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MONTESSORI TEACHER ALBUMS - Children's House (3-6) - Math
READ: The Absorbent Mind: Chapter 17, Discovery of the Child: Chapters 18 & 19

Table of Contents

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NUMBERS TO TEN

Number Rods

Purpose: To learn to count to ten and understand the value of each number. To learn the names one to ten in association with the quantities.

Materials: Ten wooden rods varying in length from 1 dm to 1 m. Each decimeter is colored alternately red and blue. The first rod is red; the second rod is red-blue; the third rod is red-blue-red, and so on. The rods correspond in length to the sensorial red rods. A mat for the floor.

Presentation 1: Invite the child to set out the mat on the floor and to help you carry the rods as before. Ask the child to arrange the rods in stair formation beginning with the 'one' rod as we want the stair to grow. When the child is familiar with the rods proceed to teach the names by the Three Period Lesson. First: Bring down the one, two and three rods towards the child. Place the one rod in front of the child and say its name a few times. This is 'one', 'one'. Remove the one rod to the side and put the 'two' rod in front of the child and name it. This is 'two', 'two'. Count it while touching each section and say: "One, two". Put the two rod to the side and place the three rod in front of the child and name it: This is 'three', 'three'. Count it while touching each section and say, 'one', 'two', 'three'. Second: Put the three rods in front of the child in mixed order, parallel and a short distance apart from each other. Ask the child to give you the 'one' rod, then the 'three' rod, then the 'two' rod. Playfully, ask for the rods again by name until you know the child knows them by name. Third: Place a rod in front of the child and ask him to name it. 'Which rod is this, count it for me.' Place that rod to the side and place another rod in front of the child and do the same with the third rod. Always end your lesson by placing the rods in order.

Exercise: The child works with the material as in the presentation.

Presentation 2: Continue to teach the names of the other rods three at a time, always reviewing the previous rods until the child knows how to count the rods using their correct names.

Control: The uneven formation of the rods, the number of sections in the rod, the teacher.

Age: 4 - 5 years

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Sandpaper Numbers

Purpose: To teach the written symbol corresponding to the numerals one to ten and to make the association between the name of the numeral and its symbol.

Materials: A set of numerals from 1 to 9 cut out of fine sandpaper and each mounted on a separate card. There is also a symbol for zero.

Presentation: Invite the child to come with you to where the Sandpaper Numerals are kept. At a table sit beside the child and take the first three number cards with the numerals 1, 2, 3 represented on them. First: Take the number card with the numeral '1' and trace it with the finger tips saying, "This is 'ONE' and this is how we write it." Repeat the name a few times. Invite the child to trace the numeral '1'. Place the numeral '1' to the side and take the numeral '2' card and do the same as with '1'. Then the numeral '3' card. Second: Place the three number cards in front of the child. Ask him to find a particular numeral and to feel it until he has recognized each of the three numerals. Third: Place one of the number cards in front of the child. Ask him to trace the numeral and tell you its name. Complete the lesson by placing the numerals learned in sequential order from left to right. On another day review the numerals already learned and teach the next three numerals. Continue on subsequent days until all numerals are learned.

Exercise: The child traces and names the numerals. He may also wish to write them.

Control: The sandpaper and the smooth surface of the card.

Age: 4 - 5 years

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NUMBER RODS & CARDS

Purpose: To associate the written symbol with the quantity. - To give further experience of the sequence of numbers.

Materials: The number rods and a set of number cards on which are printed the numerals 1 to 10. - One or two mats may be used depending on the space available.

Presentation 1: Using two mats place the number rods in mixed order on one mat and the number cards stacked in mixed order on the other mat. Show the child a card and ask him to name the numeral. Ask the child to find the number rod from the other mat to correspond with the number card and place the card on the correct section of the number rod. The child continues to work in this way until all the number cards and number rods have been matched. Always finish the lesson by placing the rods and numerals in correct sequence.

Presentation 2: (reverse exercise of the previous presentation) Using two mats place the number cards in mixed order on one mat and the number rods in mixed order but parallel on the other mat. Show the child a rod and ask him to name it and to count it. Ask the child to find the number card from the other mat to correspond with the rod and place it on the correct section of the rod. The child continues to work in this way until all the rods and number cards have been matched. Always finish the lesson by placing the rods and numerals in correct sequence.

Presentation 3: Composition and Decomposition of Numbers 1-10: a) The number rods are arranged in sequential order. Isolate the 'ten' rod at the top of the mat by shifting the other rods down to the bottom of the mat. Say to the child that you are going to make tens with the number rods. Place the 'nine' rod against the 'ten' rod aligning both at the left hand side and ask the child which rod should we add to the 'nine' rod to make the 'ten' rod. By a process of trial and error the child finds that the 'one' rod and the 'nine' rod make ten. Count the 'nine' rod and say: "Nine and one make ten." The child continues making all the combinations of ten until he reaches the five rod. Show him how to flip it to the right hand side and say, 'If we had another five we could make 'ten'. Note: Number cards may be used to state equation together with the 'plus' symbol. Show the child how to make nines, eights, etc. b) The child makes tens as in Exercise 'a'. Begin at the bottom and work up. Review 5 and if we had another five it would make 10. Take away 5 by flipping the five rod back. Ask what is left. State equation: 10 take away 5 is 5. Repeat for the other combinations of tens and then the nines, eights, etc.

Control: The colored sections guide the child's memory. Counting the sections carefully.

Age: 4 - 5 years

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SPINDLE BOXES

Purpose:
To give the idea that symbols also represent a quantity of separate objects whereas in the exercise with the number rods the quantity was fixed and the symbols were loose. To clarify the meaning of 'zero' by having an empty compartment.

Materials:
Two plainly varnished wooden boxes with ten compartments. The first compartment has the symbol for 'zero' printed on the back, the second compartment has the numeral 'one' and so on up to nine. Forty five spindles in a basket or box. Eight green pipe cleaners (optional).

Presentation:
Invite the child to help you carry the spindle boxes, then the basket of spindles and the pipe cleaners to the table. Ask the child to identify the printed numerals 1-9 in non sequential order. Say to the child that you are going to read the numerals on the back of the compartments and that you will count the correct number of spindles into your hand before you place them in the particular compartment. He may bind them with the pipe cleaners before placing them in the compartments if he wishes. When all the forty-five spindles have been used, draw the child's attention to the empty compartment and say, "This is 'zero'. Zero means 'nothing'."

Exercise: The child works with the materials as shown.

Control: Correct number of spindles.

Age: 4 - 5 years

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MEMORY GAME OF NUMBERS

Purpose:
To train the memory by keeping the image of a number in the mind over a period of time. - To show that all objects may be counted.

Materials:
Slips of paper, on each is written a single numeral (0 - 10). - A container for the slips. - A box of identical objects like buttons, beads, or marbles.

Presentation:
Invite a group of eleven children to play the game with you. At a mat, give a child one slip, ask him to look at his numeral carefully, place the slip on the mat and go off to find that number of specified objects. Repeat for each child. When all the objects have been gathered, ask each child, individually, to identify their numeral and count their objects. When you reach the child who has '0', playfully reinforce the concept by saying, "You didn't bring anything - you must have the zero!"

Exercise: As in the presentation.

Control: The teacher and the child check by reading the slip of paper with the numeral written on it and counting the objects.

Age: 4 - 5 years

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CARDS AND COUNTERS

Purpose:

Direct:
To recognise the numerals 1 - 10 and their correct sequence. - To understand how many separate units form each number. - To give visual and muscular impression of odd and even numbers.

Indirect:
Preparation for the divisibility of numbers, therefore of multiples and submultiples.

Materials:
Number cards with numerals from 1-10 and 55 counters of one color.

Presentation:
Invite the child to bring the box of counters and the number cards to a table. Place the cards in disarray and place the box of counters near the child. Ask the child to find the numeral '1' and to place it at the top left corner of the table. Then ask, "How many counters will we put under '1'?" If the child says 'one', say, "Yes, we put one counter below the number card with the numeral '1'." Ask what comes after 'one'. If the child says 'two', ask him to find the numeral 'two' and show him how to place it, find the correct number of counters and place them as a pair. Continue in this way placing the even numbers in pairs and the odd numbers with the odd counter on its own below the pairs. Explain the concept 'odd' and 'even'.

Exercise: As in the presentation.

Control: The exact number of counters.

Age: 4 - 5 years

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DECIMAL SYSTEM

PRESENTATION WITH THE GOLDEN BEAD MATERIAL

Purpose:
To familiarise the child with the names of the different categories and to acquaint him with the relative difference in size of the categories, e.g. the difference between the quantity of six units and six thousands.

Materials:
One unit bead; one bar of ten beads; one square of a hundred beads; one cube of one thousand beads all on a tray. A supply of units, tens, hundreds and thousands. Empty trays, castors.

Presentation 1:
Individual exercise with presentation tray. Take the presentation tray to the table. Position the tray so that the unit is always to the right. Tell the child that this is the 'Golden Beads'. Hold and experience the unit, stating its name, "This is a unit." Give the child the bead. Encourage the child to repeat the name. Continue in the same manner experiencing a ten, a hundred, a thousand respectively. Proceed into the second period and eventually the third period of the Three Period Lesson.

Exercise: The child works with the material as shown.

Presentation 2:
Individual exercise with a tray of Golden Beads.

Materials: 9 Unit Beads, 9 Ten bead bars, 9 Hundred Square, 1 Thousand Cube

Count through each hierarchy in order from units to thousands. Each time nine is reached, state that if we had one more we would have ten. Instead of having ten loose beads (or tens, or hundreds) we can have a ten bead bar (or a hundred square, or a thousand cube).

Exercise: The child works with the material as shown.

Presentation 3: A small group exercise. Show the tray of Golden Beads to the children. Each child gets an empty tray with a castor. Ask each child, individually, to fetch a quantity of beads from the tray - one quantity per child. Upon their return ask each child, "What amount did you bring me?" The children will replace the beads in the tray. Continue as above according to the interest of the children. On another day the quantity may include two hierarchies until all the hierarchies have been included. Establish that when counting the category the name is included ie. five tens, six hundred, etc.

Control: The teacher and the child's counting of the beads.

Age: 4 - 5 years

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PRESENTATION WITH THE LARGE NUMBER CARDS

Purpose: - To acquaint the child with the written symbols for the new quantities he has learned.

Materials: - A box containing four sets of large number cards - one set from 1 - 9, with numerals in green - one set from 10 - 90, with numerals in blue - one set from 100 - 900, with numerals in red - one set from 1000 - 9000, with numerals in green - a tray of Golden Beads for review

Presentation 1: Individual exercise.

Bring a box of Large Number Cards to a table/mat. Remove the number card for 1, 10, 100, 1000. Familiarise the child with the color coding and review 1 and 10 which he has learned previously by referring to the Golden Beads. Place them as headers at the top of the mat. Give a Three Period Lesson with the new numerals, i.e., 100 and 1000. At the end of the lesson put the number cards in correct sequence with the thousand on the left and the one on the right.

Exercise: The child works with the material as shown.

Presentation 2: Individual exercise.

Remove all the Large Number Cards from the box. Stack the cards for each category in sequential order with the thousands on the left and the units on the right at the bottom of the mat. Place the '1' at the top right corner and continue placing and counting the unit cards in a vertical column until nine is reached. Then place the 10 - 90 to the left of the units, the 100 - 900 to the left of the tens and the 1000 - 9000 to the extreme left of the hundreds. As you place the number cards count them using their category name, i.e., one unit, two units, three units, etc. Show the child how to replace the number cards by stacking them in their categories with all the 'ones' showing on top. Bind them with a well fitting elastic band and replace them in the box.

Exercise: The child works with the material as shown.

Presentation 3: Small group exercise.

Ask the child to help you lay out the number cards as in exercise 2. Give each child an empty tray. Ask each of them to bring you certain number cards beginning with the unit category and building up as in Presentation 3 with the Golden Beads.

Control: The teacher.

Age: 4 - 5 years

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FORMATION OF LARGE NUMBER CARDS WITH BEADS

Purpose: Direct: - To make the child more familiar with the different categories of numbers, especially in regard to reading the written form. - To give him the language for large numbers.

Indirect: Preparation for working with: - The hierarchy of numbers, i.e. that while the numerals are always from 1-9, it is the place they occupy in the large number that gives them their value. - In a number '0' holds a place for a specific category. - Only 1-9 cards of each category is necessary to form any number.

Materials: - 9 unit beads, 9 ten bars, 9 hundred squares, 1 thousand cube. - A set of large number cards 1 - 9, 10 - 90, 100 - 900, 1000. - Three empty trays, three castors, two large mats.

Presentation: Small group exercise.

Layout: Set out two mats on the floor. Bring a tray of beads to one mat. Lay out the beads, similar to the layout for the number cards. Beginning with the units, lay out the beads for each category, individually, in a straight column working from top to bottom. Count each bead using the category name as you lay it out. On the other mat the children lay out the number cards as in the previous exercise. Note: only 1000 card corresponding to only 1 thousand cube is necessary at this stage. Each child gets a tray with a castor for holding the unit beads.

1. Single Category - Cards to Beads e.g. Units Place one number card of a single category on each child's tray. Ask the child to identify it and to obtain that quantity of beads from the other mat. When the child returns, he reads the card and counts the beads. The child returns the card and the beads to their mats.

2. Single Category - Beads to Cards Place quantity of beads from one category on the child's tray. The child counts the beads and obtains the corresponding card from the other mat. Check.

3. Two Adjacent Categories - Cards to Beads As in presentation 1a except place two cards which represent two adjacent categories on to the tray. Show the child how to superimpose the number cards and how to read the combined number, e.g., two tens and eight units.

4. Two Adjacent Categories - Beads to Cards As in presentation 1b except place a quantity of beads from two adjacent categories on to the tray.

Continue in this manner until all four categories have been included and proceed to non-adjacent categories as the child progresses.

Control: The teacher and the child's counting of the beads.

Age: 4 - 5 years

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COMBINATION OF GOLDEN BEAD AND LARGE NUMBER CARDS

Materials: - A set of Large number Cards. 1 - 1000. - A quantity of Golden Beads: 45 units 45 ten bead bars 45 hundred squares 1 thousand cube

Purpose: To enable the child to see the number system laid out with the numerals and their corresponding quantities.

Presentation:

The teacher shows the child how to place the number card '1' on the top right-hand corner of the mat and how to place the quantity, one golden unit, to the right of the card. The child continues in this way until he has used all the golden bead material for the units. The teacher shows the child how to place the number card for '10' and the ten bead bar parallel to the number card and the child continues until he reaches '90'. He then continues with the hundreds and the thousand.

Control: Visual and counting the number cards and the Golden Beads. All the material should be used.

Age: 4 - 5 years

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TEENS AND TENS

TEENS, FORMATION OF QUANTITIES 11-19 WITH GOLDEN BEADS

Purpose: - To teach the names of 11 - 19 in association with the correct quantity.

Materials: - A box with nine golden ten bead bars and a set of colored Bead Stair. - A small, neutral color, felt mat.

Presentation:

The lessons on Teens and Tens are usually given after the introduction to the decimal system exercises, but may be presented when the need for the terminology arises.

At a table place the colored bead stair in disarray on the felt mat. Ask the child to count the beads on each bar and place them in stair formation. Give the child time to acquaint himself with the material.

1. Take three of the ten bead bars from the box. Place one of the ten bead bars in a vertical position in front of the child. Place the one red unit bead touching the 'one' on the ten bead bar and say, "This is eleven." Repeat the name a few times. Bring the 'eleven' to the left side and place another ten bead bar as before with the green, two bar touching the first two units of the ten bead bar and say, "This is twelve - twelve ... twelve." Do the same for thirteen. 2. Place the three quantities in front of the child and ask him to point to each quantity in non-sequential order. 3. Place one quantity in front of the child and ask him to name it. Always end with your material in order.

Proceed with 14, 15, 16 and then 17, 18, 19 always reviewing the previous step.

Exercise: As in the Presentation.

Control: The pattern formed by the beads - the tens remain constant while the units increase as the child builds 11-19 in a vertical column.

Age: 4 - 5 years

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FORMATION OF 11-19 WITH TEEN BOARD

Purpose: - To associate the names eleven to nineteen with the symbols.

Materials: - A number frame consisting of two boards numbered with nine tens. - A set of number cards, with the numerals 1 - 9, which will slide into the boards to cover the 0 in each ten.

Presentation:

At a mat on the floor place the boards in a vertical column. Stack the cards from 1 - 9 at the top right corner of the boards. The child identifies the 10's. Using the THREE PERIOD LESSON teach the names of the numerals by sliding the unit cards over the zeros to create the numerals 11-19. At first teach 11, 12, and 13 and continue if the child shows interest until the child knows 11-19. Playfully, ask the child to name the numerals in non-sequential order. End the lesson by asking the child to read the numerals in sequential order. Show how to replace the number cards and the boards and to return them to the shelf.

Exercise: As in the presentation.

Control: The teacher.

Age: 4 - 5 years

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COMBINATION OF TEEN BOARD AND BEADS TO FORM 11 TO 19

Purpose: - To associate the quantities with the numerals 11 to 19.

Materials: - Teen boards, a box containing nine golden ten bars and a set of colored bead stair 1 to 9.

Presentation:

Invite the child to bring the Teen Boards to a mat on the floor. Show the child how to lay out the boards as before and then to arrange the beads to the left of the board. Slide in the first card and ask the child to read the numeral and to make the quantity. He makes eleven with the beads directly to the left of the numeral. Ask him what comes after eleven and if he says twelve he may proceed to build the quantity for twelve.

Exercise: The child continues in this way until he has all the numerals from 11 -19 in place and their corresponding quantity in the beads.

Control: The teacher. The child sees the pattern of the beads and recognises the sequence of the numeral on the board.

Age: 4 - 5 years

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FORMATION OF 10 - 90 WITH THE GOLDEN BEADS

Purpose: - To teach the names 'twenty', 'thirty', etc., with the ten bead bars.

Materials: - Forty-five golden ten bead bars. - A felt mat.

Presentation: Invite the child to place the mat on the table and to bring the box of forty-five ten bead bars. Remove six ten bead bars. Place one ten bar in front of the child and say, you know what this is. Yes it is 'ten'. Remove the ten to the left and place two ten bars in front of the child and say, This is 'twenty' ... 'twenty'. Remove the two ten bars and place three ten bars in front of the child and say: This is 'thirty' ... 'thirty'. Proceed with the Three Period Lesson until the child understands the new concept. On another day teach 40, 50, 60 and then 70, 80, 90 until the child is quite familiar with how we make the quantity 10 - 90 and the correct terminology.

Exercise: The child works with the material as shown.

Control: The teacher and the child as he counts the tens and learns the new names.

Age: 4 - 5 years

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FORMATION OF 10 - 90 WITH THE TEN BOARD

Purpose: - To teach the names 'twenty' to 'ninety' in association with the written numeral on the Ten Board.

Materials: - The Ten Board - A floor mat

Presentation:

The child has already learned the names in association with the quantities and therefore he can be taught the names of of all the numerals in one lesson through the Three Period Lesson.

Exercise: The child reads the numerals on the Ten Board.

Control: The teacher and the sequence of number.

Age: 4 - 5 years

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COMBINATION OF 10 - 90 WITH GOLDEN BEADS AND TEN BOARDS

Purpose: - To teach the names 'twenty, 'thirty', etc. and to associate the name with the quantity and the numeral. - Preparation for skip counting.

Materials: - Two boards with the numerals 10, 20, to 90 printed on them. - A box of forty-five ten bead bars. - A floor mat.

Presentation:

Invite the child to place the boards in order on the floor and to place the box with the bead bars to the left of the board. Ask the child to read the numerals on the board beginning with '10'. Show him how to place one ten bead bar immediately to the left of the numeral. Ask him to read the next numeral. He will probably say 'two tens'. Say, "Yes, you are quite right but remember it has a special name - it is 'twenty'." "Can you say 'twenty'?" "Now we are going to make 'twenty' with the ten bead bars." "How many will we need?" "Yes, two." "Two tens make 'twenty'." Proceed in this way saying, "This is how we write thirty." "Can you make thirty for me?"

Exercise: The child works with 10, 20, 30 and may continue to 40, 50, 60 and then 70, 80, 90 with the teacher naming the new names forty, fifty, sixty, etc. for the numerals. When he has reached '90' he will see very clearly how the 'tens' have grown from 'one ten' to 'nine tens' (90) and, as in the previous exercise, he can also see and associate the correct name with the numeral and the quantity.

Control: The teacher and the child as he balances the quantity with the written symbol.

Age: 4 - 5 years

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FORMATION OF 11 - 99 WITH GOLDEN BEADS AND TEN BOARD

Purpose: - To learn the terminology for 11 - 99 with the aid of the bead material and the written symbol.

Materials: - The Ten Boards - A Box of Golden Beads containing nine golden ten bead bars and nine golden units. - A floor mat.

Presentation:

Two children can work together at this stage with the teacher's guidance. Ask the child to place the Ten Boards in position and to place the nine ten bead bars together with the nine golden units to the far left of the '10' section leaving space for the quantities 11 - 99 immediately to the left of the Board. Ask the other child to stack the number cards in order 1 to 9. Ask one child to make 'eleven' with one ten bead bar and one unit - the unit touching the 'one' bead in the ten bead bar. Ask the other child to make the numeral 11 by slipping the '1' unit card over the zero in 10. Say, "Ten and one make eleven. This is how we write 11." Continue in this way until the children have reached '19' and say if we had one more golden unit we would have ten. Instead of having ten loose unit beads we can replace the units to the left and exchange them for a ten bead bar. Each time a number card is removed, place it face downward in a pile. Bring down the ten bar from the 10 section and with the 'new' ten place them to the left of the 20 section and say this is 'twenty'. Now let us make 'twenty one' by placing a golden unit to the right of the ten bead bars touching the 'one' bead in the ten bead bar nearest the Board. Say, "Now we have made 'twenty-one'." Turn the number cards to show the '1' at the top and proceed to slip the unit cards over the '0' in '20' as the other child makes twenty-one to twenty-nine with the Golden Bead material. Continue in this manner until '99' is reached when the nine ten bead bars and the nine Golden Units show at the left of the Board and the numeral 99 appears on the last section of the Board.

Control: The teacher and the material.

Age: 4 - 5 years

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CHANGING EXERCISE

Purpose: - To give experience in the changing process, i.e., 10 of one category make 1 of the next higher category. - To give practice in changing as a preparation for further exercises.

Materials: - A tray containing a supply of units, tens, hundreds and thousands. - A dish large enough to hold a supply of units. - Castors, large mat.

Presentation: A small group exercise.

- Units to Tens A good supply of units and tens on a tray and a castor for each child. Invite the children to help you count the units. Begin counting the units into a castor - stop when you reach ten. Remind the children that there are ten units in one ten. Explain that when you count to 10 you must exchange the 10 units for 1 ten. When all the exchanges have been made, read the quantity of beads, e.g., seven tens and four units/seventy-four.

- Tens to Hundreds Using the same procedure exchange 10 tens for 1 hundred.

- Hundreds to Thousands Using the same procedure exchange 10 hundreds for 1 thousand.

EXERCISE 1: The child works with the material as shown.

Presentation: At a later date do the reverse process (before starting subtraction).

- Thousands to Hundreds A good supply of thousands at a mat. Remind the children that 10 hundreds are exchanged for a thousand. Explain that you can reverse this procedure: exchanging 1 thousand for 10 hundred. Children exchange thousands in a systematic manner.

- Hundreds to Tens Using the same procedure exchanging 1 hundred for 10 tens.

- Tens to Units Using the same procedure exchanging 1 ten for 10 units.

EXERCISE 2: The child works with the material as shown.

Presentation:

Place a quantity of units, tens, hundreds and thousands all together in a pile on a mat. Playfully, say to the children, "I wonder how much we have here?" "How can we find out?" Hopefully the children will suggest sorting the quantity into their hierarchies and then count them beginning with the units and changing when necessary until they have all been counted. Get the child to use the Large Number Cards to show the quantity in written form.

EXERCISE 3: The child works with the material as shown.

Control: The teacher.

Age: 4 - 5 years

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ADDITION

Purpose: - To give the children the experience of the process of addition. Addend + Addend = Sum

Materials: - A supply of golden bead material: 50 unit beads, 45 tens, 45 hundred and 9 thousands kept neatly on a tray. - One set of large number cards 1 to 9000. - Three sets of small number cards 1 to 3000. - Three trays and three castors to hold the units.

Presentation: A small group exercise.

This work can be done at a large table or at a mat on the floor. Invite one child to lay out the large number cards, another sets out three sets of small number cards in the same formation as the large number cards. Ask another child to take care of the tray of Golden Bead Material acting as a 'banker'. Give a tray to each of three children with a castor to hold the unit beads.

- Static Addition (Without changing) Place a low four digit number on each of the three trays using the small number cards - each tray should have a different number. The children read the number on their tray. Show them how to approach the 'banker' and to ask politely for that amount of golden bead. The children return to the work area where the teacher removes the golden beads in hierarchical order from each tray placing them on the mat in three separate quantities (addends). Place the small number cards in problem formation at the left corner of the mat (representing addends). Add the units by bringing the three amounts together in the center of the mat. Ask the child in charge of the large number cards to give you the numeral which represents the answer. Place it below the units. Proceed to work in the same way with the tens, hundreds and thousands. Superimpose the large number cards and place them below the small number cards (sum). Review the process and say, "Now we have done 'addition'."

______ brought me ______ ______ brought me ______ and ______ brought me ______ and when we added them together we found that we had _________________.

Point to each of the small number cards and say, "This is an addend, this is an addend and this is an addend." Point to the Large Number Cards and say, "This is the sum." The children will pick up the terminology quite naturally.

- Dynamic Addition (With changing) Follow the same procedure as for the static addition but now you select small number cards whose sum will require exchanging.

- Individual Work When the children have understood the process of addition they may work on their own taking a tray of golden bead to their table or taking what they need from a 'pool' of golden beads. Later, they may progress from using the number cards to having the teacher write problems in their work book or taking a sheet of prepared problems and working with them. However, this needs some preparation. The children need to be shown how to write problems in their work books and how to space the problems and use a ruler to draw straight lines. Neat work habits are fostered in this way.

Exercise: As in the presentation. The teacher helps to set up the exercise and will act as a guide until the children can work together confidently.

Control: The teacher. However the emphasis is on the process not on the exact result.

Age: 5 years

Note: The process is the focus of the decimal system. When working with the golden beads do not correct the answer.

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MULTIPLICATION

Purpose: - To give the children the experience of the process of multiplication. Multiplicand x Multiplier = Product.

Materials: - The same as for addition.

Presentation: The same procedure as for static and dynamic addition. Ensure that each of the children have the same number; and when reviewing the children will note that the same number was brought so many times! Name the process - 'multiplication'.

Exercise: As in the presentation.

EXERCISE 2: Introduce multiplier card: After the children have completed several examples, review the completed equation. Explain that it is not necessary to lay the multiplicand out so many times - a multiplier card may be used (use a small unit card which is placed under the multiplicand as in a written problem).

Control: The teacher. However the emphasis is on the process not on the exact result.

Age: 5 years

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SUBTRACTION

Purpose: - To give the children the experience of the process of subtraction. MINUEND - SUBTRAHEND = DIFFERENCE

Materials: - The same as for addition.

Presentation: A small group exercise. Layout as for addition and multiplication.

Static Subtraction: Select a high number in the large number cards and the corresponding amount of golden bead material (minuend). Place the golden bead in hierarchical order in the center of the work area. Place the large number cards at the top right corner of the mat. Select a low number in the small number cards and place them on a child's tray (subtrahend). Tell him he may take that quantity from the golden bead on the table, beginning taking from the units. He proceeds in this way until he has taken the quantity specified in the small number cards on his tray. The teacher then states the quantity that is left on the table and asks a child to find the small number cards and place them below the beads (difference). Review the process and place the small number cards from the child's tray (subtrahend) below the large number cards (minuend) and place the small number cards representing the difference, below the subtrahend. State, now we have done 'subtraction'.

Dynamic Subtraction: This can follow when the child understands the process of subtraction.

Individual Work: The child can take the tray of golden bead and work at his table or he may take what he needs from a 'pool' of golden bead material.

EXERCISE 1: As in the presentation.

Presentation & EXERCISE 2: Minuend of 9000 (When the child has mastered Exercise 1.) Select a minuend of 9000. Proceed as for dynamic subtraction showing the child how to exchange a thousand for hundreds; a hundred for tens and a ten for units before he can begin subtracting.

Control: The teacher. However, the emphasis is on the process not on the correct answer.

Age: 5 years

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DIVISION

Purpose: - To give the experience of the process of division by units. - Dividend ¸ Divisor = Quotient

Materials: - The same as for addition.

Presentation: A small group exercise. Layout as for addition, multiplication and subtraction.

a) Static Division: Select a high number in the large number cards and the corresponding amount of golden bead material (quotient). The amount selected must be divisible evenly by the number of children who represent the divisor. Place the golden bead in hierarchical order in the center of the work area. Place the large number cards at the top right corner of the mat. Explain that you are going to share the quantity of golden beads equally between the number of children. Begin with the thousands by placing one thousand in each tray and proceeding in this way until all the thousands have been used. The child in charge of the small number cards gives each child the small card which represents the amount of thousands on their tray. Proceed in this way sharing hundreds, tens and units until all the golden bead is used. Review the process and state that in division our answer is what one person gets and therefore we need only take a set of small number cards from one tray. Place these small number cards representing the dividend above the large number cards. State that now we have done 'division'.

b) Dynamic Division (no remainder)

c) Dynamic Division (with remainder)

EXERCISE 1: As in the presentation

Presentation & EXERCISE 2: Individual Work: Introduce the 'Divisor' card and green skittles to represent the number of children. Share the golden bead between the skittles and the answer will be what one unit skittle gets.

Control: The process is more important than the correct answer.

Age: 5 years

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THE STAMP GAME

Note: The process is still important, however if answers are frequently incorrect re-present.

This material is used by the children for individual work with the Decimal System, following the group exercises done with the golden bead material.

Purpose: To give the child the opportunity of carrying out the four operations as individual exercises.

Materials: Small colored 'tiles' or 'stamps': green with '1' written on them to represent units; blue with '10' to represent tens; red with '100' to represent hundreds; green, again, marked '1000' to represent thousands. Skittles: 1 large green, 9 of each red, blue and green representing the decimal system categories. Some red, blue and green plastic discs to represent decimal system categories. Squared paper, pencil and ruler.

Presentation: Individual exercise. Place the stamp game, writing materials and presentation tray (golden beads) at a table. The teacher removes one stamp from each category and asks the child to identify the numeral. Ask the child to align the stamps with their corresponding golden beads. Explain that the stamps may be used, individually, for the same exercises as the golden beads. The child returns the presentation tray to the shelf and the stamps to the box. The number cards will no longer be necessary and instead we will write our numbers. The teacher writes a four digit number beginning with the highest category. The child reads and makes the quantity with the stamps. Repeat for a few examples. Introduce some numbers with zero.

Note: Use correct terminology with each operation. Addition - addends, sum. Multiplication - multiplicand, multiplier, product. Subtraction - minuend, subtrahend, difference. Division - dividend, divisor, quotient. Introduce the signs used to symbolise, e.g., + for addition; - for subtraction; x for multiplication and ¸ for division.

EXERCISE 1: Static Addition With the child's input, write two addends which will not require carrying. Draw a line under the addends and include a plus sign. Point out the use of a plus sign denotes this is addition. Read the problem with the child. The child lays out the appropriate stamps for the first addend. Encourage the child to check by reading the quantity made with the stamps. Place a ruler under the first addend and have the child lay out the second addend. Check. Remove the ruler. Remind the child of the necessary process to find the answer - combine categories and count beginning with the units. To combine the categories push the stamps up towards the top of the table until they form a double column per category. Count stamps using the category name. As each stamp is counted move it slightly toward you. The child records the answer in the units place - below the equal line. Have the child repeat the process for the other categories: tens, hundreds, thousands respectively. Review the problem with the child.

Dynamic Addition Follows the same procedure except when counting, exchange the categories as necessary by removing one stamp of the next higher category from the box and replacing the ten stamps, which have been counted, into their appropriate place in the box.

Presentation & EXERCISE 2: Multiplication With the child's input, write a multiplicand. The child chooses a multiplier of 2 or 3, which is written in the units column below the multiplicand. Introduce the multiplication sign. Read the equation with the child. Proceed as in dynamic addition. The child lays out the multiplicand the appropriate number of times combines the categories and counts exchanging as necessary and records the answer for each category as he counts.

After some experience use '0' in the multiplicand.

Presentation & EXERCISE 3: Static Subtraction With the child's input, write a minuend. Write a subtrahend which does not necessitate exchanging. Introduce the subtraction sign. Read the problem with the child. The child lays out the appropriate stamps for the minuend. Beginning with the units the child takes away the necessary number of stamps and replaces them into the box. The child counts the remaining number of units and records the answer. He repeats this process for the remaining categories in their respective order. Review the problem with the child.

Dynamic Subtraction Follows the same procedure except when subtracting exchange categories as necessary by replacing one stamp of the next higher category into the box and removing ten stamps of the needed category (using a ruler for spacing purposes).

Presentation & EXERCISE 4: Static Short Division With the child's input write a dividend as a statement and as in a process. Introduce the division sign and read the problem with the child. Remind the child that the skittles represent the divisor. Set out the appropriate number of skittles horizontally. Stack the appropriate stamps for the dividend to the left of the skittles in hierarchical order. Review the procedure for division: we start with the highest category and we give an equal number of stamps to each skittle. Share out the stamps underneath the skittles. Remind the child that the answer is what one unit received. The child counts the stamps under one skittle and records the quotient above the dividend. Read the problem with the child.

Dynamic Short Division (No remainder) Follows the same procedure except to exchange categories as necessary.

Dynamic Short Division (With remainder) Follows the same procedure as dynamic short division, except to introduce 'remainder'. Write the remainder to the right of the quotient with a small case 'r' before it. Explain that the 'r' is an abbreviation of remainder.

Control: The teacher. For each process the checking can be taught e.g., in addition one of the addends can be subtracted from the sum to find the other addend. In subtraction the subtrahend can be added to the difference to find the minuend. In multiplication the product can be divided by the multiplier to find the multiplicand. In division the quotient can be multiplied by the divisor to find the dividend.

Age: 5 - 5.5 years

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LINEAR COUNTING: CHAINS OF 100 AND 1000 GOLDEN BEADS

Purpose: To consolidate the child's knowledge of counting.

Materials: The golden 100 chain of 10 ten bars. A box (red, if possible) containing labels: 1 to 9 in green; tens in blue; one red 100. Golden bead square of 100. The golden 1000 chain of 100 ten bars. A box (green, if possible) containing these labels: 1 to 9 in green; tens in blue; hundreds in red, numbers between hundreds, ie. 130, in blue; one green label for 1000. This is a total of 10 green, 90 blue and 9 red. Ten golden hundred squares and one golden thousand cube. A long piece of carpet or felt the length of 1000 chain.

Presentation: Set out a mat for 100 chain. Introduce the bead cabinet and the 100 chain. Stress the careful handling of the chains. Show how to carry the chain with palms up, lay the chain over the child's hands. At the mat, fold the chain in a zigzag manner. Pull the right end to straighten the chain in a horizontal manner. Invite the child to fold the chain. With the child bring a hundred square from the cabinet to the mat. Superimpose the hundred square on the folded chain. The child straightens the chain, placing the square above the last ten bar, at the right. With the child bring the appropriate arrows (color coded) from the shelf to the mat. Lay out the arrows with the numbers facing up, sorted by color. Count the beads from left to right, placing an arrow for each bead in the first bar. For the remaining bars label the last bead in each bar. When the chain has been counted, note the number of beads in the chain. Read the arrows in sequence.

Exercise: As in the presentation.

EXERCISE 2: Introduce the special mat and set it out. Introduce the 1000 chain. Show how to carry the chain. Note the chain's rings. Hold the left hand perpendicular to the floor, close to the cabinet. Lift the chain from the left hook and over on to your hand, so that the ring lies on top of the index finger. Proceed as above for each ring, then return the chain to the cabinet one ring at a time. Invite the child to remove the chain from the cabinet and carry it to the mat. Lay out the chain horizontally on the mat. With the child, fold the chain as above. The ring denotes the end of each square which leaves a space. After each square is folded get a hundred square from the cabinet and superimpose it on to the folded chain. When the chain is folded count the hundreds. Get the 1000 cube from the cabinet - compare by stacking the hundreds. Pull out the chain - place a square above the last bar of each section at the right, place a cube at the end of the chain. Lay out the arrows, sorting by color and by hundreds. Count as above, establishing the repeating process. When the chain has been counted, note the number of beads in the chain. Read the arrows in sequence. Show the child how to pick up the chain by every other ring.

Control: The labels and the teacher's guidance.

Age: 5 - 6 years

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SKIP COUNTING

Purpose: To give the child a means of counting other than one by one. Preparation for multiplication. Preparation for squaring and cubing. To show the difference in quantity between the square and the cube of a number and the difference, or relationship, between the squares and the cubes of different numbers.

Materials: A cabinet with a shelf for the squares, the square chains and the cubes of each number and the corresponding tickets. Hooks for the cube chain of each number. Four red unit beads: 1, 1 squared, 1 and 1 cubed; red labels marked '1'. Long and short chains of 2 - green: green labels marked 1,2,4 for the square chain; 1, 2, 4, 6, 8 for the cube chain. Long and short chains of 3 - peach: peach labels marked 1,2,3,6,9 for the square chain; to 27 for the cube chain. Long and short chains of 4 - yellow: yellow labels marked 1,2,3,4,8 to 16 for the square chain; to 64 for the cube chain. Long and short chains of 5 - light blue: light blue labels marked 1,2,3,4, 5,10 to 25 for the square chain; to 125 for the cube chain. Long and short chains of 6 - mauve: mauve labels marked 1,2,3,4,5,6,12 to 36 for the square chain; to 216 for the cube chain. Long and short chains of 7 - white: white labels marked 1,2,3,4,5,6,7,14 to 49 for the square chain; to 343 for the cube chain. Long and short chains of 8 - brown: brown labels marked 1,2,3,4,5,6,7,8,16 to 64 for the square chain; to 512 for the cube chain. Long and short chains of 9 - blue: blue labels marked 1,2,3,4,5,6,7,8,9,18 to 81 for the square chain; to 729 for the cube chain. Corresponding number of bead squares for each square chain. A cube of each number. A felt mat.

Presentation: After the 100 and 1000 chains have been counted. Begin with the square of five chain (short chain). Proceed in the same manner as the 100 chain but 'skip-count' at the end of the exercise. Note the arrows are the same color as the beads on the chain.

Exercise: As in the presentation, the child skip-counts all the square chains at the end of the exercise.

EXERCISE 2: Begin with the cube of five chain (long chains). Proceed in the same manner as the 1000 chain but 'skip-count' at the end of the exercise. Encourage the child to skip-count the other cube chains in the same manner.

Control: The labels and the teacher's guidance.

Age: 5 - 6 years

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EXPLORATION AND MEMORIZATION OF TABLES

ADDITION SNAKE GAME

Purpose: To familiarize the child with all the possible number combinations that make ten. To give practice in memorising addition.

Materials: Five colored bead stairs (1 - 9) in a box. Twenty-five golden ten bars in a box. A set of 1 to 9 black and white bead stair in a black and white box. A red felt mat. A small notched card ('bridge'). A box or tray to hold the bars of the colored stair removed from the snake after they have been counted (the empty black and white bead stair box may be used for this).

Presentation: At a table, unroll a small mat and introduce the material laying them out in order at the top of the mat (box of tens, box of colored bead bars, black and white box). Build the black and white stair in the upper right hand corner. Place the notched card below the stair. State the use of the card, "This card is to mark where we finished counting." Make a colored snake which will come out even in tens (one of each bead bar plus an extra five). Close the box. The bead bars are placed at random, touching in a zigzag pattern. Change the colored snake to a golden one by counting to 10. Place the notched card on the right side of the tenth bead. Place a ten bar directly above the counted beads. Use the black and white bead bars to represent the remaining beads. Count remaining beads on the bead bar. Place the corresponding black and white bar above the counted beads. Remove the card and place the counted colored bead bars into the empty black and white box. Join up the snake. When counting the next ten beads begin with the black and white bar, if there was a remainder. Repeat the above process until all the bars have been counted. Count the ten bars. Arrange the ten bars side by side at the left. Remove the colored bead bars from the black and white box - arrange in diminishing size from left to right in the center. Isolate a ten bar in the center of the mat; make a matching ten using colored bead bars beginning with the largest, ie. 9 + 1. Repeat until all the tens and colored bead bars are matched.

EXERCISE 2: The child uses any combination of colored bead bars to make the snake. Use a black and white bar for any remaining colored beads less than 10. When checking the child may need to change to make the appropriate combination - use the supply of colored bead bars in the box. Note: use only two colored bead bars to make a ten. Match one colored bead bar to the black and white bead bar in remainder.

EXERCISE 2: Isolating combinations After child has worked well with the snake game, proceed as before. To count, pull down two bead bars isolating them at the bottom of the mat. Count to 10 as above using the black and white bar for remainder. Compare two sets of beads visually. Place colored bars into box. Move ten bar up into snake and pull down next colored bead bar. If two bead bars are less than 10 exchange them for a black and white bar. Proceed until snake is counted. Check as above.

EXERCISE 3: Showing multiples - second check Proceed as above, this time make a snake using a number of the same colored bead bars (choose three of your favourite numbers or colors). To check - Arrange tens and remainder as before. Arrange the colored bead bars horizontally, grouping like beads bars parallel to one another from left to right, in diminishing size. Beginning with the largest set of colored bead bars, note how many there are in the set. Count into tens - laying tens and remainder vertically below (use tens and colored bead supply). Repeat this process for the other sets of colored beads. Note: that there should be the same number of beads in each of the three areas. Check by placing the tens (from colored beads) below the other tens (from the golden snake). Change the remaining colored bars into tens. Child counts top 10's and the bottom 10's to see if they match.

Control: After each exercise the child is shown how to check the result.

Age: 5 - 6 years

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ADDITION STRIP BOARD

Purpose: To give the child all the possible combinations in addition. The red line has the same purpose as the notched card in the snake game: to show how many went to make ten and how many units were left over. The red line teaches how many numbers are composed of a ten and a quantity of units bringing us that much further toward another ten. This is the mechanism of addition that has to be learned. The child is helped to see the entire structure of addition and to memorise the combinations.

Materials: A board divided into 18 squares across from left to right and 12 squares from top to bottom. Each square is 2 x 2 cm.. Above the grid are the numerals from 1 to 18. Numerals 1 to 10 are in red, then a red line divides the board vertically; the numerals from 11 to 18 are in blue. Two sets of numbered strips. One set is blue with a numeral in red (1 to 9) at the end of each strip; the other set is red, divided into squares by blue lines, with a numeral in blue (1 to 9) at the end of each. Squared paper and pencil. Control Charts 1 & 2.

Presentation: At a table, ask the child to read the numbers along the top of the board in random order. Note the red line, the color of the numbers and the grid. Remove the blue strips - arrange like the number rods to the left of the board. In the same manner arrange the red strips to the right of the board. The child chooses a blue strip ie. 6 or 7. Place on the top line of the grid, aligned at left side. Place the 'one' red strip directly to the right of the blue strip. Read the board - ie. 7+1=8. The answer is found directly above the last section of the red strip. Write the equation. Remove and replace the red strip into the stair. Repeat for each red strip, 2 through 9. By the third or fourth equation the child may take over. When finished, show the child how to check his work with the Addition Control Chart # 1.

Exercise: As in the presentation the child works through all the tables.

EXERCISE 2A: Ask the child to choose a number, ie. 6 or 7. Write the number centerd on the page. State the goal: to find all the pairs of numbers which make up that amount. Begin with the 'one' blue strip - laying it on the grid as before. Ask the child what is needed to add to '1' to make the chosen amount. (The child may count the squares to find the answer.) The child places the appropriate strip to the right of the blue '1'. Read the equation, ie. 1 + 5 = 6. The child writes the equation. Repeat the procedure until all the possible combinations are on the board. Check the work with the Addition Control Chart #1. The child continues through all sums 2 - 18.

EXERCISE 2B: When the child has completed the above exercise, have him make a chart of all the numbers and their combinations on graph paper. ie. 2 = 1 + 1 3 = 1 + 2, 2 + 1 4 = 1 + 3, 2 + 2, 3 + 1 Check chart with Addition Control Chart #1.

EXERCISE 3A: Commutative Law Write an equation, ie. 2 + 4 =. The child uses the board to find the answer (the blue strip for the first addend, the red for the second addend) and records it. Write another equation reversing the above addends, ie. 4 + 2 =. The child uses the board as above to find and record the answer. Ask if the same numbers were used to make the sum. Check by comparing the strips (place equivalent strips underneath each other). Note that it does not matter which side of the plus sign the addends are on the answer is the same. Repeat with a few more examples.

EXERCISE 3B: Using the chart the child made in Exercise 2b make a second chart which does not include duplicates. ie. 2 = 1 + 1 3 = 1 + 2 4 = 1 + 3, 2 + 2 5 = 1 + 4, 2 + 3 Check chart with Addition Control Chart #2.

Control: Addition Control Chart #1. Exercise 3 - Addition Control Chart #2.

Age: 5 - 6 years

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ADDITION CHARTS 3, 4, 5 AND 6 (BLANK)

Purpose: To give practice in the memorization of addition combinations. The various charts give the possibility of repetition which helps fix the combinations in the memory. Each chart is a variation on the preceding one. The variations are there to sustain the child's interest thereby extending his work with the addition combinations.

Materials: Four baskets or boxes, each with red tickets for all of the addition problems of chart #1 written without answers. Paper and pencil for Charts 3 to 5. A box of tiles printed in answers to all problems used with Chart #6.

CHART 3 Presentation: At the table, examine the chart with the child. Ask the child to identify the numerals and the color across the top and repeat for the numerals down the left hand side. Note that this chart is for the memorization of the addition facts. Open the box of problems and select one, placing it on the top of the board. Read the equation and write it. Ask the child to guess the answer. Fingering - Find the first addend in the red column and mark it with your left index finger. Find the second addend in the blue row and mark it with your right index finger. Move the right finger down until it is in the same row as the left finger. Move the left finger across until it meets the right finger. Where they meet is the answer. Record the answer. Repeat two more times then invite the child to do the fingering. Check the work with Addition Control Chart #1.

Exercise: As in the presentation.

CHART 4 Presentation: Examine the chart, note it is only half the size of Chart 3. There is no top blue row. Yet it has the same number of problems. As before, the child selects, reads and writes a problem. Encourage the child to guess the answer. Fingering - Place the left index finger on the larger addend and the right index finger on the smaller addend in the red column. Move the right finger across to the end of the row. Move the left finger across its row until it is in the same column as the right finger. Move the right finger down the column until it meets the left finger. Where they meet is the answer. Record answer. Note: When the addends are the same, locate the numeral in the red column with both index fingers and move across to the end of the row to find the answer. Repeat two more times then invite the child to do the fingering. Check work with Addition Control Chart #2.

Exercise: As in the presentation.

CHART 5 Presentation: Examine the chart. There are very few answers, however, we have the same number of problems. As before, the child selects, reads and writes a problem. Encourage the child to guess the answer. Fingering - Find addends along the red column as in Chart 4. Slide both fingers to the end of their respective rows. For an even answer: hop fingers simultaneously toward each other, on the diagonal square by square. Where they meet is the answer. Record the answer. For an odd answer: hop fingers as above. They eventually land into squares which are touching. The answer lies in the square which has become isolated by the fingers. Record the answer. Repeat a few more times then invite the child to do the fingering. Check work with Addition Control Chart #2.

Exercise: As in the presentation.

CHART 6 Presentation: Examine the chart. It is the same as Chart 3 except it is blank. State, "This time you are going to give the answer instead of finding it." Introduce the tiles. Lay out the tiles in an ordered arrangement - tiles progressing from 2 to 18 with like numbers together. Note arrangement. The child selects and reads a problem. He finds the answer in the tiles and isolates the appropriate tile to the side of the chart. Fingering is done in the same manner as for Chart 3. Place the tile where the fingers meet. Repeat until the chart is complete. Check the work with Addition Chart 3.

Exercise: As in the presentation.

Control: Addition Control Charts 1 & 2. For Chart 6 - Chart 3.

Age: 5 - 6 years

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NEGATIVE SNAKE GAME

Purpose: To familiarize the child with subtraction combinations. Indirect preparation for algebra as from time to time equal quantities of the opposite sign cancel each other out, as in algebraic formulae.

Materials: Boxes containing: colored bead bars (about 3 of each); 20 golden ten bars; black and white stair; negative bead stairs; one box or little tray to contain the beads that have been counted; small notched card; and a green felt mat.

Presentation: (After the child has mastered the positive snake game.) At a table lay out the materials as in the positive snake game. Introduce the grey bead stair and say, "These are the negative bead bars. They tell us how many to take away." The child identifies the grey bead bars by counting. The child builds a snake as before and places three or four grey bead bars interspersed among the colored bead bars. (Note: in placing the grey bead bars, the snake should not go down to zero or into a minus number.) Proceed to count as before. When a grey bead bar is reached, in order to take away, count back that number of beads. Count back on only black and white bead bars or golden bead bars. Before counting back, any colored bead bars (less than 10) to the left of the grey bar must be changed into one black and white bead bar. To count back - The child counts the beads on the grey bar. Then counts back into the black and white and/or golden beads, counting from right to left. Mark with the notched card. Count any remaining beads on the left side of the notched card and represent this remainder with the corresponding black and white bead bar. Place the grey bead bar into the box with the other counted colored bead bars. Replace the black and white bead bar into the stair. Replace the golden bead bar into its box. Rejoin the snake and note that it is getting shorter. To check - Arrange the golden snake and colored bead bars as before. Arrange the grey bead bars vertically to the right of the colored bead bars in diminishing order. Note: The colored bead bars represent the minuend, the grey beads represent the subtrahend and the golden bead bars represent the difference. Match the colored bead bars to the golden and grey bead bars changing as necessary.

Exercise: As in the presentation.

EXERCISE 2: Isolating subtraction - Proceed as above. When the child has counted back and (at a later stage) found the remainder, pull down all the bead bars involved in the counting back process to the bottom of the mat. Arrange the bead bars horizontally, so that the bead bars counted back on, match the grey bar and the black and white remainder. Note the quantity of the golden bead bar and/or the black and white bead bar, ie. 10 and 4 make 14. Note the quantity of the grey bead bar and the remainder, ie. 7 and 7 make 14.

"So 14 (pick up golden bead bar and/or black and white bead bar) take away 7 (pick up grey bead bar) is 7 (leave remaining black and white bar on the table)." Place the bead bars in your hand in their respective places as before. Place the remaining black and white bead bar back into the snake and rejoin the snake. Continue in the same manner. Check as in the presentation.

Control: The child is shown how to check his answer with the control.

Age: 5.5 - 6 years

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NEGATIVE STRIP BOARD

Purpose: Memorization of the subtraction combinations.

Materials: The Negative Strip Board is a replica of the Addition Strip Board in size and number of squares. The only differences are that a blue line divides the board vertically after the ninth square, the first 9 numerals are blue and the numerals from 10 to 18 are in red. Two sets of colored strips, one set in red and the second set in blue. A third set of plain wooden strips, not numbered. There are seventeen of these ranging from one square in length to seventeen squares. Squared paper and pencil. Subtraction Control Chart #1.

Presentation: At a table, examine the board. Note the blue line. Introduce the wooden strips and build a stair with them. The wooden strips are placed vertically above the board from longest to shortest respectively placed above the numerals 1 to 17. The child builds the blue stair to the right of the board. Note: To take away, always begin with the largest blue strip, never record an answer in red. Point to the 18 on the board and ask the child to identify it. State that the blue strips are used to take away. They represent the subtrahend. Place the 9 blue strip on the board. Align it on the right, beginning with the number 18 representing the minuend and cover up the numerals along the top of the board with the plain wooden strip. The answer is found directly to the left of the blue strip. Write the equation, 18 - 9 = 9. Repeat using the next strip in diminishing order. The answer is in red, therefore, the child will not write down the equation. Move across the numerals on the board, from right to left, to locate the next minuend, ie. 17. We no longer need 18. Cover the numeral(s) with the appropriate wooden strip directly above the current minuend, ie. 17. Proceed as above. When the child reaches 9 as the minuend he may need some help as he will get zero for an answer, 9 - 9 = 0. When 8 is the minuend, tell the child not to record an answer where the blue strip goes beyond the left barrier of the grid. The child checks his work with the Subtraction Control Chart #1.

Exercise: As in the presentation, the child works through all the tables.

EXERCISE 2: Set out the materials as before. The child sets out the red strips in stair formation to the left of the board. The child chooses a number, ie. 6 or 7. Using the wooden strip cover the extra numerals to the right of the minuend. The child writes the minuend centerd at the top of the page and builds a table. Beginning with the red strips build a table as in the Addition Strip Board Exercise 2a beginning with the largest addend. The last row will be the minuend represented by one blue strip. This is only possible for minuends less than 10. "Now we're going to take away the blue strips." Using the board, note the minuend, ie. 6. The child writes '6'. Take away the first blue strip, ie. 6, sliding it to the bottom right of the board. 'What's left? - Nothing.' The child finishes the equation, ie. 6 - 6 = 0. Note: the difference is indicated by the red strip or its absence. Proceed in the same manner, working up through the rows. Check the work with Subtraction Control Chart #1.

Control: Subtraction Control Chart #1.

Age: 5.5 - 6.5 years

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SUBTRACTION CHART 2 AND 3 (BLANK)

Purpose: To memorize the subtraction combinations.

Materials: A basket or box containing green slips of paper for all subtraction combinations on Chart 1, written without answers. Subtraction Chart 2. Subtraction Chart 3 - blank. A green box marked with the subtraction sign, with tiles, nine of each number 0 to 9 (the tiles are white, numerals are in green). The blank chart is like Chart 2, except that all answer squares are blank.

CHART 2

Presentation: At a table, examine the board. The child identifies the numbers in red and then the numbers in blue. Note the minus signs. The child selects, reads, writes and guesses solution of a problem. Fingering - Find the minuend among the numbers in red - mark it with the right index finger. Find the subtrahend among the numbers in blue - mark it with the left index finger. Move the right finger down until it is in the same row as the left finger. Move the left finger across until it meets the right finger. Where they meet is the answer. Record the answer. Repeat two more times then invite the child to try the fingering. Check work with Subtraction Control Chart #1.

Exercise: As in the presentation.

CHART 3

Presentation: Examine the chart which is the same as Chart 2 except it is blank, just like Addition Chart 6. Introduce the tiles. Lay out the tiles in an ordered arrangement progressing from 0 to 9 with like numbers together. Note the arrangement. The child selects and reads a problem and finds the answer in the tiles, which he isolates to the side of the chart. Fingering is done in the same manner as for Chart 2. Place the tile where the fingers meet. Repeat until the chart is complete. Check work with Subtraction Chart 2.

Exercise: As in the presentation.

Control: Subtraction Charts 1 and 2.

Age: 5.5 - 6.5 years

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MULTIPLICATION BEAD BAR LAYOUT

Purpose: To show by this geometrical form of multiplication that the multiplier is never a constant as is the multiplicand. It is only indicative of how many times a number is taken or a given quantity is repeated. To show that a succession of lines creates a surface that is why it is called geometrical multiplication. Preparation for square root and for factoring. Preparation for division by helping the child to visualize the divisibility of numbers. The geometrical formation is an indirect preparation for exercises that follow later in connection with geometry and algebra.

Materials: Nine boxes marked 1 x, 2 x, 3 x, etc. each containing 55 bars of each number from 1 to 9. A box with tens and colored bead bars. A felt mat.

Presentation: Start with the table of 7. Bring the answer box, a yellow mat and a box marked '7x' to a table or a mat. Introduce the materials. Place one seven bar from the '7x' box horizontally at the upper left of the mat (multiplicand). Count the number of beads on the bar, i.e. 7. Remove a seven bar from the answer box (product) and place it vertically below the multiplicand. State the equation, 7 one time is 7. Proceed in the same manner, increasing the multiplier by one each time up to 9 inclusively. Count the multiplicand in the same manner as the snake game - counting to ten and placing a ten bar, any remaining beads are represented by the appropriate bead bar from the answer box. When complete, go over the table verbally with the child.

Exercise: As in the presentation, the child works through the tables 1 to 9.

EXERCISE 2: Multiplying by 10 At a table, the child chooses a multiplicand, ie. 4. The child writes the multiplicand centerd on a page and takes out 10 four bars from the answer box and arranges them horizontally. The child counts as in the presentation, and lays out the product vertically. Say, "When we have 10 fours it's the same as 4 tens or forty." "When you multiply by 10 all you have to do is add a zero to the multiplicand." Repeat with a few more examples. Then child may continue on his own.

EXERCISE 3: Divisibility of product Select a number, ie. 12. Find the number of ways to make 12. Begin with 1 and see if you can make 12 by counting, adding each bar on one at a time. The use of 1 works however at this point note that the multiplier should be less than 10. Continue in the same manner for 2 through 9. Leave out the combinations which make 12. Child may write down combinations.

EXERCISE 4: Commutative Law At a table, using the answer box layout place 7 five bars horizontally as before. Then 5 seven bars. The child counts the beads placing the answer vertically below. Compare the answers. They are the same. Turn the beads to compare. Review and write the equations. 7 x 5 = 5 x 7 Try a few more examples. Conclude that it does not matter which side of the multiplication sign the multiplicand and the multiplier are on - the answer is the same.

EXERCISE 5: Making the decanomial Build the decanomial square (see the sensorial album). Set out two mats. On a mat set out the supply of bead bars. Use the other mat to build the decanomial square, in the same manner as above, using the bead bars. When the decanomial is complete visually explore the layout. Note the squares on the diagonal and have the child exchange them with the squares from the bead cabinet. Note: the number of bars in the band is the number squared; the number of beads in the band is the number cubed.

Control: The teacher.

Age: 5.5 - 6.5 years

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MULTIPLICATION BOARD

Purpose: The memorization of the multiplication tables.

Materials: A perforated board with 100 holes in rows of ten. On the left side of the board there is a little window with a slot for the insertion of a card. A set of cards, 1 to 10. A counter. 100 red beads. Squared paper and pencil. Multiplication Control Charts 1 & 2.

Presentation: At a table, examine the materials with the child who identifies the numbers along the top of the board. Place a disc in indent of board. Lay out the number cards in random order. The child chooses either a 6 or a 7. The child slides the number card into the slot and returns the remaining number cards to the box. Place the disc above numeral '1' and read the board, ie. 7 x 1 =. Child writes the equation. Lay out the appropriate number of beads under '1', ie. seven. To find the answer the child counts the beads and writes the answer. Repeat the process until the table is complete. Note: To count the beads start from the answer to the last problem. When the table is complete, introduce the Multiplication Control Chart #1 for the child to check his work.

Exercise: As in the presentation, the child works through tables 1 to 10.

Control: Multiplication Control Charts 1 & 2.

Age: 5.5 - 6.5 years

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MULTIPLICATION CHARTS 3, 4 AND 5 (BLANK)

Purpose: The memorization of the multiplication tables.

Materials: Charts 3, 4 and 5. A basket or box for each chart, each containing yellow tickets of all the combinations of the multiplication tables from 1 times 1 to 10 times 10 (without the answers). For the blank chart - a yellow box (marked X) with wooden tiles printed with answers for each combination of the multiplication tables as in Chart 3. Squared paper and pencil.

CHART 3

Presentation: At the table, examine the chart. The child identifies the numbers in blue and then in red. The child selects , reads, writes and guesses the solution to the problem. Fingering - In the same manner as Addition Chart 3. Find the multiplicand among the numbers in red and mark it with the left index finger. Find the multiplier among the numbers in blue and mark it with the right index finger. Note: when the multiplicand or the multiplier is one the answer will be found along either the red or the blue numbers. Record the answer. Repeat two more times and then invite the child to do the fingering. Check the work with Multiplication Control Chart #1.

Exercise: As in the presentation.

CHART 4

Presentation: Examine the board and note the differences between Charts 3 and 4. The child selects, reads, writes and guesses the solution to the problem. Fingering - In the same manner as Addition Chart 4. Note: when the multiplier or the multiplicand is one the answer is found along the red numbers. Record the answer. Check the work with Multiplication Control Chart #2.

Exercise: As in the presentation.

CHART 5

Presentation: Examine the chart which is the same as Chart 3 except it is blank. Lay out the tiles in an ordered arrangement - smallest to largest from left to right above the board, with like numbers together. Note the arrangement. Proceed as in Addition Chart #6. The child selects and reads a problem, then finds the answer among the tiles and isolates it. Fingering is done in the same manner as for Chart 3. Place the tile where the fingers meet. Repeat until the chart is complete.

Exercise: As in the presentation.

Control: Multiplication Control Charts 1 & 2. For Chart 5 - Chart 3.

Age: 5.5 - 6.5 years

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UNIT DIVISION BOARD

Purpose: To make the child familiar with the ways in which numbers may be divided. To show that not every number is evenly divisible and some only by a few numbers. To show the relationship between multiplication and division.

Materials: The Unit Division Board. 9 green skittles. 81 green beads in a jar. Castor or a cup. Red and lead pencils. Multiplication Bead Board for Exercise 2.

Presentation: Individual exercise. At a table examine the board, note the numbers along the top and the side. Skittles will again represent the divisor. Choose a dividend - one which is evenly divisible by 9, ie. 45. Invite the child to count out the appropriate number of beads into the castor. State that we always begin with a divisor of 9. Set out 9 skittles across the top of the board. Write the equation, ie. 45 _: 9 =. Share out the beads, moving from left to right, placing an equal amount of beads under each skittle. Record the answer which is what one unit receives. Note: underline the equation in red if it divides out evenly. Remove one skittle (divisor is now 8) and redistribute the extra beads under the remaining skittles. Those beads which will not share out evenly are placed into the castor and represent the remainder. Note: the remainder cannot be as large or larger than the divisor. Record the equation. Repeat the process until the quotient is 9. The quotient can be 9 or less -no larger than 9 - due to the limitations of the board (only 9 spaces under each skittle). Then remove one bead from the dividend replacing it into the castor and begin again starting with a divisor of 9. Continue until the child is able to take over for himself.

Exercise: As in the presentation, the child explores with dividends up to and including 81. The child may compile a chart of all the equations underlined in red to show those numbers which are evenly divisible.

EXERCISE 2: Bring both the unit division and the multiplication bead board to the table. Write an equation, ie. 7 x 3 =. The child uses the multiplication board to find the answer.

Write the corresponding division equation, ie. 21 ¸ 7 =. The child uses the unit division board to find the answer. Compare the results with the child to show what you build in multiplication, you take apart in division (division is the inverse process of multiplication). Explore with a few more examples.

Control: The teacher.

Age: 5.5 - 6.5 years

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DIVISION CHARTS 1 AND 2

Purpose: To teach the child division combinations. Exercise 2 is a preparation for finding the Highest Common Factor and the Lowest Common Multiple.

Materials: Division Charts 1 and 2. Boxes or baskets for each Chart, containing blue tickets on which are written all short division problems with a dividend of 81 or less that are exactly divisible. A blue box of answer tiles for Chart 2 with the division sign on the box. Squared paper and pencil.

CHART 1

Presentation: At a table, examine the board with the child who identifies the numbers along the top row. Note the two colors, the prime numbers are in white. The child also reads the numbers along the diagonal and notes the division signs. The child selects, reads, writes and guesses the solution to a problem. Fingering - Find the dividend along the top row and mark it with the right index finger. Find the divisor along the diagonal numbers with the division signs and mark it with your left index finger. Move the right finger down until it is in the same row as the left finger. Move the left finger across until it meets the right finger. Where they meet is the answer. Record the answer. Repeat two more times then invite the child to try the fingering. Check the work with the child's own chart or Multiplication Control Chart #1.

Exercise: As in the presentation.

CHART 2

Presentation: Examine the chart which is the same as Chart 1 except it is blank. Introduce the tiles. Lay out the tiles in an ordered arrangement progressing from 1 to 9 with like numbers together. Note the arrangement. The child selects and reads a problem and finds the answer in the tiles and isolates it. Fingering is done in the same manner as for Chart 1. Place the tile where the fingers meet. Repeat until the chart is complete. Check work with Division Chart 1.

Exercise: As in the presentation.

EXTENSION: The teacher explains prime numbers and factoring. When the child inquires about the numbers in white explain they are 'prime numbers'. "That means they can be only divided by 1 and themselves." The child gets the chart he compiled and paper and pencil. Select a quotient, ie. 56, the child writes it in the center of the page. Look it up on the chart and record the two numbers which make it up, ie. 7 and 8, below and a bit apart with arrows pointing down from 56. Look on the chart again to find out the numbers which make up 7 and then 8. Continue on (showing factoring). Since 7 and 2 are prime numbers they do not have any factors. Repeat according to the interest of the child. 56

7 8 2 4 2 2

Control: Multiplication Control Chart 1 or child's own chart. For Chart 2 - Chart 1.

Age: 5.5 - 6.5 years

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PASSAGE TO ABSTRACTION

THE DOT GAME

Purpose: To illustrate the mechanism of the decimal system. To focus the child's attention on the mechanism of changing. This game prepares the child for abstract addition.

DESCRIPTION OF MATERIAL: Paper which is squared and has columns headed 1, 10, 100, 1000 and 10,000. The columns are divided into small squares so that there are ten in each horizontal row. At the foot of each column are two spaces, the upper one for indicating the changing process with dots, and lower one for the result. There is a blank column on the right side in which the problem to be done is written. A lead pencil, an orange colored pencil and a ruler.

Presentation: At a table, the child writes his name and date in the appropriate places. Note the column headings. Have the child identify the numbers 1 - 1000, introduce 10,000. Write 4 or 5 four digit addends in the blank column with the child. The child reads the equation. State that we put a dot for each number in its proper category. Begin with the first addend starting with the units. Fill up the rows from left to right - when the row is complete continue on to the next row below. To count dots: Begin at the units, upon reaching 10 - put a line through the ten dots and add an orange dot to the next higher category. Continue counting. When reaching a number less than 10, record that number in the bottom box of that column. Read the answer and write the sum below the addends.

Exercise: As in the presentation.

EXERCISE 2: Begin as before. This time put out the dots according to the category starting with the units (ie. put out the dots for all the units in all the addends - count as above, only now the orange dot(s) will appear at the beginning of the top row - repeat the process for tens, hundreds, thousands).

Control: The teacher.

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Age: 5 to 5.5 years

SMALL BEAD FRAME

Purpose Direct: The exercises of numeration recapitulate the function of the decimal system by making the child realize once again the following: 1) That ten of one category make one of the next higher category and then in each category there can be no more than 9. 2) The value of the numerals is determined by the place they hold. 3) The function of zero is that of a place holder. The whole set of exercises brings the child to the realization that when he writes down addition and subtraction problems, he must place all the figures belonging to the same category in the same vertical column. The exercises also provide the children with opportunities to apply all they have learned before and thus prepare them further for abstraction.

Indirect: Both the exercises on numeration and the multiplication exercises prepare the child for the distributive law of multiplication where it becomes necessary to analyze the numbers into their hierarchical values.

Materials: A frame with support to enable it to stand. The frame has four wires across, each strung with ten beads. Top wire: 10 green beads for units; second wire: 10 blue beads for tens; third wire: 10 red beads for hundreds; fourth wire: 10 green beads for units of thousands. On the left side of the frame the numeral category of each wire is written. Background for hierarchy of units is white; background for thousands is grey. Notation paper for small frame is lined in the same colors as the numerical categories. Pencil and ruler.

Presentation: Introduce the Small Bead Frame Individual exercise. At a table, examine the frame. Ask the child to identify the numerals along the left-hand side of the frame. Note the new background color at 1000, it denotes where we write a comma. Note the beads correspond in color to the stamp game. Introduce the notation paper to the child. The child reads the headings at the top of the paper. Explain that as we count each bead we can write its number directly on the color coded line under the appropriate column. Start with the units counting the beads by sliding them individually across the wire from left to right while recording the number down in its appropriate place on the notation paper. Upon counting 10 units remind the child that we must exchange. Slide one ten bead to the right and slide the 10 units back over to the left. To record - write a 1 on the blue line and note there are no units therefore we write a zero in the units category. Continue counting beads and noting the numbers on paper up to 1000, stressing exchanges. Make large numbers. The teacher writes a four digit number, the child reads and makes the quantity on the frame. Note: we read the number on the right-hand side on the frame. Repeat for a few examples - reverse process and use zero.

Exercise: On the same or on another day. Static Addition - Write an equation on the notation paper which does not require exchanging. The child reads. The child sets out the first addend on the frame. Remind the child to begin adding at the units. The child reads the number of units in the second addend and counts the corresponding number of unit beads to the right. The child counts and records the sum of the unit beads which are at the right of the frame. Repeat the process for the remaining categories, tens to thousands respectively. Read the equation. Dynamic Addition - follows immediately. Write an equation which will require exchanging. The child proceeds as above, reads the equation and sets out the first addend. Beginning with the units, the child reads the number and counts the corresponding number of unit beads to the left. He runs out of unit beads - 10 unit beads lie on the right-hand side of the frame - therefore he must exchange the 10 units for a ten. Slide one ten to the right and the 10 units back to the left. The child continues to count the necessary unit beads to the right. The child records the number of beads that sit on the right. Repeat the process for the remaining categories, exchanging as necessary. Read the problem.

EXERCISE 2: Dynamic Addition - Proceed as above except this time add all the units first, then the tens, hundreds and thousands. The child reads the numbers in the units column - sets out the first, then adds on the second, exchanging as necessary and records the sum. Repeat for the remaining categories. Read the problem.

EXERCISE 3: Static Subtraction - Write a problem on the notation paper that does not require exchanging. The child reads and sets out the minuend. Remind the child to begin with the units. The child reads the units in the subtrahend and takes them away by counting the appropriate number of beads from right to left. The child reads the number of beads remaining at the right of the frame, which is the difference and records. Repeat for the remaining categories, tens to thousands respectively. Read the problem.

Dynamic Subtraction - Write an equation that necessitates exchanging. The child reads and sets out the minuend. Beginning at the units, the child notes the number of beads which must be taken away. The child counts the corresponding number beads from right to left, however the child will run out of beads. Remind the child of how to get more units. Slide one ten bead to the left and slide the 10 unit beads to the right. The child continues to count the appropriate number of beads to the left. The child reads the number of beads remaining at the right of the frame which is the difference and records it. Repeat for the remaining categories. Read the problem.

Control: The child may be shown how to check his work.

Age: 5 - 6 years

Note: Multiplication is done as for Addition. The child can be shown how to take the multiplicand a certain number of times (multiplier).

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