Reflection: Connection to Prior Knowledge More with Solving Two Variable Equations - Section 2: Investigation

My students did well with this investigation.  This lesson was really a benchmark lesson for me because it represents a shift in my thinking about mathematics.  I have always been a "y=mx+b" person when it comes to graphing lines.  Then I would worry when students were faced with difficult equations where getting y alone was challenging.  Now students are understanding that a line is made up of the infinite number of solution points to its equation.  Because of this, all they have to do is find coordinates that make the equation true.  I feel that this is a much more rich understanding than simply finding y-intercept, plotting slope, etc.  Students are really beginning to understand what a linear equation (or any equation for that matter) in two variables represents.

I did notice that students are still struggling with part "e" on question 1.  They are able to plot the points that make an equation true but are still unsure why the points should be connected with a line.  In my observations and discussions with students this is due to their lack of understanding of rational numbers in general.  The real reason the points are connected with a line is because there are an infinite number of inputs and outputs (coordinates) in between each set of integer coordinates.  Students were having trouble grasping this because they are not comfortable "playing around" with x-values such as 0.1 or 0.25 or 1.375 or 4.3245.  They were great with finding integer values that work but their lack of comfort with decimal values made it difficult to put down a solid explanation for question "e".

I was pleasantly surprised how well students did on the last question of the investigation (4x+5y=0).  I thought that the values would give the students trouble  but they were very strategic about picking x and y-values that would yield integer results.  Once again, students intentionally stayed away from any decimal values that would make the equation true.  In the past, this would be one of those equations that students would have difficulty graphing because they were locked into the y=mx+b mode.  Letting students find values that make the equation true offers them some flexibility when they have equations in this form.

Connection to Prior Knowledge: Deciding on a new approach

More with Solving Two Variable Equations

Unit 8: Extending Equations
Lesson 11 of 13

Big Idea: Students will formalize their understanding of two variable equations by examining their graphs.

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Standards:
Subject(s):
Math, two variables, Algebra, solve, solution, ordered pairs, equation, equation
40 minutes

James Bialasik

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