September 03, 2013
Notes – Eliminating Possibilities
Eliminating possibilities is a powerful problem solving strategy. It is somewhat of an indirect method. In other words, we don’t immediately go directly for the solution, but determine what it _______________________________________. September 03, 2013
We use _________________________ in the same manner as Sherlock Holmes did to solve mysteries. In the book Sign of Four Sherlock Holmes says: “… once you have eliminated the _________________________________, then whatever is left, no matter how improbable, must be the ___________________________________.” Even when we cannot Eigure out how to get to the exact answer, _______________________________________________ can help us make a better, educated guess. September 03, 2013
Eliminating Possibilities Example
Let’s say that we were playing a 20-­‐questions type math game where I could answer only “yes” or “no” to any question you asked. In this game I have picked a number between 1 and 100. How could we use the strategy of eliminating possibilities along with logic to determine the value of the number?
Each question you ask should be designed to eliminate possibilities. Which of the following would be the most effective Eirst question?
1.
Is the number 29?
2.
Is the number even?
3.
Is the number greater than 50?
Between 1 and 100 there are _____________ possible whole numbers. Question 1 eliminates one possibility. You have a ___________% chance of guessing correctly, but this will not be an effective or efEicient approach
September 03, 2013
Questions 2 and 3 both eliminate ________________% of the possible numbers. These are _____________________Eirst questions. Combined, they will eliminate _____________% of the possibilities.
Notice that questions 2 and 3 have _______________% chance of giving you the actual solution but they help you focus your search. That is the effectiveness and power of eliminating possibilities.
September 03, 2013
Example Problem – Penny’s Dimes Part II Penny’s favorite coin is the dime, as we saw when we were studying systematic lists. Since we worked the problem about Penny and her dimes, she has spent some of her dimes and acquired some more. She did not keep track of how many she has. She knows that it less than 100. One day she was arranging them in piles on her desk in different ways. The results were peculiar.
1.
When she put them into piles of 2 there was 1 left over. 2.
When she put them into piles of 3 there was 1 left over.
3.
When she put them in piles of 4 there was 1 left over.
4.
When she put them in piles of 5 there were none left over.
5.
How many dimes does Penny have? September 03, 2013
Try This Problem on with your partner. Use the space below. Use additional paper if necessary.
Solution:____________________
Perhaps you used the ideas below as part of your logic:
1.
The clue within the problem statement says that she has less than 100 dimes, so we need not consider any number above ______________.
2.
We need not consider _____________________ values since Penny cannot have negative dimes.
3.
We need not consider ___________________ because it says she was stacking dimes. __________________dimes cannot be stacked.
September 03, 2013
Final Thoughts
Eliminating possibilities falls into the category of ____________________ information. After organizing the information given in a problem, you can eliminate possibilities. Again, to solve a problem, _____________________ strategies are often used.
You use the eliminating possibilities strategy in your every day life. For example, when you decide what you want for breakfast, what to do after school or what you will write about for an essay you must eliminate possibilities.