Outcomes
A student:

 MA5.11WM
uses appropriate terminology, diagrams and symbols in mathematical contexts

 MA5.13WM
provides reasoning to support conclusions that are appropriate to the context

 MA5.16NA
determines the midpoint, gradient and length of an interval, and graphs linear relationships
Related Life Skills outcomes: MALS32MG, MALS33MG, MALS34MG
Content
 Students:
 Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (ACMNA294)
 determine the midpoint of an interval using a diagram
 use the process for calculating the 'mean' to find the midpoint, M, of the interval joining two points on the Cartesian plane
 explain how the concept of mean ('average') is used to calculate the midpoint of an interval (Communicating)
 plot and join two points to form an interval on the Cartesian plane and form a rightangled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point
 use the interval between two points on the Cartesian plane as the hypotenuse of a rightangled triangle and use the relationship \( \textrm{gradient} = \dfrac{\rm{rise}}{\rm{run}} \) to find the gradient of the interval joining the two points
 describe the meaning of the gradient of an interval joining two points and explain how it can be found (Communicating)
 distinguish between positive and negative gradients from a diagram (Reasoning)
 use graphing software to find the midpoint and gradient of an interval
 Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (ACMNA214)
 use the interval between two points on the Cartesian plane as the hypotenuse of a rightangled triangle and apply Pythagoras' theorem to determine the length of the interval joining the two points (ie 'the distance between the two points')
 describe how the distance between (or the length of the interval joining) two points can be calculated using Pythagoras' theorem (Communicating)
 use graphing software to find the distance between two points on the Cartesian plane
 Sketch linear graphs using the coordinates of two points (ACMNA215)
 construct tables of values and use coordinates to graph vertical and horizontal lines, such as \(x=3\), \(x=1\), \(y=2\), \(y=3\)
 identify the \(x\) and \(y\)intercepts of lines
 identify the \(x\)axis as the line \(y= 0\) and the \(y\)axis as the line \(x = 0\)
 explain why the \(x\) and \(y\)axes have these equations (Communicating, Reasoning)

graph a variety of linear relationships on the Cartesian plane, with and without the use of digital technologies, eg
\( y = 3  x \), \( y = \frac{x+1}{2} \), \( x + y = 5 \), \( x  y = 2 \), \( y = \frac{2}{3}x \)
 compare and contrast equations of lines that have a negative gradient and equations of lines that have a positive gradient (Communicating, Reasoning)
 determine whether a point lies on a line by substitution
 Solve problems involving parallel lines (ACMNA238)
 determine that parallel lines have equal gradients
 use digital technologies to compare the graphs of a variety of straight lines with their respective gradients and establish the condition for lines to be parallel (Communicating, Reasoning)
 use digital technologies to graph a variety of straight lines, including parallel lines, and identify similarities and differences in their equations (Communicating, Reasoning)
Background Information
The Cartesian plane is named after the French philosopher and mathematician René Descartes (1596–1650), who was one of the first mathematicians to develop analytical geometry on the number plane. He shared this honour with the French lawyer and mathematician Pierre de Fermat (1601–1665). Descartes and Fermat are recognised as the first modern mathematicians.
National Numeracy Learning Progression links to this Mathematics outcome
When working towards the outcome MA5.16NA the subelements (and levels) of Number patterns and algebraic thinking (NPA7) describe observable behaviours that can aid teachers in making evidencebased decisions about student development and future learning.
The progression subelements and indicators can be viewed by accessing the National Numeracy Learning Progression.