Triangular Pastimes from Another Universe

Show diagrams (several for each game) and a few sentences of explanation for each.  Also, if you can, relate your strategy to some of the centers we talked about in class:  angle bisectors, perpendicular bisectors, altitudes, midpoints, etc.

Game 1:

The referees define a triangular field having three goal lines (sides).  The field could be equilateral, scalene, right, obtuse – any kind of triangle.  A caller calls which side is the goal for that round, and you try to race to be the first one to reach that side.  Your current theory is that the best way to win is to find the place that is an equal distance from each side.  Where is this point?  How can you identify it on different shaped fields?

Game 2:

Again, the referees define a triangular field, this time by placing three cones.  Your goal is to reach the cone that is called on any given round and you’d like to be in a good position for a play at any cone.  You want to be the same distance from each cone.  Where do you stand?  How do you find this spot?  How is it different in different shaped fields?

Print a color copy of your golden rectangles and attach it to the answers to these questions.

Type your answers.  Include a typed heading.  Each answer should be about one paragraph.

1.  What is the significance of adding squares to create larger rectangles?  How is this related to the Fibonacci sequence?

2.  Are the golden rectangles similar? Explain.

3.  What is the proportion of the width to the length?  How does this change as the rectangles get larger?

The Trains Problem

Your 2-3 page solution is due on Friday.  Show all your thinking and reasoning clearly.  Remember that the most important part of the solution is explaining why the pattern is what it is.  Use a tree diagram, words, table, and anything else you find helpful to make your reasoning clear.  Be sure to proofread your work.  Spelling and grammar count.

As we talked about in class, the difference between the number of arrangements when order matters and the number of arrangements depends on how many items you are choosing at once.  When you are choosing 2 items at once, you divide by 2 to find the number when order doesn’t matter.  If you are choosing 3 items, you divide by 6.  If you are choosing 4 items, you divide by 24.

Answer the following questions in about one page (typed):

What is the significance of these numbers?

How do they relate to the situation?

How can you predict how many arrangements there will be for any situation (when order doesn’t matter)?

Hint:  How many ways are there to arrange 2 different items (A and B, for example)?  How many ways are there to arrange three different items (such as A, B, and C).  How many ways are there to arrange four items (such as A, B, C and D)?

1. What do you need to do to find the number of variations if order doesn’t matter?
2. How is this process the same/different from the number of the number of variations when order DOES matter?

* As you are testing possibilities, use different numbers of “choices” and choose different amounts at a time.  Otherwise you won’t get a full picture.  (For example, try 5P2 and 5P3 but also 4P2 and 4P3).  You can use tree diagrams and a table to keep track of your experiments.  Keep your numbers small to avoid having a “stabby” feeling. :)