Challenges, Explorations with Lines, and Explorations with

Parabolas

Jeff Morgan Chair, Department of Mathematics

Director, Center for Academic Support and Assessment

University of Houstonjmorgan@math.uh.edu

http://www.math.uh.edu/~jmorgan

Geometry ChallengeSomething 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?

4 1 5 3 3 5 2 43 2 2 5 1 5 2 52 4 2 1 3 4 2 33 5 4 3 2 3 3 31 1 1 3 5 5 5 51 2 1 5 5 5 3 3

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.

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?

Probability ChallengeSomething to Sleep On

Quick Challengewarm 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 1warm up #2

Three lines are graphed below. Use a ruler to determine equations for the lines.

Exploration 2A 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 3Try 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 4Create 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 f (n)3

4

5

6

7

8

9

Exploration 4

The line 8 is graphed and the point 1, 2

is chosen on this line. A new point is formed by adding to the coordinate of and to the coordinate of .

Discuss the relation

3

between h

3

t

2

2

e l

x y P

Qx P y P

ine segment and the line 8.

Discuss any possible generalizatio

2

n

3

.

PQx y

Exploration 5

1

2 1 3 2

1Graph both 1 and . Find their point of3

intersection, and then explore the sequence of values 0 ,

, , ... etc.

Let 1 1. Graph both 1 and . Find

their po

f x x g x x

a f

a f a a f a

m f x m x g x x

1 2 1 3 2

int of intersection, and then explore the sequence of values 0 , , , ... etc.a f a f a a f a

Exploration 3

A rectangle with sides parallel to the and axes has its lower left hand vertex at the origin and its upper right hand vertex in the first quadrant along the line 10 2 . Give the dimensions of

x y

y x 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 3 passes through the point 2 , 4 1 . Give the

equation of the l

a a a

ine in slope-intercept form.

III. A line with slope 3 passes through the point 2 , 4 1 . Is there a

value of for which 1, 2 is on the line?

IV. For each real number , a line is created wia

a a a

a

a L

th slope 3 that passes

through the point 2 , 4 1 . Are there any points that fail to be on any of

these lines?

a

a a

Exploration 4 - Figure

Exploration 12

21Graph the parabola . Then draw the vertical line segment2

from the point , 20 to the point where it intersects the parabola

for several values of between 4 and 4. Now imagine thateach of

y x

a

a

these vertical line segments is a path of a laser beam that

is shown towards the parabola, and then reflects off of the parabolatowards the axis. Discuss the points of intersection of the reflectedylaser with the axis, and the total length of the beam's path from its origin to the axis.

yy