Mastering Long Division with Decimals: A Step-by-Step Guide Using Montessori Racks and Tubes (The Test Tube Game)

Mastering Long Division with Decimals: A Step-by-Step Guide Using Montessori Racks and Tubes (The Test Tube Game)

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✏️ Introduction:

Teaching long division—especially with decimals—can be one of the most intimidating concepts for young learners. But with the Montessori Racks and Tubes, also known as the Test Tube Game, division becomes not just manageable, but concrete, visual, and hands-on. This powerful material turns abstract numbers into something students can touch, move, and understand deeply.

In this guide, you'll discover how to use the Montessori Racks and Tubes to break down even complex decimal division problems into clear, logical steps. Whether you're a classroom teacher introducing this concept for the first time, or a homeschooling parent looking to bring the Montessori approach into your math instruction, this article offers everything you need: materials, setup, a detailed example (12.34 ÷ 2.7), and a scaffolded explanation using authentic Montessori strategies.

Let’s turn long division into an “aha!” moment your students will never forget.

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Would you like a printable version of this guide for parent handouts or Montessori shelf work cards?

Let's walk through dividing 12.34 by 2.7 step-by-step using the Montessori Test Tube Division method (Racks and Tubes), which is brilliant for teaching long division with decimal numbers in a tactile way. We’ll make it 100% hands-on and Montessori-aligned.

PRINTABLE STAMP GAME 

PRINTABLE STAMP GAME 

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🌟 Problem:

12.34 ÷ 2.7

🎯 Goal:

Use Racks and Tubes (Test Tubes) to model and solve the division problem.

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🔢 Step 1: Normalize the Divisor

To work with whole numbers using the Racks and Tubes method, you need the divisor to be a whole number.

12.34 ÷ 2.7 becomes:

➡️ 123.4 ÷ 27
(We moved the decimal one place to the right in both numbers.)

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🧪 Step 2: Prepare Your Materials

Montessori Test Tube Setup:

• 1 test tube for each place value: hundreds, tens, units, tenths.

• 9 skittles (color-coded people markers) for divisors — we'll place 27 skittles.

• Beads (green = units, blue = tens, red = hundreds, golden = tenths or decimals).

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🧠 Step 3: Place the Dividend Beads

We model 123.4 using beads:

• 1 red bead in the hundreds tube

• 2 blue beads in the tens tube

• 3 green beads in the units tube

• 4 golden beads in the tenths tube

This setup visually shows 123.4.

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👣 Step 4: Begin Grouping (Division Process)

We're dividing the beads evenly among 27 skittles (the divisor).

A. Can we divide 1 red bead (100) among 27?

No — not enough to give even 1 to each skittle.

➡️ So we combine it with the tens (bring down the 1 red and convert to 10 tens → now we have 12 tens total).

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B. Divide 12 tens among 27 skittles?

Still not enough — can't give even 1 ten to each skittle.

➡️ Convert 12 tens → 120 units, combine with 3 units already there = 123 units.

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C. Divide 123 units by 27:

Now we divide:

• 27 × 4 = 108

• 27 × 5 = 135 (too much)
  So we give 4 units to each skittle

➡️ Quotient so far: 4

Remainder: 123 - 108 = 15 units

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🔄 Step 5: Bring Down the Tenths

We still have 4 tenths (from the original 123.4).
Let’s convert the 15 leftover units + 4 tenths into tenths:

• 15 units = 150 tenths

• Add 4 tenths = 154 tenths

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D. Divide 154 tenths by 27:

• 27 × 5 = 135

• 27 × 6 = 162 (too much)

So we give 5 tenths to each skittle.

Quotient now: 4.5

Remainder: 154 - 135 = 19 tenths

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🧮 Step 6: Add a Zero to Bring Down More Decimal Places

Let’s continue one more digit:

Convert 19 tenths = 190 hundredths

• 27 × 7 = 189

• So we can give 7 hundredths to each skittle

Final Quotient: 4.57

✅ So, 12.34 ÷ 2.7 = 4.57

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🔍 Summary with Montessori Language:

• We shifted the decimal to make the divisor whole.

• We built the dividend using place value beads.

• We grouped beads evenly using skittles (the divisor).

• We traded and regrouped whenever we couldn't divide evenly at that level.

• We continued into decimals by converting place values down.

What We’re Solving For:

Let’s use this example problem:

Example:

$\frac{15.6}{1.2}$

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🧠 Montessori Manipulative of Choice:

✅ Test Tube Division Set (best for long division, especially with multi-digit divisors and decimals)

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👣 Step-by-Step Montessori Procedure:

🔢 Step 1: Normalize the Divisor

We can't divide by a decimal in Montessori concrete materials. So we slide the decimal point in the divisor to make it a whole number.

• $\frac{15.6}{1.2}$ becomes $\frac{156}{12}$

• Think of this as multiplying both numbers by 10 (same rule as in the algorithm: if you move the decimal 1 place in the divisor, do the same in the dividend).

💡 Use a place value chart or decimal board to physically move place values over one column.

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🔍 Step 2: Set Up the Test Tube Division Board

• Place 12 beads (green unit beads) in each cup of the divisor row.

• You will divide 156 (the dividend) across the test tube set.

• Use color-coded beads:

  • Red = thousands,

  • Blue = hundreds,

  • Green = tens,

  • Yellow = units

• Set up 1 hundred bead + 5 ten beads + 6 unit beads in the dividend tray.

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👷 Step 3: Distribute Beads by Place Value

• Start dividing from left to right, one place value at a time:

  • Can you divide 100 (1 red bead = 100) by 12? No. So regroup.

  • Combine the hundred with tens: 100 + 50 = 150

    • 150 ÷ 12 = 12 R6

• Place 1 bead in the tens place of the quotient rack.

• Regroup remaining 6 tens with 6 unit beads = 66

• 66 ÷ 12 = 5 R6

• Place 5 beads in the units place of the quotient rack.

• The remainder 6 becomes a decimal extension:

  • Add a zero to the dividend → 60 tenths

  • 60 ÷ 12 = 5

✅ Quotient: 13

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🧠 Visual of Sliding the Decimal:

Use a place value chart with sliders (Decimal Board) to model this:

Place	Original Dividend (15.6)	After shifting (156)
Tens	1	1
Ones	5	5
Tenths	6	—
Units	—	6

This helps students understand that shifting decimals is multiplying by a power of 10, not “magic.”

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🎮 Optional Manipulative Alternatives:

🔹 Montessori Stamp Game (Decimal Variation)

Use decimal stamp tiles (tenths, hundredths) and place value layout to model regrouping and distribution of values during division. Works best for simpler 2-digit ÷ 1-digit decimal problems.

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✅ Division Rule Students Should Learn:

1. Shift the decimal in the divisor to make it a whole number.

2. Shift the decimal in the dividend the same number of places.

3. Use hands-on materials to model the new division problem.

4. Track the decimal placement in the final quotient.

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✏️ Example Problems for Practice:

Problem	Normalized Form	Quotient
$\frac{3.45}{0.15}$	$\frac{345}{15}$	23
$\frac{7.2}{0.08}$	$\frac{720}{8}$	90
$\frac{9.36}{0.72}$	$\frac{936}{72}$	13
$\frac{25.5}{2.5}$	$\frac{255}{25}$	10.2

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🖼️ Graphic: Step-by-Step Visual with Montessori Test Tube Division

“Simplifying Decimal Division with the Montessori Stamp Game: A Hands-On Guide for Teachers and Homeschoolers”

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✏️ Introduction:

Division with decimals can be a confusing leap for many students—until they’re given the tools to see and manipulate numbers physically. The Montessori Stamp Game offers an ideal bridge between abstract algorithms and concrete understanding. With this material, students gain clarity by breaking down division step by step and learning how to shift decimals to simplify the process without altering the value of the problem.

This guide will walk you through how to teach decimal division using the Stamp Game with an easy-to-follow example, focusing on the importance of working with whole numbers in the divisor. By using a hands-on approach, learners can experience division not just as a rule to memorize, but as a logical, meaningful process they understand deeply.

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🔍 Why Eliminate Decimals from the Divisor?

• It simplifies the process: Working with whole numbers allows learners to perform long division in familiar, predictable steps.

• It avoids confusion with decimal placement: Shifting the decimal point in both the dividend and divisor ensures accurate placement in the quotient.

• It preserves the value of the expression: Multiplying both numbers by the same power of 10 doesn’t change the overall value of the problem.

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🧠 How to Eliminate Decimals from the Divisor (Before Using the Stamp Game):

1. Identify how many decimal places are in the divisor.
   Example: In 2.7, there’s 1 decimal place.

2. Multiply both the divisor and the dividend by a power of 10 to make the divisor a whole number.
   Example:
   12.34 × 10 = 123.4
   2.7 × 10 = 27

3. Now use the Stamp Game to divide 123.4 ÷ 27, just like a standard long division problem—with concrete materials and place-value alignment.

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📸 Visual Example: