Arithmetic I/1

The Decimal System

Numeration and Operations

General Plan for Math

Age / Subject Area / Description / Comment
6 / Numeration within 10 / Idea of operations
Snake for Research of 10
6-7 / Decimal System / Use of the Decimal system
Functions of all operations
6-7 / Numeration above 10 / Seguin tables
Chains of 100 and 1000
6-8 / Stamps / All operations / This is a long work because of all the subtraction and division cases
6-8 / Memorizations / All memorizations
Related exercises and snakes
6-9 / Games and Exercises for memorization / For addition
For Subtraction
Skip counting
Squares of numbers 1-10
Squares of binomials and trinomials up to decanomial
6-7 / Small Bead Frame
7 / Dot Game
7 / Multiplication by 10, 100, 1000
7-8 / Hierarchical Material
Large Bead Frame
Flat Bead Frame
7-9 / Long Division with all Particular Cases
7-9 / The Bank Game
7-9 / Checkerboard / First exercise through cross multiplication
7-10 / Fractions / All exercises up to passage to abstraction
8-10 / Decimal Numbers / All operations up to abstraction
8-10 / Metric system
8-10 / Powers
8-10 / Multiples / Research of factors
Divisors
Research of GCD and LCM
8-10 / Divisibility
8-10 / Last Passages on Binomial and Trinomial over 10 / Preparation for square root
9-11 / Square Root
10-11 / Cubing / Preparation for cube root
11-12 / Cube Root
10-12 / Interest
Distance, Rate, Time / These come after fractions but are not part of the main thread and may be presented at the discretion of the teacher

The Decimal System

Introduction

With these materials, the child can form any number knowing only the numerals from one to nine. It is important to arouse and keep the child’s interest. Keep the vital force alive!

Materials

Boxes of golden beads consisting of:

1. loose unit beads

2. bars of 10 beads bound as a unit

3. squares of 100 beads bound as a unit (10 ten bars)

4. cube (1) of 1000 beads bound as a unit (10 squares of 100)

Note: These become analogs to the point, line, surface, and solid later in the study of geometry.

Outline of the Decimal System

Introduction

  1. Quantities
  2. Presentation of a piece for each category- 1, 10, 100, 1000
  3. Deepening of the knowledge of quantities using a great number of beads.
  1. Symbols
  2. introduction of a card for each category
  3. deepening of knowledge of symbols using large cards- 9 units, 9 tens, 9 hundreds, 9 thousands
  1. Union of Symbols and Quantities
  2. put together quantities and symbols forming numbers with symbols and cards
  3. Materials: 9 units, 9 tens, 9 hundreds, and 1 thousand for large cards and bead material (9 is the highest quantity that can stay loose.)

Functions

1. We are not teaching calculation at this point.

2. Each operation is introduced in a static way, then dynamic.

3. When introducing operations, we will introduce 3 sets of small cards

4. With division, we will introduce decurion and centurion division, where ribbons are used

5. Before operations are introduced, we must teach the child how to perform exchanges.

Operations

  1. Explanation of Exchanges
  2. Static addition
  3. Dynamic addition
  4. Static subtraction
  5. Dynamic subtraction
  6. Dynamic multiplication
  7. Dynamic Division

First Presentation/ Introduction to Quantities

First we will give the child the fundamental names of each quantity.

Take a tray and place in it one of each bead type: 1, 10, 100, 1000

Part I

Say to the child, “This is one, this is ten, this is one hundred, this is one thousand.”

Take the one in your hand. “See, it’s so small it nearly gets lost.”

Take the ten in your hand. “See, it is long, but I can close my hand around it.”

Take the 100 in your hand. “See, it covers my open hand.”

Take the 1000 in two hands. “See, I need two hands to hold it.”

Repeat the name each time after you show it in your hand.

Then, with the beads displayed before the child: “Show me ______” [10, 100, 1, 1000] Repeat a few times if the enthusiasm of the child warrants.

Finally, holding up one at a time: “What is this?” [Once for each hierarchy]

Note: To this point, the child has simply learned the names of the pieces. Now we want to show how they are formed.

Part II

Take the ten bar. “How many unit beads are there here?” Have the child count.

Take the 100 square. “How many 10 bars are there here?” Have the child count.

Take the 1000 cube. “How many 100 squares are there here?” Have the child count.

It is important that the child verifies that we are always in the Kingdom of TEN.

Then:

“How many units in this ten?” [Ten!]

“How many tens in this hundred?” [Ten!]

“How many hundreds in this thousand?” [Ten!]

This must be very clear. Re-present if necessary at another time. When the child has learned these names and quantity representations well, the games are introduced.

First Game

Distribute trays to four children. Ask one to bring four 100’s. “Let us see you count them.” Ask another to bring six 10’s. “Let’s see you count them.” [Teach the child to say “one ten, two tens, etc.” It’s not yet time to say twenty, thirty, etc.

Now ask for a single large quantity (carefully omitting the tens quantity) such as “One thousand four hundred six [1406]”. Verify the child’s count aloud.

Direct Aim

To make the child sure of different groups of numbers.

To give the child the relative sizes of each quantity.

An introduction to the sense of hierarchy.

Indirect Aim:

Preparation for geometry (point, line, surface, solid)

Geometry should be introduced a few days later as an activity parallel to arithmetic.

Second Presentation/ Introduction to Symbols

Now that the child knows the various quantities we may continue and introduce their symbolic representations.

Materials

Box containing large cards of units (1 to 9), tens (10 to 90), hundreds (100 to 900) and thousands (1000 to 9000)

The width of cards is determined by hierarchy: units have one place, tens two places, hundreds three places, thousands four places.

The color-coding is uniform throughout all materials: units are green, tens are blue, and hundreds are red, and the sequence repeats. Thousands are green as they begin a new family of numbers.

Part I

Have the four cards 1, 10, 100, 1000. (The child will likely already know at least 1 and 10.)

Hold up 10. “How many zeros are there here?” Response: “One.” Teacher: “It says ‘ten’.”

Hold up 100. “How many zeros are there here?” Response: “Two.” Teacher: “It says ‘one hundred’.”

Hold up 1000. “How many zeros are there here? Response: “Three.” Teacher: “It says ‘one thousand’.”

Repeat, pointing to numeral cards: “This has one zero. It says ‘ten’ “This has two zeros. It says ‘one hundred’.” “This has three zeros. It says ‘one thousand’.”

Mix the cards and lay out: “Show me ______.” [100, 1, 10, 1000]

Hold up a card: “What is this?” [Hold up all.]

Conceal the cards: “How many zeros in ______?” [100, 1000, 10]

When you feel the child knows that the increasing of the number is due to increasing the zeros, you may pass to the next part.

Note: Although it is not explained now, be aware that these numerals also represent 100,10 1,102,103. In base 2 (binary), the same numerals are used for 20, 21, 22, 23.

Part II

Take all the cards and ask the child to lay them out in order.

Then emphasize to the child what is displayed. “Here are all the units, here are all the tens, here are …, etc.”

Ask the child to select some numbers. Initially ask for amounts represented by single cards. Always ask the number of zeros. “Bring me the card for eight hundreds. Howmanyzeros does it have?”

Then move on to quantities such as twothousandfivehundredfour (2504).

Note: If a child refers to sixtens as “sixty”, don’t correct. But your language should refer to “six tens” for emphasis of the hierarchy.

Idea: One strategy for replacing card in the array is to collect the cards in a tray as you gather them and then to pass the tray, asking each child to take one and return it to its proper place in the array.

Direct aim: Giving the child the sense that the value changes with the number of zeros.

Indirect aim: Preparation for the concept of powers of ten.

Third Presentation/ Union of Quantities and Symbols

Materials:Large cards (delete 2000 to 9000 for this exercise)

Golden beads: 9 units, 9 tens, 9 hundreds, 9 thousands

The teacher lays out on separate tables the beads and cards in the same pattern

Part I

Do not yet give the idea of quantities or symbols. The point here is that the largest amount of each loose quantity is nine.

“See that the highest number of ones, tens, and hundreds is nine?”

Here count the units to nine.

“What would come next?” [Ten]

“Now we go to the next column.” Count these: “One ten, two tens, etc.….nine tens.” “What would come after nine tens?” [Ten tens] “We call ten tens one hundred.”

Repeat to10 hundreds, which we call one thousand.

This all brings consciousness of the decimal system.

Part II

Place quantities on a tray and ask the child to get the corresponding symbol cards.

If the child makes an error, cover the zeros on the card and ask the child to count the beads and compare.

Part III

Place some cards on a tray and ask the child to bring the corresponding amount of beads. Elementary children start with three cards, such as 300, 40, and 5. Have the child place each card on top of the matching beads. The child should count from the top of the bead displays, as you did previously.

Here, the child recognizes that each card is singular, but there are many possible quantities.

Part IV

Ask the child to bring you cards with quantities. Omit the 10’s place.

“Please bring me one thousand three hundred five.”

Then ask the child to put the cards together:

Point to the one (thousand): “How many zeros are hidden under here?”

Reveal the zeros. Continue with the hundreds and then the units.

This shows the child the number can be read exactly as written. (Which is why we omitted the 10’s, whose names we don’t know).

Remind the child that when the number is written, the zero in the 10’s position must be included. It is there as a reminder that there were no tens. Prove this by getting the quantities and cards for 135 and comparing with 1305.

This exercise shows the importance of zeros. It is very important for the child to be conscious that the value depends on the number of zeros and the value of the numeral (also called a cipher in many parts of the world).

Direct aims

1. Making the child familiar with the different categories and symbols of the numbers.

2. Providing the opportunity to write large numbers knowing only 9 numerals and zero.

3. Illustrating how zero occupies the place of any missing hierarchy.

4. Creating as a point of consciousness a mental synthesis for the child which is an understanding of the decimal system.

Note: Each symbol is only a symbol, but the possible quantities are many.

Operations with the Decimal System

In general there are two types of operations:

STATIC- which does not require exchange, e.g., 242+333

DYNAMIC- which does require exchange, e.g., 954+487

In the Children’s House, the child remains a long time with static operations. In the Elementary school, the child passes quickly to DYNAMIC.

Before presenting operations, we must give the show the child how to make exchanges. That is the key to the decimal system. Make sure the child understands that ten units of any order make one unit of the following order. Through this the child can explore the world of numbers, infinitely great and small.

It is important for the child to understand the rules guiding the decimal system:

When a group of numbers greater than 9 exists, we must transform them into the following category.

Every 3 powers of numbers forms a new class (family of numbers):

Explanation of the Exchange Process (Cards and bead material needed)

Have the child bring a quantity of golden beads (in excess of ten): 15 units, 18 tens, 13 hundreds.

“Can we represent 15 units with only a single card? No, the most is 9. So when we come to the tenth, they can no longer stay loose. So, change the ten loose beads for a bar of 10.

Then have the child verify that the bar of ten plus five loose beads still add up to 15.

Next: “The ten bar must now go with the other ten bars. Can you find the card for the remaining loose unit beads?” [The 5 card is placed.]

Continue and repeat this procedure for the 10’s and the 100’s.

At the end, there will be a group of beads and cards representing them. These should be displayed in hierarchical order.

Six year olds will usually grasp the idea of exchange quickly, but when it is introduced into addition and subtraction it will be a difficulty.

Pass next to STATIC ADDITION.

Static Addition

Material

This requires all of the decimal system material. The more you have the better.

Large cards (from 1 to 9000)

Three series of small cards (from 1 to 3000)

Four trays

All of the golden beads

Layout: The large cards are placed on one table, the beads on another, and all three series of small cards on another.

Personnel: This work requires four children. One is the cashier. Three will bring the material.

Note: The Seguin boards are a parallel work to the early decimal system material. In that work, the child has isolated and learned the numbers twenty, thirty, etc., and should know them by this point.

Presentation

The teacher prepares three trays. On each there are four cards from the small card series (representing each hierarchy), e.g.,

2432

1214

3132

Ask each child to bring you the bead quantities represented by the cards in his/her tray.

Confirm the count on each tray.

Place the small cards in a heap on each other and show the child some “magic”. This is tapping the cards on the zeros end so that the actual number shows.

Say, “Now we must make an addition. Each of you dump all your beads onto the rug.”

Pull the four corners of the rug and shake to “pull them all together.” (The teacher performs this action.)

Say, “Now that they are put together, we must count what we have. So, put the groups into ones, tens, hundreds, and thousands.” (The words “ones” and “units” are used interchangeably at this point.)

Count each category, starting with units, and have the children get the corresponding large cards.

Stack the large cards and do “magic. Read the result.

Note: If a child asks why we use small and large cards, explain that the small cards are for separate quantities and the large are when they are considered as a whole.

The children may use pre-made cards to write the problem just completed in linear fashion. E.g., 2342 + 1214 + 3132 =6688

With Elementary children we will pass quickly to dynamicaddition.

The Exchange Game

Materials:A tray with ten units, nine tens, nine hundreds.

A container for the unit beads

One thousand cube (left on the shelf with the decimal layout material) will also be needed.

Aims:

Direct:

1. Practice in exchange procedure

2. Reinforcement of relationships between ones, tens, hundreds, and thousands

Indirect:

To exchange ten units for a ten, ten tens for a hundred, ten hundreds for a thousand

Previous Learning:

DirectBuilding tray

Number composition

IndirectLinear counting

Age:5 years

Teacher’s Work

PreparationThe material for this presentation will be borrowed from the decimal layout material.

Presentation

1.Set out a rug with the child and walk with him to the decimal layout material. Give the child the tray, and ask him to take ten units and place them in the container on the tray. When that is done, ask the child to place nine tens on the tray. Finally, ask him to place nine hundreds on the tray.

  1. Bring the tray and beads to the rug. Sit down at the rug and place the bead on the rug: units on the right, tens to the left, then hundreds.

3.Ask the child to help count the units: “one, two, three, four, five, six, seven, eight, nine. If we add one more what do we get? What does ten units equal? We must exchange these ten units for one ten.” Have the child exchange the ten units for one ten bar and put it with the other ten bars.

4.The child assists in counting the tens: “ten, twenty, thirty, forty….ninety. If we add one more ten, what do we get? What does 10 ten bars equal? We must exchange these ten bars for one hundred square.” Have the child exchange the tens for a hundred square.