Understand the links between equations and graphs, and use graphing skills to complement, enhance and extend algebra skills Describe 2-D spatial relationships using a Cartesian coordinate system and solve problems Understand, represent and analyse patterns and relationships using tables, equations and graphs Begin to develop an understanding of linear functions as tools for mathematical modeling Effectively use the GDC to solve problems with one or two equations (including quadratics)
Conceptual Framework
Form system, space in the context of Scientific and Technical Innovation
Statements of Inquiry
Students will appreciate the need to develop mathematical systems to explore spatial relationships
Factual
How can we develop a system to locate one's position?
What are the advantages and disadvantages of a grid system?
Conceptual
What relationships exist between points and lines?
Can one line be represented by different equations?
Debatable
How reliable are mathematical models in describing real-life phenomena?
What situations give rise to straight-line equations?
Description
We begin by creating a need for students to develop a system to locate one's position in a 2-D space. This is abstracted to the idea of a Cartesian coordinate system to locate points which have both positive and negative values. An algorithm to find the distance between points is based on the Pythagorean theorem while the midpoint is based on the average of each pair of coordinates. Points can then be placed on our grid in linear patterns for which students can begin to describe with general rules. They soon learn that the lines which join these points can be described with the same rule. Students then inquire into the steepness of different lines and discuss the concept of slope as the rate of change of y per unit change in x. We then move into a more algebraic approach to finding the equation of a line and begin to look algebraically at the relationship between slopes of lines and the points where lines intersect. Students will develop fluency in moving between the different forms or representations of a linear relationship including tables, graphs, and algebraic rules. Many naturally occurring phenomena can be described using the equation of a line which we learn is a linear model.
Learning Outcomes
Identify the following in a Cartesian plane: axes, origin, quadrants, ordered pairs
Find and calculate the distance and midpoint between two points in the Cartesian plane
Identify patterns and develop understanding of linear sequences, predict the next term, and begin to develop the rules which describe them
Calculate and interpret gradients (slopes) as rates of change
Investigate and develop understanding of different techniques that can be used to draw the graph of a linear relation including: table of values, intercepts
The generalised form of \(y=mx+c\) and a clear understanding of why c is the y-intercept and the gradient m as the ratio of the change in y to the change in x
Investigate and develop understanding of characteristics of special lines including vertical, horizontal, parallel, and perpendicular
Investigate and develop understanding of the relationship between the graph of a line and its equation and write the equation of a line in \(y=mx+c\) form
Transform the equation of line written in slope-intercept form into standard form \(ax+by=c\)
Determine whether a given point satisfies the equation of a line geometrically and algebraically
Construct graphs by hand (as required for science) when necessary including the selection of suitable scales, and understanding that uniform or equal scales are not necessary but will affect the appearance of the graph
Understand how and why the graphs of any two functions will meet at the solution point to the two simultaneous equations
Use Technology (GDC, Autograph) to find the solution of two simultaneous equations by graphing including readjusting the window or zooming out
As above extended to one linear and one non-linear equation
Graphing of simple quadratic equations
Using the GDC to identify all three intercepts and the vertex co-ordinates for a quadratic function
Understand the core features of quadratic graphs and use the appropriate terminology (concave up/down, vertex, zeroes, maximum, minimum)
Use other methods of solving on the GDC such as the PLYSMT app
Use a variety of methods for formulating and solving problems with simultaneous equations and to select the most appropriate method
Model and solve contextualized real-life problems that involve linear relationships