**Exploration and Memorisation of Numbers – (Group IV)**

These exercises aim to help the child with basic operation, preparing for Mental Mathematics. This group can come after the Decimal System (Group II), parallel to Linear Counting (Group III) or before it. The child must have acquired the ability to make dynamic shifts and have the confidence and capacity to manipulate different categories in a variety of materials. The child continues to work with essential combinations taking them unconsciously into her memory.

As 9 is the maximum number the decimal system recognises;

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- In basic addition a total of forty five is the maximum figure
- In basic subtraction the minuend can be a maximum of 18 as there cannot be a subtrahend (difference) of more than 9
- In basic multiplication there can only be 45 multiplications with 9 as the largest multiplicand and multiplier
- For basic division memorisation is not used, instead multiplication tables are used to support the child’s learning.

**Positive Snake Game**

**Material Description:**

- 3 red boxes on a tray, containing;
- 5 sets of coloured bead stairs
- 25 golden beads
- 1 set of black and white beads from 1-9 and a notched card for the bridge
- A red felt cloth set out on a Working Mat

**Display**

The three boxes are displayed left to right in the above order. This material begin a new shelf.

**Presentation:**

Laying out the material

- Invite the child, saying, “I will show you another way to do addition”, the name of the material is not said here, but the child is asked to identify the shape made by the coloured beads once the ‘snake’ is formed.
- Bring the material to the Working Mat, unroll the felt mat.
- Open the boxes, show the coloured beads and replace the lid allow the child to handle them if they wish. Show the golden ones, ask how much they are and allow the child to handle if she wishes and replace the lid. Introduce the black and white bead stair and show how it is laid out as an inverted isosceles triangle.
- Take out a pile of coloured beads and put them in a pile on the Working Mat and FIRMLY CLOSE THE BOX, ask the child if she can count them all, she will say, no
- Make a snake with the coloured beads along the felt mat, the first combinations should add up to pairs of ten, e.g. 3, 7 – 9, 1 – 8, 2 – 6, 4 – 5, 7 – 3, 8

The First Presentation

- Count aloud, pointing to each bead with your left finger, when you reach ten take the bridge from the box and put it in between the chains, after the tenth bead. Tell the child, “We do not count more than ten”.
- Take a ‘bar of ten’ and put it above the counted beads, place the counted beads in the empty box that held the black and white bead stair.
- Put the ‘bar of ten’ n the place of the coloured beads at an angle and continue counting 1 to 10.
- Proceed in the same way, until you reach ten in the middle of a bead bar, then place the bridge directly after the tenth bead
- Ask the child to count the remaining beads on the bar, aid her by pointing with the right index, take a black and white bead bar of this quantity and put it near to the bridge.
- Put the coloured bead bar in the box as usual and link in the black and white bar.
- Count the first black or white bead as one and continue till ten, when you replace the beads with a ‘bar of ten’, replace the black and white bead in the bead stair
- Follow the procedure until the coloured beads have been replaced by golden ones with a black and white stair
- Place the bars of ten’ parallel and the remainder close by and count them

Note:

After working with the snake the child memorises the colours of the bead bars and begins to ‘skip count’ the first bar

The Second Presentation

- Prepare the material and proceed as above encouraging the child not to count the first coloured bar but to ‘see’ the amounts, using the colours as a guide and then linearly count the second bar.

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

- The child is shown how to check her work in the next exercise.

**Direct Aim:**

- To give the child the experience of combining units to make tens
- Memorisation of combinations used in basic addition
- To sensorially give the unconscious knowledge that no two basic numbers can give a sum greater than 18
- To reinforce the knowledge that addition and multiplication are part of the same concept

**Indirect Aim:**

- Introduction to the memorisation of multiplication combinations (2nd Control)

**Age at Presentation:**

Five to six years

**Footnote:**

- Here addition is shown as ‘making tens’
- ‘Seeing’ the value of the first bead bar prepares the child for using the blue slips with the charts

**Addition Snake Game **

**Material Description:**

- 3 red boxes on a tray, placed from left to right, containing;
- 5 sets of coloured bead stairs
- 25 golden beads
- 1 set of black and white beads from 1-9 and a notched card for the bridge
- A red felt cloth set out on a Working Mat

After sufficient exposure to the Snake Game the child may request verification of her process, if she does not give this presentation after a few weeks.

**Presentation:**

First Proof

- Layout the material and continue as usual until counting the ‘bars of ten’, again place them vertical and parallel, but now with a gap between. Match the coloured beads used to make the original snake placing them adjacent to the bars of ten. One at a time, starting with the longest. Ask the child what would be needed to ‘complete’ the bar of ten’ and add it from the coloured beads. It may be necessary to exchange them sometimes. At the end there will be no spare Golden beads. The child sensorially experiences addition.

Second Proof

- Layout the material and continue as usual until counting the ‘bars of ten’. This time separate the coloured beads from the Golden ones and any black and white bead stair ones. Place the coloured beads horizontally in groups of the same unit, beginning with whichever is the largest. Count all of the beads of one type until you reach a tens number and represent them with a Golden ‘bar of ten’. It may be necessary to place the bridge and exchange some ‘bars of ten’ with unused coloured beads of the units if you have a remainder. Place the ‘bars of ten’ vertically to make a T-shape, if the number is less than ten exchange a ten bar for the coloured beads and represent it in the same way. At the end there will be no spare Golden beads.

**Indirect Aim:**

- Introduction to the memorisation of multiplication combinations (2nd Control)

**Age at Presentation:**

Five to six years, after positive Snake Game

**Footnote:**

Depending on the child’s work with the materials of Group 3 the child may be able to count all of the coloured beads in the second proof or may need to put the bridge after reaching ‘ten’ and put a ‘bar of ten’ before proceeding.

**Addition Strip Board**

**Material Description:**

- 2 chowkis
- A board with 18 horizontal and 12 vertical squares (2cm
^{2}). The top squares are numbered 1 to 10 in red, 11 to 18 in blue. A vertical line divides the board after ’10’. - Nine graded blue strips, marked 1 to 9 at the end.
- Nine graded red strips with blue lines dividing them into units, marked 1 to 9 at the end
- Printed addition booklets and a pencil
- Control Chart 1: lists all the possible addition combinations and the answers in red
- Control Chart 2: lists the addition combinations with the answers eliminating all reverse combinations

**Presentation:**

Introduction

- Invite the child, saying, “I will show you another way to do addition”
- Bring the material to the chowkis
- Place the strips in sequence, 1 to 9, blue on the left, above or to the side of the board, so they look like the Red Rods
- Ask the child to read the numbers on the board and highlight the red line
- Place any blue strip on the top horizontal line, under the numeral 1
- Place any red strip on the top horizontal line, beside the blue strip
- State the value of each strip and show the child that the sum of the numeral is above the last unit
- Replace the strips
- Continue doing at least four more, some should be greater than ten and others less

Addition

- Take the addition booklet and read the first sum, 1 +1 = 2
- Show this using the strips, saying the numbers out loud, read the sum
- Record it in the booklet and replace the red strip and read the next sum, putting the red strip of the next sum, 2, in it’s place, next to the blue strip
- Read and record the answer
- Continue until 1+9=10
- Reread the booklet with the child, take the Addition Control Chart 1 and show the cild how to check her word reading the answers on the chart and booklet

Addition Chart 1 and Control

- This shows all the combination for basic addition, so that the child can control her work
- After the child has completed a few sums show the numeral on paper as the first addend in the chart

Let the child do other pages of the booklet in order of her interest, repeat the booklet and make her own booklets

**Exercises:**

- The child’s own work
- How many ways can a number be made (as Control Chart 2)?

- After sufficient experience, record a set of additions either horizontally or vertically along a sheet of long paper.
- Under each addition list other ways to reach the same sum
- Begin by using the strips on the board to make 2 and record ‘1+1=2’
- Then say, “How many other combinations can be made to get 2?”
- The child may realise this is impossible, or help her, then remove the red ‘1’
- Put the red ‘2’ by the blue ‘1’ and say, “How many other combinations can be made to get 3?”, record ‘1+2 = 3’, then continue, writing ‘2+1=3’ below it (if the chart is horizontal) or to the side (if the list is vertical).
- Return both strips and continue until ‘9+9=18’
- When the duplicates are removed the list looks like Control Chart 2

Variation:** **The child can list identical addends only, e.g. ‘1+1=2’, ‘2+2=4’…’9+9=18’, now she will have made Control Chart 2

- How many ways can sum number be made?

- Ask the child how many ways the number ‘9’ can be made
- Place the blue strip 1 and invite the child to find a single red strip to complete the number 9
- Record the combination and continue
- Check the work with Control Chart 1, showing her how to look diagonally to find the combinations

- 4a, Eliminating the Duplicates

- Make combinations of a number e.g. 9
- Read them, e.g. ‘1+ 8=9’
- Ask the child, “Can you see one and eight anywhere else?”, the child identifies the inverse duplicate. The Director says, “We don’t need to learn this one”, indicating the duplicate and slides it down to the lowest line on the chart
- Continue looking for duplicates.
- Continue with combinations of 8, for this number and other evens after eliminating the duplicates put the strip with identical addends in between the remaining sums and the duplicates

4b, Eliminating and writing the duplicates

- Let the child find all the duplicates by reading the top line and asking her if she can identify others which are the same.
- Write all the possibilities and let her cross out the duplicates or copy only those that remain after isolating the duplicates

Variation:** t**he child can cancel the duplicates from the long list of additions she made in Exercise 2

Addition Chart 2 and Control

- Introduce Chart 2, point out that only the essential combinations are given, the sums are along the same line as in Chart 1. Now the child can use this to verify her work.

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

- Charts 1 and 2

**Direct Aim:**

- To give the child the experience working out all of the possible basic addition combinations
- The child realises that for any combination above ten a ten and a number of units above ten is required; the red vertical line aids this understanding
- To help the memorisation of the addition table

**Indirect Aim:**

- Introduction to the commutative aspect of addition (that in addition the sequence of addends does not matter)

**Age at Presentation:**

Five to six years

**Footnote:**

- With the addition strip board this work also clarifies that in basic addition the sum cannot be greater than 18
- The red line sensorially indicates that numbers greater than ten are ten plus a unit

**Addition Charts 3, 4, 5 and 6**

**Material Description:**

Working Chart 3 (full) with the answers for all addition combinations

Working Chart 4 (half) with half of the answers, eliminating the reversible

Working Chart 5 (hopping chart) with answers 1 to 18

For Working Charts 3,4 and 5;

Control Charts 1 and 2

A small tray with squared paper and a pencil

For Working Chart 6, (blank chart), separate wooden tiles with all the answers for all of the combinations and Control Chart 3

For all Charts, a set of addition strips without answers in a red colour coded container and a separate container for solved slips

**Presentation:**

Chart 3

- Take any slip from the red box, show it to the child and copy the question onto the squared paper, e.g. ‘9 + 7 =’
- Tell her; “Now we are going to find out what happens when we put these numbers together”
- Ask her to identify the first written number and put the right index finger on the blue strip
- Ask her to identify the second written number and put the left index finger on the red strip
- Move the right index in line with the second and bring the second to meet the first finger (or vice versa)
- The square where the fingers meet is the sum; ’16’, record this on the squared paper
- Place the used slip in the empty box
- Take another slip and let the child continue for as long as she is interested, checking her work with the Control Chart 1 or 2
- Read the whole problem again after using the charts

Criteria of Perfection (Control of Error):

- Control Charts 1 and 2

Age at Presentation:

Five to six years, after Chart 2

Chart 4

- Take any slip from the red box, show it to the child and copy the question onto the squared paper, e.g. ‘9 + 7 =’
- Tell her; “Now we are going to find out what happens when we put these numbers together”
- Ask her to identify the first written number and put the right index finger on the red vertical strip
- Ask her to identify the second written number and put the left index finger on the same red strip
- Move the right index horizontally to the extreme right and then vertically until it is in line with the second, then bring the second finger to meet it (or vice versa)
- The square where the fingers meet is the sum; ’16’, record this on the squared paper
- Place the used slip in the empty box
- Take another slip and let the child continue for as long as she is interested, checking her work with the Control Cards 1 or 2
- Read the whole problem again after using the charts

Criteria of Perfection (Control of Error):

- Control Charts 1 and 2

Age at Presentation:

Five to six years, after Charts 1 and 2

Chart 5

- Take any slip from the red box, show it to the child and copy the question onto the squared paper, e.g. ‘5 + 3 =’
- Tell her; “Now we are going to find out what happens when we put these numbers together”
- Ask her to identify the written numbers and put both your index fingers on the red strip
- Move them horizontally to the end of the line, then ‘hop’ the fingers an equal number of times along the diagonal, towards each other.
- The square where the fingers meet is the sum; ‘8’, if the answer is an odd number it will be on the row below, move one finger one square ‘into’ the chart
- Record the sum on the squared paper
- Place the used slip in the empty box
- Take another slip and let the child continue for as long as she is interested, checking her work with the Control Cards 1 or 2
- Read the whole problem again after using the charts

Criteria of Perfection (Control of Error):

- Control Charts 1 and 2

Age at Presentation:

Five to six years, after Chart 4

Chart 6

- Take any slip from the red box, e.g. ‘7 + 5 = ‘, show it to the child
- Lay out the wooden squares in groups of the same numeral
- Tell her; “Now we are going to find out what happens when we put these numbers together”
- Ask her to identify the written numbers, mentally process the question and select the wooden square with the sum ’12’
- Proceed as for Chart 3, placing the square where the fingers meet
- Place the used slip in the empty box and take another slip and corresponding wooden square
- The child takes another slip and continues for as long as she is interested or until the board is complete checking her work with the Control Card 3

Criteria of Perfection (Control of Error):

- Control Charts 1 and 2

Age at Presentation:

Five to six years, after Chart 4

**Exercises**

- The Child’s own activity with the material
- Pick up any tile and the child thinks of suitable addends and places it
- The child takes all the tiles with the same numeral and places them on the board

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

- Control Charts 1, 2 and 3

**Direct Aim:**

- The memorisation of all the addition combinations, chart 6 serves as a 3rd Period

**Age at Presentation:**

Five to six years, after Charts 1 and 2

**Footnote:**

The various charts give the child the opportunity to repeat, helping her to memorise the combinations

**Subtraction Snake Game **

**Material Description:**

- 4 green boxes on a tray, containing;
- 5 sets of coloured (positive) bead stairs
- 20 golden beads
- 1 set of black and white beads from 1-9 and a notched card for the bridge
- 3 sets of grey (negative) bead stairs
- A green felt cloth set out on a Working Mat

After sufficient exposure to the Snake Game the child may request verification of her process, if she does not give this presentation after a few weeks.

**Presentation:**

Laying out the material

- Invite the child, saying, “I will show you how to do the Snake Game to Subtract”
- Bring the material to the Working Mat, unroll the felt mat. From left to right place the boxes containing the coloured beads, golden beads, bead stair and finally negative beads
- Open the box with the coloured beads, then the grey ones and then the golden beads. Allow her to familiarise herself with them before closing each box. Ask the child to lay out the black and white bead stair and lay it out as an inverted isosceles triangle, remove the bridge and put the lid under the box
- Remove a pile of golden beads and two grey bars of negative beads say, “We will take these (indicate grey beads) away from these (indicate coloured beads)” Then ask the child if she can guess how many beads there will be left, she will say ‘no’. Then FIRMLY CLOSE BOTH BOXES
- Make a snake with at least 50 coloured beads use the grey ones towards the end separated by a coloured bead bar or two, on the, felt mat

The First Presentation

- Count aloud, pointing to each bead, replacing them with a ‘bar of ten’ after the tenth bead as with the Addition Square Game
- When you reach a grey bead bar count the number of beads and then the same number of previous coloured beads, place the bridge directly after the last one
- Ask the child to count the remaining beads on the bar, aid her by pointing with the right index, take a black and white bead bar of this quantity and put it near to the bridge.
- Put the coloured, grey and ‘bar of ten’ bead bars away and link in the black and white bar.
- Say, “The snake just got shorter”
- Continue from the first black and white bead bar and follow the procedure until the coloured beads have been replaced by golden ones with a black and white stair
- Place the bars of ten’ parallel and the remainder close by and count them

The Second Presentation

- Isolate the bar previous to the negative bar and subtract directly.

Proof

- After converting the snake put the used coloured and ‘bars of ten’ bars on the right and the negative bars on the left. Match the negative beads with the coloured ones, placing them parallel, and then the golden beads with the golden beads with the coloured ones, Ask the child what would be needed to ‘complete’ the bar of ten’ and change the coloured beads when necessary with others from the box with a remainder. The child sensorially experiences subtraction.

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

- The child is shown how to check her work in the next exercise.

**Direct Aim:**

- Memorisation of combinations used in basic subtraction

**Indirect Aim:**

- Indirect preparation for algebra as sometimes equal positive and negative numbers cancel each other

**Age at Presentation:**

Five to six years

**Footnote:**

- For the first presentation prepare a long snake which will result in zero when the negative bars have been subtracted to create the biggest sensorial contrast with the Addition Snake Game
- Separating each subtraction, as in the second presentation makes a great impression on the memory. However, do not be tempted to do this without the snake as the ‘whole’ snake holds the child’s attention, her interest is in seeing the result of so much counting, while the shortened exercise of subtraction will tire her

Both notes Mario Montessori

**Subtraction Strip Board (Blue strips only)**

**Material Description:**

- 2 chowkis
- A board with 18 horizontal and 12 vertical squares (2cm
^{2}). The top squares are numbered 1 to 9 in blue, 10 to 18 in red. A vertical line divides the board after ’9’. - Nine graded blue strips, marked 1 to 9 at the end.
- Nine graded red strips with blue lines dividing them into units, marked 1 to 9 at the end
- A blank, varnished wooden set of strips the same lengths as for the coloured strips 1 to 17
- Printed subtraction booklets and a pencil
- Subtraction Chart 1: lists all the possible basic subtraction combinations

**Presentation:**

Introduction

- Invite the child, saying, “I will show you another way to do subtraction”,
- Bring the material to the chowkis
- Place the blue strips in sequence, 1 to 9, blue on the left, above or to the side of the board
- Ask the child to read the numbers on the board and highlight the red line, tell her, “These numbers show both the quantity from which we subtract and the difference that is left when we take away…The difference will always be found in the blue numbers”

Subtraction

- For the subtraction ‘7 – 3 = 4’
- Take a subtraction and find a varnished strip which will cover the row of numerals from the number above the minuend to 18, draw attention to the numbers that remain visible, so that ‘7’ is the largest visible number
- Place a blue strip with the value of the subtrahend on the top horizontal line, adjacent to the varnished strip, as you place it, covering 5, 6 and 7 say, ‘minus three’ and then indicate the numbers 1 to 4 which are still visible, ‘is four’.
- The combination is complete with the difference revealed subtrahend and minuend
- State the value of each part and show the child that the difference is the sum of the remaining squares
- Make it clear to the child that we will only record answers with blue numbers
- Replace the strips
- Continue doing several other examples in order to show the function of each part of the subtraction and to reinforce that we only record basic subtractions
- Writing is optional at this stage

Continued Practice

- Take the addition booklet and read the first subtraction, copy it
- Prepare the board with the varnished strip and using a blue strip, saying the numbers out loud, read the answer
- Record it in the booklet and replace the strip

Show the Control Chart before replacing the material

Let the child do other pages of the booklet in order of her interest

**Exercises:**

Exercise 1 The child’s own work

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

- Subtraction Chart 1

**Direct Aim:**

- To give the child the experience working out all of the possible basic subtraction combinations, leading to memorisation
- Exercise 2 shows the child the connection between addition and subtraction; as the reverse processes

**Indirect Aim:**

- Preparation for negative numbers

**Age at Presentation:**

Five and a half to six years

**Subtraction Strip Board Exercise 2 (Blue and Red strips)**

**Presentation:**

- Invite the child, saying, “I will show you another way to do subtraction another way”,
- Bring the material to the chowkis
- Place the blue and red strips in sequence, 1 to 9, blue on the left, red above or to the side of the board
- Choose a number to build, e.g. 4 and cover the unwanted numerals with the varnished strips .
- Place a red strip under the numbers 1 to 4
- Say, “I can make 4 another way”
- Ask the child to make all the possible combinations below the red strip this time place the red strips to the left (e.g. 3) and the blue to the right (e.g. 1)
- Record, showing the child any combinations she has missed
- Tell the child that the blue strip tells us the amount to be subtracted (the subtrahend)
- Beginning at the top, point a the single red strip, write “4 – 0 = 4”, saying, “There is no blue strip, four take away zero is four”
- On the row below slide the blue strip of ‘1’ from the red strip of ‘3’ to the far right side, saying, “Four take away one is three”
- The child continues working through all the combinations systematically

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

- Subtraction Chart 1

**Subtraction Charts 2 and 3**

**Material Description:**

Working Chart 1(full) Subtraction Control Chart

A small tray with squared paper and a pencil

For Working Chart 3 (blank chart) with separate wooden tiles with all the answers for all of the combinations

For all Working Charts, a set of subtraction slips without answers in a green colour coded container and a separate container for solved slips

**Presentation:**

Chart 2

- Take any slip from the green box, show it to the child and copy the question onto the squared paper, e.g. ‘9 – 7 =’
- Tell her; “Now we are going to find out what happens when we take seven from nine”
- Ask her to identify the first written number and put the right index finger on the top row
- Ask her to identify the second written number and put the left index finger on the blue strip
- Move the right index in line with the second and bring the second to meet the first finger (or vice versa)
- The square where the fingers meet is the difference; ’2’, record this on the squared paper
- Place the used slip in the empty box
- Take another slip and let the child continue for as long as she is interested, checking her work with the Control Chart 1
- Read the whole problem again after using the charts

Chart 3

- Take any slip from the green box, e.g. ‘7 – 5 = ‘, show it to the child
- Lay out the wooden squares in groups of the same numeral
- Tell her; “Now we are going to find out what happens when we take five from seven”
- Ask her to identify the written numbers, mentally process the question and select the wooden square with the difference ’2’
- Proceed as for Chart 2, placing the square where the fingers meet
- Place the used slip in the empty box and take another slip and corresponding wooden square
- The child takes another slip and continues for as long as she is interested or until the board is complete checking her work with the Control Chart 2

**Exercises:**

- The Child’s own activity with the material
- Take any tile and ask her to place it
- Pick up any tile and the child thinks of suitable minuend and subtrahend and places it, she can place all the tiles with the same number at one time.

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

- Control Charts 1 for Chart 2 and Chart 2 for Chart 3

**Direct Aim:**

- The memorisation of all the subtraction combinations, chart 3 serves as a 3rd Period

**Age at Presentation:**

Five and a half to six years, after the subtraction strip board and when the child knows some combinations

**Footnote:**

The various charts give the child the opportunity to repeat, helping her to memorise the combinations

**Multiplication Bead Bar Layout**

**Note:**

This is an introduction to the Memorisation of multiplication tables, just as the addition and subtraction snakes introduce the child to respective combinations.

**Material Description:**

- For presentation – box with 45 seven bars and 27 ten bars plus bead stair
- For multiplying any number 1 to 9

The Coloured Bead Bars of Multiplication, kept together in a box marked yellow

- 45 of each bead bar, 1 to 9
- For the answers – 4 one-bars, 12 two-bars, 4 three-bars, 12 four-bars, 9 five-bars, 12 six-bars, 4 seven-bars, 12 eight-bars, 4 nine-bars, and 166 golden ten-bars.

- For all the tables with he answers 1 to 10

- same material as for (2) plus 100 more tens

- Pencil and squared paper for Exercise 2

**Presentation:**

- Invite the child saying, “I will show you another way to do multiplication”
- Choose any number, e.g. ‘7’
- Place a bar of 7 horizontally on the left side of the felt mat, say, “seven taken once” and count the bead bar. Place a bar of seven beneath it to form a ‘T’ shape
- Adjacent to it, place two bars of seven, saying, “seven taken twice”, count the bead bars, when the counting reaches ten place the bridge, represent the ten with a Golden Bar of ten placed under the two ‘seven bars’ to for a ‘T’
- Continue counting the ‘seven bars’ placing a ‘four bar’ by the Golden Bar
- Continue placing, counting and exchanging until you place seven ten times.

**Exercises:**

- Over a period of time the child lays out all the tables

- Reoccurrence of ‘0’ (multiplying by 10)

- Write ‘2’ on squared paper and lay out ten ‘two bars’ parallel with each other
- Count the beads then show that the product is ’20’ by recording a ‘0’ on the paper to the right of the ‘2’
- The child continues with the other numbers

- How many ways can a number be made (using the product)?

To emphasis the application of the Commutative Law to multiplication use ‘bars of ten’ and coloured bead bars to show the essential combinations for building basic numbers

- Ask the child how many ways the number ’12’ can be made
- Place 6 two bars, one at a time, parallel. Beneath this place 4 three bars and then 3 four bars, attempt and acknowledge that bars of 5 cannot make ‘12’, then use 2 six bars.
- Say “Twelve is made with the bar of two taken six times”

“Twelve is made with the bar of three taken four times”

“Twelve is made with the bar of four taken three times”

“Twelve is made with the bar of six taken twice”

- Later make the link between different groups of bead bars, e.g. 2×6 converts to 6×2, to make the Commutative Law explicit

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

- The Child’s ability to count

**Direct Aim:**

- Memorisation of multiplication tables
- To show the geometric form of multiplication
- To show that the multiplier is not a concrete quantity but an indicator of how many times any number is taken
- The application of the Commutative law in multiplication

**Indirect Aim:**

- Showing that lines moving together in space make shapes
- Preparation for division
- Indirect preparation for geometry, algebra, square root and factoring

**Age at Presentation:**

Five and a half to six and a half years, parallel to Bead Frame

**Multiplication Board**

**Material Description:**

- 2 chowkis
- A large board with 100 shallow holes in rows of ten. Numbers 1 to 10 are written in a horizontal row along the top, at the left there is a slop for the multiplicand card
- A set of small white cards, 1 to 10
- A skittle
- 100 red beads
- Printed multiplication booklets and a pencil
- Multiplication Control Chart 1: lists all the possible basic multiplication combinations and the answers

**Presentation:**

Introduction

- Invite the child, saying, “I will show you another way to do multiplication”
- Bring the material to the chowkis
- Show the child the cards with the multiplicands, let her choose one and use it’s booklet
- Place a card representing the multiplicand in the slot ‘3’ and a skittle for the multiplier ‘1’ above the printed number ‘1’ on the horizontal line
- Count three beads and place them in the first vertical row, saying “one is taken three times”
- Record the product on paper
- Leave these three beads and get three more, place a skittle for the multiplier ‘2’ above the printed number ‘2’ on the horizontal line
- Complete the next vertical strip, counting from 3. Then count all the beads together, finally stating, saying “two is taken three times”
- Continue until 3×10, after a few beads let the child remember the product from the previous exercise and begin counting from there if she initiates this

- The child works with the Multiplication Control Chart to complete all of the tables

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

- The Multiplication Control Chart

**Direct Aim:**

- Memorisation of multiplication tables

**Age at Presentation:**

Five and a half to six and a half years, after Bead Bar layout

**Footnote:**

- This material builds on the child’s work with the chain counting in Group 3

**Multiplication Charts 2, 3, 4 and 5 **

**Material Description:**

Multiplication Control Chart

Control Chart 2 Second Multiplication Control Chart with half the answers for all combinations

Working Chart 3 (full) with all the answers for all combinations

Working Chart 4 (half) with half of the answers, eliminating the reversible

For Working Charts 3 and 4; a small tray with squared paper and a pencil

For Working Chart 5, (blank chart), separate wooden tiles with all the answers for all of the combinations and Working Chart 3

For all Working Charts, a set of addition slips without answers in a yellow colour coded container and a separate container for solved slips

**Presentation:**

Chart 3

- Take any slip from the yellow box, show it to the child and copy the question onto the squared paper, e.g. ‘3 x 7 =’
- Tell her; “Now we are going to find out what happens when we take three seven times”
- Ask her to identify the first written number and put the right index finger on the blue strip
- Ask her to identify the second written number and put the left index finger on the red strip
- Move the right index in line with the second and bring the second to meet the first finger (or vice versa)
- The square where the fingers meet is the product; ’21’, record this on the squared paper
- Place the used slip in the empty box
- Take another slip and let the child continue for as long as she is interested, checking her work with the Control Chart 1
- Read the whole problem again after using the charts

Criteria of Perfection (Control of Error):

- Control Chart 1

Chart 4

- Take any slip from the yellow box, show it to the child and copy the question onto the squared paper, e.g. ‘9 x 7 =’
- Tell her; “Now we are going to find out what happens when we take nine seven times”
- Ask her to identify the first written number and put the right index finger on the red vertical strip
- Ask her to identify the second written number and put the left index finger on the same red strip
- Move the right index horizontally to the extreme right and then vertically until it is in line with the second, then bring the second finger to meet it (or vice versa)
- The square where the fingers meet is the product; ’63’, record this on the squared paper
- Place the used slip in the empty box
- Take another slip and let the child continue for as long as she is interested,
- Demonstrate to the child how to check her work with the Control Chart 2
- Read the whole problem again after using the charts

Criteria of Perfection (Control of Error):

- Control Chart 2

Chart 5

- Take any slip from the yellow box, e.g. ‘7 x 5 = ‘, show it to the child
- Tell her; “Now we are going to find out what happens when we take seven five times”
- Ask her to identify the written numbers, mentally process the question and select the wooden square with the product ’35’
- Proceed as for Chart 3, placing the square where the fingers meet
- Place the used slip in the empty box and take another slip and corresponding wooden square
- The child takes another slip and continues for as long as she is interested or until the board is complete checking her work with the Control Chart 1

**Exercises**

- The Child’s own activity with the material
- Pick up any tile and the child thinks of suitable multiplier and multiplicand and places it
- The child picks any time and places

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

- Control Charts 1, 2 and 3

**Direct Aim:**

- The memorisation of all multiplication, chart 5 serves as a Third Period

**Age at Presentation:**

Five and a half to six and a half years, after Charts 1 and 2 and the child knows many tables

**Footnote:**

The various charts give the child the opportunity to repeat, helping her to memorise the combinations

Also give Mental Arithmetic Word Problems

**Unit Division Board**

Note: As learning division tables happens automatically when multiplication tables are learnt there are fewer activities here. The inverse relationship between division and multiplication is shown sensorially.

**Material Description:**

- 2 chowkis
- A large board with 81 shallow holes in rows of nine. A green horizontal row along the top of the board has indentations for skittles, along the left side the numbers 1 to 9 are printed vertically
- 9 green skittles
- 81 green beads
- Printed division booklets with 81 pages and graphite and coloured pencils
- The above are kept in boxes marked with blue

**Presentation:**

Introduction

- Invite the child, saying, “I will show you another way to divide”,
- Bring the material to the chowkis
- Remind the child that division mean ‘to split numbers EQUALLY”
- Write the numbers to be divided, e.g. ’27 – 9 = ‘
- Take nine skittles, representing the divisor in the indentations
- Count twenty-seven beads, the dividend and share them horizontally in the holes of the first three rows
- Read what each skittle has and record the quotient, reminding the child that the answer is always how many one skittle has
- Remove the beads
- Write the next puzzle, ’27- 8 = ‘ and share the beads in the same way as above, record the quotient and the remainder, ‘3 r.3
- Continue until the quotient would exceed 9, say, “The answer must be in the units, so we cannot do it with this board”

Presentation

- Referring to the first page of the 81 page division booklet begin with ’81 – 9 = ‘
- Remind the child that division mean ‘to split numbers EQUALLY”
- Place the skittles, take the beads and share them equally, read and record the quotient and clear the board of beads
- Next try ’81 – 8 = ‘, remove one skittle, take the beads and share them equally, but then discover that the board is full, leaving a remainder of 10. Explain that the remainder must not be larger than the divisor and must be a single unit, so this puzzle cannot be solved using this board, hence the quotient is not recorded.

- Referring to the second page of the 81 page division booklet begin with ’80 – 9 = ‘Replace the skittle and remove one bead from the 81 and share them equally, read and record the quotient and remainder and clear the board of beads
- Next try, ’80 – 8 = ‘, remove one skittle, take the beads and share them equally, read and record the quotient and clear the board of beads

**Exercises:**

- The child continues over a length of time until the booklet is complete
- Take a coloured pen and underline all the combinations in the completed booklet which have no remainders
- Later, if the child has not realised, draw her attention to the relationship between the dividend, divisor and quotient, 12 -4 = 3 and 12- 3 =4
- Relationship with Multiplication

- Bring the coloured bead bars of multiplication. Choose a common dividend and divisor (e.g. 12 – 3)
- Let the child discover the answer using the Division Board, then lay out the beads, here the bar of 3 four times. Verbalise the procedure, “3 taken four times is 12”
- Then show the reverse, using the bar of 4 three times
- Give further examples

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

- There is no external control of error, the child’s careful counting, knowledge of multiplication tables and the process help to guide her

**Direct Aim:**

- Memorisation of multiplication tables

**Age at Presentation:**

Five and a half to six and a half years, when the child knows her multiplication tables well

**Footnote:**

89 beads can be used instead of 81 because it is the largest dividend in basic division, giving a quotient of 9 r.8

**Division Chart 1 and Chart 2 **

**Material Description:**

Division Working Chart 1 (full)

Division Working Chart 2, (blank chart) separate wooden tiles with all the answers for all of the quotients

A set of division slips without answers in a blue colour coded container and a separate container for solved slips

**Presentation:**

Chart 1

- Take any slip from the blue box, show it to the child and copy the question onto the squared paper, e.g. ’54 – 9 = ’
- Tell her; “Now we are going to find out what happens when we split fifty-four equally among nine”
- Ask her to identify the first written number and put the right index finger on the blue strip
- Ask her to identify the second written number and put the left index finger on the red strip
- The square where the fingers meet is the quotient; ’6’, record this on the squared paper
- Place the used slip in the empty box
- Take another slip and let the child continue for as long as she is interested, checking her work with the Division Control Chart
- Read the whole problem again after using the charts

Note

The numbers in the horizontal line in white are the basic prime numbers (3,5, and 7)

Chart 2

- Take any slip from the blue box, show it to the child and copy the question onto the squared paper, e.g. ’21 – 3 =’
- Lay out the wooden squares in groups of the same numeral, or in stacks
- Tell her; “Now we are going to find out what happens when we share twenty-one equally among three”
- Ask her to identify the written numbers, mentally process the question and select the wooden square with the quotient ‘7’
- Proceed in the same way with the other slips, placing the square where the fingers meet
- Place the used slip in the empty box and take another slip and corresponding wooden square
- The child takes another slip and continues for as long as she is interested or until the board is complete checking her work with the Division Control Chart
- If the child is unsure of an answer she can put it to one side and later use the Division Control Chart or invert the Division Control Chart while she is working and rotate it to check her working or find the answer when she pleases

**Exercises**

- The Child’s own activity with the material
- Take any tile and ask her to place it
- Pick up any tile and the child thinks of suitable dividend and divisor and places it

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

- Division Control Chart, the child’s completed division booklet, the child’s answers using the Unit Division Board or the Multiplication Control Chart

**Direct Aim:**

- The teach the child the division combinations, chart 2 serves as a Third Period

**Age at Presentation:**

Five and a half to six and a half years onwards

**Footnote:**

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