Problems and Problem Solving - Ministry of Education

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What about DAILY and SALMON? Predict the number of ways in which MATHEMATICS could be written if it were presented in a similar grid. •y PrOBlemS mAy Be PreSenTeD SO ThAT They PrOVIDe SCAffOlDInG Or SOme GuIDAnCe TO STuDenTS. Providing scaffolding means that we supply students with information about the problem and give a framework for deriving further solutions. This is demonstrated in the way the conversation problem is presented. A conversation relationship occurs when two persons in a group share information. For example, if there are 2 persons (A and B) in the group, then there is only one possible conversation relationship – A and B will converse with each other. However, if there are 3 persons (A, B and C) in the group, then the following 3 conversation relationships are possible: o A with B o A with C o B with C This is shown right: Now, if there are 4 persons in the group, there are 6 possible conversation relationships. Complete the diagram below to verify that this is so. Now, determine the number of conversation relationships possible if there were 30 persons in a group. 30 PROBLEMS AND PROBLEM SOLVING ProblemSolving.indd 30 8/24/12 6:55:49 PM
As we can see from the example, the information provided gives the students “a way into the problem”. Let us consider the following. •y •y •y The first few situations were solved for the students – they now know how many conversation relationships are possible when two, three and four persons are in the relationship. They can start looking for patterns at this point and use this pattern to predict the number of conversation relationships possible among five persons. Students were shown how to represent the solution for three persons. They were also given an insight into how to proceed for determining the number of possible relationships. The answer for a four-person relationship was provided so as to present students with the opportunity to practise the use of the method of their choice. Students are, therefore, given the opportunity to check the accuracy of their methods by ensuring that they, indeed, end up with six possible relationships. It is important to decide on how much information you will share with the students, as well as how the questions will be structured when designing a problem. For example, in exploring the conversation problem, we expect students to obtain the following results: Students can fill in the unknown cells by realizing that the value in each cell is obtained by adding consecutive numbers to the number on its left. Using this idea, the following table can be obtained. No. of Persons 1 2 3 4 5 6 7 8 9 10 No. of Conversations 0 1 3 6 10 15 21 28 36 45 The question asked students to say how many conversation relationships are possible when 30 persons are present. This could be tedious if they had to continue adding consecutive numbers to count up to 30 persons. We used the number 30 as we wanted to demonstrate to students that this method of ‘counting up’ to a particular number is time consuming. Here we hope they will realize that they must now think of relationships between the number of persons and the number of conversation relationships (vertical and not horizontal relationships). This is shown on the following page. Ministry of Education 2011 31 ProblemSolving.indd 31 8/24/12 6:55:49 PM
Page 1 and 2: Ministry of Education Problems and
Page 3 and 4: Co n t e n t s Introduction........
Page 5 and 6: In t r o d u c t i o n Overview Acc
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 21 and 22: Model solutions Essentially, the ap
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Ministry of Education 2011 85 Probl
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Problem 29 - Frog Hopping Grade: 4
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Problem 38 - The Trapezium Problem
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Problem 41 - Stand Out Grade: 3 or
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Problem 45 - Circling Numbers Grade
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Problem 48 - Factors Grade: 3 or ab
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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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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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Heuristics Strategies employed in p
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Scaffolding A teaching strategy in
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Problems and Problem Solving - Ministry of Education

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