Developing Problem Solving Skills Discussion Board Directions: For this activity, you will find a collection of short problems on the discussion board that are designed to strengthen the critical-thinking and problem-solving skills. Unit 2C of the text discusses problem solving by elaborating on Polya's four-step method:

Step one: Understand the problem Step two: Devise a strategy Step three: Carry out the strategy and revise

Step four: Check, interpret, and explain

You will use these steps to complete one of the problem below on the discussion board. Every student must work a unique problem and students will pick problems first come first pick. You may place your name as a reply to a problem to show you are working on it. However, any problems without a complete solution within 24 hours of "claiming" it will be released to the class. When you post your solution, cut and paste the problem into the discussion response with your solution.

There is a sample solution below to assist you. Example: A race. Stan placed exactly in the middle among all runners in a race. Dan was slower than Stan, in 10th place, and Van was in 16th place. How many runners were in the race?

Solution:

Step one: Understand the problem and organize the given information. We are given three pieces of information: (1) Stan placed exactly in the middle among all runners; (2) Dan was slower than Stan, in 10th place; and (3) Van was in 16th place. We might draw a picture to organize this information: 16 10 Last place First place Van Dan Stan

Step two: Devise a strategy. Of the three given facts, which one gives us an opening on a solution? In fact (1), the words "Stan placed exactly in the middle" means that there were as many runners faster than Stan as slower than Stan. Here is the key insight: There must have been an odd number of runners in the race. Just to get a feeling for the possibilities, let's draw another picture. Let's try a race with five runners.

5 16 10 3 1 | | |

Van Dan Stan

In this race, Stan would place in the middle at 3rd, but there could not be a 10th and 16th place (because there are only five runners).Clearly this scenario doesn't work, but it leads to a solution strategy: Investigate other races with odd numbers of runners to see if we can find a case that is consistent with all of the given facts.

Step three: Carry out the strategy. A few more diagrams show that there could not have been 7, 9, 11, 13, or 15 runners because there cannot be a 16th place in these races. What about 17 runners? We now get the following diagram with Stan in the middle in 9th place: 17 16 10 9 1 | | |

Van Dan Stan

In this race, Dan is slower than Stan in 10th place and there are enough runners for a 16th place. This solution meets all of the conditions of the problem. So we seem to have a solution: The race has 17 runners.

Step four: Check, interpret, and explain. Could there be other solutions? What about 19 runners? Now Stan places in the middle in 10th place, but we know that Dan is in 10th place, slower than Stan, so that solution doesn't work. Similarly, with more than 19 runners, Stan could not have place in the middle with Dan running slower than Stan. So we have one solution: The race has 17 runners.