Transcript Teaching Problem-solving Skills

Word Problems, Investigations &
Thinking Skills
Mathematics CPD course
04/05
Nigel Davies
1
An actual question

“A shepherd owns 19 sheep and 13 goats. How
old is the shepherd?”
Answer given was 32

“A 27-year-old farmer owns 25 sheep, 30 cows,
12 pigs and 10 goats. How old is the farmer?”
Answers given included 77 and 104
2
A problem!
The anchovy is a small sea-fish, much used by
way of sauce or seasoning. It is caught in the
months of May, June and July, in the
Mediterranean, on the coasts of Catalonia, in
Spain, and the late province of Provence, now
forming the departments of Var, the Lower Alps,
and the Mouths of the Rhone, in France; and
particularly near the small island of Gorgona, in
the Tuscan sea, adjacent to Leghorn. In years of
plenty 20,000 quintals of anchovies have been
exported a year from Malaga. What is the value
of three pounds of anchovies at £2 per pound?
3
A problem!
The anchovy is a small sea-fish, much used by
way of sauce or seasoning. It is caught in the
months of May, June and July, in the
Mediterranean, on the coasts of Catalonia, in
Spain, and the late province of Provence, now
forming the departments of Var, the Lower Alps,
and the Mouths of the Rhone, in France; and
particularly near the small island of Gorgona, in
the Tuscan sea, adjacent to Leghorn. In years of
plenty 20,000 quintals of anchovies have been
exported a year from Malaga. What is the value
of three pounds of anchovies at £2 per pound?
4
Not a bolt-on!
Mechanical skills introduced,
practised & developed through a
variety of concepts
5
You’re never too young!
Build word problem-solving skills from an early
age by practising number mechanics in as many
different contexts as possible.
6
Key Questions
What is the problem asking?
 What information do I need to
answer the question?
 What operations should I use to
solve it?
 Have I answered the original
question?
 Is my answer sensible?

7
Problem Solving
Strategies
Skills
Ideas
Facts
What can
you
DO?
What do you
REALISE?
What can you
REMEMBER?
Building
awareness of:
• the number system
• the relative value of
numbers
• number relationships
Recalling facts
such as:
• halves and
doubles
• number
bonds
• multiplication
facts
Developing
skills such as:
• counting
• partitioning
and
recombining
numbers
Construct a
strategy
8
Advice to pupils

Read and make sense of the problem

Recognise keywords, relevant information
and redundant information

Find parts of the problem that can be
handled

Put in easier numbers if this would help
9
… More Advice
Decide which # operation(s) to
perform and in which order
 Try to find an efficient way of
calculating
 Change any measurements to the
same units before calculating
 Check that your answer makes
sense

10
Types of problems
Single operation + or –
 Single operation x or 
 Mixed operations
 Choice of relevant data
 Redundant data

11
Act out the problem

Whenever possible, act out the
situation the problem describes :
Sam has saved 93p, Amy has 55p. How much
more money does Sam have than Amy?
12
… to a number line …
93p
55p
55p
13
93p
Single operation
Sam has saved 93p, Amy has 55p. How much
more money does Sam have than Amy?
Ideas? Easier numbers? Pictures? I understand that …
⇨ What do I know? Facts
⇨ Which operation? What can I do? Skills
+5
55
+30
60
+3
90
93
38p more
14
Another single operation
432 school children are going on an outing.
If each bus takes 15 passengers, how many buses
will be needed?
Ideas? Easier numbers? Pictures? I understand that …
⇨ What do I know? Facts
⇨ Which operation? What can I do? Skills
15
Ideas? Facts? Skills?
40 school children are going on an outing.
If each bus takes 8 passengers, how many buses
will be needed?
Ideas? Group the 40 into 8s ⇨ “How many 8s in 40?”
⇨ “How many 15s in 432?”
⇨ “I can take groups of 15s from 432”
⇨ “This will be the number of buses”
Facts? 15 x 10 = 150 ⇨ 15 x 20 = 300
Skills? “I can find other groups of 15 to take away”
16
432  15
432
-300
132
(20 buses x 15)
- 90
(6 buses x 15)
42
-30
12
(2 buses x 15)
(people left)
So we need 29 buses or 28 buses and some
cars!
17
Take-away and Count-on
Look at the word problems on the
following page.
Which one implies taking away and which
one implies counting on?
18
Take away or Count on?
A box holds 70 biscuits. How many biscuits are
left if you eat 17?
A family sets off for a 524 miles drive. After 267
miles, how much further do they have to go?
Ideas? Easier numbers? Pictures? I understand that …
⇨ What do I know? Facts
⇨ Which operation? Which method? What can I do? Skills
19
Take away
A box holds 70 biscuits. How many biscuits are
left if you eat 17?
-17
53
20
70
Count on
A family sets off for a 524 miles drive. After 267
miles, how much further do they have to go?
+200
+33
267
300
33 + 200 + 24 = 257 miles
21
+24
500
524
Calculator Problems
185 people go to the school concert.
They pay £1.35 each.
How much ticket money is collected?
Programmes cost 15p each.
Selling programmes raises £12.30
How many programmes are sold?
Ideas? Easier numbers? Pictures? I understand that …
⇨ What do I know? Facts
⇨ Which operation? What can I do? Skills
22
How much ticket money?
8 people go to the school concert.
They pay £2
each.
How much ticket money is collected?
Ideas? 8 lots of £2 ⇨ 185 lots of £1.35
Facts? “I know ‘lots of’ means ‘multiply’ ”
Skills? Using the calculator to multiply decimals
Interpreting calculator displays regarding money
23
How many programmes?
Programmes cost 10p each.
Selling programmes raises £1.80
How many programmes are sold?
Ideas? “How many 10p in £1.80” ⇨ “How many 15p in £12.30”
⇨ “I would need to change £1.80 into pence”
⇨ “I need to change £12.30 into pence”
Facts? “I know ‘how many in …’ means ‘divide’ ”
Skills? Using the calculator to divide whole numbers
Interpreting calculator displays
24
More problems
To make a box pieces of wood 135mm long have to be cut
from a 2.5m length.
How many lengths of wood can be cut?
Train fares cost £14.50. I have £52.
How many people can I take on the journey?
Ideas? Easier numbers? Pictures? I understand that …
⇨ What do I know? Facts
⇨ Which operation? What can I do? Skills
25
Task

Devise four simple problems, one
per operation, in different contexts
For example, money, measurement
etc.
26
45 ÷ 6
:
Easy!
How many games of rounders, each lasting 6
minutes, can be played during a 45 minute P.E.
lesson?
7
45 sweets are to be shared between 6 pupils.
How many sweets will each pupil receive & how
many will be left over?
7 sweets each with 3 left over
£45 is to be shared between 6 pupils. How much
will each pupil receive?
£7.50
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45 ÷ 6
:
Still Easy!
45 pupils are going on a school trip. Each minibus
can carry 6 pupils. How many minibuses are
needed?
8 minibuses
A piece of string 45cm long is to be cut into 6
equal pieces. What will be the length of each
piece?
7.5 cm
28
Multiple operations
Jon has £10. Saturday afternoon he spent
£3.50 going to the cinema and £2.99 on
burger and chips. How much change
does he have after his outing?
Ideas? Easier numbers? Pictures? I understand that …
⇨ What do I know? Facts
⇨ Which operation? What can I do? Skills
29
Multiple Operations
Contd.
Welcome to Kingston Theme Park
Entry:Child
£8.50
Adult £13.75
Family £30.50
(2 adults and 2 children)
Alun and his family are going to the Theme Park. There are 3
adults and 4 children in the group. What is the entry fee for
the group?
Ideas? Easier numbers? Pictures? I understand that …
⇨ What do I know? Facts
⇨ Which operation? What can I do? Skills
30
Extracting relevant
information
£0.68
£0.87
£0.35
£0.79
£0.93
31
Ignoring redundant
information
This morning I woke up at 6 a.m. in order to
take my 2 dogs for a walk. We walked for 3
miles & then arrived back home at 7.15 a.m. for
breakfast. I had a bowl of porridge (it says on
the box that a typical bowl of porridge
contributes 300 kcal) & each dog had 125g of
dog food.
I then walked the 6 miles to work, burning off
450kcal in the process.
32
Task
Devise three simple problems :
One
should involve mixed operations
Another,
A
a choice of relevant data
third, redundant data
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Suggested Approaches

Ask children to make up some problems
for themselves or for other children

Give the answer to a problem, then
make up questions with that answer

Give one or more operations and some
numbers, then make up story problems

Choose a topic, e.g. sports day, then
make up problems on the topic
34
What do
I have to
find out?
What do I do
now?
What maths
should I use?
Have I
solved
the
problem
?
Is my
answer
reasonable?
Have I got
an
answer?
35
What else
can I find
out from
the
problem?
Would
pictures
or
number
lines
help?
What do
I have to
find out?
What do I do
now?
What maths
should I use?
Have I
solved
the
problem
?
Is my
answer
reasonable?
Have I got
an
answer?
36
What else
can I find
out from
the
problem?
Would
pictures
or
number
lines
help?
Watching the video …
Investigations in KS1
37
An Investigations checklist
Understand the problem
Try smaller or easier examples
Use a systematic approach
Use diagrams
Make a table of results
Look for patterns
38
Checklist (contd.)
Extend your table
Find a rule
Check the rule
Answer the original question
Explore something new
39
An activity
This is a 3-snake.
It takes seven straws to make this snake.
How many straws would it take to make a 100-snake?
40
Working through the
activity
1-Snake
3 Straws
4-Snake
9 Straws
2-Snake
5 Straws
3-Snake
7 Straws
5-Snake
11 Straws
41
Working on …
Number of triangles
Number of straws
1
3
2
5
3
7
4
9
5
11
42
Choosing investigations
Does the investigation have a low access point?
Is differentiation built in?
Will it practise & apply previously learned skills?
… or will it be used to introduce a new topic?
Are there opportunities for pupils to extend the
investigation?
43
The magic toy box
For your birthday you have been given a magic toy box.
Each day there are new presents inside it.
On day 1 there are 2 presents
On day 2 there are 4 presents
On day 3 there are 6 presents
On day 4 there are 8 presents
Guess how many presents there will be in the box on day
10.
Find out whether your guess was right.
44
Frogs
These frogs want to swap places so that the girl frogs end
up on the boys lily pads & vice versa.
A frog can make one of two moves :
A slide to an adjacent empty lily pad.
A jump over one other frog on to an empty lily pad.
What is the least number of moves required to swap places?
45
Handshakes
Everyone in this room shakes hands with
everyone else.
How many handshakes will there be?
46
Watching the video …
Investigations in KS2
47
“Out of the box”
Creativity
Imagination
48
Puzzles  Challenges
Teacher as ‘prompt’
Higher-order questioning :
“What would happen if …?”
“How can you tell that …?”
“Why does that work for this example, but not
that one?”
49
3-Part Processing : 1
Explore the task
Make a guess
Extract relevant information
Organise
Plan
50
3-Part Processing : 2
Search for patterns & relationships
Record
Analyse
Find properties of results
Adjust guess
Hypothesise further
Make links
51
3-Part Processing : 3
Check results
Look back
Communicate
Extend
52
Tom, Dick & Harry
Tom, Dick & Harry have £575 to be divided
between them.
They agree to divide it so that Tom receives £19
more than Dick, & Dick receives £17 more than
Harry.
How much does each man receive?
53
Kingston Theme Park
Revisited
A coach operator
organised a trip to the
theme park.
The travel prices were
£3.25 for adults &
£1.70 for children.
He sold 50 tickets for a total amount of £128.40.
How many children’s tickets did he sell?
54
Not 64!
How many squares are there on a chess board?
55
Puzzle or Challenge?
4
3
24
7
2
3
8
28
Using the first 2 circles as examples, fill in the empty segment
56
Brainteaser 1
A spider falls down a 30 metre well.
Each day, it climbs 3m up the wall, but then slides
back 2m.
How many days will it take for the spider to get out
of the well?
57
Brainteaser 2
Move one coin to form two rows with four coins in
each row.
58
Brainteaser 3
With nothing to hand except a three pint jug & a
five pint jug, how can you measure exactly one pint
of water?
59
Brainteaser 4
A farmer’s wife took 99 eggs to market in four
baskets, carrying an odd number of eggs in each
basket.
How was she able to do that?
60
Brainteaser 5
Can you write down five odd
numbers so that added together
they make 14?
61
Brainteaser 5 : solution
1
1
1
+ 11
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Brainteaser 6
Move one of the numbers
so that the totals of the
two columns are equal.
63
Brainteaser 6
64
Brainteaser 6
65
Brainteaser 7
Add one straight line to the following to change it
into a correct number sentence.
5 + 5 + 5 = 550
66
Brainteaser 7
Add one straight line to the following to change it
into a correct number sentence.
5 + 5 + 5 = 550
67
Brainteaser 8
What is the next number in this list :
1
2 4 7 12 …
68