xMath :: 69 :: Multiplication  
xTable of Contents:  
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xStamp Game Multiplication  
Presentation: [top] 

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. Materials: [top] MULTIPLICATION BOARDS (Bead
Boards) 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] MULTIPLICATION BOARDS (Bead
Boards) Materials: Exercise: 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. [top] MULTIPLICATION BOARDS (Bead
Boards) Materials: Exercise: Control of error: When the child has finished his work, he controls with Chart I. This control reinforces memorization of the combinations. [top] SKIP COUNTING (Linear Counting) Materials: b. Initial Presentation: c. Short Chain: Activity: d. Long Chain: Materials: same as before Presentation: Direct Aim: comparison of quantities sensorially Indirect Aim: preparation for the powers of numbers (exponential increase) [top] BEAD BAR MULTIPLICATION Materials: Presentation: Direct Aim: memorization of the multiplication
tables Indirect Aims: to understand that a number when
multiplied by ten results in the same number of tens and zero
units [top] BEAD BAR MULTIPLICATION Materials: Presentation: Direct Aims: memorization of multiplication [top] MULTIPLICATION CHARTS AND
COMBINATION CARDS 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) [top] MULTIPLICATION CHARTS AND
COMBINATION CARDS Materials: Exercise: Control of Error: Chart I or II [top] MULTIPLICATION CHARTS AND
COMBINATION CARDS Materials: Exercise: Control of Error: Chart I or II Note: At his point, to verify
memorization, the child may be given command cards.
[top] MULTIPLICATION CHARTS AND COMBINATION CARDS d. The Bingo Game of Multiplication (using Chart V) Materials: A. Exercise Control: Chart I for combinations, Chart II for placement. B. Exercise Control: Charts I and III. C. Exercise Note: What shape is made when the stacks of tiles are lined up in order? No special figure is made this time. Group Games: Age: from 67 (this work lasts for one year) [top] MULTIPLICATION BY 10, 100, 1000 (Note: This activity is a prerequisite for the small bead frame) Materials: Presentation: Aim: to be sure that the child has understood the concept of change By ten By one hundred By one thousand 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 [top] 

xCheckerboard  Geometrical Analysis of Multiplication  
Materials: The Checkerboard: Presentation: Games: 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 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 Materials: Presentation:
Aim: to understand the process of multiplication using the board.
[top] MULTIPLICATION AND DRAWING Materials: Presentation:
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. Age: 78 years [For all checkerboard work] Aim: to reinforce the concept of hierarchies [top] 

xBead Frame Multiplication  
Materials: Presentation: 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 [top] SMALL BEAD FRAME Materials: 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.
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. Experiment: 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: 67 years Aim: realization of the importance of the position of each digit [top] LARGE BEAD FRAME  MULTIPLIERS
OF 2 OR MORE DIGITS Materials: Presentation:
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 threedigit multiplier there will be 5 decompositions of which only the 1st, 3rd, 5th will be used for the multiplication on the frame. [top] LARGE BEAD FRAME  MULTIPLIERS
OF 2 OR MORE DIGITS Presentation: The child by this time should have
reached a level of abstraction with column addition.
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: 78 years, or when the child is adding abstractly [top] HORIZONTAL GOLDEN BEAD FRAME Materials: (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: [top] HORIZONTAL GOLDEN BEAD FRAME Materials: 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: 78 years [top] HORIZONTAL GOLDEN BEAD FRAME Materials: 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. 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. [top] 

xMore Memorization Exercises  
Materials: same as for addition snake Exercise: 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 [top] VARIOUS WAYS OF CONSTRUCTING A PRODUCT Materials: Presentation: 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: [top] SMALL MULTIPLICATION Materials: Presentation: 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: [top] INVERSE PRODUCTS Materials: same as above Presentation: Direct Aims: memorization of multiplication understanding of the commutative property of multiplication [top] CONTRUCTION OF A SQUARE Materials: Presentation: Direct Aim: Indirect Aim: preparation for the powers of numbers [top] MULTIPLICATION OF A SUM Materials: Binomial Presentation: On a different day:
Trinomial Presentation: On yet another 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. Direct Aim: memorization of multiplication Indirect Aim: preparation for the square of the polynomial [top] ANALYSIS OF THE SQUARES  BINOMIAL Materials:
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:
[top] ANALYSIS OF THE SQUARES  TRINOMIAL Materials:
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 100square on top. These are the two ways to prove that this equals one hundred. [top] PASSAGE FROM ONE SQUARE TO A SUCCEEDING SQUARE Materials: Presentation:
Aim: indirect preparation for the square of a binomial [top] PASSAGE FROM ONE SQUARE TO A NONSUCCESSIVE SQUARE Materials: Presentation:
[top] SKIP COUNT CHAINS  FURTHER
EXPLORATION Materials: [top] SKIP COUNT CHAINS  FURTHER
EXPLORATION Materials: short chains Exercise: 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 [top] SKIP COUNT CHAINS  FURTHER
EXPLORATION Note: This material is presented parallel to memorization of multiplication, in three different presentations. Materials: [top] SKIP COUNT CHAINS  FURTHER
EXPLORATION 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. [top] SKIP COUNT CHAINS  FURTHER
EXPLORATION 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 row10
x 10 [top] SKIP COUNT CHAINS  FURTHER
EXPLORATION 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 2bars. Outline the formation
with a finger to help the child to observe the square that was
formed. Control of Error: visible arrangement; number of bead bars in the box Direct Aim: Indirect Aim: preparation for the Decanomial [top] SKIP COUNT CHAINS  FURTHER
EXPLORATION Materials: Part One Part TwoBuilding the Tower Variation on the 1st Method
2nd Method Part ThreeDecomposition of the Tower [top] SKIP COUNT CHAINS  FURTHER
EXPLORATION Materials: Presentation: Age: between 6 1/2 and 81/2 years Aim: [top] SPECIAL CASES Materials: Presentation: 1 Calculate the Multiplier 3 Inverse of Case ZeroCalculate
the Product 4 Inverse of The FirstCase,
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 Activity: [top] 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: Presentation:
The operation continues in this way. When all of the digits have been multiplied the childrenassembles 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. Age: 7 years Aim: development of mental flexibility to prepare for mental calculating [top] 

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: The child writes his answer:
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