NSW Syllabuses

# Mathematics K–10 - Stage 4 - Number and Algebra Algebraic Techniques

## Algebraic Techniques 1

### Outcomes

#### A student:

• MA4-1WM

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

• MA4-3WM

recognises and explains mathematical relationships using reasoning

• MA4-8NA

generalises number properties to operate with algebraic expressions

Related Life Skills outcome: MALS-18NA

### Content

• Students:
• Introduce the concept of variables as a way of representing numbers using letters (ACMNA175)
• develop the concept that pronumerals (letters) can be used to represent numerical values
• recognise that pronumerals can represent one or more numerical values (when more than one numerical value, pronumerals may then be referred to as 'variables') (Communicating, Reasoning)
• model the following using concrete materials or otherwise:
• expressions that involve a pronumeral, and a pronumeral added to a constant, eg a, a + 1
• expressions that involve a pronumeral multiplied by a constant, eg 2a, 3a
• sums and products, eg 2a + 1, 2(a + 1)
• equivalent expressions, eg
\begin{align} x + x + y + y + 3 \, &= \, 2x + 2y + 3 \\ &= \, 2(x + y) + 3 \end{align}
• simplifying expressions, eg
\begin{align} \left( a + 2 \right) + \left( 2a + 3 \right) \, &= \, \left( a + 2a \right) + \left( 2 + 3 \right) \\ &= \, 3a + 5 \end{align}
• recognise and use equivalent algebraic expressions, eg
\begin{align} y + y + y + y &= 4y \\ w \times w &= w^2 \\ a \times b &= ab \\ a \div b &= \frac{a}{b} \end{align}
• use algebraic symbols to represent mathematical operations written in words and vice versa, eg the product of x and y is xy, $$x+y$$ is the sum of x and y
• Extend and apply the laws and properties of arithmetic to algebraic terms and expressions (ACMNA177)
• recognise like terms and add and subtract them to simplify algebraic expressions, eg $$2n + 4m + n = 4m + 3n$$
• verify whether a simplified expression is correct by substituting numbers for pronumerals (Communicating, Reasoning)
• connect algebra with the commutative and associative properties of arithmetic to determine that  $$a + b = b + a$$ and $$\left( a + b \right) + c = a + \left( b + c \right)$$ (Communicating)
• recognise the role of grouping symbols and the different meanings of expressions, such as $$2a+1$$ and $$2(a+1)$$
• simplify algebraic expressions that involve multiplication and division, eg $$12a \div 3$$, $$4x \times 3$$, $$2ab \times 3a$$, $$\frac{8a}{2}, \; \frac{2a}{8}, \; \frac{12a}{9}$$
• recognise the equivalence of algebraic expressions involving multiplication, eg $$3bc = 3cb$$ (Communicating)
• connect algebra with the commutative and associative properties of arithmetic to determine that $$a \times b = b \times a$$ and $$\left( a \times b \right) \times c = a \times \left( b \times c \right)$$  (Communicating)
• recognise whether particular algebraic expressions involving division are equivalent or not, eg $$a \div bc$$ is equivalent to $$\frac{a}{bc}$$  and  $$a \div \left( b \times c \right)$$, but is not equivalent to $$a \div b \times c \,$$ or $$\, \frac{a}{b} \times c$$ (Communicating)
• translate from everyday language to algebraic language and vice versa
• use algebraic symbols to represent simple situations described in words, eg write an expression for the number of cents in x dollars (Communicating)
• interpret statements involving algebraic symbols in other contexts, eg cell references when creating and formatting spreadsheets (Communicating)
• Simplify algebraic expressions involving the four operations (ACMNA192)
• simplify a range of algebraic expressions, including those involving mixed operations

### Background Information

It is important to develop an understanding of the use of pronumerals (letters) as algebraic symbols to represent one or more numerical values.

The recommended approach is to spend time on the conventions for the use of algebraic symbols for first-degree expressions and to situate the translation of generalisations from words to symbols as an application of students' knowledge of the symbol system, rather than as an introduction to the symbol system.

The recommended steps for moving into symbolic algebra are:

• the variable notion, associating letters with a variety of different numerical values
• symbolism for a pronumeral plus a constant
• symbolism for a pronumeral times a constant
• symbolism for sums, differences, products and quotients.

So, if = 6, a + = 6 + 6, but 2= 2 × 6 and not 26.

To gain an understanding of algebra, students must be introduced to the concepts of pronumerals, expressions, unknowns, equations, patterns, relationships and graphs in a wide variety of contexts. For each successive context, these ideas need to be redeveloped. Students need gradual exposure to abstract ideas as they begin to relate algebraic terms to real situations.

It is suggested that the introduction of representation through the use of algebraic symbols precede Linear Relationships in Stage 4, since this substrand presumes that students are able to manipulate algebraic symbols and will use them to generalise patterns.

#### Purpose/Relevance of Substrand

Algebra is used to some extent throughout our daily lives. People are solving equations (usually mentally) when, for example, they are working out the right quantity of something to buy, or the right amount of an ingredient to use when adapting a recipe. Algebra requires, and its use results in, learning how to apply logical reasoning and problem-solving skills. It is used more extensively in other areas of mathematics, the sciences, business, accounting, etc. The widespread use of algebra is readily seen in the writing of formulas in spreadsheets.

### Language

For the introduction of algebra in Stage 4, the term 'pronumeral' rather than 'variable' is preferred when referring to unknown numbers. In an algebraic expression such as 2+ 5, x can take any value (ie x is variable and a pronumeral). However, in an equation such as 2+ 5 = 11, x represents one particular value (ie x is not a variable but is a pronumeral). In equations such as = 11, x and y can take any values that sum to 11 (ie and y are variables and pronumerals).

'Equivalent' is the adjective for 'equal', although 'equal' can also be used as an adjective, ie 'equivalent expressions' or 'equal expressions'.

Some students may confuse the order in which terms or numbers are to be used when a question is expressed in words. This is particularly apparent for word problems that involve subtraction or division to obtain the required result, eg '5x less than x' and 'take 5x from x' both require the order of the terms to be reversed to – 5x in the solution.

Students need to be familiar with the terms 'sum', 'difference', 'product' and 'quotient' to describe the results of adding, subtracting, multiplying and dividing, respectively.

## Algebraic Techniques 2

### 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

• MA4-8NA

generalises number properties to operate with algebraic expressions

Related Life Skills outcome: MALS-18NA

### Content

• substitute into algebraic expressions and evaluate the result
• calculate and compare the values of x2 for values of x with the same magnitude but opposite sign (Reasoning)
• generate a number pattern from an algebraic expression, eg

• Extend and apply the distributive law to the expansion of algebraic expressions (ACMNA190)
• expand algebraic expressions by removing grouping symbols, eg
\begin{align} 3\left( a+2 \right) &= 3a + 6 \\ -5 \left( x+2 \right) &= -5x - 10 \\ a \left( a+b \right) &= a^2 + ab \end{align}
• connect algebra with the distributive property of arithmetic to determine that $$a \left( b+c \right) = ab + ac\,$$ (Communicating)
• factorise a single algebraic term, eg $$6ab = 3 \times 2 \times a \times b$$
• factorise algebraic expressions by finding a common numerical factor, eg
\begin{align} 6a + 12 &= 6 \left( a+2 \right) \\ -4t - 12 &= -4\left( t+3 \right) \end{align}
• check expansions and factorisations by performing the reverse process (Reasoning)
• Factorise algebraic expressions by identifying algebraic factors
• factorise algebraic expressions by finding a common algebraic factor, eg
\begin{align} x^2 - 5x &= x\left( x-5 \right) \\ 5ab + 10a &= 5a \left( b+2 \right) \end{align}

### Background Information

When evaluating expressions, there should be an explicit direction to replace the pronumeral with the number to ensure that full understanding of notation occurs.

### Language

The meaning of the imperatives 'expand', 'remove the grouping symbols' and 'factorise' and the expressions 'the expansion of' and 'the factorisation of' should be made explicit to students.