Division 1 | Multiplication and division | Arithmetic | Khan Academy

Summary

This educational script explores the concept of division through relatable examples, such as splitting money or objects like quarters and bell peppers. It illustrates division as the process of evenly distributing items into groups and emphasizes its relationship with multiplication. The script uses visual aids to demonstrate division with remainders and encourages mental math strategies for solving division problems, aiming to clarify and solidify the understanding of this fundamental arithmetic operation.

Takeaways

• 😀 Division is the process of splitting a group into equal parts or distributing a quantity among a number of recipients.
• 📚 The word 'divide' is used in everyday language, such as dividing money between people.
• 👦👧 An example given is dividing 4 quarters between two people, resulting in each person getting 2 quarters.
• 🤔 Division can be thought of as the opposite of multiplication, where multiplication finds the total of groups and division finds the number of groups or the size of each group.
• 🍒 The script uses the example of dividing 6 bell peppers into 3 equal groups to explain division, resulting in each group having 2 bell peppers.
• 🔢 Division can be represented mathematically, such as 6 divided by 3 equals 2, showing the number of groups when the total is divided by the number of groups.
• 🍓 Another way to think about division is dividing a number into groups of a certain size, like dividing 6 raspberries into groups of 3, resulting in 2 groups.
• 🔄 The relationship between division and multiplication is highlighted, showing that 6 divided by 3 equals 2 and 6 divided by 2 equals 3, reflecting the commutative property of multiplication.
• 🍎 Division can result in a whole number or a quotient with a remainder, as shown with 9 divided by 4, which equals 2 with a remainder of 1.
• 🍏 The script explains that division can be done mentally by using multiplication tables to find how many times the divisor fits into the dividend before subtracting to find the remainder.
• 📝 Practice is emphasized as a way to become proficient in division, suggesting that over time, one may not need to draw out the problem but can solve it mentally.

Q & A

What does it mean to divide something?

To divide something means to cut it up into smaller parts or groups. For example, dividing money between two people means each person gets an equal share of the total amount.

How can you divide 4 quarters between 2 people?

You can divide 4 quarters between 2 people by giving each person 2 quarters. Mathematically, this is written as 4 divided by 2 equals 2.

What is the opposite operation of division?

The opposite operation of division is multiplication. For example, if you have 2 groups of 2 quarters, multiplying 2 by 2 gives you 4 quarters.

How can you illustrate 6 divided by 3?

You can illustrate 6 divided by 3 by drawing 6 objects and dividing them into 3 equal groups. Each group will have 2 objects, so 6 divided by 3 equals 2.

How does dividing by groups differ from dividing into groups?

Dividing by groups means you are separating objects into a specific number of groups. Dividing into groups means you are creating groups with a specific number of objects in each group.

What is the relationship between multiplication and division in the context of 6 divided by 2?

The relationship is that 6 divided by 2 equals 3 because 2 times 3 equals 6. Similarly, 6 divided by 3 equals 2 because 3 times 2 equals 6.

How can you handle division with remainders, such as 9 divided by 4?

For 9 divided by 4, you can create two groups of 4, with 1 object left over. So, 9 divided by 4 equals 2 with a remainder of 1.

What is the result of 12 divided by 4?

The result of 12 divided by 4 is 3, because you can divide 12 objects into 3 equal groups of 4.

How do you find 14 divided by 5 using multiplication tables?

You determine how many times 5 goes into 14 without exceeding it. Since 5 times 2 equals 10, and 5 times 3 equals 15 (which is too much), 5 goes into 14 two times, with a remainder of 4.

What does 8 divided by 2 equal, and how can it be visualized?

8 divided by 2 equals 4. You can visualize this by drawing 8 objects and dividing them into 4 groups of 2.

Outlines

00:00

🔢 Introduction to Division

This paragraph introduces the concept of division by illustrating the division of 4 quarters between two people. It explains division as cutting something into equal parts and provides a mathematical representation of dividing 4 by 2. The paragraph emphasizes that division is a familiar concept used in everyday activities like splitting money.

05:02

🍅 Dividing Bell Peppers and Raspberries

The paragraph extends the concept of division with examples using 6 bell peppers and raspberries. It explains two perspectives: dividing 6 objects into 3 equal groups, and dividing 6 objects into groups of 3. The examples reinforce understanding of division by demonstrating different grouping methods.

10:03

🍏 Exploring Remainders in Division

This section introduces the concept of remainders in division, using the example of dividing 9 objects by 4. It explains how to handle leftover objects that don't fit into equal groups and introduces the idea of representing division with remainders. The paragraph also revisits the division of 12 objects by 4 and by 3 to reinforce the relationship between multiplication and division.

15:04

🎲 Practical Division Examples

The final paragraph provides additional practice with division, including examples of 14 divided by 5 and 8 divided by 2. It demonstrates the process of determining how many times a divisor fits into a dividend and how to handle remainders. The paragraph aims to solidify the reader's understanding of division through repeated examples and alternative representations.

Keywords

💡Divide

Divide is a mathematical operation that separates a quantity into equal or proportional parts. In the video, it is the central theme, used to explain how to evenly distribute items like quarters or bell peppers among people or into groups. For example, 'divide the money between you and your brother' or dividing 4 quarters between two people results in each person getting 2 quarters.

💡Quarters

In the context of the video, quarters refer to the coins representing a quarter of a dollar, often featuring George Washington's image. They are used as an example to illustrate the concept of division, as in dividing 4 quarters between two people to demonstrate how division works.

💡Equal Groups

Equal groups are sets of items that are identical in number, resulting from the division process. The video emphasizes this concept by showing how to create equal groups of items, such as dividing 6 bell peppers into 3 equal groups of 2, to help understand the division operation.

💡Multiplication

Multiplication is the mathematical operation of scaling a number by another, which is presented in the video as being conceptually linked to division. It is mentioned as the opposite of division, as seen when explaining that 2 groups of 3 quarters equal 6, which is the same as 6 divided by 2 resulting in 3.

💡Remainder

Remainder refers to the leftover part of a number that is not completely divisible by another number. In the video, the concept is introduced when dividing 9 by 4, which results in 2 groups of 4 with a remainder of 1, indicating that not all items can be evenly distributed.

💡Bell Peppers

Bell peppers are used in the script as a visual aid to represent the division of 6 items into 3 equal groups. The term helps to create a relatable context for the division concept, showing that each group would receive 2 bell peppers.

💡Groups

Groups are collections of items formed during the division process. The video script uses the term to describe the result of dividing a set of items, such as creating groups of 3 or 4 from a larger set, to illustrate the division operation.

💡Division by

Division by is the process of splitting a number into a certain number of equal parts. In the video, phrases like 'divided by 3' or 'divided by 4' are used to describe the action of dividing a total quantity into groups of a specified size.

💡Raspberries

Raspberries are used in the script as an example to illustrate another division scenario, where 6 raspberries are divided into groups of 3. This example helps to further clarify the concept of division by showing different ways to group items.

💡Multiplication Tables

Multiplication tables are a tool used to quickly recall the products of multiplication for numbers. In the video, the script refers to using multiplication tables in one's head to determine how many times 5 fits into 14, which is part of the division process.

💡Division Notation

Division notation refers to the mathematical symbol and format used to represent division, such as '8 divided by 2' or '14 ÷ 5'. The video explains how to use this notation to express division problems and solve them conceptually without drawing.

Highlights

Introduction to the concept of division as the act of splitting or distributing items equally among a group.

Illustrative example using 4 quarters to demonstrate division by 2, resulting in 2 quarters each.

The importance of emphasizing the number of groups in division, as shown with the 4 quarters divided by 2.

Visual representation of division through drawings of people and objects to enhance understanding.

Explanation of division as the process of cutting a group into equal parts, using the 4 quarters example.

Mathematical notation of division with the formula 4 divided by 2 equals 2.

The relationship between division and multiplication, highlighting the inverse nature of the operations.

Further examples using 6 divided by 3 and 6 divided by 2 to illustrate different division scenarios.

The concept of remainders introduced with the division of 9 by 4, resulting in 2 groups with a remainder of 1.

Division of 12 by 4, showing how to divide objects into groups with no remainder.

Exploration of division with remainders in the example of 14 divided by 5, resulting in 2 groups with a remainder of 4.

The multiplication table method to determine how many times a number divides into another without drawing.

Division of 8 by 2 as a straightforward example to reinforce the concept of equal distribution.

The significance of the order of factors in multiplication and its relevance to division.

Practical application of division in everyday scenarios such as splitting money or sharing objects.

Emphasis on the intuitive understanding of division through multiple relatable examples.

The use of visual aids and drawings to clarify the process and results of division.

The transition from concrete examples to abstract mathematical notation to deepen the understanding of division.

Transcripts

I think you've probably heard the word divide before, where

someone tells you to divide something up.

Divide the money between you and your brother or between

you and your buddy.

And it essentially means to cut up something.

So let me write down the word divide.

Let's say that I have 4 quarters.

Do my best to draw the quarters.

If I have 4 quarters just like that.

That's my rendition of George Washington on the quarters.

And let's say there's two of us and we're going to divide

the quarters between us.

So this is me right here.

Let me try my best to draw me.

So that's me right there.

Let's see, I have a lot of hair.

And then this is you right there.

Do my best.

So you're bald.

You have side burns.

Maybe you have a little bit of a beard.

I want to get too focused on the drawing.

So that's you, that's me, and we're going to divide these 4

quarters between the 2 of us.

So notice, we have 4 quarters and we're going to divide

between the 2 of us.

There are 2 of us.

And I want to stress the number 2.

So we're going to divide 4 quarters by 2.

We're going to divide it between the 2 of us.

And you've probably done something like this.

What happens?

Well, each of us are going to get 2 quarters.

So let me divide it.

We're going to divide it into 2.

Essentially what I did do is I take the 4 quarters and I

divide it into 2 equal groups.

And that's what division is.

We cut up this group of quarters into 2 equal groups.

So when you divide 4 quarters into 2 groups, so this was

4 quarters right there.

And you want to divide it into 2 groups.

This is group 1.

Group 1 right here.

And this is group 2 right here.

How many numbers are in each group?

Or how many quarters are in each group?

Well, in each group I have 1, 2 quarters.

Let me use a brighter color.

I have 1, 2 quarters in each group.

1 quarter and 2 quarters in each group.

So to write this out mathematically, I think this is

something that you've done, probably as long as you've been

splitting money between you and your siblings and your buddies.

Actually, let me scroll over a little bit, so you can

see my entire picture.

How do we write this mathematically?

We can write that 4 divided by-- so this 4.

Let me use the right colors.

So this 4, which is this 4, divided by the 2 groups, these

are the 2 groups: group 1 and this is group 2 right here.

So divided into 2 groups or into 2 collections.

4 divided by 2 is equal to-- when you divide 4 into 2

groups, each group is going to have 2 quarters in it.

It's going to be equal to 2.

And I just wanted to use this example because I want to show

you that division is something that you've been

using all along.

And another important, I guess, takeaway or thing to realize

about this, is on some level this is the opposite

of multiplication.

If I said that I had 2 groups of 2 quarters I would multiply

the 2 groups times the 2 quarters each and I would say

I would then have 4 quarters.

So on some level these are saying the same thing.

But just to make it a little bit more concrete in our

head, let's do a couple of more examples.

Let's do a bunch of more examples.

So let's write down, what is 6 divided by-- I'm trying to

keep it nice and color coded.

6 divided by 3, what is that equal to?

Let's just draw 6 objects.

They can be anything.

Let's say I have 6 bell peppers.

I won't take too much trouble to draw them.

Well, that's not what a bell pepper looks like,

but you get the idea.

So 1, 2, 3, 4, 5, 6.

And I'm going to divide it by 3.

And one way that we can think about that is that means I want

to divide my 6 bell peppers into 3 equal groups

of bell peppers.

You could kind of think of it if 3 people are going to share

these bell peppers, how many do each of them get?

So let's divide it into 3 groups.

So that's our 6 bell peppers.

I'm going to divide it into 3 groups.

So the best way to divide it into 3 groups is I can have 1

group right there, 2 groups, or the second group right there,

and then, the third group.

And then each group will have exactly how many bell peppers?

They'll have 1, 2.

1, 2.

1, 2 bell peppers.

So 6 divided by 3 is equal to 2.

So the best way or one way to think about it is that you

divided the 6 into 3 groups.

Now you could view that a slightly different way,

although it's not completely different, but it's a good

way to think about it.

You could also think of it as 6 divided by 3.

And once again, let's say I have raspberries

now-- easier to draw.

1, 2, 3, 4, 5, 6.

And here, instead of dividing it into 3

groups like we did here.

This was 1 group, 2 group, 3 groups.

Instead of dividing into 3 groups, what I want to do is

say well, if I'm dividing 6 divided by 3, I want to

divide it into groups of 3.

Not into 3 groups.

I want to divide it into groups of 3.

So how many groups of 3 am I going to have?

Well, let me draw some groups of 3.

So that is one group of 3.

And that is two groups of 3.

So if I take 6 things and I divide them into groups of 3, I

will end up with 1, 2 groups.

So that's another way to think about division.

And this is an interesting thing.

When you think about these two relations, you'll see a

relationship between 6 divided by 3 and 6 divided by 3.

Let me do that right here.

What is 6 divided by 2 when you think of it in this

context right here?

6 divided by 2, when you do it like that-- let me

draw 1, 2, 3, 4, 5, 6.

When we think about 6 divided by 2 in terms of dividing it

into 2 groups, what we can end up is we could have 1 group

like this and then 1 group like this, and each group

will have 3 elements.

It'll have 3 things in it.

So 6 divided by 2 is 3.

Or you could think of it the other way.

You could say that 6 divided by 2 is-- you're taking 6

objects: 1, 2, 3, 4, 5, 6.

And your dividing it into groups of 2 where each

group has 2 elements.

And that on some level is an easier thing to do.

If each group has 2 elements, well, that's the 1 right there.

They don't even have to be nicely ordered.

This could be one group right there and that could be the

other group right there.

I don't have to draw them all stacked up.

These are just groups of 2.

But how many groups do I have?

I have 1, 2, 3.

I have 3 groups.

But notice something, it's no coincidence that 6 divided by 3

is 2 and 6 divided by 2 is 3.

Let me write that down.

We get 6 divided by 3 is equal to 2 and 6 divided

by 2 is equal to 3.

And the reason why you see this relation where you can kind of

swap this 2 and this 3 is because 2 times 3

is equal to 6.

Let's say I have 2 groups of 3.

Let me draw 2 groups of 3.

So that's 1 group of 3 and then here's another group of 3.

So 2 groups of 3 is equal to 6.

2 times 3 is equal to 6.

Or you could think of it the other way if I

have 3 groups of 2.

So that's 1 group of 2 right there.

I have another group of 2 right there.

And then I have a 3 group of 2 right there.

What is that equal to?

3 groups of 2-- 3 times 2.

That's also equal to 6.

So 2 times 3 is equal to 6.

3 times 2 is equal to 6.

We saw this in the multiplication video that

the order doesn't matter.

But that's the reason why if you want to divide it, if you

want to go the other way-- if you have 6 things and you

want to divide it into groups of 2, you get 3.

If you have 6 and you want to divide into groups

of 3, you get 2.

Let's do a couple of more problems.

I think it'll really make sense about what

division is all about.

Let's do an interesting one.

Let's do 9 divided by 4.

So if we think about 9 divided by 4, let me draw 9 objects.

1, 2, 3, 4, 5, 6, 7, 8, 9.

Now when you divide by 4, for this problem, I'm thinking

about dividing it into groups of 4.

So if I want to divide it into groups of 4, let

me try doing that.

So here is one group of 4.

I just picked any of them right like that.

That's one group of 4.

Then here's another group of 4 right there.

And then I have this left over thing.

Maybe we could call it a remainder, where I can't put

this one into a group of 4.

When I'm dividing by 4 I can only cut up the

9 into groups of 4.

So the answer here, and this is a new concept for you maybe, 9

divided by 4 is going to be 2 groups.

I have one group here and another group here, and then

I have a remainder of 1.

I have 1 left over that I wasn't able to do with.

Remainder-- that says remainder 1.

9 divided by 4 is 2 remainder 1.

If I asked you what 12 divided by 4 is, so let me do 12.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

So let me write that down.

12 divided by 4.

So I want to divide these 12 objects-- maybe

they're apples or plums.

And divide them into groups of 4.

So let me see if I can do that.

So this is one group of 4 just like that.

This is another group of 4 just like that.

And this is pretty straightforward.

And then I have a third group of 4 just like that.

And there's nothing left over like I had before.

I can exactly divide 12 objects into 3 groups of 4.

1, 2, 3 groups of 4.

So 12 divided by 4 is equal to 3.

And we can do the exercise that we saw on the previous video.

What is 12 divided by 3?

Let me do a new color.

12 divided by 3.

Now based on what we've learn so far we say, that should just

be 4 because 3 times 4 is 12.

But let's prove it to ourselves.

So 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 11, 12.

Let's divide it into groups of 3.

And I'm going to make them a little strange looking just

so you see that you don't always have to do it into

nice, clean columns.

So that's a group of 3 right there.

12 divided by 3.

Let's see, here is another group of 3 just like that.

And then, maybe I'll take this group of 3 like that.

And I'll take this group of 3.

There was obviously a much easier way of dividing it up

then doing these weird l-shaped things, but I want to show

you it doesn't matter.

You're just dividing it into groups of 3.

And how many groups do we have?

We have one group.

Then we have our second group right here.

And then we have our third group right there.

And then we have-- let me do it in a new color.

And then we have our fourth group right there.

So we have exactly 4 groups.

And when I say there was an easier way to divide it, the

easier way was obviously, maybe not obviously-- if I want to

divide these into groups of 3 I could have just done 1,

2, 3, 4 groups of 3.

Either of these I'm dividing the 12 objects

into packets of 3.

You can imagine them that way.

Let's do another one that maybe has a remainder.

Let's see.

What is 14 divided by 5?

So let's draw 14 objects.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 14.

14 objects.

And I'm going to divide it into groups of 5.

Well, the easiest thing is there's one group right there,

two groups right there.

But then this last one, I only have 4 left, so I can't

make another group of 5.

So the answer here is I can make 2 groups of 5 and I'm

going to have a remainder-- r for remainder-- of 4.

2 remainder 4.

Now, once you get enough practice, you're not always

going to be wanting to draw these circles and dividing

them up like that.

Although that would not be incorrect.

So another way to think about this type of problem is to

say, well, 14 divided by 5, how do I figure that out?

Actually, another way of writing this and no

harm in showing you.

I could say 14 divided by 5 is the same thing as 15 divided

by-- this sign right-- divided by 5.

And what you do is you say, well, let's see.

How many times does 5 go into 14?

Well, let's see.

5 times and you kind of do multiplication

tables in your head.

5 times 1 is equal to 5.

5 times 2 is equal to 10.

So that's still less than 14, so 5 goes at least two times.

5 times 3 is equal to 15.

Well that's bigger than 14, so I have to go back here.

So 5 only goes two times.

So it goes 2 times.

2 times 5 is 10.

And then you subtract.

You say 14 minus 10 is 4.

And that's the same remainder as right here.

Well, I could divide 5 into 14 exactly two times, which

would get us 2 groups of 5.

Which is essentially just 10.

And we still have the 4 left over.

Let me do a couple of more just to really make sure you get

this stuff really, really, really, really well.

Let me write it in that notation.

Let's say I do 8 divided by 2.

And I could also write this as 8-- so I want

to know what that is.

That's a question mark.

I could also write this as 8 divided by 2.

And the way I do either of these, I'll draw the circles in

the second, but the way I do it without drawing the circles, I

say, well, 2 times 1 is equal to 2.

So that definitely goes into 8, but maybe I can think of a

larger number that goes into-- that when I multiply it

by 2 still goes into 8.

2 times 2 is equal to 4.

That's still less than 8.

So 2 times 3 is equal to 6.

Still less than 8.

2 times-- oh, something weird happened to my pen.

2 times 4 is exactly equal to 8.

So 2 goes into 8 four times.

So I could say 2 goes into 8 four times.

Or 8 divided by 2 is equal to 4.

We can even draw our circles.

1, 2, 3, 4, 5, 6, 7, 8.

I drew them messy on purpose.

Let's divide them into groups of 2.

I have one group of 2, two groups of 2, three groups

of 2, four groups of 2.

So if I have 8 objects, divide them into groups of 2,

you have four groups.

So 8 divided by 2 is 4.

Hopefully you found that helpful.