 # The Order of Operations

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Lesson Plans and Worksheets for Grade 6
Lesson Plans and Worksheets for all Grades

Examples, videos, and solutions to help Grade 6 students learn what is the order of operations and how to use the order of operations to evaluate expressions.

New York State Common Core Math Module 4, Grade 6, Lesson 6

Lesson 6 Student Outcomes

Students evaluate numerical expressions. They recognize that in the absence of parentheses, exponents are evaluated first.

Example 1: Expressions with Only Addition, Subtraction, Multiplication, and Division

Multiplication can be thought of as repeated addition.
Division is repeated subtraction.

Multiplication and division are more powerful than addition and subtraction, which led mathematicians to develop the order of operations in this way. When we evaluate expressions that have any of these four operations, we always calculate multiplication and division before doing any addition or subtraction.

Since multiplication and division are equally powerful, we simply evaluate these two operations as they are written in the expression, from left to right.

Addition and subtraction are at the same level in the order of operations and are evaluated from left to right in an expression.

In what order do we evaluate
3 + 4 × 2

What operations are evaluated first?
What operations are always evaluated last?

Exercises
4 + 2 × 7
36 ÷ 3 × 4
20 - 5 × 2

Example 2: Expressions with Four Operations and Exponents

In the last lesson, you learned about exponents, which are a way of writing repeated multiplication. So, exponents are more powerful than multiplication or division. If exponents are present in an expression, they are evaluated before any multiplication or division.

When we evaluate expressions, we must agree to use one set of rules so that everyone arrives at the same correct answer. These rules are based on doing the most powerful operations first (exponents), then the less powerful ones (multiplication and division, going from left to right), and finally the least powerful ones last (addition and subtraction, going from left to right).
4 + 92 ÷ 3 × 2 - 2

What operation is evaluated first?
What operations are evaluated next?
What operations are always evaluated last?

Example 3: Expressions with Parentheses

The last important rule in the order of operations involves grouping symbols (usually parentheses). These tell us that in certain circumstances or scenarios, we need to do things out of the usual order. Operations inside grouping are always evaluated first, before exponents.

Consider a family of 4 that goes to a soccer game. Tickets are \$5.00 each. The mom also buys a soft drink for \$2.00.
How would you write this expression?
How much will this outing cost?

Consider a different scenario: The family goes to the game like before, but each of the family members wants a drink.
How would you write this expression?
Why would you add the 5 and 2 first?
How much will this outing cost?

The last complication that can arise is that if two or more sets of parentheses are ever needed, evaluate the innermost parentheses first, and work outward.
2 + (92 - 4)
2 • (13 + 5 - 14 ÷ (3 + 4))

Example 4: Expressions with Parentheses and Exponents

Let’s take a look at how parentheses and exponents work together. Sometimes a problem will have parentheses and the values inside the parentheses have an exponent.
2 × (2 + 42)

What do you think will happen when the exponent in this expression is outside of the parentheses?
2 × (2 + 4)2
Will the answer be the same?
Which should we evaluate first?
What happened differently here than in our last example?

Closing

When we evaluate expressions, we use one set of rules so that everyone arrives at the same correct answer. Grouping symbols like parentheses tell us to evaluate whatever is inside them before moving on. These rules are based on doing the most powerful operations first (exponents), then the less powerful ones (multiplication and division, going from left to right), and finally the least powerful ones last (addition and subtraction, going from left to right).

Lesson Summary

Numerical Expression: A numerical expression is a number, or it is any combination of sums, differences, products or divisions of numbers that evaluates to a number.

Statements like, “3 +” or “3 ÷ 0” are not numerical expressions because neither represents a point on the number line. Note: Raising numbers to whole number powers are considered numerical expressions as well, since the operation is just an abbreviated form of multiplication, e.g., 23 = 2 • 2 • 2

Value of a Numerical Expression: The value of a numerical expression is the number found by evaluating the expression. Examples and Exercises

Exercises
4. 90 - 52 × 3
5. 43 + 2 × 8
6. 2 + (92 - 4)
7. 2 • (13 + 5 - 14 ÷ (3 + 4))
8. 7 + (12 - 32)
9. 7 (12 - 3)2 Problem Set
Evaluate each expression.
1. 3 × 5 + 2 × 8 + 2
2. (\$1.75 + 2 × \$0.25 + 5 × \$0.05) × 24
3. (2 × 6) + (8 × 4) + 1
4. ((8 × 1.95) + (3 × 2.95) + 10.95) × 1.06
5. ((12 ÷ 3)2 - (18 ÷ 32)) × (4 ÷ 2)

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