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xMath :: 6-9 :: Mulitplication
xTable of Contents:
  • Introducton
  • Numeration
  • Addition
  • Multiplication
    • Stamp Game Multiplication
    • Memorization Exercises
      • Multiplication Boards (Bead Boards)
        • Introduction and List of Materials:
        • Initial Presentation:
        • Multiplication Booklets
        • Combination Cards
      • Skip Counting (Linear Counting)
        • List of Materials
        • Initial Presentation:
        • Short Chain:
          • Activity:
        • Long Chain:
        • Games of Comparison
      • Bead Bar Multiplication
      • Multiplication Charts and Combination Cards
        • Passage from Chart I to Chart II
        • Passage from Chart II to Chart III (the Whole Chart)
        • Passage from Chart III to Chart IV (the Half Chart)
        • The Bingo Game of Multiplication (using Chart V)
          • Exercise
          • Exercise
          • Exercise
            • Group Games:
      • Multiplication By 10, 100, 1000
        • Presentation:
        • By ten
        • By one hundred
        • By one thousand
    • Checkerboard - Geometrical Analysis of Multiplication
      • Introduction to the Checkerboard
        • Presentation:
        • Games:
      • Multiplication with the Checkerboard
        • 1st level
        • 2nd level-Small Multiplication
        • 3rd level-Partial Products (this passage can be skipped)
        • 4th level-Mental Carrying Over
      • Multiplication and Drawing
    • Bead Frame Multiplication
      • Small Bead Frame
        • Multiplication By 10, 100, 1000
        • Multiplication with a One-Digit Multiplier
      • Large Bead Frame-Multipliers of 2 or More Digits
        • The Whole Product
        • Partial Products
      • Horizontal Golden Bead Frame
        • The Whole Product
        • Partial Products
        • Carrying Mentally
    • More Memorization Exercises
      • The Snake Game
      • Various Ways of Constructing a Product
      • Small Multiplication
      • Inverse Products
      • Construction of a Square
      • Multiplication of a Sum
      • Analysis of the Squares-Binomial
      • Analysis of the Squares-Trinomial
      • Passage From One Square to a Succeeding Square
      • Passage From One Square to a Non-Successive Square
      • Skip Count Chains-Further Exploration
        • List of Materials
        • Construction of Geometric Figures
        • Decanomial (a polynomial having ten terms) & The Construction of Chart I
          • List of Materials
          • Vertical Presentation
          • Horizontal Presentation
          • Angular Presentation
          • Commuted Decanomial
          • Part One
          • Part Two-Building the Tower
          • Variation on the 1st Method
          • 2nd Method
          • Part Three-Decomposition of the Tower
      • Skip Counting Chains-Further Explorations
        • Decanomial (a polynomial having ten terms) & The Construction of Chart I
        • Numeric Decanomial
      • Special Cases
        • 0- Calculate the product
        • 1- Calculate the Multiplier
        • 2- Calculate the Multiplicand
        • 3- Inverse of Case Zero---Calculate the Product
        • 4- Inverse of The First---Case, Calculate the Multiplier
        • 5- Inverse of the Second Case ---Calculate the Multiplicand
        • 6- Calculate the Multiplier and the Multiplicand
        • Activity:
      • The Bank Game
    • Word Problems
  • Subtraction
  • Division
  • Fractions - COMING SOON
  • Decimals - COMING SOON
  • Pre-Algebra - COMING SOON

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xStamp Game Multiplication


Materials:
...wooden stamps of four types:
xx...green unit stamps printed with the numeral 1,
xx...blue tens stamps printed with the numeral 10,
xx...red hundred stamps printed with the numeral 100, and
xx...green thousand stamps printed with the numeral 1000
...box with three compartments each containing 9 skittles and one counter in the hierarchic colors;
...four small plates

Presentation:
Given a multiplication problem, the child prepares several like quantities, puts them together, makes the necessary changes and records the problem and the result in his notebook.

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xMemorization Exercises


MULTIPLICATION BOARDS (Bead Boards)

a. Introduction and List of Materials:

The child has encountered multiplication before. The first impression was given with the number rods, finding that the double of 5 is 10. Later with the decimal system material, the child learned that multiplication is a special type of addition. In the exercises that follow this concept will be reinforced and the child will be given the chance to memorize the necessary combinations.

Materials:
...Bead Board, and corresponding Green box which contains:
xx...100 green beads (the product), numeral cards of 1-10 (the multiplicand), and
xx.xone green counter (this, when placed by one of the xx...numerals 1-10 across the top, indicates the multiplier).
...Booklet of Combinations (10 pages of 10 combinations each)
...Box of Multiplication Combinations (same combinations as are found in the booklet)
...Box of green tiles for bingo game
...Multiplication Charts I-V (for control)

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MULTIPLICATION BOARDS (Bead Boards)

b. Initial Presentation:

To familiarize the child with the materials, the teacher suggests a problem and writes it down, i.e. 3 x 4 = (three taken 4 times). The 3 numeral card is placed in the slot. The counter is placed over 1 as 3 beads are placed in the first column...(Attempt to get children at this point to be counting by threes up to whatever level they are capable, in place of counting every bead.) ...'three taken one time...' As the three beads are placed in each column, the counter is moved along, until '...three taken four times...' We've taken 3 four times, what is the product? The beads are counted and the result is recorded.

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MULTIPLICATION BOARDS (Bead Boards)

c. Multiplication Booklets

Materials:
...Bead Board, and box with beads, cards, counter
...Combination booklet
...Chart I (for control; a summary of the combinations found in the booklet)

Exercise:
(Starting with 1 is a problem because it doesn't give the concept of multiplication) Start with any other unit like the number 3. The numeral card is placed in the slot. The child reads first combination 3 x 1 Three beads are placed in the first column with the counter over the numeral 1. The child records the answer in the booklet. The next combination is read: 3 x 2 =. The counter is moved over and three more beads are added. The product is recorded in the booklet. At 3 x 3 = the child should notice the geometrical form created when the multiplier and multiplicand are equal. If he doesn't, the teacher may set up a situation wherein he may easily make the discovery himself forming several doubles in a row.

Control of error: Chart I

Note: In the material the child ends his work with the table of 10, rather than 9 as was the case in addition and subtraction. This is to show the simplicity of our decimal system. The table of 1 is very similar to the table of 10. It differs only in the presence of zero.

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MULTIPLICATION BOARDS (Bead Boards)

d. Combination Cards

Materials:
...Bead Board and box with beads, cards, counter
...box of loose combinations
...Chart I (for control)

Exercise:
To facilitate the child's work in this exercise, the number cards used to indicate the multiplicand are arranged in a row or column on the table. The child fishes for a combination, 3 x 9 =, reads it, and writes it on his paper. The number card 3 is placed in the slot, the counter is placed over column 1; 3 beads are placed in the first column. The counter is moved to column 2, as three more beads are placed in that column, making a subtotal of 6. This continues up to 9. The result of 3 taken 9 times is 27. This product is written on the paper.

NOTE: In the beginning the teacher should supervise the child's work to see that he skip counts the beads as he goes along3, 6, 9, 12, 15, 18, 21, 24, 27. For if the child counts the beads one at a time when he is finished, he will never memorize the combinations.

The child removes the beads, number card and counter and fishes for another combination.

Control of error: When the child has finished his work, he controls with Chart I. This control reinforces memorization of the combinations.

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SKIP COUNTING (Linear Counting)

a. List of Materials

Materials:
...Board of powers (though it is not named as such at this point)
...Cubes, long chains, squares, short chains
...Two Boxes:
xxx...arrows for short chains, i.e. for 5 we have 1, 2, 3, 4, 5, 10, 15, 20, 25
xxx...arrows for long chains, i.e. for 3 we have 1, 2, 3, 6, 9, 12, 15, 18, 21, 24, 27

b. Initial Presentation:
In the first presentation of the material the nomenclature should be given in a three period lesson, i.e. cube of seven, long chain of seven, short chain of seven, square of seven.

c. Short Chain:
The chain is lain stretched out on the table, and the child identifies it as, i.e. the short chain of 5. The teacher folds it up like a fan, and the child identifies it as a square of 5. This is proven by placing a square on top of the folded chain to see that they are equal. The child is given the arrows to lay face up on the table. Together the teacher and the child put the arrows in their respective places as the beads are counted...1,2,3,4,5...; the counting continues by ones up to 10, and the arrow for 10 is placed there, and likewise counting by ones to 15. From 15, we add five more to reach 20, and place an arrow and add five more to 25, The square of 5 is placed at the end of the chain since the chain is equal to the square.

Activity:
Materials
: same as before
The child lays out the arrows as before. Little by little he works from counting one by one using the arrows face up, to skip counting as he lays the arrows out, and then skip counting with all of the arrows face down.
When the child is able to skip count well with the arrows face down, he may also skip count regressively.

d. Long Chain:
The chain is lain out one the table or floor (if necessary), and the child identifies it as, i.e. the long chain of three. The child lays out the arrows appropriately as he skip counts. At 9, as the square of three is placed, the child is reminded if they do not see it, that this part of the chain makes a square. (It is equal to the short chain.) A square is placed by the chain at 9. The skip counting continues placing a square at 18 and finally at 27. The three squares are stacked up to see that they make a cube of 3, thus this chain is also equal to the cube of three. The cube is placed at the end of the chain.
In successive activities the child works up to counting progressively and regressively with the labels turned over.

e. Games of Comparison

Materials: same as before

Presentation:
The short chains are arranged as the pipes of the organ. Here the child sees the progression of the quantities which is the same as that seen on the shelves of the frame, in the hanging chains and in the cubes,
The visualization of the difference between the quantities becomes more apparent when the long chains are lain out in the same arrangement. In the long chains, the jump from one quantity to the next is more drastic.

Direct Aim: comparison of quantities sensorially

Indirect Aim: preparation for the powers of numbers (exponential increase)

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BEAD BAR MULTIPLICATION

Materials:
...box containing colored bead bars 1-10, 55 of each
...Chart I (for control)

Presentation:
We are going to represent the table of a certain number with bead bars. The child is invited to choose a number, i.e., 8. We start with 8 taken one time. One 8 bar is lain horizontally 8 x 1 = ?8. The product is also represented by an 8 bar (lain vertically below the first) The child writes 8 x 1 = 8.
Now take 8 two times. The two 8 bars are lain horizontally 8 x 2 = ?16. A ten bar and a six bar to represent the product are lain vertically, thus making a double row, The child writes the equation in his notebook. This continues until 8 x 10 = 80. Observe the geometric figures which have been formed with the 8 bars: 8x1 is a line; 8x2 is a rectangle; and so on; 8x8 represents a square, etc.

NOTE: Notice the rectangles that come before the square have a base longer than the height. The rectangles that come after the square have a base which is shorter than the height. 8x8 produced a square, which is when the number was multiplied by itself.

Afterward the child does the other tables.

Direct Aim: memorization of the multiplication tables
to bring the child to awareness of the functions of the multiplier and the multiplicand

Indirect Aims: to understand that a number when multiplied by ten results in the same number of tens and zero units
to realize that a number multiplied by itself results in a square to give the concept of forming surfaces, starting with a line, progressing to rectangles

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BEAD BAR MULTIPLICATION

Adjunct: Multiplication by Ten

Materials:
...same box of colored bead bars 1-10
...Chart I
...small white strips of paper
...black pen

Presentation:
The child is invited to take 10 bars of any color he wants. Lay them in a column, skip counting as you go. When he is finished and has found the total, lay out the corresponding number of 10 bars vertically in a row beneath the column (note the resulting rectangles should be congruent)
The number we took was 4 (place a label for 4 beneath the row of beads), 10 times. The result was 40. How do we write 40? The 4 will now indicate tens, and we put a zero for the units. (place a 0 label next to the 4 to make 40)
Repeat this with other numbers until the child realizes by himself that when you multiply a number by 10, the product will be the multiplicand with a zero tacked on in the units place.

Direct Aims: memorization of multiplication
independent realization of the fact stated previously: n x 10 = n0

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MULTIPLICATION CHARTS AND COMBINATION CARDS

a. Passage from Chart I to Chart II

The child copies Chart I. Later with the teacher or a group of children, they try to find those combinations which can be eliminated, that is, those which have like factors and equal products. Look at the first column 1 x 1 = 1 must remain. 1 x 2 = 2 and 2 x 1 = 2 are the same. 2 x 1 = 2 is crossed out. ( Or the combinations to be eliminated may be covered with green strips of the appropriate size) As in addition we can change the order of the multiplier and multiplicand, eliminating many combinations. At the end we find that half of the chart is eliminated giving us Chart II. The combinations of two equal factors were not eliminated1 x 1 = 2, 2 x 2 = 4, 3 x 3 = 9This was the same case in addition. Chart II has only 55 combinations to be memorized. (We can make the child see that only 45 of these must be memorized, as the table of ten is simply a repetition of 1)

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MULTIPLICATION CHARTS AND COMBINATION CARDS

b. Passage from Chart II to Chart III (the Whole Chart)

Materials:
...Chart III (products only- the numbers in pink serve as the multiplicand, blue as the multiplier; the one should be colored violet.
-xAlong the diagonal are found the squares of the numbers)
...box of combination cards
...Chart I or II

Exercise:
The child fishes for a combination, writes it down on a piece of paper 5 x 7 =. A finger of the left hand is placed on the 5 (pink) while a finger of the right hand is placed on the 7 (blue). Where the fingers meet, the product is found. This is written on the paper to complete the equation. The child fishes for another combination, and so on.

Control of Error: Chart I or II

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MULTIPLICATION CHARTS AND COMBINATION CARDS

c. Passage from Chart III to Chart IV (the Half Chart)

Materials:
...Chart IV (half of Chart III)
...box of combination cards
...Chart I or II

Exercise:
The child fishes for a combination and writes it down 8 x 3 =. I know that 8 x 3 gives me the same result as 3 x 8. One finger is placed on 3, another on 8 (both on the pink column) The top finger goes to the end of the row, then the two fingers come together. Where they meet, we find the product. This is written down. The child fishes for another combination, and so on.

Control of Error: Chart I or II

Note: At his point, to verify memorization, the child may be given command cards.
Example:

Find these products:

2x3=
5x6=
4x7=

etc.

and their inverses

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MULTIPLICATION CHARTS AND COMBINATION CARDS

d. The Bingo Game of Multiplication (using Chart V)

Materials:
...Chart V
...box of tiles with products
...(a) box of combination cards
...Chart I and III

A. Exercise
All of the tiles are lain out on the table face up. The child fishes for a combination and writes it down 8 x 4 =. He thinks of the answer and writes it down. The tile with the product is found and placed on the board. The child fishes for another combination.

Control: Chart I for combinations, Chart II for placement.

B. Exercise
With all of the tiles in the box, the child fishes for a tile. On the paper he writes down a combination that will yield that product, 18 = 3 x 6. The tile is placed on the chart appropriately.

Control: Charts I and III.

C. Exercise
All of the tiles are placed in stacks of common products. The child chooses one stack, i.e. 12's. On the paper he writes down a combination that will yield 12, 12 = 2 x 6. One of the '12' tiles is placed on the chart where 2 and 6 meet. The child thinks of another, and continues until all of the tiles in the stack are used. Control: With Chart 1, we can check to see that all possible combinations have been considered.
Chart III controls placement.

Note: What shape is made when the stacks of tiles are lined up in order? No special figure is made this time.

Group Games:
As before the teacher may read a combination, the child responds with the correct product; or the teacher picks a tile. The children give all of the possible combinations. These games should be done frequently, as they encourage the child to go back to these exercise if he needs more practice.

Age: from 6-7 (this work lasts for one year)

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MULTIPLICATION BY 10, 100, 1000

(Note: This activity is a prerequisite for the small bead frame)

Materials:
...decimal system materials
...paper
...black and red pencils

Presentation:
The teacher isolates a 10-bar. How many units are there in 10? 10. The teacher isolates a hundred square. How many tens are there in 100? How many units? Isolate the cube. How many hundreds are there in 1000? How many units? tens?
We can say that 10 tens is the same as 100, 100 tens is the same as 1000 and so on. With the child draw relative conclusions of all the changes possible.

Aim: to be sure that the child has understood the concept of change

By ten
Write down a multiplication problem and ask the child to lay out the problem, using the golden bead material i.e. ( 21 x 10 =). The child, knowing the function of multiplication, combines these quantities and makes the necessary changes. With the answer - two hundreds, one ten, and the zero is written in red. 21x10 = 210 Observe that the product is simply 21 (the multiplicand) with a zero after it.
Do many examples of this type, including: 30 x 10 = 300

By one hundred
Write down a multiplication problem such as 23 x 100 = and ask the child to lay out the material.. We can't put out 23 one hundred times, we would run out of beads! We can multiply each unit by 100. Isolate one bead from the 23.
1 x 100 = 100 Substitute the bead for a hundred square. Repeat for the other two units. Then 10 x 100 = 1000. Replace each ten bar with a thousand cube, and so on. Record the product. 23 x 100 = 2300. Notice that the product has the same number of zeros as the multiplier.

By one thousand
Write the problem 4 x 1000 =. As before, multiply each unit by 1000, replacing each bead with a thousand cube. Record the product 4 x 1000 = 4000. In this case we jumped from the units, past the tens, past the hundreds, to the thousands. For each hierarchy that we increased, one zero was added. Observe as before that the number of zeros in the product is the same as the number of zeros in the multiplier. The product is simply the number of zeros in the multiplier.

Direct Aim: ease of multiplying by powers of ten, and understanding of the characteristic patterns of such multiplication.

Indirect Aim: preparation for multiplication using the bead frames

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xCheckerboard - Geometrical Analysis of Multiplication


INTRODUCTION TO THE CHECKERBOARD

Materials: The Checkerboard:
each square has sides of 7 cm so that the longest bar will fit;
squares of the same color on the diagonal represent the same value;
the numbers along the side and the bottom are printed in hierarchic colors the bottom right square, which is green and represents simple units is the square having the least value;
whereas the square in the opposite corner, which is also green and represents units of billions, is the square having the greatest value
also, box of bead bars, 55 of each (remove the 10-bars)
box containing numeral cards 1-9
3 series printed in black on white cards (multiplicand)
3 series printed in black on gray cards (multiplier)

Presentation:
Familiarize the children with the board, noting the value of each square, the hierarchic colors, and the pattern of the values along the diagonal. Afterward the child may draw his own board.

Games:
A) Place a bead bar such as (5) on the unit square and ask the child to identify its value. Move the bead bar to the left to the tens square. Identify its value. (50) Move the bead bar along the diagonal to the next tens square. Identify its value. (50) Place the bead bar on the bottom row-hundred square. Continue moving the bead bar, and identifying its value as it changes its position.

B) Place a bead bar on the unit square and identify its value. As it moves up the column, identify its value. Note that the value increases by 10 each time. Repeat the procedure moving the bead bar down the column, noting that the value decreases by 10 each time. Move the bead bar to the ten square at the bottom and repeat the game. Again we notice that the value increases by 10 as it goes toward the top, and it decreases by 10 as it moves toward the bottom again.

C) Place two bead bars on two different squares and read its value. Place two bead bars in such a way that an inferior hierarchy is left blank.-430,403.

D) Place four bead bars on four different squares along the bottom row. Identify the number. Move one bar to the second row and identify the value; it is the same. Continue moving one bead bar at a time along the diagonal, identifying the number; it stays the same.

Aim: to familiarize the child with the board
to emphasize that squares on the diagonal have the same value

Note: With the bead frames and the hierarchic materials (blocks) we gave the concept of the hierarchies. With this material we will reinforce that concept. Since the concept is presented in a different way, we must be sure that the child understands how this work is organized.

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MULTIPLICATION WITH THE CHECKERBOARD

Materials:
...checkerboard
...box of numeral cards 1-9, gray and white
...box of bead bars 1-9, 55 of each

Presentation:
Propose a problem: 4357 x 23 =

a. 1st level
Form the multiplicand by using the white cards placed on the appropriate numerals
on the bottom edge of the board. (7 is placed on 1, 5 is placed on 10, etc.) Form the multiplier using the gray cards placed on the appropriate numerals on the right edge of the board.
Begin multiplying with the units. First we take 7 three times. Place 3 seven bars on the unit square.
5 x 3place 3 five bars on the tens square
3 x 3place 3 three bars on the hundreds square
4 x 3place 3 four bars on the thousands square
Keep a finger on the digit of the multiplicand to remember your place. Notice that there are three of each quantity in this row. Why? because the multiplier is 3. Since we have finished multiplying by the units, we can turn over the gray card. Continue multiplying by the tens noting the value of each square (this emphasis is important): tens multiplied by units give tens, tens multiplied by tens gives hundreds, etc. Notice that 2 dominates the row. Turn over the card.
Move the bead bars of the upper row along the diagonal to the bottom row. Beginning with units make changes to total the product, carrying over as necessary, i.e. the bead bars in the ten square total 3131 tens. How do we express 31 tens in conventional language? Three hundred ten. So, place a unit bead in the ten square, and a 3 bar in the hundred square.
Read the total and record the product.

Aim: to understand the process of multiplication using the board.

b. 2nd level-Small Multiplication
Propose a problem: 4357 x 423 = and set up the board with the numeral cards.
Begin multiplying with the units, but this time only put out the bead bars for the product.
7 x 3 = 21 put a unit bead in the unit square, 2-bar in tens
5 x 3 = 15 5-bar in tens square, unit in hundreds
3 x 3 = 9 9-bar in the hundred square
4 x 3 = 12 2-bar in thousands, unit in ten thousands
Turn over the gray card. Continue with the tens. Move the bead bars along the diagonal in the end.
Make the necessary changes and read the final product.

c. 3rd level-Partial Products (this passage can be skipped)
Multiply in the same way as before (2nd level). After everything in the multiplicand has been multiplied by the units, make the necessary changes in that row and record the partial product,
Continue with the tens, etc. After all of the partial products have been recorded, move the bead bars along the diagonal to the bottom row.
Make the changes and read the total product.

d. 4th level-Mental Carrying Over
The procedure is different from the 3rd level only in that the child carries mentally. 7 x 3 = 21 put the unit bead down, remember 2...5 x 3 = 15plus 2 = 17. etc. The partial product is read without making any changes.

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MULTIPLICATION AND DRAWING

Materials:
...checkerboard
...box of bead bars 1-9, 55 of each
...box of numeral cards 1-9, gray and white
...graph paper
...colored pencils in red, green, and blue

Presentation:
The multiplication is done in the same manner as the first level of checkerboard multiplication. Propose a problem and write it down. Set up the numeral cards on the board and begin multiplying by the units of the multiplier (place 3 bars of 2 for 3 x 2, etc.) Draw the result of this partial product - 1 square for each bead. Color the rectangles and squares in the appropriate hierarchic colors: units multiplied by units gives units, so color it green, etc. Write each product in the rectangles.
Analyze the first partial product: units times units gives us units (write 1 x 1 = 1); 2 x 3 = 6; these are 6 units tens. Tens times units gives us tens (10 x 1 = 10); 3 x 3 = 9; 9 tens = 90. Continue analyzing the partial product in this way.
Go on to multiply by the tens, placing the bars on the checkerboard. Draw the result and write the products of the small multiplications. Analyze the partial product in the same way as before.
Make the necessary changes in the first row to obtain the partial product. Verify this by adding the column of products (in the analysis of the first partial product) Write this partial product under the multiplication problem. Repeat the procedure for the second partial product.
Move the bars along the diagonal to the bottom row. Make the changes to get the total. Add the two partial products to control and verify. Record this final product under the original problem.

3432
x_43
10296
137280
147576

u x u = u
t x u = t
h x u = h
th x u = th

u x t = t
t x t = h
h x t = th
th x t = tth

1 x 1 =
10 x 1 =
100 x 1 =
1000 x 1=

1 x 10 =
10 x 10 =
100 x 10=
1000 x 10=

1
10
100
1000

10
100
1000
10,000

2 x 3 = 6
3 x 3 = 9
4 x 3 = 12
3 x 3 = 9

2 x 4 = 8
3 x 4 = 12
4 x 4 = 16
3 x 4 = 12

6
90
1200
+ 9000
10296
80
1200
16000
+120000
+137280

147,576

Having completed and understood this activity, the child should have realized what multiplication must be done to change from one hierarchy to another: to obtain hundreds, he has three possibilities as indicated on the checkerboard: 100 x 1, 10 x 10, 1 x 100.
He also should realize that the quantities are moved along the diagonal to add quantities of the same hierarchy. This is a change from adding in vertical columns on the forms for the bead frame. The colors, however, aid the understanding of this difference.

This drawing activity allows the child to visualize all multiplication geometrically as rectangles and squares. Even the square of a number with 2 or more digits is composed of smaller squares and rectangles. Thus this work is remotely indirect preparation for square roots and the study of perfect squares.

Age: 7-8 years [For all checkerboard work]

Aim: to reinforce the concept of hierarchies
to visualize multiplication in its geometric form

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xBead Frame Multiplication


SMALL BEAD FRAME


a. Multiplication By 10, 100, 1000

Materials:
...small bead frame
...paper and pencil

Presentation:
Write down the problem 2 x 10 =. Perform this on the frame by sliding forward groups of two beads, changing as necessary. Record the product 2 x 10 = 20. Repeat the process in the problem 20 x 10 =. Record the product 20 x 10 = 200. Try this problem
2 x 100 =. It would take all day and most of the night to bring forward 100 groups of two beads. Recall from the multiplication game-Multiplication by 10, 10, 1000 the simple way to do this. Slide forward the 2 beads to correspond to the multiplicand. We can multiply 1 x 100 and get 100-slide the one unit back, and slide one hundred forward. Repeat for the second unit. Record the product 2 x 100 = 200. We went from units to the hundreds. How many jumps did we have to take? 2 How many zero's are in the product? 2 The two zero's indicate that we have passed two hierarchies.
Do many other examples, i.e. 2 x 1000 = to be sure that the child is very comfortable and familiar with this process.

Note: We are limited by the frame to having a multiplicand of only one digit when the multiplier is 1000 and vice versa.

Aims: to understand of the use of the small bead frame in performing multiplication
to reinforce the concepts of the relative positions of the hierarchies, and changing from one hierarchy to another

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SMALL BEAD FRAME

b. Multiplication with a One-Digit Multiplier

Materials:
...bead frame
...small form

Presentation: To isolate the difficulty of decomposing the multiplicand, we begin with a static multiplication. From then on the child will work with dynamic problems.

2321
x     3

  Write the problem on the left side of the form.



The first thing we must do is to decompose the multiplicand. There are how many units? 1, we write 1 on the right side under units. All of this we must multiply by 3. On the bead frame, perform the multiplication. 1 x 3 = 3, move forward three units beads.
2 x 3=6, but 6 what? 6 tens! Move forward 6 ten beads, etc. (By this time the child should have memorized the combinations and should bring forward the product of the small multiplication) Read the product and record it on the left side of the form.

Try a dynamic multiplication

2463
x     4

Decompose the multiplicand in the same way as before.


Perform the multiplication 3 x 4 = 12, 12 is 2 units and 1 ten...6 x 4 = 24, 24 what? 24 tens4 tens and 2 hundreds, etc.
Read the product on the frame and record it.

Experiment:
Try performing any one of these multiplications out of order, i.e. 6 x 4 = 24 tens,
2 x 4 = 8 thousands, 3 x 4 = 12 units and 4 x 4 = 16 hundreds. The product is still the same.

Note: Maria Montessori said, "When you go to the theater, you find that people are all sitting in different areas; some are in the balcony, some are in the boxes. Why? Each person has chosen a seat by buying a certain type of ticket. In the same way, these units must be in the top row of the bead frame. That is their fixed place."

Age: 6-7 years

Aim: realization of the importance of the position of each digit

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LARGE BEAD FRAME - MULTIPLIERS OF 2 OR MORE DIGITS

a. The Whole Product

Materials:
...large bead frame
...corresponding long form
...red and black pencils

Presentation:
Write a multiplication problem on the form

8457
x   34

Decompose the multiplicand in the same way as before.


On the right side decompose the multiplicand as before. First decompose the number for multiplication by 4 units.

7
30
400
8000

x 4

We must also multiply the multiplicand by 30; decompose the number a second time below the first. We know that we cannot multiply by such a large number on the bead frame. The rule is that we must always multiply by units. 7 x 30 is the same as 70 x 3. ( 7 x 30 = 7 x 3 x 10 = (commutative property) 7 x 10 x 3 = 70 x 3 ) So we can write this decomposition in a different way. For our work we will use the first and third decompositions.
Note: By decomposing the multiplicand we have reduced the problem to a series of small calculations at the level of memorization.

Begin multiplying

7 x 4 = 28 28 units move forward 8 units, 2 tens
3 x 4 = 12 12 tens. move forward 2 tens and 1 hundred
4 x 4 = 16 16 hundreds. move forward 6 hundreds and 1 thousand
8 x 4 = 32 32 thousands move forward 2 thousands and 3 ten thousands

Continue with the second decomposition in the same way.

Read the final product and record it.

Note : This multiplication can be shown on an adding machine in the same way, though as a repeated addition. Calculators operate on the same principle of moving the multiplicand to the left and adding zeros.

The child may go on to do multiplication with multipliers of 3 or more digits as well. With a three-digit multiplier there will be 5 decompositions of which only the 1st, 3rd, 5th will be used for the multiplication on the frame.

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LARGE BEAD FRAME - MULTIPLIERS OF 2 OR MORE DIGITS

b. Partial Products

Presentation: The child by this time should have reached a level of abstraction with column addition.
The procedure is exactly like the first, except that the child will stop after each multiplier and record the partial product, clearing the frame he begins multiplying with the next multiplier. When the child records all of the partial products, he adds them to find the total product.

4387
x 245
21935
175480
877400
1074815

Here we can observe that the first partial product which was the result of multiplying the units has its first digit under the units column. The first digit (other than zero) of the second partial (which was the result of multiplying by the tens) is under the tens column, etc.

Age: 7-8 years, or when the child is adding abstractly

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HORIZONTAL GOLDEN BEAD FRAME

a. The Whole Product

Materials:
...The Horizontal Bead Frame, which lies flat on the table.
...It is less sensorial in that hierarchic colors and spaces between the classes have been eliminated.
...box of 4 series of gray cards on which 1-9 is written in black (to serve as multiplier)
...strips of white paper on which multiplicand will be written.

(note: the black lines are drawn on the board beneath the wires; they will indicate where to begin the multiplication when multiplying by units, tens, hundreds or thousands.)

Presentation:
All of the previous operations can be done with this material, but we will do the most interesting: multiplication with a two-digit multiplier.
Write down a problem, 6542 x 36 = and show the child how to set up this problem. Place a white strip over the zeros and secure it with a rubber band or tape. Write the multiplicand on the strip so that the digits correspond to the correct wires. Find among the gray cards the digits needed to form the multiplier. Place the 6 over the lowest green dot which represents the units, and the 3 over the blue dot for the tens. The beads should be at the top to start.
Begin multiplying 2 x 6 = 12, bring down 2 units and 1 ten. 4 x 6 = 24 tens - bring down 4 tens and 2 hundreds and so on. After the multiplicand has been multiplied by the units we can turn over the card '6'.
In order to multiply by the 3 tens, 30, we must move the multiplicand to the left one space to let one red zero show. This is just like multiplying the number by 10.
The black line indicates that we start with the row of tens. Continue multiplying, making changes as necessary. In the end we read the product and record it.

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HORIZONTAL GOLDEN BEAD FRAME

b. Partial Products

Materials:
...The Horizontal Bead Frame
...box of 4 series of gray cards on which 1-9 is written in black
...strips of white paper

The procedure followed here is exactly the same, except that when the child has finished with one multiplier he turns over the card, reads the partial product, writes it and clears the frame before beginning with the next multiplier. In the end he adds abstractly to total the partial products.

Age: 7-8 years

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HORIZONTAL GOLDEN BEAD FRAME

c. Carrying Mentally

Materials:
...The Horizontal Bead Frame
...box of 4 series of gray cards on which 1-9 is written in black
...strips of white paper

The child sets up the multiplication problem on the frame.

2443
x 46

 

3 x 6 = 18 move down 8 units, remember one ten in your head.

5 x 6 = 30...+1 = 31 move down 1 ten, remember 3 hundreds

4 x 6 = 24...+3 = 27 hundreds-move down 7 hundreds, etc.

Record the partial product and clear the frame before beginning multiplication by the tens.

Age: 8 years

Note: The work done with this frame is on a higher level of abstraction than the work with the hierarchic frames. In both activities the tens, hundreds and thousands of the multiplier were reduced by a power of 10, while the multiplicand increased by a power of 10. The same work was done in two different ways.
At the end of this work the child should understand that when he starts multiplying with a new digit of the multiplier, he must move over one hierarchy. The partial products must start from the same hierarchy as the corresponding digit of the multiplier.
This activity forms the basis for an understanding of the function of multiplication with a multiplier of two or more digits, and a preparation for abstract solution. The child doing this activity will be stimulated to invent his own problems.

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xMore Memorization Exercises


THE SNAKE GAME

Materials: same as for addition snake

Exercise:
This time in constructing the snake, it is not important to use many different colors; rather several bead bars of each of a few colors should be used. A resulting snake may be: 2 + 4 + 8 + 4 + 2 + 4 +8 + 4 + 2. The snake is counted as in addition.

Control of Error: To one side the ten bars and black and white bars are grouped together. At the other side the original bead bars are grouped according to color. How many times do we have 8? We can say 8x2; the equation is written on a piece of paper. The beads to represent the product 16 are placed below the group of 8 bars. The same is done for 4x4 and 2x3. The three products are added on paper and/or with the second row of bead bars to show that the snake was counted correctly.

Age: from 6 years onwards

Note: The child does this work after completion of the exercises for memorization of multiplication

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VARIOUS WAYS OF CONSTRUCTING A PRODUCT

Materials:
...box of colored bead bars 1-10
...Chart I

Presentation:
The teacher writes a product on a piece of paper and places it on the table (it will be the heading). Let's see how many ways we can make 12? Shall we try with one? The child should see that any number can be made using one, therefore it will be omitted for this game.
Beginning with two, the child tries to make 12 by repeating 2, skip counting as he takes out the bars. Stopping at 12, he finds that 2 taken six times equal 12. These bars are placed in a column under the strip. the child writes the combination in his notebook,
12 = 2 x 6, which is read 12 is constructed by taking 2, 6 times.
Go on trying with 3 and 4 following the same steps. When the child tries with 5, he will find that 15 is greater than 12, so the bead bars are put back in the box. Go on with 6. Trying with 7, the child finds that it won't work, because at the second bar we've already exceeded 12. Thus we must stop at 6.

Control of error: The child looks on Chart I, finds the combinations he has made, and the absence of 12 in the 5 column, 7 column and so on.

Direct Aim: memorization of multiplication

Indirect Aims:
...preparation for memorization of division
...preparation for decomposition into prime factors
...preparation for the study of multiples

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SMALL MULTIPLICATION

Materials:
...box of colored bead bars 1-10
...Chart I

Presentation:
The child is invited to take any number (not to exceed 10) of any color of bead bars. From this pile the child begins to lay them out in a column; saying 4 taken 1 time, putting out a bar and writing either the equation or just the product in his notebook, He continues until the whole pile has been lain out.

Control of error: Chart I. If the child wrote the equation then he has written a table. If only the products were recorded, then he has done progressive numeration (skip counting).

Direct Aim: memorization of multiplication

Indirect Aims:
...preparation for memorization of division
...preparation for decomposition into prime factors
...preparation for the study of multiples

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INVERSE PRODUCTS

Materials: same as above

Presentation:
The child is asked to take the bar of 7, 6 times. The bars are lain in a column. How do we express this? 7 x 6 = 42 is written down. Now take the bar of 6, 7 times. These are lain next to the others. When the child writes down 6 x 7 =42, he should observe that the product was the same. Are the multiplication problems the same? Notice that the two rectangles are equal, each having 42. even though the order of the factors is not the same. The child should observe in the written form that the factors are the same, just reversed in their positions.
Give several examples of this.

Direct Aims: memorization of multiplication understanding of the commutative property of multiplication

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CONTRUCTION OF A SQUARE

Materials:
...box of colored bead bars 1-10
...one each of the squares 1-10

Presentation:
Let's try to multiply all of the numbers by themselves. During this activity the child writes the equations as he goes along. He may also draw this on graph paper as he progresses or when he is finished.
Start with one. One taken one time is one. Put out one bead. Write down 1 x 1 = 1. Two taken two times is four. Place two 2- bars in a column next to one. Write down
2 x 2 = 4. Continue in this manner until 10 x 10, resulting in 9 columns in a row. If the child doesn't remember a combination he may check Chart I.
We have multiplied all the numbers by themselves. What have we formed? squares. Because these are only bars, we can substitute them with the real squares. With one, there is no square because 1 x 1 is just 1. Replace the 2 bars of 2 with the 2 square and so one up to 10.

Direct Aim:
...memorization of multiplication
...realization that a number taken by itself n = makes a square.

Indirect Aim: preparation for the powers of numbers

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MULTIPLICATION OF A SUM

Materials:
...box of colored bead bars 1-10
...box of signs for the operations
...Chart I
...pieces of white paper
...pen

Binomial Presentation:
The teacher writes on a strip ( 5 + 2 ) x 3 =, which is read, take 5 plus 2, 3 times. This is a multiplication problem. The problem is prepared as before with bead bars for the addends, signs, parentheses and the multiplier, written on a little card. Recall that the operation inside the parentheses is done first. The child places the 7-bar for the sum under the parentheses, places signs, multiplier and the product, represented in bead bars. The work is recorded.

On a different day:
When you find a problem of this kind, you can also multiply one term at a time by the multiplier. The other way will be put aside for now. (The equation in beads and cards:
7 x 3 = 21 is placed off to the side, leaving the slip of paper and the original layout of beads)
First take 5, 3 times, 5 x 3 is written on a strip and 3 bars of 5 are placed below the original 5 barplus (put out the sign)2 taken 3 times, 2 x 3 is also written on a strip and 3 bars of 2 are lain out. Now we must find these products. The products are placed below the group in a perpendicular position. Add 15 + 6 and put out the result, The result is the same as the equation we put aside. The child writes in his notebook:

( 5 + 2 ) x 3 =
( 5 x 3 ) + ( 2 x 3 ) =
15 + 6 =

 

21

Trinomial Presentation: On yet another day
The teacher writes a problem on a strip such as: ( 5 + 2 + 3 ) x 4 =. The child lays out the corresponding beads for 5, 2, and 3, the signs, parentheses and a little card for 4. As before we must multiply each term by the multiplier. Then, for control, the child may add the addends within parentheses and multiply the sum by 4. When his work is written in his book it should be:

( 5 + 2 + 3 ) x 4 =
( 5 x 4 ) + ( 2 x 4 ) + ( 3 x 4) =
20 + 8 + 12 =

(__+__+__) x 4 =
beads
40 beads

Note: After the child has learned to multiply such a problem term by term, he should not go back to the first way of adding first, then multiplying. In this way the following aims will be achieved.

Direct Aim: memorization of multiplication
understanding of the distributive property of multiplication over addition

Indirect Aim: preparation for the square of the polynomial

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ANALYSIS OF THE SQUARES - BINOMIAL

Materials:
...box of colored bead bars 1-10
...one each of the squares of 1-10
...rubber bands

Presentation:
The teacher presents the hundred square, and the child identifies it as the square of 10, or the hundred square. Observe that it has 10 beads on one side and ten on the other. Write 10 x 10 = 100 on a strip of paper.
Because we know this square so well, we are going to perform a small division of the square. The child is asked to count 6 beads along one side. A rubber band is placed after the 6th bead around the square, (Note: the result will be two perpendicular rubber bands.)
How many parts has the square been divided into? Let's see how the 4 parts are composed: 6 on one side, 6 on the other, 6x6; 4 by 6 or 4 taken 6 times , 6x4, 4x4. Let's write this down and while writing, we can reconstruct the square with colored bead bars, As each combination is written, the product is recorded, as the appropriate beads are lain out.

 6 X 6 = 36

4 X 6 = 24

6 X 4 = 24

4 X 4 = 16

60 + 40= 100

Add the products. Push the bead bars together and place the 100 square over it to verify sensorially that the decomposition was done correctly.

Note: Later, after the passing of a year and much work with the decomposition of a square and the powers of numbers, the child will learn the exact way of writing this:


10

2
=

( 6 + 4 )
2
= ( 6 + 4 ) x ( 6 + 4 ) = ( 6 x 6 ) + ( 6 x 4 ) + ( 4 x 6 ) + ( 4 x 4 )

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ANALYSIS OF THE SQUARES - TRINOMIAL

Materials:
...box of colored bead bars 1-10
...one each of the squares of 1-10
...rubber bands

Presentation:
This time we'll decompose the square in a different way. Count 3 along one side, and place the rubber band around the square. Count 3 along the other side and put on a rubber band. Now continue counting 5 more beads, and put a rubber band around. Do the same on the other side.
Into how many parts have we decomposed the square? Observe how each of the 9 parts is composed. Notice the three squares which all lie on the diagonal, and the various rectangles formed.
As before, we'll construct the squares as we write it all down.

3 x 3 = 09
3 x 5 = 15
3 x 2 =
06

5 x 3 = 15
5 x 5 = 25
5 x 2 = 10

2 x 3 = 06
2 x 5 = 10
2 x 2 =
04

 

30

+ 50

+ 20

= 100

Control of Error: Compute the sums of the three columns and add them together. Then slide all the bead bars toward the center, and place the 100-square on top. These are the two ways to prove that this equals one hundred.

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PASSAGE FROM ONE SQUARE TO A SUCCEEDING SQUARE

Materials:
...box of colored bead bars 1-10
...one each of the squares of 1-10
...rubber bands

Presentation:
The child identifies the square chosen by the teacher, i.e. the square of 4. Let's use this square to build a square of 5. Allow the child to suggest and discover ways of doing this. Guide the work, giving guidelines such as: 'The sides must be built upon.'
Four bars are placed to the right and the bottom of the square. A one-bead fills in the hole left in the lower right hand corner. Superimpose the square of five to verify successful completion. What was added to the square of 4 to make the square of 5? two bars of 4 and one unit.

square of 4
+ what was added
= square of 5

4 x 4 = 16
4 + 4 + 1 = 9
5 x 5 = 25

Aim: indirect preparation for the square of a binomial

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PASSAGE FROM ONE SQUARE TO A NON-SUCCESSIVE SQUARE

Materials:
...box of colored bead bars 1-10
...one each of the squares of 1-10
...rubber bands

Presentation:
The teacher volunteers to assist in constructing a square of 9 from a square of 5. What is the difference when you take 5 from 9? So, how many more bars of 5 must be added to this side to make 9? Four bars of 5 are added to the right side. In the same way 4 5-bars are added to the bottom. Again, there is a hole. Count the number of beads on the sides of the hole...4 by 4. Four 4-bars can be replaced by the square, Use the square of 9 to control:

 

5 x 5

+ 5 x 4 + 5 x 4 + 4 x 4
What was added?

25 +

20 +
20 + + 16 = 81
    9 x 9 = 81  

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SKIP COUNT CHAINS - FURTHER EXPLORATION

a. List of Materials

Materials:
...Board of powers (though it is not named as such at this point)
...Cubes, long chains, squares, short chains
...Two Boxes:
......arrows for short chains, i.e. for 5 we have 1, 2, 3, 4, 5, 10, 15, 20, 25
......arrows for long chains, i.e. for 3 we have 1, 2, 3, 6, 9, 12, 15, 18,21, 24, 27

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SKIP COUNT CHAINS - FURTHER EXPLORATION

b. Construction of Geometric Figures

Materials: short chains

Exercise:
Using the short chains the child tries to construct regular polygons. With the short chain of 2, he cannot make anything, though with 3, he can make a triangle; with 4, a square; with 5, a pentagon, and so on.

Direct Aim: reinforcement of law: the smallest possible polygon must have three sides

Indirect Aim: preparation for perimeters of polygons: preparation for multiples and divisibility

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SKIP COUNT CHAINS - FURTHER EXPLORATION

c. Decanomial (a polynomial having ten terms) & The Construction of Chart I

List of Materials

Note: This material is presented parallel to memorization of multiplication, in three different presentations.

Materials:
...box of bead bars, 1-10, 55 of each
...square of the numbers 2-10
...Multiplication (memorization) Charts I and III

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SKIP COUNT CHAINS - FURTHER EXPLORATION

c. Decanomial (a polynomial having ten terms) & The Construction of Chart I

Vertical Presentation

Beginning with the ones table, reconstruct Multiplication Chart I, 1 x 1 =...2 The child states the product and puts out one bead, 1 x 2 = ...2 The child takes a unit bead (1) two times and places them in a column, 1 x 3 = ...3 The child puts out three unit beads in a column and so on to 1 x 10 = 10.
Construct the twos table 2 x 1 =...2 Place the two bar at the beginning of a new column, so that it lines up with 1 x 1. Go on making a column for the table of 2...2 x 2, 2 x 3, 2 x 4, 2 x 5, 2 x 6...making new columns for each multiplicand, ending the last column with 10 x 10. Use Chart I for control, as you go along, if the child needs it for recalling products. The result will be a square, made up of 10 vertical strips each strip being the width and color of that bead bar.

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SKIP COUNT CHAINS - FURTHER EXPLORATION

c. Decanomial (a polynomial having ten terms) & The Construction of Chart I

Horizontal Presentation

This time, the multiplier will remain constant, as we progress along the rows of Chart I. Begin with 1 x 1=...1 Place the unit bead 2 x 1 = 2 Place the two bar next to it forming a row. Go on to 10 x 1 =...10. Beginning the second row with 1 x 2, each of the bars must be taken twice. Place the unit beads in a column forming the beginning of the second row. Continue in this way up to the end of the last row-10 x 10
The result of this work is the same square as before. We can't tell by looking at the finished product whether it was made the first way or the second way. This square was formed by 10 horizontal strips, varying in width as before, but now with each having the same multi-colored pattern.

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SKIP COUNT CHAINS - FURTHER EXPLORATION

c. Decanomial (a polynomial having ten terms) & The Construction of Chart I

Angular Presentation

As always we begin with 1 x 1 = 1. Go on to 2 x 2 = 2; put out a two bar, making a row, 2 x 1 gives us the same as 1 x 2; put out two unit beads making a column 2 x 2 =...4; put out two 2-bars. Outline the formation with a finger to help the child to observe the square that was formed.
Continue with 3 x 1, then 1 x 3...3 x 2, then 2 x 3 and finally 3 x 3 to fill in the arrangement to make a larger square. Continue in this way with the child stating the products for each combination and stopping to observe each square that is formed.
The result of this work is the same square as before. Notice the various geometric forms; point, lines, rectangles, and squares. Substitute real squares for the bars: 1 x 1 is still 1 so we leave the bead. Replace 2 x 2, 3 x 3 and so on up to 10 x 10. We can see that the squares are placed on the diagonal just as they are on Chart III. (outlined in bold black lines)

Control of Error: visible arrangement; number of bead bars in the box

Direct Aim:
...memorization of multiplication
...development of mental flexibility

Indirect Aim: preparation for the Decanomial

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SKIP COUNT CHAINS - FURTHER EXPLORATION

c. Decanomial (a polynomial having ten terms) & The Construction of Chart I

Commuted Decanomial

Materials:
...Two boxes of bead bars 1-10, 55 of each
...(actually 1 box of 55 each, and nine 8-bars, twenty six 9-bars, and forty five 10-bars)
...squares and cubes of the numbers 2-10
...Multiplication Charts I and III

Part One
Presentation:
The Decanomial is lain out already with the squares substituted along the diagonal.
"We're going to change its beautiful colors. As usual we start with 1x1. It stays the same. Here we have 2x1 which is a green bar. 1x2, which is the same thing as 2x1, is a poor imitation of a bar. We'll exchange this fake bar for a real 2-bar. Then we have 2x2, which is a square; it remains the same."
Continue with 3 x 1, and 1 x 3 which are the same thing. Therefore, we'll make them look the same by placing a pink 3-bar in place of these unit beads. We have 3 x 2 and 2 x 3 which are both equal to 6, because changing the order of the factor doesn't affect the product. Change the 2-bars for two 3-bars. Then there is 3 x 3, which is the square -9; we have it there.
Continue changing the beads following the same order as the angular presentation that preceded. After we have reached 4 x 4 (see diagram) the changing pattern of colors begins to appear. The strip of yellow outlines two sides of the square, which as before becomes bigger and bigger. After four we're going to make the right angle blue. Continue with 5 up to 10, where the angle is gold.

Part Two-Building the Tower
The square has changed its colors, but we haven't finished here. We can see that the diagonal is made up of several squares. The diagonal is like the spinal cord of the square, because, like your spinal cord, it supports everything and keeps it straight.

(Note: present one of these methods to the children)

1st Method: 1 x 1 = 1 It remains the same. Look at the green angle. We have 2 x 2 and
1 x 2. When we place these 2-bars together we make a square of 2. Exchange these 2 bars for a square of 2, since there is no room for it in the places vacated by the 2-bars.
Starting at the column of three, slide the bars of 33 x 3 towards the bar of 3+ ( 3 x 1 ) to see that when combined they form a square of three. Substitute the bars for a square which is placed in the vacant column. Going to the row of three, do the same3 x 2 + 3 x 1 = a square of three.
Continue with the column of four. Push one bar from 4 x 2 down to meet 4 x 3:
4 x 3 + 4 x 1 forms a square. Replace these bars with a square. On the row repeat the procedure: 4 x 3...+...4 x 1...forms a square. Replace the bars with a square. At the head of the row and the column combine the single bars to form 4 x 2 and 4 x 2. Combine these two groups to make a square which is placed on top of the one on the diagonal.
(cont.)
Continue with the column of five: 5 x 4 + 5 x 1 a square. Repeat on the row. The on the column: 5 x 2 + 5 x 25 x 1 = a square. Repeat on the row.
As a result the square will be transformed into an uneven arrangement of squares on the diagonal, with 2 squares stacked at each even number. Superimpose all of the squares to make an oblique line of stacks. These are then transformed into cubes and stacked to make a tower.
Our spinal column has been transformed into a tower, just like the pink tower. Everything that was spread out on this table has been used to construct the tower. Our Pythagorean table has been transformed into a beautiful multi-colored tower just like the ugly duckling became a swan.

Variation on the 1st Method
Since the newly obtained square of two will not fit in the vacated places, we can move the original square and place one square in the column and one in the row. When the problem arises at each even number, re-arrange the portions of the square slightly so that half are in the column and half are in the row. The result will be a broken diagonal, having a vacant space at each even number.

2nd Method
As before, 1 x 1 is left alone, and 2 x 1 is combined with 2 x 1 to form a square which is stacked on top of the other square. The combination of groups of bead bars will always be with reference to the existing square.
For three, choose the bar on the column which is farthest from the square and combine it with the group of bars on the row which are nearest to the square: 3 x 1 + 3 x 2 = a square. Place the new square on top. Continue combining: 3 x 2+3 x 1
gives a square. The third square completes the stack.
For four, start with the bar on the column furthest from the square, combining it with the group of bars on the row nearest the square4 x 1 + 4 x 3 = a square. Then 4 x 2...+...4 x 2... = a square. Continue until all the squares are stacked forming an oblique line. Proceed as before.

Part Three-Decomposition of the Tower
This collective activity is similar to the bank game for the decimal system. One child acts as the banker, while the others change the quantities.
Begin by dismantling the tower of cubes to make a diagonal of cubes. Starting with two, the child recalls that the cube is made from 2 squares of two. The cube is replaced by a stack of squares. Continue until all of the cubes are transformed into stacks.
Take one square of two and ask, 'What is the square made of?' two 2-bars. Exchange the square. Where will we put them? They correspond to 2x1 and 1x2; therefore place them accordingly.
Continue to break down squares in a way that is the opposite of the way you chose to construct them. Notice again the angles of color being formed, with the square always serving as the vertex. The result will be the original Pythagorean table from which we began.
This activity is complicated only in the sense that the tasks must be divided among the numbers of the group.
Note: ( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 )2 = 552 = 3025
13 + 23 +33 +43 +53 +63 +73 +83 +93 +103 = 3025

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SKIP COUNT CHAINS - FURTHER EXPLORATION

c. Decanomial (a polynomial having ten terms) & The Construction of Chart I

Numeric Decanomial

Materials:
...a control chart
...envelopes numbered 0-9: 0 contains 10 blue squares, 1-9: rectangular pieces on which products are written (i.e. 5 has products of
...5 x 6, 5 x 7, 5 x 8, 5 x 9, 5 x 10, 6 x 5, 7 x 5, 8 x 5, 9 x 5, 10 x 5)

Presentation:
Take out the contents of envelope 0 and lay them out in order on the diagonal. If the child hasn't had powers, simply explain that 102 is one way we can write 10 x 10.
Examine the contents of envelope 1, forming pairs of like rectangles. Mix them up and fish for one. The child reads the product, thinks of the combination and places the rectangle in the formation, just as in the Bingo Game. Allow the child to continue choosing envelopes in any order. At a later stage, eliminate the envelopes 1-9 and mix the pieces in a basket from which the child fishes.
Later the child will realize that this puzzle like the table of Pythagorean is symmetrical, having equal products on opposite sides of the diagonal.

Age: between 6 1/2 and 81/2 years

Aim:
...to incarnate the geometric figures formed by multiplication
...an indirect preparation for square roots and the square of polynomials

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SPECIAL CASES

Materials:
...multiplication combination booklet
...chart
...black and red pens
...special combination cards

Presentation:
The procedure is as for special cases in addition and subtraction, resulting in a chart as below: (The words in parentheses do not appear on the chart, but are given orally)

0- Calculate the product
2 x 3 = ? ( two taken three times; what does it give me?)

1- Calculate the Multiplier
2 x ? = 6 ( two taken how many times will give me six?)

2- Calculate the Multiplicand
? x 3 = 6 ( what number taken three times will give me six?)

3- Inverse of Case Zero---Calculate the Product
? = 2 x 3 ( what number do I obtain when I take two, three times?)

4- Inverse of The First---Case, Calculate the Multiplier
6 = 2 x ? ( six is equal to two taken how many times?)

5- Inverse of the Second Case ---Calculate the Multiplicand
6 = ? x 3 ( six is equal to what number taken three times?)

6- Calculate the Multiplier and the Multiplicand
6 = ? x ? ( six is the number I obtain when I take a certain number, a certain number of times; what are these numbers?)

Note: In cases 1, 4 and 7, the child performs multiplication, but in all others division is indirectly involved

Activity:
The seven special combination cards are left at the disposition of the children, combining these with the cards given previously for special cases of addition and subtraction.

Aim: further understanding of the concept of multiplication

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THE BANK GAME

This activity is the culmination of the many skills the child has developed: addition, multiplication at the level of memorization, multiplication and division by powers of 10 and changing from one hierarchy to another. This work is parallel to the large bead frame.

Materials:
...box containing 9 series of white cards - product
...4 series of colored cards - multiplication
...3 series of gray cards - multiplier
...box of signs for the operations

Presentation:
Invite the children to lay the cards out in columns, naming them as they go. The child should recognize the numerals up to 9 million. If necessary identify for him 10 million and 100 million.
One child must go to the bank. (For the first demonstration use a one-digit multiplier. In all other work a two digit multiplier will be used) Write the problem on a piece of paper. The child takes this to the bank and sets up the multiplication, using colored cards for the multiplicand, signs for the operation and a gray card for the multiplier.

4876 x 6 =

We then decompose the multiplicand, just as was done on the bead frame for multiplication, and begins multiplying:

      6 x 6 =
    7 0      
  8 0 0      
4 0 0 0      


6 x 6 = 36. He asks the banker for 36. The two cards are placed separately at the near edge of the table. Move the cards for the operation and the multiplier to align them with the next digit - 70. 7 x 6 = 42, 42 tens. He asks the banker for 420. There are placed with the previous product cards so that columns are being formed.

  20  
400 30 6

The operation continues in this way. When all of the digits have been multiplied the children-assembles the multiplicand. To find the product, he begins with the lowest hierarchy, combining and making changes. The product cards are assembled and placed by the equal sign. The children record the equation.

Let's try a different one. Write the problem on a piece of paper:
6835 x 48 =. The child sets up the problem with the colored cards and the gray cards. We don't have a gray card for forty, so we place a 4 next to a 0 to make 40, then place the 8 on top of the 0 to make 48.
As before, the child decomposes the multiplicand. We only want to multiply by one digit of the multiplier at a time. Remove the 40 and set it aside, yet together for later retrieval
Begin multiplying as before: 5 x 8 = 40. The banker gives 40, etc. After the multiplicand has been multiplied by the units of the multiplier, we can begin multiplying by the tens. ( 8 and 40 switch places, 8 is turned over) However, just as with the bead frame we have the rule: the multiplier must be units. Transfer the zero from the multiplier to the multiplicand.
Continue multiplying. The child will realize that 8 x 4 = 32 and 3 zeros after it- makes 32,000. In the end make changes in the product, carrying mentally, The product is assembled, the equation is recorded.

Notes: Since there is only one set of cards for the product much changing will be involved, calling for quick mental addition and subtraction at the level of memorization, on the part of the banker.
In multiplication with a two-digit multiplier, the child never stops in this game to record partial products, because the aim here is to develop agility in changing from one hierarchy to another. With only one set of cards it would be difficult to set aside partial products.
It is interesting to perform the same problem using the bank game, the checkerboard and the large bead frame to recognize the similarities and differences in the work. Be aware of the limits of the bead frame.

Age: 7 years

Aim: development of mental flexibility to prepare for mental calculating

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xWord Problems

As for addition and subtraction, word problems are prepared on cards which deal with each of the seven special cases. These are mixed in with those given for addition and subtraction.

Example:
Steve has 48 stamps in his collection at school. Each day he brought a certain number of stamps, for a certain number of days. How many stamps did he bring each day? and how many days did he bring stamps?

The child writes his answer:

48 = 6 x 8
48 = 8 x 6

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