form when given the line's parametric equations.

**Graphing Calculator Activity:** Using the parametric mode
on the TI-82 and graphing two sets of parametric equations simulanteously,
students simulate the linear relationship of a car race. This proves to
be quite entertaining. The race is 500 miles. One car averages 105 mph and
the other averages 120 mph but is delayed 30 minutes at the start of the
race due to electrical problems. Which of the two cars will finish before
the other, assuming they both finish the race?

First, students write a set of parametric equations to represent each car's
position after **t** hours. Car 1 is on track one so its equations
are

Car 2, on track two, is represented by

Discussion ensues for establishing the values of t-min and t-max, x-min,
x-max, y-min and y-max. What does** t **represent in this activity?
How much time is enough for this simulation? What does **x **represent?
Having answered these questions, students enter the values t-min=0, t-max=5,
x-min=0, x-max=500, y-min=0, y-max=5....and its off to the races....graphically.
The results of the visualization of the race can be confirmed by finding
the time that each car finished the race. John Woody, a junior, says, "I
want to do this algebraically", and so he did. By comparing the paths
of each car, students could tell when car 2 passed car 1...and then find
that time exactly using algebra. Excitement ran high for this activity....Mathematics
can be fun!

**Assessment: **In order to evaluate how well students met
the objectives of this lesson the following problem is assigned.

"Two semi-trucks are driving loads for Chicago to Denver, a distance
of 1125 miles. The first truck leaves at 8:00am and averages 50 mph. The
second truck leaves at 9:00am. Since it has a lighter load, the second truck
averages 54 mph. Set up two sets of parametric equations to model this situation
and use a graphing calculator to analyze the model.

a. How long is it until the second truck overtakes the first truck?

b. How far are they from Chicago when the second truck overtakes the first
truck?

c. If each of the drivers stops for meals for a total of 3 hours, what time
will it be in Chicago when each truck reaches Denver?

d. How much would the driver of the first truck have to increase her speed
in order to arrive in Denver first?" p447, *Merril Advanced Mathematical
Concepts*