NSW Syllabuses

# Mathematics K–10 - Stage 5.2 - Number and Algebra Equations

## Outcomes

#### A student:

• MA5.2-1WM

selects appropriate notations and conventions to communicate mathematical ideas and solutions

• MA5.2-2WM

interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems

• MA5.2-3WM

constructs arguments to prove and justify results

• MA5.2-8NA

solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques

Related Life Skills outcome: MALS-19NA

## Content

• solve linear equations, including equations that involve grouping symbols,
eg $$\, 3\left(a+2\right) + 2\left(a-5\right) = 10, \,\,$$  $$\,\, 3\left( 2m-5 \right) = 2m+5$$
• solve linear equations involving one or more simple algebraic fractions,
eg $$\, \dfrac{x-2}{3} + 5 = 10, \,\,$$  $$\,\, \dfrac{2x+5}{3} = 10, \,\,$$  $$\,\, \dfrac{2x}{3} + 5 = 10, \,\,$$  $$\,\, \dfrac{x}{3} + \dfrac{x}{2} = 5, \,\,$$  $$\,\, \dfrac{2x+5}{3} = \dfrac{x-1}{4}$$
• compare and contrast different algebraic techniques for solving linear equations and justify a choice for a particular case (Communicating, Reasoning)
• solve simple quadratic equations of the form $$\, ax^2 = c$$, leaving answers in exact form and as decimal approximations
• explain why quadratic equations could be expected to have two solutions (Communicating, Reasoning)
• recognise and explain why $$x^2 = c$$ does not have a solution if $$c$$ is a negative number (Communicating)
• solve quadratic equations of the form $$ax^2 + bx + c = 0$$, limited to $$a=1$$, using factors
• connect algebra with arithmetic to explain that if $$p \times q = 0$$, then either $$p=0$$ or $$q=0$$ (Communicating, Reasoning)
• check the solution(s) of quadratic equations by substitution (Reasoning)
• Substitute values into formulas to determine an unknown (ACMNA234)
• solve equations arising from substitution into formulas, eg given $$P=2l+2b$$ and $$P=20$$, $$l=6$$, solve for $$b$$
• substitute into formulas from other strands of the syllabus or from other subjects to solve problems and interpret solutions, eg $$\, A = \dfrac{1}{2}xy, \,\,$$ $$\,\,v=u+at, \,\,$$ $$\,\,C = \dfrac{5}{9}\left(F-32\right), \,\,$$ $$\,\,V=\pi r^2h$$ (Problem Solving)
• Solve problems involving linear equations, including those derived from formulas (ACMNA235)
• translate word problems into equations, solve the equations and interpret the solutions
• state clearly the meaning of introduced pronumerals when using equations to solve word problems, eg '$$n$$ = number of years' (Communicating)
• solve word problems involving familiar formulas, eg 'If the area of a triangle is 30 square centimetres and the base length is 12 centimetres, find the perpendicular height of the triangle' (Problem Solving)
• explain why the solution to a linear equation generated from a word problem may not be a solution to the given problem (Communicating, Reasoning)
• represent simple inequalities on the number line, eg represent $$x \ < -3$$ on a number line
• recognise that an inequality has an infinite number of solutions unless other restrictions are made
• solve linear inequalities, including through reversing the direction of the inequality sign when multiplying or dividing by a negative number, and graph the solutions, eg solve and graph the inequalities on a number line of $$3x-1 > 9, \,\,\,\, 2(a+4) \ge 24, \,\,\,\, \dfrac{t+4}{5} < -3, \,\,\,\, 1-4y\le 6$$
• use a numerical example to justify the need to reverse the direction of the inequality sign when multiplying or dividing by a negative number (Reasoning)
• verify the direction of the inequality sign by substituting a value within the solution range (Reasoning)
• Solve linear simultaneous equations, using algebraic and graphical techniques, including with the use of digital technologies (ACMNA237)
• solve linear simultaneous equations by finding the point of intersection of their graphs, with and without the use of digital technologies
• solve linear simultaneous equations using appropriate algebraic techniques, including with the use of the 'substitution' and 'elimination' methods, eg solve $$\left\{ \begin{array}{l} 3a+b=17 \\ 2a-b=8 \end{array} \right.$$
• select an appropriate technique to solve particular linear simultaneous equations by observing the features of the equations (Problem Solving)
• generate and solve linear simultaneous equations from word problems and interpret the results

### Background Information

Graphing software and graphics calculators allow students to graph two linear equations and to display the coordinates of the point of intersection of their graphs.

The 'substitution method' for solving linear simultaneous equations involves substituting one equation into the other. This may require the rearranging of one of the equations to make one of its pronumerals the subject, in order to facilitate substitution into the other equation. The 'elimination method' for solving linear simultaneous equations involves adding or subtracting the two equations so that one of the pronumerals is eliminated. This may require the multiplication of one or both of the given equations by a constant so that the pronumeral targeted for elimination has coefficients of equal magnitude. Students should be encouraged to select the most efficient technique when solving linear simultaneous equations.

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA5.2-8NA the sub-elements (and levels) of Number patterns and algebraic thinking (NPA8-NPA9) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.