14 Unit 1 Thinking and reasoning
Problem solving in early thinking skills
exams was firmly founded on these three
basic processes. The BMAT and TSA syllabuses
still refer to them explicitly. In the Cambridge
examinations, the three basic processes have
been expanded into a wider range of skills
which are tested at AS Level using multiple-
choice questions and at Advanced Level with
longer, more open-ended questions which
can draw on several of the basic skills. For
example, the problem-solving category of
‘searching for a solution’ is one of the strands
of ‘finding procedures’.
Unit 3 of this book is entitled ‘Problem
solving: basic skills’ and deals with these
extended skills. The chapter structure is firmly
based on the problem-solving skills defined in
the Cambridge syllabus. Unit 5, ‘Advanced
problem solving’, deals with the extension to
Advanced Level and wider-ranging questions.
Questions at this level will generally include
the use of several of the basic skills. This covers
the analysis of more complex data sets, and
mathematical modelling and investigation.
These questions have open, rather than
multiple-choice, answers. Unit 6, ‘Problem
solving: further techniques’, deals mainly with
mathematical techniques which may be useful
in examinations at all levels.
The end-of-chapter assignments have often
been left open-ended rather than framed as
multiple-choice questions. This is so you will
have to solve the problem, rather than
eliminating answers or guessing. Some of the
activities and questions are marked as ‘harder’
and are intended to stretch candidates.
Here is a ‘taster’ problem to start with. It is
certainly not trivial, but illustrates the essence
of problem solving. The problem contains
only three relevant numbers and the only
mathematics required is the ability to add,
subtract and divide some small two-digit
numbers. Solving the problem requires no
specialised knowledge, either of techniques or
skills, just clear thinking.Marina is selling tickets on the door for a
university play. It costs $11 for most people to
buy a ticket, but students only have to pay $9.
Just after the play starts, she remembers that
she was supposed to keep track of the number
of students in the audience. When she counts
the takings, there is a profit of $124.
How many people in the audience are
students?A 2 B 3 C 4 D 5 E 6Activity
Commentary
The $124 is made up of a number of $11 tickets
plus a number of $9 tickets. We need to find
out what multiples of 11 and 9 will add to 124.
We can do this systematically by subtracting
multiples of 11 and dividing the remainder by- For example, if there were one audience
member paying the full ticket price, there
would have been $113 from students. This is
not a multiple of 9, so cannot be correct. We
can list the possibilities in a table:
Number of
full-fee payersAmount paid Remainder
from $1 $11 $2 $22 $3 $33 $4 $44 $5 $55 $6 $66 $7 $77 $8 $88 $9 $99 $10 $110 $