Math Misconceptions & Considerations

7.EE.1 7.EE.2

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.

This standard introduces students to linear expressions with more operations. Students will need to use their understanding about the order of operations, the properties of operations, and the rules for performing operations with negative numbers to transform more complex expressions. MISCONCEPTION:

WHAT TO DO:

Students need experience with multiple ways of writing an expression. Each student may see the expression 7 – 2(3 – 8x) in a different way. This image demonstrates 3 different ways of looking at the same expression. This is a valuable lesson for students to experience because it helps develop flexibility in their thinking.

In the first example, the student has tried to follow order of operations by performing the computation in parentheses first even though no simplification is possible. In the second example, the student has removed the -2 from the multiplication. In the last example, the student does not have an understanding of the rules for multiplying negative numbers.

Innately students want to solve each problem by giving a single answer. When students see an answer like 2m + 6 many think it is unfinished because of the operation symbol. Many students see an operation symbol, +, -, x, or ÷, as an invitation to do something. Logic tells them that if they can still do something then they must not be done. Therefore, 2m + 6 becomes 8m. MISCONCEPTION:

WHAT TO DO: The left side of the equation below looks like a mathematical question, “What is 3 divided by 40?” While the right side simply looks like a fraction giving no real indication that a mathematical question is looming.

In reality, both of these mean the same thing. This is an example of two different ways of writing the same thing. The fraction appears to be an answer, but it is also another way to tell us to perform a mathematical calculation. This realization can help students understand that some expressions lead a dual existence as both process and product. 2m + 6 can be seen as a set of instructions for calculating a numerical value, and/or it can be seen as a mathematical object in its own right.

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. NOTE: Students in grade 7 do not need to recognize when a problem has infinite solutions, one solution, or no solution.

Often when students are working with percent increase or decrease they incorrectly assume that the percent of increase or decrease equals the amount paid. When students do this they are forgetting to include the cost of the item itself. Putting problems into real world contexts and drawing models helps students to remember to include the original amount of the item in their solution. MISCONCEPTION:

WHAT TO DO: A sweatshirt costs $45. Sales tax is 7%. What is the total amount you will pay for the sweatshirt? The model below represents the cost, c, of the sweatshirt before sales tax is calculated. The sales tax is an additional 7% of the cost that must be paid. This can be rewritten as c + 0.07c which could also be written as 1c + 0.07c. Students should see that this can be simplified further by combining like terms so c + 0.07c = 1.07c. Percentage of decrease works the same way but with subtraction. For this problem you can say the sweatshirt is on sale for 20% off. For ease you might want to say that there is no sales tax on this sweatshirt. This expression can be rewritten as c – 0.20c = 1c – 0.20c = 0.80c. By using a model with percentage of decrease students can also begin to see the correlation between a 20% discount and paying 80% of the cost.

100% of the cost

c

+ 7%

0.07c

100% of the cost

c 0.20c

20% -