Multiplication and Drawing

MULTIPLICATION AND DRAWING

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

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

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.

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