365
+21
386
Addition
&
Subtraction
1 2
3 4
5 6
11 1
2 13
7 8
14 1
9 1
5 16
21 2
0
2 23
17 1
8 19
24 25
31 3
20
26 2
2 33
7 28
34 3
29 3
5 36
41 42
0
37 3
43 4
8 39
4 45
51 5
40
4
6
2 53
47 4
8 49
54 5
5 56
61 6
50
2 63
57 5
8 59
64 6
5 66
60
71 7
2 73
67 6
8 69
74 7
5 76
81 8
70
2 83
77 7
8 79
84 8
5 86
80
91 9
2 93
87 8
8 89
94 95
90
96 9
7 98
99 10
0
65 = 100 -
8
1
© 2000 Andrew Harris
Addition & Subtraction
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© 2000 Andrew Harris
The Complexities of Learning about Addition and Subtraction
Symbolic Encoding and Abstraction
It is important for any teacher to realise that an apparently simple mathematical statement such as
3 + 2 = 5 has a multiplicity of meanings. For a start, the number 3 in the statement could mean many
things, for example, 3 objects (toy cars or multilink cubes or shells etc.), 3 groups of objects, a
position on a number line or simply an abstract number. The same is true of the numbers 2 and 5 in
this statement. Thus, the numbers in the statement are generalisations of many different types of 3s,
2s and 5s found in different contexts.
In addition, the complete statement 3 + 2 = 5 itself is a generalisation which can represent many
different situations.
e.g.
John has 3 sweets and Jill has two more, five in total;
A plant that is 3cm tall grows by an extra 2cm and is now 5cm tall;
Spending £3 and then spending another £2 is equivalent to spending £5 in one
payment.
Thus, to young children the statement 3 + 2 = 5 often has little meaning when no context is given. A
similar argument applies to simple subtraction statements. Consequently, all early addition and
subtraction work should be done by means of practical tasks involving children themselves, ‘real’
objects or mathematical apparatus in which the context is entirely apparent. Similarly, recording of
early addition and subtraction work should also, for the most part, contain some representation of the
operations attempted. This can be in the form of the objects themselves or in pictorial form.
Moreover, the addition symbol, +, and the subtraction symbol, −, also both have a multiplicity of
meanings. This can be seen in the conceptual structures for addition and subtraction which follow.
The manipulative actions undertaken by a child using real or mathematical objects, and which are
represented by the 3 + 2 = 5 addition statement, vary according to context of the task. The outcomes
of these potentially different physical processes resulting from adding 3 + 2 in different contexts can
all be represented by the number 5 and so mathematicians can use the statement 3 + 2 = 5 to
represent many different types of addition. This efficiency and economy of expression is one of the
attractions of mathematics. However, for young children, the representation of very different
physical manipulations of objects by the same set of mathematical symbols (3 + 2 = 5) is often
confusing because of the high degree of generalisation and abstraction involved. Therefore,
mathematical symbols at this early stage should only be used alongside other forms of representation
such as pictures or actual objects. A similar complexity applies to subtraction which also has a
multiplicity of meanings which are dependent upon context.
The different conceptual structures of addition and subtraction that children will encounter are
explained on the following pages. Note that the model of addition or subtraction adopted for a
calculation depends on the context and phrasing of the question asked.
Addition & Subtraction
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Conceptual Structures in Addition
These are illustrated here for the addition statement 3 + 2 = 5.
1. Combining Two or More Quantities
Two or more discrete quantities are combined to form a larger quantity. This is the inverse of the
‘partitioning’ type of subtraction.
Some examples of this model of addition are:
The dice total
is 5 (spots).
This can also occur in word problems, a typical example of which is the following:
“John had 3 sweets and Tom had 2 sweets. How many did they have
altogether?"
2. Augmentation of One Quantity
This model of addition involves adding to an existing quantity thereby augmenting it. Note that,
because this type of addition only involves one quantity, it is different from the ‘combining’ model
above (which involves at least two quantities). This is the inverse of the ‘reduction’ type of
subtraction.
Some examples of this are:
0
1
2
3
4
5
6
7
This type of addition is also found in word problems such as:
“John had 3 sweets and bought 2 more. How many does he have
now?”
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3. Comparative Addition
This involves a comparison of equivalent situations at least one of which involves addition.
Some examples might be:
3
2
5
A 3-rod and a 2-rod added together are equivalent to a 5-rod.
0
1
2
3
4
5
7
6
A jump of 3 followed by a jump of 2 is equivalent to a larger jump of 5.
A typical scenario in words might be:
“John has 3 sweets in one pocket and 2 sweets in the other. Tom has 5 sweets in one pocket. They
both have the same number of sweets.”
At a more complex level, this model of addition encompasses harder comparative additions such as
2 + 2 + 2 = 3 + 3:
2
2
2
3
0
1
3
2
3
4
5
6
7
Three jumps of size 2 is equivalent to two jumps of size 3.
Addition & Subtraction
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Conceptual Structures for Subtraction
These are illustrated for the subtraction statement 5 − 2 = 3.
1. Partitioning
This involves splitting one quantity into two or more sub-quantities. This is the inverse of the
‘Combining’ model for addition.
Some examples of this are:
and
A typical word problem for this kind of subtraction is:
“In a box are 5 cars. Two are John’s and the rest are Tom’s. How
many are Tom’s?”
2. Reduction
This type of subtraction involves reducing the value of one quantity. This subtraction structure is
often known as ‘take away’. It should be evident from this that ‘take away’ is not an appropriate
description for other types of subtraction and therefore the commonly-held view that ‘subtraction’
and ‘take away’ are interchangeable terms with the same meaning is mistaken. This is a common
misconception among children who may have been incorrectly taught or who have rarely
encountered any other form of subtraction. The ‘reduction’ form of subtraction is the inverse of the
‘augmentation’ model for addition.
Examples of subtraction in the form of reduction are:
-2
0
1
2
3
4
5
6
7
'take away'
3 are left.
2 cubes
Word examples of this type of subtraction often take the form:
“John had 5 sweets and ate two of them. How many were left?”
Addition & Subtraction
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3. Comparative Difference
This form of subtraction involves comparison of 2 quantities and the assigning of a numerical value
to the difference between them. This is the inverse of the ‘comparative addition’ model for addition.
Note that ‘find the difference between 5 and 2’ is equivalent to ‘find the difference between 2 and 5’.
This is not true for other types of subtraction such as reduction (i.e. 5 take away 2 = 2 take away 5).
Some comparative difference examples are:
There are 3 more red cubes than green.
The difference is 3 cubes.
3 cubes
"How many more sweets than Tom does John have?"
This form of subtraction in some contexts can also be considered as being a form of ‘Complementary
Addition’ or ‘Additive Difference’ i.e. finding the difference or complement by adding instead of
subtracting. Some examples of this are:
5
2
?
?
0
1
2
3
4
5
6
7
A typical word problem of this kind of subtraction is:
“I have 2 sweets. How many more must I buy in order to have 5
sweets?”
This additive difference or complementary addition model of subtraction is the one commonly used
by shopkeepers when giving change (counting on from the cost of the items bought up to the amount
tendered in payment).
Addition and Subtraction as Inverse Operations
Since addition and subtraction are inverse operations (i.e. one is the mathematical ‘opposite’ of the
other they should be taught alongside each other rather than as two separate entities. It is important
that children are taught to appreciate and make use of this mathematical relationship when
developing and using mental calculation strategies. By comparing the different conceptual structures
for addition and subtraction we can see that each addition model has a corresponding subtraction
model as its inverse:
Addition Model
Combining
Augmentation
Comparative Addition
Addition & Subtraction
Corresponding (Inverse) Subtraction Model
Partitioning
Reduction
Comparative Difference
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© 2000 Andrew Harris
The Complexities of Addition and Subtraction Language
Additional confusion can be caused by the use of language. The large number of different ways in
which addition or subtraction tasks can be phrased in words means that children become unsure what
operation is required. Also, some words have different meanings in different syntactic constructions.
Language-related problems can be due also to inappropriate or imprecise use of language. Be careful
to use correct mathematical language when talking to children. In particular, be aware that the + sign
should be read as ‘add’ or ‘plus’, the −sign as ‘subtract’ or ‘minus’ and the = sign as ‘equals’.
Children often read + as ‘and’ and = as ‘makes’ which seems acceptable until they become
confronted with a statement of the form 5 = 3 + 2 whereupon reading it as ‘5 makes 3 and 2’
doesn’t make much sense.
Similarly, in subtraction children often interpret the − sign as ‘take away’ which really only applies
to reduction-type subtractions. In subtraction also, the = sign is often read by children as ‘leaves’ (5
take away 3 leaves 2) which can cause difficulties when, at a later date, children are confronted by a
statement of the form 2 = 5 − 3 whereupon reading it as ‘2 leaves 5 take away 3’ makes no sense at
all.
Progression in Teaching Calculation Strategies for Addition
and Subtraction
The National Numeracy Strategy recommends that teachers observe the following progression in
teaching calculation strategies for addition and subtraction:
1
2
3
4
Mental counting and counting objects;
Early stages of mental calculation and learning number facts (with recording);
Working with larger numbers and informal jottings;
Non-standard expanded written methods, beginning in late Year 3, first whole numbers
then decimals;
5 Standard written methods, beginning in Year 4;
6 Use of calculators, beginning in Year 5.
This summary of the required progression in learning for addition and subtraction is outlined in more
detail below.
Preparation for Addition and Subtraction
It is important that children are able to count securely up to at least 10 and preferably higher before
being taught basic ideas about addition and subtraction. In particular, children should know:
1 counting involves attaching one number-label to one object (one-one correspondence);
2 the order of the counting numbers, and that this order is always the same;
3 that the value of the set being counted is the number-label associated with the last object to
be counted;
4 that a number always has the same value whatever the child happens to be counting,
that is, the number 5 is a valid representation of the value of a set of 5 elephants or,
equally, of a set of 5 multilink cubes;
5 that the order in which the child counts a set of objects does not matter. Thus, rearranging
the objects does not alter the value of the set.
Addition & Subtraction
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The process of introducing basic addition and subtraction ideas goes hand-in-hand with refining
children’s counting strategies.
Progressing from Counting to Addition
Initially, this involves teaching children to refine their counting strategies as follows:
1
2
3
Counting all
A child calculating 2 + 3 counts out 2 bricks, then counts out 3 bricks and finally counts
all the bricks to find the total.
Counting on from the first number
To calculate 3 + 5 the child counts on from the first number (i.e. 3) : ‘four, five, six,
seven, eight’.
Counting on from the larger number
The child selects the larger number, even when it is not the first number and counts on
from this.
After much repeated teaching of simple addition facts in many contexts the child becomes
able to recall from memory certain common facts. In particular, it is important for the
child to master doubles of the numbers from 1 up to 10 (4 + 4, 5 + 5 etc.), number pairs
whose sum is 10, and facts involving addition of numbers 0 - 10. Since counting becomes
inefficient (and eventually almost impossible) as a strategy when numbers grow in size
the mastery of such number facts is crucial. The progression in learning continues:
4
5
6
Using a known addition fact
The child gives an immediate response for facts known by heart.
Using a known fact to derive new facts
The child is able to use a known fact to calculate one that is unknown e.g. using
knowledge of 7 + 7 = 14 to work out 7 + 8 = 15 or 6 + 7 = 13.
Using knowledge of place value
The child works out facts such as 20 + 30 = 50 from knowing 2 + 3 = 5 or calculates that
37 + 12 = 49 from knowing that 37 + 10 = 47.
Once the child understands how place value knowledge (especially the idea of
partitioning) can be used as an aid, it is then possible to begin the process of developing
written algorithms (i.e. methods/routines) for tackling harder addition calculations, as
follows:
7
Using informal paper jottings, such as an ‘empty number line’, as an aid to mental
calculation of 2 digit additions .
8 Adding in columns, most significant digits first.
9 Adding by rounding the number to be added and then compensating.
10 The standard algorithm i.e. adding in columns, least significant digits first (initially with
no carrying, then with carrying).
11 Addition of decimals.
Addition & Subtraction
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Progressing from Counting to Subtraction
A similar kind of learning process is involved in developing understanding of subtraction. The
process begins by refining counting techniques but then progresses to learning of key facts and
making use of these for harder calculations:
1
2
3
4
5
6
7
Counting out
The child calculating 6 − 4 holds up six fingers, folds down four and counts the remaining
upright fingers.
Counting back from larger number
To calculate 6 − 4 the child counts back four numbers, thus: ‘five, four, three, two’.
Counting back to the smaller number
The child working out 6 − 4 counts back from six to four, keeping a tally using the fingers
of the numbers that are uttered: ‘five, four’ (holding up 2 fingers).
Counting up from smaller to larger number
To work out 6 -4 the child counts up from 4 to 6 (‘five, six’), keeping count of the spoken
numbers using fingers. Children often find this confusing because of their mistaken
perception that subtraction is solely about ‘taking away’.
Using known facts
Rapid responses are offered to known facts e.g. 7 − 4 or 20 − 5.
Using a known fact to derive new ones
e.g. the child knows 20 − 5 = 15 and uses this to calculate 20 − 4 = 16 or 21 − 5 = 6.
Using knowledge of place value
e.g. a child who knows that 34 − 10 = 24 uses this when working out 34 − 9 = 25.
As in addition, using place value knowledge previously acquired (especially the idea of
partitioning), it is then possible to begin the process of developing written algorithms (i.e.
methods/routines) for tackling harder subtraction calculations, as follows:
8
Using informal paper jottings, such as an ‘empty number line’, as an aid to mental
calculation of 2 digit subtractions (using complementary addition or compensation
methods).
9 Using an expanded form of the standard decomposition algorithm.
10 The standard algorithm using decomposition.
11 Subtraction of decimals.
Addition & Subtraction
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Learning Basic Addition and Subtraction Facts
As shown above, the initial emphasis is on progressing from merely counting to knowledge (i.e.
instant recall) of certain frequently used number facts which will be of great assistance in harder
calculations at a later stage in the learning process. It is important that children are given
opportunities to use a wide variety of equipment in learning these basic number facts. This enables
the teacher to provide learning opportunities of the same number facts in different contexts. This
also allows the child to understand (eventually) that 2 + 6 = 8 is true whether he/she is using
multilink cubes, pebbles or spots on a domino. In this way, the child learns to discard all irrelevant
features of the equipment being used (e.g. colour, shape) and begins to recognise over a period of
time the relevant feature which is common to all the apparatus used (in this case, the number of
cubes, pebbles or spots) thus abstracting the number fact 2 + 6 = 8 from the range of contexts
experienced.
In particular, children should be expected to:
•
•
•
•
add/subtract 1 or 2 from other numbers up to 10
double numbers up to 10
halve even numbers up to 20
use the commutative law of addition
e.g. knowing that 2 + 6 = 6 + 2 = 8.
Familiarity with this law (the child would not be expected to know its name) immediately
reduces the number of addition facts to be learned. Note that there is no corresponding
commutative law for subtraction i.e. 9 − 2 = 2 − 9.
• know the pairs of numbers that add up to 10:
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10 etc.
• relate corresponding addition and subtraction facts:
4+3=7
3+4=7
7−4=3
7−3=4
• add 10 to any number from 0 - 10
• use different methods for calculating:
e.g. 9 + 3 = 9 + 1 + 2
or 9 + 3 = 10 + 3 − 1
• number pairs which add up to 20 (and the corresponding subtraction facts)
• begin to understand place value
• recognise patterns in calculations
e.g. 97 − 11 = 86
87 − 11 = 76
77 − 11 = 66 etc.
The National Numeracy Strategy gives examples of those addition and subtraction facts that
children should be able to recall rapidly in Section 4 pages (Reception), Section 5 pages 30 - 31
(Years 1- 3) and Section 6 pages 38 - 39 (Years 4 - 6).
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Some common activities for acquiring familiarity with these frequently-used facts are given below:
The ‘Story’ of a Number
Children are asked to find many ways of ‘making’ a given number, initially for numbers up to 5 and
then extending this to numbers up to 10 and numbers up to 20. It is possible to use a wide variety of
apparatus to do this.
For example:
Using Unifix or Multilink ‘Trains’
'The Story of 5'
0+5=5
1+4=5
2+3=5
3+2=5
4+1=5
This can then be extended to include, for example:
1+2+2=5
1+1+1+1+1=5
This is the ‘combining’ model of addition. It is, of course, possible (and desirable) to use this for the
equivalent subtraction facts using the ‘partitioning’ model of subtraction, i.e. 5 − 0 = 5, 5 − 1 = 4,
5 − 2 = 3, 5 − 3 = 2, 5 − 4 = 1, 5 − 5 = 0 etc.
Similar constructions can be made with Cuisenaire rods when building a ‘wall’ for a given number:
5
4
1
3
2
2
1
Addition & Subtraction
‘Building a wall’ for the number 5
3
4
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© 2000 Andrew Harris
Hand games
Take a given number of multilink cubes (or pebbles, buttons etc.). Ask the children to close their
eyes while you conceal some of them in one hand leaving the remainder visible in the other hand.
The children can then be asked how many have been hidden.
This is a good way of looking at addition tasks of the form 4 +
= 7 and at subtraction facts of the
form 7 −
= 4.
Partitioning a Given Number of Beads on a String
e.g. partitioning 7 beads
For addition: 3 + 4 = 7
For subtraction: 7 - 3 = 4
Dominoes
For addition, find all the dominoes with a given total number of spots.
both have a total of 7 spots.
For subtraction, find the dominoes with a given difference between the number of spots on each
half.
Using an Equaliser Balance
Put a ‘weight’ on the peg for a given number on one side of the balance and investigate on which
pegs on the other side of the balance you need to put ‘weights’ in order to balance the equaliser
balance.
e.g. a ‘weight’ on the 7 peg on the left hand side of the equaliser balance can be balanced by putting
‘weights’ on the 2 and 5 pegs on the right hand side so 7 = 2 + 5 or 2 + 5 = 7.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
Using 0-9 Digit cards or 0-9 Number Petals
Find pairs of numbers that add up to 10. Order the pairs to observe the pattern:
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10 etc.
Find pairs of numbers with a given difference instead. Again, order the pairs to observe the pattern.
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Teaching Mental Strategies
Once some basic facts are known children should be encouraged to work out slightly harder facts
from those which they already know. In this way, the number of known facts that children can
instantly recall will gradually increase.
However, in order to derive new facts from known facts children will need to be aware of a range of
possible strategies for making use of what they already know. While some more mathematicallyable children will develop their own strategies without input from the teacher, it is important to note
that, for many children, such mental strategies must be taught explicitly; it is not adequate merely to
‘do some mental mathematics’ and hope that all children will recognise the strategies involved.
Consequently, the teacher must be pro-active in order to ensure that children acquire the necessary
understanding of, and familiarity with, appropriate strategies. It is important that the child should
be taught mental calculation methods prior to those for written calculation.
Some ways of teaching mental strategies are listed below. Using a mixture of these methods over a
period of time should enable children to acquire the ability to use successfully those strategies which
are taught.
Teachers can:
• explain to the children how to do something;
• use a child’s error or a less efficient strategy as the starting point for a
demonstration of a better strategy;
• encourage children to improve on their strategies;
• talk with the children about choosing strategies and the merits of each;
• show the children how to use something they already know in developing a new
strategy;
• provide the children with a ‘prompt’ (a way of learning and then remembering a
strategy or number fact);
• use apparatus to model a strategy;
• support the initial development of a strategy with rough jottings;
• build mental visualisation of a strategy.
The mental/oral starter and the plenary sections of daily mathematics lessons are good times to
discuss and teach mental strategies with the whole class. In this way children will encounter the
calculation methods used by other children as well as their own and those of their teacher.
Characteristics of mental addition and subtraction strategies
In order for children to achieve success in mental addition and subtraction tasks they will
increasingly need familiarity with aspects of place value. In particular, it is crucial that children
understand, and are able to, partition numbers into their constituent parts by recognising the value of
the digits within the numbers, thus
36 = 30 + 6; 147 = 100 + 40 + 7 and, at later stages, 5·23 = 5 + 0·2 + 0·03.
For further information on how to teach partitioning see the Place Value Booklet.
They will also need to be familiar with the relative sizes of numbers (from ordering activities in
place value work) and with the relationship between addition and subtraction.
It should be noted that mental methods often differ from pencil-and-paper methods. Many mental
Addition & Subtraction
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© 2000 Andrew Harris
methods involve:
• working from left to right (starting with the most significant digits) whereas standard
written methods often work from right to left (starting with the least significant digits);
• remembering the place value of each of the digits so 83 is thought of as 80 + 3. In written
methods 83 is often (though wrongly) thought of as ‘an 8 and a 3’.
• tackling an easier calculation and then adjusting the answer appropriately, the which rarely
occurs in written standard methods.
In a similar vein, it should also be noted that most calculations can be performed by more than one
method. The method adopted by a child will depend on the particular numbers involved and the
strategies with which the child is familiar and ‘comfortable’. Different children will inevitably adopt
different methods as the following example shows:
Year 3 children's methods for 19 + 19
1. 10 + 10 = 20
5 + 5 = 10
4+4= 8
20 + 10 + 8 = 38
2. 19 + 10 = 29
29 + 9 = 38
3. 20 + 20 = 40
40 − 2 = 38
4. 19 + 20 = 39
39 − 1 = 38
5. 19 + 19 = 18 + 20
18 + 20 = 38
6. 19 × 2 = 38
(from NNS Five Day Course materials)
All of these are valid methods, though some are more efficient than others.
Children, when calculating mentally, should be taught to be selective about the strategies they
employ.
Any strategy used should be:
• understood by the child
• reliable - the child can apply the strategy accurately and consistently
and, preferably,
• efficient
The National Numeracy Strategy’s Framework for Teaching Mathematics lists many examples of
mental strategies for addition and subtraction in Section 4 pages 14 - 17 (Reception), Section 5 pages
32 - 41 (Years 1-3) and Section 6 pages 40 - 47 (Years 4 - 6).
Addition & Subtraction
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Common Mental Strategies for Addition and Subtraction
1 Rearranging Numbers
e.g. putting the larger number first when adding using the commutative law
so 7 + 23 , for example, is more easily tackled as 23 + 7.
2 Using Repeated Operations
e.g. subtracting 300 by subtracting 100 three times.
3 Compensation/Adjustment
e.g. calculating 56 + 19 by working out 56 + 20 and then subtracting 1.
4 Using Near Doubles or Halves
e.g. calculating 16 + 17 as ‘double 16 and add 1’ or ‘double 17 and subtract 1’
or
calculating 24 − 13 as (24 − 12) − 1
5 Partitioning
e.g. 27 + 34 = (20 + 30) + (7 + 4) = 50 + 11 = 61
or
27 + 34 = ( 20 + 30) + 7 + 4 = 50 + 7 + 4 = 57 + 4 = 61
6 Bridging to the next Multiple of 10 or 100
e.g. calculating 158 + 17 by splitting 17 into 2 + 15
so 158 + 17 = (158 + 2) + 15 = 160 + 15 = 175
7 Using Inverse Relationships
e.g. calculating 100 − 60 by knowing that 60 + 40 = 100 and that addition and
subtraction are inverse operations.
8 Rounding (for the purpose of approximating answers)
e.g. checking that 24 + 67 = 91 by rounding 24 to 20, 67 to 70
and then adding 20 + 70 = 90
9 Using Equivalences
e.g. calculating 101 − 35 by noting that this is equivalent to calculating 100 − 34
or
calculating 5 + 8 by noting that this is equivalent to 5 + 5 + 3 = 10 + 3 = 13
The National Numeracy Strategy recommends that the focus should be on mental methods,
supported by informal jottings where necessary, up to the end of Year 3. When written methods are
introduced, mental skills should continue to be kept sharp and further developed. Any calculation
should be done mentally if at all possible. Failing this, a written method may be adopted.
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Using recording to assist mental calculation
Children can:
• jot down numbers part-way to the answer;
• use a blank number line to help them through the steps:
e.g. for 27 + 18:
+20
–2
27
45
47
e.g. for 87 − 26:
+ 50
+4
26
30
+7
80
87
so 87 - 26 = 50 + 7 + 4 = 61
(using complementary addition in order to subtract)
• read number sentences (equations) and record answers in a box,
e.g.
5+
= 9;
• record and explain mental steps using numbers and symbols,
e.g. for 27 + 18, calculated as a two-stage process in which the results of intermediate steps
are recorded so as to reduce the number of items to be remembered:
27 + 20 = 47
47 - 2 = 45.
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Developing Written Algorithms (Methods) for
Addition and Subtraction
Ultimately, the aim, as far as written methods are concerned, should be to ensure that as many
children as possible can, by the age of 11, carry out a standard written method for both addition and
subtraction. The key principles to consider when guiding children through this developmental
process are:
• Ensure that recall skills are established first so children can concentrate on a written
method without reverting to first principles.
• Remember that children must understand partitioning and the idea of ‘zero as a place
holder’ from place value work before beginning formal written methods.
• Once written methods are introduced, ensure that children continue to look out for and
recognise the special cases that can be done mentally.
• Cater for children who progress at different rates; some may grasp a standard method
readily while others may never do without considerable help.
• Recognise that children tend to forget a standard method if they have no understanding of
what they are doing.
• Ensure that those written methods to be taught are derived clearly from mental methods
which are already established.
• Introduce expanded formats for written algorithms and gradually refine/contract these into
the more compact standard format.
• Tackle examples without carrying (in addition) and without exchanging/decomposing (in
subtraction before examples that do involve these processes.
• Eventually extend written methods developed for whole numbers to calculations involving
decimals.
Progression in Written Methods for Addition
The National Numeracy Strategy recommends the following progression of methods:
1 Counting on in Multiples of 100, 10, or 1
86 + 57 = 86 + 50 + 7 = 136 + 7 = 143
+50
+4
86
136
or
+3
140
143
356 + 427 = 356 + (400 + 20 + 7)
= 756 + 20 + 7
= 776 + 7
= 783
+400
356
Addition & Subtraction
+20
756
+4
+3
776 780 783
17
© 2000 Andrew Harris
2 Horizontal Format using Partitioning, adding most significant digits first
The most significant digits are added first since this is how children will be used to adding
two-digit numbers mentally
e.g.
67 + 24 = (60 + 20) + ( 7 + 4) = 80 + 11 = 91
3 Expanded Vertical Layout, adding most significant digits first
e.g. adding 2 two-digit numbers
e.g. adding 2 three-digit numbers
36 8
+ 493
––––
700
150
11
––––
86 1
47
+
76
––––
110
13
––––
123
This algorithm (calculation) method can be modelled using Dienes’ base 10 apparatus as
shown (on the following pages) for the example 47 + 76 = 123:
hundreds
tens
ones
(a) place appropriate Dienes' blocks for each of
the quantities on a HTU board.
This uses the idea of partitioning from place
value:
47 = 40 + 7 = 4 tens + 7 ones
and 76 = 70 + 6 = 7 tens + 6 ones.
Addition & Subtraction
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(b) Add the most significant (right - most) column first and exchange where possible.
exchanging 10 tens
for 1 hundred
+
4 tens
40
=
+
+
7 tens
70
=
=
tens
hundreds
+
=
11 tens
110
=
=
1 hundred
100
+ 1 ten
+
10
ones
(c) Place the resulting Dienes' blocks
in the appropriate columns
of the intermediate answer box.
(d) Add the next most significant column and exchange where possible.
exchanging 10 ones
for 1 ten
7 ones
7
+
=
+
+
6 ones =
6
=
Addition & Subtraction
=
13 ones
13
+
= 1 ten + 3 ones
= 10 +
3
19
© 2000 Andrew Harris
hundreds
tens
ones
(e) Place the resulting Dienes' blocks
in the appropriate columns
of the intermediate answer box.
(f) Add up each of the columns in the intermediate answer box.
hundreds
tens
ones
(g) Place the resulting Dienes' blocks
in the appropriate columns
of the final answer box
thus giving an answer of 213.
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The addition of two three-digit numbers by this method can be modelled in the same way
but using an intermediate answer box with 3 rows instead of two (one for the addition of
the hundreds, one for the addition of the tens and one for addition of the ones).
4 Vertical Format, adding least significant digits first
The change to least significant digits first is in preparation for ‘carrying’.
e.g. adding 2 two-digit numbers
e.g. adding 2 three-digit numbers
36 8
+ 493
––––
11
150
700
––––
86 1
47
+
76
––––
13
110
––––
123
This adapted algorithm can be modelled using Dienes’ apparatus in the same way as
described for the algorithm in section 3 above but starting with the least significant (leftmost) columns first when working out the contents of the intermediate answer box.
5 The Standard (Contracted) Format, using ‘carrying’ where appropriate
Note that ‘carrying’ involves the use of place value knowledge when converting a given
number of ones (greater than 9) into tens and remaining ones or when converting a given
number of tens into hundreds and remaining tens.
i.e. in the first example below, the total of the ones column is 13. The child, using
knowledge of place value relationships, decides that 10 of the 13 ones can be exchanged
for 1 ten leaving 3 ones unexchanged (13 = 10 + 3 = 1 ten + 3 ones). So the digit 3 is
entered in the ones column of the answer line and 1 ten is ‘carried’ in the tens column
below the answer line.
e.g. adding 2 two-digit numbers
47
+
76
–––– contracts to
13
110
––––
123
Addition & Subtraction
e.g. adding 2 three-digit numbers
47
+
76
––––
123
––––
11
36 8
+ 493
––––
11 contracts to
150
700
––––
86 1
21
36 8
+ 493
––––
86 1
––––
11
© 2000 Andrew Harris
This can be modelled with Dienes’ base 10 apparatus as follows:
hundreds
tens
ones
(a) place appropriate Dienes' blocks for each of
the quantities on a HTU board.
This uses the idea of partitioning from place
value:
47 = 40 + 7 = 4 tens + 7 ones
and 76 = 70 + 6 = 7 tens + 6 ones.
hundreds
tens
ones
(b) Add the least significant (right-most) column
Addition & Subtraction
22
© 2000 Andrew Harris
hundreds
tens
ones
(c) Add the least significant (left-most) column
and exchange if possible.
After any exchange, put the remaining ones
in the answer box and 'carry' any ten produced
by the exchange process below the answer box.
Exchanging 10 ones
for 1 ten
hundreds
tens
ones
(d) Add the next least significant
column and exchange if possible.
After any exchange, put the remaining tens
in the answer box and 'carry' any hundred
produced by the exchange process
below the answer box.
Exchanging 10 tens
for 1 hundred
Addition & Subtraction
23
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hundreds
ones
tens
(e) Continue to add the columns moving
from right to left in the same manner.
When the contents of all columns have
been added and entered in the answer box
read the value represented by the Dienes'
materials as the answer (in this case, 123).
6 Extend to numbers with more digits
7 Extend to decimal numbers
These are tackled using the same algorithms as illustrated above for whole numbers.
To use Dienes’ base 10 apparatus to model the algorithm simple relabel the value of each
of the Dienes’ blocks so that the small cube represents the value of the least significant
column, the ‘long’ block as the value of the next-to-least significant column and so on.
The most significant change for children is that they now have to align the numbers to be
added so that the decimal points of the respective numbers are above/below each other as
shown in Example (i) below. It is quite common for children, when adding whole
numbers, to perceive (incorrectly) the required alignment of the digits of the respective
numbers as ‘aligning of the digits in the right-most columns above/below each other’ and
to continue to do this for decimals at which point errors become apparent (as in Example
(ii) below).
Example (i)
3 2.4
+ 2.5 6 7
Example (ii)
+
3 2.4
2.5 6 7
It follows that children should be taught, even for whole number additions, to align the
digits of the second number (and any additional numbers) to be added underneath those of
the first number according to the place value of each of the digits, that is, tens under tens,
ones under ones, tenths under tenths etc. . In this way, the misconception outlined above,
Addition & Subtraction
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which only becomes apparent when additions of decimal numbers of the type shown in
example (i) are encountered, may be avoided.
Further details about this progression for written addition methods can be found in the National
Numeracy Strategy Framework for Teaching Mathematics in Section 5, page 43 (Year 3) and
Section 6, pages 48 - 49 (Years 4 - 6).
Progression in Written Methods for Subtraction
The National Numeracy Strategy recommends the following progression in written methods:
1 Counting Up from the Smaller to the Larger Number
This combines use of the additive difference (or complementary addition) subtraction
structure with use of the ‘empty number line’ as an elementary written format.
for subtraction of two-digit numbers
84 – 56
56 + 4 + 20 + 4 = 84
56
60
80
+20
+4
+4
60
56
80
84
4
20
4
84
28
for subtraction of three-digit numbers
783 – 356
+4
356
+300
+40
360
400
+83
700
783
783
- 356
4
40
300
83
427
to
to
to
to
360
400
700
783
2 Compensation (take too much, add back)
for subtraction of two-digit numbers
84 – 56 = 84 – 60 + 4 = 24 + 4 = 28
– 60
+4
24
28
Addition & Subtraction
84
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for subtraction of three-digit numbers
783 – 356 = 783 – 400 + 44
= 383 + 44
= 427
783
– 356
383
+44
300
120
7
427
ta ke 400
a d d 44
3 Expanded Form of Standard Algorithm (with no decomposition necessary)
This stage is the beginning of the standard algorithm but in a much expanded format. The
purpose of this expansion is to allow the child to understand the algorithm, rather than
merely achieving rote learning of a method. Initially, the expanded format is introduced for
subtractions in which the digits of the first number are all larger than the corresponding
digits of the second number thus allowing the child to become familiar with the new
algorithm without the complications of decomposition (for explanation of ‘decomposition’
see below). Notice that this algorithm relies heavily on the child’s ability to partition
numbers into their constituent parts e.g. 88 = 80 + 8, 563 = 500 + 60 +3.
for subtraction of two-digit numbers: 88 − 57
88 = 80 + 8
– 57
50 + 7
30 + 1 = 31
for subtraction of three-digit numbers: 563 − 241
-
500 + 6 0 + 3
200 + 40 + 1
–––––––––––––
300 + 20 + 2 = 322
This algorithm can be modelled using Dienes’ base 10 apparatus as follows:
e.g. for 253 − 121
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hundreds
ones
tens
(a) partition the first number into hundreds,
tens and ones: 253 = 200 + 50 + 3.
Place the corresponding Dienes' blocks
in the relevant columns.
200
hundreds
50
ones
tens
200
3
Notice that for addition with Dienes'
apparatus both numbers are represented
but for subtraction only those blocks
representing the first number are needed
(a source of confusion for some children).
50
3
(b) Partition the second number into
hundreds, tens and ones:
121 = 100 + 20 + 1.
Remove Dienes' blocks equivalent to
the second number from the first, one
column at a time.
100
Addition & Subtraction
20
1
27
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hundreds
ones
tens
200
50
100
20
3
1
(c) Place the blocks which remain from the
first number after the subtraction has been
performed in the answer box and combine
the values of these blocks to produce the
required answer:
100 + 30 + 2 = 132.
100
30
2
If the algorithm is to be modelled using Dienes’ base 10 apparatus, it is recommended that
the algorithm is practised initially using just the Dienes’ apparatus, then using the
apparatus alongside written recording and, finally, using written recording without the
apparatus. The aim of using Dienes’ apparatus should be to ensure understanding of the
algorithm and not as a prop for calculation.
4 Expanded Form of Standard Algorithm (where decomposition is necessary)
This is the same method as above but with the complication that some of the digits in the
first number are smaller than the corresponding digits in the second number. This makes
the use of decomposition necessary.
Decomposition can be seen as a different way of partitioning numbers. In the example
below the 63 part of the number 563 has been decomposed as 63 = 50 + 13. Note,
however, that the 563 has been partitioned in the standard way (563 = 500 + 60 + 3) before
decomposing the 60 + 3 into 50 + 13. It is important that the child understands that the
overall value of the first number in the subtraction has not changed; it is still 563.
(a) decomposition of tens into ones (using the exchange 1 ten = 10 ones)
for subtraction of two-digit numbers:
81 - 57
81 = 80 + 1 = 70 + 11
– 57
50 + 7
50 + 7
20 + 4 = 24
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for subtraction of three-digit numbers:
563 - 258
500 + 60 + 3
200 + 50 + 8
–––––––––––––
-
500 + 50 + 13
200 + 50 + 8
––––––––––––––
300 + 0 + 5 =
305
(b) decomposition of hundreds into tens (using the exchange 1 hundred = 10 tens)
569 - 278
500 + 60 + 9
200 + 70 + 8
–––––––––––––
-
400 + 160 + 9
200 + 70 + 8
––––––––––––––
200 + 90 + 1 =
291
(c) decomposition of hundreds into tens and then tens into ones
566 - 278
500 + 60 + 6
200 + 70 + 8
–––––––––––––
(decomposing the hundreds)
-
400 + 160 + 6
200 + 70 + 8
––––––––––––––
(decomposing the tens)
-
400 + 150 + 16
200 + 70 + 8
––––––––––––––
200 + 80 + 8 =
288
This algorithm can be modelled using Dienes’ base 10 apparatus as follows:
e.g. for the subtraction 243 − 179
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hundreds
ones
tens
(a) partition the first number into hundreds,
tens and ones: 243 = 200 + 40 + 3.
Place the corresponding Dienes' blocks
in the relevant columns.
40
200
hundreds
ones
tens
200
3
40
3
(b) Partition the second number into
hundreds, tens and ones:
179 = 100 + 70 + 9.
100
Addition & Subtraction
70
9
A decision needs to be made at this point
as to whether or not the subtraction of
individual columns is immediately feasible.
In this example, the tens and ones columns
are not.
30
© 2000 Andrew Harris
hundreds
ones
tens
(c) One of the hundreds blocks is decomposed
into 10 tens in order to make subtraction
of the tens digits feasible.
However, the ones column is still infeasible
so a further decomposition is necessary.
100
140
3
100
70
9
hundreds
ones
tens
(d) This time, one of the tens is decomposed
into 10 ones. This finally makes the
subtraction column by column feasible.
100
130
13
100
70
9
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hundreds
ones
tens
100
130
13
(e) Column by column, blocks equivalent to the
values of the digits in the second number
are removed from the first number.
100
hundreds
70
9
ones
tens
100
130
13
100
70
9
(f) The blocks remaining from the first number
after the subtraction has been performed
are now placed in the answer box and their
values combined to produce the answer:
60 + 4 = 64.
60
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5 Contraction of Expanded Standard Algorithm to Standard Algorithm
As this is just a contracted form of the algorithm outlined in section 4 above, the contracted
forms given here can be modelled using Dienes’ base 10 apparatus in the same way as
described in section 4.
(a) with no decomposition necessary
80 + 8
- 50 + 7
30 + 1 = 31
88
– 57
31
contracts
to
(b) with decomposition of tens into ones (using the exchange 1 ten = 10 ones)
for subtraction of two-digit numbers:
81 - 57
80 + 1 = 70 + 11
7811
contracts
50 + 7
50 + 7
– 57
to
20 + 4 = 24
24
for subtraction of three-digit numbers:
-
500 + 6 0 + 3
200 + 40 + 1
–––––––––––––
contracts
to
56 3
- 24 1
––––
322
(c) with decomposition of hundreds into tens (using the exchange 1 hundred = 10 tens)
-
500 + 60 + 9
200 + 70 + 8
–––––––––––––
-
40 0 + 160 + 9
20 0 + 70 + 8
––––––––––––––
200 + 90 + 1 =
4516 9
contracts
- 2 78
to
291
291
(d) with decomposition of hundreds into tens and then tens into ones
(decomposing the hundreds)
-
500 + 60 + 6
200 + 70 + 8
–––––––––––––
400 + 160 + 6
- 200 + 70 + 8
––––––––––––––
(decomposing the tens)
400 + 150 + 16
- 200 + 70 + 8
––––––––––––––
200 + 80 + 8 = 288
4 15 1
contracts
to
56 6
- 278
28 8
6 Extend to larger numbers
Use the same algorithms (as outlined above) for subtractions of larger numbers with
expanded formats being introduced first, then contracting these to the standard algorithm
when the expanded format has been understood and used accurately over a period of time.
7 Extend to Decimal numbers
There is the potential for children to continue to align the right-most (least significant)
digits of each number vertically even when this is not appropriate for the decimal numbers
involved. The remarks made earlier on this subject regarding addition also apply to
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© 2000 Andrew Harris
subtraction calculations. It is important that children are taught, even for whole number
subtractions, to align the digits of the second number underneath those of the first number
according to the place value of each of the digits, that is, tens under tens, ones under ones,
tenths under tenths etc.. In this way, the misconception outlined above, which only
becomes apparent when subtractions of certain decimal numbers are encountered, may be
avoided.
Further details about this progression for written subtraction methods can be found in the National
Numeracy Strategy Framework for Teaching Mathematics in Section 5, page 45 (Year 3) and
Section 6, pages 50 - 51 (Years 4 - 6).
Subtraction by the Equal Addition Method
While the National Numeracy Strategy and also most schools and published schemes advocate the
teaching of decomposition as the standard algorithm for harder subtractions (in which the some of
the digits of the first number are less than the corresponding digits of the second number), other
methods have been used in the past.
In particular, the ‘equal additions’ method was frequently used. It differs from the decomposition
method in that
• it does not use the principle of exchange (i.e. 1 hundred exchanged for 10 tens or 1 ten for
10 ones etc.);
• it is not easily modelled by mathematical equipment;
• it involves making adjustments to both numbers in the subtraction calculation and not
merely the first;
• the value of the two numbers involved is changed. Contrast this with the decomposition
algorithm where the value of each number remains unchanged throughout.
For these reasons, and because it is hard to teach with understanding (as opposed to rote learning of
the method), it has largely been abandoned in schools. It is based on the notion that the relative
difference between the two numbers involved remains unchanged if the same number is added to
each. It can be viewed as being a shift of both numbers by the same amount (ten) along a number
line:
e.g. 56 - 37 = (50 + 6) - (30 + 7)
= (50 + 6 + 10) - ( 30 + 7 + 10)
('equal addition' of 10 to both numbers)
= (50 + 16) - (40 + 7)
The difference is 19
37
56
+10
+10
addition of 10 to both numbers
('equal addition')
The difference is still 19
37
47
Addition & Subtraction
56
66
34
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The equal addition column-based algorithm looks like this:
for subtraction of two digit numbers:
56
- 37
and for three-digit numbers:
4 5 3
- 1 6 6
1
56
4
- 37
4 5 13
- 17 6 6
7
1
56
4
- 37
19
4 15 13
- 217 6 6
8 7
4 15 13
- 217 6 6
2 8 7
The process can be summarized as working right to left (column by column) and, when the digit of
the first number is smaller than the corresponding digit of the second number, adding ten to the digit
of the first number and also to digit of the second number in the next column.
The equal addition algorithm was often accompanied by the phrase ‘borrow and pay back’. It
appears to have little meaning since nothing is borrowed or paid back in reality. It is, unfortunately,
also quite common to hear pupils (and teachers!) use this phrase when using the decomposition
algorithm even though it has no application to the methodology of the decomposition algorithm
which is based upon the principle of ‘exchanging one of these for ten of those’.
Errors and Misconceptions in Addition and Subtraction Calculations
Diagnosing Difficulties: General Points
Difficulties with calculations can occur for a number of reasons. Incorrect responses may be owing
to:
• computational error/careless mistake
The child uses the correct operation and procedure but incorrectly recalls a basic number
fact(s).
• misconceptions
The child has not grasped the concept of the operation being used (addition or subtraction)
or, in the case of formal, vertically-written algorithms, fails to understand aspects of place
value required in order to understand the algorithm being attempted.
• lack of understanding of relevant vocabulary
The child misinterprets the language used in the question or task.
• wrong operation
The child uses the wrong operation for the question.
• defective procedure or method
The correct operation is chosen and number facts are recalled correctly but there are errors
in the use of the procedure (algorithm) adopted.
• over-generalisation
the child has learned a pattern, ‘rule’, or method and then has applied it to situations where
it is not appropriate (hence the importance of real understanding and not just mechanical
learning of a procedure).
e.g. the child, having been introduced to decomposition in subtraction of two-digit
numbers, then uses decomposition in every subtraction calculation even when
decomposition is not necessary.
• under-generalisation
The child has encountered insufficient examples or examples which have insufficient
Addition & Subtraction
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© 2000 Andrew Harris
variation. A sufficiency of both examples and of variation in examples is required in order
for the child to be able to abstract the conceptual or procedural understanding required and
so, without this sufficiency, the child may generalise on the basis of inadequate knowledge
or experience.
• random response
There is no discernible relationship between the question and the response given.
(adapted from ‘Guide for Your Professional Development Book 2, NNS)
It is important to distinguish between different kinds of difficulties experienced by children (as
evidenced in their work) since the help offered to address such difficulties must be appropriate
otherwise the difficulties will persist (and possibly worsen). The National Numeracy Strategy
guidance materials offer the following general points about diagnosing errors and misconceptions:
• Children's errors are often due to misconceptions or misunderstood rules, rather than
careless slips.
• It is important to diagnose each misconception rather than simply to re-teach the method.
• To make the diagnosis, ask the child to explain how they worked out the answer ...
• ... then deal with the misconception, helping the child to use another method or a simpler
approach that she or he is confident with (e.g. an expanded layout for a written
calculation).
(from ‘Guide for Your Professional Development Book 3’, NNS)
The plenary of the daily mathematics lesson is often a good time to address common misconceptions
and errors because there may be several children who have the same difficulty. That said, it is
incumbent upon the teacher to teach concepts and strategies in ways which will pre-empt or avoid
misconceptions and errors developing.
The examples on the following pages indicate common misconceptions and errors in addition and
subtraction calculations involving two-digit numbers. Many of the difficulties illustrated stem from:
• poor understanding of place value,
or
• possessing a mechanical knowledge of the algorithm involved but not a conceptual
understanding of the processes involved (e.g. carrying, exchange, decomposition), or
• poor understanding about when an algorithm is applicable and when it is not.
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Common Errors and Misconceptions in
Addition of Two-digit Numbers
A.
54
+3
84
Failure to understand the significance of the
place value of the digit 3.
B.
42
+ 9
411
Insertion of a column which does not exist.
Lack of awareness of place value columns
and that 11 = 10 + 1 = 1 ten + 1 unit.
C.
34
+ 9
61
3
D. 6 8
+ 71
39
E. 6 8
+ 34
92
1
F. 2 7
+ 93
210
1
G. 7 0
+ 15
80
H.
99
+ 32
1211
Reversal of the tens and ones digits when
carrying
Failure to understand that the hundreds column
exists even when no digits reside in it initially.
Forgetting to add the carrying digit.
Reversal of hundreds and tens digits in the
answer.
Confusion between multiplication by 0 and
addition of 0.
Failure to appreciate place value columns (as
in B) and that 11 = 10 + 1 = 1 ten + 1 one,
12 tens = 10 tens + 2 tens = 100 + 20.
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Common Errors and Misconceptions in
Subtraction of Two-digit Numbers
A.
36
-42
14
Wrong positioning of the two numbers when
writing down. Frequently occurs when the
problem is presented in words.
B.
58
- 9
51
Failure to appreciate that 8 - 9 = 9 - 8 or
confusion from being told "start with the bigger
number and subtract the smaller number".
C.
60
- 21
40
5
61
- 23
32
E. 611
"Can't do 0 -1 so the answer must be 0".
Perception of the subtraction as 0 - 1 and 6 - 2
instead of 60 - 21.
D.
- 23
48
F.
G.
Partial decomposition (due to mechanical
knowledge of algorithm but without real
understanding of decomposition and of
exchanging 1 ten for 10 ones).
4 1
59
- 28
211
Decomposition when unnecessary (inadequate
understanding of reasons for, and the process
of, decomposition).
25
+ 18
113
Confusion between addition and subtraction
algorithms (i.e. between decomposition and
carrying - both involve exchanging between
tens and ones).
There are also specific difficulties in using the decomposition algorithm which are caused by the
presence of zero digits in the first (top) number of the subtraction. The various types are shown
overleaf.
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Specific Subtraction Errors owing to
the Presence of Zero Digits
H.
411
505
- 186
329
34 1
1
I.
505
- 186
229
J.
K.
Decomposition from hundreds to tens has been
correctly applied but the subsequent
decomposition from tens to ones has only been
partially carried out since the tens digit has not
been reduced by one.
Decomposition of the hundreds twice instead of
hundreds into tens and then tens into ones.
91
505
- 186
419
Decomposition of tens is incorrect. The child's
thinking is 'decomposing a ten leaves 9' (one
less than ten). Failure to understand the notion
of zero as a place-holder is evident.
11
505
- 186
429
Decomposition without decreasing the size of
the tens digit (which is zero). Thus, the
decomposition of tens to ones has only been
applied partially.
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© 2000 Andrew Harris
Some Addition and Subtraction Activities
The following activities are possible ideas to try with children. Many of them can be adapted for use
with either for addition or for the corresponding inverse form of subtraction. In a similar way, many
of them can be adapted for use with decimal addition or subtraction simply by changing the size and
type of the quantities involved.
Addition Activities
1. Two-sided Beans/Counters
Colour one side of all the counters or beans differently from the other. Toss the beans or counters
and record the relevant number sentence which the beans or counters
demonstrate. For the example shown opposite, this could be:
or
2+3=5
5−2=3
for addition
for subtraction.
2. Domino Squares
Make a domino hollow square so that the dots on each side add up to 12.
How many different such squares can you make?
What happens if you change the target number?
Or if you make a bigger domino square?
3. Card Pairs
Play number pairs with a double set of 0 - 20 digit cards turned face down
on the table. Take it in turns to turn a pair of cards over and add. If they add
up to the target number (between 10 & 20) the player keeps the cards as a
pair. If not, the player must turn them face down again and play passes to the
next player. Look at the patterns of card pairs for a given target number.
6
+
11
7
+
10
8
+
9
Number pairs for a
target number of 17
4. Investigating Sums of Consecutive Numbers
(a) Add consectutive number pairs: 0 + 1, 1 + 2, 2 + 3, 3 + 4 etc. Note down the numbers you
produce. Also, colour these numbers on a hundred square. What pattern emerges?
(b) Repeat this adding sets of three consective numbers: 0 + 1 + 2, 1 + 2 + 3, 2 + 3 + 4, ... etc.
Again note the numbers down, colour them on the hundred square in a different colour. What pattern
is produced this time?
(c) Do the same for sets of 4 consective numbers and for sets of 5 consecutive numbers.
(d) Which numbers on the hundred square can never be made by adding consecutive numbers and so
will never be coloured? What pattern do these numbers provide?
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5. Three Number Investigation
(a) Give the children a two-digit number total. Ask them to locate 3 horizontally adjacent numbers
on a hundred square which, when they are added together, produce the required total.
e.g. 36 = 11 + 12 + 13.
Which two-digit totals can be made by adding 3 such numbers? Is there a pattern in the totals which
are possible?
(b) Try the same idea but this time locating 3 vertically adjacent numbers which, when added, give
the required total. What totals are possible? Is there a pattern involved?
(c) Predict what happens with totals of 3 numbers which are diagonally adjacent. Test your
prediction.
6. Rectangles on Number Squares
Locate a 2 × 2 square within the hundred square like the one shown and add up the numbers in the
corners of the square.
2
3
12
13
2 + 13 + 3 + 12 = 30
Try this with several 2 × 2 squares. What do you notice about the relationship between the numbers
in the corners and the total of the corner numbers? What happens for larger squares e.g. 3 × 3, 4 × 4?
What happens with rectangles within the hundred square?
7. Boxes
Each player has a set of shuffled, face down 0 - 9 digit cards and a base board with boxes this:
+
Players are given a target e.g. make the largest total you can. Players take it in turn to turn over one
of their digit cards and place it in a box. Once placed, the digit card may not be moved. Play
continues until each player has filled all four boxes on his/her board. The winner is the one whose
addition produces an answer closest to the required target. This completes one ‘round’ of the game.
Then change the target. Other possible targets include:
• the smallest number;
• a given number e.g. 56;
• an odd/even number;;
• a multiple of 5;
• a prime number;
• some combination of these.
The targets can be set by the teacher (e.g. as a whole-class game or when working with a group), by
means of a set of cards with targets written on them or by rolling 2 ten-sided dice labelled 0 - 9 and
using the numbers produced as the tens and ones digits of the target number.
The game requires children to use properties of numbers, understand aspects of place value and to
estimate the outcomes of adding various combinations of numbers and digits.
The game can be adapted by:
• changing the sign on the board from add to subtract;
• changing the board to include hundreds as well as tens and ones;
• putting a decimal point between the boxes in each row in order to provide children with
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opportunities to explore decimal addition/subtraction.
8. Changing the Number
Use a place value base mat with hundreds, tens and ones columns. Each player takes a handful of
Dienes’ base 10 blocks and places them in the appropriate columns on his/her board. The number of
each type of block each player has at the beginning is unimportant.
Three ten-sided dice (labelled 0-9, preferably different colours) are rolled and the numbers obtained
become the hundreds, tens and ones digits of the target number. Alternatively, obtain a target number
by turning over three cards from a set of 0-9 digit cards.
The target number is placed centrally where all players can see it. Players then take it in turn to either
add one block of any size (from a central pool of Dienes’ blocks) to the appropriate column on their
place value mat or to subtract one block of any size from their collection and return it to the central
pool of blocks. The aim of the game is to change the number of blocks on a player’s place value mat
until the value of his/her blocks matches the target number. The first player to transform their
number of blocks into the target number is the winner.
The game involves children changing one of the hundreds, tens or ones columns on their place value
mat each time it is their turn. The adding or subtracting of a block of any size can be mirrored for
each player on (a) a spike abacus, and (b) a calculator display. The children should be encouraged to
state what operation they are performing e.g. ‘I’m adding/subtracting ten and my number is now 200
+ 50 + 3 = 253’.
Difference Games
1. Clixi Track Difference Game (For 2 players/teams)
Use Clixi to build a long row of linked squares.
You need an odd number of squares. Place one
The difference is 2 so move 2 squares
counter (to be shared by both players) on the
middle square to begin. Each player in turn
throws two dice and finds the difference
between the two numbers obtained. This player
can then move the counter that number of
squares towards his/her end of the row of
squares. Play then passes to the next player.
The winner is the player who manages to reach
his/her end of the row of squares first.
2. Race to One Hundred (For 2-6 players).
Each player starts with a 10x10 grid of squares e.g. squared paper. Take it in turns to throw
2 dice and find the difference between the numbers. On finding the difference that player is allowed
to colour in the corresponding number of squares on their grid. Any disputes can be settled using a
calculator to check. The winner is the player who completes their grid first.
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3. Card Pairs (For 2-6 players)
Make a pack of number cards containing 2 of every number between 0 and 20.Spread the
cards out face down on the table. Players take it in turn to turn over
two cards.
Version 1: If the player can calculate the difference he/she can keep the
pair of cards. The winner is the player with the most pairs
when all the cards have been claimed.
Version 2: If the difference is the same as a previously agreed target
number (between 0 and 19) then the player can keep the pair
of cards. If the difference does not match the target number
then they have to be replaced face down. The winner is the
player with the most pairs at the end. This version is useful
for looking at the pairs of numbers which produce a
particular difference in a structured way. Simply order the
pairs of numbers at the end of the game to emphasise that
20 - 8 = 12,
19 - 7 = 12,
18 - 6 = 12,
17 - 5 = 12 etc.
17
5
16
4
15
3
14
2
Number pairs
for a target
number of 12
4. Pam’s Game (for 2 players)
Each player has a board (or a strip of 2cm squared paper) like this:
Each player throws a dice and places that number of multilink cubes on his/her board. They look at
the difference between the amounts. The player with the largest number takes the difference and
‘keeps it’. This process continues until one player has 10 multilink cubes which he/she has ‘kept’.
That player is the winner.
5. Domino Trains and Loops
Build a straight line of dominoes (a domino ‘train’) such that the difference between the adjacent
halves of each pair of dominoes has a given difference of 0.
e.g.
Try this activity with a different difference between adjacent dominoes e.g. a difference of 1, 2 or 3.
Which is easiest/hardest and why?
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To make this slightly harder, build a domino loop (using a complete set of dominoes) so that the last
domino to be placed has the correct difference between it and the fist half of the initial domino.
Both domino trains and domino loops can be made collaboratively (pool a complete set of dominoes
and help each other complete the task) or competitively by sharing out the dominoes before starting
and taking turns to place a domino at either end of the chain of dominoes. If playing the competitive
version, a player misses his/her turn if he/she cannot put down a domino which gives the required
difference between it and the previous domino.
6. Cover the Number
Each player needs a strip of 6 squares (e.g. drawn on squared paper) numbered from 0 to 5.
Players take it in turns to roll two dice marked 1 to 6 and find the difference between the two
numbers rolled. This difference number is located on the player’s strip of squares and is then covered
with a counter. The first player to cover all the numbers on their strip of squares wins.
7. Sum or Difference?
A blank 5 × 5 grid of squares is drawn on squared paper and shared between 2 players. Each player
in turn throws the two dice and calculates either the sum or the difference of the two numbers
thrown. This sum or difference is entered anywhere on the grid of squares. A point is scored each
time a line (horizontal, vertical or diagonal) totals 15. The game can be played co-operatively or
competitively.
8. Multilink Breakages
Start with a rod of ten multilink cubes. Break the rod into 2 smaller rods and find the difference
between the number of cubes in each smaller rod.
Investigate to find out what differences can be made in this way?
What differences are possible if I start with a rod of 9 cubes instead?
9. Cuisenaire Staircases
Build a staircase from Cuisenaire rods so that the difference between adjacent steps is 1. then try with
differences of 2 or 3 or 4 ... etc.. Find a way of recording which rods you used in each staircase.
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Glossary of Mathematical Terms associated
with Addition and Subtraction
Inverse Operation
Commutativity
The operation which is ‘opposite’ mathematically to that
being considered. Thus, subtraction is the inverse of addition
and vice versa.
The commutative law applies to addition but not to
subtraction. It states that, for any addition statement, the
numbers to be added may be interchanged around the
addition sign without altering the sum of the two numbers:
e.g. 3 + 2 = 2 + 3
Associativity
The associative law applies to addition but not to
subtraction. It states that addition operations may be
interchanged without altering the outcome. In the example
below, the operation in the brackets are performed before
those outside the brackets.
e.g. 3 + (2 + 4) = (3 + 2) + 4
Terms associated with Addition and Subtraction
Number Sentences
3 + 2 = 5
augend
addend
sum or
total
Note that it is, strictly speaking, mathematically incorrect to refer to all calculations as ‘sums’. In
mathematics, the term ‘sum’ means the result obtained by performing an addition calculation. For
example, 37 is the sum of 33 and 4.
5 - 2 = 3
difference
minuend
subtrahend
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Terms associated with Addition and Subtraction
Calculation Methods
Partitioning
Splitting a number, usually into the sum of multiples of powers of ten
e.g. writing 157 as 100 + 50 + 7 because 157 = (1×102) + (5×101) + (7×100).
Exchanging
Changing 10 of a given power of ten for 1 of the next power of ten (or
vice versa
e.g. exchanging 10 ones for 1 ten
exchanging 10 ones
for 1 ten
=
13 ones
13
Decomposition
+
= 1 ten + 3 ones
= 10 +
3
Splitting a multiple of a power of ten into the next smallest whole
number multiple of this power of ten and ten of the next smallest
power of ten (using the principle of ‘exchange’ defined above).
e.g. using knowledge of 1 hundred = 10 tens to decompose one of the 7 hundreds in 700 below into
6 hundreds + 10 tens:
decomposing 1 hundred
into 10 tens
+
7 hundreds
=
6 hundreds
+
10 tens
Decomposition is employed in the standard algorithm for subtraction in cases where a digit of the
minuend is smaller than the corresponding digit of the subtrahend. Decomposition is a special type of
partitioning:
e.g. 67 -—--> 50 + 17 (decomposition)
67 –-----> 60 + 7 (the usual partitioning into multiples of powers of ten).
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