15152908Topic9Problem-solvingProcesses

7/28/2019 15152908Topic9Problem-solvingProcesses

1/7


2/7

TOPIC 9 PROBLEM-SOLVING PROCESSES W 119

HEURISTICS IN PROBLEM-SOLVING

When a scenario describes a problem to be solved, the next logical activity is to

guide learners through the steps in problem-solving, known as heuristics.

Heuristics direct ones path towards the solution of a problem. You are taught tofollow these heuristics in every problematic situation you face. Although describeddifferently by different mathematicians, it is generally accepted that there are stageswhich an individual passes through when solving problems.

According to Polya, there are four stages of heuristics in problem-solving. Theframework in Figure 9.1 illustrates the dynamic nature of the stages.

Figure 9.1: Four stages of heuristicsSource: Polya, G. (1957). How to solve it. USA: Princeton University Press.(a) Understanding the problem

(i) What are you required to find/solve?

(ii) What facts do you have in hand?

(iii) Is there sufficient information provided to solve the problem?

(iv) The problem is translated into the language of the reader.

(b) Making a plan(i) What strategies do you have?

(c) Carrying out the plan(i) Use computational skills.

(ii) Use algebraic skills.

(iii) Use geometric skills.

9.1


3/7

X TOPIC 9 PROBLEM-SOLVING PROCESSES120

(d) Looking back(i) Did you answer the question asked?

(ii) Does your answer make sense?(iii) How different is the answer compared to your estimation?

(iv) Is there any other alternative solution?

Although the steps mentioned provide a road map towards a solution to aproblem, they do not guarantee success in solving the problem. That depends onindividual skills.

Try the following exercise to test your understanding.

STRATEGIES FOR PROBLEM-SOLVING

Basically, strategies focus on helping students become more competent in problem-solving. There are a number of common strategies used for solving problems, whichwill be discussed in the following topics.

9.2.1 Looking for a PatternPatterns appear in nature, drawing and design as well as in mathematics.Identifying patterns requires a careful observation of common characteristics,variations and repetitive properties and shapes.

Example: Find the sum of all odd numbers less than 100. GAnswer:1 + 3 = 4 = 2 u 2

1 + 3 + 5 = 9 = 3 u 31 + 3 + 5 + 7 = 16 = 4 x 4

1 + 3 + 5 + 7 +... + 99 = 50 u 50 = 2500

9.2

SELF-CHECK 9.1

Briefly describe the stages involved in problem-solving.


4/7


9.2.2 Working Backwards

Working backwards is a strategy used when the outcome of a problem is known

and the initial conditions are required. When using this strategy, the operationsrequired by the original action will be replaced by inverse operations.

Example: The fine schedule for overdue books at a universitys library is asfollows: 20 sen per day for the first three consecutive days and 10 senper day thereafter. If Chong paid a fine of RM1.30, how many dayswas his book overdue?

Answer: The final outcome of RM1.30 was known. Chong paid 20 sen for eachof the first three days or 60 sen, which leaves 70 sen. At 10 sen per day,

this represents seven days. Therefore, his book was overdue for 10days.

9.2.3 Using BeforeAfter ConceptThis strategy is simple: Make sure you determine the order of operation in theproblem.

Example: Alice has 5kg of sugar. She used 1kg of the sugar to bake a cake. Then,she used half the remaining sugar to make cookies. Find out theamount of sugar she has left.

Answer: Initially, Alice had 5kg of sugar. After baking a cake, she had 4kg ofsugar (5kg1kg). Then, she used half the remaining sugar to makecookies, that is, 2kg. So, she has 2kg of sugar left.

9.2.4 Making SuppositionsAs the first step, you need to think of an answer and then verify the answer with the

information given in the problem. This process is repeated until you get the correctanswer.

Example: Farah bought 70 packets of chocolates and candies for her birthdayparty. She spent RM60 on chocolates and RM40 on candies. If eachpacket of chocolates cost twice as much as each packet of candies, howmany packets of chocolates and how many packets of candies did she

buy?


5/7


Answer 1: Suppose there were 20 packets of chocolates and 50 packets of candies.Each packet of chocolates costs RM3 (60/20) and each packet ofcandies costs RM 0.80 (40/50).

This is inconsistent with the information that each packet of chocolates costs twiceas much as a packet of candies.

Answer 2: Suppose there were 30 packets of chocolates and 40 packets of candies.Each packet of chocolates costs RM2 (60/30) and each packet ofcandies costs RM1 (40/40).

This is consistent with the given information.

Therefore, Farah had bought 30 packets of chocolates and 40 packets of candies.

9.2.5 Restating the ProblemThe problem is rephrased according to the information given, leading to a betterunderstanding of the solution.

Example: Mrs Yap spent RM400 on two toys, three dolls and three teddy bears.Each teddy bear costs twice as much as one doll and four times asmuch as each toy. What was the cost of each toy, doll and teddy bear?

Answer: Since,1 teddy bear = 2 dolls1 teddy bear = 4 toys3 teddy bears = 12 toys2 dolls = 4 toys

Therefore,1 doll = 2 toys3 dolls = 6 toys

2 toys + 3 dolls + 3 teddy bears = RM400

This can be restated as,2 toys + 6 toys + 12 toys = RM400

1 toy = RM201 doll = RM401 teddy bear = RM80


6/7


9.2.6 Simplifying the Problem

This strategy is often used in conjunction with other strategies. Simpler cases of the

same problem are solved first, leading to an understanding of the solution strategyto be used. The students focus on what to do and how to do it, and can then transferthis understanding to the original problem.

Example: There are 120 pages in a Mathematics book. What is the first digit ofthe sum of all even page numbers?

Answer: 2 + 4 + 6 + 8 +... + 114 + 116 + 118 + 120There are 30 pairs of 120 and an extra 120.

30 u 120 + 120 = 3720

The first digit of the sum is 3.

9.2.7 Using Equations

In order to write an equation that represents the problem, you must have a goodunderstanding of the problem. If you have knowledge of algebra, it helps you toovercome the problem you may face using this strategy.

Example: A teacher is standing in the middle part of the stairs to the secondfloor. He moved up three steps and then moved down five steps.Finally, he went up 10 steps to the second floor. Find out how manysteps there are from the first floor to the second floor.

Answer: Let n = the number of stepsThen, n/2 is the place where the teacher is standingn/2 + 3 5 + 10 = nn/2 + 8 = nn n/2 = 82n n = 16n = 16

Try the following exercise to test your understanding.


7/7


x Problem-solving is the means by which an individual uses previously acquiredknowledge, skills and understanding to satisfy the demands of an unfamiliarsituation.

x The process can be represented as a series of steps called heuristics.

x The four stages of heuristics are: understanding the problem, devising a plan,carrying out the plan and looking back.

x Patterns appear in nature, drawing, design as well as in mathematics.

x Identifying patterns requires a careful observation of common characteristics,variations and repetitive properties and shapes.

Heuristics

Patterns

Problem-solving

Strategies

Supposition

Krulik S., & Rudnick J.A. (1987). Problem solving: A handbook for teachers. NewtonMA: Allyn & Bacon.

Polya, G. (1957). How to solve it. USA: Princeton University Press.

1. What are the differences between heuristics and strategies inproblem-solving?

2. List the strategies involved in problem-solving. Briefly explaineach step in your own words.

SELF-CHECK 9.2

15152908Topic9Problem-solvingProcesses

Documents

Transcript of 15152908Topic9Problem-solvingProcesses