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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
Math
4th grade foundations (Eureka Math/EngageNY)
Module 3: Foundations
Topic B & C: Foundations
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Intro to associative property of multiplication

Google Classroom
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!
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, equals, 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

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.
5, times, 4, times, 2
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
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
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

Let's try a few problems

Problem 1
Which expressions are equal to 6, times, 3, times, 4?
Choose all answers that apply:

Now let's try evaluating an expression two different ways.
Problem 2
Fill in the missing information to solve the expression left parenthesis, start color #7854ab, 3, times, 2, end color #7854ab, right parenthesis, times, 5, point
left parenthesis, start color #7854ab, 3, times, 2, end color #7854ab, right parenthesis, times, 5space, equals, space
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
times, 5
empty spacespace, equals, space
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Now solve the same expression that has been grouped in a different way.
Problem 3
Fill in the missing information to solve the expression 3, times, left parenthesis, start color #1fab54, 2, times, 5, end color #1fab54, right parenthesis.
3, times, left parenthesis, start color #1fab54, 2, times, 5, end color #1fab54, right parenthesisspace, equals, space, 3, times
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
empty spacespace, equals, space
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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
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
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
So our original expression, 2, times, 2, times, 5, is also equivalent to 4, times, 5 and 2, times, 10.
Problem 4
Which expressions are equivalent to 8, times, 2, times, 4?
Choose all answers that apply:

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
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

Let's try a problem

Problem 5
How can we group the expression 2, times, 3, times, 9?
Choose all answers that apply:

Problem 6
If we don't want to multiply a two digit number to get the final product how should we group the numbers?
Choose 1 answer:

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