NSW Syllabuses

# Mathematics K–10 - Stage 4 - Number and Algebra Equations

## 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

• MA4-10NA

uses algebraic techniques to solve simple linear and quadratic equations

Related Life Skills outcome: MALS-19NA

## Content

• distinguish between algebraic expressions where pronumerals are used as variables, and equations where pronumerals are used as unknowns
• solve simple linear equations using concrete materials, such as the balance model or cups and counters, stressing the notion of performing the same operation on both sides of an equation
• solve linear equations that may have non-integer solutions, using algebraic techniques that involve up to two steps in the solution process, eg
\begin{align} & x - 7 = 15 \\ & 2x - 7 = 15 \\ & 7-2x = 15 \\ & \frac{x}{7} = 5 \\ & \frac{2x}{7} = 5 \end{align}
• compare and contrast strategies to solve a variety of linear equations (Communicating, Reasoning)
• generate equations with a given solution, eg find equations that have the solution x = 5 (Problem Solving)
• Solve linear equations using algebraic techniques and verify solutions by substitution (ACMNA194)
• solve linear equations that may have non-integer solutions, using algebraic techniques that involve up to three steps in the solution process, eg
$$\begin{array} {llll} 3x+4 = 13 \qquad& 3x+4 = x-8 \qquad& \dfrac{x}{3}+5 = 10 \qquad& \dfrac{2x}{3}+5 = 10 \qquad \\ 3\left(x+4\right) = 13 \qquad& 3x+4 = 8-x \qquad& \dfrac{x+5}{3} = 10 \qquad& \dfrac{2x+5}{3} = 10 \qquad \end{array}$$
• check solutions to equations by substituting
• determine that if $$c > 0$$ then there are two values of $$x$$ that solve a simple quadratic equation of the form $$x^2 = c$$
• explain why quadratic equations could be expected to have two solutions (Communicating, Reasoning)
• recognise that $$x^2 = c$$ does not have a solution if $$c$$ is a negative number (Communicating, Reasoning)
• solve simple quadratic equations of the form $$x^2 = c$$, leaving answers in 'exact form' and as decimal approximations

### Background Information

The solution of simple equations can be introduced using a variety of models. Such models include using a two-pan balance with objects such as centicubes and a wrapped 'unknown', or using some objects hidden in a container as an 'unknown' to produce a number sentence.

The solution of simple quadratic equations in Stage 4 enables students to determine side lengths in right-angled triangles through the application of Pythagoras' theorem.

#### Purpose/Relevance of Substrand

An equation is a statement that two quantities or expressions are equal, usually through the use of numbers and/or symbols. Equations are used throughout mathematics and in our daily lives in obtaining solutions to problems of all levels of complexity. 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.

### Language

Describing the steps in the solution of equations provides students with the opportunity to practise using mathematical imperatives in context, eg 'add 5 to both sides', 'increase both sides by 5', 'subtract 3 from both sides', 'take 3 from both sides', 'decrease both sides by 3', 'reduce both sides by 3', 'multiply both sides by 2', 'divide both sides by 2'.