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НазваниеMontessori teacher albums
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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

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Table of Contents

  • NUMBERS TO TEN

    • Number Rods

    • Sandpaper Numerals

    • Number Rods and Number Cards

    • Spindle Boxes

    • Memory Game of Numbers

    • Cards and Counters

  • DECIMAL SYSTEM

    • Presentation with the Golden Bead Material

    • Presentation with the Large Number Cards

    • Formation of Large Number Cards with Beads

    • Combination of Golden Bead and Large Number Cards

  • ^ TEENS AND TENS

    • Teens, Formation of Quantities 11-19 with Golden Beads

    • Formation of 11-19 with Teen Board

    • Combination of Teen Board and Beads to Form 11 to 19

    • Formation of 10-90 with the Golden Beads

    • Formation of 10-90 with the Ten Board

    • Combination of Quantities 10-90 with Golden Beads and Ten Boards

    • Formation of 11-99 with Golden Beads and Ten Board

    • Changing Exercise

    • Addition

    • Multiplication

    • Subtraction

    • Division

    • The Stamp Game

    • Linear Counting: Chains of 100 and 1000 Golden Beads

    • Skip Counting

  • ^ EXPLORATION AND MEMORIZATION OF TABLES

    • Addition Snake Game

    • Addition Strip Board

    • Addition Charts 3, 4, 5 and 6 (Blank)

    • Negative Snake Game

    • Negative Strip Board

    • Subtraction Chart 2 and 3 (Blank)

    • Multiplication Bead Bar Layout

    • Multiplication Board

    • Multiplication Charts 3, 4 and 5 (Blank)

    • Unit Division Board

    • Division Charts 1 and 2

  • ^ PASSAGE TO ABSTRACTION

    • The Dot Game

    • Small Bead Frame

[top]

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

[top]

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

[top]

^ 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

[top]

^ 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

[top]

^ 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

[top]

^ 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

[top]

^ 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

[top]

^ 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

[top]

^ 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

[top]

^ 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

[top]

^ TEENS AND TENS

TEENS, FORMATION OF QUANTITIES 11-19 WITH GOLDEN BEADS

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