Hello All!
I got this week's problem of the week from community member, Connie Rivera. Hope you enjoy this and be sure to identify the CCR Math Standards and which of the 8 Mathematical Practices is being used. Let us know if you did this in class and what were your successes and struggles. Have a great week!
Brooke
Problem of April 9: http://screencast.com/t/k49zOkCsZ1TV
Take two quarters, place one heads up and the other tails up, keep the quarter with the heads motionless. Rotate the other quarter around it, never slipping and always tangent. When the rotating quarter has completed a turn around the stationary quarter, how many turns has it made around its own center point? Explain your reasoning. Please try and post your answer below.
Comments
Has anyone tried this problem?
Brooke
I'm working on it!
I think it is two from rotations, but mathematically I can't figure out why???
My first guess was one rotation. I figured that since the two quarters have the same circumference, you're basically just rotating the tails quarter one time. It's the same as if you had a line the length of the circumference and rolled it along the line. The only difference in this case is that the line is not straight.
When I tried it, though, I see that it is two rotations (this is easier to see if you mark one point on the rotating coin with white-out or something similar). My observation is that when you get to the opposite side of the stationary coin, the mark on the rotating coin is in the same place relative to the rest of the coin (i.e. if the mark pointed up at the beginning, it points up again halfway through the rotation). The points of the coins that were farthest apart at the beginning are now touching. Then you can complete the rotation, and the outside coin makes another full revolution.
I don't know how to explain this any more mathematically than this, but I did notice that this is a good example of two opposites undoing each other. When the opposite side of the rotating coin gets to the opposite side of the stationary coin, the rotating coin is back to its original orientation.
This might be a good activity to use with my evening class. We finished talking about circles a couple weeks ago. It's also a really good incentive to actually test your hypotheses...
If anyone gets any further on the math, please post!
Rachel
I agree that it is 2 rotations. The center of the second circle creates another circle outside the perimeter of the first circle. That circle would have a diameter twice the size of the stable quarter; therefore, it has a perimeter twice the size of the stable quarter.
The problem description is essentially of having two gears of the same diameter and they stay meshed all the time. Let's suppose they each have a hole in the center for a pin and that they are held together by a link with a hole near each end which has the pins holding the gears so they can turn. Let's mark a radius on each gear pointing to where they touch under the center of the link. If we then hold the link still but turn gear1 counterclockwise through an angle A, the marked radius on gear1 now makes an angle A with the link, and gear2 has turned clockwise through the same angle A and the marked radius of gear 2 now also makes an angle A with the link. Of course the same must be true if we hold gear1 fixed and allow the link and gear2 to turn. Suppose that we start with the link horizontal, radius1 pointing to the right, and radius2 pointing to the left. Now turn the link clockwise through angle 90 degrees, then the link is vertical with gear2 at the bottom and radius 2 is horizontal also but now pointing to the right So gear2 has already turned 180 degrees around its center. More generally, for any angle A <= 90, if the link has turned clockwise through angle A, then the link makes an angle A with a horizontal line through the center of gear2, and radius2 makes an additional angle A to the link, so that the angle from the horizontal to radius2 is 2*A. Just draw a little diagram with two horizontal lines through the centers of the gears, the link line intersecting both at angle A, and the radius2 making angle A with the link. Of course this continues to be true for the whole link sweep of 360 degrees; gear2 turns through 2*360 degrees..