Two Digit Addition & Subtraction Jump Strategies

Fostering the Development of Jump Strategies

Jump strategies involve working from one of the numbers and partitioning the second in order to make several jumps forward. In subtraction, jump strategies involve making several jumps backward or alternatively, jumping forward from the smaller number to the larger, and keeping track to work out the total jump, jump strategies have the advantage that they are not significantly more difficult to use when the addition of the numbers in the ones exceeds nine. Working out 26 + 38 is not necessarily much more difficult than working out 26 + 32. Being facile with addition and subtraction in the range 1 to 20 is a prerequisite for learning jump strategies. Children learning to use jump strategies should not be limited to strategies involving counting-by-ones and should not incorporate counting-by-ones into their developing jump strategies.

The Role of the Empty Number Line (ENL)

The empty number line is an instructional device that is regarded as particularly suited to fostering the development of jump strategies. Children use the ENL to make a written record that serves to summarize their particular strategy. Making a record for themselves, for future reference, about how they solved particular tasks. Children can use an ENL to communicate their method to their colleagues and the teacher. Recording strategies in this way is referred to as 'notating'.

Other Means of Notating Jump Strategies

Two other means of notating jump strategies are arrow notation and horizontal number sentences. The use of arrow notation is demonstrated below in '‘Fostering the Development of a Range of Strategies'. Using horizontal number sentences involves writing a horizontal number sentence to describe each step in the strategy. A significant strength of notating with horizontal number sentences is that it can be used for virtually any strategy.

ENL as a Notating Device versus ENL as a Means of Solving a Task

The most productive way for children to use the ENL is as a means of notating a mental strategy. Children should be strongly encouraged to solve tasks mentally, write their answers, and finally to use the ENL to notate their method. This contrasts with the use of the ENL in a procedural way to solve the task. This approach is problematic, because children will develop a reliance on using the ENL to solve addition and subtraction tasks. The goal is for children to develop flexible mental strategies. Not to develop an alternative written method for solving addition and subtraction tasks.

Alternative Means of Notating for Split Strategies

Teachers should be particularly mindful that in spite of instruction aimed at fostering jump strategics, children might sometimes use a split strategy and a few children might have a continuing, strong preference for split strategies. As a general rule, it is not productive to use an ENL to notate a split strategy. The alternative is to use horizontal number sentences or branching notation to notate the child's strategy. Notating split strategies in these ways is described in Chapter Two Digit +/- Split Strategies.

Jump versus Split

Becoming facile with jump strategies is particularly useful because jump strategies tend to be more versatile and flexible than split strategies. Some children will find split strategies difficult in cases where the combination (addition or subtraction) in the ones is beyond the range 1 to 10, that is, calculations that, in terms of the traditional algorithm, involve regrouping (carrying) in the case of addition, or regrouping (renaming, borrowing) in the case of subtraction. It is quite common for children who mainly develop and use split strategies to have particular difficulty when they first encounter subtraction with regrouping. What typically happens in the case of these children is that they do the following: 62 - 25: 60 - 20 = 40; 5 - 2 = 3; answer 43.

Some Children's Initial Preference for Split Strategies

Children who have difficulty with mental strategies will tend to find the initial split strategy easier to use than the corresponding jump strategy. A child having general difficulty with this might when working out 43 + 35 find it easier to use a split strategy than a jump strategy. One reason for this is that using a jump strategy involves incrementing (for addition) or decrementing (for subtraction) off the decuple (a multiple of 10 such as 10, 20, 30 ...). Thus a child might have difficulty using a jump strategy to work out 47 + 32 because they have difficulty with saying, for example, 47, 57, 67, 77, and simultaneously keeping track of the increments. In order to stop at 77, it is necessary to monitor the increments of ten, and to realize that three increments have been made (corresponding to adding 30 to 47). Instruction focusing on incrementing and decrementing 2-digit numbers on and off the decuple is shown below. The Jump Strategy and learning to add and subtract through a decuple, constitute important building blocks for the development of jump strategies.

Learning to Add and Subtract through a Decuple

Children who are facile at adding and subtracting in the range 1 to 20 are ready to extend this to the range 1 to 100. Learning to add and subtract through a decuple is an important initial step for addition and subtraction involving two 2-digit numbers. The following are two groups of tasks related to this topic, which are important because they can constitute one or more steps of the jump strategy.

Adding and Subtracting Involving a Decuple and a Number from 1 to 9

This includes cases such as:

• 60 + 5, 40 + 6. For example. 60 + 5 might arise as the final step when using a jump strategy to work out 38 + 27.
• 52 - o = 50, 74 - o = 70. For example, 74 - □ = 70 might arise as an interim step when using a jump strategy to work out 94 - 26.
• 27 + o = 30, 58 + o = 60. For example, 58 + □ = 60 might arise as an interim step when using a jump strategy to work out 38 + 27.
• 80 - 6, 50 5. For example, 50-5 might arise as the final step when using a jump strategy' to work out 72 - 27.

Adding and Subtracting Involving a Non-decuple and a Number from 1 to 9

This includes cases such as:

• 62 + 5, 43 + 6. For example, 62 + 5 might arise as the final step when using a jump strategy to work out 32 + 35.
• 56 - 3, 48 - 5. For example, 48-5 might arise as the final step when using a jump strategy to work out 98 - 55.
• 67 + 8, 88 + 4. For example, 88 + 4 might arise as the final step when using a jump strategy to work out 68 + 24.
• 53 - 6, 41 - 5. For example, 41-5 might arise as the final step when using a jump strategy to work out 71 - 35.