Topics
1. Integers
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United States of America
Grade 7

1.07 Evaluate expressions with integers

Lesson
Practice

Introduction

We are now familiar with the rules in  addition and subtraction  as well as  multiplication and division of integers  . We have already learned the different  properties of operations  and so we are now prepared to evaluate expressions with integers.

Evaluate expressions with integers

In life, the order in which we do things is important. For example, we put on socks then shoes, rather than shoes and then socks.

The order of operations tells us the steps to evaluate expressions with multiple operations. The order goes:

  1. Complete all operations within grouping symbols such as brackets [\ldots], parentheses (\ldots) or absolute values |\ldots|. If there are grouping symbols within other grouping symbols, do the innermost operation first.
  2. Evaluate all exponents, such as squares and cubes.
  3. Multiply and/or divide in order from left to right.
  4. Add or subtract in order from left to right.

Let's look through some examples of questions involving integers and the order of operations.

Examples

Example 1

Evaluate: (-14) \div 2 - 18 \div (-2)

Worked Solution
Create a strategy

We evaluate any multiplication and division before addition or subtraction.

Apply the idea
\displaystyle \left(-14\right) \div 2 - 18 \div \left(-2\right)\displaystyle =\displaystyle -7 - (-9)Evaluate division from left to right
\displaystyle \text{ }\displaystyle =\displaystyle -7 + 9Combine adjacent signs
\displaystyle \text{ }\displaystyle =\displaystyle 2Evaluate the addition

Example 2

Evaluate: 100 - 9 \times \left(-4\right) + 18 \div \left(-6\right)

Worked Solution
Create a strategy

We evaluate the multiplication and division (going from left to right), then evaluate the addition and subtraction (going from left to right).

Apply the idea
\displaystyle 100 - 9 \times (-4) + 18 \div (-6)\displaystyle =\displaystyle 100 - (-36) + (-3)Evaluate multiplication and division
\displaystyle \text{ }\displaystyle =\displaystyle 100 + 36 -3 Combine adjacent signs
\displaystyle \text{ }\displaystyle =\displaystyle 136 - 3Evaluate the addition
\displaystyle \text{ }\displaystyle =\displaystyle 133Evaluate the subtraction
Reflect and check

We can work through order of operations problems with any type of real number such as integers, fractions, or decimals.

Example 3

Evaluate: [48 \div (-12) - 35 \div 5] \times 3^{2}

Worked Solution
Create a strategy

We need to simplify the problem by using our order of operations rules.

Apply the idea
\displaystyle [48 \div (-12) - 35 \div 5] \times 3^{2}\displaystyle =\displaystyle (-4-7) \times 3^{2}Evaluate the divisions within the brackets
\displaystyle \text{ }\displaystyle =\displaystyle (-11) \times 3^{2}Complete operations within parentheses
\displaystyle \text{ }\displaystyle =\displaystyle (-11) \times 9Evaluate the exponent
\displaystyle \text{ }\displaystyle =\displaystyle -99Evaluate the multiplication
Idea summary

Order of operations:

  1. Complete all operations within grouping symbols such as brackets [\ldots], parentheses (\ldots) or absolute values |\ldots|. If there are grouping symbols within other grouping symbols, do the innermost operation first.
  2. Evaluate all exponents such as squares and cubes.
  3. Multiply and/or divide in order from left to right.
  4. Add or subtract in order from left to right.

Outcomes

7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

7.NS.A.1.D

Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.A.2

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

7.NS.A.2.C

Apply properties of operations as strategies to multiply and divide rational numbers.

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