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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) CCT
  • 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) CCT
  • 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) CCT
  • 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) CCT
  • 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) CCT
  • 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) CCT
  • solve a variety of problems by applying coordinate geometry formulas
  • derive the formula for the distance between two points (Reasoning) CCT
  • show that three given points are collinear (Communicating, Reasoning) CCT
  • use coordinate geometry to investigate and describe the properties of triangles and quadrilaterals (Communicating, Problem Solving, Reasoning) CCT
  • use coordinate geometry to investigate the intersection of the perpendicular bisectors of the sides of acute-angled triangles (Problem Solving, Reasoning) CCT
  • show that four specified points form the vertices of particular quadrilaterals (Communicating, Problem Solving, Reasoning) CCT
  • prove that a particular triangle drawn on the Cartesian plane is right-angled (Communicating, Reasoning) CCT