Math Misconceptions

8.EE.5 8.EE.6

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.

What does an equal sign mean? For many students an equal sign is a signal to perform a given computation. For example, when given “5 + 2 =” students will give the answer 7, but what happens when students are given “5 + 2 = x + 3”. It is important for students to recognize that an equal sign is NOT a signal to perform a given computation nor is it a signal that the answer to a problem comes next. Equal signs have two purposes. First, an equal sign is a way to indicate that two expressions are equivalent.

11x – 4x = 7x 15x + 20 = 10x + 50

When we use the equal sign to indicate that two expressions are equivalent, we may be using variables as unknowns. In both examples above, the variable x is an unknown quantity and we can find the unknown value(s), if any exist, in which these expressions have the same value. The idea of a variable as a changing quantity is an important concept to develop as it helps students understand relationships in mathematical and real-world situations.

11x – 4x = 7x 7x = 7x

Infinite solutions Finding the value, if any exist, in which the expressions are the same can also help build understanding. The example below could represent the conditions under which one membership might be a better value than another.

15x + 20 = 10x + 50

x = 6 One solution

Second, an equal sign is a way to name an expression.

d = 10t + 20 In this case the t indicates a varying quantity with many possible values. In a real-world situation this equation might represent a relationship in which distance is proportional to the amount of time someone ran. This understanding leads to the study of functions; the value of one variable is defined in terms of the other. Understanding the difference between these two uses of the equal sign is fundamental in the study of algebra.

When students learn about slope, they often learn the ratio of “rise over run” and memorize the slope formula. As a result, students think of slope in terms of how it is calculated without understanding it as a rate of change. Many students simply substitute values for the coordinates of points into the slope formula without understanding the meaning of the formula or why it works. MISCONCEPTION: Some students confuse vertical and horizontal change, thus reversing the ratio to run over rise. Other students substitute the wrong coordinates into the formula, sometimes using the coordinates of points in a different order. WHAT TO DO: By making a strong connection between what slope actually describes and the process for determining slope, students can use this understanding to prevent common errors.