Teaching Models

Equations and Functions

Evaluating Expressions
An arithmetic expression consists of numbers and operations using parentheses, exponents, multiplication, division, addition, and subtraction. An algebraic expressionis like an arithmetic expression, but contains at least one variable. A variable is a letter that represents a number.

When evaluating the expressions in this chapter, students are told to first evaluate within the parentheses and then complete the evaluation from left to right. Students also use the Commutative, Associative, Distributive, Identity, and Zero properties when evaluating expressions. A review of these properties is necessary as students progress through this chapter. The summary below provides a verbal statement of each property followed by an arithmetic example and an algebraic example.

Associative Property of Addition
When numbers or variables are added, the addends can be grouped in different ways without changing the result.
(2 + 3) + 4 = 2 + (3 + 4)
(a + b) + c = a + (b + c)
Associative Property of Multiplication
When numbers or variables are multiplied, the factors can be grouped in different ways without changing the result.
(2 × 3) × 4 = 2 × (3 × 4)
(a × b) × c = a × (b × c)
Commutative Property of Addition
When numbers or variables are added, the order of the addends can be changed without changing the result.
2 + 3 = 3 + 2
a + b = b + a
Commutative Property of Multiplication
When numbers or variables are multiplied, the order of the factors can be changed without changing the result.
2 × 3 = 3 ×
a × b = b × a
Distributive Property
When two addends are multiplied by a factor, the answer is the same as if each addend is multiplied by the factor and then the sum of the products is found.
2 × (3 + 4) = (2 × 3) + (2 × 4)
a × (b + c) = (a × b) + (a × c)
Identity Property for Addition
When 0 is added to a number or variable, the result is the same number or variable.
2 + 0 = 2
a + 0 = a
Identity Property for Multiplication
When a number or variable is multiplied by 1, the result is the same number or variable.
2 × 1 = 2
a × 1 = a
Zero Property for Multiplication
When a number or variable is multiplied by 0, the result is always 0.
2 × 0 = 0
a × 0 = 0

Using the Commutative, Associative, and Distributive properties may simplify the process of evaluating expressions or make it possible to complete the evaluation mentally.

Examples:

5 + 87 + 3 = 92 + 3 = 95
Work from left to right
5 + 87 + 3 = 5 + (87 + 3)
= 5 + 90 = 95
Use the Associative Property.
76 × 97 + 3 × 76 = 7,372 + 228
= 7,600
Multiply first. Then add.
76 × 97 + 3 × 76 = 76 × 97 + 76 × 3
Use the Commutative Property.
= 76 × (97 + 3)
Use the Distributive Property.
= 76 × 100 = 7,600
(This may look longer, but can be done mentally.)

Equations
The equality of two expressions gives an equation. To solve an equation means to find the value of the variable that will make the equation true. The equations in this chapter are simple enough to be solved by inspection or by using a guess-and-check strategy. Explain to students that not all equations are as simple as the equations in this chapter, so it is necessary to learn to use inverse operations to solve simple equations before learning to solve more difficult ones.

Examples:

4p = 24
p = 24 ÷ 4
p = 6
 
← Use inverse operations. →
 
x ÷ 3 = 9
x = 9 × 3
x = 27

In future chapters, students will learn more about evaluating expressions and solving equations.

An equation can be used to define a function. A function is a rule that associates one and only one value of one variable with each value of another variable. The function y = 2x expresses y in terms of x. For each value of x there is one and only one value of y. An equation determines y as a function of x, if for each x, the equation can be solved to give exactly one value of y. In this chapter, students are introduced to functions by making function tables. Students will learn more about functions later.


Teaching Model 21.2: Write and Solve Equations


Houghton Mifflin Math Grade 5