Mathematics Education

EMAT 6680, Professor Wilson

Final Assignment by Ursula Kirk

A.                   Bouncy Barney

Barney is in the triangular room shown here. He walks from a point on BC parallel to AC. When he reaches AB, he turns and walks parallel to BC. When he reaches AC, he turns and walks parallel to AB. Prove that Barney will eventually return to his starting point.

How many times will Barney reach a wall before returning to his starting point?

Explore and discuss for various starting points on line BC, including points exterior to segment BC.

Discuss and prove any mathematical conjectures you find in the situation.

Barney’s Path

Here is the path that Barney takes when he starts from the STAR point on the picture. He comes back to his starting point after 6 steps.

There are three points from where Barney will be able to complete his route in only 3 steps. If Barney starts from any of the middle points of the sides of the triangle, he will be done in 3 steps.

Barney will be done in 3 steps also, if he starts from any of the vertices of the triangle.

If he starts from a point outside the triangle, he will be done in 6 steps. From V, W, X, Y, Z or U, Barney will be back to the starting point in 6 steps.

Therefore, my conjecture is that Barney will always complete his route in 6 steps with the following exceptions:

·         Barney starts from any of the vertices of the triangle; in this case Barney will complete the route in only 3 steps.

·         Barney starts from the middle point of any of the sides of the triangle; in this case he will be done in 3 steps.

B.     Complete a Write-up on your Web Page for one additional investigation, chosen from Exploration 0 through Exploration 12.

·         I have chosen to complete my write up from exploration 0, investigation 7.

Graph the equation:

Where n=1

Our graph at n=1 has circles with centers at (0,0). The circles get closer together as we approach infinity. Also, we have vertical and horizontal hyperbolas symmetric about the x-axis and the y-axis. Like the circles, the hyperbolas are getting closer as they move away from the center.

As we change the value on n, our graph also changes. As n approaches 0, we start getting bigger gaps in between the graphs. The circles and the hyperbolas are gone. Still our graph is center at (0,0) and it is symmetric about the x-axis and the y-axis.

Finally when n=0, the graph is totally gone.

If we start increasing from 1 to 2 the gaps start getting also smaller and at n=2 the graph is totally gone.

Here, we do not have a center piece any more. The shapes have change, but still the graph is symmetric over the x-axis and the y-axis.

As we get closer to 2, the gaps are getting bigger and the graph is disappearing.

Here is a movie for the equation when n varies from 0 to 2

MOVIE