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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 3 2 3 1 1 1 2 4 5 1 2 5 2 2 4 1 1 3 5 1 3 3 5 3 1 3 2 5 5 5 5 4 3 5 5 2 2 2 3 5 3 4 5 3 3 5 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.