Notes & FAQs

At first glance, the division might appear to be due to solving single variable equations versus solving equations with two variables; however, this approach is more than that.  The idea was to approach abstract equations conceptually first through context, manipulatives, and visual models including using algebra tiles and bar models. If systems were explored in Unit 2 for depth, then algebra tiles are helpful in understanding the solving systems by elimination while bar models are helpful in understanding solving systems by substitution.  Both can be revisited here, time permitting, and connected to the graph of a linear system as further depth in this unit (see Question 3).  The idea was that solving systems graphically was even more abstract than solving systems algebraically.  As students struggle with graphs in general, the preference was to make them the attention of the entire unit rather than just a piece of other instruction.  This also serves to decrease the demand on students working memory by not expecting them to learn multiple new concepts at once. 

Arguably, this unit could be sequenced in a variety of ways.  The sequence suggested by the ordering of the standards at the left involves understanding linear functions, applying them to data as an extension & a check for understanding, and then comparing two proportional relationships, which can lead into linear systems, time permitting.

While linear functions are exlored here, functions are not formalized until the next unit.  Linear functions are contrasted with non-linear functions in the next unit to provide greater depth and additional time on this major work of the grade level.

The design team felt that plotting points from a linear realtionship followed by applying a line of best fit over a set of plotted points (scatter plot) would prepare students best for graphing a linear system with two different relationships, time permitting.  Data is an ever increasing part of our world and a focus of our new CA framework; it should not be neglected.  Remember too that Algebra 1 will emphasize systems.

Comparing two proportional relationships on the same coordinate plane would lead to a linear system with a single solution at (0,0).  If unit rates where the same, but expressed differently like $5 dollars per t-shirt or $25 for 5 t-shirts, then they would be collinear & have infinite solutions. Comparing a proportional relationships with a non-proportional relationship (different y-intercepts with the same slope), would lead to parallel lines, a system with no solution. Finally, comparing a proportional relationships with a non-proportional relationship (different y-intercepts with different slopes), would lead to linear systems with a single solution other than (0.0).

Yes, they are "essential" for 8th grade; however, through vertical articulation with high school.  8th grade will emphasize linear equations while Algebra 1 will emphasize linear systems of equations.  That said, a system is merely two linear equations; therefore, instruction could move to linear systems before mastering linear equations as systems provide double the chance to practice linear equations.