Transcript ppt - Department of Mathematics

Challenges, Explorations with
Lines, and Explorations with
Parabolas
Jeff Morgan
Chair, Department of Mathematics
Director, Center for Academic Support and
Assessment
University of Houston
[email protected]
http://www.math.uh.edu/~jmorgan
Geometry Challenge
Something to Sleep On
Is it possible to cut a circular disk into 2 or
more congruent pieces so that at least one
of the pieces does not “touch” the center of
the disk?
Probability Challenge
Something to Sleep On
Pick a value in the first 2 rows. Then move forward that number from left to right
and top to bottom. Keep going until you cannot complete a process.
4 1 5 3 3 5 2 4
3 2 2 5 1 5 2 5
2 4 2 1 3 4 2 3
3 5 4 3 2 3 3 3
1 1 1 3 5 5 5 5
1 2 1 5 5 5 3 3
In this case, you will always land on the 4th entry in the last row.
Question: Create an 8 by 6 grid of values from 1 to 5, with the values chosen
randomly. Repeat the process above. What do you observe?
Quick Challenge
warm up #1
A set of line segments is shown below. Believe it or not, they all have the same
length. What do you think you are looking at?
Exploration 1
warm up #2
Three lines are graphed below. Use a ruler to determine equations for the lines.
Exploration 2
A hexagon is shown below. Draw lines through each pair of opposite sides and
mark the point of intersection. What do you observe?
Do you think this happens with every hexagon?
Exploration 3
Try to plot more than 4 noncollinear points so that if a line passes through any 2
of the points then it also passes through a third point.
Exploration 4
Create a special function f. The domain of this function is the set of natural
numbers larger than 2. The range of this function is the set of nonnegative
integers. Given a value n in the domain of f, the value f (n) can be found by
determining the largest number of distinct lines that can be drawn in the xy plane,
along with n distinct points in the xy plane, so that each line passes through
exactly 3 of the points. Complete the chart below.
n
3
4
5
6
7
8
9
f (n)
Exploration 4
The line 2 x  3 y  8 is graphed and the point P  1, 2 
is chosen on this line. A new point Q is formed by adding
2 to the x coordinate of P and 3 to the y coordinate of P.
Discuss the relation between the line segment PQ and the
line 2 x  3 y  8.
Discuss any possible generalization.
Exploration 5
1
Graph both f  x   1  x and g  x   x. Find their point of
3
intersection, and then explore the sequence of values a1  f  0  ,
a2  f  a1  , a3  f  a2  , ... etc.
Let  1  m  1. Graph both f  x   1  m x and g  x   x. Find
their point of intersection, and then explore the sequence of values
a1  f  0  , a2  f  a1  , a3  f  a2  , ... etc.
Exploration 3
A rectangle with sides parallel to the x and y axes
has its lower left hand vertex at the origin and its
upper right hand vertex in the first quadrant along
the line y  10  2 x. Give the dimensions of the
rectangle so that it has the largest possible area.
Exploration 3 – Figure
Exploration 4
I. A line with slope  3 passes through the point  2,3 . Give the equation
of the line in slope-intercept form.
II. A line with slope  3a passes through the point  2a, 4a  1 . Give the
equation of the line in slope-intercept form.
III. A line with slope  3a passes through the point  2a, 4a  1 . Is there a
value of a for which  1, 2  is on the line?
IV. For each real number a, a line La is created with slope  3a that passes
through the point  2a, 4a  1 . Are there any points that fail to be on any of
these lines?
Exploration 4 - Figure
Exploration 12
1 2
Graph the parabola y  x . Then draw the vertical line segment
2
from the point  a, 20  to the point where it intersects the parabola
for several values of a between  4 and 4. Now imagine that
each of these vertical line segments is a path of a laser beam that
is shown towards the parabola, and then reflects off of the parabola
towards the y axis. Discuss the points of intersection of the reflected
laser with the y axis, and the total length of the beam's path from
its origin to the y axis.