# 4.6: Mental Math to Evaluate Products

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

**Practice**Mental Math to Evaluate Products

Have you ever had to use mental math to solve a problem? Well, sometimes it doesn't make sense to use paper and a pencil. It is easier to use mental math.

At the Science Museum, three groups of students went into an exhibit show on lightening. Given the structure of the seating, each group was able to sit together. There are eight students in each group, but right before the exhibit show started three more students joined. An additional person joined each group of eight.

Here is an expression to show the groupings. What is the total?

\begin{align*}3(8 + 1)\end{align*}

We could use the distributive property to solve this, but mental math is probably just as simple.

**This Concept will help you to practice this skill. Then you can come back to this problem at the end of the Concept.**

### Guidance

Some of you may have found that while the Distributive Property is useful, that sometimes it is easier to simply find the products by using mental math.

Some of you may have found that you did not need to write out the distribution of the number outside of the parentheses with the number inside of the parentheses to find the sum of the products.

The Distributive Property is a useful property, especially as you get into higher levels of mathematics like Algebra. There it is essential, but sometimes, you can use mental math to evaluate expressions.

\begin{align*}2(1 + 4)\end{align*}

Now this is a problem where you could probably add and multiply in your head.

You know that you can add what is in parentheses first, so you add one and four and get five. Then you can multiply five times two and get a product of 10.

**Our answer is 10.**

When you have larger numbers, you can always use the Distributive Property to evaluate an expression. When you have smaller numbers, you can use mental math.

Now let's practice.

#### Example A

\begin{align*}4(2 + 3)\end{align*}

**Solution: 20**

#### Example B

\begin{align*}6(2 + 7)\end{align*}

**Solution: 54**

#### Example C

\begin{align*}5(2 + 6)\end{align*}

**Solution: 40**

Ready to use mental math? Here is the original problem once again.

At the Science Museum, three groups of students went into an exhibit show on lightening. Given the structure of the seating, each group was able to sit together. There are eight students in each group, but right before the exhibit show started three more students joined. An additional person joined each group of eight.

Here is an expression to show the groupings. What is the total?

\begin{align*}3(8 + 1)\end{align*}

We could use the distributive property to solve this, but mental math is probably just as simple.

**Using mental math, our answer is 27.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Numerical expression
- a number sentence that has at least two different operations in it.

- Product
- the answer in a multiplication problem

- Sum
- the answer in an addition problem

- Property
- a rule that works for all numbers

- Evaluate
- to find the quantity of values in an expression

- The Distributive Property
- the property that involves taking the product of the sum of two numbers. Take the number outside the parentheses and multiply it by each term in the parentheses.

### Guided Practice

Here is one for you to try on your own.

Use mental math to solve this problem.

\begin{align*}12(8 + 1)\end{align*}

**Using mental math, our solution is \begin{align*}108\end{align*}.**

### Video Review

Here is a video for review.

Khan Academy The Distributive Property

### Practice

Directions: Use mental math to evaluate the following expressions.

1. 2(1 + 3)

2. 3(2 + 3)

3. 3(2 + 2)

4. 4(5 + 1)

5. 5(3 + 4)

6. 2(9 + 1)

7. 3(8 + 2)

8. 4(3 + 2)

9. 5(6 + 2)

10. 7(3 + 5)

11. 8(2 + 4)

12. 9(3 + 5)

13. 8(3 + 2)

14. 9(10 + 2)

15. 7(9 + 2)

16. 9(7 + 1)

17. 12(8 + 2)

18. 12(9 + 3)

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, .Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.Product

The product is the result after two amounts have been multiplied.Property

A property is a rule that works for a given set of numbers.Sum

The sum is the result after two or more amounts have been added together.### Image Attributions

Here you'll use mental math to evaluate products.

## Concept Nodes:

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, .Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.Product

The product is the result after two amounts have been multiplied.Property

A property is a rule that works for a given set of numbers.Sum

The sum is the result after two or more amounts have been added together.