## Separating and combining

To solve a question such as “There are 28 children at a party and 13 more arrive. How many are there all together?” a range of solution methods is possible. Apart from using unitary counting or visualising the question as a vertical algorithm and carrying out the sum, students’ solution methods fall into two broad groups. Some methods will build on one number, say, 28 + 10 + 3 = 41, whereas others will partition both numbers. That is, 28 = 20 + 8 and 13 = 10 + 3, before combining the tens and ones separately. This standard (or canonical) separating of two-digit numbers into a “tens part” and a “ones part” underpins front-end mental addition methods. The approach of building-on one number relies on knowing how numbers can be broken into parts. However, when building-on one number, this partitioning of the number being added is not always a standard partition, based on positional values. For example, 28 + 6 can involve going through the next multiple of ten, 28 + 2 + 4 = 34. This relies upon part-whole knowledge to 10 — breaking both the decade and the six into parts. Knowledge of partitioning numbers to 10 supports the process of bridging to tens.

Partitioning numbers to 10 is not all that we need to learn about partitioning for subtraction. Subtraction within 20 refers to subtraction requiring regrouping with the minuends (number being subtracted from) between 10 and 20, such as 12 – 6 or 15 – 7. All of the subtraction problems with larger numbers involving trading, such as 52 – 37, are transformed into problems involving subtraction within 20 (12 – 7 in this example).

To successfully carry out subtraction with trading, students need to be able to do the following subtractions within 20 by using number combinations.

11 – 2 =, 11 – 3 =, 11 – 4 =, … 11 – 9 =,

12 – 3 =, … 12 – 9 =,

13 – 4 =, … 13 – 9 =,

14 – 5 =, … 14 – 9 =,

15 – 6 =, … 15 – 9 =,

16 – 7 =, 16 – 8 =, 16 – 9 =, 17 – 8 =, 17 – 9 =, 18 – 9 =

In addition to using bundling of objects to model addition and subtraction with trading (NS1.2) students need to count on by tens starting from any number and be able to recall the above number combinations. However, these number combinations need not be taught in isolation from solving subtractions.