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
- Multiplication Boards (Bead
Boards)
- 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
- Introduction to the Checkerboard
- 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
- Small Bead Frame
- 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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xMemorization Exercises | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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.
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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. [top]
c. Multiplication Booklets
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d. Combination CardsMaterials:
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a. List of MaterialsMaterials: b. c.
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.
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Adjunct: Multiplication by Ten
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a. Passage from Chart I to Chart IIThe 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) [top]
b. Passage from Chart II to Chart III (the Whole Chart)
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c. Passage from Chart III to Chart IV (the Half Chart)
Note: At his point, to verify
memorization, the child may be given command cards.
[top] MULTIPLICATION CHARTS AND COMBINATION CARDSd. The Bingo Game of Multiplication (using Chart V)
Control: Chart I for combinations, Chart II for placement.
Control: Charts I and III.
Note: What shape is made when the stacks of tiles are lined up in order? No special figure is made this time.
Age: from 6-7 (this work lasts for one year) [top]
(Note: This activity is a prerequisite for the small bead frame)
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xCheckerboard - Geometrical Analysis of Multiplication | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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.
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. [top] MULTIPLICATION WITH THE CHECKERBOARD
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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.
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xBead Frame Multiplication | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Materials:
Note: We are limited by the frame to having a multiplicand of only one digit when the multiplier is 1000 and vice versa.
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b. Multiplication with a One-Digit Multiplier
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
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.
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."
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a. The Whole ProductMaterials:
On the right side decompose the multiplicand as before. First decompose the number for multiplication by 4 units.
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 : 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. [top]
b. Partial Products
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.
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a. The Whole Product
(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.)
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b. Partial Products
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.
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c. Carrying Mentally
The child sets up the multiplication problem on the frame.
3 x 6 = 18 move down 8 units,
remember one ten in your head. Record the partial product and clear the frame before beginning multiplication by the tens.
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
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xMore Memorization Exercises | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Materials: same as for addition snake
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On a different day:
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.
[top] ANALYSIS OF THE SQUARES - BINOMIAL
60 + 40= 100 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:
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a. List of Materials
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b. Construction of Geometric Figures
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c. Decanomial (a polynomial having ten terms) & The Construction
of Chart IList of MaterialsNote: This material is presented parallel to memorization of multiplication, in three different presentations.
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c. Decanomial (a polynomial having ten terms) & The Construction
of Chart IVertical PresentationBeginning 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. [top]
c. Decanomial (a polynomial having ten terms) & The Construction
of Chart IHorizontal PresentationThis 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 [top]
c. Decanomial (a polynomial having ten terms) & The Construction
of Chart IAngular PresentationAs 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.
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c. Decanomial (a polynomial having ten terms) & The Construction
of Chart ICommuted Decanomial
Variation on the 1st Method
2nd Method
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c. Decanomial (a polynomial having ten terms) & The Construction
of Chart INumeric Decanomial
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1- Calculate the Multiplier 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 Note: In cases 1, 4 and 7, the child performs multiplication, but in all others division is indirectly involved
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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.
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: 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.
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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.
The child writes his answer:
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