Multi-digit Numerical Long Division 1 © 2013 Meredith S. Moody.

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The Number SystemMulti-digit Numerical Long Division

© 2013 Meredith S. Moody

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Divide numbers with 2 or more digits using a variety of methods for long division

Divide numbers with 2 or more digits using the standard algorithm for long division

Objective: You will be able to…


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Division is determining how many groups of one number can be made out of another number

For example, I have the number 15 and I want to make 3 groups; how many will be in each group? The answer would be 5

That is the same as dividing 15 by 3

What is division?


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What if there are not a whole number of groups?

Let’s say I have 15 cookies and I want to make 4 bags (equal groups) of cookies.

If I divide 15 into 4 equal groups, I would have 3 cookies in each bag, but I would have 3 cookies left over.

3 cookies would ‘remain’ In other words, 3 is my remainder if I want

to divide 15 by 4

Remainders


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Mathematical operations come in pairs Which operations do you think are pairs? Addition and subtraction are a pair Multiplication and division are a pair In order to divide, you have to understand

multiplication

Inverse operations


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Multiplication is repeated addition◦ 3 x 5 = 15◦ 3 + 3 + 3 + 3 + 3 = 15

Division is repeated subtraction◦ 15 ÷ 5 = 3 (3 groups of 5, none left over)◦ 15 – 5 – 5 – 5 = 0

Relationships


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If I have 15 cookies and want to make 5 equal bags of cookies, there must be 3 cookies in each bag

I can make 5 bags of 3 cookies. 5 x 3 = 15 15 ÷ 3 = 5 15 ÷ 5 = 3 Division and multiplication are inverse

operations

How dividing & multiplying are related


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What happens if the numbers are too large to divide mentally?

What if I want to divide 487 by 32? How could I do that? I could use a calculator, yes, but what if I

don’t have one? Let’s look at three different methods of

dividing by hand◦ Repeated subtraction◦ Standard algorithm◦ Scaffold division

Example 1


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Division is actually repeated subtraction How many times can I subtract 32 from

487? 487–32=455–32=423–32=391–32=359

359-32=327-32=295-32=263-32=321 321-32=199-32=167-32=135-32=103 103-32=71-32=39-32=7

How many times did we subtract 32? 15 How many is left over? 7 Wow! That took a long time. Is there

another way?

Example 1, repeated subtraction


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An “algorithm” is a step-by-step procedure for calculations

We can use a division algorithm for multi-digit division

In this method, there are specific parts with universal names

Knowing these names are important so everyone can discuss division without becoming confused

Standard algorithm


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The division bracket is the “box” into which we put the dividend


Example 1, standard algorithm, continued

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487 is the dividend, it goes in the “box”

32 is the divisor, it goes outside the “box”

The answer is called the “quotient”

The left over amount is called the “remainder”



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The most efficient way to divide multi-digit numbers by hand is called ‘long division’

How many groups of 32 are in the number 4? 0. 32x0=0. subtract 4-0=4. ‘Bring down’ the next digit (8)

How many groups of 32 are in the number 48? 1. 32x1=32. subtract 48-32=16. ‘Bring down’ the next digit (7)

How many groups of 32 are in the number 167? 32x5=160. subtract 167-160=7



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Wow, that standard algorithm doesn’t make sense to me

Is there another way? Yes Instead of trying to divide 487 by 32, we

can break up our steps into smaller chunks This is called scaffold division

Example 1, scaffold division


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We can break up large numbers using the place value system

487 becomes 400+80+7 How many groups of 32 can I make out of

400? Well, I know 3x4=12; I should be able to make about 12 groups of 32 out of 400

Well, if I make 12 groups of 32, how much of the 400 have I ‘used’? 12x32=384

How much of the 400 do I still have to ‘use’? 400-384=16; I have 16 ‘left over’

Example 1, scaffold division, continued


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Now I work with the number 80 How many groups of 32 can I make out of

the number 80? I know 3x3=9, but 90 is too much; I should

be able to make 2 groups of 32 out of 80 If I make 2 groups of 32, how much of the

80 have I ‘used’? 32x2=64 How much do I have left to ‘use’? 80-

64=16; I have 16 ‘left over’



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Now I have to look at the number 7 How many groups of 32 can I make out of 7? None Let’s use our ‘leftovers’ I had 16 left over from the 400, 16 left over

from the 80, and 7 left over from my original work

16 + 16 + 7 = 39 How many groups of 32 can I make out of 39? I can make 1 group of 32 out of 39, with 7 left

over



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Now I just add my groups together: I had 12 groups in the 400 I had 2 groups in the 80 I had 1 group in the ‘leftovers’ 12+2+1=15 I have 7 ‘left over’ now, so the answer to my

problem: what is 487÷32, is 15 remainder 7 That was a little hard to follow; is there an

easier way to write this? Yes



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Let’s put our scaffold method into an easy-to-read structure:

487 = 400 + 80 + 7 400÷32 = 12

◦ 32 x 12 = 384◦ 400-384 = 16

80÷32 = 2◦ 32 x 2 = 64

◦ 80-64 = 16 7÷32 = 0 16+16+7 = 39 39÷32 = 1

◦ 32 x 1 + 32◦ 39-32 = 7

12+2+1 = 15 487÷32 = 15 r7

Example 1, scaffold division, visual


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The scaffold method took quite a while, too Is there a more efficient way to scaffold? Yes

Example 1, efficient scaffold division


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Let’s try another together Two individuals are to equally share an

inheritance of $860. How much should each receive?

To solve the problem, we want to divide 860 by 2

Let’s look at the three ways we could solve (no calculators!)

Example 2


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Trying to repeatedly subtract 2 from 860 would take a LONG time

It makes sense to use a faster method

Example 2, repeated subtraction


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Let’s use the extended scaffold division method

First, we break up 860 using place values:

800 + 60 = 860 We can easily divide 800

by 2. 800÷2=400. Each person would get

$400 so far 400+400=800. Since we

have ‘used’ $800, we subtract 860-800 = 60. We still have $60 to share.

Example 2, scaffold division


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Next, we share the $60. Dividing $60 by 2 is easy.

Each person would get $30.

We need to add another 30 to our quotient.

Notice we place the 30 in the proper place value above the 400. We have ‘used’ the last $60, 60-60 = 0.

We have no money left to share.



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The last step is to sum the two partial quotients to obtain the final quotient 400+30=$430

Each person would each receive $430



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Let’s use the standard algorithm

The dividend is 860 The divisor is 2 There are 4 groups of 2

in 8 ‘bring down’ the 6 There are 3 groups of 2

in 6 ‘bring down’ the 0 There are 0 groups of 2

in 0 The quotient is 430

Example 2, standard algorithm


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What if we had three people and they needed to split $986 evenly among them?

Repeated subtraction would take too long The extended scaffold division method

would take a long time, too Let’s start with the efficient scaffold division

method

Example 3


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Example 3, scaffold division How much money

would each person receive if 3 people had to split $986 evenly?

Each person would receive $328

There would be $2 left over


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Let’s use the standard algorithm to solve



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If 14 children had 239 cookies, what is the highest number of cookies each child could receive if each one had to have the same number?

Repeated subtraction would take too long. The extended scaffold method would take

too long Let’s start with the efficient scaffold method

and then try the standard algorithm

Example 4


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Example 4, efficient scaffold division


239 ÷ 14 = 17 r 1

Each child would receive 17 cookies

There would be 1 cookie left over

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Use either the traditional scaffold division, efficient scaffold division, or standard algorithm method to solve: 236÷4

You try


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Traditional scaffold solution


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Efficient scaffold solution


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Standard algorithm solution


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Use any long division method (repeated subtraction, standard, scaffold, or extended scaffold) to solve 193 ÷ 11

Repeated subtraction solution: 193-11=182-11=171-11=160-11=149-11=138

138-11=127-11=116-11=105-11=94-11=83-11=72 72-11=61-11=50-11=39-11=28-11=17-11=6

17 remainder 6

You try


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Traditional scaffold solution


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Efficient scaffold solution


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Standard algorithm solution


Multi-digit Numerical Long Division 1 © 2013 Meredith S. Moody.

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Transcript of Multi-digit Numerical Long Division 1 © 2013 Meredith S. Moody.