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Somebody asked over in the comment thread about what techniques of solving linear equations were required by 8.EE.8 (the comment says 8.EE.7 but I think that’s a typo). The standard does not require any specific method, but it does require an algebraic solution. So I read this as saying that students should have at least one algebraic method of solution to them, but do not necessarily have to know all methods. Since this is the first appearance of a topic that will re-appear in high school, and since there are many other things to learn in Grade 8, an excessively thorough treatment does not seem warranted here, as long as students have an algebraic method available. As for which method to teach first, substitution or elimination, I have discovered this is a matter of (sometimes passionate) opinion. So I think that’s a choice left up to the curriculum writers. (And, obviously, the assessment writers can’t insist on one method over another.)
The standard (8.EE.8b) simply says “solve systems of two linear equations in two variable algebraically,” but I find it hard to believe that the authors’ intent is simply for students to perform a procedure. I am trying to formulate a statement of what it is that I want students to understand about solving systems algebraically. Combining two linear equations in two variables using substitution or linear combinations results in a single equation in one variable. The solution of that equation is a partial solution of the system. Why? What sense making am I looking for from the students?
If you look at all of 8.EE.8, you will see both procedural and conceptual components, in particular in the stem of the standard, which says “Analyze and solve pairs of simultaneous linear equations.” Students should both learn procedures and understand why they work; different curricular approaches might handle this in different ways, and the standards don’t dictate a particular curricular approach.
To your specific question, consider, for example, the following (abbreviate) description of the method of substitution.
If both equations are true, then the equation you obtain by isolating one of the variables in one of the equations is also true, and therefore so is the equation you get by substituting in the other equation. You can solve this equation for one of the coordinates, and you can use that to find the other coordinate.
Is this a description of a procedure, or is it a passage of reasoning? Really it’s both. Whether you focus on getting students to understand it as a procedure first, or on getting them to understand the reasoning process, is a curricular choice.
A wondering… How is the high school standard A-REI.6 different from the 8th grade standard 8.EE.8b?
A-REI.6 reads, “Solve systems of linear equation exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.”
I can’t find any substantive difference in the language of the standards, but I can’t imagine that there was no intended difference. Thanks.
The Grade 8 standard explicitly limits to systems of two equations in two variables, whereas the high school standard only suggests a focus on such systems, but allows for bigger systems as well. There is also a general uptick in fluency and complexity expectations from Grade 8 to high school. In general, there is some overlap in the topics studied in Grade 8 and high school, with greater depth expected in high school.
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