Teaching and Learning Mathematics In Rural Schools
Mathematical Problem Solving
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Problem solving is an integral part of all mathematics learning. This is because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics - functional, logical and aesthetic (Taplin, 2004).  In everyday life and in the workplace, being able to solve problems can lead to great advantages. However, solving problems is not only a goal of learning mathematics but also a major means of doing so (NCTM, 2004).

NCTM STANDARDS FOR PROBLEM SOLVING

The set standards for problem solving from the National Council of Teachers of Mathematics are categorized as reasoning and proof, communication, connections, and representation.  They specifically state that instructional programs from pre-kindergarten through grade 12 should enable all students to:

 

  • Organize and consolidate their mathematical thinking through communication. 
  • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
  • Analyze and evaluate the mathematical thinking and strategies of others.
  • Use the language of mathematics to express mathematical ideas precisely.
  • Recognize reasoning and proof as fundamental aspects of mathematics.
  • Make and investigate mathematical conjectures.
  • Develop and evaluate mathematical arguments and proofs.
  • Select and use various types of reasoning and methods of proof.
  • Organize and consolidate their mathematical thinking through communication.
  • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
  • Analyze and evaluate the mathematical thinking and strategies of others.
  • Use the language of mathematics to express mathematical ideas precisely.  
  • Recognize and use connections among mathematical ideas.
  • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
  • Recognize and apply mathematics in contexts outside of mathematics.
  • Create and use representations to organize, record, and communicate mathematical ideas.
  • Select, apply, and translate among mathematical representations to solve problems.
  • Use representations to model and interpret physical, social, and mathematical phenomena (NCTM, 2004).

 

GOOD PROBLEM SOLVING

 

Problem solving means engaging in a task for which the solution is not known in advance. Good problem solvers analyze situations carefully in mathematical terms and naturally come to pose problems based on situations they see.  Good problems give students the chance to solidify and extend their knowledge and to stimulate new learning. Most new mathematical concepts can be introduced through problems based on familiar experiences coming from student’s lives or from mathematical contexts (NCTM, 2004). In this way students can identify with the problem due to its relevance and their creative thought is stimulated as they attempt to devise a strategy for solving the problem.  

 

TEACHERS’S ROLE

 
Teachers play an important role in developing students' problem-solving dispositions. They must choose problems that engage students. They need to create an environment that encourages students to explore, take risks, share failures and successes, and question one another. In such supportive environments, students develop the confidence they need to explore problems and the ability to make adjustments in their problem-solving strategies.  Additionally, students need to develop a range of strategies for solving problems, such as using diagrams, looking for patterns, or trying special values or cases, in order to be successful with problem solving. Thus, it is the responsibility of the teacher to provide instruction in the use of these strategies to their students (NCTM, 2004).

SAMPLE PROBLEMS

 

 

Problem 1

 

Problem:           What number comes next?

                        37, 46, 56, 67, ______

 

Solution:           Notice that to go from 37 to 46 means you add 9.  To go from 46 to 56, add 10.  To go from 56 to 67, add 11.  To get the next number we need to add 12 to 67.  The answer is 79.

 

 

Problem 2 

 

Problem:           Look at the chart and fill in the numbers.

4

5

9

 

 

 

16

 

7

2

2

4

 

5

7

 

1

 

2

3

5

6

 

8

 

 

4

4

6

20

42

25

 

63

2

 

 

Solution:           The key was to identify the pattern.  Notice that in column 1 the values in rows 2 and 3 can combine as a sum, product, or exponent to give you the values in rows 1 and 4.  We need to examine column 2 for a similar pattern.  Notice that the sum of rows 2 and 3 results in the value in row 1, and the product of rows 2 and 3 result in the value in row 4. By checking the pattern for column 3 we see that we have identified the correct pattern.  Now it is a case of calculating the missing values.

 

4

5

9

13

10

15

16

3

7

2

2

4

7

5

7

7

1

3

2

3

5

6

5

8

9

2

4

4

6

20

42

25

56

63

2

12

 

Problem 3 

 

Problem:           Sherry, Betty and Jackie are daughters of Mr. Smith, Mr. Mouland and Mr. Budgell.  Four of these people are playing tennis doubles.  Mr. Budgell’s daughter and Mr. Smith are partners.  Sherry’s father and Mr. Mouland’s daughter are also partners.  There are not any father-daughter combinations.  Who is Sherry’s father?

 

 Solution:

  • Sherry is not Mr. Moland’s daughter.
  • Because Mr. Mouland’s daughter is playing on one side and Mr. Smith is on the other, Mr. Mouland cannot be playing.
  • Sherry’s father is on a team with Mr. Mouland’s daughter.
  • Sherry’s father is Mr. Budgell

 

Court A                                                                  Court B

Mr. Smith

Mr. Mouland’s Daughter

Sherry

Mr. Budgell

Not Playing:            Mr. Mouland

 

Problem 4 

 

Problem:           How many five-digit positive integers, divisible by 9, can be written using only the digits 3 and 6?

 

Solution:           For a number to be divisible by nine, the sum of its digits must be a multiple of 9.  This means that there are only two cases to consider.  If students do not remember this they will be listing off all of the combinations which will take much more time.

                       

Case 1 – One 6 and four 3’s (Sum of 18)

                                    63 333, 36 333, 33 633, 33 363, and 33 336

                        Case 2 – One 3 and four 6’s (Sum of 27)

                                    36 666, 63 666, 66 366, 66 636, and 66 663

                        So, in total there are 10 possibilities.

 

Problem 5

 

Problem:     The product of the digits of a four-digit number is 810.  If none of the digits is repeated, then what is their sum?

 

Solution 1:        Suppose that the 4-digit number has digits abcd, so the product abcd = 810.  We need to write 810 as the product of 4 different digits, none of which is 0 (If a digit was 0 then the product would equal 0). 

 

                  When we factor 810 we get, 810 = 10 x 81 = 2 x 5 x 34.

 

One of the digits is 5 since the only non-zero digit that is a factor of 5 is 5. Next, we have to find 3 different digits whose product is 2 x 34.  These numbers are 3, 6 and 9. 

 

                  Thus, the digits are 3, 6, 9 and 5, giving a sum of 23.

 

Solution 2:                    Try eliminating possibilities.

·        0 cannot be a digit since product would have to be 0.

·        1 cannot be a digit since no combination of remaining digits will be greater than 504 (9 x 8 x 7).

·        4, 7, and 8 will not divide into 810 evenly thus eliminating them.

·        The remaining digits are 2, 3, 5, 6, and 9.

 

The possible products are:

 

Numbers

Product

2, 3, 5, 6

180

2, 3, 5, 9

270

2, 3, 6, 9

324

2, 5, 6, 9

540

3, 5, 6, 9

810

 

The digits are 3, 5, 6, 9 giving a sum of 23 (University of Waterloo, 2003). 

Problem 5

 

Problem:     John has fish in two aquariums.  In one aquarium, the ratio of the number of guppies to the number of goldfish is 2:3.  In the other, this ratio is 3:5.  If John has 20 guppies in total, what is the least number of goldfish that he could have?

 

Solution 1:  The following tables give the possible numbers of fish in each aquarium.  The results which give a total of 20 guppies are 2 + 18, 8 + 12 and 14 + 6.  The corresponding numbers of goldfish are 33, 32 and 31.  The least number of goldfish that he could have is 31.

 

1st                                                                                                  2nd

 

Number of

Guppies

Number of

Goldfish

  

Number of

Guppies

Number of

Goldfish

2

3

 

3

5

4

6

 

6

10

6

9

 

9

15

8

12

 

12

20

10

15

 

15

25

12

18

 

18

30

14

21

 

 

 

16

24

 

 

 

18

27

 

 

 

 

 

Solution 2:        In the first aquarium, the ratio of the number of guppies to goldfish is   2:3 so let the actual number of guppies be 2a and the actual number of goldfish be 3a.  In the second aquarium, the actual number of guppies is 3b and the actual number of goldfish is 5b.  In total, there are 20 guppies so we have the equation 2a + 3b = 20.  We can use a chart to consider the different possibilities and to find the number of goldfish.

 

 

 

2a + 3b

a

b

3a + 5b

20

1

6

33

20

4

4

32

20

7

2

31

 

The smallest possible number of goldfish is 31 (University of Waterloo, 2001).

  

Problem 6

 

Problem:           What letter comes next?

 

                  A, B, D, O, P, Q, _______

 

Solution:           The answer is R. Start with A and proceed alphabetically keeping the letters that have an enclosed area (Carter & Russell 2005, p. 8).

  

Problem 7

 

Problem:           You are given 5 matches.  How can you use them to form a cube if you are not allowed to bend or break them?

 

Solution 1:              If cube is taken in the numerical sense then there are various answers.  Examples include 1, 27, Roman Numeral 8 (VIII), or 1 to the exponent 3.

 

Solution 2:              If the matches are straight, then an arrangement can be made that forms a small cube at the center. 

Problem 8

 

Problem:     A secretary types 4 letters to 4 people and addresses the 4 envelopes.  If she inserts the letters into the envelopes at random, each in a different envelope, what is the probability that exactly 3 letters will go into the right envelopes?

 

Solution:     The answer is 0.  If 3 letters match the envelopes then the fourth will as well (Gardner 1978, p. 138).

 

Problem 9

 

Problem:     The integers 1, 3, 8 and 120 form a set with a remarkable property: the product of any two integers is 1 less than a perfect square.  Find a fifth number that can be added to the set without destroying this property.

 

Solution:     The answer is 0.  The question then can be asked, is there a positive number that can be added to the set?  Research shows that any answer would have to be more than 1, 700, 000 digits and no known answer has been found yet (Gardner 1978, p. 210).

 

PROBLEM 10

 

Problem:     A man with some time to kill in a particular town decided to get a hair cut.  There were two barbers to choose from.  The first kept an untidy shop, he needed a shave, his clothes were unkempt, and his hair badly cut.  The second barber had a tidy shop, was freshly shaven, dressed clean, and had his hair neatly trimmed.  Who cut the man’s hair and why?

 

Solution:     Since each barber would have to have cut the others hair, the first was the better barber.  Thus, this is who the man chose to give him a haircut (Gardner 1978, p. 138).

 

REFERENCES

 

Active Minds (2005).  Math Booster: Grades 3/4.  Toronto: Chapters Inc.

Carter, P. & Russell, K. (2005).  The Great IQ Puzzle Book.  Toronto: Indigo Books.

Gardner, M. (1978).  Mathematical Magic Show.  New York: Vintage Books.

Math Counts (1999).  Problem Solving Strategies.  Retrieved July 11, 2005 from http://206.152.229.6/Problems/strategies.html.

National Council of Teachers of Mathematics (2004).  Appendix: Table of Standards and Expectations.  Retrieved July 2, 2005 from http://standards.nctm.org/document/appendix/process.htm.

Taplin, M. (2004).  Mathematics Through Problem Solving.  Retrieved July 2, 2005 from http://www.mathgoodies.com/articles/problem_solving.html

University of Waterloo (2003).  Cayley Math Contest.  Retrieved July 2, 2005 from http://www.cemc.uwaterloo.ca/english/contests/index.shtml.

University of Waterloo (2001).  Gauss Math Contest.  Retrieved July 2, 2005 from http://www.cemc.uwaterloo.ca/english/contests/index.shtml.