## Properties and patterns for multiplication | 3rd grade | Khan Academy

We want to figure out how

many balloons we have here. And obviously, we

could just count these. But now, we have other

ways of thinking about it, especially because

they’re arranged in this nice array, this

nice grid pattern here. And the reason why it’s

useful to not just always have to count it, but to be able

to use little multiplication with the number of rows

and the number of columns is that you might

run into things and you will run

into things where it’s very hard to count each

of the objects individually, but it might be a little

bit easier to count the rows and to count the columns. So for example,

right over here, we see that we have

1, 2, 3, 4 rows. And we have 1, 2, 3,

4, 5, 6, 7 columns. So you could view this

4 as an array of objects where we have 4 rows. Let me write that down. We have 4 rows and

we have 7 columns. And you might already

remember that we can calculate the

total number of objects by multiplying the rows

times the columns– 4 rows times 7 columns. Now, why does this work? Why will this give us the

actual number of objects? Well, we could view

this– we have 4 rows. So we have 4 groups of things. And how many things are

in each of those rows? Well, the number of columns. We have 7 things in each of

those 4 rows– so 4 groups of 7. Or you could view it

the other way around. You could view that

each column is a group. So then you have 7 groups. And how many objects

do you have in each? Well, that’s what

the rows tell you. You have 4 things in

each of those columns. And we already know that

both of these quantities are going to find the exact same

number, the number of things that we have right over here. So these two things

are equivalent. 4 times 7 is equal to 7 times 4. And there’s a bunch

of ways that we can calculate

either one of these. We can skip count by 4. We say 4 times 4, 8,

12, 16, 20, 24, 28. And let’s see. Let me make sure that that’s 7. So 4 times 1, 2, 3, 4,

5, 6, 7– so we get 28. We could just calculate that

there are 28 objects here. And likewise, we

could skip count by 7. So we could go 7– 7 times 2

is 14, times 3 is 21, times 4 is 28, just adding

seven every time. And so we could get

28 the other way. Let’s do that in

that same color. We could get 28 the other way. But if you had a situation

where you didn’t know– where you either didn’t want

to do these techniques or it had been hard

to do these techniques or you didn’t know what 4 times

7 was off the top of your head, which you should know at

some point in the very, very near future. Is there any way to break

this down to something that maybe you do know or maybe

that’s a little bit easier to compute? Well, you could

realize that 7 columns is the same thing as 5

columns and then 2 columns. So you could view 7

columns as 5 columns– so this is 5 columns right

over here– plus 2 columns. So that’s just like

saying that 4 times 7 is the same thing

as 4 times 5 plus 2. And all I do is I replace

the seven with a 5 plus 2. 7 has been replaced

with a 5 plus 2. Now, why is this interesting? Well, now, I can break this

up into two separate arrays. So I could say,

well, look, there’s the array that has 4 rows and

2 columns right over here. And then there’s the array that

has 4 rows and 5 columns right over here. So how many objects

are in this one, in the yellow one

right over here? Well, there’s 4 times 5 objects. So there’s 4 times 5

objects in the yellow grid or yellow array. And how many in this

orange-ish looking thing? Well, there’s going

to be 4 times 2. And if we take the sum

of the 4 times the 5 and the 4 times the 2,

what are we going to get? Well, we’re going to

get the 4 times the 7. We’re going to get the

4 times the 5 plus 2. So if we take sum

of these things– and we want to do the

multiplication first, so I’ll just put a

parentheses around that to emphasize that–

this is going to be the same thing as

these things up here. And so you might say, oh,

well, I know what 4 times 5 is. 4 times 5 is 20. 4 times 2 is 8. 20 plus 8 is 28. And you might say,

OK, Sal, I get it. 4 times 7 is 28, which is the

same thing as 4 times 5 plus 2. And I see that that’s the same

thing as 4 times 5 plus 4 times 2. And actually, this is called

the distributive property– that 4 times 5 plus 2 is

the same thing as 4 times 5 plus 4 times 2. But I could just do one

of these first techniques you talked about. Why is this

distributive property that you just showed me

useful for computing or doing multiplication problems? Well, let me give you a

slightly more difficult one. Let’s imagine you wanted

to multiply 6 times 36. Actually, I don’t need to

write that parentheses. So how could you do this? Well, you could decompose

36 into two products or into two numbers

where it’s easier to find the product

of that and 6. So for example 36, is the

same thing as 30 plus 6. So this is going to be

equal to 6 times 30 plus 6. And what’s this going to be? Well, we just saw. 6 times these two things

added together first, this is going to be equal to

6 times 30 plus 6 times 6. Notice, we distributed the

6– 6 times 30 plus 6 times 6. Now, why is this useful? Why was this useful at all? I’m going to put

parentheses to emphasize. We’re going to do the

multiplication first. In general, when you see

multiplication and addition in a row like this,

or division, you want to do your multiplication

and division first, then do your addition

and subtraction. So what’s 6 times 30? Well, this is

easier to calculate. 6 times 3, we know to be 18. So 6 times 30 is

going to be 180. And 6 times 6– well, we

know that’s going to be 36. So this is going

to be 180 plus 36. And what’s that going to be? 180 plus 36– well,

0 plus 6 is 6. 8 plus 3 is 11. 1 plus 1 is 2. So you just figured out that

6 times 36 is equal to 216. And what we just did with

the distributive property, this is actually

going to be how you’re going to multiply all sorts

of larger numbers, way larger than what we just saw. So the distributive property,

which hopefully you’re pretty convinced by, based

on how we broke things up, is a super useful

thing as you want to compute larger

and larger numbers. And you’re going to find

it even more useful when you go even further in

your mathematical career.

## Tweasel

Dec 12, 2014, 12:35 amHI

## pacrii

Sep 9, 2015, 8:31 pm+Khan Academy You got "3th grade" in the title. You mean "3rd grade."

## Bethany Hershkowitz

Sep 9, 2015, 8:36 pmi remember the 3th grade

## Thant Lynn

Sep 9, 2015, 10:13 pm3th grade is hard

## ewelina s

Nov 11, 2015, 6:04 pmGreat video. Very helpful as Im working on creating a lesson plan. Can anybody tell me what computer program it's used in that video?

Thank you

## Fantastic Sammi

Mar 3, 2016, 5:53 pmi learned multiplication in like 2nd grade

## Fantastic Sammi

Mar 3, 2016, 5:54 pmnice drawing by the way

## MochiHamster35 AJ

May 5, 2016, 1:58 pmWhen I was in the 3th grade I thought that I was gay

## Finexity

Mar 3, 2017, 2:21 amIf you think about about each time you multiply 8 the second digit subtracts 2 each time as you can see 8,16,24,32,40,48,56,64 and if you add up the digits of each multiple you subtract by 1 from 8 each time 8=8 16=7 24=6 32=5 40=4 48=12 12= 3 56 =11 11=2 64=10 10 = 1 the pattern is actually pretty interesting

## Latina Thomas

Mar 3, 2017, 12:40 am28

## Tava T

Apr 4, 2017, 2:29 amgood

## WENXIN ZHENG

Feb 2, 2018, 4:47 pm3rd grade is easy and i learn multiplication in 2nd grade

## Jacob C

May 5, 2018, 2:23 am>THIRTH

## Danerys Cepeda

Mar 3, 2019, 9:07 pm3rd grade Math’s are easy

## danial david

Oct 10, 2019, 12:55 ami think you should do 6 groups of 7