Mathematics Problem Presentation
for EMAT 7050 Fall 2009 (Dr. A. Conner)
-- Which ball to shoot?
In Sep. was the Snooker Shanghai Masters 2009 held. I got the news that Ronnie O'Sullivan, nicknamed "the Rocket" due to his rapid playing style, an English professional snooker player, booked his place in the semi-finals of the Shanghai Masters with a 5-3 win over Ding Junhui, a talented Chinese snooker player. O'Sullivan said: "Ding was first in the balls in every frame. He'll be disappointed because he might feel he deserved to win." Anyway, this gave me the inspiration to make up a problem like the following one.
One of the rules of snooker is to hit one color ball between two red balls. Now O'Sullivan needs to use the cue ball to shoot a color ball after a red ball. By the way, color balls worth more points (Black 7; Pink 6; Blue 5; Brown 4; Green 3; Yellow 2), so he would be glad to do that.
1) In the situation shown in the above figure, is it highly possible to pot a color ball without hitting any cushion?
2) If not, then, O¡¯Sullivan tries to hit the center of the front surface of the cue ball to push one color ball to hit the rubber cushions once or more to go into one of the pockets. But it¡¯s hard to make a 2-or-3-cushion shot in snooker. I want to see a fancy shot so I will let O¡¯Sullivan play on an ordinary table, which has larger pockets. And we ignore the friction, air resistance, etc. Now, in your opinions, which one has the highest possibility to be potted in? How would you estimate it? What will the route of your object ball be? Note the name ¡°cue ball¡± does help you start.
3) Can you explain why these two angles (of incidence & of reflection) are equal? Do you need a protractor to construct the angles?
4) Of course, you can use straight edge and compass. Essentially, in order to ¡°estimate¡± the route of the object ball, you can use a marked ruler, and estimate by just looking whether a line is perpendicular to another line. (Why?)
5) If we suggest O¡¯Sullivan choosing the blue ball, do you have any other way to check your suggestion? What is it?
6) Have you noticed two parallel lines? Why?
You may want to check out the GSP script tools I created for this problem:
By the way, in billiards sports professional players use their rich experience and fantastic skills and sometimes the ¡°diamond system¡± (though not as precise as you may think; just a reference object) on the table to help them square the direction of their cue.
Figure cited from http://www3.sympatico.ca/eric.perreault/diamond_system_en.html
The main mathematical knowledge involved in the problem of finding the route of object ball:
If we hit the center of the front surface of the ¡°cue¡± ball, namely, O here, then O, A, B are ¡°collinear¡±. Both physically and mathematically if we ignore the friction, air resistance, etc., then the angle of incidence equals the angle of reflection (force decomposition), in symbols, and . How can we construct these angles? We construct AC perpendicular to BD at D such that AD=CD. Draw ray CB meeting GE at E; similarly, construct BF perpendicular to GE at E such that BG=FG. Draw ray FE theoretically passing through point H, as desired. By construction , so . But ¡¡since they are vertical angles. So we have ¡¡by transitivity of angle equality. Similarly, . The mathematical idea of this construction also works for some other problems like ¡°the shortest distance problem¡± or ¡°the image of mirror problem¡±.
To prove line OB is parallel to line HE (in Euclidean geometry):
We have several theorems in Euclidean geometry to prove two lines are parallel. We can choose to prove the two corresponding angles along the transversal are equal or to prove the sum of the two interior angles on the same side of the transversal equals 180 degrees. Both of them use essentially the same mathematical reasoning. I will show the proof using the latter parallel principle as follows.
¡¡since the sum of interior angles of any triangle is and ¡¡is this problem is the corner of the pool table which can be abstracted as a right angle.
¡¡since , , ¡¡all together compose a straight angle DBG, and as shown above, and similarly, .
By distributivity, .
Hence, , as desired.
Where the problem fits in Georgia Performance Standards
v M6G1. Students will further develop their understanding of plane figures.
v a. Determine and use lines of symmetry.
v M7G1. Students will construct plane figures that meet given conditions.
v a. Perform basic constructions using both compass and straight edge, and¡¡¡¡¡¡¡¡ appropriate technology. Constructions should include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
v b. Recognize that many constructions are based on the creation of congruent triangles.
v M8G1. Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.
v a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.
v b. Apply properties of angle pairs formed by parallel lines cut by a transversal.
v d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.
M8P1. Students will solve problems (using appropriate technology).
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
M8P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
d. Select and use various types of reasoning and methods of proof.
M8P3. Students will communicate mathematically.
M8P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M8P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
c. Use representations to model and interpret physical, social, and mathematical phenomena.