## Tens and ones (Level 2)

When students have developed knowledge of tens and ones, ten is treated as a special unit. The use of ten as a special unit can manifest itself in one of two different ways. One method involves the tens and units being split off and handled separately. For example, in the following response to 48 + 26 the answer is achieved by splitting the tens and units.

In combining the units (8 + 6) the student has also made use of double 6 by partitioning the 8 into 6 and 2. This is an example of the split method as is the response below.

The following response shows the split method used with subtraction.

The second of the two methods arrives at the total by taking one number as the starting point and increasing by jumps of tens and ones (48 + 20 + 6). Sometimes the ones are broken into smaller hops as in the following response.

There are several variations of the jump method. The following example starts with 48 and goes through 50 with a hop of 2 before jumping the remaining 24 to achieve the answer.

For simplicity we consider any solution method that effectively adds on or subtracts from one number that has not been split, as an example of the jump method. Consequently, subtracting 7 or jumping back 7 followed by subtracting 20 is also considered to be an example of using tens and ones with the jump method.

Compensation (48 + 26 as 50 + 26 = 76 followed by 76 – 2 = 74) can also be considered to be an extension of the jump method.

To use this understanding of tens and one without relying on counting by ones, students usually need to develop part-whole knowledge of number combinations to at least twenty. When students learn to use trading with the traditional subtraction algorithm, subtraction problems such as 53 – 27 are transformed into problems involving subtraction within 20 (13 – 7 in this example).

The subtraction algorithm has a standard way of regrouping the 53:

When this standard regrouping is used, students need part-whole knowledge of numbers to 20.