OUTCOME
A student:
MA4-11NA:
creates and displays number patterns; graphs and analyses linear relationships; and performs transformations on the Cartesian plane
TEACHING POINTS
When describing number patterns algebraically, it is important that students develop an understanding of the use of pronumerals as algebraic symbols for numbers of objects rather than for the objects themselves.
In Linear Relationships, the study of patterns focuses on those that are linear. However, teachers may include a few simple non-linear patterns so that students realise that not all patterns are linear.
The Cartesian plane (commonly referred to as the ‘number plane’) is named after the French philosopher and mathematician René Descartes (1596–1650), who was one of the first to develop analytical geometry on the number plane. On the Cartesian plane, the coordinates of a point refer to an ordered pair (x,y) describing the horizontal position x first, followed by the vertical position y.
Students are introduced to the four quadrants of the Cartesian plane in Stage 3. However, they may not be familiar with the terms ‘Cartesian plane’, ‘x-axis’ and ‘y-axis’, as in Stage 3 these are generally referred to as the ‘number plane’, ‘horizontal axis’ and ‘vertical axis’, respectively.
LANGUAGE
In Stage 3, students were introduced to patterns involving one operation and used the terms ‘position in the pattern’ and ‘value of the term’ when describing a rule for a pattern from a table of values, eg ‘The value of the term is three times the position in the pattern’.
Students will need to become familiar with and be able to use new terms, including ‘coefficient’, ‘constant term’ and ‘intercept’.
Students should be aware that ‘gradient’ may be referred to as ‘slope’ in some contexts.
PURPOSE RELEVANCE
Linear relationships are very common in mathematics and science. The graph of two quantities that have a linear relationship is a straight line. A linear relationship may be a direct relationship or an inverse relationship. In a direct relationship, as one quantity increases, the other quantity also increases, or as one quantity decreases, the other quantity also decreases. In an inverse relationship, as one quantity increases, the other quantity decreases. Examples of linear relationships familiar in everyday life include the distance travelled and time taken, the conversion of one currency to another, the cost of printing involving an initial set-up cost and a dollar rate per item, the cost of taxi fares involving a hiring charge and a dollar rate per kilometre, and the cost of catering involving a base amount for a set number of people plus a rate for each extra attendee. Coordinate geometry facilitates the exploration and interpretation of linear relationships.
Given coordinates, plot points on the Cartesian plane, and find coordinates for a given point (ACMNA178)
plot and label points on the Cartesian plane, given coordinates, including those with coordinates that are not whole numbers
identify and record the coordinates of given points on the Cartesian plane, including those with coordinates that are not whole numbers
Describe translations, reflections in an axis, and rotations of multiples of 90° on the Cartesian plane using coordinates (ACMMG181)
use the notation ${P}^{\prime}$
“>P′
to name the ‘image’ resulting from a transformation of a point $P$
“>P
on the Cartesian plane
plot and determine the coordinates for ${P}^{\prime}$
“>P′
resulting from translating $P$
“>P
one or more times
plot and determine the coordinates for ${P}^{\prime}$
“>P′
resulting from reflecting $P$
“>P
in either the $x$
“>x
or $y$
“>
$y$-axis
investigate and describe the relationship between the coordinates of $P$“>P and ${P}^{\prime}$
“>P′
following a reflection in the $x$
“>x
– or $y$
“>y
-axis, eg if $P$
“>P
is reflected in the $x$
“>$x$
“>
$x$
-axis, ${P}^{\prime}$
“>P′
has the same $x$
“>$x$
“>
$x$
-coordinate, and its $y$
“>$y$
“>
$y$
-coordinate has the same magnitude but opposite sign (Communicating)
recognise that a translation can produce the same result as a single reflection and vice versa (Reasoning)
plot and determine the coordinates for ${P}^{\prime}$
“>P${\prime}^{}$ resulting from rotating $P$
“>P
by a multiple of 90° about the origin
investigate and describe the relationship between the coordinates of $P$“>$P$
“>
$P$
and ${P}^{\prime}$
“>P′
following a rotation of 180° about the origin, eg if $P$
“>P
is rotated 180° about the origin, the $x$
“>x
– and $y$
“>y
-coordinates of ${P}^{\prime}$
“>P′
have the same magnitude but opposite sign (Communicating)
recognise that a combination of translations and/or reflections can produce the same result as a single rotation and that a combination of rotations can produce the same result as a single translation and/or reflection (Reasoning)
Plot linear relationships on the Cartesian plane, with and without the use of digital technologies (ACMNA193)
use objects to build a geometric pattern, record the results in a table of values, describe the pattern in words and algebraic symbols, and represent the relationship on a number grid, eg
check pattern descriptions by substituting further values (Reasoning)
replace written statements describing patterns with equations written in algebraic symbols, eg ‘You multiply the number of pentagons by four and add one to get the number of matches’ could be replaced with ‘ $m=4p+1$“>m=4p+1 (Communicating, Reasoning)
determine whether a particular pattern can be described using algebraic symbols (Problem Solving)
represent the pattern formed by plotting points from a table and suggest another set of points that might form the same pattern (Communicating, Reasoning)
explain why it is useful to describe the rule for a pattern in terms of the connection between the top row and the bottom row of the table (Communicating, Reasoning)
recognise a given number pattern (including decreasing patterns), complete a table of values, describe the pattern in words and algebraic symbols, and represent the relationship on a number grid
generate a variety of number patterns that increase or decrease and record them in more than one way (Communicating)
determine a rule in words to describe the pattern by relating the ‘position in the pattern’ to the ‘value of the term’ (Communicating, Problem Solving)
explain why a particular relationship or rule for a given pattern is better than another (Communicating, Reasoning)
distinguish between graphs that represent an increasing number pattern and those that represent a decreasing number pattern (Communicating, Reasoning)
determine whether a particular number pattern forms a linear or non-linear relationship by examining its representation on a number grid (Problem Solving)
use a rule generated from a pattern to calculate the corresponding value for a larger number
form a table of values for a linear relationship by substituting a set of appropriate values for either of the pronumerals and graph the number pairs on the Cartesian plane, eg given y = 3x + 1, form a table of values using x = 0, 1 and 2 and then graph the number pairs on the Cartesian plane with an appropriate scale
explain why 0, 1 and 2 are frequently chosen as x-values in a table of values (Communicating, Reasoning)
extend the line joining a set of points on the Cartesian plane to show that there is an infinite number of ordered pairs that satisfy a given linear relationship
interpret the meaning of the continuous line joining the points that satisfy a given number pattern (Communicating, Reasoning)
read coordinates from the graph of a linear relationship to demonstrate that there are many points on the line (Communicating)
derive a rule for a set of points that has been graphed on the Cartesian plane
graph more than one line on the same set of axes using digital technologies and compare the graphs to determine similarities and differences, eg parallel, pass through the same point
identify similarities and differences between groups of linear relationships, eg
determine which term of the rule affects the gradient of a graph, making it increase or decrease (Reasoning)
use digital technologies to graph linear and simple non-linear relationships, such as $y={x}^{2}$
“>y=x^{2}
recognise and explain that not all patterns form a linear relationship (Communicating)
determine and explain differences between equations that represent linear relationships and those that represent non-linear relationships (Communicating)
recognise that each point on the graph of a linear relationship represents a solution to a particular linear equation
use graphs of linear relationships to solve a corresponding linear equation, with and without the use of digital technologies, eg use the graph of $\phantom{\rule{thinmathspace}{0ex}}y=2x+3\phantom{\rule{thinmathspace}{0ex}}$
“>y = 2x + 3
to find the solution of the equation $\phantom{\rule{thinmathspace}{0ex}}2x+3=11\phantom{\rule{thinmathspace}{0ex}}$
“>2x + 3 = 11
graph two intersecting lines on the same set of axes and read off the point of intersection
explain the significance of the point of intersection of two lines in relation to it representing the only solution that satisfies both equations (Communicating, Reasoning)