The Montessori Racks and Tubes Division

The Montessori Racks and Tubes Division material teaches long division with dividends into the millions. It is sometimes called Test Tube Division, Division Hierarchical Material, or simply Long Division Material. It is one of my Top Ten Math Materials for elementary.

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The Tubes

The first thing you might notice about the Montessori Racks and Tubes Division material are the tubes containing colorful beads of green, blue, and red. They are used to represent the dividend (the amount to be divided or shared) in a division equation.

Green beads represent units/ones
Blue beads represent tens
Red beads represent hundreds

The tubes sit in racks that are colored according to the numerical hierarchy (simple family, thousands family, millions family).

The Racks

Three white racks represent the simple family. Each rack contains tubes filled with either green beads (units/ones), blue beads (tens), or red beads (hundreds) representing the place value category in the simple family.

Three grey racks represent the thousands family. Each rack contains tubes filled with either green beads (units/ones), blue beads (tens), or red beads (hundreds) representing the place value category in the thousands family.

One black rack represents the millions family. The rack contains tubes with green beads (units/ones) representing the units in the millions family.

Creating Dividends

The child creates dividends (amounts to be shared) by filling little cups with beads to be shared among a divisor represented by wooden pegs called “skittles.”

Creating Divisors

The Montessori Racks and Tubes Division material also includes bead boards for dividing/sharing beads. There is a unit board (which is green), a ten board (which is blue), and a hundred board (which is red) allowing for a 3-digit divisor. Each board allows for up to 81 beads distributed to as many as 9 groups (represented by wooden “skittles”).

Preceding Lessons

The Racks and Tubes Division material is preceded by Stamp Game Division and the Division Bead Board.

The Lessons

Introduction to Racks and Tubes Division
Building Dividends and Divisors
One-digit Divisor
One-digit Divisor with Recording
Two-digit Divisor with Recording
Three-digit Divisor with Recording
Abstraction

Introduction to the Montessori Racks and Tubes Division Material

When introducing a new material, we strive to move from the known to the unknown. The child should have some experience with the unit Division Bead Board. She should also have experience with the Bead Frames that show the hierarchy of numbers using white for the simple family, grey for the thousand’s family, and black for the million’s family. She should have experience with The Stamp Game Division lesson.

Compare and contrast these materials with the Montessori Racks and Tubes Division material. What does she notice about the beads, the racks, the bowls, and the boards?

Building Dividends and Divisors

Dividends

Allow the child to explore the material by building dividends (amounts to be distributed/shared) with the beads and bowls. Place the bowls out in front of her and ask her to build a number using the beads from the tubes.

This activity will reinforce place value through the bead colors and inside color of the bowls and number hierarchy through the corresponding color of racks and outside color on the bowls.

For example, make the number 3,458,267 by filling the appropriate cups with beads from the appropriate racks and tubes. Allow her to build and read other numbers.

Divisors

Explore the Montessori Racks and Tubes Division material by building divisors using the green unit division board and wooden peg “skittles.” For example, use the green unit division board and represent a divisor of 2 by placing two green, wooden “skittles” at the top. Note that the unit board can only represent a divisor of nine or fewer. Allow her to create other divisors with units.

Introduce divisors for the blue ten board and skittles and the red hundred board and skittles later when introducing two and three-digit divisor lessons.

One-Digit Divisor Division

Setting Up

Begin with one-digit division so the child can become confident in the process of division. Choose an introductory equation that does not require exchanges or result in a remainder. For example, 4,884 ÷ 4.

Using the green unit division board, place 4 green skittles at the top.
Next, fill the appropriate cups with beads to represent 4,884.

The Distributive Division Process

*NOTE: Now that the equation is set up it is important to note that we begin dividing with the largest place value. This is different from addition, subtraction, and multiplication where we always begin with the lowest place value.

Division is done by sharing an equal amount with a specified group. We want to share 4,884 among 4 groups. In our equation, 4,884 is the dividend (the amount to be shared) and 4 is the divisor (the number of groups to be shared with.

The child begins with the thousands place by sharing the 4 beads from the thousands cup, placing beads on the board equally under each of the 4 wooden skittles.
When the cup is empty, the child counts the number of beads under one of the skittles. The answer (quotient) is what each skittle received. In our example, the share each skittle received is one bead. The quotient in the thousands place is one.
The child clears the board of beads and begins sharing beads from the hundreds place value cup.
The child continues the process until all beads have been shared in each place value category and a final quotient is reached.

Subsequent lessons will be equations with exchanges, remainders, and zeros in the dividend and zeros in the quotient with the Montessori Racks and Tubes Division material.

No recording is done at first so the child can focus on the long division process.

Exchanges

Any beads not shared evenly in a place value category are regrouped by exchanging each left over bead for ten beads of a lesser category.

In the pictures above, one green thousand bead could not be shared evenly among a divisor of seven.

The remaining one thousand bead is regrouped into the hundreds place by exanching it for ten hundred beads.

One-Digit Divisor Division with Recording

Montessori Racks and Tubes One Digit Recording

Once a child is familiar with the process of division using the Racks and Tubes material she can begin to record the long division steps. The steps are: Division, Multiplication, Subtraction, and Bring down. It is helpful to teach the steps using a Mnemonic device. For example:

Dirty, Monkeys, Smell, Bad,

or

Dad, Mom, Sister, Brother.

The child does the equation using the Montessori Racks and Tubes Division material but records each mathematical step on paper resulting in a long division notation.

For example, in our earlier equation, the child will share 4 thousand beads among 4 groups and record the quotient of 1 as the division step.
She then multiplies 1×4 (4), which can also be seen visually as an array of beads laid out on the board. She records the product 4 underneath the 4 in the thousands place of the dividend as the multiplication step.
She then subtracts 4 – 4, which equals zero in the subtraction step. This is visualized in that the share was equal and there were no leftover beads to exchange.
Finally, she brings down the 8 to the right of the zero as the Bring down step.
The new number to work with is 8, which is in the red hundreds bowl. She then begins the steps again, dividing, multiplying, subtracting, and bringing down.

Recording the long division process is a step toward abstraction (no materials, pencil and paper only).

Two-Digit Divisor Division with Recording

Montessori Racks and Tubes Two Digit Divisor

When a child is comfortable with one-digit division and long division, she is ready for a two-digit divisor. A two-digit divisor requires the use of the blue tens board and blue skittles in addition to the green unit board and green skittles. The process is entirely the same. The difference is in the blue skittle, which represents a group of ten.

If a child shares a green bead from the thousands bowl with a blue skittle this means that each of the ten in the group gets 100 because there are ten hundreds in a thousand.

She must then share a red hundred bead with each green skittle so that each skittle receives an equal amount of 100 each.

The child records the long division work on paper as she works the problem.

Three-Digit Divisor Division with Recording

Montessori Racks and Tubes Three Digit Divisor

Three-digit divisor division requires the use of the red hundred board and red skittles in addition to the blue and green boards and skittles.

Each red skittle represents a group of 100.

If a child shares a green thousand bead with a red skittle, this means that each of the hundred in the group gets 10, because there are 10 hundreds in a thousand.

To be fair and give an equal share, she then must share a red hundred bead with each blue skittle so that each of the 10 in the group will receive ten, because there are 10 tens in 100.

For an equal share, she must then share a blue ten bead with each green skittle so each one gets ten.