Math Misconceptions & Considerations 6.EE.1 6.EE.2 6.EE.3 6.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 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
Secondly, 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.
Translating words into algebraic symbols, equations, or inequalities can be challenging. We use letters to represent numerical quantities. Students have been using variables since the early grades. Students have seen problems like this since first grade.
Write the missing number: 8 + ___ = 10 How is that different from a first-year algebra text?
Solve for x: 8 + x = 10 Which one requires algebraic thinking? In both examples we fill in the missing number, or solve for x, the value of the blank/x is not variable in the literal sense as something that varies. It is a specific unknown value, which we determine to be two. While the blank/x is a variable by name and definition, its value cannot possibly vary. The more accurate term that should be used is unknown. The blank/x is a specific unknown value. In the area formula for a rectangle, A = lw, a relationship is expressed between three variables - A, l, and w. In this case these are variables in the true sense of the meaning. Their values are not only unknown, but can take on infinitely different possibilities as long as the equation is true.
Area = length x width Many students do not understand that a variable represents a number of items rather than an object. Students may interpret 3t as “three trucks” instead of “three times the number of trucks.” This is understandable because students have been working with measurement since 2nd grade; a room 10 meters wide is noted as 10 m. Why would 3t be any different? The key to understanding variables is to recognize that a variable is a quantity, not a thing, an object, or a unit of measure. Students will do better in writing expressions, solving equations, and solving problems if they understand this simple fact.
The order of operations tells us how to interpret expressions. For example, the P in PEMDAS indicates that parentheses come first. When reading the expression, 8(5 + 1), order of operations tells us this means 8 times the sum of 5 + 1. It is important for students to use the appropriate mathematical language when they are asked to write verbal expressions from algebraic expressions. Students write expressions from verbal descriptions using letters and numbers, understanding that order is important. Students read algebraic expressions: • r + 21 is read as “some number plus 21” or “r plus 21” • 6n as “6 times some number” or “6 times n” s • as “some number divided by 6” or “s divided by 6” 6 It is imperative for students to read algebraic expressions in a manner that reinforces that the variable represents a number. Students write algebraic expressions: • 7 less than 3 times a number means 3x – 7 • Twice the difference between a number and 5 means 2(z – 5) • 3 times the sum of a number and 5 means 3(x + 5) • It costs $100 to rent the skating rink plus $5 per person. Write an expression to find the cost for any number (p) of people. What is the cost for 25 people? x+4 • The quotient of the sum of x plus 4 and 2 means 2 Students should identify the parts of an algebraic expression including variables, coefficients, constants, and the names of operations. They can describe expressions such as 2(z – 5) as the product of two factors: 2 and (z – 5). The quantity (z – 5) is viewed as a factor consisting of two terms.
WHAT TO DO: Consider the expression x2 + 5y + 3x + 6. Ø The variables are x and y. Ø There are four terms, x2, 5y, 3x, and 6. Ø There are three variable terms, x2, 5y, and 3x. Ø There are three coefficients, 1, 5, and 3. The coefficient of x2 is 1. Ø There is one constant term, 6. Ø The expression represents the sum of all four terms.
The order of operations tells us how to interpret expressions, but does not necessarily dictate how to calculate them. The expression 8(5 + 1) can be solved using two different methods. Using order of operations:
Using the distributive property:
8(5 + 1) 8(6) 48 8(5 + 1) 8x5+8x1 40 + 8 48
Students need practice simplifying expressions using order of operations as well using the properties and comparing the methods looking for similarities and differences.
WHAT TO DO: Evaluate 5(n + 3) – 7n, when n =
1 2
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: 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.
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.
Properties are introduced throughout elementary grades; however, it is not until 6th grade that students identify the properties by name, and use them to justify a solution method. Using a students understanding of area models from elementary school can be powerful to illustrate the distributive property with variables. Students should identify the factors of the area of the flower garden below as 2.5 and (x + 3). The area can be then expressed as 2.5(x + 3).
x 2.5
Roses
3
Pansies
Students should also be able to find the factors when given an expression. The expression 9x + 12 represents the area of the figure below. Students need to find the width and length of the figure. They start by finding the greatest common factor of 3 to represent the width and then use the distributive property to find the length of (3x + 4). The factors of the figure are 3(3x + 4).
9x
12
Students use their understanding of multiplication to interpret 4(3 + x) as 4 groups of (3 + x). They use a model to represent x, and make an array to represent 4(3 + x). The model helps students explain why 4(3 + x) is equal to 12 + 4x. An array with 4 columns and (3 + x) in each column: