If MATH-8 teachers ever doubted that they teach algebra, diving into the Common Core State Standard 8.EE.C.8 would certainly end the debate. Students are required to solve and analyze pairs of simultaneous linear equations. The depth of this standard involves understanding the meaning of solution(s) of a system, solving a system graphically and algebraically, and using this knowledge to solve real-world problems that lead to a system of two linear equations.
The prerequisite for this content includes solving one-variable equations in Grade 7 and previously in Grade 8 as outlined in this Solutions of Linear Equations post. The tiling activity towards the end of the post would be a great review of types of solutions to set the stage for systems.
In a recent Genius Hour, we focused on how to teach 8th grade students to solve a system of linear equations algebraically. While it may be common to ask student to solve each equation for the variable "y" and set the two resulting expressions equal to solve for the variable "x", this process begs the question "Why?". Are we helping our students make connections so that they understand the origin of the algebraic method and know this process is valid? Do they understand the concept behind solving algebraically?
We looked at an introduction activity that extends from the "Balance This!" Activity in the Solutions of Linear Equations post. The visual of the balanced scale was used to discuss solutions and the adjustments necessary if the conditions are represented with two different scales. In this activity, students reason with a balance of shapes and a given value to determine the value of each shape.
|Solving Systems of Linear Equations|
We started with a school store scenario in which two different "store specials" were offered. One special involved a purchase of three items from the top shelf and two items from the bottom shelf for a total of $16. The second special involved a purchase of two items from the top shelf and three items from the bottom shelf for a total of $14.
How much do the items on each shelf cost? Which shelf has the more expensive items? Justify your reasoning. Many students will use trial and error to begin solving the puzzle, but beware of students who verbalize a strategy that is actually substitution! The activity opens several paths, one of which is solving numerically and then making sense of how that cumbersome numeric approach connects to a systematic algebraic approach. The activity extends to all types of solutions and coordinates with several support activities that were reviewed.
- What are some possible values that keep the firsts scale balanced? How many different combinations can you find? How many combinations are possible?
- What about the second scale? How many total combinations are possible?
- What if you consider both scales simultaneously...meaning the possible values that keep the first scale balanced must also be the same values that keep the second scale balance? How many different combinations can you find? How many combinations are possible?
These activities highlight the Common Core State Standard 8.EE.C.8 that is included in MATH-8 and Accelerated Algebra 1.