# 1.10: Numerical Expression Evaluation with Basic Operations

**At Grade**Created by: CK-12

**Practice**Numerical Expression Evaluation with Basic Operations

You wouldn't think that an aviary would have mathematics in it, but this aviary has a problem and solving numerical expressions using the four operations is the way to solve it. Have you ever thought about this?

Keisha loves the birds in the aviary at the city zoo. Her favorite part of the aviary is the bird rescue. Here the zoo staff rescues injured birds, helps them to heal and then releases them again. Currently, they have 256 birds in the rescue. Today, Keisha has a special visit planned with Ms. Thompson who is in charge of the bird rescue.

When Keisha arrives, Ms. Thompson is already hard at work. She tells Keisha that there are new baby birds in the rescue. Three of the birds have each given birth to five baby birds. Keisha can’t help grinning as she walks around. She can hear the babies chirping. In fact, it sounds like they are everywhere.

“It certainly sounds like a lot more babies,” Keisha says.

“Yes,” Ms. Thompson agrees. “We also released two birds from the rescue yesterday.”

“That is great news,” Keisha says smiling.

“Yes, but we also found three new injured birds. Our population has changed again.”

“I see,” Keisha adds, “That is \begin{align*}256 + 3 \times 5 - 2 + 3\end{align*} that equals 1296 birds, I think. I’m not sure, that doesn’t seem right.”

Is Keisha’s math correct? How many birds are there now? Can you figure it out? This is a bit of a tricky question. You will need to learn some new skills to help Keisha determine the number of birds in the aviary.

**Pay attention. By the end of the Concept, you will know all about the order of operations. Then you will be able to help Keisha with the bird count.**

### Guidance

This Concept begins with evaluating numerical expressions. Before we can do that we need to answer one key question, **“What is an** *expression***?”** To understand what an ** expression** is, let’s compare it with an

**.**

*equation*
**An equation is a number sentence that describes two values that are the same, or equal, to each other. The values are separated by the "equals" sign. An equation may also be written as a question, requiring you to "solve" it in order to make both sides equal.**

\begin{align*}3 + 4 = 7\end{align*}

This is an equation. It describes two equal quantities, "3+4", and "7".

**What is an expression then?** **An expression is a number sentence without an equals sign. It can be simplified and/or evaluated.**

\begin{align*}4 + 3 \times 5 \end{align*}

This kind of expression can be confusing because it has both addition and multiplication in it. **Do we need to add or multiply first?** **To figure this out, we are going to learn something called the** *Order of Operations.***The Order of Operation is a way of evaluating expressions. It lets you know what order to complete each operation in.**

**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**

*Take a few minutes to write these down in a notebook.*

Now that you know the order of operations, let’s go back.

\begin{align*}4 + 3 \times 5\end{align*}

Here we have an expression with addition and multiplication. We can look at the order of operations and see that multiplication comes before addition. We need to complete that operation first.

\begin{align*}& 4 + 3 \times 5\\ & 4 + 15\\ & = {20}\end{align*}

When we evaluate this expression using order of operations, our answer is 20.

**What would have happened if we had NOT followed the order of operations?**

\begin{align*}4 + 3 \times 5\end{align*}

We probably would have solved the problem in order from left to right.

\begin{align*}& 4 + 3 \times 5\\ & 7 \times 5\\ & = 35 \end{align*}

**This would have given us an incorrect answer. It is important to always follow the order of operations.**

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

#### Example A

\begin{align*} 8 - 1 \times 4 + 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

**Solution: 7**

#### Example B

\begin{align*}2 \times 6 + 8 \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}

**Solution: 16**

#### Example C

\begin{align*}5 + 9 \times 3 - 6 + 2 =\underline{\;\;\;\;\;\;\;}\end{align*}

**Solution: 28**

Here is the original problem once again. Let’s look back at Keisha and Ms. Thompson and the bird dilemma at the zoo.

Keisha loves the birds in the aviary at the city zoo. Her favorite part of the aviary is the bird rescue. Here the zoo staff rescues injured birds, helps them to heal and then releases them again. Currently, they have 256 birds in the rescue. Today, Keisha has a special visit planned with Ms. Thompson who is in charge of the bird rescue.

When Keisha arrives, Ms. Thompson is already hard at work. She tells Keisha that there are new baby birds in the rescue. Three of the birds have each given birth to five baby birds. Keisha can’t help grinning as she walks around. She can hear the babies chirping. In fact, it sounds like they are everywhere.

“It certainly sounds like a lot more babies,” Keisha says.

“Yes,” Ms. Thompson agrees. “We also released two birds from the rescue yesterday.”

“That is great news,” Keisha says smiling.

“Yes, but we also found three new injured birds. Our population has changed again.”

“I see,” Keisha adds, “That is \begin{align*}256 + 3 \times 5 - 2 + 3\end{align*} that equals 1296 birds, I think. I’m not sure, that doesn’t seem right.”

We have an equation that Keisha wrote to represent the comings and goings of the birds in the aviary. Before we figure out if Keisha’s math is correct, let’s underline any important information in the problem. As usual, this has been done for you in the text. Wow, there is a lot going on. Here is what we have to work with.

256 birds

\begin{align*}3 \times 5\end{align*} - three birds each gave birth to five baby birds

- birds were released
- injured birds were found.

Since we started with 256 birds, that begins our equation. Then we can add in all of the pieces of the problem.

\begin{align*}256 + 3 \times 5 - 2 + 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

This is the same equation that Keisha came up with. Let’s look at her math. Keisha says, “That is \begin{align*}256 + 3 \times 5 - 2 + 3\end{align*} that equals 1296 birds, I think. I’m not sure, that doesn’t seem right.” It isn’t correct. Keisha forgot to use the order of operations.

According to the order of operations, Keisha needed to multiply \begin{align*}3 \times 5\end{align*} BEFORE completing any of the other operations. Let’s look at that.

\begin{align*}256 + 3 \times 5 - 2 + 3 = \underline{\;\;\;\;\;\;\;}\\ 256 + 15 - 2 + 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

Now we can complete the addition and subtraction in order from left to right.

\begin{align*}256 + 15 - 2 + 3 = 272\end{align*}

**The new bird count in the aviary is 272 birds.**

### Vocabulary

Here are the vocabulary words that are used in 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

### Guided Practice

Here is one for you to try on your own.

\begin{align*}6 + 8 \times 4 - 11 + 6 =\underline{\;\;\;\;\;\;\;}\end{align*}

**Answer**

33

### Video Review

Here are two videos that will help you to review this concept.

Khan Academy Introduction to Order of Operations

James Sousa Example of Order of Operations

James Sousa Example 2 of Order of Operations

### Practice

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

1. \begin{align*}2 + 3 \times 4 + 7 = \underline{\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}4 + 5 \times 2 + 9 - 1 = \underline{\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}6 \times 7 + 2 \times 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

4. \begin{align*}4 \times 5 + 3 \times 1 - 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}5 \times 3 \times 2 + 5 - 1 = \underline{\;\;\;\;\;\;\;}\end{align*}

6. \begin{align*}4 + 7 \times3 + 8 \times2 = \underline{\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}9 - 3 \times 1 + 4 - 7 = \underline{\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}10 + 3 \times 4 + 2 -8 = \underline{\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}11 \times 3 + 2 \times 4 - 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}6 + 7 \times 8 - 9 \times 2 = \underline{\;\;\;\;\;\;\;}\end{align*}

11. \begin{align*}3 + 4^2 - 5 \times 2 + 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}2^2 + 5 \times 2 + 6^2 - 11 = \underline{\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}3^2 \times 2 + 4 - 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}6 + 3 \times 2^2 + 7 - 1 = \underline{\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}7 + 2 \times 4 + 3^2 - 5 = \underline{\;\;\;\;\;\;\;}\end{align*}

Equation

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

Here you'll learn how to evaluate numerical expressions using the four operations.

Equation

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