Saturday, May 3, 2025

MONTESSORI STAMP GAME DECIMAL FRACTIONS: Task Cards, "Command Cards," and Control Cards

The Montessori Math Process: Task Cards, "Command Cards," and Control Cards

MONTESSORI STAMP GAME DECIMAL FRACTIONS

In Montessori mathematics education, the journey from teacher demonstration to independent learning follows a carefully structured process that emphasizes hands-on learning, independence, and self-correction. Let me unpack this unique approach.

The Learning Sequence

  1. Teacher Presentation: The process begins with the teacher giving a demonstration to a small group of students, showing how to use specific math manipulatives to understand mathematical concepts.

  2. Independent Work: After the demonstration, students collect the necessary manipulatives and work independently or with partners rather than continuing with whole-class instruction.

Task Cards (Command Cards)

Task cards serve as guides for student practice after the teacher's demonstration. They:

  • Provide step-by-step instructions for mathematical operations
  • Allow students to work at their own pace
  • Include various problems for students to solve using manipulatives
  • Progress in complexity as students master concepts
  • Enable independent learning without constant teacher direction

Task cards might include instructions like "Build all combinations of 7 using the colored bead bars" or "Use the stamp game materials to show 3254 - 1362."

Control Cards

Control cards are a distinctive and powerful element of the Montessori approach:

  • They contain the answers to problems on the task cards
  • More importantly, they explain why or how the answer is derived
  • Facilitate self-correction without teacher intervention
  • May include visual representations of the problem-solving process
  • Support metacognition by showing the reasoning behind solutions

For example, a control card might show not just that 43 × 5 = 215, but illustrate the multiplication process using visual representations of the manipulatives.

Benefits of this System

This approach offers several educational advantages:

  • Self-paced learning: Students progress according to their readiness
  • Immediate feedback: No waiting for teacher assessment
  • Development of independence: Students learn to check their own work
  • Deeper understanding: Focus on process, not just correct answers
  • Intrinsic motivation: Success comes from personal discovery rather than external validation

The control card system transforms error from something to be avoided into a learning opportunity. When students discover their own mistakes through comparison with control cards, they can revisit their work, identify misconceptions, and correct their understanding without feeling judged.

This full cycle of demonstration, independent practice with task cards, and self-verification with control cards represents a comprehensive approach to mathematics that builds both procedural skills and conceptual understanding. 

Montessori Stamp Game Command Cards Bundle

4th, 5th, and 6th Grade Mathematics

WHOLE NUMBER OPERATIONS (4th Grade)

ADDITION COMMAND CARD 1

Problem: 4,325 + 2,763 Materials: Stamp game material, place value mat Steps:

  1. Place 4 thousand squares, 3 hundred squares, 2 ten bars, and 5 unit stamps in the thousands, hundreds, tens, and units columns.
  2. Place 2 thousand squares, 7 hundred squares, 6 ten bars, and 3 unit stamps in the same columns.
  3. Begin with the units column: Count 5 units + 3 units = 8 units.
  4. Move to the tens column: Count 2 tens + 6 tens = 8 tens.
  5. Move to the hundreds column: Count 3 hundreds + 7 hundreds = 10 hundreds. Exchange for 1 thousand square.
  6. Move to the thousands column: Count 4 thousands + 2 thousands + 1 thousand (from exchange) = 7 thousands.
  7. Read the sum: 7,088

Dynamic/Static: Dynamic – involves regrouping in the hundreds column

ADDITION COMMAND CARD 2

Problem: 5,856 + 4,997 Materials: Stamp game material, place value mat Steps:

  1. Place 5 thousand squares, 8 hundred squares, 5 ten bars, and 6 unit stamps.
  2. Place 4 thousand squares, 9 hundred squares, 9 ten bars, and 7 unit stamps.
  3. Start with units: 6 + 7 = 13 units. Exchange 10 units for 1 ten. Place 3 units.
  4. Move to tens: 5 + 9 + 1 (from exchange) = 15 tens. Exchange 10 tens for 1 hundred. Place 5 tens.
  5. Move to hundreds: 8 + 9 + 1 (from exchange) = 18 hundreds. Exchange 10 hundreds for 1 thousand. Place 8 hundreds.
  6. Move to thousands: 5 + 4 + 1 (from exchange) = 10 thousands.
  7. Read the sum: 10,853

Dynamic/Static: Dynamic – involves multiple regrouping across columns

SUBTRACTION COMMAND CARD 1

Problem: 8,534 - 5,321 Materials: Stamp game material, place value mat Steps:

  1. Place 8 thousand squares, 5 hundred squares, 3 ten bars, and 4 unit stamps.
  2. Remove 5 thousand squares, 3 hundred squares, 2 ten bars, and 1 unit stamp.
  3. Read the difference: 3,213

Dynamic/Static: Static – no trading required

SUBTRACTION COMMAND CARD 2

Problem: 7,302 - 4,158 Materials: Stamp game material, place value mat Steps:

  1. Place 7 thousand squares, 3 hundred squares, 0 ten bars, and 2 unit stamps.
  2. Need to subtract 8 units, but only have 2. Exchange 1 ten for 10 units. Now have 12 units.
  3. Subtract 8 units from 12 units = 4 units.
  4. Need to subtract 5 tens, but now have 0 - 1 = -1 tens. Exchange 1 hundred for 10 tens. Now have 9 tens.
  5. Subtract 5 tens from 9 tens = 4 tens.
  6. Need to subtract 1 hundred, but now have 3 - 1 = 2 hundreds.
  7. Subtract 1 hundred from 2 hundreds = 1 hundred.
  8. Subtract 4 thousands from 7 thousands = 3 thousands.
  9. Read the difference: 3,144

Dynamic/Static: Dynamic – involves trading across columns

MULTIPLICATION COMMAND CARD 1

Problem: 246 × 5 Materials: Stamp game material, place value mat Steps:

  1. Place 2 hundred squares, 4 ten bars, and 6 unit stamps.
  2. Multiply each column by 5.
  3. Start with units: 6 × 5 = 30 units. Exchange for 3 tens.
  4. Move to tens: 4 × 5 = 20 tens + 3 tens (from exchange) = 23 tens. Exchange for 2 hundreds and 3 tens.
  5. Move to hundreds: 2 × 5 = 10 hundreds + 2 hundreds (from exchange) = 12 hundreds. Exchange for 1 thousand and 2 hundreds.
  6. Read the product: 1,230

Dynamic/Static: Dynamic – involves multiple exchanges

MULTIPLICATION COMMAND CARD 2

Problem: 124 × 13 Materials: Stamp game material, place value mat, paper and pencil Steps:

  1. Multiply 124 × 3:
    • 4 × 3 = 12 units. Exchange for 1 ten and 2 units.
    • 2 × 3 = 6 tens + 1 ten (from exchange) = 7 tens.
    • 1 × 3 = 3 hundreds.
    • First partial product: 372
  2. Multiply 124 × 10:
    • 4 × 10 = 40 units. Exchange for 4 tens.
    • 2 × 10 = 20 tens + 4 tens (from exchange) = 24 tens. Exchange for 2 hundreds and 4 tens.
    • 1 × 10 = 10 hundreds + 2 hundreds (from exchange) = 12 hundreds. Exchange for 1 thousand and 2 hundreds.
    • Second partial product: 1,240
  3. Add both partial products: 372 + 1,240 = 1,612

Dynamic/Static: Dynamic – involves exchanges and addition of partial products

DIVISION COMMAND CARD 1

Problem: 693 ÷ 3 Materials: Stamp game material, place value mat, skittles or dividers Steps:

  1. Place 6 hundred squares, 9 ten bars, and 3 unit stamps.
  2. Place 3 skittles to represent dividing into 3 equal groups.
  3. Start with hundreds: Distribute 2 hundreds to each skittle.
  4. Move to tens: Distribute 3 tens to each skittle.
  5. Move to units: Distribute 1 unit to each skittle.
  6. Read the quotient by counting what each skittle has: 231

Dynamic/Static: Static – no trading required

DIVISION COMMAND CARD 2

Problem: 827 ÷ 4 Materials: Stamp game material, place value mat, skittles or dividers Steps:

  1. Place 8 hundred squares, 2 ten bars, and 7 unit stamps.
  2. Place 4 skittles to represent dividing into 4 equal groups.
  3. Start with hundreds: Each skittle gets 2 hundreds (8 ÷ 4 = 2).
  4. 0 hundreds remain.
  5. Move to tens: Cannot distribute 2 tens among 4 skittles. Exchange 2 tens for 20 units.
  6. Now have 7 + 20 = 27 units.
  7. Distribute 6 units to each skittle (24 units total), with 3 units remaining.
  8. Read the quotient: 206 remainder 3, or 206¾

Dynamic/Static: Dynamic – involves exchanging tens for units

DECIMAL AND FRACTION OPERATIONS (5th Grade)

DECIMAL ADDITION COMMAND CARD 1

Problem: 24.65 + 13.82 Materials: Stamp game material, decimal place value mat Steps:

  1. Place 2 tens, 4 units, 6 tenths, and 5 hundredths on the mat.
  2. Place 1 ten, 3 units, 8 tenths, and 2 hundredths on the mat.
  3. Start with hundredths: 5 + 2 = 7 hundredths.
  4. Move to tenths: 6 + 8 = 14 tenths. Exchange 10 tenths for 1 unit. Place 4 tenths.
  5. Move to units: 4 + 3 + 1 (from exchange) = 8 units.
  6. Move to tens: 2 + 1 = 3 tens.
  7. Read the sum: 38.47

Dynamic/Static: Dynamic – involves exchanging tenths for units

DECIMAL SUBTRACTION COMMAND CARD 1

Problem: 56.37 - 29.48 Materials: Stamp game material, decimal place value mat Steps:

  1. Place 5 tens, 6 units, 3 tenths, and 7 hundredths on the mat.
  2. Need to subtract 8 hundredths, but only have 7. Exchange 1 tenth for 10 hundredths. Now have 17 hundredths.
  3. Subtract 8 hundredths from 17 hundredths = 9 hundredths.
  4. Need to subtract 4 tenths, but now have 3 - 1 = 2 tenths. Exchange 1 unit for 10 tenths. Now have 12 tenths.
  5. Subtract 4 tenths from 12 tenths = 8 tenths.
  6. Need to subtract 9 units from 6 - 1 = 5 units. Exchange 1 ten for 10 units. Now have 15 units.
  7. Subtract 9 units from 15 units = 6 units.
  8. Subtract 2 tens from 5 - 1 = 4 tens = 2 tens.
  9. Read the difference: 26.89

Dynamic/Static: Dynamic – involves multiple exchanges across decimal places

DECIMAL MULTIPLICATION COMMAND CARD 1

Problem: 3.46 × 5 Materials: Stamp game material, decimal place value mat Steps:

  1. Place 3 units, 4 tenths, and 6 hundredths on the mat.
  2. Multiply each column by 5.
  3. Start with hundredths: 6 × 5 = 30 hundredths. Exchange for 3 tenths.
  4. Move to tenths: 4 × 5 = 20 tenths + 3 tenths (from exchange) = 23 tenths. Exchange for 2 units and 3 tenths.
  5. Move to units: 3 × 5 = 15 units + 2 units (from exchange) = 17 units. Exchange for 1 ten and 7 units.
  6. Read the product: 17.30

Dynamic/Static: Dynamic – involves exchanges across decimal places

DECIMAL DIVISION COMMAND CARD 1

Problem: 14.4 ÷ 4 Materials: Stamp game material, decimal place value mat, skittles or dividers Steps:

  1. Place 1 ten, 4 units, and 4 tenths on the mat.
  2. Place 4 skittles to represent dividing into 4 equal groups.
  3. Distribute 1 ten: Cannot distribute evenly. Exchange 1 ten for 10 units. Now have 14 units.
  4. Distribute 3 units to each skittle (12 units total), with 2 units remaining.
  5. Exchange 2 units for 20 tenths. Now have 24 tenths.
  6. Distribute 6 tenths to each skittle.
  7. Read the quotient: 3.6

Dynamic/Static: Dynamic – involves exchanging units for tenths

FRACTION ADDITION COMMAND CARD 1

Problem: 2⅜ + 1¾ Materials: Stamp game material, fraction cubes, place value mat Steps:

  1. Convert to improper fractions: 2⅜ = 19/8 and 1¾ = 7/4
  2. Find common denominator: LCD = 8
  3. Convert 7/4 to equivalent fraction: 7/4 = 14/8
  4. Place 19 unit stamps on the mat for 19/8.
  5. Place 14 unit stamps on the mat for 14/8.
  6. Add: 19/8 + 14/8 = 33/8
  7. Convert to mixed number: 33/8 = 4⅛
  8. Read the sum: 4⅛

Dynamic/Static: Dynamic – involves converting between mixed numbers and improper fractions

FRACTION SUBTRACTION COMMAND CARD 1

Problem: 5¾ - 2⅜ Materials: Stamp game material, fraction cubes, place value mat Steps:

  1. Convert to improper fractions: 5¾ = 23/4 and 2⅜ = 19/8
  2. Find common denominator: LCD = 8
  3. Convert 23/4 to equivalent fraction: 23/4 = 46/8
  4. Place 46 unit stamps on the mat for 46/8.
  5. Remove 19 unit stamps for 19/8.
  6. Count remaining stamps: 27 stamps = 27/8
  7. Convert to mixed number: 27/8 = 3⅜
  8. Read the difference: 3⅜

Dynamic/Static: Dynamic – involves converting between mixed numbers and improper fractions

ADVANCED OPERATIONS (6th Grade)

MULTI-DIGIT MULTIPLICATION COMMAND CARD 1

Problem: 326 × 48 Materials: Stamp game material, place value mat, paper and pencil Steps:

  1. Multiply 326 × 8:
    • 6 × 8 = 48 units. Exchange for 4 tens and 8 units.
    • 2 × 8 = 16 tens + 4 tens (from exchange) = 20 tens. Exchange for 2 hundreds and 0 tens.
    • 3 × 8 = 24 hundreds + 2 hundreds (from exchange) = 26 hundreds. Exchange for 2 thousands and 6 hundreds.
    • First partial product: 2,608
  2. Multiply 326 × 40:
    • 6 × 40 = 240 units. Exchange for 2 hundreds and 4 tens.
    • 2 × 40 = 80 tens. Exchange for 8 hundreds.
    • 3 × 40 = 120 hundreds. Exchange for 12 thousands.
    • Second partial product: 13,040
  3. Add both partial products: 2,608 + 13,040 = 15,648

Dynamic/Static: Dynamic – involves multiple exchanges and addition of partial products

LONG DIVISION COMMAND CARD 1

Problem: 3,864 ÷ 12 Materials: Stamp game material, place value mat, skittles or dividers Steps:

  1. Place 3 thousand squares, 8 hundred squares, 6 ten bars, and 4 unit stamps.
  2. Can we distribute thousands into 12 groups? No. Exchange 3 thousands for 30 hundreds.
  3. Now have 38 hundreds. Distribute 3 hundreds to each group (36 total), with 2 hundreds remaining.
  4. Exchange 2 hundreds for 20 tens. Now have 26 tens.
  5. Distribute 2 tens to each group (24 total), with 2 tens remaining.
  6. Exchange 2 tens for 20 units. Now have 24 units.
  7. Distribute 2 units to each group (24 total), with 0 units remaining.
  8. Read the quotient: 322

Dynamic/Static: Dynamic – involves multiple exchanges across columns

DECIMAL MULTIPLICATION COMMAND CARD 2

Problem: 4.26 × 3.5 Materials: Stamp game material, decimal place value mat Steps:

  1. Note: When multiplying decimals, we'll first multiply as if they are whole numbers, then adjust the decimal point afterward.
  2. Multiply 426 × 35:
    • First multiply 426 × 5:
      • 6 × 5 = 30 units. Exchange for 3 tens.
      • 2 × 5 = 10 tens + 3 tens (from exchange) = 13 tens. Exchange for 1 hundred and 3 tens.
      • 4 × 5 = 20 hundreds + 1 hundred (from exchange) = 21 hundreds.
      • First partial product: 2,130
    • Then multiply 426 × 30:
      • 6 × 30 = 180 units. Exchange for 1 hundred and 8 tens.
      • 2 × 30 = 60 tens + 8 tens (from exchange) = 68 tens. Exchange for 6 hundreds and 8 tens.
      • 4 × 30 = 120 hundreds + 6 hundreds (from exchange) = 126 hundreds. Exchange for 12 thousands and 6 hundreds.
      • Second partial product: 12,780
    • Add both partial products: 2,130 + 12,780 = 14,910
  3. Count decimal places: 4.26 has 2 decimal places, 3.5 has 1 decimal place. Together they have 3 decimal places.
  4. Place the decimal point 3 places from the right in the result: 14.910 = 14.91

Dynamic/Static: Dynamic – complex with multiple exchanges and decimal point placement

DECIMAL DIVISION COMMAND CARD 2

Problem: 45.36 ÷ 2.4 Materials: Stamp game material, decimal place value mat Steps:

  1. First, convert the divisor to a whole number by multiplying both numbers by 10:
    • 45.36 × 10 = 453.6
    • 2.4 × 10 = 24
    • New problem: 453.6 ÷ 24
  2. Place 4 hundreds, 5 tens, 3 units, and 6 tenths on the mat.
  3. Can we distribute hundreds into 24 groups? No. Exchange 4 hundreds for 40 tens.
  4. Now have 45 tens. Can we distribute into 24 groups? Yes, 1 ten to each group (24 total), with 21 tens remaining.
  5. Exchange 21 tens for 210 units. Now have 213 units.
  6. Distribute 8 units to each group (192 total), with 21 units remaining.
  7. Exchange 21 units for 210 tenths. Now have 216 tenths.
  8. Distribute 9 tenths to each group (216 total), with 0 tenths remaining.
  9. Read the quotient: 18.9

Dynamic/Static: Dynamic – involves conversion of divisor and multiple exchanges

FRACTION MULTIPLICATION COMMAND CARD 1

Problem: 2⅔ × 1¼ Materials: Stamp game material, fraction cubes, place value mat Steps:

  1. Convert to improper fractions: 2⅔ = 8/3 and 1¼ = 5/4
  2. Multiply numerators: 8 × 5 = 40
  3. Multiply denominators: 3 × 4 = 12
  4. Result is 40/12, which simplifies to 10/3
  5. Convert to mixed number: 10/3 = 3⅓
  6. Read the product: 3⅓

Dynamic/Static: Static – involves calculation with fraction values

FRACTION DIVISION COMMAND CARD 1

Problem: 4½ ÷ 1⅕ Materials: Stamp game material, fraction cubes, place value mat Steps:

  1. Convert to improper fractions: 4½ = 9/2 and 1⅕ = 6/5
  2. To divide by a fraction, multiply by its reciprocal: 9/2 × 5/6
  3. Multiply numerators: 9 × 5 = 45
  4. Multiply denominators: 2 × 6 = 12
  5. Result is 45/12, which simplifies to 15/4
  6. Convert to mixed number: 15/4 = 3¾
  7. Read the quotient: 3¾

Dynamic/Static: Static – involves calculation with fraction values

OPERATION GUIDES & REFERENCE

ADDITION PRINCIPLES:

  • Static Addition: When no exchanges are needed
  • Dynamic Addition: When exchanges are required (carrying)
  • Key Steps:
    1. Place the first number on the mat
    2. Place the second number on the mat
    3. Combine stamps column by column, starting from the smallest place value
    4. Exchange 10 stamps of one place value for 1 stamp of the next higher place value when needed

SUBTRACTION PRINCIPLES:

  • Static Subtraction: When no exchanges are needed
  • Dynamic Subtraction: When exchanges are required (borrowing)
  • Key Steps:
    1. Place the minuend (larger number) on the mat
    2. Remove stamps according to the subtrahend (smaller number)
    3. If there aren't enough stamps in a column, exchange 1 stamp from the next higher place value for 10 stamps of the needed place value
    4. Continue until all columns are processed

MULTIPLICATION PRINCIPLES:

  • Key Steps:
    1. Place the multiplicand on the mat
    2. Multiply each column by the multiplier
    3. Start from the smallest place value
    4. Exchange as needed
    5. For multi-digit multipliers, create partial products and add them

DIVISION PRINCIPLES:

  • Key Steps:
    1. Place the dividend on the mat
    2. Use skittles to represent the divisor (groups)
    3. Distribute stamps equally among the skittles
    4. Exchange higher place values for lower place values when needed
    5. The quotient is what each skittle receives

DECIMAL OPERATIONS:

  • Use the decimal place value mat with columns for ones, tenths, hundredths, etc.
  • Follow the same principles as whole numbers, paying attention to the decimal point
  • For multiplication and division with decimals, follow special rules for decimal point placement

FRACTION OPERATIONS:

  • Use fraction cubes to represent numerators and denominators
  • For addition and subtraction, find common denominators
  • For multiplication, multiply numerators and denominators
  • For division, multiply by the reciprocal of the divisor

These command cards provide a structured approach to the Montessori stamp game across grades 4-6, incorporating whole numbers, decimals, and fractions. Each card includes step-by-step instructions and indicates whether the operation is static or dynamic.



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