Two Digit Addition & Subtraction Split Strategies

There is a need to develop of a range of strategies which are referred to as split strategies. The child splits each of the two addends into tens and ones and then separately combines, tens with tens and ones with ones. This includes the topic of higher decade addition and subtraction, a forerunner to 2-digit addition and subtraction. There is a focus on extending strategies for adding and subtracting involving two 2-digit numbers, to working with two 3-digit numbers. The focus on jump strategies can precede the focus on split strategies or vice versa.

Split Strategies can be a sweet treat for students

Higher Decade Addition and Subtraction

Higher decade addition and subtraction involves a 2-digit number and a number in the range 1 to 10. This is because each addition or subtraction of this kind can be linked to a corresponding addition or subtraction respectively, in the range 1 to 10 or 1 to 20. For example, 43 + 5 can be linked to 3 + 5, 75 - 2 can be linked to 5 - 2. 76 + 9 can be linked to 6 + 9, and 52-7 can be linked to 12 - 7. Thus solving a higher decade addition or subtraction task can involve using the corresponding addition or subtraction respectively. For example, 3 + 4 is used to work out 53 + 4, 7 + 8 is used 10 work out 37 + 8, 7 - 2 is used to work out 87-2 and 14 - 9 is used to work out 44 - 9. Second, by starting with addition (or subtraction) in the range I to 10 or 1 to 20, a sequence of additions (or subtractions) can be determined. For example, 3+4, 13 + 4, 23 + 4, and so on; 7 + 8, 17 + 8, 27 + 8, and so on 7-2, 17 - 2. 27 - 2, and so on; 14 - 9. 24 - 9, 34 - 9, and so on. Children can come to see the simple pattern in these kinds of sequences involving addition or subtraction.

Within the Decade - Addition

This involves cases such as 47 + 2, 83 + 5, 64 + 4, and so on. Children should be able to solve these tasks using the corresponding addition in the range 1 to 10, for example using 7 + 2 to work out 47 + 2. Tasks of this kind can be an important building block for adding two 2-digit numbers. For example, when a child works out 27 + 22 by first adding 27 and 20, the next step involves adding 47 and 2.

Within the Decade - Subtraction

This involves cases such as 49 - 2, 88 - 5, 68 - 4, and so on. Children should be able to solve these tasks using the corresponding subtraction in the range 1 to 10. for example using 9 - 2 to work out 49 - 2. Tasks of this kind can be an important building block for subtraction involving two 2-digit numbers. For example when a child works out 69 - 22 by first working out 69 - 20, the next step involves working out 49 - 2,

Beyond the Decade - Addition

This involves cases such as 46 + 7, 77 +- 5, 29 + 6. and so on. Two ways that children might work out this problem are: (a) using the corresponding addition in the range 1 to 20, for example, using 6 + 7 to work out 46 + 7; and (b) working the addition out directly without using 6 + 7, for example, first solving 46 + □ = 50, then partitioning 7 into 4 and 3, and finally adding 50 and 3. Tasks of this kind can be an important building block for adding two 2-digit numbers. For example, when a child works out 26 + 27 by first adding 26 and 20. the next step involves adding 46 and 7.

Beyond the Decade - Subtraction

This involves cases such as 53 - 7, 82 - 5, 35 - 6, and so on. Two ways that children might work out this problem are: (a) using the corresponding subtraction in the range 1 to 20, for example, using 13 - 7 to work out 53 - 7; and (b) first solving 53 - □ = 50, and then partitioning 7 into 3 and 4, and finally solving 50 - 4. Tasks of this kind can be an important building block for subtracting two 2-digit numbers. For example, when a child works out 73 - 27 by first working out 73 - 20, the next step involves working out 53 - 7.

Sequences in Higher Decade Addition and Subtraction

Children can work with and notate sequences such as the following:

5+2 = 7, 15 + 2= 17, 25 + 2 = 27, 35 + 2 = 37, and so on.
8 - 3 = 5, 18 - 3 = 15, 28 - 3 = 25, 38 - 3 = 35, and so on.
8 + 4 = 12, 18 + 4 = 22, 28 + 4 = 32, 38 + 4 = 42, and so on.
11 - 6 = 5, 21 - 6 = 15, 31 - 6 = 25, 41 - 6 = 35, and so on.

Working with these sequences can help children make connections from a higher decade addition or subtraction to a corresponding addition or subtraction in the range 1 to 10 or 1 to 20.

Fostering the Development of Split Strategies

Split strategies involve calculating separately with tens and ones. First, the child calculates with the tens, then with the ones, and finally the resultant tens and ones are combined. The split strategy is easier to use when the calculation in the ones does not involve numbers larger than 9. Like: 42 + 25, 36 + 21, 36 + 53, and in the case of subtraction: 86 - 32, 44 - 23, 97 - 55.

Prior Knowledge for the Split Strategy

An important point to keep in mind is that when children are working these examples, they should be facile with adding and subtracting in the range 1 to 10. When a child works out 36 + 53, they should be able to calculate each of 30 + 50 (using 3 + 5), and 6 + 3 almost instantaneously, and similarly calculate 80 + 9 almost instantaneously. The child who is using strategies involving counting-by-ones to calculate 3 + 5 and 6 + 3 is not ready to learn 2-digit addition to 100 This applies similarly in the case of subtraction. In using a split strategy to work out 86 - 32, the child should be able to calculate each of 80 - 30 and 6 2 almost instantaneously.

Base-ten Blocks and Bundling Sticks

Settings such as bundling sticks or base-ten blocks are very useful for fostering the development of split strategies. These can be used by the teacher working with a group or whole class using an image projector. Also, the teacher can gradually develop situations in which the materials are screened (hidden from view) to encourage visualization on the part of the children. An important point is that simply giving base-ten materials to each child with instructions to work examples such as those above is unlikely to foster split strategies. In this situation our instructional goal is to foster mental strategies. When children are given the materials they are likely to use several strategies that are not conducive to developing more facile strategies. Strategies that tend to be counterproductive at this point include: counting a collection of blocks from one, counting-by-ones, and counting blocks that can be seen.

Means of Notating for Split Strategies

There are three means of notating in the case of jump strategies; the empty number line, arrow notation and horizontal number sentences. As well, we described how horizontal number sentence involve writing a number sentence for each step in the strategy and can be used for virtually any strategy. This includes jump, split, split-jump, and a variation of the split strategy involving working from right to left. Because of its similarity to the standard written algorithm we refer to this strategy as a quasi vertical algorithm strategy. Use of horizontal number sentences to notate this strategy is shown below.

36 + 47:

7 + 6 = 13
13 = 10 + 3
40 + 10 = 50
50 + 30 = 80
80 +30 = 83

The figure below shows an alternative method of notating split strategies, referred to as splitting, branching or drop-down notation.

Extending to 3-digit Addition, Subtraction and Place Value

Children who have developed facile and flexible jump and split strategies for adding and subtracting two 2 digit numbers are ready to extend these strategies to working with 3-digit numbers. In order to do so, children need:

to be facile with number word sequences and numerals in the range 100 to 1,000, and
to be able to increment and decrements 3 digit numbers, by 10 and by 100.

Number Words and Numerals to 1,000

This refers to learning to name and write 3-digit numerals and learning to say number word sequences forward and backward in the range 100 to 1,000. The latter topic refers to saying the sequence forward or backward from a given 3-digit number, or knowing what comes after or before a given 3-digit number. These include:

being able to count on or back from 810, 701, 297, and so on;
knowing what comes after 246. 310, 700, 899, and so on; and
knowing what comes before 416, 800, 920, and so on.

Developing children's knowledge of number word sequences and numerals in the range 100 to 1,000 should be developed in parallel with the development of jump and split strategies in the range 1 to 100.

Incrementing and Decrementing by Tens

Extending to 3-digit addition and subtraction involves several steps: first, the strategies of incrementing and decrementing by tens on and off the decuple in the range 1 to 100 can be extended beyond 100. For example,

going forward by tens from 350, from 287, and from 790;
going backward by tens from 650, from 289, and from 800.

Incrementing and Decrementing by Hundreds

Second, children can learn to increment and decrement by hundreds both on and off the decuple and on and off the hundred. For example,

going forwards by hundreds from 100, from 260, from 407, from 165;
going backwards by hundreds from 1,000, from 920, from 608, and from 482.

Becoming facile with number word sequences in this way provides a basis for using strategies akin to jump and split strategies to do addition or subtraction involving two 3-digit numbers. For example, children might solve tasks such as the following using a split strategy and a jump strategy:

634 + 211 459 + 203 175 + 282 367 + 255

889 - 236 372 - 106 416 - 121 723 - 235

Developing Strategies for 3-digit Addition and Subtraction

In similar vein to the case of 2-digit addition and subtraction, children's development of flexible jump and split strategies involving 3-digit addition and subtraction will support a corresponding development of an understanding of place value to 1,000. As in the case of 2-digit numbers, place value knowledge is developed in conjunction with, rather than prior to, the development of flexible strategies for adding and subtracting. Base-ten materials (hundreds, tens and ones) are particularly suited to the development of split strategies. Further, situations which involve low-level counting strategies should generally be avoided. As before, this can be done by displaying and then screening base-ten materials to encourage visualization or imaging on the part of the children. The empty number line can be used to support the development of jump strategies. The ENL enables children to notate, discuss explain and reflect on their strategies.