**The Decimal System (Group Two)**

The decimal system is a numeral system which organises and classifies numerical quantities into different hierarchies of units. In the Casa it is offered when the child can count to ten with complete understanding; with the knowledge of the symbols 1-9 and can recognise zero. The child is given the total decimal system – clearly, simply, harmoniously and with its unlimited, universal applicability. More knowledge at this stage (such as knowledge of the teens and words used to describe the tens) distracts from the enjoyment of the minimalist aesthetic. At this stage the child knows what is necessary and sufficient to see and apply the laws governing the decimal system (that ten units can be dynamically exchanged for one of the category above etc.) The young child’s Sensitive Period for Order and Classification ensures a greater thrill for handling large quantities at this stage.

Geometrical entities are used by Montessori as Material Abstractions for the decimal system of numeration

- 1 Golden Bead is a unit (point)
- 10 Golden Beads make a ‘bar of ten’
- 10 ‘bars of ten’ make a ‘hundred square’
- 10 ‘hundred squares’ make a ‘thousand cube’

Laws of the decimal system

- There are only nine in each category
- There are three hierarchies in each level
- The ratio between one category and the next is1:10
- The ratio between one level and the next level is 1:1000

**Introduction to Beads (Quality)**

This presentation is given in a ‘Three Period Lesson’

**Material Description:**

- A small tray with 1 bead, 1 bar of 10 beads, 1 square of 100 composed of 10 bars of 10 and a cube of 1,000 composed of 10 squares of 100.
- A large tray with a supply of beads from each category

**Presentation:**

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First Period

- Invite the child, introduce the material and bring it to a Working Mat
- Begin with the unit, placing it in front of the child and saying, “How much is this?”, the child answers, “one”, then set it out on the mat to your far right
- Place the bar of 10 in front of the child, count the number of beads aloud and say, “This is the bar of ten”, place it to the left of the unit
- Place the square of a hundred in front of the child and say, with wonder, “This is one hundred, there are so many tens in one hundred, lets count how many”. Hold the bar of ten over each similar bar of ten in the square of one hundred and count, then place both in sequence on the mat
- Lightly drop the cube of one thousand in the child’s palm, then place it on the mat in front of her and say, “There are so many hundreds in this thousand, let’s count them”, do so holding the square, then say, “Ten hundreds is one thousand”

Second Period

- Ask the child for a category giving the various commands, including asking the child to count the categories.

Third Period

- Ask the child to identify and name each category.

Note: Leave the child with 9 units, 9 bars of ten and 9 hundreds and the thousand to explore the material more fully

**Exercises:**

This additional presentation is more enjoyably given with two or three children, it can be done on the same day as the first one

- Ask for a quantity
- Place the objects randomly and ask each child to name them
- Give each child an empty tray and bowl and ask them to being a quantity from one category at a time
- When the child returns ask her to remind you how many she was asked to bring and count the tray’s contents with her

- Ask for a name
- Place one or more object from one category on the child’s tray, asking each child to count and name them

**Criteria of Perfection (Control of Error):**

- The Director

**Direct Aim:**

- To give the names of the different categories
- To show the relationship between one category and the next
- To offer the child a sensorial experience of the relative increments between categories
- To extend the sensorial experience of different categories and the difference in bulk; i.e. between six units and six hundred units

**Age at Presentation:**

Four and a half years, after all of the activities of Group 1

**Presentation with Cards**

This presentation is given in a ‘Three Period Lesson’

**Material Description:**

Presentation

- 1 mat show 1, 10, 100 and 1000 and bring other cards for child to use after presentation

- The numbers are on a white background in decimal colours
- from 1 to 9 in green
- from 10 to 90 in blue
- from 100 to 900 in red
- 1000 in green

- The length of the cards
- 10 is twice the length of 1
- 100 is three times the length of 1
- 1000 is four times the length of 1

Therefore when cards of different categories are placed on top of each other, aligned to the right, the number of each category is clearly given in it’s hierarchical position.

Exercises

- A tray for the first exercise

**Presentation:**

First Period

- Invite the child, introduce the material and bring it to a Working Mat
- Begin with the card of one, placing it in front of the child and saying, “How much is this?”, the child answers, “one”. Mention to the child that it is written in green and the set it out on the mat to your far right
- Place the card of 10 in front of the child and saying, “How much is this?”, the child answers, “ten”, then ask her to count the number of zeros aloud. Mention to the child that it is written in blue and place it to the left of the unit
- Place the card of 100 in front of the child and saying, “This is one hundred”, then ask her to count the number of zeros aloud. Mention to the child that it is written in red and place it to the left of the ten
- Place the card of 1000 in front of the child and saying, “This is one thousand”, then ask her to count the number of zeros aloud. Mention to the child that it is written in green and place it to the left of the ten.

Second Period

- Ask the child for a category giving the various commands, including asking the child to count the zeros.

Third Period

- Ask the child to identify and name each category.

**Exercises:**

This additional presentation is more enjoyably given with two or three children, it can be done on the same day as the first one

- Ask for a card
- Place the cards at random, within groups of the same category on the working mat, naming each numeral.
- Ask the child to bring a card from any one category at a time, e.g. “eight hundred” or “five tens”
- Check the card with her on her return, read it back to her giving the full name, e.g. “fifty” and ask her to replace it

- Identify a given card
- On another occasion arrange the cards as above, pick one and ask the child to give it’s name

**Criteria of Perfection (Control of Error):**

- The child’s knowledge of the numbers 1 to 9

**Direct Aim:**

- To acquaint the child with the written symbols for the new quantities she has learnt previously

**Age at Presentation:**

Four years, after all of the activities of Group 1

**Formation of Large Numbers**

This activity can be done with three or four children or an individual

**Material Description:**

- A tray with golden beads: 9 units, 9 tens,9 hundred and 1 thousand
- A set of Large number Cards from 1 to 1,000
- The wooden dynamic materials for 2,000 to 9,000 can also be shown at this point and then put away
- Three small trays with small dishes
- Two mats placed at a distance

**Presentation:**

The layout of the material

- Unroll one mat, arrange the beads vertically in categories along the top
- Count the units as you place them, one below the other until you have a vertical row of nine.
- Then ask what comes next, and indicate the bar of ten, count with the child in tens, placing the bars of then beneath in a vertical row to the left of the units
- Continue doing this with the hundreds and acknowledge that after we have 900 we have 1,000. Indicate the cube of 1,000.

- Unroll a second mat at a distance, arrange the top categories of the Large Cards along the top of the mat. Put them in order with the units on the farthest right. Lay the other cards out as for the beads.
- Ask the child to identify the card of ‘1’ then ask her, “what comes after ‘1’?’, and place the cards in sequence to ‘9’. At nine ask, what comes next and where the ’10’ card can be found.
- Continue to count in tens while placing the cards vertically
- At ’90’ ask her what comes next, if she says, ‘ten tens’ ask her what ‘ten tens’ is, and then where ‘100’ is. Do the same for 900 and indicate the final card of 1,000

Association of quantity and symbol

The Director gives the Card

- The Director give a card from one category and place it on the child’s tray
- Ask the child to fetch the corresponding quantity
- When she returns verify it, asking her what she was asked for and what she has brought
- Ask her to return the card and quantity and continue giving cards

The Director gives the Quantity

- The Director give a quantity from one category and place it on the child’s tray
- Ask the child to fetch the corresponding card
- When she returns verify it, asking her what she was asked for and what she has brought
- Ask her to return the card and quantity and continue giving quantities

The Director gives more than one category

- The Directress brings cards from more than one category and arranges them on the tray so that each figure is revealed.
- Ask the child to fetch the quantity, showing her how to stack quantities of the same category
- On her return reread the card and count the qualities with her, then superimpose the Large Cards stacking then to the left sliding them to the right to reveal the figure. Read the figure out-loud so the child hears the correct wording of a large number

**Exercises:**

- Later give the child cards from three and then four categories. When using four for he first time, wait till the child returns, then have your cards to that the digits are placed to the right, let the child see them then hold them together and tip then towards the floor so they ‘spill’ to the left and the numbers can be seen in their correct places

**Criteria of Perfection (Control of Error):**

- In the ability of the child to count the qualities and read the symbol

**Direct Aim:**

- To make the child familiar with the different categories of numbers both in bead quantities and in symbols
- To give the wording of large numbers
- To show the function of zero as a place holder in a large number
- To show that only 9 are necessary in each category to form any number

**Age at Presentation:**

Four and a half to five years, after working with the Beads and cards separately

**Footnote:**

- Alternative Presentation

After introducing the quantity with the Beads and Card material, link it to the wooden Dynamic material or introduce the wooden Dynamic material before beginning the operation

- When the child brings he quantity, cards or completes the operations do not give as such importance to the correctness of the answer as to her counting and understanding of the processes
- The decimal system is abstracted here in such a way that the child experiences the laws of place value – by the geometrical abstractions, which collectively offer a unique way to introduce this outstanding cultural endeavour.

**Decimal System**

**Dynamic Part: Addition**

After awareness of the dynamic relationship between categories in the decimal system children are given further experiences of handling huge qualities with other children to support them.

Note: It is recommended to have two complete sets of materials for the collective exercises in a class

**Material Description:**

- For each set:
- Golden bead material: 50 unit-beads, and a supply of tens and wooden material for hundreds and thousands
- Large Number Cards: 1 set of 1 to 9000 and three small sets of 1 to 3,000
- One large tray and a medium dish
- 3 small trays and 3 small dishes
- 2 Working Mats for Large Cards and quantities and three chowkis
- 1 large fabric square

**Presentation:**

Static with three children

- Prepare a number which will not require a dynamic change
- Lay out the materials on the mats, displaying the Large Cards, beginning with the units on the right of one and on the other the boxes with the quantities
- Ask the children to prepare the small cards on their chowkis
- If you have not done so before, link the Bead material with the wooden material, then return the bead material
- Give each child a tray and bowl and ask her to make a number with the small cards, check it and then ask her to bring the corresponding quantity
- Spread the fabric on the mat with the quantities
- When the children return verify the amount asked for and the amount they have brought, then ask the children to dump the quantity onto the fabric square
- Ask each child to do the same
- When all of the quantities have been dumped bring up the corners of the fabric ad lift it, say how heavy the sum of what they brought is to give the sensorial impression of addition
- Open the fabric and ask each child to separate the quantities into their various categories, say, “Lets see how much we have here”, invite each child to count one category and fetch the corresponding Large Card from the mat
- When the children bring the Large Cards place it above or below the quantity it represents, then superimpose all the large cards to show the total
- Ask each child to show their small cards, read the number and place the small cards one below each other on the right of the mat and the Large cards underneath

1342

2410

136

- Say to the children, “All that you have brought has been put together, this is called addition”

For the static presentation the directress gives the children the number for the addends. Later move to the dynamic presentation where the children choose their own addends, here is is likely that they will come across dynamic addition

Dynamic (changing) with three children

- Prepare the material a for the static presentation
- Invite each child to make a number with the small cards and fetch the corresponding quantity continue as before
- When the children count the categories they may find more than nine of any one category, if this happens say, “Every time we have TEN of an one category, we change it for ONE from the next highest category”

**Criteria of Perfection (Control of Error):**

- At this stage, the exact answer is not important, the focus is on the child’s understanding of the process

**Direct Aim:**

- To give the child a sensorial impression of the nature of addition, which is adding smaller quantities to make up one larger quality
- To show the function of the Decimal System
- To understand the process of changing from 10 of a lower category to 1 of the next highest category

**Indirect Aim:**

- To experience the ordering effect of the laws of the decimal system
- To become familiar with the mechanism for changing from one hierarchy to the next (the dynamic)

**Age at Presentation:**

Four and a half years onwards

**Footnote:**

- In these operations signs are not used
- Make sure that the chidden bring a large card before counting the next category, if the subsequent one is higher than ten another dynamic shift occurs in which the previous large card just be changed. After a few turns if the children do not realise that it is easier to start with the units, ask if thy would like to know a way to change the large card only once and demonstrate counting from the units up. “The rule that in addition we should count from the units will make sense to the child and later they will have no problem remembering it (Mario Montessori)

**Dynamic Part: Subtraction**

Note: Present after dynamic material for addition since the principle of presenting contrasting impressions enables the child to clearly grasp the fundamental mathematical concepts of merging, joining and bringing together (addition and multiplication) and splitting, taking-away (subtraction and division)

**Material Description:**

- For each set:
- Golden bead material: 50 unit-beads, and a supply of tens and wooden material for hundreds and thousands
- Large Number Cards: 1 set of 1 to 9000 and three small sets of 1 to 9,000
- One large tray and a medium dish
- 3 small trays and 3 small dishes
- 2 Working Mats for Large Cards and quantities and a chowki for each child

**Presentation:**

Static with one child

- Prepare a number which will not require a dynamic change
- Lay out the materials on the mats, displaying the Large Cards, beginning with the units on the right of one and on the other the boxes with the quantities
- Ask the children to prepare the small cards on their chowkis
- Give each child a tray and bowl
- Make a large number e.g. 5,987 in Large Cards and quantities and display it on your mat (this is the whole from which subtractions will be made)
- Give each child a small number and ask her to take this from your mat and represent it with her small cards. Explain, “I had 5,987, you came and asked to take away 3,546, now I have 2,441. To take away is called subtraction”
- Lay out the cards verbalising the process
- Gather new cards

Dynamic (changing with one child)

- When the child seems ready begin the dynamic with a number that involves one change only
- Prepare the material a for the static presentation
- Invite one child take away a certain number from your quantity and make the corresponding number with her small cards
- When the child needs to take away from one category more than you have explain that, “Every time we have ONE of an one category, we change it for TEN from the next lowest category”. When she does this she will have sufficient to get the number she needs

Dynamic (changing with two children)

- Prepare the material a for the static presentation
- Make a large number with the Large Cards
- Invite the children to make a number with their small cards and one at a time to take away a certain number from your quantity and make the corresponding number with her small cards
- When the first child does this pt your total face down on the mat and count what is left of the original (the minuend), represent this with large cards to show that even though some has been taken some remains
- The second child takes her portion ant the Director writes the amount she has left in small cards
- The Director turns over the cards and summarises what has happened, “I had 5,487, John took 3,546 and Kate took 1,629, now I have 312”, arranging the children’s Small Cards and Your Large Cards and the Quantities to illustrate this.
- Later do a presentation in which the quantities and the remainders come together to total the original number

**Criteria of Perfection (Control of Error):**

- At this stage, the exact answer is not important, the focus is on the child’s understanding of the process

**Direct Aim:**

- To give the child a sensorial impression of the nature of subtraction; which is taking a smaller number(s) away from a larger one, what remains is less than the original figure.
- To show the function of the Decimal System
- To understand the process of changing from 1 of a higher category to 10 of the next lowest category

**Indirect Aim:**

- To experience the ordering effect of the laws of the decimal system
- To become familiar with the mechanism for changing from one hierarchy to the next (the dynamic)

**Age at Presentation:**

Four and a half years onwards, after addition

**Footnote:**

- In these operations signs are not used
- Make sure that the children bring a large card before counting the next category, if the subsequent one is higher than ten another dynamic shift occurs in which the previous large card just be changed. After a few turns if the children do not realise that it is easier to start with the units, ask if thy would like to know a way to change the large card only once and demonstrate counting from the units up. “The rule that in addition we should count from the units will make sense to the child and later they will have no problem remembering it (Mario Montessori)
- Be dramatic to show the loss that is subtraction
- If you run out of any materials ask the child what she thinks should be done, cards can be written using the adult’s tray if necessary

**Dynamic Part: Multiplication**

**Material Description:**

- For each set:
- Golden bead material: 50 unit-beads, and a supply of tens and wooden material for hundreds and thousands
- Large Number Cards: 1 set of 1 to 9000 and three small sets of 1 to 3,000 and small cards 1 to 9 for the multiplier
- One large tray and a medium dish
- 3 small trays and 3 small dishes
- 2 Working Mats for Large Cards and quantities and three chowkis

**Presentation:**

Static with three children

- Prepare a number which will not require a dynamic change
- Lay out the materials on the mats, displaying the Large Cards, beginning with the units on the right of one and on the other the boxes with the quantities
- Ask the children to prepare the small cards on their chowkis
- Give each child a tray and bowl
- Whisper to each child the same number e.g. 3,231 and ask her to collect small cards, putting them face down and the quantity
- When the children return verify the amount asked for and the amount they have brought, then ask the children to place their quantity on the mat
- Ask each child to do the same
- When all of the quantities have been placed, separate the quantities into their various categories, say, “Lets see how much we have here”, invite each child to count one category and fetch the corresponding Large Card from the mat
- Remind the children about what has happened, and ask them to show you their cards – they will be surprised when they realise they all had the same number
- Count the number of times 3,231 has been taken and say, “When the same number is taken three times it is called multiplication”
- Turn over two sets of the small cards and replace it with a small card ‘3’
- Verbalise the procedure 3,231 x 3 =

For the static presentation the directress gives the children the number for the addends. Later move to the dynamic presentation where the children choose their own addends, here is is likely that they will come across dynamic addition

Dynamic (changing) with three children

- Prepare the material a for the static presentation
- Invite each child to make a number with the small cards and fetch the corresponding quantity continue as before
- When the children count the categories they may find more than nine of any one category, if this happens say, “Every time we have TEN of an one category, we change it for ONE from the next highest category”

**Criteria of Perfection (Control of Error):**

- At this stage, the exact answer is not important, the focus is on the child’s understanding of the process

**Direct Aim:**

- To give the child a sensorial impression of the nature of multiplication, which is the addition of equal numbers

**Indirect Aim:**

- To experience the ordering effect of the laws of the decimal system
- To become familiar with the mechanism for changing from one hierarchy to the next (the dynamic)

**Age at Presentation:**

Four and a half years onwards, after subtraction

**Footnote:**

- Emphasis the fact that multiplication is putting together the SAME quantity several times is emphasised by keeping the fact that the children have been asked for the same amount as a surprise.

**Dynamic Part: Division**

Note: The concept of taking away is linked by sharing equally to sensorially portray the process of division

**Material Description:**

- For each set:
- Large Number Cards: 1 set of 1 to 9000 and three small sets of 1 to 3,000 and small cards 1 to 9 for the divisor
- Green, blue and red bows
- One large tray and a medium dish
- 3 small trays and 3 small dishes
- 2 Working Mats for Large Cards and quantities and a chowki for each child

**Short Division, with a one digit divisor**

**Presentation:**

Static with three children

- Prepare a number which will not require a dynamic change
- Ask the children to prepare the small cards on their chowkis
- Give each child a tray and bowl
- Make a number in Large Cards which is exactly divisible by the number of children, e.g. 6,936 and ask a child to fetch it
- Explain that that this amount, which is put on a tray will be SHARED EQUALLY among them
- Sit with space for the children to walk around you, sharing out one of the highest quality as they go round, when the quality has run out ask the children to take Small Number Cards to represent it
- Continue till the units have been distributed
- Point out that each child now has an equal amount, say, “How much have you received?” to each child in turn, then, “So you have received the same amount”
- Ask for the small cards and lay them out below each other on the mat

2312

6936 2312

2312

- Say “six thousand and nine hundred and thirty six has been shared among the three of you”, turn over two copies of ‘2312’, place the small card ‘3’ in their place and explain that one of the children’s numbers is enough to show what they received
- Say “So we have done a division”

Dynamic (changing with three children using a one digit divisor)

- When the child seems ready begin the dynamic with a number that involves one change only, e.g. 6536
- Prepare the material as for the static presentation
- Distribute the 6000 and ask the children to get the small cards and 300 as for the static, the children get 100 in small cards ad the two ‘hundred squares’ are exchanged for twenty ‘bars of ten’ which are added to the three ‘bars of ten’ on the tray, distribute twenty one of the ‘bars of ten’, then exchange the two remaining ones for twenty units. Distribute eighteen units and the two units are the remainder
- You can remind the child that, “Every time we have ONE of an one category, we change it for TEN from the next lowest category”.
- Layout the cards and verbalise the process as for the static, refer to the two remaining units as a remainder

**Criteria of Perfection (Control of Error):**

**Direct Aim:**

- To show the function of the Decimal System
- To give the child a sensorial impression of the distributive aspect of division: the sharing out of quantities equally, one category at a time and finding the answer in the amount one unit gets
- To reinforce the role of changing from a higher category to the next lowest category

**Indirect Aim:**

- To experience the ordering effect of the laws of the decimal system
- To become familiar with the mechanism for changing from one hierarchy to the next (the dynamic)

**Age at Presentation:**

Five to five and a half years onwards, after multiplication

**Dynamic Part: Division**

Note: The concept of taking away is linked by sharing equally to sensorially portray the process of division

**Material Description:**

- For each set:
- Large Number Cards: 1 set of 1 to 9000 and three small sets of 1 to 3,000 and small cards 1 to 9 for the divisor
- Green, blue and red bows
- One large tray and a medium dish
- 3 small trays and 3 small dishes
- 2 Working Mats for Large Cards and quantities and a chowki for each child

**Long Division, with a two digit divisor**

**Presentation:**

Dynamic – Long Division (using a two digit divisor)

- Prepare the material a for the static presentation
- Make a large number with the Large Cards, e.g. 3864
- Ask the children to represent this in Large Cards and quantities
- Gather the number of equivalent number of children to the number you wish to divide by, here e.g. 12
- Explain, “I will share the quantities amongst all of you equally, as there are so many of you one child will represent ten children and collect their share”, pin a blue bow onto the child representing ten, green bows on the other two children and ask the other nine children to continue their work
- Say, “The child with the blue bow will get ten times the amount of the children with the green bows.
- Divide the quantities accordingly, ending by asking how much each child has received
- Remind the children that the child with the blue bow has been collecting for nine other children as well as herself, the nine original children come and receive their share
- The children count the quantity they have each received again and find they have the same number,
- Summarise the procedure representing what has happened with the small cards

3864 12 322

After a few times the children understand that the child with the blue bow represents ten children

**Criteria of Perfection (Control of Error):**

**Direct Aim:**

- To show the function of the Decimal System
- To show how quantities are distributed in long division, but that in the end the result is the same. The answer in division is the amount that each unit of the divisor receives. This is not verbalised at this stage, it is sufficient for the child to experience the fact
- To reinforce the role of exchanging from a higher category to the next lowest category.

**Indirect Aim:**

- To experience the ordering effect of the laws of the decimal system
- To become familiar with the mechanism for changing from one hierarchy to the next (the dynamic)

**Age at Presentation:**

Five to five and a half years onwards, after short division

**Footnote:**

- The children could do the Stamp Game before or after this activity

**Stamp Game**

**Material Description:**

- Small wooden squares, equal in size: Green ones with ‘1’, blue ones with ’10’, red ones with ‘100’ and green ones with ‘1000’
- Skittles: 10 green, 9 blue, 9 red and one large green
- Small counters in green, blue and red
- A box for the above
- Two chowkis
- A tray with squared paper, a graphite pencil and a coloured one and a writing pad (for writing and making numbers and operations)
- Golden Bead material (for introduction only)

Note: The children must work on individual sums but can share the same box of stamps, they can write on squared paper or squared blackboards, or in booklets

**Presentation:**

Introduction

- Link the golden material to that of the stamps, make sure the child can connect the two with ease. Replace the Golden Bead material on the shelf and the stamps into their box

Making Numbers

- Ask several times for a number from a singe category
- Show the child how to;

form a number with two or more categories

place the stamps onto the chowki, separating the categories

read the number

- Ask the child to make a new number with stamps from all the categories and to read it
- After sufficient practice forming and reading the numbers ask for a number with a ‘0’ in one category

Making and Writing Numbers

- Ask the child to make and read any number e.g. 4861
- Beginning with the highest category, thousands ask the child, “How many thousands are there?…Write 4 on the left”
- Then ask the child, “How many hundreds are there?…Write 8 on the right of 4”, continue if the child requires support, saying;

“How many tens are there?…Write 6 on the right of 8”

“How many units are there?…Write 1 on the right of 6”

Making and Writing Numbers in Columns

- Ask the child to use the stamps to write two numbers, e.g. 4872 and 986, place the second number below the first
- Read both numbers out loud
- Write the first number
- Ask, “How many hundreds are there?”, indicating the second number, write ‘6’ below ‘8’, forming a hundreds column and continue to write the second number
- When the child is confident with this process give numbers containing a ‘0’

**Addition**

Addition (static)

- The Director gives a sum, laying out the addends one below the other, with an exaggerated gap between them.
- Write the addends in columns, without an addition sign
- Draw lines beneath the numbers and say, “We call this an equals sign”
- Returning to the stamps, push the stamps of the lower number up to physically combine them with the higher addend, sensorially they become one.
- Beginning with the units count each category and complete the addition on paper

Signs

- Tell the child, “When we look at these numbers there is nothing to show that we have done an addition”.
- Introduce the sign by writing + in coloured pencil in the customary place, saying “This is a plus sign”
- Give the child other examples to try, supporting her with the first one or two e.g.

2,342 7,215 4,682

+3,426 +1,362 +5,217

5,768 8,577 9,899

Addition (dynamic)

- The Director gives a sum which when added will require a dynamic change.
- Proceed as for static
- Show how to exchange ten stamps of one category for one stamp of the next largest, add this one to the appropriate category and continue

1 1 1 1

4,652 7,815 4,682

+3,476 +1,362 +5,257

8,128 9,177 9,939

- The child begins to make her own sums

Note: Give several presentations of the static, when the child works independently give one presentation of the dynamic and later more if necessary.

**Age at Presentation:**

After plenty of experience with the decimal system using Golden Beads, when the child can write (unless specific learning problem)

**Subtraction**

Subtraction (static)

- The Director gives a minuend, and the child forms it with stamps
- The Directress gives the subtrahend and the child writes it below the minuend in stamps
- Tell the child, “This is how much we will take away”
- Write the numbers in columns, without a minus sign but with an equals sign
- Return to the stamps, beginning with the units subtract each category and complete the work on paper

Signs

- Tell the child, “When we look at these numbers there is nothing to show that we have done an subtraction”.
- Introduce the sign by writing – in coloured pencil in the customary place, saying “This is a minus sign”
- Give the child other examples to try, supporting her with the first one or two e.g.

8,925 3,492 8,639

-5,714 -1,271 -4,215

3,211 2,221 4,424

Subtraction (dynamic)

- The Director gives a problem which when added will require a dynamic change.
- Proceed as for static
- Show how to exchange ten stamps of a lower category, when it is no longer possible to subtract for one stamp of the next largest, add this one to the appropriate category and continue
- Later introduce subtraction problems with ‘0’

6 3 7 7

7,342 7,815 7,682

-3,426 -1,362 – 5,257

3,916 6,453 2,225

- The child begins to make her own sums

Note:

- You can write the subtrahend on paper before taking away
- The original amount no longer exists after the subtraction
- Help the child to realise that when she has less than she can give she can exchange before giving the numbers away

**Age at Presentation:**

After the child has done addition and subtraction with the beads and addition with the stamp game

**Multiplication**

Multiplication (static)

- The Director gives a number for the child to write in stamps and say, “We are going to take this three times”
- Write the figure, it’s multiplier and the equals sign on squared paper
- The child finds the number ‘3’ in stamps and is asked to place it well below the other number
- Returning to the stamps, push the multiplier up to physically combine it with the figure to be multiplied, sensorially they become one.
- Beginning with the units take each number the given amount of times, and complete the multiplication on paper

Signs

- Tell the child, “When we look at these numbers there is nothing to show that we have done an multiplication”.
- Introduce the sign by writing x in coloured pencil in the customary place, saying “This is a multiplication sign”
- Give the child other examples to try, supporting her with the first one or two e.g.

2,342 1,213 3,122

x 2 x 3 x 3

4,684 3,639 9,366

Multiplication (dynamic)

- The Director gives a sum which when added will require a dynamic change.
- Proceed as for static
- Show how to exchange ten stamps of one category for one stamp of the next largest, add this one to the appropriate category and continue

1,342 1,515 2,382

x 3 x 6 x 4

4,026 9,090 9,528

- The child begins to make her own sums

Note: Prepare a few examples of static multiplication before the child moves to dynamic

**Age at Presentation:**

After the child has done addition and subtraction with the beads and addition with the stamp game

**Division**

Short Division (static, one digit divisor)

- Introduce the skittles, write on a piece of paper a problem which does not require a dynamic shift when split into a single divisor, highlight the division symbol y using the contrast colour pen and show it to the child.
- The child forms the dividend with stamps and the divisor is prepared using the small green skittles
- Starting with the thousands column share each figure equally among the skittles
- The child counts and writes the quotient
- Show the child how to conventionally record the division process on the squared paper
- Introduce the sign of division at this point for the child’s reference

- Give the child other examples to try, supporting her with the first one or two e.g.

3,122 2,314 1,221

3 9,366 2 4,628 4 4,848

Short Division (dynamic)

- The Director gives a problem which when added will require a dynamic change.
- Proceed as for static
- Show how to exchange ten stamps of a lower category, when it is no longer possible to subtract for one stamp of the next largest, add this one to the appropriate category and continue
- Later introduce subtraction problems with ‘0’

3,055 613 r1 371

3 9,165 4 2,453 6 2,226

- The child begins to make her own sums

Long Division with a two digit divisor

- Give a number
- Divide the dividend equally among the skittles
- Write the example on paper
- Start sharing the stamps from the highest category, emphasise that each skittle is given the amount appropriate for it’s category, saying, “If the tens skittle receives one hundred, how much will the unit skittle receive?…ten times less than a hundred is ten.”

3657-23=159

Long Division with a three digit divisor and a tens ‘0’

- Give a number
- Divide the dividend equally among the skittles, represent the missing tens number with a blue counter
- Write the example on paper
- Start sharing the stamps from the highest category, emphasise that each skittle is given the amount appropriate for it’s category, saying, “If the hundred skittle receives one thousand, how much will the ten skittle receive?…, but there are no hundred so ten times less than a hundred is ten…so share the thousands between the hundreds and the unit skittles?”

5259-203=25 r184

Long Division with a three digit divisor and a ‘0’as the final digit

- Give a number
- Divide the dividend equally among the skittles, represent the missing tens number with a green counter
- Write the example on paper
- Start sharing the stamps from the highest category, emphasise that each skittle is given the amount appropriate for it’s category, saying, “If the hundred skittle receives one thousand, how much will the ten skittle receive?…,What would a unit skittles receive if it was there?…As there are no green skittles nothing is given?”
- As the quotient is what one unit skittle receives divide the number of the blue skittles’s stamps among ten green ones.
- The child counts and records the result

5289-230=22 r229

Note:

- You can write the dividend on paper before beginning to divide
- The original amount no longer exists after the division
- Help the child to realise that when she has less than she can give she can exchange before dividing the numbers

**Age at Presentation:**

Five and a half to six

**Criteria of Perfection (Control of Error):**

- No control of error, other than careful attention to the process

**Direct Aim:**

- To reinforce and consolidate the understanding acquired previously in the collective exercises, by means of individual work.
- To further the sensorial experience of place value
- To teach the child how to write a problem

**The Dot Game**

**Note:**

After the group work with the operations the children have worked individually with the decimal system. Both the golden bead material and the stamp game provide opportunities to understand the decimal system;

- How the numbers are written
- How the operations are carried out

The Dot Game focuses on addition an area which the child has some experience, to give further opportunity to practice the dynamic more fully and independently, giving greater security, further extension and the opportunity to rectify misunderstandings. The Dot Game isolates the process of moving from one category to the next step-by-step.

The Dot Game is more symbolic, the beads have become stamps and the stamps dots- moving the child from the concrete towards the abstract with the continued support of colours and dots.

**Material Description:**

A Laminated Card and markers or paper and pencils or chalks and sponge with a prepared slate, each showing columns each up to 10,000 (written in blue). Each column has a rectangle with the category written in appropriate colours. The columns have ten rows of ten horizontal squares. Below the squares are two larger rectangles, the upper one is for placing the dots, the lower one for the final answer. A blank column on the right side is for writing the problem.

One writing instrument should be in colours other than those used to represent the decimal system

**Presentation:**

Introduce the child to the material

- Show her the numbers on the board
- Introduce 10,000, counting the 10, show the remainder looks like 1,000 but now it is 10,000
- Show that each column has ten squares
- Show where the sum is written in the far right column by writing a sum with many large addends

Addition

- Starting with the first addend record the number of units, writing dots, in graphite, in the squares in the unit column, then cancel the digit
- Continue for the tens, hundreds and thousands in the same way
- Continue for the second addend, filling the first line of squares before moving to the second
- When all of the addends have been processed begin counting the units. After counting the first ten, cancel the complete lines of squares in the units column in pencil. Mark dots from the top left corner of the rectangle immediately below the squares to represent each line of canceled units.
- Count the incomplete line of unit squares (if there is one) and record their value in the lowest rectangle (if there isn’t place a ‘0’).
- Now ask the child what each dot represents, say it is, “One ten of ones”, indicating one of the cancelled rows, “As there are three dots we have made ‘three tens’, so we take them to the ‘tens’ house (column). Write the number 3 in the ‘tens’ column adjacent to the dots, then with a contrast colour mark three dots in the small tens boxes above, saying, “We have made ‘three’ new ‘tens’, to remember that we put them here”. Now cancel the figure ‘3’

- Proceed as above for the other digits and transcribe the sum in the far right column under the addends

**Exercises:**

- When the child is confident count one category of addends at a time
- Take a number from any category and cancel that figure and mark it with dots

**Criteria of Perfection (Control of Error):**

- No control of error, other than careful attention to the process

**Direct Aim:**

- To illustrate the mechanism of the decimal system, that is, the relationship between one category and the next in a more abstract form
- To focus the child’s attention on carrying

**Indirect Aim:**

- Preparation for abstract addition

**Age at Presentation:**

At around five and a half years of age onwards, when the child knows the language for teens, tens.

**Footnote:**

It is not the exactness of the result but achieving the aim that matters most

**Word Problems**

**Material Description:**

- Prepared word problems using the four operations on colour coded paper (addition -red, subtraction – green, multiplication – yellow; division – blue) and a set of mixed problems on another colour
- The child’s choice of materials; golden beads, the Stamp Game or Dot game for addition
- Paper, pencil and ruler

**Presentation:**

- Individually or in a small group ask the children to read a word problem check their understanding and allow them to find and check their answers. Ask them how they solved the problem.

Addition

Jane has seven apples, if she feeds the horse two how many will she have?

Subtraction

John has five Euros, if he spends two Euros, how many does he have?

Multiplication

Mary, Kate and Ian all have two packets of biscuits, how many do they have all together?

Division

I have a packet of 10 balloons, I give my sister five, how many do we have each?

**Criteria of Perfection (Control of Error):**

- The child may check her answers with the Control Chart for that operation

**Direct Aim:**

- To give the opportunity for the child to apply the operations to everyday situations

**Age at Presentation:**

Five to six years onwards

**Footnote:**

This is an opportunity to remind the child about how we use the symbols for the operations

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