
xMath :: 69 :: Division 
xTable of Contents: 
 Introducton
 Numeration
 Addition
 Multiplication
 Subtraction
 Division
 Stamp
Game Division  Single Digit
 Memorization
Exercises
 Division Boards (Bead Boards)
 Introduction and List of Materials
 Initial Presentation
 Preparation for Presentation
 Presentation
 Division Booklets
 Division Charts and Combination
Cards
 Chart I
 Bingo Game (Chart II)
 Exercise A:
 Exercise B.
 Exercise C.
 Group Game 1.
 Group Game 2.
 Division
By More Than One Digit
 Double Digit Division
 Intro. to Double Digit Division
(on The Change Game)
 Double Digit Division on The
Stamp Game
 Triple Digit Division
 Intro. to Triple Digit Division
(on The Change Game)
 Triple Digit Division on The
Stamp Game
 Division
Involving Zeros
 Division with Zero in the
Dividend
 Division with Zero in the
Divisor
 Group
Division
 Single Digit Group Division
 Double Digit Group Division
 More
Memorization Exercises
 Special Cases
 0 Calculate the quotient
 1 Calculate the Divisor
 2 Calculate the Dividend
 3 Inverse of Case Zero: Calculate
the Quotient
 4 Inverse of The First Case:
Calculate the Divisor
 5 Inverse of the Second Case:
Calculate the Dividend
 6 Calculate the Divisor and
the Dividend
 Search for Quotients
 Prime Numbers
 Division
With Racks & Tubes
 Introduction:
 Materials: racks and tubes:
 Small Division (1digit divisor,
4digit dividend)
 1st level
 2nd level
 3rd LevelGroup Division
 Big Division (2 or more digits
in the divisor)
 1st level:
 2nd level:
 3rd levelGroup Division with
a two digit divisor
 Long Division
 Division with Zero in the
Divisor
 Word
Problems
 From Combination Cards
 Distributive vs. Group Division
 Divisibility
 1. Introduction
 2. Div. by 2
 3. Div. by 4
 4. Div. by 5
 5. Div. by 25
 6. Div. by 9
 Synthesis
of Multiples & Divisibility
 Fractions  COMING SOON
 Decimals  COMING SOON
 PreAlgebra  COMING SOON
[top]

xStamp Game Division  Single Digit 
STAMP GAME DIVISION  SINGLE DIGIT
Materials:
...wooden stamps of four types:
......green unit stamps printed with the
numeral 1,
......blue tens stamps printed with the
numeral 10,
......red hundred stamps printed with
the numeral 100, and
......green thousand stamps printed with
the numeral 1000
...box with three compartments each containing 9 skittles and
one counter in each of the hierarchic colors;
...four small plates
Presentation:
The child is given a division problem on a work card. Since this
is division, we put the stamps in these little dishes. 'How many
must this quantity be distributed to?' Instead of calling children
we can use these green skittles, one skittle for each child .
Now you begin the division.
The skittles are set out in a row. Beginning with thousands,
the child distributes equally to the skittle. When the child
runs out of i.e. hundreds before each skittle has received one,
the hundreds are returned to the dish, and the child goes on
to tens. If he doesn't have enough tens (or if he runs out) one
of the remaining hundreds is changed to ten tens. This continues
until all of the hundreds have been changed.
When the child has finished distributing, he reads the resultwhat
one skittle received. The problem is recorded in his notebook.
Note: If necessary to simplify counting, the stamps of one skittle
may be arranged in hierarchic order after the distribution is
finished.
Age: 67
Aims: to perform the operations at a more abstract
level (here there is no visual difference in value according
to size)
Note: These materials are used
parallel to operations in the decimal system
[top]

xMemorization Exercises 
DIVISION BOARDS  BEAD BOARDS
a. Introduction and List of Materials
Introduction:
The memorization of division is the synthesis of the four operations.
For this reason the child must precede this work with a great
deal of work with the other operations, especially multiplication.
It is very important that the child know multiplication really
well before going on to division.
The child has encountered division before via many materials
and regarding many cases: distributive and group division, division
with a 1 or 2 digit divisor. In order to go on, the child must
memorize certain combinations in division.
Materials:
...Division Bead Board: the numerals 19 across the top on a
green background represent the divisors; the numerals 19 down
...the left side represent the quotients;
the 1 in the green circle in the upper left indicates that the
numbers 19 below it represent
...units; 81 holes.
...Orange box containing 9 green skittles
...Orange box containing 81 green beads
...forms
...Booklet of Combinations (36 pages)
...Box of Multiplication Combinations (same combinations as are
found in the booklet)
...Box of orange tiles for bingo game
...Charts I, II (for control)
[top]
DIVISION BOARDS  BEAD BOARDS
b. Initial Presentation
Preparation for Presentation
Beforehand, the teacher prepares pads of problems which consist
of 81 forms. On each form, in the spaces at the top, is written
a number, which is the constant dividend for that form. The forms
are arranged on the strip in order according to the dividends,
starting with 81, ending with 1. Under the forms on which a combination
with remainders appears, the ordinal number of the form is written
in red. On each form are written all of the combinations possible
to perform on the bead board, beginning always with 9 (except,
of course, when the dividend is less than 9) All of the combinations
which have a remainder of zero are underlined in red.
Presentation
In this box are 81 beads which we will distribute to these 9
skittles. The nine skittles are placed along the top green strip
of numbers 19. These 81 beads are the dividend, so we write
81 under the column 'dividend.' Then write the sign for division.
The beads must be divided among 9 skittles; nine is our divisor,
so we write 9 under 'divisor.' Now give out the beads in rows
until all the beads are given out equally.
Each skittle received nine beads, (note the 9 at the end of the
row), so we write 9 under 'quotient.' The last column is for
the remainder, but here we have no remainder, so we write zero
in that column. Whenever we have a combination that has no remainder,
it is very important for our work, so we underline 'it' in red.
Let's try 81 ÷ 8. Remove one skittle and the extra beads
to their respective boxes. What is the quotient and the remainder?
81 ÷ 8=9 r 9 In this game there are two rules to be followed:
1) The quotient should never be greater than 9.
2) The remainder may never be equal or greater than the divisor.
Therefore, we cannot have this combination because the remainder
is too big.
Go on to 80. Change to a new form. Remove one bead and place
it back in a box. Start with 80 ÷ 9=. Write the combination
on the form, distribute the beads and count the remainder. 80
÷ 9 = 8 r 8
Try 80 ÷ 8. Remove one skittle, redistribute the beads,
and count the remainder.
80 ÷ 8 = 9 r 8. This cannot be used because the remainder
is too big, being equal to the divisor. Erase or cross out this
combination.
Go on this way until 72 so that the children see another page
on which combinations can be underlined. At this point bring
out the prepared roll of forms. On this strip we can see all
the combinations that have zero remainder. All of the forms that
have at least one combination with remainder zero, have been
reproduced into booklet form for your work.
Aim: to understand how the combination booklet was
formed
[top]
DIVISION BOARDS  BEAD BOARDS
c. Division Booklets
Materials:
...Division Bead Board
...box with beads
...box of skittles
...Combination booklet
...Chart I
Exercise:
The child chooses a form in the booklet, i.e. 7 Since 7 is the
dividend, count out 7 beads into the box cover. The first combination
is 7 ÷ 7 =, so 7 skittles are put out; the 7 beads are
divided among them. Each skittle receives one; 1 is the quotient.
There is no remainder. 7 ÷ 7 = 1 r 0
The next combination is 7 ÷ 6 =. Only six skittles are
needed. Each skittle receives one and the remaining bead is placed
in the bottom row, or in a box cover. 7 ÷ 6 = 1 r 1. Continue
until the form is completed.
Control of error: Chart I. Find the dividend along
the top and the divisor in red at the left. Go down and across
to find the quotient where the fingers met. If there is a remainder,
the box will be empty, thus move along the row to the right until
a box is full. There you will find the quotient. To check the
remainder subtract the dividend at the top of that column, from
your original dividend. The difference is your remainder.
Note: When presenting the chart
to the children, we identify the prime numbers as well, since
they are shaded in red. 7,5,3,2 and 1 are special numbers because
they only have as divisors, themselves and one. 7 can only be
divided by 7 and by 1, and so on. These special numbers are called
prime numbers.
[top]
DIVISION CHARTS AND COMBINATION
CARDS
a. Chart I
Materials:
...box of loose combination cards, only those that have a remainder
of zero
...Chart I (as a control)
Exercise:
In this box are only the even division combinations: those having
a remainder of zero. The child fishes for a combination, reads
it and copies it into his notebook. On the chart he finds the
dividend at the top and the divisor on the left side. The place
where the two fingers meet is where the answer is found. He writes
the quotient to complete the equation.
Later the child can do the combinations in his head, write down
the quotient and use Chart 1 only as a control.
[top]
DIVISION CHARTS AND COMBINATION
CARDS
b. Bingo Game (Chart II)
Materials:
...box of tiles
...box of combination cards
...Chart II and Chart I (for control)
Note: Much practice should
have preceded these exercises.
Exercise A:
Spread out the tiles face up. The child fishes for a combination,
writes it down including the quotient, and finds the corresponding
tile. The tile is placed on Chart II appropriately.
Exercise B.
With all of the tiles in the box, the child fishes for a tile.
He thinks of a combination that would yield that quotient and
writes the equation in his book, i.e. 8 = 56 ÷ 7. The
tile is placed on Chart II appropriately.
Exercise C.
All the tiles are stacked as usual (this time forming a parallelopiped)
The child chooses a stack, and one at a time he thinks of all
the possible combinations that will yield that quotient, writes
them down and places the tiles on Chart II appropriately. This
continues until all the tiles of the stack which was chosen,
are used. The child uses Chart I to check if he found all of
the possible combinations and if they were placed correctly.
Note: For many children the
aim of this work can be to fill up the entire board.
Group Game 1.
The teacher or a child leading a group of children draws a combination
and reads it. One child responds.
Group Game 2.
The teacher draws a tile and reads the quotient. One child may
try to give all of the possible combinations, or each child in
turn may give one until all of the possibilities are exhausted.
[top]

xDivision By More Than One Digit 
DOUBLE DIGIT DIVISION
a. Intro. to Double Digit Division (on The Change Game)
Materials:
...golden bead materials and numeral cards
...ribbons: green, blue, (also red for later)
Presentation:
The teacher prepares the numeral cards and asks a child to bring
the corresponding quantity. Now I would like to distribute this
quantity among twelve children. Twelve children are called, but
this creates such confusion. How can we solve this problem? Twelve
is made up of ten and two units. Two children can represent the
two units and can be given green ribbons. These ten children
must choose one who will represent ten. What color ribbon do
we give the representative?
Now we are ready to distribute this quantity. If we give one
thousand to the child who represents ten children, what will
each of the other children get? 100 (because 1000 is 10 hundreds)
also each of these children receives 100another thousand for
the group of ten, another hundred for this one, and another hundred
for this one. This continues, making all of the necessary changes,
until all has been distributed (perhaps leaving a remainder)
What is our result? The result is what one unit receives. This
child who represents ten children has enough on his tray so that
each of these ten children will receive what one child received.
(This quantity may be distributed if necessary.)
Control of error: The quantity may be added together
again, making the necessary changes to form the original number.
Age: 67
Aim: to learn the concept of twodigit division:
that if the ten receives a certain quantity, the unit receives
1/10 of that quantity.
[top]
DOUBLE DIGIT DIVISION
b. Double Digit Division on The Stamp Game
Materials: stamp game work cards
Presentation:
The first quantity for this problem is formed with the stamps
and placed in little dishes. We need to divide this among twelve.
This blue skittle can be used to represent 10, and these green
skittles will represent the two units. Now we must give out this
quantity. One thousand is given to the ten, so how much does
each unit receive? 100. The child distributes and changes as
necessary, 'one hundred to the ten, ten to the units, another
hundred to the tenand so on.' What did one receive? The child
records the problem in his notebook.
Control of error: The quantity may combine the quantities
distributed, count them, change them, to obtain the original
number.
Age: 67
Aim: to practice twodigit division
[top]
TRIPLE DIGIT DIVISION
a. Intro. to Triple Digit Division (on The Change Game)
Materials:
...golden bead materials and numeral cards
...ribbons: green, blue, (also red for later)
Presentation:
Given a division problem, the child can see that it would be
impractical to distribute one by one to over a hundred people.
The child representing 100 wears a red ribbon. When red (100)
receives 1000, blue (ten) will receive 100, and units (green)
will receive 10.
Control of error: The quantity may be added together
again, making the necessary changes to form the original number.
Age: 67
Aim: to learn the concept of threedigit division.
[top]
TRIPLE DIGIT DIVISION
b. Triple Digit Division on The Stamp Game
As in the previous presentation,
skittles are used. Here one red skittle represents 100.
[top]

xDivision Involving Zeroes 
DIVISION INVOLVING ZEROS
Materials: stamp game work cards
Presentation:
When the child places the stamps in the dishes to show the dividend,
the hierarchy that has zero is indicated by an empty dish. When
the child needs to change to that hierarchy, the procedure is
the same.
[top]
DIVISION INVOLVING ZEROS
Materials:
...stamp game
...work cards
Presentation:
The child lays out the first quantity in the dishes. This quantity
must be divided among 104. I don't need any tens skittles, but
so as not to forget the tens, we place a blue counter in its
place. The child begins distributing: 1000 to the hundred, the
ten would get 100, but I give it nothing, and the unit received
10and so on.
By this time the child should realize that any remainder must
be less that the divisor.
[top]

xGroup Division 
SINGLE DIGIT GROUP DIVISION
Materials:
...stamp game
...work cards (divisor up to 9)
Presentation:
The division problem says 24 divided by 4 =____. How many groups
of four are we able to make with this number. The four skittles
are placed in a bunch and groups of four units are in rows in
front of this group. Changes are made as necessary. When the
distribution is complete, how many groups of four were we able
to make from 24? The rows are counted. Six is the result of this
group division.
The child then does the same problem using distributive division
to see that the result is the same.
Age: 67
Aim: to understand a different aspect of division
Note: This is presented parallel
to abstract division. Memorization has begun.
[top]
DOUBLE DIGIT GROUP DIVISION
Materials:
...stamp game
...work cards
Presentation:
1. Given a division problem,
the child prepares the stamps and the skittles. Since we want
to do a group division, we put the skittles together in a group.
How many groups of 26 can be made from this number? The child
places two tens and six units in a row, continuing his distribution
by making all horizontal rows of 26 in a column. (The stamps
are always placed in hierarchic order) Here, the skittles serve
only as a reminder of the number in the group.
2. This time we will first
make groups only of tens. Groups are made of two tens and lain
out in rows. How many groups of ten did I make? So that each
group has 26, I must make the same number of groups of units.
Groups of six units are made in rows that line up with the groups
of ten, yet in a separate column. When the child finds that more
units are needed, one group of tens is returned to the dish so
that they may be changed. How many groups of tens do I have ?
4 How many groups of units? 4 is our answer.
Age: 7
Aims:
...to learn the concept of group division
...to continue towards further abstraction in division
[top]

xMore Memorization Exercises 
SPECIAL CASES
As for the other operations,
we examine the special cases using as a starting point the combination
that is most familiar. A chart will be constructed as follows:
0 Calculate the quotient
72 ÷ 9 = ? ( 72 divided by 9 gives me what number?)
1 Calculate the Divisor
72 ÷ ? = 8 ( 72 divided by what number gives me 8?)
2 Calculate the Dividend
? ÷ 9 = 8 ( What number when divided by 9 gives me 8?)
3 Inverse of Case ZeroCalculate
the Quotient
? = 72 ÷ 9 ( what number will we obtain by dividing 72
by 9)?
4 Inverse of The FirstCase,
Calculate the Divisor
8 = 72 ÷ ? ( We obtain 8 as a quotient when dividing 72
by what number?)
5 Inverse of the Second Case
Calculate the Dividend
8 = ? ÷ 9 ( We obtain 8 as a quotient when dividing what
number by 9?)
6 Calculate the Divisor and
the Dividend
8 = ? ÷ ? ( We obtain 8 by dividing a certain number by
another number. What is the first number and the second number?)
Note: Here we also see the
relationship between multiplication and division. In cases 2
and 5 the child must multiply to find the dividend.
[top]
SEARCH FOR QUOTIENTS
Materials:
...Chart II
...bingo tiles for multiplication
Presentation:
Have the child find one bingo tile to match all the dividends
along the top of Chart II. These are placed in a box cover or
something.
The child fishes for a tile, i.e., 24. Let's try to find all
the quotients with zero remainders that can be made with this
dividend. Start with 24 ÷ 9 =. It won't work so leave
it blank, and go on24 ÷ 8 = 3 and so on with the child
giving the correct quotients. At the end erase those that would
not yield zero remainders, thus leaving space to correspond with
Chart I. Notice how the column of quotients matches the column
under 24 on the chart.
These are the combinations I wanted, because now we can do 3
x 8 = 24. Write this to the right of 24 ÷ 8 = 3. 24 was
my dividend: now it is my product. Go on in the same way for
the other combinations making a second column.
The child will realize that if 24 ÷ 8 = 3, then 24 ÷
3 = 8. It is a sort of game where the numbers change positions.
Aims:
...to find quotients with zero remainders
...to realize the relationship between multiplication and division
memorization of division
Indirect Aim: indirect preparation for the divisibility
of numbers
[top]
PRIME NUMBERS
Materials: Chart I
Presentation:
Recall the child's attention to the numbers in pink on the chart,
which were called prime numbers, and which can be divided only
by themselves and 1. These are very important numbers because
they form all of the other numbers. We can see that this is true.
(refer to the chart) 1, 2, and 3 are prime numbers. 4 is not
a prime number, but is made of 2 x 2, and 2 is a prime number.
Go on to 6 which is not prime. It is made up of 3 x 2; 3 and
2 are prime numbers.
If you try to decompose any number, you will find that it is
made up of prime numbers. Invite the child to choose one of the
dividends, and think of one combination: 24 = 3 x 8. 3 is a prime
number, but not 8, 8 is made up of 2 x 4. 2 is a prime number,
but not 4, 4 is made up of 2 x 2. 2 is a prime number.
Try another combination of
24 to check.
24 = 6 x 4 neither 6 nor four is prime
6 = 2 x 3 both are prime
4 = 2 x 2 both are prime
Direct Aim: to realize the importance of prime
numbers
Indirect Aim:
...to prepare for divisibility of numbers, LCMleast common multiple
GCDgreatest common divisor and
...reduction of fractions to lowest terms
Aims:
...(for all of division) to memorize the combinations necessary
for division.
...to stimulate an interest that will help him to use the experiences
acquired previously.
[top]

xDivision With Racks & Tubes 
INTRODUCTION
The operations of addition,
subtraction and multiplication can be performed with the large
bead frame. But since the beads are connected to the wires, the
beads frame cannot be used for division. The hierarchic material
for division consists of loose beads.
Up to this point division has been done with the decimal system
material to give the concept, including division with a 2or
3digit divisor, and group division. These concepts were reinforced
with the stamp game. Following a research of the combinations
necessary for memorization, where the quotient was limited to
a maximum of 9. Division was dealt with indirectly in many of
the multiplication activities. Using this material the dividend
may have up to 7 digits and a divisor of 1, 2 or 3digits may
be used.
Maria Montessori referred to this material "as an arithmetical
pastime for the child." This work clarifies the analytic
procedure for the development of the operation. The fundamental
difficulty of division is obtaining the digits of the quotient,
recognizing their values and placing them in their proper hierarchical
position.
At this level more importance is given to the quotient, that
is, what each unit receives, and not so much to the quantity
to be divided.
[top]
MATERIALS: RACKS & TUBES
Materials:
...7 test tube racks: 3 white, 3 gray and 1 black
...each rack contains 10 test tubes
deposit each tube contains 10 loose beads
[These are the "deposit" from which quantities are
drawn. The racks are white for the simple class, gray racks for
thousands and 1 black rack of green beads for units of millions.]
7 bowls  1 to correspond to each rack:
dividend the exterior of the bowl corresponds to the color
of the rack the interior of the bowl corresponds to the color
of the beads The dividend is formed in these bowls, just as was
done with the stamp game.
3 bead boards: in hierarchical colors for units, tens, hundreds
For a 1digit divisor the green board is used
divisor For a 2digit divisor the green and blue boards
are used
For a 3digit divisor the green, blue and red boards are used.
[As in memorization, the distribution is done on the boards.]
Box with three compartments containing nine skittles of each
of the three hierarchic colors that represent the divisor.
Also: A large tray to hold the racks while they are not in use.
[top]
SMALL DIVISION (1DIGIT
DIVISOR, 4DIGIT DIVIDEND)
1st level
Isolate the racks that are needed to form the dividend. Place
the other racks on the tray. Pour the quantities into the respective
bowls. Place the green skittles on the green board for the divisor.
Begin distributing "bringing down" the units of thousands,
that is moving the rack and bowl closer to the board. After distributing
the units of thousands, record the first digit of the quotient
in its hierarchical color, reading the number at the left side
of the board.
Remove the beads from the board and place them back into the
tubes. There is one units of thousand bead remaining in the bowl,
which can't be distributed as is. Change it for 10 hundreds (pour
the hundreds into the hundreds bowl). Having finished with the
units of thousands, place the rack and the bowl out of the way
on the tray.
Continue in the same way for hundreds, tens and units.
9764 ÷ 4
= 2441 
2441 
49764 
Note: Here also the operation
is reduced to the level of memorization.
Recall the problem 81 ÷
8 = which couldn't be done before. This does not mean that it
couldn't be done, just that it couldn't be done with that material.
Try it using this new material.
It is important to emphasize that every time a hierarchy is considered,
a digit must be placed in the quotient. If there are not enough
beads to distribute, we must still record a zero. This is where
the child could easily make a mistake.
2nd level
Set up the problem as before
and begin distributing the units of thousands. Record the first
digit of the quotient using the hierarchical color green. There
are no beads in the bowl, so the remainder is zero. Write the
remainder under the 9. Put the thousands away.
Bring down the hundreds, that is, move the bowl closer to the
board and write the 2 next to the 0. Since we can't distribute
these two beads, write the next digit in the quotient and write
the remainder. Remove the beads.
9216 ÷ 3
= 
30 
39216 
02xnnnnnni 
02xi 
02xinnnnnn 
02xi 
The two hundreds must be changed
for tens. Put away the hundreds. Bring down the tens. Now we
must distribute 21 tens. Continue in this way.
9216 ÷ 3
= 
3072 
39216 
02xnnnnnni 
02xii 
021innnnnn 
021ii 
06xnnnni 
06 
0xnnnni 
0 
Note: It is important to work through a problem such as 1275
÷ 3 = to demonstrate the grouping of the first two digits.
In a case such as this we do not record a zero in the quotient
for the first hierarchy
[top]
SMALL DIVISION
3rd LevelGroup Division
Note: The child will never
reach abstraction using the distributive division technique.
Introduction:
Recall the concept of group division. Take out 15 loose golden
beads. Invite the child to find out how many groups of three
can be formed.
Relate word problems to demonstrate the difference between distributive
division and group division:
1. I have 12 pencils which
I must give to 6 children. How many pencils will each child get?
2. What kind of division did we do? distributive division.
2. I have 25¢. I want to buy pencils costing 5¢ each.
How many pencils may I buy? 5. This time I had to think of how
many groups of 5 are in 25. This is group division.
Note: The Difference here is
mostly in language, for this is an important step in the development
toward abstraction. At this level the child incorporates the
other operations in a conscious way. The quotient is no longer
written in the hierarchical colors, because by this time, the
concept should be firm in the child's mind.
Set up the materials as before.
We must see how many times this group of 5 (indicate the skittles)
is contained in this 7 (the units of thousands). Distribute the
beads. Record the quotient. One group of 5, that is 5 x 1 = 5.
Write 5 under the 7 and subtract. This is our remainder. Check
to see if the number of beads in the bowl matches the difference.
Change the remaining two units of thousands to hundreds and bring
down the hundreds.
Now we must find how many groups of 5 are contained in 26.
Age: from 6  7 years ( at
7 years old, the child should reach abstraction)
Note: When the child has reached
abstraction of division, he has actually reachedabstraction for
all the operations, since division involves all of the operations.
Before progressing to Big Division (having a divisor of more
than one digit) the child should have reached abstraction with
small division, that is, without the materials.
[top]
BIG DIVISION (TWO OR MORE
DIGITS IN THE DIVISOR)
Introduction:
Recall with the child the presentation of the concept of division
with a twodigit divisor, using the decimal system materials
and the arm ribbons. Introduce the bead board for tens. Recall
that each blue skittle represents 10 units. If I give 10 to the
blue skittle, what must I give to the green skittle? After this
concept is already recalled, begin division.
1st level:
Set up the material as before.
Bring down the tens and units of the thousands, one for each
board. Distribute the beads: one 10,000 for the tens; one thousand
for the units. The first digit of the quotient is 1, but 1 what?
The result is what one unit receives, so it is one thousand.
Record the digit in color.
37,464 ÷
24 = 1 
1 
2437464 
Remove the beads from the board.
Change the 10,000 to ten units of thousands and out away the
ten thousands. Move the rack and bowl of units of thousands to
the left, to the tens board. Bring down the hundreds. Distribute.
When the bowl of the lesser of the two hierarchies being considered
is emptied, continue changing and distributing. However, when
the bowl of the greater hierarchy is emptied, we must stop, record
the digit of the quotient, and move on to another hierarchy.
37,464 ÷
24 = 1561 
1561 
2437464 
2nd level:
As for the second level of
the small division, record the remainder and bring down the digits
of the dividend.
3rd levelGroup Division with
a two digit divisor
Recall the meaning of group
division in the same way as before.
Set up the problem as usual.
Bring down the first two hierarchies. How many times is 24 contained
in 88? First, we must find how many times 2 is contained in 8.
Distribute the beads 4 times. Now we must see if 4 is contained
in 8, 4 times also. Distribute the beads. It doesn't work. Take
off one row of beads from the board and place them in the bowl.
Change a thousand bead to 10 hundreds and distribute.
Since we want to make groups of 24, there must be the same number
of groups of 2, as there are groups of 4. Thus 3 groups of 24
were made. 3 what? Refer to what one unit received  3 hundreds.
Multiply 3 x 24, carrying mentally and recording the product
beneath 88 in the dividend. (The number that we had to carry
in the small multiplication, corresponds to the number of changes
that were made while distributing.) Subtract; the difference
should match the quantity that remained in the bowls. Remove
the beads. Change. Bring down a new hierarchy and continue as
before.
[top]
LONG DIVISION WITH A THREE
DIGIT DIVISOR
Note: If the child has reached
abstraction in division with a 2digit divisor, he will encounter
very little difficulty here because the mechanics of the operation
are the same. Thus, the material will be used less by the child.
The material is used for the presentation to be certain that
the child has understood the concept. At this level, group division
is used immediately.
Presentation:
Recall the activity done with the decimal system material and
arm ribbons. Note the difference that the centurion received
over the decurion and the unit. Present the materials.
56,438 ÷
234 = 241 r.44 
241

r. 44 
23456438


The procedure follows the pattern
set down previously, now using 3 bead boards, and bringing down
3 hierarchies at a time. Remember that the first digit tells
what all of the others must receive: How many times is 2 contained
in 5? 2 We must see if 3 is contained in 6 2 times and if 4 is
contained in 4 2 times.
[top]
DIVISION WITH ZERO IN THE
DIVISOR
51,252 ÷ 207 =
Recall the similar case in
the stamp game where a counter took the place of zero for the
skittles. Here the board without any skittles reminds us of the
zero. Move down the three hierarchies one for each board. The
hierarchy by the empty board reminds us of what the tens would
receive if there were any. Distribute as before using group division.
19,293 ÷ 370 = 676 ÷
300 =
Make the child conscious of
what the units would have received, if there were any , in order
to determine the value of the digit of the quotient. Hierarchic
colors can be used for recording the quotient.
Dividend: 70,569 ÷ 229
=
Place the dividend in the bowls,
leaving one empty. Do not put it back on the tray since it will
be needed for making changes. Bring down the hierarchies as usual,
ignoring the fact that one bowl is empty; that hierarchy corresponds
to one of the digits in the divisor.
Age: 9 years (by 9 1/2 the child should reach abstraction)
Aims:
...mastery of long division
...knowledge of the reasons for every aspect of the procedure
[top]

xWord Problems 
FROM COMBINATION CARDS
The teacher prepares seven
special combination cards and mixes them with the other 21. The
child fishes for one, solves it, writing it down substituting
the ? for the answer in red.
Also, word problems are prepared and mixed with the others. The
child copies the text of the word problem, writes an equation
with the answer in red, and writes a conclusion, that is, a complete
sentence which answers the question stated in the word problem.
Aim: further understanding of the concept of multiplication
[top]
DISTRIBUTIVE VS. GROUP DIVISION
These word problems are to
aid the child's understanding of the difference between distributive
division and group division:
Example: (distributive) Mother
has 24 cookies. She wants to give these out to her three children
equally. How many cookies will she give to each child?
Example: (group) Mother has
24 cookies. She wants to make up packages with three cookies
in each package. How many packages can she make?
[top]

xDivisibility 
DIVISIBILITY BY 2
Materials:
...decimal system materials (wooden)
...blackboard or blank chart
Presentation:
Write a number and invite the child to bring that quantity with
the materials. Try to make two equal groups from this quantity,
making changes as necessary .If it is possible to make two equal
groups, write "yes" next to the number and underline
the last digit. If not, write "no". Add or subtract
one unit, and repeat the process. Examine many numbers in the
same way. At a certain point the child will realize the rule:
When the last digit is an even number
or zero, the number is divisible by 2.
If the child does not reach this point of consciousness on her
own, ask questions to call her attention to the relationship
between the yes or no and the oddness or evenness of the last
digit.
1126 yes
1125 no
1124 yes
1123 no
1122 yes
78 yes
79 no
80 yes
12 yes
[top]
DIVISIBILITY BY 4
Materials: Same as above
Presentation:
The procedure is the same as before, except that the last two
digits are underlined. Now it is no longer a matter of oddness
or eveness.
Rule: When the last two digits are divisible by 4, or they are
both zeros, the number is divisible by 4.
816 yes
817 no
818 no
819 no
820 yes
[top]
DIVISIBILITY BY 5
Materials: Same as above
Presentation:
The procedure is the same as before, only the last digit is underlined.
Rule: When the last digit is 5 or 0, the number is divisible
by 5.
125 yes
126 no
127 no
130 yes
45 yes
100 yes
[top]
DIVISIBILITY BY 25
Materials:
...ten bars and 40 or more golden unit beads
...small white square pieces of paper,
...pen
Presentation:
Put out one group of 25 with a little card over it. Place another
group next to it in an inverse position to make it easy to visualize
the group as 50. Place a little card over it ("50").
Continue placing groups of 25 (7 tens, 5 units for 75) to the
left of the previous group, the respective card is placed over
the top. Continue up to 8 groups of 25, substituting 100 squares
after 100.
Invite the child to speculate
on the next few multiples of 25.
Rule: When the last two digits of a number are 23, 50, 75 or
two zeros, the number is divisible by 25.
[top]
DIVISIBILITY BY 9
Materials:
...Peg board
...Box of pegs in hierarchic colors
...Small white square pegs
...pen
Presentation:
9 is a very important number, Why? It is the last digit that
can remain loose in our system. It is the square of 3, and 3
is a perfect number. Therefore in this work we will consider
9 as the square of 3.
On the peg board use 9 green pegs to construct a square of 3
by 3. This is a square of 3. Dissolve the square into a column
at the top left corner of the board, labeling it 9.
Form two more squares of 3 side by side. Remove one unit from
one square and add it to the other. Now one group has ten, change
the ten green unit pegs for one blue ten peg. Dissolve the pegs
into two columns next to the previous column. Label 18.
Repeat the procedure with 3 squares. Add one peg to each of the
first two squares, taking the pegs from the last square. Change
each group of 10 to a blue peg and dissolve. Continue in this
way up to 10 squares = 90.
Observe that as the units decrease, the tens increase. At one
extreme there are 9 units, at the other, 9 tens. This special
pattern exists only in the table of 9. The sum of each pair of
digits is 9. i.e. 27, 2 = 7 9.
Rule: A number is divisible by 9 when the sum of its digits equals
9 or a multiple of 9. If a number is divisible by 9, it is also
divisible by 3.
Note: This characteristic may
have been noticed in the multiplication booklet for memorization.
[top]
PROOF WITH MULTIPLICATION
WITH RULE OF DIVISIBILITY BY 9
Materials: Decimal system materials
Presentation:
Take the thousand cube and try to make 9 equal groups. 1000 is
not divisible by 9, but take away one unit and try again. 999
is divisible by 9. Write 1000 1 = 999. Repeat the procedure
for 100 and 10. Write 100 1 = 99, 10 1 = 9. Choose
a number: 5643. Convert it to expanded notation:
3..................................................................................r.3
40 = 4 x 10............10 1 = 9...........40
 4 = 36..............r.4
600 = 6 x 100
.......100 1 = 99 .......600  6 = 594...........r.6
5000 = 5 x 1000
...1000 1 = 999 ....5000  5 = 4995....+
r.5
.................................................................................... 18
10 1 = 9, but we have
4 tens. We must multiply the whole equation by 4, which gives
us 36. 36 from our original 40 leaves us 4 as a remainder. Continue
for the others. Add all the remainders in the end. Since 18 is
divisible by 9, the whole number is divisible by 9. (cont.)
Show an example of multiplication: 643 x 1527 = 981,861. Make
sums of 9 in the digits  643 x 1527 = 981,861. Total the remaining
digits (4 from the multiplicand, 1 and 5 from the multiplicand)
multiply the sums ( 6 x 4 = 24 ), and add the digits of the product
( 2 + 4 = 6 ). This number should correspond to the sum of the
remaining digits in the original product  6. 6 is also the remainder
when the original product ( 981,861 ÷ 9 = 109,095 r. 6)
is divided by 9. The "remaining sums" from above multiplicand
and multiplier (4 and 6) will also be the remainder when divided
by 9: ( 643 ÷ 9 = 71 r. 4 and 1527 ÷ 9 = 169 r.
6).
Age: Ten years
[top]

xSynthesis of Multiples & Divisibility 
SYNTHESIS OF MULTIPLES AND DIVISIBILITY
Refer back to the multiples
work and the charts constructed to find the multiples of numbers
up to 10 (circling the numbers in different colors). Repeat this
work making new observationsi.e. A number is a multiple of (is
divisible by) 6 if it is also a multiple of 2 and 3. This is
noticed when the charts for 2, 3, and 6 are done simultaneously.
A multiple of 6 intersects the lines of multiples of 2 and 3.
This work really shows the close relationship of multiples and
divisibility. Knowledge of one reinforces the other.
Take Table C with the prime factors. Here also we can find, for
example, by what numbers 18 is divisible, by making all possible
combinations of the prime factors:
18 = 2 x 3 x 3.
18 is divisible by 2, 3, 6, and 9.
18 is even, thus it is divisible by 2.
1 + 8 = 9, thus 18 is divisible by 9, which means it is also
divisible by 3.
Since 18 is divisible by 2 and 3, it is also divisible by 6.
[top]



