Mathematical challenges for able pupils

112 Pages·2000·3.01 MB·English

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:
(cid:1) grasp new material quickly;
(cid:1) are prepared to approach problems from different directions and persist in finding
solutions;
(cid:1) generalise patterns and relationships;
(cid:1) use mathematical symbols confidently;
(cid:1) 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).
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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.
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© Crown copyright 2000 Mathematical challenges for able pupils
Medium-term plan: Year 2
Part of autumn term with extension and enrichment work
Unit Topic Extension Enrichment
(days) Objectives: children will be taught to… Objectives from Y3/4 suggestions
1 (3) Counting, properties of numbers and
number sequences
Say the number names in order to at Snakes and
least 100. ladders (no. 4)
Count reliably up to 100 objects by Estimate up to 100 Cross-road
grouping them, for example in tens. objects. (no. 17)
Count on or back in ones or tens from any Extend to three-digit
two-digit number. numbers.
Recognise two-digit multiples of 10. Recognise three-digit
multiples of 10.
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 Read/write numbers
in figures and words. to 1000/10000.
Know what each digit in a two-digit number Extend to three-digit Ben’s numbers
represents, including 0 as a place-holder. numbers. (no. 24)
Partition two-digit numbers into a multiple of ten Partition three-digit
and ones (TU). Use the = sign. numbers.
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 Number lines
subtraction; recognise that addition can be (no. 11)
done in any order, but not subtraction.
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 Extend to 19, 29…;
and adjusting by 1. 21, 31…; 18, 22…
Money and ‘real life’ problems
Recognise all coins. Find totals. Give change.
Choose an appropriate operation and calculation Solve word problems. Ride at the fair
strategy to solve simple word problems. (no. 8)
Explain methods orally… …and in writing. Gold bars (no. 7)
Making decisions, checking results
Check sums by adding in a different order.
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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 Topic Extension Enrichment
(days) Objectives: children will be taught to… Objectives from Y7 suggestions
1 (3) Place value, ordering and rounding Understand and use
Multiply and divide decimals by 10 or 100, and decimal notation and
integers by 1000, and explain the effect. place value.
Using a calculator
Develop calculator skills and use a calculator Millennium
effectively. (no. 81)
2–3 (10) Understanding multiplication and division Express simple functions
Understand and use the relationships between at first in words and then Maze
the four operations, and the principles of the in symbols, and use (no. 62)
arithmetic laws. simple function machines.
Mental calculation strategies (× and ÷)
Use related facts and doubling or halving: Shape puzzle
e.g. halve an (even) number, double the other; (no. 72)
multiply by 25, e.g. by ×100, then ÷4. Make five
Extend mental methods (to decimals). numbers
(no. 61)
Pencil and paper procedures (× and ÷)
Approximate first. Use informal pencil and paper Alternative
methods to support, record or explain × and ÷. multiplication
Extend written methods to ThHTU × U and Extend to decimals (see p. 14)
short multiplication involving decimals. with 2 d.p.
Money and ‘real life’ problems
Use all four operations to solve money or Spendthrift
‘real life’ problems. (no. 79)
Choose appropriate operations/calculation Franco’s fast
methods. Explain working. food (no. 67)
Making decisions, checking results
Check by estimating. Flash Harry
Use inverse operation, including with a calculator. (no. 64)
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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:
(cid:1) a common task, starting from the common experience of pupils, leading to different
outcomes – this is typical of open investigations;
(cid:1) 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;
(cid:1) 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?’
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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:
(cid:1) books of investigations and ‘open’ problems;
(cid:1) mathematics magazines and booklets produced for pupils;
(cid:1) mathematical posters and topic books that stimulate discussion and investigation;
(cid:1) computer access to the Internet;
(cid:1) calculators to solve challenging and investigative activities;
(cid:1) software;
(cid:1) ‘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
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© Crown copyright 2000 Mathematical challenges for able pupils
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
(cid:1) Problem solving with challenges and simplifications: This illustrates how
activities linked to problem solving and reasoning can challenge able pupils.
(cid:1) Using a calculator: Although the activities are designed for teachers, some of
them can be used to extend able pupils.
(cid:1) Laying the foundations for algebra: This contains a range of examples involving
reasoning and explanation.
(cid:1) Fractions, decimals, percentages, ratio and proportion: Able pupils can
develop this work, especially the inter-relationships and examples involving ratio
and proportion.
(cid:1) 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.
(cid:1) 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:
(cid:1) Solving word problems
(cid:1) Fractions, decimals, percentages, ratio and proportion
(cid:1) Shape and space
(cid:1) Calculators
(cid:1) Graphs and charts
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Mathematical challenges for able pupils © Crown copyright 2000