New NSW Syllabuses

Mathematics K–10 - Stage 1 - Number and Algebra Addition and Subtraction

Outcomes

A student:

• MA1-1WM

describes mathematical situations and methods using everyday and some mathematical language, actions, materials, diagrams and symbols

• MA1-2WM

uses objects, diagrams and technology to explore mathematical problems

• MA1-3WM

supports conclusions by explaining or demonstrating how answers were obtained

• MA1-5NA

uses a range of strategies and informal recording methods for addition and subtraction involving one- and two-digit numbers

Content

• Students:
• Represent and solve simple addition and subtraction problems using a range of strategies, including counting onpartitioning and rearranging parts (ACMNA015)
• use the terms 'add', 'plus', 'equals', 'is equal to', 'take away', 'minus' and the 'difference between'
• use concrete materials to model addition and subtraction problems involving one- and two-digit numbers
• use concrete materials and a number line to model and determine the difference between two numbers, eg
• recognise and use the symbols for plus (+), minus (–) and equals (=)
• record number sentences in a variety of ways using drawings, words, numerals and mathematical symbols
• recognise, recall and record combinations of two numbers that add to 10
• create, record and recognise combinations of two numbers that add to numbers up to and including 9
• model and record patterns for individual numbers by making all possible whole-number combinations, eg
$$\begin{array}{c}5 + 0 = 5\\4 + 1 = 5\\3 + 2 = 5\\2 + 3 = 5\\1 + 4 = 5\\0 + 5 = 5\end{array}$$ (Communicating, Problem Solving)
• describe combinations for numbers using words such as 'more', 'less' and 'double', eg describe 5 as 'one more than four', 'three combined with two', 'double two and one more' and 'one less than six' (Communicating, Problem Solving)
• create, record and recognise combinations of two numbers that add to numbers from 11 up to and including 20
• use combinations for numbers up to 10 to assist with combinations for numbers beyond 10 (Problem Solving)
• investigate and generalise the effect of adding zero to a number, eg 'Adding zero to a number does not change the number'
• use concrete materials to model the commutative property for addition and apply it to aid the recall of addition facts, eg 4 + 5 = 5 + 4
• relate addition and subtraction facts for numbers to at least 20, eg 5 + 3 = 8, so 8 – 3 = 5 and 8 – 5 = 3
• use and record a range of mental strategies to solve addition and subtraction problems involving one- and two-digit numbers, including:
• counting on from the larger number to find the total of two numbers
• counting back from a number to find the number remaining
• counting on or back to find the difference between two numbers
• using doubles and near doubles, eg 5 + 7: double 5 and add 2
• combining numbers that add to 10, eg 4 + 7 + 8 + 6 + 3: first combine 4 and 6, and 7 and 3, then add 8
• bridging to 10, eg 17 + 5: 17 and 3 is 20, then add 2 more
• using place value to partition numbers, eg 25 + 8: 25 is 20 + 5, so 25 + 8 is 20 + 5 + 8, which is 20 + 13
• choose and apply efficient strategies for addition and subtraction (Problem Solving)
• use the equals sign to record equivalent number sentences involving addition, and to mean 'is the same as', rather than as an indication to perform an operation, eg 5 + 2 = 3 + 4
• check given number sentences to determine if they are true or false and explain why,
eg 'Is 7 + 5 = 8 + 4 true? Why or why not?' (Communicating, Reasoning)

Language

Students should be able to communicate using the following language: counting on, counting back, combine, plus, add, take away, minus, the difference between, total, more than, less than, double, equals, is equal to, is the same as, number sentence, strategy.

The word 'difference' has a specific meaning in this context, referring to the numeric value of the group. In everyday language, it can refer to any attribute. Students need to understand that the requirement to carry out subtraction can be indicated by a variety of language structures. The language used in the 'comparison' type of subtraction is quite different from that used in the 'take away' type.

Students need to understand the different uses for the = sign, eg 4 + 1 = 5, where the = sign indicates that the right side of the number sentence contains 'the answer' and should be read to mean 'equals', compared to a statement of equality such as 4 + 1 = 3 + 2, where the = sign should be read to mean 'is the same as'.

Outcomes

A student:

• MA1-1WM

describes mathematical situations and methods using everyday and some mathematical language, actions, materials, diagrams and symbols

• MA1-2WM

uses objects, diagrams and technology to explore mathematical problems

• MA1-3WM

supports conclusions by explaining or demonstrating how answers were obtained

• MA1-5NA

uses a range of strategies and informal recording methods for addition and subtraction involving one- and two-digit numbers

Content

• Students:
• Explore the connection between addition and subtraction (ACMNA029)
• use concrete materials to model how addition and subtraction are inverse operations
• use related addition and subtraction number facts to at least 20,
eg 15 + 3 = 18, so 18 – 3 = 15 and 18 – 15 = 3
• Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)
• use and record a range of mental strategies to solve addition and subtraction problems involving two-digit numbers, including:
• the jump strategy on an empty number line
• the split strategy, eg record how the answer to 37 + 45 was obtained using the split strategy
$$\begin{array}{r}30 + 40 = 70\\7 + 5 = 12\\ \text{so }70 + 12 = 82\end{array}$$
• an inverse strategy to change a subtraction into an addition, eg 54 – 38: start at 38, adding 2 makes 40, then adding 10 makes 50, then adding 4 makes 54, and so the answer is 2 + 10 + 4 = 16
• select and use a variety of strategies to solve addition and subtraction problems involving one- and two-digit numbers
• perform simple calculations with money, eg buying items from a class shop and giving change (Problem Solving)
• check solutions using a different strategy (Problem Solving)
• recognise which strategies are more efficient and explain why (Communicating, Reasoning)
• explain or demonstrate how an answer was obtained for addition and subtraction problems,
eg show how the answer to 15 + 8 was obtained using a jump strategy on an empty number line
(Communicating, Reasoning)

Background Information

It is appropriate for students in Stage 1 to use concrete materials to model and solve problems, for exploration and for concept building. Concrete materials may also help in explanations of how solutions were obtained.

Addition and subtraction should move from counting and combining perceptual objects, to using numbers as replacements for completed counts with mental strategies, to recordings that support mental strategies (such as jump, split, partitioning and compensation).

Subtraction typically covers two different situations: 'taking away' from a group, and 'comparison' (ie determining how many more or less when comparing two groups). In performing a subtraction, students could use 'counting on or back' from one number to find the difference. The 'counting on or back' type of subtraction is more difficult for students to grasp than the 'taking away' type. Nevertheless, it is important to encourage students to use 'counting on or back' as a method of solving comparison problems once they are confident with the 'taking away' type.

In Stage 1, students develop a range of strategies to aid quick recall of number facts and to solve addition and subtraction problems. They should be encouraged to explain their strategies and to invent ways of recording their actions. It is also important to discuss the merits of various strategies in terms of practicality and efficiency.

Jump strategy on a number line – an addition or subtraction strategy in which the student places the first number on an empty number line and then counts forward or backwards, first by tens and then by ones, to perform a calculation. (The number of jumps will reduce with increased understanding.)

Jump strategy method: eg 46 + 33

Jump strategy method: eg 79 – 33

Split strategy – an addition or subtraction strategy in which the student separates the tens from the units and adds or subtracts each separately before combining to obtain the final answer.

Split strategy method: eg 46 + 33

\begin{align} 46 + 33 &= 40 + 6 + 30 + 3\\ &= 40 + 30 + 6 + 3 \\ &= 70 + 9 \\ &= 79 \end{align}

Inverse strategy – a subtraction strategy in which the student adds forward from the smaller number to obtain the larger number, and so obtains the answer to the subtraction calculation.

Inverse strategy method: eg 65 – 37

start at 37