In the past two sections, you reviewed addition, subtraction, multiplication, and division. In this section, you will notice that when simplifying expressions, there is a specific mathematical order.

To begin this part of the lesson, use a calculator or app to simplify the following expression. Once you have done so, continue the lesson to see how other students simplified the expression.

Matt and Josey simplified the same expression below and got different results. Click on each box below to see how they simplified the expression, and compare your solution to theirs.

Knowing How to Determine the Correct Answer

As you can see, it is possible to get different answers if we don’t have directions to follow. The directions used for simplifying expressions are called the order of operations.

In the above example, four operations were performed in the expression: subtraction, multiplication, addition, and division. How do you know which one to do first? Matt started on the left and worked straight through to the right. Josey started multiplying first. Click below to see the correct order of operations.

The order of operations was developed to provide consistent directions for solving math problems. Basically, start with grouping, and then do the most powerful to the least powerful operation.

Look at Matt and Josey’s work again, and then answer the following questions.

• Describe how they both started and if their first steps would make a difference in their answer.

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  What was the first thing Matt did? What was the first thing Josey did? If they did something different, would it make a difference in the next step?
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  Check Your Answer

  Matt subtracted 3 from 21, and Josey multiplied 3 × 4 first. According to the order of operations, the multiplication would be done first, so Matt’s answer would be incorrect starting from the first step.
• Explain why it is important to use order of operations when solving problems.

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  Look at Matt and Josey’s first step. Would it be possible for them to agree on an answer by starting the way they did?
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  Check Your Answer

  As shown in Matt and Josey’s work, if the order of operations isn’t followed, it is difficult for everyone to agree on a correct answer. The order of operations offers consistency, or specific directions, for everyone to follow, suggesting to do the most powerful operation first.
• Is it possible to simplify an expression and obtain the same result by not following the order of operations?

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  Think of an expression that can be done where everyone would agree if the order of operations wasn’t followed.
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  Check Your Answer

  The order of operations offers consistency, or specific directions, for everyone to follow. If the expression is written so that the order of operations would naturally occur, it would be possible. An example would be 36 ÷ 3 × 2 − 14 + 5. Whether you start from the left and move to the right or follow the order of operations, you would still start with division followed by multiplication, subtraction, and finally, addition.

Practice

Simplify each expression below using the order of operations.

1. 131 + 26 ÷ 2 − 16 × 6

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   Need a hint?

   Follow the order of operations. Multiply and divide first.
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   Check Your Answer

   131 + 26 ÷ 2 − 16 × 6
   131 + 13 − 96
   144 − 96
   48
2. 21 × 3 − 24 ÷ 8 − 8 × 5

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   Need a hint?

   Follow the order of operations. Multiply and divide first.
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   Check Your Answer

   21 × 3 − 24 ÷ 8 − 8 × 5
   63 − 3 − 40
   60 − 40
   20
3. 25 × 3 ÷ 5 + 15 × 4 ÷ 3 − 5 × 7

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   Need a hint?

   Follow the order of operations. Multiply and divide first.
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   Check Your Answer

   25 × 3 ÷ 5 + 15 × 4 ÷ 3 − 5 × 7
   75 ÷ 5 + 15 × 4 ÷ 3 − 5 × 7
   15 + 15 × 4 ÷ 3 − 5 × 7
   15 + 60 ÷ 3 − 5 × 7
   15 + 20 − 5 × 7
   15 + 20 − 35
   35 − 35
   0