8.14 - Algebraic Expressions

8.14  The student will

a)  evaluate an algebraic expression for given replacement values of the variables; and

b)  simplify algebraic expressions in one variable.

• I can determine the total cost of a family trip to the movies when the number of people and the cost of tickets varies for each age group, and I can figure my total score on a video game where different tasks give different points.
  

• I will always get the same result as everyone else when evaluating an expression by following the set of rules called order of operations.
  

·  An expression is a representation of a quantity. It may contain numbers, variables, and/or operation symbols.  It does not have an “equal sign (=)” (e.g.,, 5x, 140 - 38.2, -18 ∙ 21, (5 + 2x) ∙ 4). An expression cannot be solved.

·  A numerical expression contains only numbers, the operations symbols, and grouping symbols.

·  Expressions are simplified using the order of operations.  

·  Simplifying an algebraic expression means to write the expression as a more compact and equivalent expression.  This usually involves combining like terms.

·  Like terms are terms that have the same variables and exponents. The coefficients do not need to match (e.g., 12x and -5x; 45 and -5 ; 9y, -51y and y.)

·  Like terms may be added or subtracted using the distributive and other properties. For example,

-  2(x - ) + 5x = 2x – 1 + 5x = 2x + 5x – 1 = 7x - 1

-  w + w – 2w = (1 + 1) w – 2w = 2w – 2w = (2 – 2) w = 0
w = 0

·  The order of operations is as follows:

-  First, complete all operations within grouping symbols*.  If there are grouping symbols within other grouping symbols, do the innermost operation first.

-  Second, evaluate all exponential expressions.

-  Third, multiply and/or divide in order from left to right.

-  Fourth, add and/or subtract in order from left to right.

* Parentheses ( ), brackets [ ], braces {}, absolute value  
(i.e.,   – 7), and the division bar (i.e., ) should be treated as grouping symbols.

·  Properties of real numbers can be used to express simplification. Students should use the following properties, where appropriate, to further develop flexibility and fluency in problem solving (limitations may exist for the values of a, b, or c in this standard):

-  Commutative property of addition: .

-  Commutative property of multiplication: .

-  Associative property of addition: .

-  Associative property of multiplication: .

-  Subtraction and division are neither commutative nor associative.

-  Distributive property (over addition/subtraction):
.

-  The additive identity is zero (0) because any number added to zero is the number. The multiplicative identity is one (1) because any number multiplied by one is the number. There are no identity elements for subtraction and division.

-  Identity property of addition (additive identity property): .

-  Identity property of multiplication (multiplicative identity property): .

-  Inverses are numbers that combine with other numbers and result in identity elements
[e.g., 5 + (–5) = 0; · 5 = 1].

-  Inverse property of addition (additive inverse property): .

-  Inverse property of multiplication (multiplicative inverse property): .

-  Zero has no multiplicative inverse.

-  Multiplicative property of zero: .

-  Division by zero is not a possible mathematical operation. It is undefined.

-  Substitution property: If , then b can be substituted for a in any expression, equation, or inequality.

·  A power of a number represents repeated multiplication of the number. For example, (–5)⁴ means (–5) · (–5) · (–5) ∙ (−5).  The base is the number that is multiplied, and the exponent represents the number of times the base is used as a factor. In this example, (–5) is the base, and 4 is the exponent. The product is 625. Notice that the base appears inside the grouping symbols. The meaning changes with the removal of the grouping symbols. For example, –5⁴ means 5 ∙ 5 ∙ 5 ∙ 5 negated which results in a product of -625. The expression – (5)⁴ means to take the opposite of
5 ∙ 5 ∙ 5 ∙ 5 which is -625. Students should be exposed to all three representations.

·  An algebraic expression is an expression that contains variables and numbers.

• What is the role of the order of operations when evaluating expressions?
  Using the order of operations assures only one correct answer for an expression.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

·  8.14a₂  Represent algebraic expressions using concrete materials and pictorial representations. Concrete materials may include colored chips or algebra tiles.

·  8.14b₁  Simplify algebraic expressions in one variable. Expressions may need to be expanded (using the distributive property) or require combining like terms to simplify. Expressions will include only linear and numeric terms. Coefficients and numeric terms may be rational.

·  8.14a₁  Use the order of operations and apply the properties of real numbers to evaluate algebraic expressions for the given replacement values of the variables. Exponents are limited to whole numbers and bases are limited to integers. Square roots are limited to perfect squares. Limit the number of replacements to no more than three per expression.  

variable, coefficient, substitution, order of operations, algebraic expression

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