Math Misconceptions & Considerations

6.EE.5 6.EE.6 6.EE.7 6.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.

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 x 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 6 do not need to recognize when a problem has infinite solutions, one solution, or no solution.

PEMDAS is a mnemonic that many teachers use to help students remember the order of operations. While PEMDAS is helpful it can also mislead students into thinking that addition must always take precedence over subtraction because the A comes before the S or multiplication over division because the M comes before the D. This can lead students to make mistakes. MISCONCEPTION:

WHAT TO DO:

Students should have practice solving and justifying expressions by plugging a random value in for the variable. If the student had plugged 7 (or any other value) in for n in the expressions above they may have recognized their error in applying the order of operations.

When students simplify an expression like above, they should justify their result. Set the two expressions equal to each other, pick a value for n, and solve it. When it doesn’t work they will have the opportunity to evaluate their process and critique their own thinking.

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; it is not a label representing “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 let 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.

This student directly translated the words in this problem and incorrectly found the solution. The correct solution is 5c = 4a. This student believes c and a are labels standing for cheesecakes and apple pies. Are c and a labels or variables? Is there a difference?

This student placed the multiplier next to the letter associated with the larger group. C and e are not labels standing for China and England rather they are variables standing for the number of people in China or England. 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?

Although this is the first formal exposure students will have to inequalities, they have been dealing with them all their lives. Any statement which includes the words, “at least,” “at most,” “more than,” or “less than” is an inequality. Inequalities appear in math, physics, chemistry, biology, economics, business, cooking, spending money, driving, etc. Students will need to learn how to figure out which numbers satisfy an inequality and how to graph them.

The symbols tell us what they mean.

< Less than The symbol is smaller on the left than on the right, so the number on the left will be smaller than the number on the right.

> Greater than The symbol is bigger on the left than on the right, so the number on the left will be bigger than the number on the right.

Every inequality is a true or false statement. Inequalities are either weak or strict. 4 > 3 True Strict inequality – it doesn’t include itself 6 < 3 False 6 < 6 True Weak inequality – it includes itself 6 < 6 False x < 6 ???? We don’t know what x is so we don’t know if it is

true or false. Instead we have to have to think about all the values of x that make the inequality true. This is called the solution set. In this case, 4, 1, and -3, are part of the solution set, while 7, 10, and 6 are not.

Inequalities can be graphed to show the solution set. Students often get confused about which direction the line should point.

Inequality A statement that shows the relationship between two or more

expressions with one of the following signs: <, >, <, >

The graph lays above all the numbers in the solution set. Ask yourself, “Is 5 less than 6?” Yes! “Is -4 less than 6?” Yes! Students should think about what the inequality means, and what is true about the inequality before they graph it.