Mathematical challenges for able pupils in Key Stages 1 and 2

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Mathematical challenges
for able pupils
in Key Stages 1 and 2
Department for Education and Employment
Sanctuary Buildings
Great Smith Street
Westminster
London SW1P 3BT
© Crown copyright 2000
Extracts from this document may be reproduced for non-commercial
educational or training purposes on condition that the source is acknowledged
ISBN 0 19 312342 8
Illustrations by Graham Round
Contents
Introduction
4
Some questions answered
5
How should we organise within the school?
5
How can I adapt my termly planning?
5
How can I use the ‘extra’ week each term?
8
How can I use the three-part lesson?
8
Where can I find enrichment activities to develop
pupils’ thinking skills?
9
Which National Numeracy Strategy materials support
the teaching of able pupils?
10
Where else can I get help?
11
Activity examples
12
Palindromic numbers (Year 4)
12
Alternative multiplication (Year 6)
14
Puzzles and problems for Years 1 and 2
15
Puzzles and problems for Years 3 and 4
41
Puzzles and problems for Years 5 and 6
69
Solutions
101
Introduction
This book supplements the National Literacy and Numeracy Strategies: guidance on
teaching able children, published in January 2000. Its purpose is to help primary
teachers cater for pupils who are more able in mathematics and likely to exceed the
expected standards for their year group.
Mathematically able pupils are in every school and among all ethnic and
socio-economic groups.
They typically:
◆ grasp new material quickly;
◆ are prepared to approach problems from different directions and persist in finding
solutions;
◆ generalise patterns and relationships;
◆ use mathematical symbols confidently;
◆ develop concise logical arguments.
The Framework for teaching mathematics from Reception to Year 6 covers the
National Curriculum for Key Stages 1 and 2 from pre-level 1 up to level 4 and parts of
level 5. The draft Framework for teaching mathematics: Year 7, published in March
2000, is based mainly on work at level 5. The yearly teaching programmes in the
Framework are expressed as ‘targets for the majority of pupils in the year group’.
Many able pupils will progress more quickly through these programmes and will need
extension and enrichment activities in mathematics.
This book addresses class organisation, planning and teaching through answers to
commonly asked questions.
The puzzles and problems in the second part of this book can be photocopied for use
in schools in England participating in the National Numeracy Strategy. The puzzles
and problems are also available on the National Numeracy Strategy website (see
page 9).
4
Mathematical challenges for able pupils
© Crown copyright 2000
Some questions answered
How should we organise within the school?
Within the class
You will probably teach able pupils in their own class for their daily mathematics
lesson. They will cover the same topics as their peers but at a level to match their
abilities. You can stretch them through differentiated group work, harder problems for
homework and extra challenges – including investigations using ICT – which they can
do towards the end of a unit of work when other pupils are doing consolidation
exercises. The planning and structure of the National Numeracy Strategy address the
needs of all pupils and help you to manage classes with wide-ranging attainment
groups. Each pupil, very able or less able, needs to be part of one of these groups for
at least some of the time and not restricted totally to individual working.
With an older year group
Pupils who are exceptionally gifted in many subjects, and who are sufficiently mature,
may be promoted to work with an older age group. For example, you could timetable
Year 3 and Year 4 mathematics lessons at the same time. An exceptionally gifted pupil
in Year 3 could be taught the subject with the Year 4 class and benefit from discussion
with other pupils working at a similar level.
Setting
Larger schools with parallel classes sometimes deal with a range of attainment by
organising ‘ability sets’ for mathematics lessons. The advantage is that your planning
can be easier if the attainment gap in a class is not too wide. You could set across,
say, Years 5 and 6, if both years are timetabled for their mathematics lessons at the
same time, although you need to ensure that when Year 5 pupils move into Year 6
they do not simply repeat the previous year’s activities. Any setting arrangements
need to be flexible to allow easy transfer of pupils between sets. The success of
setting depends on very careful monitoring, close teamwork and co-operative planning
among teachers to make sure that expectations for all pupils are suitably high and that
lower expectations are not justified simply because pupils are in a ‘lower set’.
How can I adapt my termly planning?
In Key Stage 1, the aim is to provide a firm foundation in mathematics for all pupils.
The needs of able pupils are best served through an accelerated programme,
spending the same amount of time as other pupils, but going further with each topic.
This approach should be supplemented by a more investigative approach to learning.
The table overleaf shows how part of an autumn term plan for an ‘average’ Year 2
class has been modified to cater for able pupils, by including objectives from the
teaching programmes for Years 3 and 4. Each unit of work concentrates on the same
topic to help you to manage the necessary differentiation. Enrichment activities
encourage pupils to develop their skills in problem solving and reasoning. The
suggestions in the plan overleaf have been drawn from the puzzles and problems in
the second part of this book.
© Crown copyright 2000
Mathematical challenges for able pupils
5
Medium-term plan: Year 2
Part of autumn term with extension and enrichment work
Unit
(days)
Topic
Objectives: children will be taught to…
1 (3)
Counting, properties of numbers and
number sequences
Extension
Objectives from Y3/4
Say the number names in order to at
least 100.
Enrichment
suggestions
Snakes and
ladders (no. 4)
Count reliably up to 100 objects by
grouping them, for example in tens.
Estimate up to 100
objects.
Count on or back in ones or tens from any
two-digit number.
Extend to three-digit
numbers.
Recognise two-digit multiples of 10.
Recognise three-digit
multiples of 10.
Cross-road
(no. 17)
Count in hundreds from and back to zero.
2–4 (15) Place value, ordering, estimating, rounding
Read and write whole numbers from 0 to 100
in figures and words.
Read/write numbers
to 1000/10 000.
Know what each digit in a two-digit number
represents, including 0 as a place-holder.
Extend to three-digit
numbers.
Partition two-digit numbers into a multiple of ten
and ones (TU). Use the = sign.
Partition three-digit
numbers.
Ben’s numbers
(no. 24)
Say the number that is 1 or 10 more or less
than any given two-digit number.
Understanding addition and subtraction
Understand the operations of addition and
subtraction; recognise that addition can be
done in any order, but not subtraction.
Number lines
(no. 11)
Use the +, – and = signs to record mental
calculations in a number sentence.
Mental calculation strategies (+/–)
Put the larger number first.
Count on or back in tens or ones.
Count on/back in 100s.
Identify near doubles, using doubles already
known.
Add/subtract 9 or 11 by adding/subtracting 10
and adjusting by 1.
Extend to 19, 29…;
21, 31…; 18, 22…
Money and ‘real life’ problems
Recognise all coins. Find totals.
Give change.
Choose an appropriate operation and calculation
strategy to solve simple word problems.
Solve word problems.
Ride at the fair
(no. 8)
Explain methods orally…
…and in writing.
Gold bars (no. 7)
Making decisions, checking results
Check sums by adding in a different order.
6
Mathematical challenges for able pupils
© Crown copyright 2000
In Key Stage 2, the accelerated programme can continue, as well as including more
challenging problems and extended pieces of work. Termly plans should still ensure
that able pupils are taught a broad, balanced mathematics curriculum.
The table below illustrates part of a ‘typical’ Year 6 termly plan for mathematics with
enhanced provision for able pupils. The extra objectives are drawn from the Year 7
draft Framework. The programme has been supplemented with enrichment activities
that develop higher order thinking and problem solving skills. These ‘challenges’ are
best linked to the main class topic.
The work on page 14 of this book illustrates how the main Year 6 teaching programme
on multiplication can be supplemented for able pupils. While most pupils are
consolidating their skills in using a written method for multiplication, able pupils might
investigate other written methods for long multiplication.
Medium-term plan: Year 6
Part of autumn term with extension and enrichment work
Unit
(days)
Topic
Objectives: children will be taught to…
Extension
Objectives from Y7
1 (3)
Place value, ordering and rounding
Multiply and divide decimals by 10 or 100, and
integers by 1000, and explain the effect.
Understand and use
decimal notation and
place value.
Enrichment
suggestions
Using a calculator
Develop calculator skills and use a calculator
effectively.
2–3 (10) Understanding multiplication and division
Understand and use the relationships between
the four operations, and the principles of the
arithmetic laws.
Millennium
(no. 81)
Express simple functions
at first in words and then
in symbols, and use
simple function machines.
Maze
(no. 62)
Mental calculation strategies (× and ÷)
Use related facts and doubling or halving:
e.g. halve an (even) number, double the other;
multiply by 25, e.g. by ×100, then ÷4.
Extend mental methods (to decimals).
Shape puzzle
(no. 72)
Make five
numbers
(no. 61)
Pencil and paper procedures (× and ÷)
Approximate first. Use informal pencil and paper
methods to support, record or explain × and ÷.
Extend written methods to ThHTU × U and
short multiplication involving decimals.
Extend to decimals
with 2 d.p.
Alternative
multiplication
(see p. 14)
Money and ‘real life’ problems
Use all four operations to solve money or
‘real life’ problems.
Spendthrift
(no. 79)
Choose appropriate operations/calculation
methods. Explain working.
Franco’s fast
food (no. 67)
Making decisions, checking results
Check by estimating.
Use inverse operation, including with a calculator.
© Crown copyright 2000
Flash Harry
(no. 64)
Mathematical challenges for able pupils
7
How can I use the ‘extra’ week each term?
For all year groups the optional termly planning grids leave a week unallocated each
term. This ‘extra’ week can be used in different ways. Some pupils may need to
consolidate and develop a previous piece of work. Able pupils could, after an
introduction by you, do a sustained piece of extension work. This might involve some
research and investigation, and could be linked to the main teaching programme for
the class or could be a new topic. It could draw on subjects other than mathematics.
As an example, the work on palindromic numbers on pages 12–13 of this book might
be suitable for Year 4 pupils.
How can I use the three-part lesson?
In the oral/mental part of the lesson, you can direct some questions towards the most
able pupils, just as you can direct some specifically towards the children who find
mathematics difficult. Able pupils can also contribute by suggesting and explaining
alternative methods of calculation.
In the main part of the lesson you will often introduce a new topic with some direct
teaching of the whole class. You will consolidate previous ideas and develop and use
the correct mathematical language. For able pupils, the amount of practice and
consolidation needed is less than that required by other pupils. Within whole-class
teaching, you can set different tasks for pupils to undertake, for example:
◆ a common task, starting from the common experience of pupils, leading to different
outcomes – this is typical of open investigations;
◆ a stepped task that helps pupils build on their own learning strategies – each step
needs to be relevant and purposeful, and able pupils can omit earlier steps;
◆ separate tasks for each group of pupils, but linked to a common theme.
You should give all pupils opportunities to apply their mathematical knowledge. Able
pupils can often move quickly beyond basic knowledge and skills and begin to use
these in a range of contexts. Problems need not involve difficult mathematics but may
require insight, reasoning and higher order thinking skills in order to reach a solution.
The plenary session gives you opportunities to extend as well as consolidate work.
Methods of solution can be compared and explanations shared.
Homework can provide the opportunity for pupils to tackle challenging questions and
puzzles. The results can form the basis of the next lesson with either the whole class
or a group. Opportunities also exist for pupils to read about mathematical topics.
In all parts of the lesson, the quality of questioning is crucial in helping pupils develop
mathematical ideas and improve their thinking skills. The National Numeracy Strategy
Mathematical vocabulary contains guidance on types of questioning appropriate to all
parts of the mathematics lesson. The range of questioning should include recalling
and applying facts, hypothesising and predicting, designing and comparing
procedures, interpreting results and applying reasoning. You can use some open
questions to allow more pupils to respond at their own level. Such questions often
provide a greater challenge for able pupils, who can be asked to think of alternative
solutions and, in suitable cases, to list all the different possibilities: ‘Can you suggest
another method you might have used?’ ‘Would it work with different numbers?’ ‘How
do you know you have included all the possibilities?’
8
Mathematical challenges for able pupils
© Crown copyright 2000
Where can I find enrichment activities to develop pupils’
thinking skills?
Puzzles and problems in this book
The second part of this book contains puzzles and problems. These are accessible to
a wide range of pupils. There are three sections covering Years 1 and 2, Years 3 and
4, and Years 5 and 6. The problems are intended to challenge pupils and extend their
thinking. While some of them may be solved fairly quickly, others will need
perseverance and may extend beyond a single lesson. Pupils may need to draw on a
range of skills to solve the problems. These include: working systematically, sorting
and classifying information, reasoning, predicting and testing hypotheses, and
evaluating the solutions.
Many of the problems can be extended by asking questions such as: ‘What if you tried
three-digit numbers?’ ‘What if there were more boxes?’ ‘What if you used triangles
instead of squares?’ Problems can also be extended by asking pupils to design similar
problems of their own to give to their friends or families.
Learning objectives appropriate to each problem are indicated so that you can target
problems by integrating them into your main teaching programme.
Solutions are given at the end of the book.
Extended tasks, problems or investigations within/beyond the main
curriculum
Resources that schools may find useful include:
◆ books of investigations and ‘open’ problems;
◆ mathematics magazines and booklets produced for pupils;
◆ mathematical posters and topic books that stimulate discussion and investigation;
◆ computer access to the Internet;
◆ calculators to solve challenging and investigative activities;
◆ software;
◆ ‘general’ books on mathematics, e.g. history of mathematics, biographies of
mathematicians.
Competitions
The Mathematical Association introduced a Primary Mathematics Challenge in
November 1999. This competition will run annually from 2000.
Websites
The puzzles and problems in this book are available on the National Numeracy
Strategy website:
www.standards.dfee.gov.uk/numeracy/
The Maths Year 2000 website contains puzzles and problems, and links to a number
of other mathematical websites:
www.mathsyear2000.org
The nrich website also provides a regular supply of problems:
www.nrich.maths.org.uk
© Crown copyright 2000
Mathematical challenges for able pupils
9
Which National Numeracy Strategy materials support the
teaching of able pupils?
Framework for teaching mathematics from Reception to Year 6
The organisation of teaching objectives and the supplement of examples signal the
progression in topics, clarifying the links between the teaching programmes of each
year group. Teachers who are planning work for able pupils should give particular
attention to the introductory section on laying the foundations for algebra.
Framework for teaching mathematics: Year 7 (draft issued in March 2000)
This extends the original Framework and is based mainly on work at level 5.
Mathematical vocabulary
The introduction to this booklet contains useful references to questioning techniques
with examples of the types of question that help to extend children’s thinking.
Sample termly plans (on CD issued with December 1999 Professional
development materials 3 and 4)
These may help you to identify ‘What comes next?’ in a particular topic. By looking
ahead one term, two terms or even further, you can incorporate objectives into your
present plan as extension work.
Sessions from the five-day training course for intensive schools
◆ Problem solving with challenges and simplifications: This illustrates how
activities linked to problem solving and reasoning can challenge able pupils.
◆ Using a calculator: Although the activities are designed for teachers, some of
them can be used to extend able pupils.
◆ Laying the foundations for algebra: This contains a range of examples involving
reasoning and explanation.
◆ Fractions, decimals, percentages, ratio and proportion: Able pupils can
develop this work, especially the inter-relationships and examples involving ratio
and proportion.
◆ Shape and space: This gives some background work on transformations.
Reflections, translations and rotations are covered separately as well as an
introduction to combining transformations.
◆ Graphs and charts: Able pupils can be encouraged to tackle problems that require
data collection and analysis. Particular emphasis needs to be given to the
interpretation of results.
Professional development materials 3 and 4 (issued to all schools in
December 1999)
This pack covers many of the same topics as the five-day course materials. Chapters
that may be of particular relevance are:
◆ Solving word problems
◆ Fractions, decimals, percentages, ratio and proportion
◆ Shape and space
◆ Calculators
◆ Graphs and charts
10
Mathematical challenges for able pupils
© Crown copyright 2000
Where else can I get help?
Other sources of support include:
◆ Local education authority advisory services
◆ Local universities
◆ The Mathematical Association
259 London Road
Leicester
LE2 3BE
◆ The Association of Teachers of Mathematics
7 Shaftesbury Street
Derby
DE23 8YB
◆ The National Association of Able Children in Education
NAACE National Office
Westminster College
Harcourt Hill
Oxford
OX2 9AT
◆ The National Association of Gifted Children
NAGC
Elder House
Milton Keynes
MK9 1LR
◆ The Royal Institution
21 Albemarle Street
London
W1X 4BS
© Crown copyright 2000
Mathematical challenges for able pupils
11
Activity examples
Palindromic numbers (Year 4)
Meanings
Look up the meaning of ‘palindrome’ in a dictionary.
Words can be palindromic, for example ‘madam’.
Dates can be palindromic too, for example 17.8.71.
Can you think of some more examples?
Palindromic numbers
8, 33, 161, 222 and 2998992 are examples of palindromic numbers.
◆ How many palindromic numbers are there between:
0 and 100?
100 and 200?
200 and 300?
0 and 1000?
1000 and 1100? 1100 and 1200?
300 and 400?
1300 and 1400?
◆ Can you work out how many palindromic numbers there are between 0 and 2000?
What about between 0 and 10000?
Backwards and forwards
Start with a two-digit number, for example:
Reverse it and add the result to the original number:
The result is palindromic after one reversal.
32
32 + 23 = 55
55
Now try it with another two-digit number, such as:
Reverse it and add the result to the original number:
Reverse and add again:
This time the result is palindromic after two reversals.
57
57 + 75 = 132
132 + 231 = 363
363
◆ Can you find two-digit numbers that are palindromic after one reversal?
After two reversals? After three reversals? After more than three reversals?
The numbers 89 and 98 take 24 reversals!
◆ Investigate the same process with three-digit numbers.
12
Mathematical challenges for able pupils
© Crown copyright 2000
Continue the pattern
Continue each of these patterns.
In each case, describe what you notice.
◆ 1×9 + 2 =
12 × 9 + 3 =
123 × 9 + 4 =
and so on.
◆ 11 × 11 =
111 × 111 =
1111 × 1111 =
and so on.
◆ 11 × 11 =
11 × 11 × 11 =
11 × 11 × 11 × 11 =
and so on.
Questions with palindromic answers
Try to make up some questions with palindromic answers.
You might need to work out what the answers should be first!
Hints and solutions (for teachers)
One-digit palindromes:
1, 2, 3, …, 9 are palindromic, so there are 9 palindromic
one-digit numbers. (But some people might want to include 0
as well!)
Two-digit palindromes:
11, 22 and so on are palindromic, so there are 9 numbers.
Three-digit palindromes: 1◆1 where ◆ stands for the digits 0 to 9
2◆2
and so on.
There are 90 three-digit palindromes.
Four-digit palindromes:
between 1000 and 1100 there is only 1001,
between 1100 and 1200 there is only 1111,
and so on.
Between 1000 and 2000 there are 10 palindromic numbers.
Here are some other calculations that have palindromic answers:
22 × 11
© Crown copyright 2000
33 × 11
44 × 11
407 × 3
1408 × 3
143 × 7
Mathematical challenges for able pupils
13
Alternative multiplication (Year 6)
Look at these methods for long multiplication.
Can you work out what is happening? Why do they work?
Try them for yourself using other numbers.
Which method do you like best?
Multiplication method 1
27 × 43
1
2
4
8
16
× 43 = 43
× 43 = 86
× 43 = 172
× 43 = 344
× 43 = 688
So 27 × 43
43
86
344
688
=
1161
× 78 = 78
× 78 = 156
× 78 = 312
× 78 = 624
156
312
624
14 × 78
1
2
4
8
So 14 × 78
=
1092
Multiplication method 2
27 × 43
27
13
6
3
1
× 43
× 86
× 172
× 344
× 688
So 27 ×
43
43
86
344
688
=
1161
38 × 47
38
19
9
4
2
1
×
47
×
94
× 188
× 376
× 752
× 1504
So 38 ×
14
47
94
188
1504
=
1786
Mathematical challenges for able pupils
© Crown copyright 2000
Puzzles and problems
for Years 1 and 2
Four-pin bowling
Which pins must Joshua
knock down to score
exactly 5?
4
3
1
Find 2 different ways:
a. to score 5
b. to score 6
c. to score 7
Teaching objectives
1
16
Solve mathematical problems or puzzles.
Know addition and subtraction facts up to 10.
© Crown copyright 2000
2
Gob-stopper
Jade bought a gob-stopper.
It cost 6p.
She paid for it exactly.
Which coins did she use?
There are 5 different ways to do it.
Find as many as you can.
What if the gob-stopper cost 7p?
Teaching objectives
Solve mathematical problems or puzzles.
Know addition and subtraction facts up to 10.
Find totals, give change, and work out which coins to pay.
2
© Crown copyright 2000
17
Pick a pair
Choose from these numbers.
1
1.
4
2
8
Pick a pair of numbers.
Add them together.
Write the numbers and the answer.
Pick a different pair of numbers.
Write the numbers and the answer.
Keep doing it.
How many different answers can you get?
2. Now take one number from the other.
How many different answers can you get now?
Teaching objectives
3
18
Solve mathematical problems or puzzles.
Know addition and subtraction facts up to 10.
© Crown copyright 2000
Snakes and ladders
16
15
14
Home
9
10
11
8
1
7
2
13
12
6
5
3
4
Start
Your counter is on 9.
You roll a 1 to 6 dice.
After two moves you land on 16.
Find all the different ways you can do it.
Now think of other questions you could ask.
Teaching objectives
Solve mathematical problems or puzzles.
Count on from any small number.
4
© Crown copyright 2000
19
Bean-bag buckets
Dan threw 3 bean-bags.
Each bag went in a bucket.
More than one bag can go in a bucket.
1
3
Score 1
2
Score 3
Score 2
1.
What is the highest score Dan can get?
2. Find three ways to score 6.
3. Find three ways to score 9.
4. What other scores can Dan get?
Teaching objectives
5
20
Solve mathematical problems or puzzles.
Know addition facts up to 10.
© Crown copyright 2000
4
Score 4
Crossword
Write the answers to this puzzle in words:
ONE, TWO, THREE, …
1
2
3
4
5
Across
Down
1. 7 – 5
2. 3 + 4 – 6
3. 2 + 5 – 1
3. 9 – 2
4. 4 + 4 + 4
4. 11 – 4 + 3
5. 13 – 4
Teaching objectives
Solve mathematical problems or puzzles.
Use known number facts and place value to add and subtract mentally.
Read and write whole numbers.
6
© Crown copyright 2000
21
Gold bars
Pete is a pirate.
His gold bars are in piles.
He can move one or more bars at a time.
He made all the piles the same height.
He made just two moves.
How did he do it?
Teaching objectives
7
22
Solve mathematical problems or puzzles.
Explain methods and reasoning.
© Crown copyright 2000
Ride at the fair
Lucy had a ride at the fair.
Her Mum asked Lucy to pay less than 20p towards it.
Lucy paid exactly three coins towards the ride.
How much did Lucy pay her Mum?
Find different ways to do it.
Teaching objectives
Solve mathematical problems or puzzles.
Find totals, give change, and work out which coins to pay.
8
© Crown copyright 2000
23
Sum up
Choose from these four cards.
2
4
8 8
6
3
Make these totals:
9
10
11
12
13
14
15
What other totals can you make from the cards?
Teaching objectives
9
24
Solve mathematical problems or puzzles.
Know addition and subtraction facts to at least 10.
Add three small numbers mentally.
© Crown copyright 2000
Birds’ eggs
You may need 19 counters.
Three birds laid some eggs.
Each bird laid an odd number of eggs.
Altogether they laid 19 eggs.
How many eggs did each bird lay?
Find different ways to do it.
Teaching objectives
Solve mathematical problems or puzzles.
Recognise odd and even numbers.
Add three small numbers mentally.
10
© Crown copyright 2000
25
Number lines
1.
Make each line add up to 16.
2
5
3
2
2
2. Make each line add up to 20.
9
4
2
3
14
3. Make up your own puzzle like this.
Ask a friend to do it.
Teaching objectives
11
26
Solve mathematical problems or puzzles.
Know addition and subtraction facts up to 20.
Add three small numbers mentally.
© Crown copyright 2000
Odd one out
1.
Here is a grid of 16 squares.
One square is different from all the others.
Mark it on the grid.
2. Now do this one.
Teaching objectives
Solve mathematical problems or puzzles.
Make and describe patterns and pictures.
12
© Crown copyright 2000
27
Line of symmetry
You need:
some squared paper,
a red pen, a green pen and a blue pen.
Gopal had six squares: two red, two green, two blue.
He put them in a line.
The squares made a symmetrical pattern.
red
blue
green green
blue
red
Arrange six squares in a line.
Make two squares red, two green and two blue.
Make the line of squares symmetrical.
How many different lines can you make like this?
Teaching objectives
13
28
Solve mathematical problems or puzzles.
Begin to recognise line symmetry.
Solve a problem by sorting, classifying and organising information.
© Crown copyright 2000
Card sharp
Take ten cards numbered 0 to 9.
0
1.
1
2
3
7
9
5
6
4
8
Pick three cards with a total of 12.
You can do it in 10 different ways.
See if you can record them all.
2. Now pick four cards with a total of 12.
How many different ways can you do it?
3. Can you pick five cards with a total of 12?
Teaching objectives
Solve mathematical problems or puzzles.
Know addition facts to at least 10.
Solve a problem by sorting, classifying and organising information.
14
© Crown copyright 2000
29
Jack and the beanstalk
Jack climbed the beanstalk.
He always went upwards.
Top
Start
He first did it like this: left, right, left, right.
Find five other ways that Jack can climb the beanstalk.
Teaching objectives
15
30
Solve mathematical problems or puzzles.
Recognise turns to the left or to the right.
Give instructions for moving along a route.
© Crown copyright 2000
Monster
Alesha bought a monster using only silver coins.
It cost her 45p.
45p
There are nine different ways to pay 45p exactly
using only silver coins.
Find as many as you can.
What if the monster cost 50p?
How many different ways are there to pay now?
Teaching objectives
Solve mathematical problems or puzzles.
Find totals.
Work out which coins to pay.
16
© Crown copyright 2000
31
Cross-road
You need 5 paper plates and 15 counters.
Put the plates in a cross.
Use all 15 counters.
Put a different number on each plate.
Make each line add up to 10.
Do it again.
This time make each line add up to 8.
Teaching objectives
17
32
Solve mathematical problems or puzzles.
Know addition and subtraction facts up to 10.
Add three small numbers mentally.
© Crown copyright 2000
Fireworks
Emma had some fireworks.
Some made 3 stars.
Some made 4 stars.
Altogether Emma’s fireworks made 19 stars.
How many of them made 3 stars?
Find two different answers.
What if Emma’s fireworks made 25 stars?
Find two different answers.
Teaching objectives
Solve mathematical problems or puzzles.
Count on in steps of 3 or 4 from zero, or from any small number.
18
© Crown copyright 2000
33
Coloured shapes
What colour is each shape?
Write it on the shape.
Clues
◆ Red is not next to grey.
◆ Blue is between white and grey.
◆ Green is not a square.
◆ Blue is on the right of pink.
Teaching objectives
19
34
Solve mathematical problems or puzzles.
Explain methods and reasoning.
© Crown copyright 2000
Ones and twos
Holly has six numbers, three 1s and three 2s.
She also has lots of + signs, x signs and = signs.
1
2
1
2
1
2
She is trying to make the biggest number possible.
Here are some she tried.
First try
1x2=2
1x2=2
1x2=2
2+2+2=6
Second try
1+2+1+2+1+2=9
Can you beat Holly’s score?
What if Holly had three 2s and three 3s?
Teaching objectives
Solve mathematical problems or puzzles.
Use known number facts to add mentally.
Carry out simple multiplication.
20
© Crown copyright 2000
35
Birthdays
Mum and Paul are talking about birthdays.
They take Paul’s age and double it.
Then they add 5.
The answer is 35.
Mum says this is her age.
How old is Paul?
Make up more problems like this.
Try to use some of these words:
double
halve
add
Teaching objectives
21
36
Solve mathematical problems or puzzles.
Use known number facts to add mentally.
Carry out simple multiplication.
© Crown copyright 2000
subtract
Christmas tree
Rudolph put four stars on a tree.
He coloured each star either red or yellow.
In how many different ways can Rudolph colour
the four stars?
Teaching objectives
Solve mathematical problems or puzzles.
Solve a problem by organising information.
Explain methods and reasoning.
22
© Crown copyright 2000
37
At the toy shop
The toy shop stocks tricycles and go-carts.
The tricycles have 3 wheels.
The go-carts have 5 wheels.
Suna counted the wheels.
He counted 37 altogether.
How many tricycles are there?
How many go-carts?
Find two ways to do it.
Teaching objectives
23
38
Solve mathematical problems or puzzles.
Recognise multiples of 3 and 5.
Add mentally a pair of two-digit numbers.
© Crown copyright 2000
Ben’s numbers
Ben has written a list of different whole numbers.
The digits of each number add up to 5.
None of the digits is zero.
Here is one of Ben’s numbers.
23
Ben has written all the numbers he can think of.
How many different numbers are there in his list?
Write all the numbers in order.
Teaching objectives
Solve a given problem by organising and interpreting data in a simple table.
Write whole numbers in figures; know what each digit represents.
Order whole numbers.
24
© Crown copyright 2000
39
Spot the shapes 1
1.
How many triangles
can you count?
2. How many rectangles
can you count?
3. Draw your own diagram to count triangles.
How many can a friend find?
Can you find more?
Teaching objectives
25
40
Solve mathematical problems or puzzles.
Visualise 2-D shapes.
Explain methods and reasoning.
© Crown copyright 2000
Puzzles and problems
for Years 3 and 4
Rows of coins
1. Take five coins: 1p, 2p, 5p, 10p, 20p.
Put them in a row using these clues.
The total of the first three coins is 27p.
The total of the last three coins is 31p.
The last coin is double the value of the first coin.
2. Take six coins: two 1p, two 2p and two 5p.
Put them in a row using these clues.
Between the two 1p coins there is one coin.
Between the two 2p coins there are two coins.
Between the two 5p coins there are three coins.
What if you take two 10p coins as well, and
between them are four coins?
Teaching objectives
26
42
Solve word problems involving money.
Explain methods and reasoning.
© Crown copyright 2000
Roly poly
The dots on opposite faces of a dice add up to 7.
1. Imagine rolling one dice.
The score is the total number
of dots you can see.
You score 17.
Which number is face down?
How did you work out your answer?
2. Imagine rolling two dice.
The dice do not touch each other.
The score is the total number of dots you can see.
Which numbers are face down to score 30?
Teaching objectives
Solve mathematical problems or puzzles.
Add three or four small numbers.
Explain methods and reasoning.
27
© Crown copyright 2000
43
Dan the detective
1. Dan the detective looked for a number.
He found a two-digit number less than 50.
The sum of its digits was 12.
Their difference was 4.
What number did Dan find?
2. Dan found a two-digit odd number.
One of its digits was half the other.
The number was greater than 50.
What number did Dan find?
Teaching objectives
28
44
Solve a given problem by organising and interpreting data in a simple table.
Write whole numbers in figures; know what each digit represents.
© Crown copyright 2000
Spaceship
Some Tripods and Bipods flew from planet Zeno.
There were at least two of each of them.
Tripods have 3 legs.
Bipods have 2 legs.
There were 23 legs altogether.
How many Tripods were there?
How many Bipods?
Find two different answers.
Teaching objectives
Solve mathematical problems or puzzles.
Count on in steps of 2 or 3.
Know multiplication facts for 2 and 3 times tables.
29
© Crown copyright 2000
45
Susie the snake
Susie the snake has up to 20 eggs.
She counted her eggs in fours.
She had 3 left over.
She counted them in fives.
She had 4 left over.
How many eggs has Susie got?
Teaching objectives
30
46
Solve mathematical problems or puzzles.
Know multiplication facts for 4 and 5 times tables.
Find remainders after division.
© Crown copyright 2000
Three monkeys
Three monkeys ate a total of 25 nuts.
Each of them ate a different odd number of nuts.
How many nuts did each of the monkeys eat?
Find as many different ways to do it as you can.
Teaching objectives
Solve mathematical problems or puzzles.
Recognise odd and even numbers.
Add three or four small numbers mentally.
31
© Crown copyright 2000
47
Card tricks
Chico’s cards are all different.
There is a number from 1 to 8 on each card.
Chico has chosen four cards that add up to 20.
What are they?
There are seven different possibilities.
Try to find them all.
What if Chico has three cards that add up to 16?
Teaching objectives
32
48
Solve mathematical problems or puzzles.
Know addition and subtraction facts up to 20.
Add three or four small numbers mentally.
© Crown copyright 2000
Neighbours
Use each of the numbers 1 to 6 once.
Write one in each circle.
Numbers next to each other must not be joined.
For example, 3 must not be joined to 2 or 4.
1 2 3 4 5 6
Teaching objectives
Solve mathematical problems or puzzles.
Order numbers 0 to 9.
Explain methods and reasoning.
33
© Crown copyright 2000
49
Queen Esmerelda’s coins
Queen Esmerelda had 20 gold coins.
She put them in four piles.
◆ The first pile had four more coins than the second.
◆ The second pile had one less coin than the third.
◆ The fourth pile had twice as many coins as the
second.
How many gold coins did Esmerelda put in each pile?
Teaching objectives
34
50
Solve mathematical problems or puzzles.
Use vocabulary of comparing and ordering numbers.
Explain methods and reasoning.
© Crown copyright 2000
Duck ponds
Use 14 ducks each time.
1. Make each pond hold two ducks or five ducks.
2. Make each pond hold twice as many ducks as the
one before.
3. Make each pond hold one less duck than the one
before.
Teaching objectives
Solve mathematical problems or puzzles.
Know multiplication facts for 2 and 5 times tables.
Add three or four small numbers.
35
© Crown copyright 2000
51
Treasure hunt
Jed and Jake are pirates.
Between them they have three precious jewels:
a ruby (R), a diamond (D) and an emerald (E).
Complete the table.
Show what jewels each pirate could have.
Jed
R
E
Jake
D
Teaching objectives
36
52
Solve a given problem by organising and interpreting data in a simple table.
Explain methods and reasoning.
© Crown copyright 2000
Stamps
Tilly’s parcel cost 55p to post.
She stuck on eight stamps.
Each stamp was either 10p or 5p.
10p
5p
How many of each stamp did Tilly stick on her
parcel?
Make up your own puzzle like this.
Ask a friend to do it.
Teaching objectives
Solve mathematical problems or puzzles.
Know multiplication facts for 5 and 10 times tables.
37
© Crown copyright 2000
53
Maisie the mouse
Maisie had between
30 and 50 breadcrumbs.
She counted the breadcrumbs in fours.
There were 2 left over.
She counted them in fives.
There was 1 left over.
How many breadcrumbs did Maisie have?
Teaching objectives
38
54
Solve mathematical problems or puzzles.
Know multiplication facts for 4 and 5 times tables.
Find remainders after division.
© Crown copyright 2000
Kieron’s cats
Kieron has three cats.
Each is a different weight.
The first and second weigh 7 kg altogether.
The second and third weigh 8 kg altogether.
The first and third weigh 11 kg altogether.
What is the weight of each cat?
Teaching objectives
Solve mathematical problems or puzzles.
Know addition and subtraction facts to 20.
Explain methods and reasoning.
39
© Crown copyright 2000
55
Next door numbers
Take ten cards numbered 0 to 9.
0 1 2 3 4 5 6 7 8 9
Arrange the cards like this.
Do it so that no two consecutive numbers are next to
each other, horizontally, vertically or diagonally.
There are lots of ways to do it.
How many ways can you find?
Teaching objectives
40
56
Solve mathematical problems or puzzles.
Order numbers 0 to 9.
Explain methods and reasoning.
© Crown copyright 2000
Nick-names
Dawn, Mark, Josh and Tina are friends.
They each have a nick-name.
Their nick-names are Spider, Curly, Ace and Fudgy,
but not in that order.
What is the nick-name of each of the friends?
Clues
◆ Josh plays tennis with Curly and goes swimming
with Ace.
◆ Tina has been on holiday with Curly but travels to
school with Fudgy.
◆ Spider, Curly and Dawn play in the football team.
◆ Spider sometimes goes to tea with Josh.
Teaching objectives
Solve mathematical problems or puzzles.
Solve a problem by organising information in a table.
Explain methods and reasoning.
41
© Crown copyright 2000
57
Stickers
The twins collected some animal stickers.
They each had the same total number.
Winston had 3 full sheets and 4 loose stickers.
Wendy had 2 full sheets and 12 loose stickers.
Every full sheet has the same number of stickers.
How many stickers are there in a full sheet?
Teaching objectives
42
58
Solve mathematical problems or puzzles.
Know multiplication facts.
Explain methods and reasoning.
© Crown copyright 2000
Odds and evens
You need 13 counters or coins.
Draw a 5 by 5 grid.
Put counters on it.
You can put only one counter in each space.
1.
Place 13 counters.
Get an odd number of them in each row and column
and the two main diagonals.
2. Place 10 counters.
Get an even number of them in each row and
column and the two main diagonals.
Teaching objectives
Solve mathematical problems or puzzles.
Recognise odd and even numbers.
Explain methods and reasoning.
43
© Crown copyright 2000
59
More stamps
Rosie spent £2 on 10p and 20p stamps.
20p
10p
She bought three times as many
10p stamps as 20p stamps.
How many of each stamp did she buy?
Teaching objectives
44
60
Solve mathematical problems or puzzles.
Begin to use ideas of simple ratio and proportion.
Explain methods and reasoning.
© Crown copyright 2000
Sandcastles
Lisa went on holiday.
In 5 days she made 80 sandcastles.
Each day she made 4 fewer castles than the day
before.
How many castles did she make each day?
Lisa went on making 4 fewer castles each day.
How many castles did she make altogether?
Teaching objectives
Solve mathematical problems or puzzles.
Add two-digit numbers.
45
© Crown copyright 2000
61
Sail away
Two men and two women want to sail to an island.
The boat will only hold
two women or one man.
How can all four of them get to the island?
Teaching objectives
46
62
Solve mathematical problems or puzzles.
Explain methods and reasoning.
© Crown copyright 2000
Straw squares
You need 20 straws all the same length.
There are 12 straws in this pattern of 5 squares.
Take 20 straws.
Arrange them to make as many squares as you can.
Don’t bend or break the straws!
How many squares did you make?
Teaching objectives
Solve mathematical problems or puzzles.
Visualise 2-D shapes.
47
© Crown copyright 2000
63
King Arnold
King Arnold sits at a Round Table.
King
Arnold
There are 3 empty seats.
In how many different ways
can 3 knights sit in them?
What if there are 4 empty seats?
In how many different ways
can 4 knights sit in them?
Teaching objectives
48
64
Solve mathematical problems or puzzles.
Solve a problem by organising information.
Explain methods and reasoning.
© Crown copyright 2000
King
Arnold
Footsteps in the snow
Little has size 2 boots.
Middle has size 3 boots.
They are one and a half times the
length of Little’s boots.
Big has size 5 boots.
A little boot and a middle boot
are the same length as a big boot.
They start with the heels of
their boots on the same line.
They each walk heel to toe.
When will all three heels be in line again?
Teaching objectives
Solve mathematical problems or puzzles.
Recognise multiples of 2, 3 and 5.
49
© Crown copyright 2000
65
Ski lift
On a ski lift the chairs are equally spaced.
They are numbered in order from 1.
Kelly went skiing.
She got in chair 10 to go to the top of the slopes.
Exactly half way to the top, she passed chair 100
on its way down.
How many chairs are there
on the ski lift?
Make up more problems like this.
Teaching objectives
50
66
Solve mathematical problems or puzzles.
Solve a problem by organising information.
Explain methods and reasoning.
© Crown copyright 2000
Lighthouses
On the coast there are three lighthouses.
The first light shines for 3 seconds, then is
off for 3 seconds.
The second light shines for 4 seconds, then is
off for 4 seconds.
The third light shines for 5 seconds, then is
off for 5 seconds.
All three lights have just come on together.
When is the first time that all three lights will be off?
When is the next time that all three lights will come on
at the same moment?
Teaching objectives
Solve mathematical problems or puzzles.
Recognise multiples of 6, 8 and 10.
Explain methods and reasoning.
51
© Crown copyright 2000
67
Circle sums
1. Use each of the digits 1 to 5 once.
Replace each letter by one of the digits.
Make the total in each circle the same.
A
B
C
D
E
2. Now use each of the digits 1 to 7 once.
Make the total in each circle the same.
A
B
C
D
E
F
G
3. What if you used five circles and the digits 1 to 9?
Teaching objectives
52
68
Solve mathematical problems or puzzles.
Add several single digits.
Know addition and subtraction facts to 20.
© Crown copyright 2000
Puzzles and problems
for Years 5 and 6
Square it up
You need six drinking straws each the same length.
Cut two of them in half.
You now have eight straws, four long and four
short.
You can make 2 squares
from the eight straws.
Arrange your eight straws to make 3 squares, all
the same size.
Teaching objectives
53
70
Solve mathematical problems or puzzles.
Visualise 2-D shapes.
© Crown copyright 2000
Joins
Join any four numbers.
Find their total.
Joins can go up, down or sideways, but not diagonally.
The score shown is 8 + 15 + 6 + 18 = 47.
8
15
6
9
14
13
18
20
18
17
2
5
3
15
19
6
Find the highest possible score.
Find the lowest possible score.
Try joining five numbers.
Now try joining five numbers using only diagonal joins.
Teaching objectives
Solve mathematical problems or puzzles.
Add and subtract two-digit numbers mentally.
54
© Crown copyright 2000
71
Money bags
Ram divided 15 pennies among four small bags.
He could then pay any sum of money from 1p to
15p, without opening any bag.
How many pennies did Ram put in each bag?
Teaching objectives
55
72
Solve mathematical problems or puzzles.
Explain methods and reasoning.
© Crown copyright 2000
A perfect match
1.
A matchbox tray slides into its outer cover.
In how many different ways can you do this?
2. Imagine a cube and an open box just large enough
to hold it.
In how many different ways can you fit the cube
into the box?
Teaching objectives
Solve mathematical problems or puzzles.
Visualise 3-D shapes.
56
© Crown copyright 2000
73
Presents
Gurmit paid £21 for five presents.
C
B
A
E
D
For A and B he paid a total of £6.
For B and C he paid a total of £10.
For C and D he paid a total of £7.
For D and E he paid a total of £9.
How much did Gurmit pay for each present?
Teaching objectives
57
74
Solve a given problem by organising information.
Explain methods and reasoning.
© Crown copyright 2000
Spot the shapes 2
1. How many triangles
can you count?
2. How many squares
can you count?
3. Draw your own diagram to count triangles.
Don’t use too many lines!
How many triangles can a friend find?
Can you find more?
Teaching objectives
Solve mathematical problems or puzzles.
Visualise 2-D shapes.
Explain methods and reasoning.
58
© Crown copyright 2000
75
Four by four
You need some squared paper.
This 4 by 4 grid is divided into two identical parts.
Each part has the same area and the same shape.
Find five more ways of dividing the grid into two
identical parts by drawing along the lines of the grid.
Rotations and reflections do not count as different!
Explore ways of dividing a 4 by 4 grid into two parts
with equal areas but different shapes.
Teaching objectives
59
76
Solve mathematical problems or puzzles.
Visualise 2-D shapes.
Find fractions of shapes.
© Crown copyright 2000
Three digits
Imagine you have 25 beads.
You have to make a three-digit number on an abacus.
You must use all 25 beads for each number you make.
How many different three-digit numbers can you make?
Write them in order.
Teaching objectives
Solve mathematical problems or puzzles.
Know what each digit represents.
Order a set of whole numbers.
60
© Crown copyright 2000
77
Make five numbers
Take ten cards numbered 0 to 9.
0
1
2
3
7
9
5
6
4
8
Each time use all ten cards.
Arrange the cards to make:
a.
five numbers that are multiples of 3
b. five numbers that are multiples of 7
c.
five prime numbers
Make up more problems to use all ten cards
to make five special numbers.
Teaching objectives
61
78
Solve mathematical problems or puzzles.
Know 3 and 7 times tables.
Recognise prime numbers.
© Crown copyright 2000
Maze
Start with zero.
Find a route from ‘Start’ to ‘End’ that totals 100
exactly.
Start
+6
x9
÷H2
+9
x7
÷3
x5
x5
–6
x3
–5
÷3
x7
–8
End
Which route has the highest total?
Which has the lowest total?
Now try some different starting numbers.
Teaching objectives
Solve mathematical problems or puzzles.
Add and subtract two-digit numbers mentally.
Multiply and divide by single-digit numbers.
62
© Crown copyright 2000
79
Jack’s book
The pages of Jack’s book are numbered from 1.
The page numbers have a total of 555 digits.
How many pages has the book?
How many of the digits are a 5?
Teaching objectives
63
80
Solve mathematical problems or puzzles.
Know what each digit represents.
© Crown copyright 2000
Flash Harry
In April Flash Harry bought a saddle for £100.
In May he sold it for £200.
In June he was sorry he had sold it.
So he bought it back for £300.
In July he got tired of it.
So he sold it for £400.
Overall, did Flash Harry make or lose money?
How much did he make or lose?
Teaching objectives
Solve mathematical problems or puzzles.
Use negative numbers.
64
© Crown copyright 2000
81
Age old problems
1.
My age this year is a multiple of 8.
Next year it will be a multiple of 7.
How old am I?
2. Last year my age was a square number.
Next year it will be a cube number.
How old am I?
How long must I wait until my age is both
a square number and a cube?
3. My Mum was 27 when I was born.
8 years ago she was twice as old
as I shall be in 5 years’ time.
How old am I now?
Teaching objectives
65
82
Solve mathematical problems or puzzles.
Know multiplication facts to 10 x 10.
Recognise square and cube numbers.
© Crown copyright 2000
Zids and Zods
Zids have 4 spots.
Zods have 9 spots.
Altogether some Zids and Zods have 48 spots.
How many Zids are there?
How many Zods?
What if Zids have 5 spots, Zods have 7 spots,
and there are 140 spots altogether?
Find as many solutions as you can.
Teaching objectives
Solve mathematical problems or puzzles.
Know multiplication facts to 10 x 10.
Add two-digit numbers mentally.
66
© Crown copyright 2000
83
Franco’s fast food
This is what food costs at Franco’s café.
1 curry and 1 tea cost £4.
2 curries and 2 puddings cost £9.
1 pudding and 2 teas cost £2.
What do you have to pay in total for
1 curry, 1 pudding and 1 tea?
What does each item cost on its own?
Teaching objectives
67
84
Solve mathematical problems or puzzles.
Explain methods and reasoning.
© Crown copyright 2000
Albert Square
36 people live in the eight houses in Albert Square.
Each house has a different number of people living in it.
Each line of three houses has 15 people living in it.
How many people live in each house?
Teaching objectives
Solve mathematical problems or puzzles.
Add several small numbers mentally.
Explain methods and reasoning.
68
© Crown copyright 2000
85
Coins on the table
N
E
PENCE
10
N
E
E L I ZA
BE
N
. II
TH
10
9
EG
AB ETH . I
10
IZ
EL
.F .D. 199
TE
PE
C
TH . I I
PENCE
ELIZA
B
ETH . II
TEN
ELIZA
BE
I
T
Anna put some 10p coins on the table.
One half of them were tails up.
D.G
.R
Anna turned over two of the coins, and then
one third of them were tails up.
How many coins did Anna put on the table?
Teaching objectives
69
86
Solve mathematical problems or puzzles.
Understand simple fractions.
Explain methods and reasoning.
© Crown copyright 2000
A bit fishy
A goldfish costs £1.80.
An angel fish costs £1.40.
Nasreen paid exactly £20 for some fish.
How many of each kind did she buy?
Teaching objectives
Solve problems involving ratio and proportion.
Choose and use efficient calculation strategies to solve a problem.
Explain methods and reasoning.
70
© Crown copyright 2000
87
Pet shop
1.
Jim bought a cat and dog for £60 each.
Later he sold them.
He made a profit of 20% on the dog.
He made a loss of 20% on the cat.
How much did he get altogether when he sold
the cat and dog?
2.
Jim bought another cat and dog.
He sold them for £60 each.
He made a profit of 20% on the dog.
He made a loss of 20% on the cat.
Did he make a profit or loss on the whole deal?
Teaching objectives
71
88
Solve mathematical problems or puzzles.
Find simple percentages.
© Crown copyright 2000
Shape puzzle
Each shape stands for a number.
The numbers shown are the totals of the line of
four numbers in the row or column.
25
20
26
Find the remaining totals.
Teaching objectives
Solve mathematical problems or puzzles.
Use a symbol to stand for an unknown number.
Explain methods and reasoning.
72
© Crown copyright 2000
89
Eggs
Mrs Choy spent exactly £10 on 100 eggs for her shop.
Large eggs cost her 50p each.
Medium eggs cost her 10p each.
Small eggs cost her 5p each.
For two of the sizes, she bought the same
number of eggs.
How many of each size did she buy?
Teaching objectives
73
90
Solve problems involving ratio and proportion.
Explain methods and reasoning.
© Crown copyright 2000
Anyone for tennis?
Two boys and two girls can play tennis.
Ali said: ‘I will only play if Holly plays.’
Holly said: ‘I won’t play if Ben is playing.’
Ben said: ‘I won’t play if Luke or Laura plays.’
Luke said: ‘I will only play if Zoe plays.’
Zoe said: ‘I don’t mind who I play with.’
Which two boys and which two girls play tennis?
Teaching objectives
Solve a problem by extracting and interpreting data.
Explain methods and reasoning.
74
© Crown copyright 2000
91
Bus routes
A
Six towns are connected by bus routes.
F
The bus goes from A back to A.
It visits each of the other towns once.
E
How many different bus routes are there?
B
D
This table shows the bus fare for each direct route.
B to A costs the same as A to B, and so on.
A to B B to C C to D D to E E to F F to A B to D B to F C to E C to F
£4
£3
£4
£4
£3
£4
£5
£3
£2
£2
Which round trip from A to A is the cheapest?
Teaching objectives
75
92
Solve a problem by extracting and interpreting data.
Add several numbers mentally.
© Crown copyright 2000
C
Slick Jim
Slick Jim won the lottery.
He spent two thirds of his
winnings on a very posh house.
He spent two thirds of what he
had left on a luxury yacht.
Then he spent two thirds of what
he had left on a hot air balloon.
He spent his last £20000 on a
flashy car.
How much did Slick Jim win on the lottery?
Teaching objectives
Solve a problem by organising information.
Find fractions of quantities.
Understand the relationship between multiplication and division.
76
© Crown copyright 2000
93
All square
On each of these grids, the counters lie at the four
corners of a square.
What is the greatest number of counters you can place
on this grid without four of them lying at the corners
of a square?
Teaching objectives
77
94
Solve a problem by organising information.
Visualise 2-D shapes.
© Crown copyright 2000
Sleigh ride
In Snow Town, 3 rows of 4 igloos are linked by
17 sleigh paths.
Each path is 10 metres long.
When Santa visits, he likes
to go along each path at least once.
His route can start and end at any igloo.
How long is the shortest route Santa can take?
What if there are 4 rows of 5 igloos?
Teaching objectives
Solve a problem by organising information.
Visualise 2-D shapes.
78
© Crown copyright 2000
95
Spendthrift
Choc bars cost 26p each.
Fruit bars cost 18p each.
Anil spent exactly £5 on a mixture of choc bars
and fruit bars.
How many of each did he buy?
Teaching objectives
79
96
Solve mathematical problems or puzzles.
Choose and use efficient calculation strategies to solve a problem.
Add sums of money.
© Crown copyright 2000
Cola in the bath
A can of cola holds 33 centilitres.
If you had a bath in cola – don’t try it! –
approximately how many cans of cola would you need?
Hint: 1 cubic centimetre is the same as 1 millilitre.
Teaching objectives
Solve mathematical problems or puzzles.
Estimate lengths and convert units of capacity.
Develop calculator skills and use a calculator effectively.
80
© Crown copyright 2000
97
Millennium
At what time of what day of what year will it be:
a.
b.
c.
d.
e.
2000 seconds
2000 minutes
2000 hours
2000 days
2000 weeks
after the start of the year 2000?
Teaching objectives
81
98
Solve mathematical problems or puzzles.
Convert smaller to larger units of time.
Develop calculator skills and use a calculator effectively.
© Crown copyright 2000
People in the crowd
Estimate how many people there are in the crowd.
Teaching objectives
Solve mathematical problems or puzzles.
Count larger collections by grouping.
Give a sensible estimate.
82
© Crown copyright 2000
99
Make 200
1 2 3 4 5 6 7 8 9
Choose four of these digits.
Each one must be different.
Put one digit in each box.
This makes two 2-digit numbers reading across
and two 2-digit numbers reading down.
Add up all four of the numbers.
In this example the total is 100.
12 + 47 + 14 + 27 = 100
1 2
4 7
How many different ways of making 200 can you
find?
Teaching objectives
83
100
Solve mathematical problems or puzzles.
Know what each digit represents.
Add several two-digit numbers.
© Crown copyright 2000
Solutions
1 Four-pin bowling
5 Bean-bag buckets
Score 5 by knocking down 1 and 4,
or 2 and 3.
1.
The highest score is 12 (3 bags in 4).
2.
Score 6 in three ways:
1 bag in 4 and 2 bags in 1, or
1 bag in 1, 1 bag in 2 and 1 bag in 3, or
3 bags in 2.
3.
Score 9 in three ways:
1 bag in 1 and 2 bags in 4, or
1 bag in 2, 1 bag in 3, 1 bag in 4, or
3 bags in 3.
4.
Besides 6, 9 and 12, other possible
scores are:
Score 6 by knocking down 2 and 4,
or 1, 2 and 3.
Score 7 by knocking down 3 and 4,
or 1, 2 and 4.
2 Gob-stopper
Five different ways to pay 6p:
5p + 1p
2p + 2p + 2p
2p + 2p + 1p + 1p
2p + 1p + 1p + 1p + 1p
1p + 1p + 1p + 1p + 1p + 1p
Six different ways to pay 7p:
5p + 2p
5p + 1p + 1p
2p + 2p + 2p + 1p
2p + 2p + 1p + 1p + 1p
2p + 1p + 1p + 1p + 1p + 1p
1p + 1p + 1p + 1p + 1p + 1p + 1p
3:
3 bags in 1
4:
2 bags in 1, 1 bag in 2
5:
2 bags in 1, 1 bag in 3, or
1 bag in 1, 2 bags in 2
7:
1 bag in 1, 2 bags in 3, or
2 bags in 2, 1 bag in 3, or
1 bag in 1, 1 bag in 2, 1 bag in 4
8:
2 bags in 2, 1 bag in 4, or
1 bag in 2, 2 bags in 3, or
1 bag in 1, 1 bag in 3, 1 bag in 4
10:
1 bag in 2, 2 bags in 4
3 Pick a pair
Adapt this puzzle by using larger numbers.
There are six different sums and
six different (positive) differences.
6 Crossword
1.
1+2=3
1+4=5
2+4=6
1+8=9
2 + 8 = 10
4 + 8 = 12
2.
2–1=1
4–2=2
4–1=3
8–4=4
8–2=6
8–1=7
Adapt the puzzle by using larger numbers.
4 Snakes and ladders
Watching out for snakes, there are four
different ways to get to 16 in two throws:
1 then 6; 3 then 4; 4 then 3; 5 then 2.
102
Solutions
1
3
2
T W O
S
N
E
T W E
L V
4
X
E
E
E
N
I
5
N I
N E
7 Gold bars
Move two bars from pile 1 to pile 3.
Move one bar from pile 4 to pile 2.
8 Ride at the fair
10 Birds’ eggs
The amounts up to 20p that cannot be
made from exactly three coins are:
There are 10 possibilities:
1p, 2p, 10p, 18p, 19p.
Lucy could have given her Mum:
1, 1, 17
1, 3, 15
1, 5, 13
1, 7, 11
1, 9, 9
3, 3, 13
3, 5, 11
3, 7, 9
5, 5, 9
5, 7, 7
3p = 1p + 1p + 1p
4p = 2p + 1p + 1p
5p = 2p + 2p + 1p
6p = 2p + 2p + 2p
11 Number lines
1.
For example:
2
7p = 5p + 1p + 1p
5
3
6
8p = 5p + 2p + 1p
2
9
9p = 5p + 2p + 2p
11p = 5p + 5p + 1p
5
12p = 5p + 5p + 2p
13p = 10p + 2p + 1p
14p = 10p + 2p + 2p
1
2
8
Other solutions are possible.
2.
For example:
15p = 5p + 5p + 5p
9
16p = 10p + 5p + 1p
5
2
4
17p = 10p + 5p + 2p
2
15
9
9 Sum up
2
3
14
1
If each number can be used only once:
9=2+3+4
10 = 2 + 8
12 Odd one out
11 = 3 + 8
1.
12 = 4 + 8
*
13 = 2 + 3 + 8
14 = 2 + 4 + 8
15 = 3 + 4 + 8
Other solutions are possible if numbers can
be repeated.
Other totals:
2.
5=2+3
6=2+4
7=3+4
17 = 2 + 3 + 4 + 8
*
Solutions
103
13 Line of symmetry
16 Monster
There are five other ways for Gopal to
arrange the squares:
Alesha can use these coins to pay 45p:
red, green, blue, blue, green, red
green, red, blue, blue, red, green
green, blue, red, red, blue, green
blue, red, green, green, red, blue
blue, green, red, red, green, blue
What if Gopal has eight squares: two red,
two blue, two green and two yellow?
How many different symmetrical lines can
he make now? (24)
14 Card sharp
1.
There are 10 different ways to choose
three cards with a total of 12:
0, 3, 9
0, 4, 8
0, 5, 7
2.
2, 3, 7
2, 4, 6
3, 4, 5
0, 2, 3, 7
0, 2, 4, 6
0, 3, 4, 5
There are 13 different ways to pay 50p
using only silver coins. First add 5p to each
of the ways for 45p. The other four
possibilities are:
two 20p and one 10p
one 20p and two 10p
five 10p
one 50p
17 Cross-road
There are 9 different ways to choose
four cards with a total of 12:
0, 1, 2, 9
0, 1, 3, 8
0, 1, 4, 7
0, 1, 5, 6
3.
1, 2, 9
1, 3, 8
1, 4, 7
1, 5, 6
two 20p and one 5p
one 20p, two 10p and one 5p
one 20p, one 10p and three 5p
one 20p and five 5p
four 10p and one 5p
three 10p and three 5p
two 10p and five 5p
one 10p and seven 5p
nine 5p
Each line adds up to 10.
1, 2, 3, 6
1, 2, 4, 5
No.
4
2
Adapt the puzzle by changing the total.
15 Jack and the
beanstalk
104
Solutions
3
1
Each line adds up to 8.
5
Jack can climb the beanstalk like this:
left, left, right, right
left, right, left, right (as shown)
left, right, right, left
right, left, right, left
right, left, left, right
right, right, left, left
5
3
1
2
4
18 Fireworks
22 Christmas tree
For 19 stars:
There are 16 different ways:
5 fireworks made 3 stars and
1 made 4 stars, or
1 way for 4 red;
1 firework made 3 stars and
4 made 4 stars
4 ways for 3 red and 1 yellow;
1 way for 4 yellow;
4 ways for 1 red and 3 yellow;
For 25 stars:
6 ways for 2 red and 2 yellow (shown
below).
3 fireworks made 3 stars and
4 fireworks made 4 stars, or
R
7 fireworks made 3 stars and
1 firework made 4 stars
R
R
Y
Y
R
R
Y
Y
Y
Y
R
Y
Y
Y
19 Coloured shapes
Red
White
R
Y
R
Pink
Blue
Green
Grey
R
R
R
Y
There are 9 tricycles and 2 go-carts, or
4 tricycles and 5 go-carts.
Some higher scores:
2+1=3
2+1=3
2+1=3
3 x 3 x 3 = 27
21 Birthdays
Most pupils will guess then try to improve.
For example, try 10:
20 + 5 = 25
24 Ben’s numbers
There are 16 different numbers in Ben’s
list:
5, 14, 23, 32, 41, 113, 122, 131, 212,
221, 311, 1112, 1121, 1211, 2111, 11111.
What if the digits add up to 4, or if they
add up to 6? How many different numbers
are there now?
Answer: Paul is 15.
10 x 2 = 20
Y
23 At the toyshop
20 Ones and twos
2x2x2=8
1+1+1=3
8 x 3 = 24
R
too small
25 Spot the shapes 1
1.
There are 9 triangles.
2.
There are 18 rectangles.
Solutions
105
26 Rows of coins
31 Three monkeys
1.
5p, 2p, 20p, 1p, 10p
There are 10 possibilities:
2.
2p, 5p, 1p, 2p, 1p, 5p, or its reverse
When two 10p coins are also used:
2p, 5p, 10p, 2p, 1p, 5p, 1p, 10p, or its
reverse
27 Roly poly
1.
2.
The total number of dots on the dice is
21. Of these dots 17 are showing, so
the face with 4 dots is face down.
The total number of dots on two dice is
42, so 12 dots are hidden. The two
hidden faces must each have 6 dots.
1, 3, 21
1, 5, 19
1, 7, 17
1, 9, 15
1, 11, 13
3, 5, 17
3, 7, 15
3, 9, 13
5, 7, 13
5, 9, 11
What if the monkeys ate 24 nuts, with
each of them eating a different even
number of nuts?
The possible answers are:
2, 4, 18
2, 6, 16
2, 8, 14
2, 10, 12
4, 6, 14
4, 8, 12
6, 8, 10
28 Dan the detective
32 Card tricks
1.
48
2.
63
Systematic working helps to make sure
that all possibilities have been considered.
Four different cards with a total of 20 are:
29 Spaceship
3 Tripods (9 legs) and 7 Bipods (14 legs), or
5 Tripods (15 legs) and 4 Bipods (8 legs).
What if Tripods with 3 legs and
Quadrapods with 4 legs are on the
spaceship?
Find two different ways to make 23 legs.
30 Susie the snake
Susie has 19 eggs.
You could make up similar problems with,
say, 21 eggs.
If you counted them in fours, there would
be 1 left over.
If you counted them in fives, there would
be 1 left over.
106
Solutions
1, 4, 7, 8
1, 5, 6, 8
2, 3, 7, 8
2, 4, 6, 8
2, 5, 6, 7
3, 4, 5, 8
3, 4, 6, 7
Three different cards with a total of 16
are:
1, 7, 8
2, 6, 8
3, 5, 8
3, 6, 7
4, 5, 7
You could try other totals. For example,
four cards with a total of 18 are:
1, 2, 7, 8
1, 3, 6, 8
1, 4, 5, 8
1, 4, 6, 7
2, 3, 6, 7
2, 4, 5, 7
3, 4, 5, 6
Explore the different totals that can be
made with four cards. (It is possible to
make any total from 10 to 26.)
33 Neighbours
Here is one possible
solution.
37 Stamps
5
3
1
Tilly stuck three 10p stamps and five 5p
stamps on her parcel.
6
4
No. of
5p stamps
No. of
10p stamps
Total
value
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
40p
45p
50p
55p
60p
65p
70p
75p
80p
2
Can you find others?
34 Queen Esmeralda’s
coins
There were 7, 3, 4 and 6 coins in each pile.
The problem can be solved by trial and
error.
To adapt the problem, change the cost of
the parcel, or use different stamps.
35 Duck ponds
2
2
1.
38 Maisie the mouse
5
5
Maisie had 46 breadcrumbs.
2.
3.
4
2
8
4
5
The problem can be solved by experiment.
3
Alternatively, list all the multiples of 4.
Add 2 to each number in the list.
2
Now list all the multiples of 5. Add 1 to
each number in the list.
You could try similar problems with other
numbers. For example, using 15 ducks and
Now look for a number lying between 30
and 50 that is common to both lists.
5 ponds
make each hold 1 more than the
one before (1, 2, 3, 4, 5)
4 ponds
make each hold twice as many as
the one before (1, 2, 4, 8)
3 ponds
make each hold 4 more than the
one before (1, 5, 9)
39 Kieron’s cats
3 ponds
make each hold 2 less than the
one before (7, 5, 3)
Kieron’s cats weigh 5 kg, 2 kg and 6 kg.
To adapt the problem, group the
breadcrumbs in 5s and 6s, or 7s and 9s.
36 Treasure hunt
Jed
Jake
D
R
E
D
E
D
E
R
R
R
R
D
E
E
E
D
R
D
R
E D
E D
R
Solutions
107
40 Next door numbers
45 Sandcastles
For example:
Over the 5 days Lisa made 24, 20, 16, 12
and 8 sandcastles.
9
1
5
7
3
8
4
4
6
0
2
She made 84 sandcastles altogether.
0
2
3
9
5
46 Sail away
6
8
7
1
41 Nick-names
Dawn is Ace.
Mark is Curly.
Josh is Fudgy.
Tina is Spider.
42 Stickers
There are 8 stickers in a full sheet.
43 Odds and evens
Several solutions are possible. For example:
1.
Two women cross the river together.
One woman stays there and one brings the
boat back.
One man crosses the river.
One woman brings the boat back.
Two women cross the river together.
One woman stays there and one brings the
boat back.
The second man crosses the river.
One woman brings the boat back.
Two women cross the river together.
47 Straw squares
You can make a maximum of 9 squares with
20 straws.
Here are two ways of doing it.
2.
For older children, try 40 straws.
With these you can make a maximum of 30
squares.
44 More stamps
Rosie bought four 20p stamps and
twelve 10p stamps.
108
Solutions
48 King Arnold
53 Square it up
Three knights can sit with King Arnold in 6
different ways.
For example:
Four knights can sit with King Arnold in 24
different ways.
49 Footsteps in the snow
Counting from zero in 2s, 3s and 5s will
first match up at 30, when Little has taken
15 footsteps.
54 Joins
Using four numbers:
the highest score is 19 + 15 + 17 + 18 = 69,
the lowest score is 6 + 5 + 2 + 17 = 30.
50 Ski lift
Using five numbers:
the highest is 20 + 18 + 13 + 17 + 18 = 86,
the lowest is 6 + 18 + 2 + 5 + 6 = 37.
The ski lift has 180 chairs.
51 Lighthouses
All three lights will be off after 5 seconds.
All three lights will next come on together
after 120 seconds.
Using five numbers and diagonal joins:
the highest is 19 + 17 + 14 + 15 + 18 = 83,
the lowest is 13 + 6 + 20 + 2 + 6 = 47.
55 Money bags
52 Circle sums
Ram put 1p, 2p, 4p and 8p in the four bags.
1.
Any sum from 1p to 15p can be made with
these amounts.
5
1
3
2
4
56 A perfect match
or its reverse
1.
A matchbox tray fits into its outer
cover in 4 different ways.
2.
A cube will fit into a box with any one
of its 6 faces uppermost.
2.
7
3
2
5
1
4
6
or its reverse
Each face can be rotated into any one
of 4 different positions.
3.
So there are 6 x 4 = 24 ways of
fitting the cube in the box.
9
2
5
4
6
1
7
3
8
or its reverse
Solutions
109
57 Presents
63 Jack’s book
Gurmit paid £2, £4, £6, £1 and £8 for the
five presents.
The book has 221 pages.
42 of the digits are a 5.
58 Spot the shapes 2
64 Flash Harry
1.
2.
Flash Harry’s bank balance looked like this.
There are 11 triangles.
There are 17 squares.
April
May
June
July
59 Four by four
– £100
+ £100
– £200
+ £200
So Harry made £200 overall.
65 Age old problems
60 Three digits
You can make six different numbers.
In order, the numbers are:
799, 889, 898, 979, 988, 997.
61 Make five numbers
For example:
a. 12, 39, 45, 60, 78.
b. 7, 42, 63, 98, 105.
c. 5, 23, 67, 89, 401.
There are other solutions.
62 Maze
There are two routes that total 100 exactly:
+6
x 7 – 6 x 3 – 8 = 100
+9
x 7 ÷ 3 x 5 – 5 = 100
The route giving the highest total is:
+9
x 7 – 6 x 7 – 8 = 391
The route giving the lowest total is:
+6
x 7 ÷ 3 x 3 – 8 = 34
110
Solutions
1.
I am 48 years old (or possibly 104).
2.
I am now 26 years old. In 38 years’
time, when I am 64, my age will be both
a square number and a cube.
3.
I am 9 years old now.
66 Zids and Zods
There are 3 Zids with 4 spots and 4 Zods
with 9 spots.
If Zids have 5 spots and Zods have 7 spots,
the possible ways of making 140 are:
28 Zids;
21 Zids and 5 Zods;
14 Zids and 10 Zods;
7 Zids and 15 Zods;
20 Zods.
67 Franco’s fast food
A curry costs £3.50, a pudding costs £1
and a tea costs 50p.
So the total cost of a curry, a pudding and
a tea is £5.
68 Albert Square
73 Eggs
For example:
Mrs Choy bought:
10 large eggs at 50p each,
10 medium eggs at 10p each,
80 small eggs at 5p each.
3
5
7
74 Anyone for tennis?
4
2
8
1
6
69 Coins on the table
Ali, Luke, Holly and Zoe play tennis.
Two boys can play.
Ben won’t play if Luke plays.
So the two boys must be Ali and Ben,
or Ali and Luke.
Anna put 12 coins on the table.
Ali will play only if Holly plays.
Holly won’t play with Ben.
So the two boys are Ali and Luke.
70 A bit fishy
Luke will play only if Zoe plays.
So the two girls are Holly and Zoe.
Nasreen bought 4 angel fish and 8 goldfish.
71 Pet shop
1.
Jim sold the dog and the cat for £72
and £48 respectively, a total of £120.
2.
The dog cost £50 and the cat cost
£75, a total of £125.
The cat and the dog were sold for a
total of £120, so Jim made a loss of
£5.
75 Bus routes
There are six different routes from A
back to A:
A B C D E F A
A B D C E F A
A B D E C F A
and the three reversals of these.
The cheapest routes are A B D E C F A
and its reversal, which each cost £21.
72 Shape puzzle
76 Slick Jim
The circle has the value 5.
The triangle has the value 8.
The club has the value 6.
Jim won £540 000.
27
77 All square
For example:
25
20
28
27
22
25
26
Solutions
111
78 Sleigh ride
81 Millennium
With 3 rows of 4 igloos, the shortest route
is 190 metres. For example:
a.
00:33:20
1 January 2000
b.
09:20:00
2 January 2000
c.
08:00
23 March 2000
d.
00:00
23 June
2005
e.
00:00
1 May
2038
82 People in the crowd
With 4 rows of 5 igloos, the shortest route
is 350 metres. For example:
There is no precise answer, but pupils can
compare their estimates and discuss how
they arrived at them.
83 Make 200
There are 22 different solutions. Eleven of
the solutions are as follows:
79 Spendthrift
Anil bought 13 choc bars and 9 fruit bars,
or 4 choc bars and 22 fruit bars.
80 Cola in the bath
A bath 1.5 metres long by 60 cm wide would
have a floor area of approximately
9000 cm2. If there was 10 cm of cola in the
bath, the volume of liquid would be about
90 000 cm3 or 90 000 ml. This would
require roughly 270 cans of cola.
112
Solutions
1 9
7 2
2 8
6 3
2 9
5 3
3 5
7 4
4 1
9 5
4 2
8 5
4 3
7 5
5 1
7 6
6 1
5 7
6 2
4 7
7 1
3 8
Eleven more solutions are formed by
changing over the two digits in the top
right and bottom left boxes.

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