1 Word Problems, Investigations & Thinking Skills Mathematics CPD course 04/05 Nigel Davies

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Word Problems, Investigations & Thinking Skills

Mathematics CPD course 04/05

Nigel Davies

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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

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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?

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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?

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Not a bolt-on!

Mechanical skills introduced, practised & developed through a

variety of concepts

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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.

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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?

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Problem Solving StrategiesSkillsWhat can youDO?

Developingskills such as:• counting• partitioning and recombining numbers

IdeasWhat do youREALISE?

Buildingawareness of:• the number system• the relative value of numbers• number relationships

FactsWhat can youREMEMBER?

Recalling factssuch as:• halves and doubles• number bonds• multiplication facts

Construct a strategy

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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

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… 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

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Types of problems Single operation + or – Single operation x or Mixed operations Choice of relevant data Redundant data

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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?

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… to a number line …

93p

55p

93p55p

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Single operationSam has saved 93p, Amy has 55p. How much

more money does Sam have than Amy?

55 60

+5

93

+3

90

+30

38p more

Ideas? Easier numbers? Pictures? I understand that …

⇨ What do I know? Facts

⇨ Which operation? What can I do? Skills

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Another single operation432 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

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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”

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432

-300 (20 buses x 15)

132

- 90 (6 buses x 15)

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-30 (2 buses x 15)

12 (people left)

So we need 29 buses or 28 buses and some cars!

432 15

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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?

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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

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Take away

A box holds 70 biscuits. How many biscuits are left if you eat 17?

70

-17

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Count onA family sets off for a 524 miles drive. After 267

miles, how much further do they have to go?

524267

+33 +200+24

300 500

33 + 200 + 24 = 257 miles

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Calculator Problems

185 people go to the school concert.

They pay £1.35 each.

How much ticket money is collected?

Ideas? Easier numbers? Pictures? I understand that …

⇨ What do I know? Facts

⇨ Which operation? What can I do? Skills

Programmes cost 15p each.

Selling programmes raises £12.30

How many programmes are sold?

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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

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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

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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?

Ideas? Easier numbers? Pictures? I understand that …

⇨ What do I know? Facts

⇨ Which operation? What can I do? Skills

Train fares cost £14.50. I have £52. How many people can I take on the

journey?

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Task Devise four simple problems, one

per operation, in different contexts

For example, money, measurement etc.

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45 ÷ 6 : Easy!

45 sweets are to be shared between 6 pupils. How many sweets will each pupil receive & how many will be left over?

£45 is to be shared between 6 pupils. How much will each pupil receive?

7 sweets each with 3 left over

£7.50

How many games of rounders, each lasting 6 minutes, can be played during a 45 minute P.E. lesson? 7

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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?

A piece of string 45cm long is to be cut into 6 equal pieces. What will be the length of each piece?

8 minibuses

7.5 cm

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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

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Multiple Operations Contd.

Welcome to Kingston Theme Park Entry:-

Child £8.50Adult £13.75Family £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

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Extracting relevant information

£0.87 £0.68

£0.35

£0.93£0.79

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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.

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Task

Devise three simple problems : One should involve mixed operations

Another, a choice of relevant data

A 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

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What else can I find out from

the problem?

What do I have to find out?

What do I do now?

What maths should I use?

Would pictures

or number

lines help?Have I got

an answer?

Is my answer

reasonable?

Have I solved

the problem

?

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What else can I find out from

the problem?

What do I have to find out?

What do I do now?

What maths should I use?

Would pictures

or number

lines help?Have I got

an answer?

Is my answer

reasonable?

Have I solved

the problem

?

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Watching the video …

Investigations in KS1

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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

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Checklist (contd.)

Extend your table

Find a rule

Check the rule

Answer the original question

Explore something new

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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?

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Working through the activity

1-Snake 2-Snake 3-Snake

4-Snake 5-Snake

3 Straws 5 Straws 7 Straws

9 Straws 11 Straws

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Working on …

Number of triangles

Number of straws

1 3

2 5

3 7

4 9

5 11

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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?

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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.

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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?

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Handshakes

Everyone in this room shakes hands with everyone else.

How many handshakes will there be?

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Watching the video …

Investigations in KS2

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“Out of the box”

Creativity

Imagination

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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?”

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3-Part Processing : 1

Explore the task

Make a guess

Extract relevant information

Organise

Plan

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3-Part Processing : 2

Search for patterns & relationships

Record

Analyse

Find properties of results

Adjust guess

Hypothesise further

Make links

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3-Part Processing : 3

Check results

Look back

Communicate

Extend

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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?

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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?

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Not 64!

How many squares are there on a chess board?

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Puzzle or Challenge?

4 3

24

7 2

28

3 8

Using the first 2 circles as examples, fill in the empty segment

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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?

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Brainteaser 2

Move one coin to form two rows with four coins in each row.

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Brainteaser 3

With nothing to hand except a three pint jug & a five pint jug, how can you measure exactly one pint of water?

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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?

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Brainteaser 5

Can you write down five odd numbers so that added together they make 14?

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1

1

1

+ 11

Brainteaser 5 : solution

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Brainteaser 6

Move one of the numbers so that the totals of the two columns are equal.

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Brainteaser 6

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Brainteaser 6

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Brainteaser 7

Add one straight line to the following to change it into a correct number sentence.

5 + 5 + 5 = 550

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Brainteaser 7

Add one straight line to the following to change it into a correct number sentence.

5 + 5 + 5 = 550

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Brainteaser 8

What is the next number in this list :

1 2 4 7 12 …