BASIC MATHEMATICSIN POLYA’S SOLVING

PROBLEM

TITTLE:IDENTIFYING THE SUB-GOALS

Work in pairs : Muhammad Zaki B. Bajuri

Rozalia Bt. Ramli

CONTENT: the meaning of identifying the sub- goals

steps in creating sub-goals using polya’s solving problem example 1 : magic square more examples

THE MEANING OF IDENTIFYING THE SUB-GOALS

As we attempt to devise a plan for solving some problems, it may become apparent that the problem could be solved if the solution to the easier or more familiar related problem could be found.

So that, we have to create sub goals in order to solve the final problem. Sub goals consists of breaking the main problem into separate sub problem in order to solve the problem step by step.

STEPS IN CREATING SUB-GOALS

1) Make a list of sub goals/sub problem that need to be solved before solving the final problem (usually the main problem).

2) Solve each individual sub problem that eventually lead to the final solution to the problem.

USING POLYA’S SOLVING PROBLEMS

understanding the problem

devising a plan

carrying out the plan

looking back

EXAMPLE 1: A MAGIC SQUARE

Arrange the numbers 1 through 9 into a square subdivided into nine smaller squares like the one shown in Figure 1. So that the sum of every row, column and main diagonal is the same.(the result is a magic square)

Figure 1

UNDERSTANDING THE PROBLEM

we need to put each of the nine numbers 1,2,3,………..,9 in the small squares, a different number in each square, so that the sum of the numbers in each row, in each column, and in each of the two diagonals in the same.

DEVISING A PLAN

First

•If we knew the fixed sum of the numbers in each row, column, and diagonal, we would have a better idea of which numbers can appear together in a single row, column, or diagonal.

second

•Thus, our sub goal is to find that fixed sum.

third

•The sum of the nine numbers, 1+2+3+……….+9, equals 3 times the sum in one row.

fourth

•Consequently, the fixed sum is obtained by dividing 1+2+3+…………+9 by 3.

• using the process developed by Gauss, we have (1+2+3+……..+9) ÷ 3 = 15, so the sum in each row, column and diagonal must be 15.

fifth

• Next, we need to decide what numbers could occupy the various squares. The number in the center space will appear in 4 sums, each adding to 15 .

sixth

• Each number in the corners will appear in three sums of 15. Why??? If we write 15 as a sum of three different numbers 1 through 9 in all possible ways, we could then count how many sums contain each of the numbers 1 through 9.

seventh

ninthThus, our new sub goal is to write 15 in as many ways as possible as a sum of three different numbers from

the set {1,2,3,……….,9}

eighth

The numbers that appear in at least four sum are candidates for placement in the center square,

whereas the numbers that appear in at least three sums are candidates for the corner squares.

CARRYING OUT THE PLAN The sums of 15 can be written systematically as

follows:

9 + 5 + 1

9 + 4 + 2

8 + 6+ 1

8 + 5 + 2

8 + 4 + 3

7 + 6 + 2

7 + 5 + 3

6 + 5 + 4

Number 1 2 3 4 5 6 7 8 9

Number of sums containing the number

2 3 2 3 4 3 2 3 2

Notice that the order in each sum is not important.(do you see why?) hence, 1 + 5 + 9 and 5 + 1 + 9, for example, are counted as the same. Notice that 1 appears in only two sums, 2 in three sums, 3 in two sums, and so on. Table 2 summarizes this pattern.

Table 2

The only number that appears in four sums is 5; hence, 5 must be in the center of the square. (why???) because 2, 4, 6, and 8 appear three times each, they must go in the corners.

suppose we choose 2 for the upper left corner.

then, 8 must be in the lower right corner.

now, we could place 6 in the lower left corner or upper right corner.

or

2

5

8

62

5

8

2

6

5

8

to place another number in the small square, we should do some calculation and the sum must be 15.

Calculation:(step 1) 2 + 5 + 8 = 15(step 2) 6 + 5 + ? = 15

15 – 11 = 4 so, the number place in the lower left

corner is 4

2 6

5

? 8

2 ? 6

? 5 ?

4 ? 8

Thus, to complete the magic square: 6 + 8 + ? = 15 2 + 6 + ? = 15 4 + 8 + ? = 15 2 + 4 + ? = 15

2

31567

49

8

LOOKING BACK we have seen that 5 was the only

number among the given numbers that could appear in the center.

another way to see that 5 could be in the center square is to consider the sums 1 + 9,

2 + 8, 3 + 7,and 4 + 6. we could add 5 to each to obtain 15. 1 2 3 4 5 6 7 8 9

10

10

10

10

EXAMPLE 2: Given that 3x – 1 = 17.

2x – 4 = ?solve equation that follow :Sub goals :1. What does x equal when 3x – 1 = 17?

[use simple algebra skills to isolate the variable and solve for x]

2. What is 2x – 4?3. Once have the value for x, the value

can be substituted in the second equation to get final answer.

3x - 1 = 173x = 17 + 1x = 18 ⁄ 3x = 6

Substitute x = 6 into 2x – 42(6) – 4 = 8

Goal : 8

TRY THIS:Given that 4x + 4 = 20Solve equation that follow:

3x + 5 = ?

SOLUTION:Sub goals :1. What does x equal when 4x + 4 = 20?

[use simple algebra skills to isolate the variable and solve for x]

2. What is 3x + 5?3. Once have the value for x, the value

can be substituted in the second equation to get final answer.

4x + 4 = 20 4x = 20 – 4 4x = 16 x = 4Substitute x = 4 into 3x + 5 3(4) + 5 = 17

Goal : 17

THAT’S ALL FROM US…~Thank you~