Problems and Problem Solving - Ministry of Education

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se c t I o n 3: pr o B l e m so l V I n g – pr o c e s s A n d st rA t e g I e s The Problem-Solving Process The problem-solving process is a set of steps that are sufficiently general so that they can be applied to any problem in order to obtain a solution. According to Hattfield, Edwards, Bitter & Morrow (2005), Polya identified four steps in the problem solving process. 1. Understand the problem 2. Devise a plan 3. Implement the plan 4. Look back These steps are described in more detail below. The Census Taker Problem (introduced before) will be used as an example to illustrate the activities that are involved in each step. A census taker comes to the house of a mathematician and asks how many children he has and what their ages are. The mathematician replies that he has three children and the product of their ages is 72. The census taker replies that he has not been given enough information to determine their ages. The mathematician adds that the sum of their ages is the same as his gate number. The census taker leaves to check the gate number but returns shortly to say that he still does not have enough information. The mathematician thinks and says the oldest one likes chocolate ice cream. The census taker replies that he now has enough information and leaves. What are the ages of the three children? Step 1 – understand the problem At this point in the process you should spend time decoding the problem to ensure that you are able to take from it the important facts and considerations. In doing so the following considerations will be necessary. • y You must understand exactly what the problem requires you to do. Let us examine the Census Taker Problem more closely, for example. It clearly asks you to find the ages of the three children. What is implied from the story is that their ages should sum to an unknown gate number and should multiply to give 72. 36 PROBLEMS AND PROBLEM SOLVING ProblemSolving.indd 36 8/24/12 6:55:52 PM
• y Familiarize yourself and pay attention to important words in the problem. While it is not absolutely necessary for obtaining a solution, you may need a bit of background information on what a ‘census’ is. Phrases such as ‘product of their ages’ ‘sum of their ages’ and ‘the same as’, however, need to be emphasized and understood in order to decode the problem. • y Restate the problems in your own words. After reading the Census Taker Problem, you should be able to identify that the mathematician has said ‘the ages of my three children give a product of 72 and a sum that is the same as my gate number.’ More importantly, you should be able to say that this information is not enough to solve the problem. •y If necessary, represent the problem using two- and/or three-dimensional models (tables, diagrams, flowcharts, physical models, etc.) The creative problem solver, for example, may decide to model the problem as follows: • y Identify the important and the missing information in the problem. You should appreciate, for example, that the census taker leaves thinking he has enough information. It was only after he checked that he realized that he still did not have enough information to determine their ages. Additionally, the fact that the oldest child liked chocolate ice cream was not as important a clue as the fact that there was an oldest child! • y Condense and compartmentalize the problem into simpler parts and tasks. Having read the problem, you should be able to identify that you need to find ways to make a product of 72 using three numbers. You should also know that there is something peculiar about the sum of the three numbers. Ministry of Education 2011 37 ProblemSolving.indd 37 8/24/12 6:55:53 PM
Page 1 and 2: Ministry of Education Problems and
Page 3 and 4: Co n t e n t s Introduction........
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Page 7 and 8: seen it that way; explore why this
Page 9 and 10: Bearing in mind the definitions abo
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Page 17 and 18: iii. Use of geometric principles an
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Page 21 and 22: Model solutions Essentially, the ap
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Page 25 and 26: Number of coins New values after ea
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Page 29 and 30: o Explore a variety of situations.
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Problem 29 - Frog Hopping Grade: 4
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2. In general, how many squares are
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Problem 33 - Chicken Combos Grades:
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2. Explain how you know the fractio
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Problem 36 - Measuring Jugs Grades:
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Problem 38 - The Trapezium Problem
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Problem 39 - Karen’s Troubles Gra
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Problem 41 - Stand Out Grade: 3 or
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Problem 43 - Mathematics Maze Grade
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Problem 45 - Circling Numbers Grade
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Problem 47 - Triangle Problems Grad
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Problem 48 - Factors Grade: 3 or ab
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Problem 50 - Target Grade: 4 or abo
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Problem 52 - Pond Borders Grade: 5
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Problem 54 - Teacher’s Banner Gra
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Grades: 4 - 6 Strand: Number Studen
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Problem 58 - Patty Please Grade: 4
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Problem 60 - Text-a-holic Grade: 3
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Problem Number 62 - Magic Squares G
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Ap p e n d i x 1 - Se l e c t e d S
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Problem 4 - Perfect squares 1. When
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Problem 11- The 400 Problem 1. Sinc
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When we look at the results recorde
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Problem 18 - Counting Rectangles 1.
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2. We can note that: a) The pattern
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As we go around the circle we may s
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For example, if a 1g weight is plac
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Problem 30 - Chess Board Squares 1.
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Problem 32 - Newspaper Math 1. 2. 3
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Problem 38 - The Trapezium Problem
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Problem 56 - Cross Out 8 The winnin
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Heuristics Strategies employed in p
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Scaffolding A teaching strategy in
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