2. Approaches to Problem Solving
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3. Problem-Solving Guidelines and Hints Unit 2C
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4. Four Step Problem-Solving Process
Step 1: Understand the problem.
Step 2: Devise a strategy for solving the problem.
Step 3: Carry out your strategy, and revise if necessary.
Step 4: Look back to check, interpret, and explain your result.
Have the students apply the steps in working out a significant problem from section 2-C.
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5. Four Step Problem-Solving Process: Step 1
Step 1: Understand the problem.
Think about the context of the problem to gain insight into its purpose.
Make a list or table of the specific information given in the problem.
Draw a picture or diagram to help make sense of the problem.
Restate the problem in different ways to clarify the question.
Make a mental or written model of the solution.
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6. Four Step Problem-Solving Process: Step 2
Step 2: Devise a strategy for solving the problem.
Obtain needed information that is not provided in the problem statement.
Make a list of possible strategies and hints that will help you select your overall strategy.
Map out your strategy with a flow chart or diagram.
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7. Four Step Problem-Solving Process: Step 3
Step 3: Carry out your strategy, and revise it if necessary.
Keep an organized, neat, and written record of your work.
Double-check each step so that you do not risk carrying errors through to the end of your solution.
Constantly reevaluate your strategy as you work. Return to step 2 if you find a flaw in your strategy.
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8. Four Step Problem-Solving Process: Step 4
Step 4: Look back to check, interpret, and explain your result.
Be sure that your result makes sense.
Recheck calculations or find an independent way of checking the result.
Identify and understand potential sources of uncertainty in your result.
Write your solution clearly and concisely.
Consider and discuss any pertinent implications of your result.
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9. Key to improving problem solving
More Practice!
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10. Problem Solving Guidelines and Hints
Hint 1: There may be more than one answer.
Example: How can society best reduce the total amount of greenhouse gases emitted into the atmosphere?
Choose several of the hints to discuss and challenge students with some thought-provoking problems.
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11. Class Notes (1)
1. Tickets for a fundraising event were priced at $10 for children and $20 for adults. Sally worked the first shift at the box office, selling a total of $130 worth of tickets. However, she did not keep careful count of how many tickets she sold for children and adults. How many tickets of each type (child and adult) did she sell?
List all the possible answers
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12. Class Notes (2)
2. A toll booth collector on a highway receives $2 for cars and $3 for buses. At the end of the 1-hour period, she collected $32. How many cars and buses passed through the toll booth during that period?
List all the possible solutions
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13. Hint Hint 2: There may be more than one strategy
Often there are more than one right answer, we should also expect more than one right strategy!
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14. Class Notes (3)
3. Jill and jack ran a 100-meter race. Jill won by 5 meters; that is Jack had run only 95 meters when Jill crossed the finish line. The decide to race again, but this time Jill starts 5 meters behind the starting line.
Assuming that both runners run at the same pace as before, who will win?
Quantitative vs. Analytical
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15. Hint Hint 3: Use appropriate tools.
For any given task there is an appropriate level of power that is needed.
Choosing the tools most suited to the job will make your task much easier.
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16. Class Notes (4)
4. Two cars, 120 miles apart, begin driving toward each other on a long straight highway.. One car travels 20 m/h and the other 40 m/h. At the same time, a canary starting on one car, flies back and forth between the two cars as they approach each other.
If the canary always flies 150 m/h and turn around instantly at each car, how far has it flown when the cars collide?
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17. Hint Hint 4: Consider simpler, similar problems.
The problem is huge and ridiculous!
Try a smaller simpler problem that will help you understand the original problem.
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18. Class Notes (5)
Suppose you have two cups in front of you: One holds coffee and one holds milk. You take a teaspoon of milk from the milk cup and sit ir into the coffee cup. Next, you take a teaspoon of the mixture in the coffee cup and put it back into the milk cup. After the two transfers there will be either (1) more coffee in the milk cup than milk in the coffee cup, (2) less coffee in the milk cup than milk in the coffee cup, (3) equal amounts of coffee in the milk cup and milk in the coffee cup.
Which of these three possibilities is correct?
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19. Hint Hint 5: Consider equivalent problems with simpler solutions.
Sometime similar (easier problem) is not good enough, you need an equivalent problem to get a numerical answer.
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20. Class Notes (6)
6. Juan is decorating for a party in a room that has ten large cylindrical posts. The posts are 8 feet high and have a circumference of 6 feet. His plan is to wrap eight turns of ribbon around each post.
How much ribbon does Juan need?
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21. Class Notes (7)
7. Eight turns fo a wire are wrapped around a pipe with a length of 20 centimeters and a circumference of 6 centimeters.
What is the length of the wire?
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22. Hint Hint 6: Approximations can be useful.
Makes problems easier and often good enough for the final answer.
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23. Class Notes (8)
Imagine a mile-long bar of metal such as the rail along railroad tracks. Suppose that the rail is anchored on both ends (a mile apart) and that, on a hot day, its length expands by 1 foot.
If the added length causes the rail to bow upward in a circular arc, about how high would the center of the rail rise above the ground?
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24. Class Notes (9)
9. Suppose a railroad rail is 1 kilometer long, and it expands on a hot day by 10 centimeters in length.
Approximately how high would the center of the rail rise above the ground?
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25. Hint Hint 7: Try alternative patterns of thought. “AHA”
Try to avoid rigid patterns of thought that tend to suggest the same ideas and methods over and over again, approach every problem with an open mind that allows innovative ideas to develop.
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26. Class Notes (10)
In an effort to reduce population growth, in 1978 China instituted a policy that allows only one child per family. One unintended consequence has been that, because of a cultural bias toward sons, China now has many more males that females. To solve this problem, some people have suggested replacing the one-child policy with a one-son policy. Suppose that the one-son policy were implemented and birth rates returned to their natural levels.
Compared to the one child policy, how would this affect the overall birth rate and the numbers of boys and girls?
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27. Hint Hint 8: Do not spin your wheels.
Don’t get bogged down, put aside for a bit. You will be surprised at what you overlooked when you return to it.
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28. Hints 1-8 1 Answers (more than 1) 2 Strategies (more than 1) 3 Tools
4 Simpler problems
5 Simpler equivalent problems
6 Approximations
7 AHA
8 Take a break
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29. Problem Solving Example
Find the total number of possible squares on the chessboard by looking for a pattern.
Solution
Start with the largest possible square:
There is only one way to make an
8 x 8 square.
It may be necessary to help students along in the problem-solving process by starting with a simpler problem in which the chessboard contains just 4, 9, or perhaps 16 squares. Emphasize the importance of searching for patterns when solving certain quantitative problems.
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30. Problem Solving Example
Find the total number of possible squares on the chessboard by looking for a pattern.
Now, look for the number of ways to make a 7 x 7 square.
There are only four ways.
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31. Problem Solving Example
Find the total number of possible squares on the chessboard by looking for a pattern.
If you continue looking at 6 x 6, then 5 x 5 squares, and so on, you will see the perfect square pattern as indicated in the following table for this chessboard problem:
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32. Problem Solving Example
Find the total number of possible squares on the chessboard by looking for a pattern.
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