NSW Syllabuses

# Mathematics K–10 - Stage 4 - Number and Algebra Computation with Integers

## Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-3WM

recognises and explains mathematical relationships using reasoning

• MA4-4NA

compares, orders and calculates with integers, applying a range of strategies to aid computation

Related Life Skills outcomes: MALS-4NA, MALS-5NA, MALS-6NA, MALS-7NA, MALS-10NA, MALS-11NA

## Content

• use an appropriate non-calculator method to divide two- and three-digit numbers by a two-digit number
• compare initial estimates with answers obtained by written methods and check by using a calculator (Problem Solving)
• show the connection between division and multiplication, including where there is a remainder, eg $$451 \div 23 = 19\!\tfrac{14}{23}$$ means that $$451 = 19 \times 23+14$$
• apply a practical understanding of commutativity to aid mental computation, eg 3 + 9 = 9 + 3 = 12, 3 × 9 = 9 × 3 = 27
• apply a practical understanding of associativity to aid mental computation, eg 3 + 8 + 2 = (3 + 8) + 2 = 3 + (8 + 2) = 13, 2 × 7 × 5 = (2 × 7) × 5 = 2 × (7 × 5) = 70
• determine by example that associativity holds true for multiplication of three or more numbers but does not apply to calculations involving division, eg (80 ÷ 8) ÷ 2 is not equivalent to 80 ÷ (8 ÷ 2) (Communicating)
• apply a practical understanding of the distributive law to aid mental computation, eg to multiply any number by 13, first multiply by 10 and then add 3 times the number
• use factors of a number to aid mental computation involving multiplication and division, eg to multiply a number by 12, first multiply the number by 6 and then multiply the result by 2
• Compare, order, add and subtract integers (ACMNA280)
• recognise and describe the 'direction' and 'magnitude' of integers
• construct a directed number sentence to represent a real-life situation (Communicating)
• recognise and place integers on a number line
• compare the relative value of integers, including recording the comparison by using the symbols $$\,<\,$$ and $$\,>\,$$
• order integers
• interpret different meanings (direction or operation) for the + and – signs, depending on the context
• add and subtract integers using mental and written strategies
• determine, by developing patterns or using a calculator, that subtracting a negative number is the same as adding a positive number (Reasoning)
• apply integers to problems involving money and temperature (Problem Solving)
• Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies (ACMNA183)
• multiply and divide integers using mental and written strategies
• investigate, by developing patterns or using a calculator, the rules associated with multiplying and dividing integers (Reasoning)
• use a calculator to perform the four operations with integers
• decide whether it is more appropriate to use mental strategies or a calculator when performing certain operations with integers (Communicating)
• use grouping symbols as an operator with integers
• apply the order of operations to mentally evaluate expressions involving integers, including where an operator is contained within the numerator or denominator of a fraction,
eg $$\frac{15+9}{6}, \,$$ $$\frac{15+9}{15-3}, \,$$ $$5 + \frac{18-12}{6}, \,$$ $$5 + \frac{18}{6} - 12, \,$$ $$5 \times \left( 2 - 8 \right)$$
• investigate whether different digital technologies, such as those found in computer software and on mobile devices, apply the order of operations (Problem Solving)

### Background Information

To divide two- and three-digit numbers by a two-digit number, students may be taught the long division algorithm or, alternatively, to transform the division into a multiplication.

So, $$356 \div 52 = \square$$ becomes $$52 \times \square = 356$$. Knowing that there are two fifties in each 100, students may try 7, obtaining 52 × 7 = 364, which is too large. They may then try 6, obtaining 52 × 6 = 312. The answer is $$6\!\tfrac{44}{52} = 6\!\tfrac{11}{13}$$.

Students also need to be able to express a division in the following form in order to relate multiplication and division: 356 = 6 × 52 + 44, and then division by 52 gives $$\frac{356}{52} = 6 + \frac{44}{52} = 6\!\tfrac{11}{13}$$.

Students should have some understanding of integers, as the concept is introduced in Stage 3 Whole Numbers 2. However, operations with integers are introduced in Stage 4.

Complex recording formats for integers, such as raised signs, can be confusing. On printed materials, the en-dash ( – ) should be used to indicate a negative number and the operation of subtraction. The hyphen ( - ) should not be used in either context. The following formats are recommended:

\begin{align} -2-3& = -5 \\ {} \\ -7+(-4)& = -7-4 \\& = -11 \\ {} \\ -2--3& = -2+3 \\& = 1 \end{align}

Brahmagupta (c598–c665), an Indian mathematician and astronomer, is noted for the introduction of zero and negative numbers in arithmetic.

#### Purpose/Relevance of Substrand

The positive integers (1, 2, 3, …) and 0 allow us to answer many questions involving 'How many?', 'How much?', 'How far?', etc, and so carry out a wide range of daily activities. The negative integers (…, –3, –2, –1) are used to represent 'downwards', 'below', 'to the left', etc, and appear in relation to everyday situations such as the weather (eg a temperature of –5° is 5° below zero), altitude (eg a location given as –20 m is 20 m below sea level), and sport (eg a golfer at –6 in a tournament is 6 under par). The Computation with Integers substrand includes the use of mental strategies, written strategies, etc to obtain answers – which are very often integers themselves – to questions or problems through addition, subtraction, multiplication and division.

### Language

Teachers should model and use a variety of expressions for mathematical operations and should draw students' attention to the fact that the words used for subtraction and division questions may require the order of the numbers to be reversed when performing the operation. For example, '9 take away 3' and 'reduce 9 by 3' require the operation to be performed with the numbers in the same order as they are presented in the question (ie 9 – 3), but 'take 9 from 3', 'subtract 9 from 3' and '9 less than 3' require the operation to be performed with the numbers in the reverse order to that in which they are stated in the question (ie 3 – 9).

Similarly, 'divide 6 by 2' and '6 divided by 2' require the operation to be performed with the numbers in the same order as they are presented in the question (ie 6 ÷ 2), but 'how many 2s in 6?' requires the operation to be performed with the numbers in the reverse order to that in which they appear in the question (ie 6 ÷ 2).