The 120-Bead Number Line: A Comprehensive Overview K-6

The 120-Bead Number Line: A Comprehensive Overview K-6

Basic Operations and Place Value Applications

120-Bead Number Line: Operations and Place Value K-6 PODCAST

This podcast gives a quick overview of the 120-bead number line as a versatile educational tool for mathematics in grades K-6. They outline specific grade-level benchmarks for using the bead line to teach fundamental operations, place value, fractions, decimals, and even early algebraic concepts. Furthermore, the text explores advanced applications for upper elementary grades, integrating the bead line with strategies like bar modeling and algebraic thinking. Finally, it provides specific guidance on using the bead line for integer operations, emphasizing visualization and conceptual understanding.

A FANTASTIC BOOK TO GET STARTED WITH BEADED NUMBER LINES!

https://www.amazon.com/Working-Beaded-Number-Line-Grades/dp/1583247386

Grade-Level Benchmarks for the 120-Bead Number Line

Kindergarten

• Basic Counting: Count up to 20 beads accurately using one-to-one correspondence on the beaded number line • Subitizing: Instantly recognize groups of 1-5 beads without counting individually • Number Comparison: Determine which of two quantities up to 10 is greater by comparing bead arrangements side by side • Simple Addition: Demonstrate combining groups of up to 5 beads each and counting the total

1st Grade

• Skip Counting: Count by 2s, 5s, and 10s up to 120 using grouped beads on the number line • Addition/Subtraction within 20: Solve addition and subtraction problems within 20 using various strategies with beaded number line • Place Value to 100: Demonstrate understanding of tens and ones by grouping beads accordingly • Number Relationships: Identify beads that are 1 more, 1 less, 10 more, or 10 less than a given number up to 100

2nd Grade

• Place Value to 120: Read, write, and model three-digit numbers using the full 120-bead number line • Addition/Subtraction within 100: Fluently add and subtract within 100 using the beaded number line, including regrouping • Even/Odd Recognition: Sort beads into groups of 2 to determine if numbers up to 100 are even or odd • Skip Counting Patterns: Identify and explain patterns when counting by 5s, 10s, and 100s using colored bead groupings

3rd Grade

• Multiplication Concepts: Represent multiplication facts up to 10×10 using rectangular arrays with beads • Division Concepts: Demonstrate fair sharing and grouping division using the beaded number line • Fraction Representation: Represent unit fractions (1/2, 1/3, 1/4, 1/6, 1/8) using appropriate sections of the 120-bead line • Rounding to Tens/Hundreds: Round numbers to the nearest 10 and 100 using the bead line as a number line model

4th Grade

• Multi-Digit Addition/Subtraction: Solve multi-digit addition and subtraction problems using the bead line combined with the stamp game • Multi-Digit Multiplication: Model multiplication of 2-digit by 1-digit and 2-digit by 2-digit numbers using rectangular arrays of beads • Fraction Equivalence: Demonstrate equivalent fractions using different partitions of the 120-bead line • Decimal Representation: Model decimals to the hundredths using proportional sections of the beaded number line

5th Grade

• Fraction Addition/Subtraction: Add and subtract fractions with unlike denominators using the beaded number line to find common denominators • Decimal Operations: Perform addition and subtraction with decimals to the hundredths using the beaded number line combined with place value concepts • Volume Concepts: Use multiple beaded number lines to model and calculate volume of rectangular prisms • Division with Remainders: Model division of up to 4-digit dividends by 2-digit divisors with remainders using the beaded number line

6th Grade

• Ratio and Proportion: Use the beaded number line to model ratio relationships and solve proportion problems • Integer Operations: Represent addition and subtraction of positive and negative integers on the beaded number line • Algebraic Expressions: Model one-variable expressions and equations using beads to represent variables and constants • Percent Problems: Solve real-world percent problems including finding percentages, wholes, and parts using the 120-bead system

Extension Benchmarks for Advanced Problem-Solving

3rd Grade Extensions

• Bar Model Representation: Create physical bar models using the beaded number line to solve one-step word problems • Square Numbers: Identify and model square numbers up to 100 using rectangular arrays of beads

4th Grade Extensions

• Fraction Multiplication: Model multiplication of a fraction by a whole number using the beaded number line • Pattern Analysis: Create, extend, and explain patterns in sequences using the beaded number line

5th Grade Extensions

• Coordinate System: Use multiple beaded number lines to create a coordinate system and plot points • Order of Operations: Demonstrate understanding of the order of operations by solving expressions using beaded representations

6th Grade Extensions

• Rational Number System: Represent and order all rational numbers (fractions, decimals, percents) on a beaded number line • Surface Area Calculation: Use multiple beaded number lines to calculate surface area of three-dimensional figures

Student-Designed Bead Systems

• Custom Base System: Design and demonstrate a non-base-10 counting system using the beaded number line • Problem-Specific Models: Create specialized bead arrangements to solve particular types of problems efficiently • Mathematical Modeling: Design bead models to represent and solve real-world mathematical scenarios • Proof Construction: Use bead systems to create visual proofs of mathematical principles and relationships

Place Value Abacus

• Grouping by Tens: Having students group beads in tens to understand place value
• Representing Numbers: Using different colored beads to represent hundreds, tens, and ones
• Trading Game: Trading 10 ones for 1 ten, 10 tens for 1 hundred
• Expanded Form: Showing numbers like 47 as 40 + 7 with bead groupings

Addition Activities

1. Direct Addition: Moving beads to combine quantities
2. Skip Counting: Using the beads to count by 2s, 5s, 10s
3. Make Ten Strategy: Finding combinations that make 10 (helpful for mental math)
4. Adding Multiple Numbers: Using beads to show associative property

Subtraction Activities

1. Take Away: Physically removing beads to show subtraction
2. Counting On/Back: Using the beads to count backward or find the difference
3. Missing Addend: Using the beads to find how many more to get to a target number
4. Difference Game: Finding the distance between two quantities

Factor and Multiple Activities

Finding Factors

1. Grouping Equal Sets: Arranging 12 beads into groups of 1, 2, 3, 4, 6, and 12
2. Factor Rainbows: Marking arcs to show factors of numbers up to 120
3. Prime Number Hunt: Finding numbers that cannot be arranged in rectangular arrays
4. Common Factor Exploration: Finding factors shared by two numbers

Multiple Explorations

1. Skip Counting Patterns: Using different colored beads to show multiples
2. Multiplication Tables: Creating physical representations of multiplication tables
3. Least Common Multiple: Finding the first point where skip counting patterns align

Fraction Concepts

Representing Fractions

• Basic Benchmarks: 0, 1/2 (one-half), and 1 (one whole) are foundational benchmarks.
• Common Fractions: 1/4 (one-quarter), 3/4 (three-quarters), 1/3 (one-third), and 2/3 (two-thirds) are also frequently used.
• Other fractions: 1/8, 3/8, 5/8, and 7/8 are also useful benchmark fractions
• Purpose: These fractions serve as reference points for comparing and estimating the values of other fractions, especially when dealing with fractions that are harder to visualize.
• Number Line: They are helpful for visualizing fractions on a number line and understanding their relative sizes.
• Estimation: Benchmark fractions are particularly useful for estimating sums and differences of fractions.

1. Equal Parts: Dividing the 120 beads into equal groups
2. Unit Fractions: Showing 1/2, 1/3, 1/4, etc. of 120
3. Non-Unit Fractions: Representing fractions like 2/3, 3/4, etc.
4. Equivalent Fractions: Demonstrating how 1/2 = 2/4 = 3/6, etc.

Adding and Subtracting Fractions

1. Common Denominators: Using the beads to find common denominators
2. Adding Like Fractions: Combining parts with the same denominator
3. Subtracting Like Fractions: Removing parts with the same denominator
4. Mixed Numbers: Converting between improper fractions and mixed numbers

Multiplying and Dividing Fractions

1. Fraction of a Set: Finding 1/3 of 12 beads
2. Repeated Addition: Showing multiplication as repeated addition (3 × 1/4)
3. Area Model: Using rectangular arrays to show multiplication of fractions
4. Fair Sharing: Dividing quantities to show division with fractions

Integration with Stamp Game

Combined Activities

1. Place Value Bridge: Using stamps alongside beads to show place value
2. Decimal Connections: Using stamps to represent decimal values alongside beads
3. Money Problems: Combining stamps and beads for money calculations
4. Algebraic Thinking: Using stamps as variables and beads as constants

Array and Multiplication Models

Visual Multiplication

1. Rectangular Arrays: Arranging beads in rows and columns to show multiplication facts
2. Area Model: Using the beads to create an area model for multiplication
3. Distributive Property: Showing how multiplication can be broken down (7 × 8 = 7 × 5 + 7 × 3)
4. Square Numbers: Creating square arrays with beads

Number Sense and Subitizing Activities

Early Number Sense (Grades 1-2)

1. Subitizing Patterns: Quick recognition of quantities without counting
2. Number Bonds: Finding different ways to make a number
3. More/Less Comparisons: Comparing quantities with visual representation
4. Even/Odd Patterns: Discovering characteristics of even and odd numbers

Advanced Number Sense (Grades 3-6)

1. Estimation Skills: Estimating quantities on the bead line
2. Ratio and Proportion: Using beads to show relationships between quantities
3. Percents: Representing percents as parts of 100
4. Number Line Jumps: Using beads for adding and subtracting on a number line

Dr. Nicky Newton's Approaches

Beaded Number Line Strategies

1. Spatial Reasoning: Using the beaded number line for spatial understanding of numbers
2. Number Talks: Facilitating discussions about strategies seen on the beaded number line
3. Strategic Grouping: Grouping beads in different ways (5s and 10s) for mental math
4. Open Middle Problems: Creating problems with multiple solution paths using the beads

Progression Through Concrete-Pictorial-Abstract

Concrete Stage

1. Physical Manipulation: Hands-on activities moving beads
2. Tactile Learning: Feeling and grouping beads to build number sense
3. Discovery Learning: Finding patterns through experimentation with beads

Pictorial Stage

1. Drawing Representations: Drawing what was done with the beads
2. Using Photographs: Taking pictures of bead arrangements
3. Diagramming: Creating diagrams that represent the bead work

Abstract Stage

1. Symbolic Notation: Moving to standard mathematical notation
2. Mental Math: Visualizing the beads without physical manipulation
3. Algorithm Development: Creating personal algorithms based on understanding

Assessment and Differentiation

Assessment Tools

1. Observation Checklists: Monitoring how students interact with the beads
2. Performance Tasks: Having students demonstrate understanding with beads
3. Student Explanations: Having students explain their thinking using beads

Differentiation Strategies

1. Scaffolded Activities: Progressive challenges with the same materials
2. Extension Problems: Complex problems for advanced learners
3. Support Structures: Visual guides for struggling students

Games and Activities

Collaborative Games

1. Bead Race: Teams compete to create specific arrangements
2. Pattern Matching: Creating and matching patterns with beads
3. Factor Find: Racing to find all factors of a number

Independent Activities

1. Pattern Creation: Creating and recording patterns
2. Number Story Illustrations: Using beads to illustrate word problems
3. Math Journal Prompts: Recording observations and discoveries

Advanced Applications of the 120-Bead Number Line for Grades 4-6

Singapore Math Model with 13 Heuristics

1. Using the Bar Model with 120-Bead Number Line

Part-Whole Model Implementation

• Concrete Representation: Students use the 120-bead number line to physically represent the parts and whole in a bar model
• Proportional Reasoning: Using the 120 beads to create proportional bar models where the length of beads represents the relative value
• Known/Unknown Quantities: Using colored beads to distinguish between known and unknown quantities in word problems

Comparison Model Applications

• Difference Visualization: Arranging two bead lines side by side to show the difference between quantities
• Ratio Representation: Using beads to create visual ratios (e.g., 3:5 shown as 3 beads of one color and 5 of another)
• Multiple Comparison: Arranging 3+ bead lines to compare multiple quantities simultaneously

2. Heuristic Applications with 120-Bead Number Line

Act It Out

• Students manipulate beads to physically act out problem scenarios
• Recording changes in bead positions to track problem-solving steps
• Using beads to demonstrate transformations in word problems

Look for Patterns

• Arranging beads to discover number patterns and relationships
• Using different colored beads to highlight pattern sequences
• Creating visual representations of growing patterns using multiple bead lines

Systematic Listing

• Using beads to systematically list all possible combinations/permutations
• Organizing beads to represent decision trees or outcome possibilities
• Creating a physical sorting system for combinatorial problems

Working Backwards

• Starting with the answer (full set of beads) and working backwards through a problem
• Reverse-engineering multi-step problems using bead manipulations
• Verifying solutions by working forward and backward

Guess and Check

• Testing hypotheses by arranging beads in different configurations
• Recording and refining guesses based on bead arrangements
• Using beads to track multiple guess-and-check iterations

Make a Drawing/Model

• Using beads as a concrete foundation before drawing the bar model
• Creating physical models that directly translate to pictorial representations
• Developing spatial reasoning by translating between beads and drawings

Make a Table

• Organizing beads into rows and columns to create physical data tables
• Using beads to represent entries in a table before recording on paper
• Finding patterns in tables using bead arrangements

Solve Part of the Problem

• Breaking complex problems into smaller parts using bead groupings
• Solving one component at a time with dedicated bead sections
• Recombining partial solutions demonstrated with beads

Simplify the Problem

• Using beads to model simplified versions of complex problems
• Gradually adding complexity by incorporating more beads
• Identifying core problem structures using minimal bead arrangements

Use Before-After Concept

• Showing initial and final states with two bead arrangements
• Tracking transformations between states using bead manipulation
• Calculating changes by comparing before/after bead configurations

Use Equation

• Translating bead arrangements directly into algebraic equations
• Using beads to demonstrate the balance concept in equations
• Representing variables and constants with different colored beads

Use a Formula

• Building physical models of formulas using bead arrangements
• Demonstrating how formulas relate to concrete quantities
• Creating visual proofs of formula relationships

Logical Reasoning

• Using beads to create logical sequences and premises
• Demonstrating if-then relationships with conditional bead arrangements
• Testing logical conclusions with bead manipulations

Advanced Arrays and Number Patterns

1. Square Numbers and Square Roots

Square Number Visualization

• Physical Squares: Using multiple 120-bead number lines to form square arrays (e.g., 12×12 using 2 bead lines)
• Square Patterns: Exploring patterns in square numbers by adding successive odd numbers
• Perfect Square Hunting: Finding all perfect squares up to 120 (1, 4, 9, 16, 25, 36, 49, 64, 81, 100)

Square Root Exploration

• Physical Square Roots: Finding the side length of a square with a given area
• Estimating Roots: Approximating square roots by creating squares with bead arrays
• Rational vs. Irrational: Distinguishing perfect squares from non-perfect squares

2. Cubic Numbers and Cube Roots

Representing Cubic Numbers

• 3D Visualization: Using multiple bead lines to create physical representations of cubic numbers
• Cubic Patterns: Exploring patterns in cubic numbers using bead arrangements
• Volume Concepts: Building physical models of volume using bead lines as edges

Cube Root Exploration

• Physical Cube Roots: Finding the side length of a cube with a given volume
• Spatial Reasoning: Developing spatial understanding through 3D bead constructions
• Dimensional Analysis: Comparing linear, square, and cubic growth patterns

3. Advanced Array Operations

Rectangular Arrays for Complex Multiplication

• Multi-Digit Multiplication: Using bead arrays to represent multiplication of 2-3 digit numbers
• Area Model Refinement: Breaking down complex multiplication into smaller arrays
• Distributive Property: Demonstrating how multiplication can be broken down into simpler parts

Array Division

• Quotitive Division: Showing how many groups of a certain size can be made
• Partitive Division: Dividing beads into a specific number of equal groups
• Division with Remainders: Physically representing remainders with leftover beads

4. Algebraic Thinking with Arrays

Linear Equations

• Physical Equations: Using beads to represent and solve linear equations
• Balance Concept: Adding or removing equal amounts from both sides
• Multiple Solutions: Exploring equations with multiple or no solutions

Quadratic Relationships

• Growing Patterns: Representing quadratic growth with bead arrays
• Quadratic Expressions: Modeling expressions like x² + 2x + 1 with beads
• Completing the Square: Physical demonstration of completing the square method

Fractions, Decimals, and Rational Numbers

1. Fraction Complexities

All Possible Fractions with 120 Beads

• Listing Fractions: All possible fractions with denominator ≤ 120 (over 600 fractions)
• Equivalent Fractions: Finding all equivalent fractions within the 120-bead system
• Simplest Form: Identifying fractions in their simplest form

Common Denominators and Fractions

• Finding LCM: Using beads to find least common multiples for denominators
• Equivalent Fractions: Converting fractions to equivalent forms with common denominators
• Simplification: Simplifying fractions using GCD concepts with beads

2. Advanced Fraction Operations

Fraction Multiplication

• Area Model: Using beads to create an area model for multiplying fractions
• Scaling: Demonstrating how fractions can scale quantities
• Mixed Numbers: Multiplying mixed numbers using bead arrays

Fraction Division

• Reciprocals: Using beads to demonstrate reciprocal relationships
• Division Meaning: Showing how many groups of a certain size exist in a quantity
• Complex Division: Dividing by fractions and mixed numbers

Fraction Exponents

• Repeated Multiplication: Using beads to show repeated multiplication with fractions
• Fraction Powers: Demonstrating what happens when a number is raised to a fractional power
• Patterns in Exponents: Exploring patterns with positive and negative exponents

3. Decimals and Percent

Decimal Representation

• Decimal Positions: Using beads to represent place value in decimals
• Decimal Equivalents: Converting between fractions and decimals
• Decimal Operations: Performing operations with decimals using beads

Percent Concepts

• Percent as Parts per 100: Using beads to physically represent percentages
• Percent Increase/Decrease: Demonstrating percentage changes with beads
• Percent Problems: Solving complex percentage problems using beads as a model

4. Rational Number Operations

Number Line Representation

• Rational Number Placement: Placing various rational numbers on the bead number line
• Ordering Rational Numbers: Comparing and ordering various fractions and decimals
• Density Property: Demonstrating that between any two rational numbers lies another rational number

Rational Number Operations

• Addition/Subtraction: Performing operations with mixed forms of rational numbers
• Multiplication/Division: Multiplying and dividing combinations of fractions and decimals
• Order of Operations: Using beads to demonstrate PEMDAS with rational numbers

Abacus Applications and Place Value

1. Extended Place Value

Multiple Bead Lines as Multi-Digit Abacus

• Place Value Columns: Using multiple bead lines to represent ones, tens, hundreds, thousands
• Large Number Operations: Performing operations with 4-6 digit numbers
• Regrouping Visualization: Physically demonstrating regrouping across multiple place values

Decimal Place Value

• Decimal Point Positioning: Using beads to represent numbers with decimal places
• Decimal Operations: Performing operations with decimals using bead representation
• Decimal Conversions: Converting between fractions, decimals, and percentages

2. Advanced Abacus Techniques

Computational Strategies

• Mental Math Enhancement: Using abacus visualization to improve mental computation
• Estimation Techniques: Developing estimation skills using bead representations
• Algorithm Development: Creating personal computational algorithms

Number Pattern Recognition

• Divisibility Rules: Using bead patterns to discover divisibility rules
• Factor Patterns: Exploring patterns in factors and multiples
• Prime Number Relationships: Discovering properties of prime numbers through bead arrangements

Problem-Solving Applications

1. Real-World Problem Solving

Multi-Step Problems

• Problem Decomposition: Breaking complex problems into manageable parts
• Strategy Selection: Choosing appropriate strategies for different problem types
• Solution Verification: Using beads to verify solutions to complex problems

Cross-Curricular Applications

• Science Connections: Representing scientific concepts with beads (e.g., molecule arrangements)
• Social Studies Applications: Using beads for population or resource distribution models
• Art Integration: Creating mathematical art using bead patterns and relationships

2. Mathematical Reasoning

Logical Proof Development

• Concrete Proofs: Using beads to demonstrate mathematical proofs
• Inductive Reasoning: Identifying patterns and making generalizations
• Deductive Reasoning: Drawing conclusions based on given information

Advanced Mathematical Thinking

• Algebraic Reasoning: Developing algebraic thinking through concrete representations
• Functional Relationships: Exploring input-output relationships with bead models
• Proportional Reasoning: Developing deep understanding of proportions and ratios

By implementing these advanced applications of the 120-bead number line, teachers can provide students in grades 4-6 with a concrete foundation for abstract mathematical concepts, creating a seamless transition from concrete to pictorial to abstract understanding in alignment with Singapore Math principles.

Using the 120-Bead Number Line for Integer Operations

Integer Number Line Applications

Representing Positive and Negative Integers

• Zero Point Setup: Designate the middle point (bead 60) as zero
• Positive Direction: Beads 61-120 represent positive integers
• Negative Direction: Beads 59-1 represent negative integers
• Color Coding: Use different colored beads or markers to distinguish positive from negative regions

Addition with Integers

Adding Two Positive Integers

• Start at zero (bead 60)
• Move right (positive direction) for the first number
• Continue moving right for the second number
• The final position indicates the sum

Adding Two Negative Integers

• Start at zero (bead 60)
• Move left (negative direction) for the first number
• Continue moving left for the second number
• The final position indicates the sum

Adding a Positive and Negative Integer

• Start at zero (bead 60)
• Move in the direction of the first number (right for positive, left for negative)
• Move in the direction of the second number
• The final position indicates the sum
• Example: To add +5 and -3, move 5 beads right from zero, then 3 beads left, landing at +2

Subtraction with Integers

Subtraction as Adding the Opposite

• Reframe subtraction as adding the opposite
• Convert "a - b" to "a + (-b)"
• Follow the addition rules above
• Example: To calculate 7 - 10, reframe as 7 + (-10), move 7 beads right from zero, then 10 beads left, landing at -3

Direct Subtraction Method

• Start at the first number (count from zero to reach it)
• Move in the opposite direction of the second number (left if positive, right if negative)
• The final position indicates the difference
• Example: For -4 - (-6), start at -4 (4 beads left of zero), then move 6 beads right, landing at +2

Multiple Bead Lines for Multiplication and Division

Multiplication with Integers

Conceptual Framework

• Multiplication represents repeated addition or scaling
• The sign rules can be demonstrated physically
• Same signs yield positive product; different signs yield negative product

Positive × Positive

• Use one bead line to represent the multiplier
• Use a second bead line perpendicular to the first to create an array
• Count the total beads in the array
• Example: For 4 × 3, create a 4×3 rectangular array, yielding 12

Negative × Positive

• Use one bead line to represent the positive number
• Use a second bead line with negative direction to show repeated addition of the positive number in the negative direction
• Count the total movement in the negative direction
• Example: For -4 × 3, show adding 3 four times in the negative direction, yielding -12

Negative × Negative

• Use one bead line to represent the negative multiplier
• Use a second bead line to show repeated subtraction of the negative number (moving in the positive direction)
• Count the total movement in the positive direction
• Example: For -4 × -3, show subtracting -3 four times (moving right each time), yielding +12

Division with Integers

Division as Repeated Subtraction

• Arrange the dividend number of beads
• Repeatedly remove groups of the divisor size
• Count how many complete groups can be removed
• Example: For 15 ÷ 3, remove three beads at a time from 15, yielding 5 groups

Division with Negative Integers

• Apply the same sign rules as multiplication
• Same signs yield positive quotient; different signs yield negative quotient
• Use directional movements to demonstrate
• Example: For -12 ÷ 3, show how many groups of 3 can be removed from -12, yielding -4

Visualizing Division with Remainders

• Show division as creating equal groups
• Any beads remaining after creating equal groups represent the remainder
• Example: For 17 ÷ 5, create three groups of 5 with 2 beads remaining, yielding 3 R2

Classroom Implementation Ideas

Interactive Activities

Integer War Game

• Students work in pairs with a shared bead line
• Draw cards with integer operations
• Use beads to solve and compare answers
• Highest result wins the round

Integer Bead Line Race

• Teams compete to reach target positions on the number line
• Draw operation cards that require integer calculations
• Use bead movements to solve and advance

Real-World Connection Activities

• Temperature changes (positive and negative)
• Elevation changes (above and below sea level)
• Financial transactions (deposits and withdrawals)
• Sports statistics (yards gained and lost)

Assessment Methods

Performance Tasks

• Have students demonstrate specific integer operations using the bead line
• Ask students to explain their reasoning while manipulating beads
• Observe patterns in student approaches to identify misconceptions

Student Teaching

• Have students teach a partner how to solve a specific integer operation
• Record explanations for peer review
• Build conceptual vocabulary through explanation

Mathematical Discussion Prompts

Pattern Recognition

• "What patterns do you notice when adding integers with the same sign?"
• "What happens when you subtract a negative number?"
• "How does the bead line help you understand why a negative times a negative equals a positive?"

Strategic Thinking

• "What's the most efficient way to calculate -15 + 23 using the bead line?"
• "How could you use the bead line to prove the commutative property with negative numbers?"
• "When would it be better to reframe subtraction as adding the opposite?"

The 120-bead number line provides a powerful concrete model for integer operations, helping students visualize abstract concepts like negative numbers and the rules governing their operations. By physically manipulating beads, students develop stronger conceptual understanding before transitioning to algorithmic approaches, creating a solid foundation for algebraic thinking in later grades.