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Reason logically
The strategy of reasoning logically when working through a
problem involves using common sense to rule out the
impossible and decide what is irrelevant. Children will need to
decide upon an appropriate strategy and to work through a
problem systematically: some problems may involve a process
of elimination, while others may include a trick question.
Children may also need other strategies such as making a list,
constructing a drawing and finding a pattern.
Process of elimination
A problem may provide clues to help find a mystery number or shape. A good
starting point is a set of all the initial possibilities; this may be on a hundred
square, a number line or in the form of a set of shapes. The numbers can be
adjusted to suit the problem.
1
2
3
4
5
6
7
8
9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Which numbers can’t our mystery number be?
Cross out those it cannot be. If it is a two-digit number, cross out all the
single-digit numbers. It is an even number, cross out the remaining odd numbers,
and so on.
Trick questions
The answer to some problems is much easier to find than it first seems. When faced
with problems incorporating numbers, children often assume they have to carry out
a calculation, whereas logical thinking may be all that is required.
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Reason logically
John is on a train journey from London to
Liverpool. He has been on the train for
30 minutes of a 3 hour journey; he has
travelled 38 miles and has 243 miles left
to go. On the return journey the train travels
at the same speed. How many miles will he
have left to travel if he has been travelling
for 2 hours 30 minutes?
There is a lot of irrelevant information in this question. After 30 minutes he had
travelled 38 miles, so with 30 minutes of the journey left he will have 38 miles
left to go.
38 miles
30 minutes
London
Liverpool
Some children need to see a diagram of this in order to clarify their
understanding. It could be illustrated using a number line and marking on the
distance and time.
Example teaching sequence 1
Find the mystery number by using the clues.
It is 2-digit number greater than 40.
It is 1 more than a square number.
It is not a multiple of 5.
What is it?
Key steps
• Consider the possibilities.
• Narrow down the possibilities.
• Check the answer fits all the criteria.
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Reason logically
Considering the possibilities
Explain to the children that the clues will lead to a mystery number.
Put the clues on an OHT or the board, but only reveal them one at a time.
Ask a child to read the first clue.
Which numbers does it include? Which numbers are not included?
Would 37 be included? Why/Why not?
Would 47 be included?
Narrowing down the possibilities
Reveal the second clue.
What numbers should we list on the board? (Square numbers greater than 40)
What is the first square number greater than 40? Which other square numbers are
greater than 40 but are still 2-digit numbers?
Write them on the board: 49, 64, 81.
What does the clue say? Which numbers are one more than square numbers
greater than 40? (50, 65, 82)
Explain that the next clue will give the answer.
What could the next clue be? If that were the next clue what would the
answer be?
Reveal the third clue.
Which of the numbers are multiples of 5? What do you think the answer is?
Checking the answer fits all the criteria
Write 82 on the board. Explain that it is important to go back and check that the
number fits all the criteria.
Is 82 a 2-digit number? Is it greater than 40? Is it one more than a
square number? Which? Is it a multiple of 5? Does the number fit all the criteria?
Could it be any other number? Why not?
Look at 65 and 50. Ask children in pairs to write a new third clue which would give
one of these answers instead. Take some suggestions.
Plenary Unpicking common difficulties
Interpreting the clues Children need to practise finding numbers that meet
given criteria so they become familiar with the language associated with
properties of number. They need to understand what factors, multiples,
primes and squares are as well as odd and even numbers and ‘greater than’ and
‘less than’. This could be done in a plenary with children showing answers on
number fans or digit cards.
Devising relevant clues Children need to experience not only finding numbers
that meet given criteria, but also writing clues. Ask children each to write one clue
to a mystery number. Take several of the clues together: can the number be
guessed from these clues?
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Reason logically
Example teaching sequence 2
A car travels at a speed of 80 miles
per hour. It takes the car two hours to reach
its destination. After 45 minutes of the
journey the car reaches a railway crossing
where it has to stop to wait for a train
to go by. The train travels at an average
speed of 75 miles per hour throughout its
journey, and is three-quarters of an hour
into a 150-mile journey. Which form of
transport will reach its destination first?
Key steps
• List the key information.
• Carry out necessary calculations.
• Use information collected to solve the problem.
Listing the key information
Ask children to read the question in pairs and jot down all the key information.
Show the question on an OHT or the board. Ask children to come and underline
the key information in the question.
How could we show all this information more clearly? How could it be displayed
in a table?
Draw a table on the board with column headings Car and Train.
Car
Train
Speed
Time travelled
Time to destination
Distance to destination
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Reason logically
Point to each cell in turn.
Do we know how fast the car is travelling?
Write 80 mph in the table.
Continue asking questions and writing the relevant information in the table.
How fast is the train travelling? (75 mph)
For how long has the car been travelling? (45 mins)
For how long has the train been travelling? (Three-quarters of an hour)
Does the question say how long it will take the car to reach its destination?
(Yes, 2 hours take away 45 minutes already travelled)
Point to bottom right cell of the table.
How far is the train travelling? (150 miles)
Explain to the children that some calculations are needed to complete the rest
of the table.
Carrying out necessary calculations
How can we use the information in the table to work out the total time of the
train journey?
Ask children to jot down possible calculation on mini-whiteboards or paper.
Explain that we know the journey is 150 miles long and that the train is travelling
at 75 mph, so in 1 hour it will travel 75 miles and in 2 hours it will travel 150 miles.
Fill in the relevant cell in the table.
Using the information collected to solve the problem.
What does the question ask?
Read back the final part of the question.
Do you have enough information to answer the question? How long does each
journey take? (Two hours). What do you notice about the length of the journey so
far for each vehicle? How long has each vehicle got until the journey is complete?
(1 hour 15 mins)
Which form of transport will reach its destination first? (They will both reach their
destination at the same time)
Plenary Unpicking common difficulties
Listing key information Identifying key information is a vital problem solving skill.
In some ‘reasoning logically’ problems not all the information is relevant.
It is useful to ask groups of children to write each piece of information on a Post-it
and then discuss as a class which are relevant to the problem and why.
Having a clear image of the problem Some children need a diagram to
understand a problem like this. Ask children to draw their own picture to illustrate
the problem, and show some of the pictures to the class. Discuss which best
illustrate the problem, and why.
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