QUANTITIES IN THE DECIMAL SYSTEM
…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.
…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.
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.
UNION OF QUANTITIES AND NUMERALS (SYMBOLS)
…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.
…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.
ADDITIONAL EXERCISES IN NUMERATION
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
ADDITIONAL EXERCISE IN NUMERATION
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.