bimc notes 1 - how to solve mathematical problems independent math circle... · bainbridge...

How to Solve Mathematical Problems

MathPerfect Discover and achieve your potential

Bainbridge Independent Math

Circle

Problem Solving Guidelines

§ Understand the Vocabulary and Symbols of Mathematics

§ Avoid Simple Computational Errors

§ Problem-Solving Strategies

§ Problem-Solving Tactics

§ What to Do When You Become Stuck

§ Golden Errors

§ How to Profit From Golden Errors

§ Solution Format for Mathematics Problems

§ Sample Problem and Solution

§ The Pólya Method

§ How to Tackle the Problem Sets

§ Problem Set 1

§ Problem Set 1 – Answer Key

Other Resources

§ Geometer’s Sketchpad®

§ Mathematica®

§ MathType™

September 24, 2012

Study Materials Competition math problem sets:

§ American Mathematics Contests

§ The Mandelbrot Competition

Bainbridge Independent Math Circle How to Solve Mathematical Problems

2 ©2012 MathPerfect Discover and achieve your potential

Understand the Vocabulary and Symbols of Mathematics

Make sure that you understand the precise meaning of every word and symbol you read in a math problem. If you are not sure then check your textbook, a math dictionary, Google, or ask your instructor.

Avoid Simple Computational Errors 1. Write down all the steps that explain your solution. There may be times during

certain mathematics contests when you are expected to use your mental math skills. On standardized, multiple-choice tests you are usually not asked to show your work either. However, in all other cases you must clearly show how you arrived at your solution. Remember that a crucial part of problem solving is communicating your ideas to your instructor and your peers.

2. Write as neatly as possible. Make sure you write all your numbers and symbols clearly and consistently. Errors are often caused by illegible handwriting.

3. Write columns of figures exactly under one another so that you don’t lose track of place values.

4. Copy problems onto your worksheet then double-check that you transcribed all the numerical values and other important information correctly.

5. Underline important words and numbers. Distinguish between important and irrelevant information.

6. Check to see if your final answer is reasonable. Don’t think your work is done just

because you came up with an answer. A bizarre answer may be the result of a computational error.

How to Solve Mathematical Problems Bainbridge Independent Math Circle


Problem-Solving Strategies

1. Draw a picture or diagram.

2. Find a pattern.

3. Make an organized list.

4. Make a table.

5. Solve a simpler problem.

6. Trial and error.

7. Experiment.

8. Act out the problem.

9. Work backwards.

10. Write an equation.

11. Use deduction.

12. Change your point of view.



Problem-Solving Tactics

1. First read the problem for overall meaning.

2. Re-read the problem for specific information and underline key

words and numbers.

3. List the important information that is given.

4. List everything that you are supposed to find.

5. Is there a hidden question?

6. If multiple steps are required, how many and what are they?

7. What does each step require?

8. Put the steps in the proper order.

9. Perform each step.

10. Check your answer.

11. Remember that hard problems are made up of smaller problems.

12. Generalize the problem.



What to Do When You Become Stuck

1. Read the problem again. Maybe you overlooked important information.

2. Review your steps. Check for arithmetic or transcribing errors.

3. Simplify the problem. For example, try using simpler numbers.

4. Move on to the next problem (if there is one) and come back to the

troublesome one later.

5. Sleep on it.

6. See if you can find a similar problem and try to solve that one first.

7. Use technology (e.g. graphing calculator or Mathematica®) to better visualize

the problem.

8. Ask for help. Post questions on the MathPerfect discussion board.

Golden Errors

On EVERY error that you make, answer three questions.

1. Where in the problem did you make the error?

2. What did you do wrong that caused you to make the error?

3. What is the correct operation that will avoid the same kind of error in the future?

How to Profit from Golden Errors 1. Locate the error.

2. Find out what caused the error.

3. Correct the cause of the error.

4. Correct the error itself.

5. Do similar problems to make sure you have corrected your technique.



Solution Format for Mathematics Problems

Your solution should be well organized and in your best handwriting (if you are writing by hand). If you are explaining your problem-solving method in words, which I encourage, then do your best to write complete sentences with correct spelling, grammar, and punctuation. The following solution layout is the one I recommend, especially if you are uncertain how to begin solving a particular problem.

Given: List all relevant information given in the problem.

Find: Make certain that you know what you are supposed to solve.

Formulas: List any formulas that you think might be helpful.

Estimate: Estimate what the result will be (this is optional but sometimes quite helpful).

Solution: Present your complete solution (or as much as you can).

In the end, if you have succeeded in completing your solution, check if your answer is reasonable. Does your answer make sense? If you calculated that Michael rode his bike to school at a speed of 2500 miles per hour, you should probably check your work. If you are still satisfied with your result, then demonstrate your confidence and write (and underline) the following: “What you had to find = what you determined the answer to be Ans.”

For example: The volume of the cylinder = 6.8 cubic meters Ans.

The “Ans.” at the end confirms that this is your final answer. The person reviewing your solution is sure that you have finished your work and not simply stopped because you were stuck. Remember to double-check that you have solved what the problem asked you to solve. If your answer contains units, make sure that you have used the correct units. Write fractions in the simplest form possible. Unless directed otherwise, round decimal answers to the nearest thousandths.

If you cannot solve the problem, try to describe where and why you think you became stuck. Ask yourself what key piece of information might help you get unstuck and then try to uncover that bit of information. Re-read the problem to see if the key information is hidden in the problem statement. You should demonstrate what you tried to do even if you were unable to complete the solution. At least lay out the problem in the way described above. Sometimes just doing that will give you an idea how to proceed.



Sample Problem and Solution

A group of snails lived at the bottom of a twenty-foot well, which suddenly ran dry. One snail began to climb to the top of the well at the rate of three inches per hour for each of the twelve daylight hours. Each night, however, he slipped back at the rate of two inches per hour. How long did it take this snail to escape from the well?

Given: Depth of well = 20 feet Rate of climb = 3 inches per hour during 12 daylight hours Rate of slippage = 2 inches per hour during 12 nighttime hours Find: The time required for the snail to climb out of the well Formulas: Conversion: 1 foot = 12 inches Estimate: No more than 20 days Solution: First we need to determine how much progress (in feet) that the snail

makes during each 24-hour period. During the 12 daylight hours the snail climbs:

During the 12 nighttime hours the snail slips:

As a result, during each 24-hour period, the snail makes of progress towards the top of the well. After 17 days (and nights) the snail will be 17 feet from the bottom or 3 feet from the top. At dawn of the start of the eighteenth day the snail begins to climb the last 3 feet, which it completes during the 12 daylight hours. Therefore, the snail reaches the top of the well at sunset of the eighteenth day.

The snail requires 17 and a half days to climb out of the well. Ans.

Ask yourself if your answer seems reasonable. Compare it to your estimate if you made one. If you have any doubts about your answer, retrace your steps, check your arithmetic computations, and make sure that you are using the correct units.



The Pólya Method

Solving problems is a practical art, like swimming, or skiing, or playing the piano: . . . if you wish to learn swimming you have to go into the water, and if you wish to become a problem solver you have to solve problems. – George Pólya

No discussion about problem solving would be complete without mentioning George Pólya who is known as the father of problem solving. He was born Pólya György on December 13, 1887 in Budapest, Austria-Hungary. He taught for many years at ETH Zürich in Switzerland before moving to the United States in 1940. He taught briefly at Brown University then moved to Stanford University where he served as professor of mathematics from 1940 to 1953, after which he served as Professor Emeritus until his death on September 7, 1985 at the age of 97. Pólya noticed that even students who were mathematically advanced had difficulty solving problems. He began to study the strategies and techniques of problem solving so that he could understand why his students were having so much trouble and also to devise a method to teach problem solving as a distinct discipline, which he called heuristics. Pólya’s most famous book on the subject of heuristics is How to Solve It, which he published in 1945.

In summary, Pólya devised a 4-step problem solving method that can be applied to any problem, not just those that are mathematical in nature.

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Look back

The first three steps of Pólya’s method are expressed in the strategies and tactics described earlier in this document. The fourth step is the one that most students ignore even though it may be the most important step in becoming a masterful problem solver. When you look back and study a problem that you have already solved, you have the opportunity to devise simpler, more elegant, or more efficient solution methods. You may also realize that the method you used to solve your current problem could also be used to solve seemingly unrelated problems. You start thinking like a mathematician. You begin to expand your problem solving tool chest, which is crucial to your development. Not having or using the right tools can make your life much more difficult.

If the only tool you have is a hammer, every problem looks like a nail. – Unknown



How to Tackle the Problem Sets

The main reasons for working on the problem sets are to learn how to analyze unfamiliar problems, devise plans for solving them, and develop your ability to present your solutions (or partial solutions) clearly and completely.

Notice that I never mentioned that you had to come up with the correct answer. In the problem solving process, correct answers are not that important. What is much more important is what you were thinking as you tried to solve the problem, whether or not you were able to finish. Therefore, even if you don’t complete your solution, you still need to show what you did and describe why you got stuck. In real life, some problems cannot be solved because you don’t have all the required information. Recognizing when you don’t have enough information is good. Determining exactly which pieces of information you need is even better. Figuring out a way to obtain the “missing” information and using it to complete your solution is best of all. A blank answer sheet is never acceptable. Try something and describe what you tried even if it didn’t work. Even the strategies that don’t work will teach you something. Guessing and checking (trial and error) is a good strategy to use to give you a start in the right direction, but guesswork should never form part of your final solution. You can say that you started guessing to look for a pattern but, in the end, you have to explain why your solution works. Approach the problems with a sense of fun. When you write your solution to a problem, you are telling a story. Just giving an answer is like jumping straight to the end of the story without telling how you got there. Imagine how unsatisfying it would be if you were only given the first and last chapters of a Harry Potter novel (or whichever tale is your favorite). You need to tell the whole story. You need to explain your solution or, as teachers are so fond of saying,

Show Your Work!



Problem Set 1

These problems are meant to be tackled without the use of a calculator, and without measuring. Answers and calculations should always be exact, using symbols, fractions, or radical signs where appropriate, not decimal approximations. Answers should also always be in fully simplified form. After working on a problem, remember to look back and see what you’ve learned.

1. Two candles have different lengths and different thicknesses. The shorter one would last

eleven hours, the longer one would last for seven hours. Both candles are lit at the same time, and after three hours both have the same length remaining. What was the ratio of their original lengths?

2. Take any two-digit number. Subtract the sum of its digits. Then divide the answer by 9.

What do you find? Explain.

3. Which of the shaded rectangles in the diagram below has the larger area?

4. Choose any starting number other than 1. To get a new number, divide the number that is

1 bigger than your number by the number that is 1 less than your number. Now do the same with this new number. What happens? Explain.

5. What is the angle between the hands of a clock at 9:30?

6. For how many two-digit numbers is the sum of the digits a multiple of 6?

7. While watching their flocks by night the shepherds managed to lose two-thirds of their

sheep. They found four-fifths of these again in the morning. What fraction of their original flock did they then have left?

8. Two cubical dice each have faces numbered 0, 1, 2, 3, 4, 5. When both dice are thrown, what

is the probability that the total score is a prime number?



Problem Set 1 (continued)

9. is a rectangle with twice as long as is a point such that is an equilateral triangle which overlaps the rectangle is the midpoint of the side How big is the angle

10. A normal duck has two legs. A lame duck has one leg. A sitting duck has no legs. Ninety-

nine ducks have a total of 100 legs. Given that there are half as many sitting ducks as normal ducks and lame ducks put together, find the number of lame ducks.

11. Moses is twice as old as Methuselah was when Methuselah was one-third as old as Moses

will be when Moses is as old as Methuselah is now. If the difference in their ages is 666, how old is Methuselah?

12. What is the sum of the four angles in the diagram below?

13. On a bike ride, Calvin starts at home and goes up a long hill for 30 minutes at 6 mph. At the top, he turns around and rides home along the same path at a speed of 18 mph. What is his average for the round-trip?

14. The water from a swimming pool evaporates at a rate of 6 gallons per hour in the shade and

19 gallons per hour in the sun. For several weeks in August, the amount of water lost to evaporation in the shade was equal to the amount lost in the sun. What was the average rate of evaporation from the pool?

15. The battery in a portable music player is guaranteed to last 17 hours without charging if it

is used with headphones and 3 hours if it is used with speakers. If you use headphones and speakers for equal amounts of time, how many minutes will a fully charged battery last?

16. It takes seven minutes for seven moles to dig seven holes. How long will it take eight moles

to dig eight holes?

17. If five moles can dig four holes in three minutes, how many minutes will it take for nine

moles to dig six holes?




18. What is the smallest 4-digit number that is divisible by 2, 3, 4, 5, 6, 8, 9, and 10?

19. Using only 1’s and 2’s, what is the smallest integer you can create which is divisible by both

3 and 8?

20. What is the remainder when 456,564,465,645 is divided by 6?

21. In the diagram below, the rectangle is cut into four pieces. The areas of three of them are

given. What is the total area of the rectangle?

22. Two numbers are such that their difference, their sum, and their product are in the ratio

1 : 4 : 15. What are the two numbers?

23. If Gilda had subtracted her two numbers as she was meant to, she would have got the

answer 2. Instead she divided the two numbers and got the answer What were her two

starting numbers?

24. When a barrel is 30% empty it contains 30 gallons more than when it is 30% full. How

many gallons does the barrel hold when full?

25. What is 30% of 40% of 50?

26. A square peg just fits inside a round hole. What fraction of the hole is occupied by the peg?

27. Evaluate

28. If four mice can eat four pounds of cheese in four minutes, how long will it take 99 mice to

eat 99 pounds of cheese?




29. In a regular octagon, what is the ratio of the length of its longest diagonal to the length of its shortest diagonal?

30. Robert was meant to take the fourth root of a number, but instead divided by four and got

the answer 4. What should his answer have been?

31. One half of the class got A’s. One-third of the rest got B’s. One-quarter of the remainder got

C’s. One-fifth of the others got D’s. What fraction of the class received grades lower that a D?

32. Precisely one of the numbers

234, 2345, 23456, 234567, 2345678, 23456789

is a prime number. Which one must it be? Explain your reasoning.

33. A paper cylinder of radius r (without a top or a bottom) has the same surface area as a

sphere of radius r. What is the height of the cylinder?

34. A two-digit number is such that, if a decimal point is placed between its two digits, the

resulting number is one-quarter of the sum of the two digits. What is the original number?

35. One millionth of a second is a microsecond. Approximately how long is a microcentury?

36. Thirty-eight children are seated, equally spaced, around a circle. They are numbered in

order from 1 up to 38. What is the number of the child opposite child number 8?

37. What is 10% of 20 plus 20% of 30 plus 30% of 40 take away 50% of 60?

38. I travel 9 miles east, then 20 miles north, then 30 miles west. How far am I then from my

starting point?

39. Two cylindrical candles, one of diameter 1 cm and height 1 cm and one of diameter 2 cm and

height 2 cm, are melted down to make a cylindrical candle of diameter 3 cm. What will its height be?

40. is a triangle and is a point on such that If divides the angle

in the ratio 5 : 4, how big is the larger of the two angles at



Problem Set 1 – Answer Key

1. 14 : 11 2. Explain 3. Areas are equal 4. Explain

5. 6. 14 7. 8.

9. 10. 32 11. 1998 12.

13. 9 mph 14. 9.12 gallons/hour 15. 306 minutes 16. 7 minutes

17. 2.5 minutes 18. 1080 19. 2112 20. 3

21. 40 square units 22. 6 and 10 23. and 24. 75 gallons

25. 6 26. 27. 28. 4 minutes

29. 30. 2 31. 32. 23456789

33. 34. 15 35. 1 hour 36. 27

37. 38. 29 miles 39. 1 cm 40.

Bainbridge Independent Math Circle

Contact Information

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Phone: 775-‐410-‐MATH (6284) 775-‐343-‐2369 (Cell)

Email: [email protected]

Web: mathperfect.com

©2012 MathPerfect Discover and achieve your potential

Submitting Solutions to Problem Sets

While we will discuss the problem sets during our sessions, the best way for me to evaluate your development is to actually read your solutions so I can see how you present your ideas. You may type up your work using a word processor, or scan your handwritten work, and email me the files. Some students even use their phones to take digital pictures of their handwritten work and email me the pictures. If you do use a word processor, I suggest you also use MathType (http://www.dessci.com/en/products/mathtype) to make your work look spectacular. The trial version works well even after the expiration period. Whichever method you choose, please email your assignments to the address below. Remember to include your name and problem set number in the subject line.

I welcome any feedback or ideas you may have for improving my website and these study notes. I especially appreciate your negative feedback. Tell me what you don’t like as well as what you do like. I will continue to make improvements based on your suggestions. Click the link below to email me your ideas.

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