Decimal System Operations

Addition

Age of great interest

5 years.

Material

  • 4 boxes, each with sets of small white cards of (1 to 9) of the following:

1) 9 units (written in green)

2) 9 tens (written in blue)

3) 9 hundreds (written in red)

4) 9 thousands (written in green)

  • 1 box with a set of large cards from 1 to 9,000.
  • An ample supply of 

1) cubes of 1,000 beads

2) squares of 100 beads

3) 10 bead bars

4) 1 bead units

  • 3 trays with 3 little bowls for loose beads.
  • 1 large tray with 1 bowl.

Direct aim

To help the child in forming a concept of addition. To show how smaller numbers added together make one large number.

Presentation for simple addition

A small group-4 children.

  1. One child acts as the banker.
  2. The banker has a large quantity of golden beads in each category.
  3. The other three children are each given a set of small cards, and these are laid out in sequence.
  4. They are then each given a tray plus a ‘unit’ container. A set of large cards are also laid out.
  5. The teacher selects a number for each child, making sure that the total of each category does not exceed 9. (Simple addition).
  6. The Directress explains to the children how to go to the ‘banker’ and ask politely for the quantity corresponding to their number.
  7. When they have each received the correct amount (the children themselves will act as control of error) they bring them to the teacher.
  8. She piles the whole lot together!
  9. The children’s numbers are each placed on the table, one under the other.
  10. The teacher then counts the quantity of units, and fetches the corresponding large card; she counts the tens, and fetches that large card, -the thousands and hundreds and the corresponding large cards.
  11. She says “Now John brought 3222, Peter brought 2114 and Jane brought 3112 and look, when we have counted them all, we find we have 8448! This is called simple addition.” Emphasize the ‘largeness’ of the number and the large quantity the children brought between them. (The large cards and small cards give a sensorial appearance, of small numbers coming together to make a large number).
  12. This is repeated again, still with simple addition and then the children may do it on their own.

Presentation for Dynamic Addition (‘carrying): The changing game

  1. Take a large quantity of each category, mix them together then say you are going to count them, starting with the units. “How many units do you need to make a ‘ten’?” “Ten” “Well, every time l count ten units I am going to take them away to the banker and receive a ‘ten’ bar in exchange.”
  2. She counts the units, reaches ten, and dramatically exchanges them for a ‘ten’ bar.
  3. She reaches ten again and receives another ‘ten’ in exchange.
  4. She does likewise with the tens receiving a ‘hundred’ in exchange and so on. This must be done very dramatically.
  5. Encourage them to start always with the units.

Control of error

Counting the beads.

Subtraction

Age of great interest

5½ plus.

Material

  • A large quantity of unit beads, ten bars, hundred squares and thousand cubes. 
  • One set of large cards 1-9000. 
  • Three sets of small cards 1-3000.

Presentation for Simple Subtraction

  1. The cards and quantities are set out as for addition.
  2. The teacher gives one child on a tray the number made up from the large cards, e.g. 5322.
  3. The child collects this quantity on his tray.
  4. He sets the cards and quantity on the mat.
  5. The teacher gives another child on the tray the number made up from small cards which is to be subtracted, e.g. 4211.
  6. The child takes the corresponding quantity of beads from the mat.
  7. The first child then counts the remaining beads on the mat and finds the corresponding numerals in the small number cards.

Control of Error

Put together the two quantities. Added together they should amount to the original number.

Presentation for Dynamic subtraction

  1. One child is given a number made of large cards on a tray, e.g. 7346.
  2. He collects the quantity and then sets the cards of the number on one side of the mat.
  3. Another child is given a tray with a number made up of small cards, e.g. 3227.
  4. He must take this away from the larger number.
  5. He finds that there are only 6 units from which he cannot take away 7, so he takes a ten bar from the large number to the banker who exchanges it for 10 units.
  6. He puts the 10 units on the mat with the other units.
  7. He now has 16 units.
  8. He can now take away the 7 and finds that 9 remain.
  9. He picks up the symbol for 9 units from the small cards and puts it aside.
  10. He now proceeds to take away the tens, then the hundreds and lastly the thousands.
  11. Whatever is left of the large number is the answer.

The cards are arranged:

(large cards) 7346

(small cards) 3227

(small cards) 4119

Direct aim

To give the child the opportunity of working subtraction problems in the decimal system.

Multiplication

Materials

  • Large number cards
  • Small number cards
  • Unlimited golden bead material

Presentation for Multiplication

  1. Make one child the banker, and also give him the set of large cards to lay out.
  2. Give a set of small cards each to three children and ask them to lay them out.
  3. Lay a mat on the table.
  4. Having already decided on a number, you say that you are going to whisper each child’s number.
  5. They are to make it with their cards and let nobody see what it is.
  6. They are then to go and ask the banker for their number of golden beads.
  7. Whisper the same number to each child, e.g. 1322.
  8. When the children have fetched their quantity on their trays, you put everything together, and then sort them into their categories.
  9. One child can count the units, another the tens, and so on.
  10. Their cards are placed on the table.

“We have added these three numbers together and we got a large number, 3966.”(Answer put in large cards). “But do you notice anything about those numbers?” “Yes, they are all the same.” “Well, there isn’t any need to write out the same number three times. If we put out the number once, and then put a small ‘3’ near it, it shows how many times we are going to add that number, or how many times we are going to multiply it. This is called multiplication.”

Exercise

  1. Repeat the exercise of multiplying a number by (4) or by (5).
  2. Use static examples.
  3. Place out the number once, putting the small 4 or 5 near it, showing that we are going to multiply it that many times.
  4. The children go and fetch the required quantity that many times.
  5. When the children are working on their own, they should be able to solve a dynamic problem themselves.
  6. They should work up as far as x9.

Division

Short Division

Materials

  • Large number cards
  • Small number cards
  • Unlimited golden beads.

Presentation for Short Division

  1. A set of large cards should be laid out.
  2. Three children should have laid out a set of small cards each, and they should each have a tray with a dish for the units.
  3. Select a number, for example 6369.
  4. Have it laid out in large cards and collect the quantity from the banker.
  5. Tell the children you are going to divide out the material amongst them and that they are all going to have exactly the same amount. Start with the thousands.
  6. Continue on in the similar manner.

“We had the number 6369, and it was divided amongst 3 children, and they each got 2123. This is called a DIVISION SUM.”

The children work at this for a while, before going on to division with remainder and then on to dynamic division.