Specify locations and describe spatial relationships using a Cartesian coordinate system Understand, represent and analyze patterns and relationships using tables, equations and graphs Begin to develop understanding of linear functions as tools for mathematical modeling
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?
Conceptual
What relationships exist between points and lines?
Debatable
How reliable are mathematical models in describing real-life phenomena?
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.
Intended Learning
Plot an ordered pair in a Cartesian coordinate system
Identify the following in a Cartesian plane: axes, origin, quadrants, ordered pairs
Calculate the distance between two points in a plane
Find the midpoint of two points in the plane
Identify patterns and develop understanding of linear sequences whose inputs are counting numbers, predict the next term, and begin to develop the rules which describe them
Calculate and interpret gradients (slopes) as rates of change (increasing/decreasing)
Investigate and develop understanding of the meaning of slopes which are positive, negative, zero, and undefined
Investigate and develop understanding of different techniques that can be used to draw the graph of a linear relation including: table of values, intercepts, \(y=mx+b\)
Investigate and develop understanding of characteristics of special lines including vertical, horizontal, parallel, perpendicular including \(m_1\times m_2=-1\)
Investigate and develop understanding of the relationship between the graph of a line and its equation and write the equation of a line in \(y=mx+b\) form
Transform the equation of line written in slope-intercept form into standard form \(ax+by=c\)
Recognize a direct proportion \(y=kx\) as a special case of a linear equation, understanding the constant of proportionality (k) is the slope
Determine whether a given point satisfies the equation of a line geometrically and algebraically
Model and solve contextualized real-life problems that involve linear relationships