Why do this
problem?
This problem offers opportunities to reinforce the language
and characteristics of numbers such as odd, even, multiple.
However it is also a problem which can be tackled in a
systematic way, or, more elegantly, through insightful reading
of the clues. Comparing the two methods can be a useful
exercise in considering what mathematical thinking looks like.
Possible approach
Put three two-digit numbers, such as $93$, $56$, $75$ on the
board and ask the children which is the odd one out and why.
Listen for explanations which include descriptions of place
value, odds and evens, and perhaps multiples.
Say that you are going to choose one of them secretly and they
can ask just one or two questions to see if they can find which
one it is. For each correct guess ask the children to justify
their answer.
Put $23$, $45$, $62$, $101$, $94$ on the board and tell then
you have chosen one of these. What questions would be good ones
to ask and why? Make a record of the questions on the board so
that the children can refer to them. Again, ask the children to
justify any conclusions they come to.
Then offer the problem to pairs of children. Say that you are
interested in the mystery number but also how they know and you
will be asking for a description of what they did and in what
order, so they may like to keep some notes.
Bring the children back together and ask a few pairs to
describe their method. Listen for those who go through the
clues in order eliminating the impossible numbers, and explain
that working systematically is a very important skill for a
mathematician.
But also look out for the children who have scanned through the
clues to find one that is more useful and saves some work - for
example
'The sum of the two digits is a multiple of five.'
which means the magic number must be 46 or 64. After that it
only needs one more carefully chosen clue to distinguish
between them:
'The digit in the tens place is greater that the digit in the
unit (or ones) place.'
Bring out the importance of scanning through to see which clues
might be the most useful.
Key questions
Which question shall we ask first? Why?
What does 'multiple of five' mean?
Possible extension
An obvious extension is for the children to make up their own
examples for each other. They could repeat the format of the
given problem, or play 'what's my number?' with a partner, where
they try to find out what the mystery number is in the minimum
number of guesses. Give an opportunity for them to describe why
some questions are more useful than others.
Possible support
Sometimes it can be confusing to be given too much information
at one time. Write the clues and the numbers out on separate
cards. Spread the number cards out and offer the clues one at a
time, encouraging discussion of the characteristics of each
number in response to the clue.