1. Challenges, Explorations with Lines, and Explorations with Parabolas Jeff Morgan Chair, Department of Mathematics Director, Center for Academic Support and Assessment University of Houston jmorgan@math.uh.edu jmorgan@math.uh.edu http://www.math.uh.edu/~jmorgan

2. 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?

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 4 th 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 Challenge Something to Sleep On

4. 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?

5. Exploration 1 warm up #2 Three lines are graphed below. Use a ruler to determine equations for the lines.

6. 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?

7. 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.

8. 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. nf (n)f (n) 3 4 5 6 7 8 9

9. Exploration 4

11. Exploration 5

12. Exploration 3

13. Exploration 3 – Figure

14. Exploration 4

15. Exploration 4 - Figure

16. Exploration 12