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 46 4 2 53 7 48 54 5 49 5 5 56 61 6 0 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
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Addition & Subtraction

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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.

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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:

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 5

A 3-rod and a 2-rod added together are equivalent to a 5-rod.

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 3

2 3

Three jumps of size 2 is equivalent to two jumps of size 3.

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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 Johns and the rest are Toms. How many are Toms? 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
'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 5 2000 Andrew Harris

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 6 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 doesnt 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: 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 7 2000 Andrew Harris 1 2 3 4

The process of introducing basic addition and subtraction ideas goes hand-in-hand with refining childrens counting strategies. Progressing from Counting to Addition Initially, this involves teaching children to refine their counting strategies as follows: 1 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 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.

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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 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.

2 3

5 6 7

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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 74=3 73=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:

4
3 2 3 2

1
Building a wall for the number 5

1
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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. Addition & Subtraction 12 2000 Andrew Harris

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 childs 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, 523 = 5 + 02 + 003. 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 13 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 Strategys 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).

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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
e.g. for 87 26:

45
+ 50

47

+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 +3
or

86

136

140

143

356 + 427 = 356 + (400 + 20 + 7) = 756 + 20 + 7 = 776 + 7 = 783

+400 +20 +4 +3

356
Addition & Subtraction

756

776 780 783

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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

47 + 76 110 13 123

36 8 + 493 700 150 11 86 1

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.

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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

= =

11 tens 110

= =

1 hundred 100

+ 1 ten + 10

hundreds

tens

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 = 13 ones 13 = +

= 1 ten + 3 ones = 10 + 3

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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

47 + 76 13 110 123

36 8 + 493 11 150 700 86 1

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 e.g. adding 2 three-digit numbers

47 + 76 contracts to 13 110 123

47 + 76 123
11

36 8 + 493 11 contracts to 150 700 86 1

36 8 + 493 86 1
11

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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

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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

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hundreds

tens

ones

(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 24 2000 Andrew Harris

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 4 20 4 28

+20 +4 +4

56

60

80

84

84

for subtraction of three-digit numbers

783 356 783 - 356 4 40 300 83 427

+4

+40

+300

+83

356

360

400

700

783

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

84
25 2000 Andrew Harris

Addition & Subtraction

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 childs 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

tens

ones

(a) partition the first number into hundreds, tens and ones: 253 = 200 + 50 + 3. Place the corresponding Dienes' blocks in the relevant columns. 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).

200

50

hundreds

tens

ones

200

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

20

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hundreds

tens

ones

200

50

100

20

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

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 + 1 3 200 + 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 + 8 400 + 150 + 16 200 + 70 + 8 200 + 80 + 8 =

(decomposing the tens)

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

tens

ones

(a) partition the first number into hundreds, tens and ones: 243 = 200 + 40 + 3. Place the corresponding Dienes' blocks in the relevant columns.

200

40

hundreds

tens

ones

200

40

3 (b) Partition the second number into hundreds, tens and ones: 179 = 100 + 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.

100

70

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hundreds

tens

ones

(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

100

70

hundreds

tens

ones

(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

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hundreds

tens

ones

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

70

hundreds

tens

ones

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 contracts to 88 57 31

(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 + 11 781 1 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 =

contracts - 2 78 to

451 9 6 291

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

contracts to

56 6 - 278 28 8

4 15 1

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 Addition & Subtraction 33 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) = (50 + 16) - (40 + 7) The difference is 19 ('equal addition' of 10 to both numbers)

37 +10

56 +10 addition of 10 to both numbers ('equal addition')

The difference is still 19

37

47

56

66

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The equal addition column-based algorithm looks like this: for subtraction of two digit numbers: 56 - 37 56 4 - 37
1

56 4 - 37 19 4 15 13 - 217 6 6 8 7 4 15 13 - 217 6 6 2 8 7

and for three-digit numbers: 4 5 3 - 1 6 6

4 5 13 - 17 6 6 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 35 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 42 + 9 411 Failure to understand the significance of the place value of the digit 3. Insertion of a column which does not exist. Lack of awareness of place value columns and that 11 = 10 + 1 = 1 ten + 1 unit. Reversal of the tens and ones digits when carrying

B.

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

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 58 - 9 51 60 - 21 40 Wrong positioning of the two numbers when writing down. Frequently occurs when the problem is presented in words. Failure to appreciate that 8 - 9 = 9 - 8 or confusion from being told "start with the bigger number and subtract the smaller number". "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.

B.

C.

5 61 - 23 32 E. 61 1 D. - 23 48 F. 4 1 59 - 28 211 25 + 18 113

Partial decomposition (due to mechanical knowledge of algorithm but without real understanding of decomposition and of exchanging 1 ten for 10 ones).

Decomposition when unnecessary (inadequate understanding of reasons for, and the process of, decomposition). Confusion between addition and subtraction algorithms (i.e. between decomposition and carrying - both involve exchanging between tens and ones).

G.

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 411 505 - 186 329 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.

H.

34 1 1 I. 505 - 186 229

J.

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. Decomposition without decreasing the size of the tens digit (which is zero). Thus, the decomposition of tens to ones has only been applied partially.

K.

11 505 - 186 429

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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. 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? 2+3=5 52=3 for addition for subtraction.

6
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.

11 10 9

7 8

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 12 3 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 Addition & Subtraction 41 2000 Andrew Harris

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 players 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. Im 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 4 3 2

16 15

14

Number pairs for a target number of 12 4. Pams 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? Addition & Subtraction 43 2000 Andrew Harris

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 players 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 minuend subtrahend difference

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Terms associated with Addition and Subtraction Calculation Methods Partitioning Exchanging
Splitting a number, usually into the sum of multiples of powers of ten e.g. writing 157 as 100 + 50 + 7 because 157 = (1102) + (5101) + (7100).

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 +

= 1 ten + 3 ones = 10 + 3

Decomposition

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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