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Numerical Expression Evaluation with Grouping Symbols

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Kira works with Keisha at the zoo. Today, Kira has gathered a group of birds for a check - up. She gathers three groups of six birds to begin with and sets these birds aside in a cage. Then she managed to coax four more morning doves and seven parrots over to her before two of the parrots flew away. Finally, she looked at all of the birds she had and began to count them one by one.

As she did this, Kira thought that there must be an easier way. She knows that she has grouped some of the birds in the cage, plus the other birds she has gathered. How can Kira do this?

This is where the order of operations including grouping symbols can come in handy. Pay close attention and you will be able to help Kira at the end of the Concept.

Guidance

In this concept, we will be working with the order of operations, but also with grouping symbols like parentheses.

Let’s review where exponents and parentheses fall in the order of operations.

Order of Operations

P - parentheses

E - exponents

MD - multiplication or division in order from left to right

AS - addition or subtraction in order from left to right

Wow! You can see that, according to the order of operations, parentheses come first. We always do the work in parentheses first. Then we evaluate exponents.

Let’s see how this works.

2 + (3 - 1) \times 2

In this problem, we can see that we have four things to look at. We have 1 set of parentheses, addition, subtraction in the parentheses and multiplication. We can evaluate this expression using the order of operations. Here is the next step.

& 2 +  (3 - 1) \times 2\\& 2 +  2 \times 2\\& 2 + 4\\& = 6

Our answer is 6.

What about when we have parentheses and exponents?

35 + 3^2 - (3 \times 2) \times 7

We start by using the order of operations. It says we evaluate parentheses first.

& 3 \times 2 =  6\\& 35 + 3^2 -  6 \times 7

Next we evaluate exponents.

& 3^2 = 3 \times 3 = 9\\& 35 + 9 - 6 \times 7

Now, we complete multiplication or division in order from left to right. We have multiplication.

& 6 \times 7 = 42\\& 35 + 9 - 42

Finally, we complete addition and/or subtraction in order from left to right.

35 + 9 & = 44\\44 - 42 & = 2

Our answer is 2.

Here are a few for you to try on your own.

Example A

16 + 2^3 - 5 + (3 \times 4)

Solution: 31

Example B

9^2 + 2^2 - 5 \times (2 + 3)

Solution: 60

Example C

8^2 \div 2 + 4 - 1 \times 6

Solution: 30

Now let's go back to Kira and her bird dilemma. Here is the original problem once again.

Kira who works at the zoo has gathered a group of birds for a check - up. She gathers three groups of six birds to begin with and sets these birds aside in a cage. Then she managed to coax four more morning doves and seven parrots over to her before two of the parrots flew away. Finally, she looked at all of the birds she had and began to count them one by one.

As she did this, Kira thought that there must be an easier way. She knows that she has grouped some of the birds in the cage, plus the other birds she has gathered. How can Kira do this?

Using what we have just learned in this Concept, we can write a number sentence using the order of operations and grouping symbols to help Kira to figure out the total number of birds. Rather than counting one by one, this will be more efficient.

We know that she gathered three groups of six birds, but she didn't add that total until the end. Walking through the birds as they were gathered, we can write the following sentence.

4 + 7 - 2 + (3 + 6)

Notice that we added the three groups of six birds at the end just like Kira did. Next we follow the order of operations and solve.

4 + 7 - 2 + 18

Our answer is 27 birds.

Vocabulary

Order of Operations
the order that you perform operations when there is more than one in an expression or equation.

P - parentheses

E - exponents

MD - multiplication/division in order from left to right

AS - addition and subtraction in order from left to right

Grouping Symbols
Parentheses or brackets. Operations in parentheses are completed first according to the order of operations.

Guided Practice

Here is one for you to try on your own.

7^3 - 3^2 + 15 \times 2 + (2 + 3)

Answer

First, we have to follow the order of operations and evaluate the powers and grouping symbols.

7^3 = 7 x 7 x 7 = 343

3^2 = 3 x 3 = 9

(2 + 3) = 5

Next, substitute these values back into the original number sentence.

343 - 9 + 15 \times 2 + (5)

Now we can solve complete the multiplication.

15 x 2 = 30

Finally, we can solve by completing the addition and subtraction in order from left to right.

343 - 9 + 30 + 5

= 369

Video Review

Khan Academy Introduction to Order of Operations

James Sousa Example of Order of Operations

James Sousa Example of Order of Operations

James Sousa Example of Order of Operations

Practice

Directions: Evaluate each expression according to the order of operations.

1. 3 + (2 + 7) - 3 + 5 = \underline{\;\;\;\;\;\;\;}

2. 2 + (5 - 3) + 7^2 - 11 = \underline{\;\;\;\;\;\;\;}

3. 4 \times 2 + (6 - 4) - 9 + 5 = \underline{\;\;\;\;\;\;\;}

4. 8^2 - 4 + (9 - 3) + 12 = \underline{\;\;\;\;\;\;\;}

5. 7^3 - 100 + (3 + 4) - 9 = \underline{\;\;\;\;\;\;\;}

6. 7 + (3^2 + 7) - 11 + 5 = \underline{\;\;\;\;\;\;\;}

7. 2^4 + (8 + 7) + 13 - 5 = \underline{\;\;\;\;\;\;\;}

8. 3 \times 2 + (2^2 + 7) - 11 + 15 = \underline{\;\;\;\;\;\;\;}

9. 8 + (6 + 7) - 2 \times 3  = \underline{\;\;\;\;\;\;\;}

10. 22 + (3^4 + 7) - 73 + 15 = \underline{\;\;\;\;\;\;\;}

11. 3^2 + (4^2 - 7) - 3 + 25 = \underline{\;\;\;\;\;\;\;}

12. 6^3 + (3^2 + 17) - 73 + 4  = \underline{\;\;\;\;\;\;\;}

13. 243 - (5^3 + 27) - 83 + 9 = \underline{\;\;\;\;\;\;\;}

14. 72 + (11^2 + 117) - 193 + 75 = \underline{\;\;\;\;\;\;\;}

15. 82 + (10^2 + 130) - 303 + 115 = \underline{\;\;\;\;\;\;\;}

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