Math Misconceptions & Considerations 6.EE.9
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 realworld 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.
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
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?
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
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? 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.
Remembering which variable is dependent or independent can be difficult for students to master. It takes lots of repetition and practice to fully understand and remember which is which.
Independent Variables • Determine the value of the dependent variable • What you do • Controls the experiment • Cause • Input
Dependent Variables • Are determined by the dependent variable • What happens • End measure • Effect • Output
EXAMPLES: • • • •
•
•
If I drop a penny, how far will it sink into the sand on the beach? o The depth is dependent on the height. I have control over the height. What is the relationship between weight and how much someone eats? o Weight is dependent on how much someone eats. How much someone eats is going to change a person’s weight. What is the impact of a drug on a disease? o The drug can affect the disease. The drug is the independent variable and the disease is the dependent variable. Jason goes to the store and buys some apples that cost $0.50 each. The equation is c = .50a. What is the relationship? o The total amount Jason spent is dependent on the number of apples he buys. The apples are independent, and the total cost is dependent. You are charged $0.75 per email you receive on your phone. Write an equation to model the cost of your total bill for email messages. o b = .75e. The number of email messages is independent because it controls the cost of the total bill for email messages. o How would the equation change if you added a set fee of $45 a month for phone service? How does this fee affect the independent and dependent variables? You are 10 miles from home. You continue to drive for 4 hours at x miles per hour. How far are you from home? o d = 10 + 4m. The speed you are driving determines how far you are from home. Speed controls distance. Distance does not control speed.
Graphs are symbolic representations of situations that pictorially show the relationship. We use graphs to clearly show the relationship between the dependent and independent variables. When graphing equations the independent variable is the xaxis and the dependent variable is the y-axis. Many students struggle when constructing graphs
Does it really matter where the independent and dependent variables are placed on the graph? To answer this question, let’s look at an example. Jason goes to the store and buys some apples that cost $0.50 each. The equation is c = .50a. What is the relationship? • •
The total amount Jason spent is dependent on the number of apples he buys. The apples are independent, and the total cost is dependent.
EXAMPLE:
This graph shows that the number of apples directly affects the total cost of Jason’s grocery bill. The total cost is dependent on the number of apples.
NON-EXAMPLE
This graph shows that the total cost directly affects the number of apples purchased. In this problem, is the total cost independent of the number of apples?
Let’s look at one more example with its counter non-example. You are 10 miles from home. You continue to drive for 4 hours at x miles per hour. How far are you from home? • • • •
The equation is d = 10 + 4m. The speed you are driving determines how far you are from home. Speed controls distance. Distance does not control speed.
EXAMPLE:
This graph shows that my speed affects how far I am from home. The speed controls my distance.
NON-EXAMPLE
This graph shows that the distance from my house determines my speed…is that right? Does my distance control how fast I drive?
It makes sense that speed controls my distance, but it does not make sense that my distance controls my speed. The goal is to find the relationship between two variables. As one variable changes, so does the other. Graphing is a useful way to visualize and describe these relationships.