8.EE.7-8 Misconceptions

Math Misconceptions 8.EE.7 8.EE.8

3 + 4 = 34 Look closely at errors in students’ work (formative assessment) to help you reflect and make instructional decisions to suit all students’ needs.

In order to graph an equation, it must be written in slope-intercept form. Some mathematical problems found in textbook are written in slope-intercept form in the “graphing systems of linear equations” section, but when dealing with realworld situations problems are typically “written” in standard form. Students need to understand how to convert from standard form to slope-intercept form. This will take a firm understanding of the commutative property of equality.

MISCONCEPTION:

WHAT TO DO:

A system of equations is a set or collection of equations that you deal with all at once. When you are solving systems, you are graphically finding an intersection of lines. The point of intersection is the solution for the system. Systems can be solved both graphically and algebraically. In eighth grade, students focus on using graphing and substitution to solve systems. The elimination method will be introduced in Algebra.

Graphing: The first method for solving systems is to solve by graphing. You have to be careful when solving graphically because if the axis and lines are not drawn neatly and accurately it can be difficult to see the solution. Also, if the solution is not nice, neat whole-number coordinates or if the lines are almost parallel it can be difficult to see the solution.

The intersection point is at a shallow angle making it difficult to tell where the lines cross.

The intersection point isn’t a pair of whole numbers.

Typically, when solving “graphing” problems, the solutions will be nice, neat whole number coordinates.

Students should recognize that when an equation is in slope-intercept form the yintercept and slope can be used to graph the lines.

However, students can also use a table of values. Once the lines are graphed, the coordinates for the point of intersection should be used to check the solution in both equations.

Substitution: The substitution method works by solving one equation for one of the variables and then substituting this into the other equation to solve for the other variable.

It doesn’t matter which equation or which variable you choose. The answer will be the same regardless; however, some choices may be better than others.

The first example uses the equations used in the previous graphing solution.

Because both equations are already written in slope-intercept form we can set the equations equal to each other, thus eliminating the y. We can now solve for x, and then substitute that value into one of the original equations to solve for y.

WARNING: The most common mistake is to forget to substitute the first value into one of the original equations to solve for the second variable.

In this second example the equations are written in standard form. It is simpler to solve the second equation for y because y is already alone. We could solve the first equation for either variable, but the result would be fractions. We could also solve the second equation for x but again the result would be a fraction, which means it would probably be more difficult.

WARNING: The most common mistake is to substitute the expression into the same equation we solved for y. For example, if I had substituted -2x + 1 into 2x + y = 1, 2x + (-2x + 1) = 1 1=1 Although this statement is true, it doesn’t help solve the system.

In this example neither equation is easier to solve than the other. Regardless of which equation is used the result will be a fraction.

WARNING: Many students think that because the answer doesn’t make sense they did something wrong. Remember, when solving systems we are looking for a point of intersection in the lines. If the answer doesn’t make sense (and all math has been checked) then we know the lines don’t intersect. The lines are parallel.

No solution means the lines are parallel!

Table of Values:

Students can also use a table of values to solve a system. Although this isn’t a very efficient method, a table of values enables students to pay attention to the ordered pairs for each equation. The table below shows the ordered pairs for the equations y = 2x – 4 and y = -3x + 1.

x

2x - 4

-3x + 1

-2 -1 0 1 2

-8 -6 -4 -2 0

7 4 1 -2 -5

The point of intersection is easy to see on the table of values because both equations equal -2 when x = 1. When you use a table of values to solve a system you run the risk of accurately guessing what x-value will result in the same y-value for both equations.