Download MATHEMATICAL PROBLEM SOLVING

M at h e m at i c a l
Problem Solving
david kennedy
AMT P u b l i s h i n g
Contents
Prefacev
Introduction1
Making An Organised List
4
Looking For A Pattern
11
Drawing A Picture Or Diagram
19
Making The Problem Simpler
26
Acting Out The Problem
31
Using Models
34
Trial And Improvement
40
Making A Table (Part 1)
53
Making A Table (Part 2)
63
Working Backwards
73
Using Logic
84
Using Algebra
89
Taking A Different Viewpoint
95
Introduction1
Introduction
In order to become an efficient problem solver, we must first appreciate the different strategies
available. Regular practice of these strategies will greatly strengthen our effectiveness as
problem solvers and should lead us to become more efficient ones.
Alternatives
Many problems can be solved using different strategies.
Example: Shaking hands
If ten people meet in a room and each person shakes hands with each of the others,
how many handshakes take place?
Alternative Solution Strategies
1.
Act It Out: Let ten people shake hands with each other and keep a tally.
2.
Draw a Diagram: Count the lines.
3.
Use Logic: Ten people each shake hands with nine others, so 10 lots of 9 = 90
but each handshake has been counted by both shakers so you need to halve this
number, that is, 45.
4. Make an Organised List: Suppose their names are A, B, C, D, E, F, G, H, I and
J, then the results could be recorded as follows (where AB represents A shaking
hands with B).
AB
AC BC
AD BD CD
AE BE CE DE
AF BF CF DF EF
AG BG CG DG EG FG
AH BH CH DH EH FH GH
AI BI CI DI EI FI GI HI
AJ BJ CJ DJ EJ FJ GJ
HJ IJ
2
Mathematical Problem Solving
5. Make the Problem Simpler and Look for a Pattern:
With 2 people there is 1 handshake.
With 3 people there are 3 handshakes.
With 4 people there are 6 handshakes.
With 5 people there are 10 handshakes.......and so on.
Students may notice that these are the triangular numbers.
Combinations
Many solutions involve a combination of different strategies.
Example: The Magic Square
Place the digits 1 to 9 in the square below so that each row, column and diagonal
has the same sum.
Solution
1.
Make the Problem Simpler: Sum of all numbers = 45 so sum of each row = 15.
2.
Make an Organised List: Find combinations of three digits that total 15.
951
951
861
852
843
762
753
654
3.
Use Logic: The 5 occurs in four combinations so
must go in the centre square.
5
Introduction3
The 8 occurs in three combinations so must go in a
corner square.
8
5
So it follows that the 2 must be in the opposite corner.
8
5
2
The 6 also occurs in three combinations so must go in
another corner.
8
6
5
2
So it follows that the 4 must be in the opposite corner.
8
6
5
4
Now, the rest are obvious.
2
8 1 6
3 5 7
4 9 2
4
Mathematical Problem Solving
Making an Organised List
A list helps us account for all possibilities and avoid repetitions. It is important to find and use a
system. Often it is simply a case of listing results in numerical or alphabetical order. However,
we need to be rigorous in the application of our system.
Example 1
How many triangles can you find in the shape below?
Solution
First label each small section with a letter.
a
e
b
c
d
Now list all the possibilities in alphabetical order.
a b c d e
abcbc cd de
abcdebe
abe
So there are 12 different triangles in the figure.
Example 2
Domino tiles are all different. Each tile consists of two halves. Each half can be
either blank or a number from 1 to 6. How many dominoes are there in a full set?
Making an Organised List
5
Solution
The possible numbers on each half are 0, 1, 2, 3, 4, 5 and 6. List the possible
combinations as ordered pairs in numerical order. You could start at 6,6 or 0,0 but
be consistent! (Note: 6,5 is the same as 5,6 in this problem!)
6,6
6,5
5,5
6,4
5,4
4,4
6,3
5,3
4,3
3,3
6,2
5,2
4,2
3,2
2,2
6,1
5,1
4,1
3,1
2,1
1,1
6,0
5,0
4,0
3,0
2,0
1,0
So there are 28 dominoes in a full set.
0,0