NSW Syllabuses

# Mathematics K–10 - Stage 5.3 - Number and Algebra Linear Relationships §

## Outcomes

#### A student:

• MA5.3-1WM

uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

• MA5.3-2WM

generalises mathematical ideas and techniques to analyse and solve problems efficiently

• MA5.3-3WM

uses deductive reasoning in presenting arguments and formal proofs

• MA5.3-8NA

uses formulas to find midpoint, gradient and distance on the Cartesian plane, and applies standard forms of the equation of a straight line

## Content

• use the concept of an average to establish the formula for the midpoint, M, of the interval joining two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ on the Cartesian plane: $$M(x,y) = \left( \frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$
• explain the meaning of each of the pronumerals in the formula for midpoint (Communicating)
• use the formula to find the midpoint of the interval joining two points on the Cartesian plane
• use the relationship $$\textrm{gradient} = \dfrac{\textrm{rise}}{\textrm{run}}$$ to establish the formula for the gradient, m, of the interval joining two points $$(x_1,y_1)$$ and $$(x_2, y_2)$$ on the Cartesian plane: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
• use the formula to find the gradient of the interval joining two points on the Cartesian plane
• explain why the formula $$m = \frac{y_1 - y_2}{x_1 - x_2}$$ gives the same value for the gradient as $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ (Communicating, Reasoning)
• Find the distance between two points located on the Cartesian plane (ACMNA214)
• use Pythagoras' theorem to establish the formula for the distance, d, between two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ on the Cartesian plane: $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
• explain the meaning of each of the pronumerals in the formula for distance (Communicating)
• use the formula to find the distance between two points on the Cartesian plane
• explain why the formula $$d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$$ gives the same value for the distance as $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$ (Communicating, Reasoning)
• Sketch linear graphs using the coordinates of two points (ACMNA215)
• sketch the graph of a line by using its equation to find the x- and y-intercepts
• Solve problems using various standard forms of the equation of a straight line
• describe the equation of a line as the relationship between the x- and y-coordinates of any point on the line
• recognise from a list of equations those that can be represented as straight-line graphs (Communicating, Reasoning)
• rearrange linear equations in gradient-intercept form ($$y=mx+b$$) into general form $$ax+by+c=0$$
• find the equation of a line passing through a point $$(x_1,y_1)$$, with a given gradient m, using:
• point-gradient form: $$y - y_1 = m(x - x_1)$$
• gradient-intercept form: $$y = mx + b$$
• find the equation of a line passing through two points
• recognise and find the equation of a line in general form $$ax+by+c=0$$
• Solve problems involving parallel and perpendicular lines (ACMNA238)
• find the equation of a line that is parallel or perpendicular to a given line
• determine whether two given lines are perpendicular
• use gradients to show that two given lines are perpendicular (Communicating, Problem Solving)
• solve a variety of problems by applying coordinate geometry formulas
• derive the formula for the distance between two points (Reasoning)
• show that three given points are collinear (Communicating, Reasoning)
• use coordinate geometry to investigate and describe the properties of triangles and quadrilaterals (Communicating, Problem Solving, Reasoning)
• use coordinate geometry to investigate the intersection of the perpendicular bisectors of the sides of acute-angled triangles (Problem Solving, Reasoning)
• show that four specified points form the vertices of particular quadrilaterals (Communicating, Problem Solving, Reasoning)
• prove that a particular triangle drawn on the Cartesian plane is right-angled (Communicating, Reasoning)