Math Misconceptions & Considerations

7.EE.3 7.EE.4

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

As students begin to build and work with expressions containing more than two operations, students tend to set aside the order of operations. This leads to students separating or detaching numbers from the operation. Sometimes this happens because one operation appears to be easier to work with. For example, a student might solve 267 – 30 + 28 by adding the 30 + 28 first. And, sometimes this happens because certain numbers are more appealing to operate with than others. They want to focus on the easier numbers in the computation, which may go against the structural rules of algebra. MISCONCEPTION:

WHAT TO DO: Students need lots of practice not only solving expressions and equations but also finding errors in given problems and correcting the errors. As students learn to identify potential errors that can be made in the solution process they learn to avoid the errors.

Each of these mistakes shows a gap in students’ understanding of how to simplify numerical expressions with multiple operations.

Solving word problems is often a difficult task for students. Translating words into algebraic symbols, equations, or inequalities can be challenging. Many students do not understand that a variable in an equation or inequality represents a number of items rather than an object. For example, P represents the number of people. Students may also have difficulty related to their inability to interpret what the word problem is asking. MISCONCEPTION: At Jessie’s restaurant, for every 4 people who order cheesecake, there are five people who order apple pie. Write an equation that represents this situation. Let c represent the number of cheesecakes and a represent the number of apple pies ordered.

4c = 5a Write an equation to represent the situation, “There are 8 times as many people in China as in England.” Let c represent China, and e represent England.

8c = e To help students work through problems they should have time to work in small groups so that they are forced to verbalize problems. Often when students hear problems read out loud they are able to make sense of information. The discussion that occurs between students sharing ideas can help students to use reason to solve problems correctly.

Success in the future job market will require students to apply mathematical knowledge to solve problems.

Based on, “Descriptions of the Misconceptions Included in Our Study” found at: http://www.bc/edu

This student directly translated the words in this problem and incorrectly found the solution. The correct solution is 5c = 4a.

This student placed the multiplier next to the letter associated with the larger group. WHAT TO DO: Have students talk through their reasoning. There are 8 people in China for every 1 person in England; which means there are more Chinese people. It takes 8 people in England to represent 1 person in China. If we use 8c = e then we are saying there are 8 people in China for every 1 person in England. That’s not right! If we change this to 8e = 1c what are we saying?

Solving linear inequalities is very similar to solving linear equations except for one important detail; flip the inequality sign when you multiply or divide the inequality by a negative number. Why do we do this? Let’s look at some examples to compare solving an inequality with an equation.

x + 4 < 3 - 4 - 4 x < -1

2 - x > 0 + x + x

2 > x or x < 2

7 + 5x < 3x - 5 - 3x - 3x 7 + 2x < - 5 -7 - 7 2x < -12 2x / 2 < -12 / 2 x < -6

- 2x > 4 - 2x / -2 < 4 / -2 x < -2

The only difference between solving this inequality and solving the equation x + 4 = 3 is the equals sign. The solution method is exactly the same.

The only difference between solving this inequality and solving the equation 2 – x = 0 is the equals sign. The solution method is exactly the same.

The only difference between solving this inequality and solving the equation 7 + 5x = 3x – 5 is the equals sign. The solution method is exactly the same.

This is different because when we divided by -2 we flipped the inequality sign. To help students understand why this is necessary look at it with numbers instead of variables. The statement 4 > 2 is true because 4 is bigger than 2. Let’s multiply this inequality by -1… -1 (4 > 2), we get -4 > -2. Is this correct? No! If we don’t flip the inequality sign we end up with an incorrect statement; however, when we flip the inequality sign we get, -4 < -2, which is a true statement.