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## 7th grade foundations (Eureka Math/EngageNY)

### Course: 7th grade foundations (Eureka Math/EngageNY)>Unit 2

Lesson 3: Topic C: Foundations

# Intro to distributive property

Practice decomposing the factors in multiplication problems and see how it affects the product.

## Breaking up multiplication

This array is made up of 3 rows with 6 dots in each row. The dots show 3, times, 6, equals, 18.
If we add a line dividing the dots into two groups, the total number of dots does not change.
The top group has 1 row with 6 dots. The dots show 1, times, 6.
The bottom group has 2 rows with 6 dots in each row. The dots show 2, times, 6.
We still have a total of 18 dots.

## Distributive property

The math rule that allows us to break up multiplication problems is called the distributive property.
The distributive property says that in a multiplication problem, when one of the factors is rewritten as the sum of two numbers, the product does not change.
Using the distributive property allows us to solve two simpler multiplication problems.
In the example with the dots we started with start color #1fab54, 3, end color #1fab54, times, start color #7854ab, 6, end color #7854ab.
We broke the start color #1fab54, 3, end color #1fab54 down into start color #1fab54, 1, plus, 2, end color #1fab54. We can do this because start color #1fab54, 1, plus, 2, equals, 3, end color #1fab54
We used the distributive property to change the problem from start color #1fab54, 3, end color #1fab54, times, start color #7854ab, 6, end color #7854ab to left parenthesis, start color #1fab54, 1, plus, 2, end color #1fab54, right parenthesis, times, start color #7854ab, 6, end color #7854ab.
The start color #7854ab, 6, end color #7854ab gets distributed to the start color #1fab54, 1, end color #1fab54 and start color #1fab54, 2, end color #1fab54 and the problem changes to:
left parenthesis, start color #1fab54, 1, end color #1fab54, times, start color #7854ab, 6, end color #7854ab, right parenthesis, plus, left parenthesis, start color #1fab54, 2, end color #1fab54, times, start color #7854ab, 6, end color #7854ab, right parenthesis
Now we need to find the two products:
6, plus, 12
And finally, the sum:
6, plus, 12, equals, 18
start color #1fab54, 3, end color #1fab54, times, start color #7854ab, 6, end color #7854ab, equals, 18 and
left parenthesis, start color #1fab54, 1, plus, 2, end color #1fab54, right parenthesis, times, start color #7854ab, 6, end color #7854ab, equals, 18
Practice problem 1
Which expressions are the same as 4, times, 9?

### Small numbers

Some numbers like 1, comma, 2, comma, 5, and 10 are easier to multiply. The distributive property allows us to change a multiplication problem so that we can use these numbers as one of the factors.
For example, we can change 4, times, 12 into 4, times, left parenthesis, start color #01a995, 10, end color #01a995, plus, start color #74cf70, 2, end color #74cf70, right parenthesis.
The array of dots on the left shows left parenthesis, start color #01a995, 4, times, 10, end color #01a995, right parenthesis. The array of dots on the right shows left parenthesis, start color #74cf70, 4, times, 2, end color #74cf70, right parenthesis.
Now we can add the expressions to find the total.
left parenthesis, start color #01a995, 4, times, 10, end color #01a995, right parenthesis, plus, left parenthesis, start color #74cf70, 4, times, 2, end color #74cf70, right parenthesis
equals, start color #01a995, 40, end color #01a995, plus, start color #74cf70, 8, end color #74cf70
equals, 48
Since 10 and 2 are both easy to multiply, using the distributive property for this problem made finding the product easier.

### Practice problem 2

The dots represent 9, times, 4.
Problem 2, part A
Which expression shows the dots above the dotted line?

Problem 2, Part B
Which expression shows the dots below the dotted line?

Problem 2, Part C
left parenthesis, 5, times, 4, right parenthesis
left parenthesis, 4, times, 4, right parenthesis, equals, start text, space, t, o, t, a, l, space, n, u, m, b, e, r, space, o, f, space, d, o, t, s, end text

### More practice

Problem 3A
• Current
The dots represent 3, times, 8.
Which expression can we use to calculate the total number of dots?

## Working with large numbers

The distributive property is very helpful when multiplying larger numbers. Look at how we can use the distributive property to simplify 15, times, 8.
We will start by breaking start color #11accd, 15, end color #11accd into start color #11accd, 10, plus, 5, end color #11accd. Then we will distribute the 8 to both of these numbers.
start color #11accd, 15, end color #11accd, times, 8, equals, left parenthesis, start color #11accd, 10, end color #11accd, times, 8, right parenthesis, plus, left parenthesis, start color #11accd, 5, end color #11accd, times, 8, right parenthesis
empty spaceequals, space80, plus, 40
empty spaceequals, space120
Problem 4
Use the distributive property to find the product.
start color #11accd, 18, end color #11accd, times, 3, equals, left parenthesis, start color #11accd, 10, end color #11accd, times, 3, right parenthesis, plus, left parenthesis, space
times, 3, right parenthesis
empty space, equals, space, 30, plus
empty space, equals, space

## Want to join the conversation?

• I thought that the distributive property was breaking up a multiplication problem?
• And it is! When you start with a product like 6 x 4 you can break up one of the factors and transform it into a sum like (4 + 2) x 4. Then you can apply the distributive property of multiplication over addition.
• If there's a simpler way to get the answer then, why come up with the "distributive property"? I think they should just stick with just plain old math.
• Because you can't use the simplier way when you get to Pre-Algebra, which you will take in 4 years. There is a process called FOIL (First Outer Inner Last), which is exactly distributing, but, as mentioned before, the simplier process doesn't work on it.

"We 'learn' something new everyday" - ShadowKing
• Who invented the Distributive property? Why did he or she invented it?
• EXACTLY it's literately so harddd
• This was very useful. This just kind of in a way shows how to check your answers when doing an order of operations problem. Thank you so much.
• Is the distributive property only in multiplication?
• Yes, the distributive property only applies to multiplication.
• If we study hard algebra isn't difficult;is it?
• Well, yes, but look at it this way if you want the upcoming subject (or new section) to be easy, well, that just isn't possible because how could you study something you don't know or know how to do? You cant. So I say just pay attention in class (not saying you don't(mainly because I don't know who you are)), and find something that you can do to refresh your mind. If you do that you can focus better and then it'll seem like it was super easy! ;)
• what are they doing to us because this is so hard.
(1 vote)
• The distributive property is a foundation for topics you will learn in algebra, such as simplifying expressions that use parentheses, multiplying polynomials, and factoring polynomials. So what you are learning now is preparing you for what is ahead.

If you look at the array models shown in this lesson, this could help you understand more fully the concept of the distributive property: multiplying a factor by a sum of two numbers is the same as multiplying that factor by each of the two numbers and then adding the two results. Literally, the factor is distributed to each of the two numbers in the sum.

Have a blessed, wonderful day!