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

Lesson 4: Topic E: Foundations

Intro to division

Use arrays and word problems to visualize division.

What is division?

Division lets us separate a number of objects into equal-size groups.
The symbol for division is divided by .
To divide, we need to know the total number of objects. We also need to know either the number of groups or the number of objects in each group.

Equal groups

Let's look at an example:
The Big Pink Bubble Gum Company is holding a bubble blowing contest. They have start color #ed5fa6, 18, end color #ed5fa6 gumballs to share equally between start color #11accd, 3, end color #11accd people.
A division problem always starts with the total number of objects.
The total number of gumballs is start color #ed5fa6, 18, end color #ed5fa6.
The gumballs will be divided equally between start color #11accd, 3, end color #11accd people. So the number of equal groups is start color #11accd, 3, end color #11accd.
In this problem, we are dividing start color #ed5fa6, 18, end color #ed5fa6 gumballs into start color #11accd, 3, end color #11accd groups. We can show this with the expression start color #ed5fa6, 18, end color #ed5fa6 divided by start color #11accd, 3, end color #11accd.

Let's try another one

The Big Pink Bubble Gum Company decides to use start color #ed5fa6, 16, end color #ed5fa6 gumballs in the competition.
They will have start color #1fab54, 4, end color #1fab54 people blowing bubbles.
Problem 1A
There are a total of
gumballs that will be divided evenly into
groups.

Problem 1B
Which expression can we use to show start color #ed5fa6, 16, end color #ed5fa6 gumballs divided into start color #1fab54, 4, end color #1fab54 equal-size groups?

Using arrays

We can use arrays to show division.
An array is an arrangement of objects in equal-size rows.
18 gumballs shared equally between 3 people can be shown with this array:
The 18 gumballs have been divided equally between 3 rows.
This array shows the expression 18, divided by, 3.
When we divide 18 gumballs into 3 groups, how many gumballs are in each group?
We can find the answer to the division problem by counting the number of dots in each row.
18, divided by, 3, equals, 6

Practice problem 2

Problem 2A
This array shows a total of
dots divided equally into
rows.

Problem 2B
Which expression can be used to represent the array?

Problem 2C
There are
dots in each row.

Problem 2D
28, divided by, 7, equals

Practice problem 3

This array has start color #e07d10, 35, end color #e07d10 dots divided into start color #11accd, 5, end color #11accd equal rows.
Problem 3
start color #e07d10, 35, end color #e07d10, divided by, start color #11accd, 5, end color #11accd, equals

Equal shares

This type of problem is similar to the ones we just solved. However, in this case, we know the number of objects in each group instead of the number of equal groups.
Let's look at an example:
Peng's Pony Rides has start color #e07d10, 20, end color #e07d10 ponies. The ponies take kids on rides all day. At the end of the day, the ponies rest in their stables. Each stable holds start color #11accd, 4, end color #11accd ponies.
We have a total of start color #e07d10, 20, end color #e07d10 ponies.
We also know the number of equal shares in each group. Each stable holds start color #11accd, 4, end color #11accd ponies.
We can use division to figure out how many stables Peng needs for all his ponies.
Problem 4

The expression for start color #e07d10, 20, end color #e07d10 ponies divided into equal groups of start color #11accd, 4, end color #11accd is start color #e07d10, 20, end color #e07d10 divided by start color #11accd, 4, end color #11accd, point

Let's try another problem

Peng's Pony Rides has a total of start color #e07d10, 20, end color #e07d10 ponies. They built bigger stables. Each stable now holds start color #7854ab, 10, end color #7854ab ponies.
Problem 5A
There are a total of
ponies. The ponies will be divided into equal-size groups with
ponies in each group.

Problem 5B
Which expression can we use to show a total of the start color #e07d10, 20, end color #e07d10 ponies divided into groups with start color #7854ab, 10, end color #7854ab ponies in each group?

Problem 5C
Which image could we use to show start color #e07d10, 20, end color #e07d10 ponies divided equally into groups with start color #7854ab, 10, end color #7854ab ponies in each group?

Problem 5D
start color #e07d10, 20, end color #e07d10, divided by, start color #7854ab, 10, end color #7854ab, equals

Connecting division and multiplication

The array shows a total of start color #7854ab, 30, end color #7854ab dots. The dots have been divided into start color #e07d10, 6, end color #e07d10 equal rows with start color #11accd, 5, end color #11accd dots in each row.
The equation start color #7854ab, 30, end color #7854ab divided by start color #e07d10, 6, end color #e07d10, equals, start color #11accd, 5, end color #11accd represents the array.
We could also say the array is made up of start color #e07d10, 6, end color #e07d10 rows of dots with start color #11accd, 5, end color #11accd dots in each row.
The equation start color #e07d10, 6, end color #e07d10 times start color #11accd, 5, end color #11accd = start color #7854ab, 30, end color #7854ab also represents the array.
In both equations, start color #7854ab, 30, end color #7854ab is the total number of dots, start color #e07d10, 6, end color #e07d10 is the number of equal-size groups, and start color #11accd, 5, end color #11accd is the number of dots in each group.

Let's try another one.

Problem 6
Which equations represent the following array?

Want to join the conversation?

• Why is the division sign like this ÷ or this /?
• The division symbol is a just a empty "fraction" with "numerator" and "denominator" represented as a simple dot.

For instance:
20 divided by 10

20 => numerator (the "top" of the fraction.)--10 => denominator (the "bottom" of the fraction.)

Some people uses a slash symbol which means the same thing.

20/10

The "divide sign" is called Obelus. It was used as a division symbol by a Swiss mathematician Johann Heinrich Rahn on his algebra book entitled Teutsche Algebra in 1659.
• What is 27 divide into 2?
• 6, remainder 1
• Why dose these problem are up and down and all at a time?
• These numbers go "up and down" or increase and decrease all the time because we are trying to "simplify". Division by itself divides big numbers into small parts. So, a big number (or any number) divided by (any other number) will give us a smaller amount.
• Is Division the opposite of Multiplication?
• Yes, just as addition is the opposite of subtraction.
• Is 98,765,432/2= 49382716. Or did I do the math wrong?
• yes your answer is correct. given I don't know what math you did you may have just coincidentally found the answer but you probably got it right
• So what if I ran into a question like 7,000 divided by 4? What would I do to figure it out?
• you could do 7/4 then multiply it by 1000
(1 vote)
• Is there going to be long division?
• Yes, you will soon learn how to do long division if you keep learning!