Thursday, November 5, 2015

Montessori Elementary Mathematics - Multiplication Checkerboard

Montessori Elementary Mathematics: Multiplication Checkerboard

Materials: white number tiles 0-9, gray numbers tiles 0-9, checkerboard[1], checkerboard bead bar box (only has bead bars 1-9)

            This is the checkerboard. We are going to learn how to do multiplication on the checkerboard.
·   Numbers: We have these numbers across the bottom (read and point to each number).
·   Multiplicand and Multiplier: This ridge is where we place our multiplicand and multiplier (indicate the ridge at the bottom and to the right, respectively). We also have some numbers along the right side (read and point to each one). We will put the multiplier here.
·   Squares – color and value: Then we have all of these colored squares. Here are the units (point to the lower left corner – green). Tens are here and here (point to each of the blue squares touching the unit square – 2). All of these are hundreds (indicate the diagonal 3 reds). Here we have thousands (indicate these 4). Here are the ten-thousands (indicate these 4). Continue with each category.
·   Squares – value of beads: What is interesting about the checkerboard is that the value of the bead bar is determined by which square upon which it is placed. I have a 3-bar; if I put it here (place on the units square), it is three units. If I put it here (place on a hundred square), it is three hundred; if I put it here (another hundred), it is still 300. What if I put it here (place on a 100,000 square). We don’t have to stick with one bead bar. If I have a 3-bar here (on tens) and a 5-bar here (on hundreds), we have 530. What if I put it here? (slide each over one to show 5,300). What if I put it here (slide up and right one place – diagonally)? It is still 5,300. Repeat until the concept is clear and the children can readily read the numbers on the board – invite them to place beads and you read the number; and to place beads to read the numbers themselves.

Exercise 1: representing each multiplication with the bead bars
Prerequisites: knowledge of the process of multiplication; Introduction to the checker-board; Ability to read hierarchical numbers; (technically the children can do the 1st exercise without knowing the multiplication facts); Can precede work on the large bead frame (this exercise only)

Notes: If the children choose very large digits, they will have a ton of bead bars in each square that will overflow into other squares. In that situation, after they have done the multiplications, exchange within each row before sliding diagonally; then finish the exchanging to reach the final answer. 
                This work can help the children learn their multiplication because they are represented by the quantities they are putting down each time. If they do know their facts and they’ve caught on to the procedure of exercise 1, move them right on to exercise 2 on another day.

Exercise 2: Using the Multiplication Facts

Purpose: Further experience in long multiplication. Indirect preparation for category multiplication.

[1] Boards: current manufacturing only go to units of millions; felt version and the presentation in AMI training goes to hundred-millions on the horizontal

why does the checkerboard have more than 3 spaces per grouping? e.g. multiples of thousands beyond hundred thousand?

Each square on the checkerboard represents the multiplication of the bottom and right-hand values. So a board that is 9 squares along the bottom and 4 squares along the side - that last upper left hand square will represent 100,000,000 taken 1,000 times. 
The checkerboard does a few things for the children - it helps them work with VERY large numbers (now we can go into billions! - and it helps them see the "why" behind shifting the numbers one over when we multiply on paper (which the large bead frame starts to do) as well as the combinations of the quantities (so goes beyond the large bead frame in these concepts). I start the children with small numbers and they challenge themselves when they are ready to do the larger numbers.

Not to be confused with the Decimal Checkerboard (some tidbits found at Montessori Trails - a Montessori Nugget will be posted soon).

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