Montessori Division with Remainders: Multi-Tiered Lesson Plan for Grades 4-6 | Small Groups 2-4

Montessori Division with Remainders: Multi-Tiered Lesson Plan for Grades 4-6 | Small Groups 2-4

Lesson Overview

Grade Levels: 4-5-6 multi-age classroom
Duration: 3 30-minute sessions: 90 minutes total
Materials:

• Montessori flat bead frames (1 per pair)
• Stamp game materials (1 set per pair)
• Recording sheets (graph paper)
• Whiteboards and markers
• Vertical problem cards (tiered by difficulty)
• Visual timer

Learning Objectives

Students will:

1. Use Montessori materials to perform division with remainders
2. Progress through increasingly complex division problems
3. Explain their mathematical thinking and reasoning
4. Collaborate effectively using Kagan and Thinking Classroom structures

Tiered Problems (By Difficulty)

Tier 1: Two-digit dividends ÷ one-digit divisors (e.g., 45 ÷ 2)
Tier 2: Three-digit dividends ÷ one-digit divisors (e.g., 217 ÷ 4)
Tier 3: Three-digit dividends ÷ one-digit divisors with decimal answers (e.g., 325 ÷ 4)
Tier 4: Four-digit dividends ÷ one-digit divisors (e.g., 3641 ÷ 7)
Tier 5: Four-digit dividends ÷ two-digit divisors (e.g., 4256 ÷ 32)
Tier 6: Four-digit dividends with decimal extensions ÷ two-digit divisors (e.g., 4256 ÷ 32 showing decimals)

Lesson Flow

Introduction (5-10 minutes)

Teacher Actions:

• Gather students in a circle around a demonstration table
• Present the flat bead frame and stamp game materials
• Frame the day's exploration: "Today we'll investigate division with remainders using Montessori materials"
• Pose an essential question: "How can we visualize and understand what happens with the 'leftover' parts in division?"

Student Actions:

• Observe materials and listen actively
• Respond to teacher prompts about prior knowledge of division

Teacher Demonstration (15 minutes)

Teacher Actions:

• Demonstrate a Tier 2 problem (e.g., 217 ÷ 4) using both the flat bead frame and stamp game
• Think aloud while working: "I need to share 217 equally among 4 groups"
• With the flat bead frame:
  • Set up 2 hundred beads, 1 ten bead, and 7 unit beads
  • Show how to distribute them equally: "I can give each group 50 from my hundreds... I have 17 left... I can give each group 4 tens with 1 remainder... I can split the 7 units into 1 each with 3 remainder"
  • Show the final quotient: 54 with remainder 1
• With the stamp game:
  • Set out 2 hundred stamps, 1 ten stamp, and 7 unit stamps
  • Show the grouping process similarly
• Show how to record the process on paper

Student Actions:

• Observe demonstrations quietly
• Note the differences between the materials
• Mentally prepare questions

Partner Setup (5 minutes)

Teacher Actions:

• Assign random pairs using numbered cards
• Distribute materials to each pair
• Assign roles: "Person with the earlier birthday will be the first Sage, the other will be the Scribe"
• Direct pairs to stand at vertical non-permanent surfaces (whiteboards) around the room

Student Actions:

• Find partners and collect materials
• Determine roles
• Move to assigned vertical workspaces

Paired Practice - Tier 1 & 2 (15 minutes)

Teacher Actions:

• Distribute Tier 1 problem cards to struggling pairs, Tier 2 to others
• Circulate and observe without intervening unless necessary
• Note interesting approaches for later discussion
• Signal role swap halfway through

Student Actions (Sage):

• Manipulate materials to solve division problems
• Verbalize thinking process to Scribe
• Demonstrate division process using materials

Student Actions (Scribe):

• Record the Sage's process on the whiteboard
• Ask clarifying questions
• Ensure accurate recording of each step

Paired Practice - Progression (20 minutes)

Teacher Actions:

• Provide higher-tier problem cards as pairs master lower tiers
• Remind students to switch roles after each problem
• Ask probing questions: "What happens when your remainder is larger than your divisor?"
• Direct focus to decimal extensions when appropriate

Student Actions:

• Progress through problem tiers at their own pace
• Switch Sage/Scribe roles regularly
• Use materials to represent increasingly complex problems
• Record processes and solutions on whiteboards

Harkness Math Seminar Groups (15 minutes)

Teacher Actions:

• Combine pairs into groups of four
• Provide each group with one challenging problem from their highest mastered tier
• Set up circular seating for discussion
• Prompt: "Share your approaches and find consensus on the most efficient method"

Student Actions:

• Discuss various solution approaches
• Compare methods used with different materials
• Reach consensus on efficient strategies
• Prepare to share insights with whole class

Whole Group Discussion (10 minutes)

Teacher Actions:

• Facilitate sharing from each group
• Highlight different approaches
• Ask connections questions: "How does the bead frame help us understand decimal remainders differently than the stamp game?"
• Guide synthesis of key concepts

Student Actions:

• Share group insights
• Compare strategies across groups
• Ask questions of other groups
• Connect division concepts across different representations

Reflection and Close (10 minutes)

Teacher Actions:

• Distribute individual reflection cards
• Prompt: "What was one insight you gained about division today? What question do you still have?"
• Collect reflection cards as exit tickets
• Preview next lesson's focus

Student Actions:

• Complete reflection independently
• Clean up and return materials
• Submit exit reflection

Assessment

• Observation notes during paired work
• Whiteboard records of process
• Participation in Harkness discussions
• Exit reflection responses

Extensions/Modifications

For Advanced Students:

• Create their own division problems that yield specific types of remainders
• Explore converting remainders to fractions and decimals
• Investigate divisibility rules

For Students Needing Support:

• Continue with lower-tier problems
• Use additional concrete materials
• Work with teacher in small group

Follow-Up Lesson Ideas

• Connecting division with fractions
• Real-world applications of remainders
• Division patterns and algorithms

Arizona Mathematics Standards Alignment

4th Grade Standards

• 4.NBT.B.6: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.
• 4.OA.A.3: Solve multistep word problems using the four operations, including problems in which remainders must be interpreted.
• MP.1: Make sense of problems and persevere in solving them.
• MP.3: Construct viable arguments and critique the reasoning of others.
• MP.5: Use appropriate tools strategically.

5th Grade Standards

• 5.NBT.B.6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.
• 5.NBT.B.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
• MP.2: Reason abstractly and quantitatively.
• MP.4: Model with mathematics.
• MP.6: Attend to precision.

6th Grade Standards

• 6.NS.B.2: Fluently divide multi-digit numbers using the standard algorithm.
• 6.NS.B.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
• MP.7: Look for and make use of structure.
• MP.8: Look for and express regularity in repeated reasoning.

The Importance of Concrete Materials in Building Mathematical Understanding

The use of concrete manipulatives in this lesson addresses a critical educational principle: students develop deeper conceptual understanding when they progress from concrete to representational to abstract (CRA) approaches. Research consistently shows that rote memorization of algorithms without conceptual understanding leads to poor retention and difficulty applying knowledge to new situations.

Benefits of Concrete Materials:

1. Visualization of Mathematics: Materials like the Montessori flat bead frame make abstract concepts visible and tangible.
2. Understanding of Place Value: Physical manipulation of place value materials reinforces the base-10 system.
3. Process Transparency: Students can observe what happens when they "regroup" or work with remainders.
4. Accessibility: Multi-sensory approaches reach diverse learners.
5. Conceptual Foundation: Manipulatives build the foundation for understanding algorithms.

The Kagan Structure and Thinking Classroom Approach:

The Sage and Scribe cooperative structure elevates learning to the highest levels of Bloom's Taxonomy:

• Teaching Others (90% retention): When students must explain concepts as the "Sage," they solidify their own understanding.
• Verbalization: Requiring students to articulate their thinking processes deepens cognitive connections.
• Accountability: Each student must master concepts to effectively teach their partner.
• Vertical Non-Permanent Surfaces: The Thinking Classroom approach increases engagement and persistence in problem-solving.

Mathematical Discourse Through Harkness Seminars:

The lesson's culmination in Harkness-style discussions allows students to:

• Compare different approaches
• Evaluate efficiency of methods
• Recognize multiple valid strategies
• Develop mathematical communication skills

This comprehensive approach addresses Arizona standards while building lasting mathematical understanding through concrete exploration, collaborative discourse, and student-led discovery—ensuring students can explain why algorithms work rather than simply executing memorized steps.