# 1.12: Numerical Expression Evaluation

**Basic**Created by: CK-12

**Practice**Numerical Expression Evaluation

Remember Keisha and Kira from the last two Concepts? Well, today they are working with the flamingos, and they have a problem.

"The flamingo count is off," Kira said to Keisha. Keisha looked over at the notebook Kira was holding and shrugged.

"What do you mean?" Keisha asked.

"Well, we have 33 flamingos, except the count in the notebook doesn't show that. Here is what I have written down," Kira explained. She showed the notebook to Keisha.

\begin{align*}27 + 2 \times 5 - 3 + 8 + 7 = 49\end{align*}

"The count here is 49, but I know that there are only 33 flamingos," Kira explained.

**The girls clearly have a problem. Somewhere in this expression a set of parentheses is missing. One set of parentheses can make all the difference in a solution. In this Concept, you will learn how to insert parentheses to make a statement true. Pay close attention so that you can help Kira and Keisha.**

### Guidance

In the last few Concepts, you used the order of operations to evaluate numerical expressions. In those Concepts, numerical expressions included all four operations and grouping symbols. We can also use the order of operations to determine whether or not a numerical expression is true. The numerical expressions that you will see in this Concept have already been evaluated.

Now you will use what you have learned to determine whether or not the statement is true. Remember, you will need to use the order of operations.

Here is the order of operations once again.

**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**

**Now, you will get to be a “Math Detective.”**

As a math detective, you will be using the order of operations to determine whether or not someone else’s work is correct. Here is a worksheet that has been completed by Joaquin. Your task is to check Joaquin’s work and determine whether or not his work is correct. Use your notebook to take notes.

If the expression has been evaluated correctly, then please make a note of it. If it is incorrect, then re-evaluate the expression correctly. Here are the problems that are on Joaquin’s worksheet.

**Did you check Joaquin’s work?** **Let’s see how you did with your answers. Take your notebook and check your work with these correct answers.** Let’s begin with problem number 1.

We start by adding \begin{align*}4 + 1\end{align*} which is 5. Then we multiply \begin{align*}7 \times 5\end{align*} and \begin{align*}7 \times 2\end{align*}. Since multiplication comes next in our order of operations. Finally we subtract \begin{align*}35 - 14 = 21.\end{align*} Joaquin’s work is correct.

Problem Number 2

We start by evaluating the parentheses. 3 times 2 is 6. Next, consider the exponents. 3 squared is 9 and 4 squared is 16. Finally we can complete the addition and subtraction in order from left to right. Our final answer is 22. Joaquin’s work is correct.

Problem Number 3

We start with the parentheses, and find that 7 minus 1 is 6. There are no exponents to evaluate, so we can move to the multiplication step. Multiply \begin{align*}3 \times 2\end{align*} which is 6. Now we can complete the addition and subtraction in order from left to right. The answer correct is 13. Uh Oh, Joaquin’s answer is incorrect. How did Joaquin get 19 as an answer?

Well, if you look, Joaquin did not follow the order of operations. He just did the operations in order from left to right. If you don’t multiply \begin{align*}3 \times 2\end{align*} first, then you get 19 as an answer instead of 16.

Problem Number 4

Let’s complete the work in parentheses first, \begin{align*}8 \times 2 = 16\end{align*} and \begin{align*}5 \times 2 = 10\end{align*}. Next we evaluate the exponent, 3 squared is 9. Now we can complete the addition and subtraction in order from left to right. The answer is 17. Joaquin’s work is correct.

Problem Number 5

First, we need to complete the work in parentheses, \begin{align*}6 \times 3 = 18\end{align*}. Next, we complete the multiplication \begin{align*}2 \times 3 = 6\end{align*}. Now we can evaluate the addition and subtraction in order from left to right. Our answer is 30. Uh Oh, Joaquin got mixed up again. How did he get 66? Let’s look at the problem. Oh, Joaquin subtracted \begin{align*}18 - 2\end{align*} before multiplying. You can’t do that. He needed to multiply \begin{align*}2 \times 3\end{align*} first then he needed to subtract. Because of this, Joaquin’s work is not accurate.

**How did you do?**

**Remember, a Math Detective can check any answer by following the order of operations.**

Sometimes a grouping symbol can help us to make an answer true. By putting a grouping symbol, like parentheses, in the correct spot, we can change an answer.

**Let’s try this out.**

\begin{align*}5 + 3 \times 2 + 7 - 1 = 22\end{align*}

Now if we just solve this problem without parentheses, we get the following answer.

\begin{align*}5 + 3 \times 2 + 7 - 1 = 17\end{align*}

How did we get this answer? Well, we began by completing the multiplication, \begin{align*}3 \times 2 = 6\end{align*}. Then we completed the addition and subtraction in order from left to right. That gives us an answer of 17. However, we want an answer of 22.

**Where can we put the parentheses so that our answer is 22?**

**This can take a little practice and you may have to try more than one spot too.**

**Let’s try to put the parentheses around \begin{align*}5 + 3\end{align*}.**

\begin{align*}(5 + 3) \times 2 + 7 - 1 = 22\end{align*}

Is this a true statement? Well, we begin by completing the addition in parentheses, \begin{align*}5 + 3 = 8\end{align*}. Next we complete the multiplication, \begin{align*}8 \times 2 = 16\end{align*}.

Here is our problem now.

\begin{align*}16 + 7 - 1 = 22\end{align*}

Next, we complete the addition and subtraction in order from left to right.

**Our answer is 22.**

**Here are a few for you to try on your own. Insert a set of parentheses to make each a true statement.**

#### Example A

\begin{align*}6 + 3 + 4 + 7 \times 2 = 34\end{align*}

**Solution: \begin{align*}6 + (3 + 4 + 7) \times 2 = 34\end{align*}**

#### Example B

\begin{align*}8 \times 7 + 3 \times 8 - 5 = 65\end{align*}

**Solution: \begin{align*}8 \times 7 + 3 \times (8 - 5)= 65\end{align*}**

#### Example C

\begin{align*}2 + 5 \times 2 + 18 - 4 = 28\end{align*}

**Solution: \begin{align*}(2 + 5) \times 2 + 18 - 4 = 28\end{align*}**

Now back to Keisha and Kira. Here is the original problem once again.

"The flamingo count is off," Kira said to Keisha. Keisha looked over at the notebook Kira was holding and shrugged.

"What do you mean?" Keisha asked.

"Well, we have 33 flamingos, except the count in the notebook doesn't show that. Here is what I have written down," Kira explained. She showed the notebook to Keisha.

\begin{align*}27 + 2 \times 5 - 3 + 8 + 7 = 49\end{align*}

"The count here is 49, but I know that there are only 33 flamingos," Kira explained.

To make the expression match with a sum of 33 flamingos, Kira and Keisha need to add a set of parentheses. If we group 3 + 8, we will have an accurate total. Take a look.

\begin{align*}27 + 2 \times 5 - (3 + 8) + 7 = 49\end{align*}

Now when we use the order of operations, our sum is 33.

**This is our answer.**

### Vocabulary

Here are the vocabulary words used for this Concept.

- Expression
- a number sentence with operations and no equals sign.

- Equation
- a number sentence that compares two quantities that are the same. It has an equals sign in it and may be written as a question requiring a solution.

- 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. Insert a set of parentheses to make this a true statement.

\begin{align*}4 + 2 \times 3 + 7 - 5 + 8^2 = 76\end{align*}

**Answer**

To figure this out, we must first look to see where we could put a set of parentheses. Since we know that multiplication and division comes before addition and subtraction, it makes sense that we will need to group some part of addition and/or subtraction to create a true expression.

Let's try 7 - 5.

\begin{align*}4 + 2 \times 3 + (7 - 5) + 8^2 = 76\end{align*}

Now we can evaluate the expression using the order of operations.

\begin{align*}7 - 5 = 2\end{align*}

\begin{align*}4 + 2 \times 3 + 2 + 8^2 = 76\end{align*}

Next, we evaluate the exponent and multiply.

\begin{align*}8^2 = 64\end{align*}

\begin{align*}2 \times 3 = 6\end{align*}

\begin{align*}4 + 6 + 2 + 8^2 = 76\end{align*}

\begin{align*}12 + 64 = 76\end{align*}

**Our answer is correct.**

### Video Review

Here are a few videos for 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: Check each answer using order of operations. Write whether the answer is true or false.

1. \begin{align*}4 + 5 \times 2 + 8 - 7 = 15\end{align*}

2. \begin{align*}4 + 3 \times 9 + 6 - 10 = 104\end{align*}

3. \begin{align*}6 + 2^2 \times 4 + 3 \times 6 = 150\end{align*}

4. \begin{align*}3 + 6 \times 3 + 9 \times 7 - 18 = 66\end{align*}

5. \begin{align*}7 \times 2^3 + 4 - 9 \times 3 - 8 = 25\end{align*}

6. \begin{align*}2 \times 3^3 + 7 \times 3 = 183\end{align*}

7. \begin{align*}4 \times 3 + 4^2 - 9 + 8 = 25\end{align*}

8. \begin{align*}3^2 \times 2^3 + 14 - 9 = 77\end{align*}

9. \begin{align*}6 \times 3^3 - 24 + 19 \times 2 - 4 = 310\end{align*}

10. \begin{align*}5 \times 2 + 5^2 - 10 \times 2 - 7 = 8\end{align*}

Directions: Insert grouping symbols to make each a true statement.

11. \begin{align*}4 + 5 - 2 + 3 - 2 = 8\end{align*}

12. \begin{align*}2 + 3 \times 2 - 4 = 6\end{align*}

13. \begin{align*}1 + 9 \times 4 \times 3 + 2 - 1 = 110\end{align*}

14. \begin{align*}7 + 4 \times 3 - 5 \times 2 = 23\end{align*}

15. \begin{align*}2^2 + 5 \times 8 - 3 + 4 = 33\end{align*}

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Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.Grouping Symbols

Grouping symbols are parentheses or brackets used to group numbers and operations.Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.### Image Attributions

Here you'll learn how to use the order of operations to identify and create numerical expressions that are true.