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## 4th grade foundations (Eureka Math/EngageNY)

### Course: 4th grade foundations (Eureka Math/EngageNY) > Unit 3

Lesson 2: Topic B & C: Foundations- Properties and patterns for multiplication
- Intro to associative property of multiplication
- Intro to commutative property of multiplication
- Distributive property
- Basic multiplication
- Multiplying 1-digit numbers by multiples of 10, 100, and 1000
- Multiply 1-digit numbers by a multiple of 10, 100, and 1000
- Multiplying by tens word problem
- Multiply by tens word problems

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# Intro to associative property of multiplication

Practice changing the grouping of factors in multiplication problems and see how it affects the product.

## Grouping numbers

The image shows start color #01a995, 3, end color #01a995 rows with start color #e07d10, 2, end color #e07d10 dots in each row. We can use the expression start color #01a995, 3, end color #01a995, times, start color #e07d10, 2, end color #e07d10 to represent the array.

This image shows the same start color #01a995, 3, end color #01a995, times, start color #e07d10, 2, end color #e07d10 array copied start color #7854ab, 4, end color #7854ab times.

We use the expression left parenthesis, start color #01a995, 3, end color #01a995, times, start color #e07d10, 2, end color #e07d10, right parenthesis, times, start color #7854ab, 4, end color #7854ab to represent the array.

If we count the dots, we get a total of 24.

### Changing the grouping

Will we get the same total if we change the parentheses so the numbers are grouped in a different way?

Let's regroup the numbers so the start color #e07d10, 2, end color #e07d10 and the start color #7854ab, 4, end color #7854ab are grouped together: start color #01a995, 3, end color #01a995, times, left parenthesis, start color #e07d10, 2, end color #e07d10, times, start color #7854ab, 4, end color #7854ab, right parenthesis.

We can also draw an array to represent this expression. Let's start with start color #e07d10, 2, end color #e07d10 rows with start color #7854ab, 4, end color #7854ab dots in each row. This array shows start color #e07d10, 2, end color #e07d10, times, start color #7854ab, 4, end color #7854ab.

Now we need to copy the array start color #01a995, 3, end color #01a995 times to represent the expression start color #01a995, 3, end color #01a995, times, left parenthesis, start color #e07d10, 2, end color #e07d10, times, start color #7854ab, 4, end color #7854ab, right parenthesis.

If we count the dots, we still get a total of 24.

Regrouping does not change the answer!

## Associative property

The math rule that allows us to regroup numbers in a multiplication problem without changing the answer is the associative property.

Let's group the numbers in the following multiplication problem two different ways and show that we get the same product both ways.

Let's start by grouping the start color #11accd, 5, end color #11accd and the start color #11accd, 4, end color #11accd together. We can evaluate the expression step by step.

empty space, left parenthesis, start color #11accd, 5, times, 4, end color #11accd, right parenthesis, times, 2

equals, start color #11accd, 20, end color #11accd, times, 2

equals, 40

equals, start color #11accd, 20, end color #11accd, times, 2

equals, 40

Now let's group the start color #7854ab, 4, end color #7854ab and the start color #7854ab, 2, end color #7854ab together.

empty space, 5, times, left parenthesis, start color #7854ab, 4, times, 2, end color #7854ab, right parenthesis

equals, 5, times, start color #7854ab, 8, end color #7854ab

equals, 40

equals, 5, times, start color #7854ab, 8, end color #7854ab

equals, 40

We got the same product even though the numbers were grouped two different ways.

All three expressions are equal:

empty space, 5, times, 4, times, 2

equals, left parenthesis, start color #11accd, 5, times, 4, end color #11accd, right parenthesis, times, 2

equals, 5, times, left parenthesis, start color #7854ab, 4, times, 2, end color #7854ab, right parenthesis

empty space, 5, times, 4, times, 2

equals, left parenthesis, start color #11accd, 5, times, 4, end color #11accd, right parenthesis, times, 2

equals, 5, times, left parenthesis, start color #7854ab, 4, times, 2, end color #7854ab, right parenthesis

### Let's try a few problems

Now let's try evaluating an expression two different ways.

Now solve the same expression that has been grouped in a different way.

left parenthesis, start color #7854ab, 3, times, 2, end color #7854ab, right parenthesis, times, 5, equals, 30 and

3, times, left parenthesis, start color #1fab54, 2, times, 5, end color #1fab54, right parenthesis, equals, 30

3, times, left parenthesis, start color #1fab54, 2, times, 5, end color #1fab54, right parenthesis, equals, 30

We got the same product even though we grouped the numbers two different ways.

### Equivalent expressions

We can use the associative property to find expressions that are equivalent.

Let's start with the expression 2, times, 2, times, 5.

We can group this expression two ways that are both equivalent to 2, times, 2, times, 5:

left parenthesis, start color #11accd, 2, times, 2, end color #11accd, right parenthesis, times, 5

2, times, left parenthesis, start color #e07d10, 2, times, 5, end color #e07d10, right parenthesis

2, times, left parenthesis, start color #e07d10, 2, times, 5, end color #e07d10, right parenthesis

By evaluating each expression step by step we can find other expressions that are also equivalent.

left parenthesis, start color #11accd, 2, times, 2, end color #11accd, right parenthesis, times, 5, equals, start color #11accd, 4, end color #11accd, times, 5

2, times, left parenthesis, start color #e07d10, 2, times, 5, end color #e07d10, right parenthesis, equals, 2, times, start color #e07d10, 10, end color #e07d10

2, times, left parenthesis, start color #e07d10, 2, times, 5, end color #e07d10, right parenthesis, equals, 2, times, start color #e07d10, 10, end color #e07d10

So our original expression, 2, times, 2, times, 5, is also equivalent to 4, times, 5 and 2, times, 10.

## Why regroup?

Regrouping can make solving a multiplication problem easier.

Let’s look at the expression, 4, times, 4, times, 5.

We can group the expression two ways:

left parenthesis, 4, times, 4, right parenthesis, times, 5

4, times, left parenthesis, 4, times, 5, right parenthesis

4, times, left parenthesis, 4, times, 5, right parenthesis

If we evaluate the first expression step by step we get:
left parenthesis, start color #11accd, 4, times, 4, end color #11accd, right parenthesis, times, 5, equals, start color #11accd, 16, end color #11accd, times, 5

If we evaluate the second expression step by step we get:
4, times, left parenthesis, start color #7854ab, 4, times, 5, end color #7854ab, right parenthesis, equals, 4, times, start color #7854ab, 20, end color #7854ab

It might be easier to find the product of 4, times, 20 than 16, times, 5.

Even though the numbers were grouped differently, both expressions have the same product.

4, times, 20, equals, 80

16, times, 5, equals, 80

16, times, 5, equals, 80

### Let's try a problem

## Want to join the conversation?

- Does associative property apply when there are 4 factors in an equation? For example, 2 x 4 x 5 x 10...are you applying associative property if you move the numbers around and group them differently, say (2 x 5) x 10 x 4 or does that demonstrate commutative property?(7 votes)
- In your example, you applied both the commutative property (to move the numbers) and the associative property to do the grouping.(3 votes)

- Hey I think I found an error in the activity (or else I am just really confused). In the activity for associative property of multiplication I had a question that asked me to find all examples that express the problem "9 x (3x2)" in a different way. I chose the answer, "(9x3) x 2." When I clicked to check the answer, it was correct but said there should be still another one correct, too, and would not allow me to progress until I clicked on "9 x 9." The first two questions total 54, but 9 x 9 = 81. Did anyone else get this? Is it an error in the platform or am I missing something?(7 votes)
- yeah, i got it too. i'm pretty sure somethings wrong with the platform(4 votes)

- how can you divide and mutly at thhe sme time(5 votes)
- If we don't want to multiply a two digit number to get the final product how should we group the numbers?(5 votes)
- If the associative property says we can group numbers and they'll come up to the same total then why is (2x3)x9 and 2x(3x9) have different answers?(3 votes)
- They both actually lead to the same answer! For (2x3)x9, (2 * 3) = 6 (PEMDAS) and 6 * 9 = 54. For the other one, you also follow PEMDAS (3 * 9) = 27 and 27 * 2 = 54. See how they both equal 54?

So that's why (2x3)x9 = 2x(3x9) and the associative property is correct! :)(4 votes)

- Why do you keep picking on Kate?(3 votes)
- what does associative property mean(2 votes)
- You can move the parentheses around. Parentheses associate numbers together.

Associative property is the property that makes (5 * 3) * 2 = 5 * (3 * 2).(3 votes)

- How do you know when to multiply the number(2 votes)
- Well that's pretty simple , first you need to know what are the number that you are dealing with , second you need to make it into simpler form , example; if 192*8 is very difficult to multiply make it easy to understand then follow this step 100*8= 800 then multiply 92*8 which equals 736 then add 800+736=1536 isn't it easy?.

HAVE WONDERFUL DAY!(2 votes)

- do you want to do some multiplication(2 votes)
- I don't like the questions, they don't tell you how to do a lot of them!(2 votes)