Topics
3. Numeric & Algebraic Expressions
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United States of AmericaVirginia
Grade 7

3.01 Order of operations with integers

Lesson
Practice

Order of operations with integers

When we work with integers, we can perform the same four operations that we use with whole numbers: addition, subtraction, multiplication, and division.

The main difference when working with integers is that we have to take into account the signs of the numbers.

To simplify expressions with mixed operations, we need to follow the order of operations:

  1. Evaluate operations inside grouping symbols.

    • Grouping symbols may include parentheses \left(\text{ }\right), brackets [ \text{ }], absolute value bars \vert \text{ }\vert, square root symbols \sqrt{\text{ }}, and fraction bars \frac{⬚}{⬚}.

  2. Evaluate any exponents.

  3. Evaluate any multiplication or division, reading from left to right.

    • When multiplying and dividing, if one of your numbers is negative and the other is positive, your answer will be negative.

    • When multiplying and dividing, if both numbers have the same sign, your answer will be positive.

  4. Evaluate any addition or subtraction, reading from left to right.

    • If you have adjacent positive (plus) and negative (minus) signs, this will become a minus sign.

    • If you have two adjacent negative (minus) signs, this will become an addition sign.

    • When adding two numbers with different signs, we can use the number line to illustrate the process.

  5. Consider and check the sign of the final answer, especially if negative values were involved in the calculation.

Examples

Example 1

Evaluate -5+ 12 \div 2^2

Worked Solution
Create a strategy

Evaluate the exponent, then the division, and finally the addition.

Apply the idea
\displaystyle -5+ 12 \div 2^{2}\displaystyle =\displaystyle -5 + 12 \div 4Evaluate the exponent
\displaystyle =\displaystyle -5+3Evaluate the division
\displaystyle =\displaystyle -2Evaluate the addition

Example 2

Evaluate \left( \left[36 - \left(10 + 10\right)\right] \div 2\right) + 14 \cdot 6

Worked Solution
Create a strategy

We first need to evaluate inside the grouping symbols. Notice there are several layers. We will start inside the innermost parentheses, then the brackets, then the outer parentheses.

Once the grouping symbols are gone we can evaluate the multiplication and the addition.

Apply the idea
\displaystyle \left( \left[36 - \left(10 + 10\right)\right] \div 2\right) + 14 \cdot 6\displaystyle =\displaystyle \left( \left[36 - 20\right] \div 2\right) + 14 \cdot 6Evaluate the innermost parentheses
\displaystyle =\displaystyle \left( 16 \div 2\right) + 14 \cdot 6Evaluate the subtraction in the brackets
\displaystyle =\displaystyle 8 + 14 \cdot 6Evaluate the division in the parentheses
\displaystyle =\displaystyle 8 + 84 Evaluate the multiplication
\displaystyle =\displaystyle 92Evaluate the addition

Example 3

Evaluate \sqrt{56-7}+12 \div 3 \cdot \left(2 + 1\right)

Worked Solution
Create a strategy

We first need to evaluate the operations inside the grouping symbols. In this expression, the groupings symbols are the parentheses and the square root. Next is the division, then multiplication, and lastly the addition.

Apply the idea
\displaystyle \sqrt{56-7}+12 \div 3 \cdot \left(2 + 1\right)\displaystyle =\displaystyle \sqrt{56-7}+12 \div 3 \cdot 3Evaluate the addition in parentheses
\displaystyle =\displaystyle \sqrt{49}+12 \div 3 \cdot 3Evaluate subtraction under the square root
\displaystyle =\displaystyle 7+12 \div 3 \cdot 3Evaluate the square root
\displaystyle =\displaystyle 7+4 \cdot 3Evaluate the division
\displaystyle =\displaystyle 7+12Evaluate the multiplication
\displaystyle =\displaystyle 19Evaluate the addition

Example 4

Evaluate \left[6 - \left(3 + 1\right)\right]^{2}

Worked Solution
Create a strategy

We first need to evaluate the addition inside the parentheses, then evaluate the subtraction inside the bracket, and finally evaluate the exponent.

Apply the idea
\displaystyle \left[6 - \left(3 + 1\right)\right]^{2}\displaystyle =\displaystyle \left[6 - 4\right]^{2}Evaluate the addition
\displaystyle =\displaystyle \left[2\right]^{2}Evaluate the subtraction
\displaystyle =\displaystyle 4Evaluate the exponent

Example 5

Evaluate \dfrac{16 - 4^{2}}{2^{3}}

Worked Solution
Create a strategy

Evaluate the exponents first, then evaluate the subtraction, and finally evaluate the division.

Apply the idea
\displaystyle \dfrac{16 - 4^{2}}{2^{3}}\displaystyle =\displaystyle \dfrac{16 - 16}{8}Evaluate the exponents
\displaystyle =\displaystyle \dfrac{0}{8}Evaluate the subtraction
\displaystyle =\displaystyle 0Evaluate the division

Example 6

Evaluate 15 - 3 \cdot \vert 2 - 5 \vert

Worked Solution
Create a strategy

We first evaluate the subtraction inside the absolute value bars, then evaluate the absolute value, followed by multiplication, then finally the subtraction.

Apply the idea
\displaystyle 15 - 3 \cdot \vert 2 - 5 \vert\displaystyle =\displaystyle 15 - 3 \cdot \vert -3 \vertEvaluate the subtraction in the absolute value
\displaystyle =\displaystyle 15 - 3 \cdot 3 Evaluate the absolute value
\displaystyle =\displaystyle 15 - 9 Evaluate the multiplication
\displaystyle =\displaystyle 6 Evaluate the subtraction
Idea summary

The order of operations is:

  1. Evaluate operations inside grouping symbols.

    • Grouping symbols may include parentheses (\text{ }), brackets [ \text{ }], absolute value bars \vert \text{ }\vert, square root symbols \sqrt{\text{ }}, and fraction bars \frac{⬚}{⬚}.

  2. Evaluate any exponents.

  3. Evaluate any multiplication or division, reading from left to right.

  4. Evaluate any addition or subtraction, reading from left to right.

  5. Consider and check the sign of the final answer, especially if negative values were involved in the calculation.

If the problem has parentheses inside another set of parentheses, simplify the inside parentheses first.

Outcomes

7.PFA.2

The student will simplify numerical expressions, simplify and generate equivalent algebraic expressions in one variable, and evaluate algebraic expressions for given replacement values of the variables.

7.PFA.2a

Use the order of operations and apply the properties of real numbers to simplify numerical expressions. Exponents are limited to 1, 2, 3, or 4 and bases are limited to positive integers. Expressions should not include braces { } but may include brackets [ ] and absolute value bars | |. Square roots are limited to perfect squares.*

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