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xMath :: 6-9 :: Numeration
xTable of Contents:
  • Introducton
  • Numeration
    • Quantities and Symbols
      • Quantities in the Decimal System
      • Numerals (Symbols)
      • Union of Quantities and Numerals (Symbols)
      • Additional Exercises in Numeration
        • The Hundred Board
        • The Seguin Boards
          • Teen Boards
          • Ten Boards
    • Introduction to Operations Using the Change Game
      • Static Operations in the Decimal System
        • Presentation of Addition:
        • Presentation of Subtraction:
        • Presentation of Multiplication:
        • Presentation of Division:
      • Dynamic Operations in the Decimal System
        • Introduction to the Change Game:
        • Presentation of Addition
        • Presentation of Subtraction
        • Presentation of Multiplication
        • Presentation of Division
    • Golden Bead Chains
      • Chain of 100
      • Chain of 1000
    • Hierarchical Material
      • Introduction:
      • Presentation:
      • Games:
    • Introductions to Other Mathematical Materials
      • Stamp Game
      • Hierarchical Bead Frames
        • Introduction:
        • Small Bead Frames
          • First Presentation:
            • Presentation:
            • The History of The Abacus
            • Introduction to the materials
            • Passage from sensorial to symbolic representation
          • Numeration Based On Position
        • Large Bead Frames
          • First Presentation
          • Numeration According To Position
            • Presentation:
            • Exercises: Formation of Numbers
        • Horizontal Golden Bead Frame
    • Introduction to Memorization
  • Addition
  • Multiplication
  • Subtraction
  • Division
  • Fractions - COMING SOON
  • Decimals - COMING SOON
  • Pre-Algebra - COMING SOON


xQuantities and Symbols


...the golden bead materials which consist of: container of loose gold beads representing units box of gold bead bars of ten beads each box of 10 gold bead squares of ten bars (representing 100) box containing 1 gold bead cube of hundred- squares (representing 1,000)
...a large tray with a dish or smaller tray, used for transferring the quantities

Individual Presentation. As a unit bead, and then a ten bar is placed on the table, the child is asked to identify the quantities. One hundred and one thousand are presented also. The teacher gives a three period lesson naming the quantities: unit, ten, hundred, and thousand. The child is then invited to examine the materials and their composition. The child may count the ten beads on the ten-bar again. "The hundred is made up of ten ten bars". The ten-bar is placed on top of the square as the child counts. "The thousand is made up of 10 hundreds". The hundred-square is placed next to each section of the cube as the child counts. The teacher gives the three period lesson defining the composition of the quantities.

Small Group exercise. The golden bead materials, now including the wooden hundred-squares and thousand-cubes are arranged at random on a rug (in a basket). Each child takes a tray. The teacher asks the child to bring a quantity. 'Bring me 3 hundreds' As each child returns with the quantity, the child identifies it, and the teacher and child count it together. At first the child is asked to bring only one hierarchy at a time. Later he will bring all four at once.

Age: 3-6

Direct Aim: develop the concept of the hierarchical orders of the decimal system: units, tens, hundreds, thousands. give the child the relative measurement of the quantities: bead, bar, square, cube.

Indirect Aim: prepare the child for geometry concepts: point, line, surface and solid.


decimal system numeral cards:
...1-9 printed in green
...10, 20...90 printed in blue on double-sized cards
...100, 200...900 printed in red on triple-sized cards
...1000, 2000...9000 printed in green on quadruple-sized cards

Presentation: 1st Part
Individual Presentation. As the one and the ten cards are placed on the table, the child reads them. One hundred and one thousand are presented in a three period lesson. The cards are arranged as in the diagram. Then the child examines the particular characteristics of each numeral, its color and the number of zeros.

1. The cards are turned face down on the table. Without turning the card face up, the child identifies the numeral indicated by the teacher. How many zeros does it have? The card is turned up to control. Another time, the teacher asks the color of each numeral.
2. 'Magician'. The teacher picks up the four cards arranging them in a pile weighted to the left. This arrangement is shown to the child. The cards are stood on end as the top cards slide into the second position. Where did all the zeros go? They seem to have disappeared, but they are still there. The cards are lifted one by one to reveal the zeros. The child performs the magic trick.

Presentation: 2nd Part:
The first four numeral cards, just previously presented, are lain in order. The remaining unit cards are placed in a column below one, the child being encouraged to read each as he lays it in position. This continues for the tens (one ten, two tens...), hundreds (one hundred, two hundred...), and thousands (one thousand, two thousand...). The three period lesson continues noting color and number of zeros as well. If the child is familiar with the names, twenty, thirty..., these may be supplemented. It is important for the child to realize that twenty (20) is two tens.

Age: 3-6

Direct Aims: understand the orders of the decimal system. turn the numerals for each of those four orders

Indirect Aim: to understand the importance of zeros in distinguishing the numerals.


...golden bead materials
...numeral cards 1-9, 10-90, 100-900 and 1000

As the teacher lays out the unit beads, the child counts: 'one unit, two units...nine units.' The teacher goes on: 'If we added one more unit, we'd have ten units. Ten units make one ten.' The tens are counted as they are lain out: 'one ten, two tens... nine tens.' 'If we add one more ten we'd have ten tens. Ten tens make one hundred.' And so on up tone thousand. Here the rule of the decimal system is stated: Only nine quantities can remain loose. When we reach ten, we move to a superior hierarchical order.

1. The teacher places the numeral cards (as in the diagram) on one table and the quantities on another. The teacher places one quantity on a tray. The child finds the corresponding numeral card and places it on top of the quantity. The teacher controls.
2. The teacher places a numeral card on a tray. The child brings the corresponding quantity.

Subsequent Presentation:
Group Presentation: The teacher places cards of different orders on the tray. The child brings the corresponding quantities with the cards placed on top. The teacher controls and hands the cards back to the child. When the child has all of the numeral cards, he does the magic (arranges the cards) and reads the numeral. The exercise continues omitting one hierarchical order to show that the place is held by zeros.

Age: 3-6

Direct Aim: understand the rule of the decimal system: only nine quantities can remain loose. familiarize the child with the hierarchical orders offer the opportunity to write complete numerals

Indirect Aim: give the understanding that zero occupies the place of a missing order.

Note: With these and all other activities involving the golden bead material, the units should remain in the small tray. This confines the loose beads in a set and makes it easier for the child to see that he has nine, one more would make ten. When counting, the beads may be dumped into the palm and counted back into the tray.


The Hundred Board


The Seguin Boards

Teen Boards

Materials: containing two boards and 9 wooden tablets for 1-9 of ten golden ten bars of 1 each of colored bead bars 1-9

Individual presentation. The teacher presents the boards side by side and the tablets ordered in a row. Indicating the first slot, the child reads the numeral 10 and places a ten-bar to the left of that slot. The teacher then adds a unit bead and the tablet - 1 to make eleven. 'This numeral is eleven: eleven is ten and one.' This continues through nineteen. When counting the beads the child counts ' ten, eleven, twelve... ten and two is twelve.' Three period lesson follows naming the quantities and in the second period forming them.
If the child questions why the last slot is blank, explain that in order to make the numeral that comes after nineteen, other materials are needed.

Age: 3-6

Aims: clarify understanding of the decimal system (11 means: 1 ten and 1 unit ) progress in counting from 10 up to 19 learn the names of numbers 11-19


The Seguin Boards

Ten Boards

Materials: containing two boards with numerals 10, 20, 30....90, and 9 wooden tablets for 1-9 of 9 gold unit beads of 45 gold ten-bars
...1 golden hundred square

Individual presentation. With these materials we will be able to make the numeral that was missing from the teen boards.
a) Only the boards and ten-bars are used for now. Pointing to the first numeral 10, the child is asked to identify it and place the correct quantity next to it. The child identifies the next numeral 20 as two tens. We call this twenty. The ten-bars are placed next to twenty, and counted 'ten, twenty.' This continues, identifying numbers by correct names and counting the ten-bars by 10's. Now we have counted by tens up to ninety. The three period lesson follows.
b) The ten-bars have been returned to their box. Again the child identifies 10 and brings out one ten-bar. After ten is eleven: the one tablet is placed in the slot and one unit bead is added 'ten, eleven.' This continues up to nineteen. After nineteen is twenty: Twenty is two tens, so we put away the nine unit beads and take another ten-bar. Both ten-bars are moved down by twenty. This one-by-one counting continues up to 99. If we added one more bead, we'd have 10 units which make another ten-bar. Then we'd have ten ten-bars which makes one hundred. After 99 comes 100. The hundred square is placed next to the blank space.

Age: 3-6

Aims: clarify understanding of the decimal system (11 means 1 ten and 1 unit ) count from 1 to 99 learn the names of numbers 20-99

Note: These materials may be presented any time after the Union of Quantities and Numerals of the Decimal System.


xIntroduction to Operations Using the Change Game


...golden bead materials including wooden hundred squares and thousand cubes
...large numeral cards
...three sets of small numeral cards
...a box containing symbols for operations +, -, x,÷
...small pieces of paper
...a thin rod to be used for the = line
...a soft cloth.

a. Presentation of Addition:
Small Group Presentation. Each of two or three children takes a tray. The teacher states a different numeral for each and they find the appropriate small numeral cards and the quantity, placing the cards on top of the respective quantity. The teacher controls. The child arranges the cards, places the numeral on the table and dumps the quantity on the cloth. When all the quantities are on the cloth, the teacher gathers up the cloth, mixing all the quantities together. The cloth is opened and the materials are sorted. The child begins with units counting the quantity and bringing the large numeral card. When all has been counted, the child arranges the cards and reads the quantity that the combination has produced. Pointing to small numeral cards: 'The children brought these small quantities. When we put them together we made this large quantity." (indicating the large numeral cards which is seperated from the addends by the thin rod) 'We have done addition.'
The numerals are arranged in a column. The plus sign and its function is presented. The line (which was formed by the thin rod) is equivalent to the = sign. The teacher reads the problem (equation) '2,512 plus 1,234 equals 3,746.'

b. Presentation of Subtraction:
Group Presentation: Initially the teacher may play the "Rich Man, Poor Man" game to demonstrate the concept of "taking away." The teacher has a large quantity from which several children take away small quantities until there is nothing left. The purpose of this game is to make the impression of taking away and nothing remaining.
The child has an empty tray. The teacher has a large quantity on his tray. The quantity is counted beginning with the units and large numeral cards are placed on the quantities. The child arranges these cards and reads the numeral. Offering the child some of this large quantity, the teacher chooses some small numeral cards. The child arranges these cards and reads what shall be taken away. The teacher counts out this quantity from what is on the tray, beginning with units. What is left? This quantity is counted and small numeral cards placed on the quantities, arranged and read. What remains on the tray is the result of subtraction. When we take away, we are subtracting. The problem is set up with the minus sign and read. The large cards tell us the large quantity; the smaller cards are for the small quantity that was taken away and the small quantity that remains.

c. Presentation of Multiplication:
Group Presentation: Each child is given a tray and is asked to get the cards and quantities for a stated number. The teacher controls each child's tray; the cards are arranged, the numeral is read and the quantity is placed on the table. As in addition the quantities are put together, sorted, counted, labeled and the sum is read. The problem is then set up as in addition with the plus sign.
Now it is observed that in this 'special' addition, all of the quantities put together (addends) are the same. This special addition is called multiplication. Taking one small numeral : 'We can say that we took this quantity three times.' The times sign is presented and the numeral three is written on a blank piece of paper. The result has not changed; this is just an easier way to write the problem.

Note: After this initial presentation, the child no longer sets up the addition problem first.

d. Presentation of Division:
Group Presentation: The children are seated in a circle. One child is asked to pick up the large numeral cards for the stated quantity, and he brings the golden bead material. 'This large quantity must be distributed to each of these other children equally. 'Starting with the thousands, one thousand for you, one thousand for you, another thousand for, another thousand for you'... until all of the quantity has been distributed. The children who received count their quantity to be sure that everyone received the same amount. One child is asked to get the small numeral cards. It is emphasized that each child received this amount. When we distribute equally to many others, we divide. The division problem is set up, using a small piece of paper for the divisor, and it is read. The result of division is what one child receives.

After each problem has been demonstrated and set up with numeral cards and symbols, the child may write this in his notebook, preferably on paper with columns and in colors for the hierarchical orders.
After all of the operations have been presented, it is important for the child to understand the function of each operation. 'What is addition?... putting together...etc.

Age: 3-7

Control of Error: The teacher checks the quantities counted.

Aims: realize the concept of addition (putting together), subtraction (taking away), multiplication (adding the same number many times), and division (distributing equally)


...golden bead material
...large and small numeral cards
...symbol cards for the operations
...problem cards for each operation

a. Introduction to the Change Game:
Individual Presentation. A large quantity is placed on the tray and the child is invited to count it. Beginning with units, the child counts, but is stopped at 10. Ten units cannot remain loose; they must be changed for a ten-bar. The ten beads are traded for one ten-bar from the bank. The child continues counting units and placing the correct large numeral cards on the try. So on to thousands. The cards are arranged and read. The child does many exercises.
Aim: to exchange equal quantities of different hierarchies
to reinforce the rule: only 9 units can remain loose
to reinforce knowledge of the composition of each hierarchy (ten tens=100)

b. Presentation of Addition:
The teacher reads a task card. The child performs each command as it is read. The teacher controls.

c. Presentation of Subtraction:
The teacher reads a task card. The child performs each command as it is read. The teacher controls.
The teacher presents the thousand cube (golden bead) and wants to take away 1 unit. This may be symbolized with the large and small numeral cards for emphasis. How can this be done? The thousand is changed to 10 hundreds. Now can we take away one unit? Not yet. So on until one unit can be taken away. The remaining quantity is counted and represented with small cards.
Aim: to realize that one unit revolutionizes a large quantity.

d. Presentation of Multiplication:
As for addition task cards are prepared.

e. Presentation of Division:
Group Presentation. As with static division the child sets about distributing. When he finds that he doesn't have enough for one hierarchy to go around, he must exchange for a lesser hierarchy.
When there is a remainder, the corresponding small numeral cards are brought and placed after a small card with the initial r to the right of the result (quotient)


Age: 4-7

Aim: further understand the concept of addition, subtraction, multiplication, and division


xGolden Bead Chains


...a chain formed of 10 ten-bars
...a hundred square envelope containing: 9 units arrows 1-9 in green, 9 tens arrows 10-90 in blue, a red hundred arrow

The chain is folded like a fan to resemble a hundred square. Do you recognize this? It looks like 100. We prove that it is 100 by placing the hundred square on or beside the folded chain. The chain is stretched out to its full length. How many tens are there in this hundred square? How many tens are in this chain? The square and the chain are exactly equal.
The child begins counting the beads placing the corresponding arrows by the bead. At 10, he begins counting by tens to 100. The red hundreds arrow and the hundred square are placed by the last bead.

1) The unit arrows are removed and the tens arrows are turned over. The child counts by 10's to 100, and then backwards by 10's.
2) The teacher asks the child to indicate a number on the chain. Then pointing to a bead, asks, 'What is this?'

Age: 3-6

Aims: represent one hundred in a line learn numeration from 1-100 count forwards and backwards by 10, s from 1-100


...a chain of 100 ten-bars with a ring after every 100 beads envelope containing: 9 green units arrows, 9 blue tens arrows 10-90, 9 red hundreds arrows, and 1 green thousands arrow
...10 hundred squares
...1 thousand cube

The chain is stretched out to show the difference between this chain and the chain of 100. It is folded like a fan to resemble hundred squares. It is proven that there are ten hundreds in this chain by placing the hundred squares on top of each section. The hundred squares are then stacked up to prove that this chain is equal to the cube. After this correspondence has been firmly established, the child begins counting, first by units, matching the arrows, then by tens, and lastly by 100's. At each hundred the child places a hundred square. At 900 the child counts by 10's again to 990. The child counts by units from 990 to 1000. We place another hundred square here, but now we have 10 hundreds. Ten hundreds make one thousand, so we can put the cube here instead.

1) The child counts by 100's to 100, forwards and backwards from 1 to 1000, with the arrows overturned
2) The teacher asks the child to point to a number on the chain. Then pointing to a bead, the teacher asks, 'What is this?'

Age: 3-6

Direct Aim: count forwards and backwards by 10, s and 100's to 1000

Indirect Aim: prepare for learning the powers of numbers


xHierarchical Material


These materials are the geometric representation of the quantities from one unit to one million-the powers of ten; 100 to 106 Having reached one million the child will easily imagine the succeeding hierarchies.

...the wooden materials made of light wood to facilitate movement, in relative proportions:
...1 - green cube - .5cm
...10 - blue rod with green lines .5 x .5 x 5cm
...100 - red square with blue lines .5 x 5 x 5cm
...1000 - green cube with red lines 5cm
...10,000 - blue rod with green lines 5 x 5 x 50cm
...100,000 - red square with blue lines 5 x 50 x 50cm
...1,000,000 - green cube with red lines 50cm
...numeral cards 1; 10; 100; 1,000; 10,000; 100,000; 1,000,000 all on white backing with numerals printed in black
...a ruler or stick or an expensive laser pen

The materials should be laid out in a row as they are presented, from right to left. Isolate the unit cube and identify it. This is one. If I had ten of these little cubes and placed them end to end, I would have this rod. This is ten. Place the cube along the side of the rod to count the ten sections. If I had 10 of these tens, I would get one hundred-this square. Count the sections of the square using the rod. Place the three pieces on the table in a row, and place the ruler on top. These all have the same height. Identify the three pieces again - 1, 10, 100, unit, ten, hundred. They are numbers of the simple class. Set the stick aside.
Isolate the thousand cube. This is still a unit, but it is a unit of the thousands. Compare its color and shape to the unit cube. Present the ten and hundred as before. Place the stick on top to see that they are all the same height. They are 1, 10, 100 but of thousands.
Present the million cube. This is still a unit, but it is a unit of millions. Imagine the ten of millions. It would be as long as ten of these side by side. Imagine also the hundred, which would be made of ten of these tens. These would make up the class of millions.
Review the first period giving the names of the classes and the names 1, 10, 100 for the cube, rod, square, to show how these three orders are repeated in each class. The dominant figure of each class is the cube for it gives us the name of the class. Compare these materials to the concepts of point, line, surface and solid-which is only a point of the next line. The point is represented bigger each time. Even the Earth, as big as it is, is just a tiny point in space. Three period lesson

Give the child the symbols to match by placing on top of the material. Identify for the child the symbols of 10,000 · 100,000 · and 1,000,000. Notice that the comma corresponds to a change in hierarchical class.

Distribute the cards to a group of children and they place the cards on the appropriate material. Give each child a piece of material and he finds the right card.
Ask the child to identify a piece of material, the class to which it belongs, the reason for the color of the lines, of what it is composed. In this way the child will be able to form definitions in his own words. Emphasize that the superior hierarchy is always formed of 10 of the preceding hierarchy.
The child draws the material in his notebook or cuts and pastes the pieces using a different scale of measurement. The cube is drawn as a three-dimensional image.

Age: 7 years


xIntroduction to Other Mathematical Materials


...wooden stamps of four types: unit stamps printed with the numeral 1, tens stamps printed with the numeral 10, hundred stamps printed with the numeral 100, and thousand stamps printed with the numeral 1000 with three compartments each containing 9 skittles and one counter in the hierarchic colors;
...four small plates

One of each stamp is presented and identified, lain in correct order-units t the far right; thousands to the left. The teacher forms a number laying out the stamps in a straight column for each hierarchy. 'Can you read this to me?' Now the child reads a number from a slip of paper and forms the number with the stamps. After the child has done many exercises of this type, he will be ready to go on to operations.



In this work, which follows memorization, the child encounters a new difficulty. He must identify quickly the value of each digit of a number as it is indicated by the place the digit occupies. The child considers the position of the digits in a number, and determines the value of each digit according to this position.
The decimal system material: the bead-1, bar-10, square-100, and cube-1000, represented constant values, values which did not change when the position of the material changed. On the bead frames beads of the same size represent the various quantities, thus eliminating the sensorial element of size. The quantities are symbolically represented on the bead frame aided by the hierarchic colors and the relative positions of the frame.
These hierarchic colors have been encountered before in the decimal system material numeral cards and the stamp game. On the frame one blue bead represents ten (unit) beads of the previous hierarchy and one-tenth of a (hundred) bead of the next superior hierarchy.


First Presentation:

...small bead frame, corresponding form
...a golden unit bead, 10-bar, 100-square, 1000-cube
...2 green beads (from the unit division board)

Introduce the child to the concept of hierarchy with an analogy: i.e. the social organization differentiates one person from the next. The same thing happens in the beads, These 4 beads could be units of the simple class or units of the thousands depending on their position.
These three colors, green, blue, and red are repeated in each class in the same sequence. Only their position will differentiate them.
Isolate two green loose beads. How many are there? 2 On the frame isolate one unit and one thousand bead. Here I also have two green beads, but I can't call them just '2 green beads.' The one at the top has the value of one; the bead on the lower wire has the value of 1000. The position makes the difference.
The absolute value is the value of the unit independent of its position (the number of beads on the wire). The relative value is the value of a digit when its relative position is taken into consideration.

The History of The Abacus
Relate the story of the abacus: A bead frame like this is used by children all over to learn to count. It is a very, very old instrument, that was used by the Chinese as far back as 500 BC. They called it 'swan-pan'. The Japanese caught on to the idea, but they called it 'soro-ban.' The Russians learned about it and began to use it in their country, calling it 's-ciot', which means calculator. Around 1812 there were French prisoners in Russia who learned about the abacus. When he was released he brought the idea back to France. This knowledge spread rapidly around Europe and to America.
Studies have shown that this design originated long, long ago. People made little grooves in the sand and placed little pebbles into the grooves. Each groove was like one of our wires, and the pebbles were like our beads.

Introduction to the materials
Our bead frame has four wires; the first three are equal distances from one another, and between the third and fourth there is a greater distance. This space separates the simple class from the class of thousands. On the right side we see these two classes indicated by two different colors. There are ten beads on each wire. The number on the left side of the frame indicate the value of each bead on that wire. Here it says 1, so each bead has the value of one... and so on to 1000.
On this form the same situation is repeated. Turn the bead frame on its side to demonstrate the corresponding colors, names of classes, and the space to divide the classes, which has been replaced by a comma.

Passage from sensorial to symbolic representation
Isolate the golden bead, and ask the child to identify its value; one unit. Isolate one green unit bead on the right side of the frame. This green bead is also one unit. Each bead on this row is a unit.
Isolate the ten-bar, and ask the child to identify its value; ten. This blue bead also has the value of ten. Each one of the beads on this row is worth ten units. Continue in the same way with the square and the cube.
By means of the three period lesson: have the child match the corresponding quantities, i.e. Give me 100. The child gives the square. Now show me 100 on the frame, or pointing to a particular bead: What is this? The child names it and gets the corresponding golden bead material.
To check the child's comprehension, isolate one unit bead and one thousand bead. These two beads are both green: do they have the same value? Why?


Numeration Based On Position

...small bead frame
...form for each child

Moving one unit bead to the right, the teacher counts one and writes the digit on the form. The numeration continues: move a bead, say the number, write it down. As the tenth unit bead is moved forward: I change these ten units for one ten bead forward. Write the digit 1 on the blue line. Move another ten bead forward- 2 tens and write a 2 in the column. The numeration continues in this way up to 90, changing 10 tens for 100. Finally the numeration ends at 1000. This is controlled by 28 lines on the form.
At the end fill in all of the zeros to bring into focus the passage from one hierarchy to the next by the placement of one more zero each time.
This work recalls the concept of changing from one hierarchy to another from the decimal system operations. This activity helps the child to fix the places which correspond to each hierarchy.

Activities: Formation of Numbers
1) The teacher forms a number on the frame. The child reads it.
2) The child reads a number from prepared cards and forms it on the frame.
3) The child forms any number on the frame, reads it and records it on the left side of the form used earlier for the presentation.

Note: Each time the child forms a number he will recall the formation 10.

Aim: familiarization with the bead frame
knowledge of the passage between hierarchies


First Presentation

...large bead frame, bearing the same characteristics as the small frame: and change of frame color to separate the classes,
...10 beads of respective hierarchic colors on each row.
...wooden hierarchic materials

Slide one green bead to the right and isolate the unit cube. This green bead has the same value as this cube. What was the value of this cube? ..unit of what class? the simple class. Every green bead on this row has the value of one unit. In the same way identify each row of beads using the hierarchic materials.

Isolate a bead and ask the child to identify the equivalent material and ask the child to isolate the corresponding bead.


Numeration According To Position

...large bead frame
...corresponding long form with 55 lines

Move one unit bead to the right, count one and write the digit 1 in the first space of the form. Continue counting and writing. When the tenth unit bead is moved forward, 'we know that 10 units make 1 ten.' The units are moved back and one ten is moved forward. Write '1' in the tens column (without a zero) Continue in this way up to 1 million; the form will be filled up. Go back and add the zeros. Notice the passage from one hierarchy to the next as indicated by the zeros. Note that the commas correspond to the spaces between classes.

Exercises;Formation of Numbers:
1) The teacher forms a number on the bead frame. The child reads the number and writes it on the form.
2) The teacher writes a number on a piece of paper and the child reads it, forms it on the frame and writes it on the form.
3) The child creates a number on the frame and writes it on the form. The child performs addition, subtraction and multiplication (with a one-digit multiplier) on this frame. This larger frame permits the child to work with larger numbers.


Materials: the frame which lies flat on the table.

It is less sensorial in that hierarchic colors and spaces between the classes have been eliminated (note: the black lines are drawn on the board beneath the wires; they will indicate where to begin the multiplication when multiplying by units, tens, hundreds or thousands.). All of the previous operations can be done with this material, but we will do the most interesting - multiplication with a two-digit multiplier.


xIntroduction to Memorization

Memorization is the key that will allow the child to continue in his development of the mathematical mind. Memorization can be defined as conservation in the memory along with the ability to recall experience and impressions. Oftentimes the exercises of memorization are boring to the child because the same thing is repeated over and over until he remembers. In order for our goals to be achieved, we must find ways to make memorization attractive and interesting.

Memorization must be taught along with the decimal system materials. The child has realized the concept of the decimal system: that only nine units can remain loose, and he has understood the function of each operation. Now we must learn to calculate. As soon as the child has memorized all of the possible combinations of 1-9, he will be able to calculate any complex addition. In order to enter the world of mathematics, the child must be given the opportunity to memorize.


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