Math :: 6-9 :: Introductionxxxxxxxxxxxxxxxxxxxxxxxxxxxxxhome
xTable of Contents:
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xIntroduction to the Decimal System
'Our aim is not only to make the child understand, and still less to force him to memorize,
but so to touch his imagination as to enthuse him to his innermost core.'
- Maria Montessori
With the first mathematics materials the child is introduced to the numbers one through ten. The materials that follow will develop the concept of the decimal system, that is, a numerical system based on ten. Maria Montessori called the decimal system the "cell of our system."
To understand the decimal system is not easy for a child. It took humans many years to realize that the value of a numeral is dependent on the position it occupies. It was much later that the concept of zero was developed, and even later that the decimal point came into existence.
Our decimal system (base 10) has nine numerals, one through nine. The presence of one or more zeroes allows us to create numbers beyond nine up to infinity. Thus, learning the numbers one through nine and their numerals, in addition to the concept of zero, is the only truly difficult part for the child. This he has already accomplished. Counting experiences (adding one more) up to 10 have preceded; now the child will learn to count beyond 10.
In the hierarchical orders-ones, tens, and hundreds of the simple class; ones, tens, and hundreds of thousands, and so on- there are nine units: one through nine. No matter in which hierarchy the numeral one appears, the absolute value of one is one. The relative value depends on its position. The limit between one hierarchical order lies in the 'secret of ten' and in the exact value of the numerals one through nine. It is necessary that the child fixes in his mind the concept of the hierarchical orders and their values. The materials that follow enable the child to avoid the confusion and difficulties he may otherwise encounter.
xThe Great Lesson
Prepare a broadly engaging, impressionistic story of the history of Mathematics. Card materials, timelines, and captivating stories will engage the child when offered with tales of early humans and their efforts to measure and quantify their universe. Explore early symbols of numeration, the history of 'zero', and prehistoric calendars. Study the precision of the Great Pyramids of Egypt. Delve into the navigation techniques of the ancient polynesians. Pursue this information enthusiastically and your students will become enthused, as well. Present this work graciously, and your students will express grace in their own studies. Their understanding of arithmetic within the context of human progress will grow.
This understanding of how arithmetic evolved, and continues to evolve today, will inform an appropriate awe of our 'language of numbers'. Continue your 'Great Lesson' throughout the years, through attention to technology & current events and an ongoing expressed passion for the measurement of our world. This passion is a 'Fundamental Need', a common thread across cultures that ties humanity together.
xMath :: 6-9 :: Numerationxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxhome
xTable of Contents:
Quantities and Symbols
Quantities in the Decimal System
Union of Quantities and Numerals (Symbols)
Additional Exercises in Numeration
The Hundred Board
The Seguin 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
Introductions to Other Mathematical Materials
Hierarchical Bead Frames
Small Bead Frames
The History of The Abacus
Introduction to the materials
Passage from sensorial to symbolic representation
Numeration Based On Position
Large Bead Frames
Numeration According To Position
Exercises: Formation of Numbers
Horizontal Golden Bead Frame
Introduction to Memorization
Fractions - COMING SOON
Decimals - COMING SOON
Pre-Algebra - COMING SOON
xQuantities and Symbols
...the golden bead materials which consist of:
...one container of loose gold beads representing units
...one box of gold bead bars of ten beads each
...one box of 10 gold bead squares of ten bars (representing 100)
...one 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.
^ : 3-6
...to develop the concept of the hierarchical orders of the decimal system: units, tens, hundreds, thousands.
...to give the child the relative measurement of the quantities: bead, bar, square, cube.
...to 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.
...to understand the orders of the decimal system.
...to 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.
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.
^ : 3-6
...to understand the rule of the decimal system: only nine quantities can remain loose.
...to familiarize the child with the hierarchical orders
...to offer the opportunity to write complete numerals
...to 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
ADDITIONAL EXERCISES IN NUMERATION
The Seguin Boards
...box containing two boards and 9 wooden tablets for 1-9
...box of ten golden ten bars
...box 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.
...to clarify understanding of the decimal system (11 means: 1 ten and 1 unit )
...to progress in counting from 10 up to 19
...to learn the names of numbers 11-19
The Seguin Boards
...box containing two boards with numerals 10, 20, 30....90, and 9 wooden tablets for 1-9
...box of 9 gold unit beads
...box 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.
...to clarify understanding of the decimal system (11 means 1 ten and 1 unit )
...to count from 1 to 99
...to 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. ^ :
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. ^ :
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. ^ :
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.
^ : 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.
^ : 3-7
Control of Error: The teacher checks the quantities counted.
...to 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. ^ :
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. ^ :
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)
...to further understand the concept of addition, subtraction, multiplication, and division