E  
PROBLEM SOLVING STRATEGIES  
   
Problem Solvers Acknowledgements

The student should acquire as much experience of independent work as possible.  But if he is left alone with his problem without any help or with insufficient help, he may make no progress at all.  If the teacher helps too much, nothing is left to the student.  The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.

--George Polya
Strategies

1-Draw a Diagram

2-Make a List

3-Guess and Check

4-Divide and Conquer

5-Look for a Pattern

6-Start at the End

7-Mixed Problems

a

STRATEGY 3: GUESS AND CHECK

Often when presented with a problem, you are encouraged to guess the answer. For example, "What's the largest city in the world? Take a guess." You guess Istanbul. "No, it's Tokyo," you are told. "How many people live in Tokyo? Take a guess." You say you don't know. "Go on, take a guess," you're told again. You guess 15 million. "No, 34 million!"

Guessing often produces the wrong answer. But the strategy called "Guess and Check" often produces the right answer. It should probably be called "Guess and Check and Guess Again," because the process of checking the accuracy of each guess and then making another, more informed guess is an essential part of the strategy.

For example, here is a problem:

Busra went to her grandfather's farm. Her grandfather has chickens and goats on his farm. She asked him how many chickens and how many goats. Because her grandfather likes mathematical puzzles, he told her that his animals had 26 heads and 68 legs and from that information she could calculate the number of chickens and the number of goats. If you were Busra, how would you solve the problem?

To use the Guess and Check strategy, you think about the problem and start by making a guess. You expect your first guess to be wrong, but it will give you some information to help you make a better guess next time. You could start by guessing 13 chickens and 13 goats. It's a good idea to keep a record of your guesses, like this:

 

Guess

 

Chickens

Goats

Number of heads

Number
of legs

1

13

13

26

78

 

 

 

 

 

 

 

 

 

 

You see that the number of legs you guessed is too high, because Busra's grandfather said that there are 68 legs. So you guess again-you have to add more chickens and subtract some goats.

 

Guess

 

Chickens

Goats

Number of heads

Number
of legs

1

13

13

26

78

2

20

6

26

64

 

 

 

 

 

Now you have 64 legs and you need four more. But you can't add any more heads, since 26 is the correct number of heads. So you take away two chickens (two heads and four legs) and add two goats (two heads and eight legs).

 

Guess

 

Chickens

Goats

Number of heads

Number
of legs

1

13

13

26

78

2

20

6

26

64

3

18

8

26

68

Now you have the correct answer: 18 chickens and 8 goats.


*
Thanks to Highline Council P.S.T.A. for this problem: http://home.blarg.net/~math/

 

Problems:

Problem 1: Jelly Beans

Problem 2: Darts

Problem 3: Movie

Problem 4: Elizabeth's Walk

Problem 1: Jelly Beans

To raise money for a charity, students in a fifth grade class organized a competition at the school's picnic. They filled a big jar of jelly beans and other students and their parents were asked to guess how many jelly beans were in the jar. The person whose guess was closest to the real number received a prize and, of course, the jar of jelly beans. To enter the competition, they charged adults 1.25 YTL and children 75 kurus. Back in their classroom, they counted the money and learned that they had earned 60.25 YTL from the comptetition. A total of 65 people had made a guess. Their teacher asked them to calculate how many parents and how many children had entered the competition. They were able to solve the problem. Can you solve it?

Solution -->

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Problem 2: Darts

dart

In a game of darts, the first person to score 150 is the winner. You must throw four darts and must score exactly 150. If you score more or less than 150 with your four darts, you must wait until your opponent tries and then you may try again. To be successful, you must know all the combinations of four numbers that add up to 150. For example, if you aim for 50 and miss and score 10 instead, what other numbers must you aim for? Make a table and enter all the possible combinations of four numbers that add up to 150.

Solution -->

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Problem 3: Movie

Ibrahim Bey decided to take all his children and grandchildren to see a movie in the mall. The tickets cost 5YTL for children and 7YTL for adults. He spent 116YTL. How many children and how many adults went to the cinema?

* Thanks to Jim Cornish for this problem: http://www.cdli.ca/CITE/guess_and_check_one.pdf

 

Solution -->

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Problem 4: Elizabeth's Walk

This problem is taken from Mathematic Teaching in the Middle School , 11 (3) p. 134.

 

Elizabeth visits her friend Andrew and then returns home by the same route. She always walks 2 kilometers per hour (km/h) when walking uphill, 6 km/h when walking downhill, and 3 km/h when walking on level ground. If her total walking time is 6 hours, then what is the total distance she walks (in kilometers)?

Solution -->

PROBLEM SOLVING STRATEGIES

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