Complete Montessori Mathematics Curriculum Map (Ages 6-9)

 Complete Montessori Mathematics Curriculum Map (Ages 6-9)

I. NUMERATION AND PLACE VALUE

1. Golden Bead Material

Extension from primary level for deeper understanding of decimal system

Manipulative Components:

• Golden bead units (1)
• Golden bead ten bars (10)
• Golden bead hundred squares (100)
• Golden bead thousand cubes (1000)
• Numeral cards (1-9, 10-90, 100-900, 1000-9000)
• Small wooden number cards

Demonstrations (9):

1. Introduction to Quantity - Review of quantities 1-9,999
2. Introduction to Symbol - Review of number symbols
3. Association of Quantity and Symbol - Matching symbols to quantities
4. Formation of Multi-Digit Numbers - Building numbers with beads and cards
5. Reading Large Numbers - How to read numbers in thousands
6. Change Game - Exchanging quantities (e.g., 10 units for 1 ten)
7. Bank Game - Fetching specific quantities from the "bank"
8. Large Number Composition - Creating numbers into the thousands
9. Place Value Exploration - Understanding the value of each position

2. Stamp Game

Concrete representation of numbers and operations with color-coded quantities

Manipulative Components:

• Wooden stamps (color-coded for units, tens, hundreds, thousands)
• Wooden trays with compartments
• Small cards with mathematical symbols

Demonstrations (8):

1. Introduction to the Material - Exploring the stamps and their values
2. Number Formation - Making numbers using the stamps
3. Static Addition - Basic addition without carrying
4. Dynamic Addition - Addition with carrying/regrouping
5. Static Subtraction - Basic subtraction without borrowing
6. Dynamic Subtraction - Subtraction with borrowing
7. Multiplication - Using stamps for multiplication problems
8. Division - Using stamps for division problems

3. Dot Game (Dot Board)

Visual representation of place value and exchanging

Manipulative Components:

• Board with colored dots representing place values
• Small counters
• Recording paper

Demonstrations (4):

1. Introduction to the Board - Understanding the place value layout
2. Addition with Regrouping - Moving counters and recording exchanges
3. Subtraction with Borrowing - Moving counters and recording exchanges
4. Mixed Operations - Solving various problems with the dot board

4. Hierarchical Material

Introduction to larger numbers beyond thousands

Manipulative Components:

• Hierarchical cards showing values up to millions
• Wooden or cardboard material showing hierarchy of numbers
• Golden beads for representation (extension)

Demonstrations (5):

1. Introduction to Hierarchy - Understanding families of numbers
2. Reading Large Numbers - How to read numbers into millions
3. Writing Large Numbers - Proper notation for large numbers
4. Comparing Large Numbers - Determining greater/lesser values
5. Operations with Large Numbers - Extending operations to larger values

II. OPERATIONS

1. Bead Frames

A. Small Bead Frame

Concrete representation of four-digit numbers and operations

Manipulative Components:

• Frame with 10 wires and colored beads for units through thousands
• Recording sheets
• Pencils

Demonstrations (7):

1. Introduction to the Frame - Setting up and understanding the layout
2. Reading Numbers - Recording numbers shown on the frame
3. Setting Numbers - Placing beads to represent given numbers
4. Static Addition - Addition without carrying
5. Dynamic Addition - Addition with carrying
6. Static Subtraction - Subtraction without borrowing
7. Dynamic Subtraction - Subtraction with borrowing

B. Large Bead Frame

Extension for larger numbers and operations

Manipulative Components:

• Frame with more wires for numbers up to millions
• Recording sheets
• Pencils

Demonstrations (7):

1. Introduction to the Frame - Understanding the expanded layout
2. Reading Larger Numbers - Recording numbers from the frame
3. Setting Larger Numbers - Placing beads for larger quantities
4. Multi-Digit Addition - Addition with larger numbers
5. Multi-Digit Subtraction - Subtraction with larger numbers
6. Multiplication on the Frame - Using repeated addition approach
7. Division on the Frame - Using repeated subtraction approach

C. Flat Bead Frame

More abstract representation for operations

Manipulative Components:

• Flat frame with beads representing place values
• Recording paper

Demonstrations (5):

1. Introduction to the Material - Understanding layout and differences
2. Addition Technique - Performing addition on the flat frame
3. Subtraction Technique - Performing subtraction on the flat frame
4. Multiplication Applications - Using for more complex multiplication
5. Division Applications - Using for more complex division

2. Multiplication

A. Bead Bars and Board

Understanding multiplication as repeated addition

Manipulative Components:

• Colored bead bars representing quantities 1-10
• Multiplication board with grid
• Number tiles

Demonstrations (5):

1. Concept of Multiplication - Repeated addition visualization
2. Building the Multiplication Table - Creating tables with bead bars
3. Commutative Property - Demonstrating that 3×4 equals 4×3
4. Skip Counting Practice - Counting by multiples
5. Finding Products - Using the board to find multiplication answers

B. Multiplication Bead Board

Practice multiplication facts concretely

Manipulative Components:

• Board with numbered columns
• Colored beads for counting
• Number cards

Demonstrations (4):

1. Introduction to the Board - Setting up for multiplication
2. Finding Products - Using beads to find answers
3. Recording Results - Writing multiplication equations
4. Memorization Practice - Using board to practice facts

C. Multiplication Checkerboard

Concrete representation of multi-digit multiplication

Manipulative Components:

• Wooden checkerboard with colored squares for place values
• Number tiles
• Colored bead bars

Demonstrations (6):

1. Introduction to the Board - Understanding place value layout
2. Simple Multiplication - One-digit multiplier
3. Multi-Digit Multiplication - Two-digit multiplier
4. Partial Products - Understanding partial products concept
5. Recording the Algorithm - Connecting to traditional algorithm
6. Word Problems - Applying board to story problems

D. Decanomial Bead Box

Geometric visualization of multiplication

Manipulative Components:

• Box with colored bead squares and rectangles
• Printed decanomial layout

Demonstrations (3):

1. Building the Decanomial - Creating the visual square
2. Identifying Products - Finding specific products
3. Connection to Algebra - Relating to algebraic expressions

3. Division

A. Division Board with Beads

Concrete representation of division process

Manipulative Components:

• Division board
• Skittles (small pegs)
• Green division cups
• Colored beads

Demonstrations (5):

1. Introduction to Division Concept - Sharing equally
2. Simple Division - Division without remainders
3. Division with Remainders - Understanding leftovers
4. Recording Process - Writing division problems
5. Word Problems - Applying to real scenarios

B. Racks and Tubes (Test Tube Division)

Long division with concrete materials

Manipulative Components:

• Wooden rack with test tubes
• Colored beads representing place values
• Recording sheets

Demonstrations (7):

1. Introduction to the Material - Understanding the components
2. Single-Digit Divisor - Simple division process
3. Two-Digit Divisor - More complex division
4. Division with Remainders - Handling remainders
5. Division with Zeros in Quotient - Special case handling
6. Long Division Recording - Connection to algorithm
7. Word Problem Applications - Real-world applications

C. Division Charts

Visualization of division relationships

Manipulative Components:

• Division charts showing patterns
• Recording materials

Demonstrations (3):

1. Introduction to Charts - Understanding the layout
2. Finding Patterns - Discovering mathematical relationships
3. Division Fact Practice - Using charts for memorization

4. Decimal System

A. Decimal Board

Introduction to decimal numbers

Manipulative Components:

• Decimal board with place value markers
• Number tiles
• Green mat for operations

Demonstrations (6):

1. Introduction to Decimals - Understanding the decimal point
2. Reading Decimal Numbers - Proper terminology
3. Writing Decimal Numbers - Proper notation
4. Addition with Decimals - Aligning decimal points
5. Subtraction with Decimals - Maintaining place value
6. Comparing Decimal Numbers - Greater/lesser relationships

B. Decimal Fraction Material

Concrete representation of decimal relationships

Manipulative Components:

• Decimal cubes, flats, bars, and units
• Cards with decimal notation

Demonstrations (5):

1. Introduction to Material - Understanding representations
2. Equivalence - Finding equivalent decimal quantities
3. Decimal Operations - Performing operations with materials
4. Converting Fractions to Decimals - Using materials to convert
5. Real-Life Applications - Measuring and money connections

III. MEMORIZATION OF MATH FACTS

1. Addition Snake Game

Sequential addition practice

Manipulative Components:

• Colored bead bars
• Control chart
• Black and white bead bars for exchanges

Demonstrations (3):

1. Building the Snake - Creating sequence of bead bars
2. Counting and Exchanging - Trading for larger units
3. Recording Results - Writing addition equations

2. Subtraction Snake Game

Sequential subtraction practice

Manipulative Components:

• Colored bead bars
• Control chart

Demonstrations (3):

1. Building the Snake - Creating initial quantity
2. Removing Quantities - Subtraction process
3. Recording Results - Writing subtraction equations

3. Colored Bead Chains

A. Short Bead Chains

Sequences of squared numbers

Manipulative Components:

• Colored bead chains for squares (1², 2², 3², etc.)
• Number arrows
• Wooden squares for layout

Demonstrations (4):

1. Counting the Chain - Sequential counting
2. Square Numbers - Understanding the pattern
3. Laying Arrows - Marking multiples
4. Recording - Writing square number equations

B. Long Bead Chains

Sequences of cubed numbers

Manipulative Components:

• Colored bead chains for cubes (1³, 2³, 3³, etc.)
• Number arrows
• Wooden cubes for layout

Demonstrations (5):

1. Introduction to Chain - Understanding the structure
2. Counting the Chain - Sequential counting by multiples
3. Cube Numbers - Understanding the pattern
4. Laying Arrows - Marking multiples
5. Recording - Writing cube number equations

4. Math Fact Charts

Visual aids for memorization

Manipulative Components:

• Addition charts
• Subtraction charts
• Multiplication charts
• Division charts

Demonstrations (5):

1. Introduction to Charts - Understanding layouts
2. Finding Patterns - Discovering number relationships
3. Filling in Blanks - Practice with missing values
4. Math Fact Games - Interactive practice
5. Memorization Techniques - Strategies for learning facts

IV. FRACTIONS

1. Fraction Circles

Introduction to fraction concepts

Manipulative Components:

• Metal insets divided into equal parts
• Fraction notation cards

Demonstrations (7):

1. Introduction to Fractions - Understanding parts of a whole
2. Naming Fractions - Proper terminology
3. Equivalent Fractions - Finding equal values
4. Comparing Fractions - Greater/lesser relationships
5. Addition with Like Denominators - Adding same-sized parts
6. Subtraction with Like Denominators - Subtracting same-sized parts
7. Finding Least Common Multiple - For unlike denominators

2. Fraction Insets with Labels

More advanced fraction concepts

Manipulative Components:

• Fraction insets
• Labels for numerator/denominator
• Recording materials

Demonstrations (6):

1. Fraction Terminology - Understanding numerator/denominator
2. Mixed Numbers - Converting between improper fractions and mixed numbers
3. Adding Unlike Denominators - Finding common denominators
4. Subtracting Unlike Denominators - With borrowing when needed
5. Multiplication of Fractions - Using areas to demonstrate
6. Division of Fractions - Using reciprocals

3. Decimal Fraction Board

Connection between fractions and decimals

Manipulative Components:

• Decimal board
• Fraction pieces
• Recording materials

Demonstrations (4):

1. Fractions to Decimals - Converting process
2. Decimals to Fractions - Converting process
3. Operations with Decimals - Addition and subtraction
4. Word Problems - Real-world applications

V. GEOMETRY

1. Geometric Cabinet

Exploration of plane figures

Manipulative Components:

• Geometric cabinet with insets
• Cards with geometric shapes
• Recording materials

Demonstrations (6):

1. Introduction to Shapes - Names and characteristics
2. Classification of Shapes - Grouping by properties
3. Congruence and Similarity - Understanding relationships
4. Lines and Angles - Types and measurements
5. Polygons - Regular and irregular
6. Area Calculation - Finding space inside shapes

2. Constructive Triangles

Exploration of triangles and their relationships

Manipulative Components:

• Boxes of triangles in various sizes and colors
• Recording materials

Demonstrations (5):

1. Types of Triangles - By sides and angles
2. Building Shapes - Creating other polygons from triangles
3. Congruence - Demonstrating identical shapes
4. Equivalence - Same area, different shape
5. Pythagorean Theorem - Visual demonstration

3. Geometry Sticks

Exploring geometric concepts

Manipulative Components:

• Wooden sticks of various lengths
• Connectors
• Protractor and ruler

Demonstrations (5):

1. Building Polygons - Creating various shapes
2. Measuring Angles - Using protractor
3. Perimeter Calculation - Measuring around shapes
4. Area Exploration - Finding space inside
5. Geometric Constructions - Basic geometric constructs

4. Geometry Nomenclature Cards

Terminology for geometric concepts

Manipulative Components:

• Cards with geometric terms
• Cards with definitions
• Cards with illustrations

Demonstrations (3):

1. Matching Terms - Connecting terms, definitions, and images
2. Classification Activities - Grouping by properties
3. Reading and Research - Using cards for deeper study

VI. MEASUREMENT

1. Linear Measurement

Understanding standard units of length

Manipulative Components:

• Measuring tools (rulers, meter sticks, tape measures)
• Objects to measure
• Recording materials

Demonstrations (5):

1. Introduction to Units - Standard units of measurement
2. Measuring Technique - Proper use of tools
3. Estimation Practice - Guessing before measuring
4. Conversion Between Units - Relationship between units
5. Real-World Applications - Practical measuring tasks

2. Area Measurement

Understanding space inside boundaries

Manipulative Components:

• Square units (cm², m²)
• Grid paper
• Various shapes to measure

Demonstrations (4):

1. Introduction to Area - Concept of covered space
2. Measuring with Unit Squares - Counting squares
3. Area Formulas - Developing understanding of formulas
4. Area of Irregular Shapes - Approximation techniques

3. Volume Measurement

Understanding three-dimensional space

Manipulative Components:

• Cubic units (cm³, m³)
• Containers of various sizes
• Liquids for filling

Demonstrations (4):

1. Introduction to Volume - Concept of filled space
2. Measuring with Unit Cubes - Counting cubes
3. Volume Formulas - Developing understanding of formulas
4. Liquid Volume - Measuring liquids in standard units

4. Weight Measurement

Understanding mass and weight

Manipulative Components:

• Balance scales
• Standard weights
• Objects to weigh

Demonstrations (4):

1. Introduction to Weight - Concept of mass
2. Using Balance Scales - Proper technique
3. Standard Units - Grams, kilograms, etc.
4. Estimation and Comparison - Relative weights

5. Time Measurement

Understanding units of time

Manipulative Components:

• Clock materials
• Calendar materials
• Timelines

Demonstrations (5):

1. Reading Analog Clocks - Hours, minutes, seconds
2. Time Duration - Calculating elapsed time
3. Calendar Use - Days, weeks, months, years
4. Timelines - Sequential events
5. Time Problem Solving - Word problems with time

VII. DATA AND GRAPHING

1. Data Collection

Gathering and organizing information

Manipulative Components:

• Tally sheets
• Recording materials
• Objects for counting and sorting

Demonstrations (3):

1. Collecting Data - Gathering information systematically
2. Organizing Data - Creating logical categories
3. Tally Marks - Efficient counting technique

2. Graphing

Visual representation of data

Manipulative Components:

• Graph paper
• Colored pencils
• Rulers

Demonstrations (5):

1. Pictographs - Simple visual representations
2. Bar Graphs - Creating and reading
3. Line Graphs - Showing changes over time
4. Circle Graphs - Showing parts of a whole
5. Interpreting Graphs - Drawing conclusions from data

3. Probability

Introduction to chance concepts

Manipulative Components:

• Dice, spinners, coins
• Recording materials

Demonstrations (4):

1. Fair and Unfair - Understanding equal chance
2. Simple Experiments - Conducting probability trials
3. Recording Results - Tallying outcomes
4. Predicting Outcomes - Using data to make predictions

IMPLEMENTATION GUIDELINES

Progression of Learning

1. Concrete to Abstract - Always begin with hands-on materials before moving to paper
2. Isolation of Concepts - Present one difficulty at a time
3. Three-Period Lesson - Introduction, recognition, recall
4. Freedom within Limits - Allow choice within appropriate developmental level
5. Individual Pacing - Progress based on mastery, not age

Assessment Practices

1. Observation - Teacher observation of material use
2. Portfolios - Collection of student work
3. Demonstrations - Student explanations of concepts
4. Written Work - Progression to paper when ready
5. Follow-up Work - Extensions and applications

Key Integration Points

1. Geometry with Art - Geometric drawing and design
2. Measurement with Practical Life - Cooking, construction projects
3. Data with Cultural Studies - Population, climate graphs
4. Operations with Science - Scientific notation, measurement
5. Fractions with Music - Note values and timing

This curriculum map provides a comprehensive overview of the Montessori mathematics program for ages 6-9, including all major manipulatives and their associated demonstrations. The program is designed to foster deep conceptual understanding through concrete experiences before moving to abstract representations.

Montessori Mathematics Curriculum Map (Ages 9-12)

1. Numeration & Operations

1.1 Hierarchical Material (Golden Bead Material)

Purpose: Reinforcement of decimal system and operations with whole numbers

Manipulatives:

• Golden Bead Material (units, tens, hundreds, thousands)
• Number Cards (1-9000)
• Place Value Trays

Demonstrations (10):

1. Review of decimal system structure
2. Reading and writing large numbers (up to millions)
3. Formation of large numbers with beads and cards
4. Addition with the golden beads (with and without exchanging)
5. Subtraction with the golden beads (with and without exchanging)
6. Multiplication with the golden beads
7. Division with the golden beads
8. Word problems using golden beads
9. Transition to abstract notation with golden beads
10. Relationship between operations using golden beads

1.2 Stamp Game

Purpose: Abstraction of operations with whole numbers

Manipulatives:

• Stamp Game (colored stamps for units, tens, hundreds, thousands)
• Operation symbols
• Recording sheets

Demonstrations (8):

1. Review of stamp game components
2. Advanced addition with stamp game (multiple addends)
3. Advanced subtraction with stamp game (larger numbers)
4. Multiplication with stamp game (multi-digit multipliers)
5. Division with stamp game (complete and partial quotients)
6. Combined operations
7. Word problems using stamp game
8. Recording formal algorithms alongside material

1.3 Bead Frames

Small Bead Frame Purpose: Abstract computation with numbers up to 9999

Manipulatives:

• Small Bead Frame
• Recording sheets

Demonstrations (7):

1. Review of bead frame structure
2. Addition with regrouping
3. Subtraction with regrouping
4. Multiplication (one and two-digit multipliers)
5. Short division
6. Recording formal algorithms
7. Problem-solving strategies

Large Bead Frame Purpose: Abstract computation with numbers up to millions

Manipulatives:

• Large Bead Frame
• Recording sheets

Demonstrations (5):

1. Reading and writing large numbers
2. Addition with large numbers
3. Subtraction with large numbers
4. Multiplication with multi-digit multipliers
5. Advanced problem-solving

Flat Bead Frame Purpose: Advanced operations and preparation for algebra

Manipulatives:

• Flat Bead Frame
• Recording sheets

Demonstrations (4):

1. Structure and use of the flat bead frame
2. Multi-digit operations
3. Equation solving preparation
4. Pattern recognition

1.4 Checkerboard

Purpose: Geometric visualization of multiplication

Manipulatives:

• Multiplication Checkerboard
• Bead bars
• Number tiles
• Recording sheets

Demonstrations (6):

1. Setting up the checkerboard
2. Multiplication of two-digit numbers
3. Multiplication of multi-digit numbers
4. Multiplication with decimals
5. Relationship to distributive property
6. Algebraic thinking preparation

1.5 Division Materials

Test Tubes Division Purpose: Concrete representation of division process

Manipulatives:

• Test Tube Division Board
• Skittles
• Number cards
• Recording sheets

Demonstrations (5):

1. Division setup with dividend and divisor
2. Long division process step-by-step
3. Division with remainders
4. Verifying results with multiplication
5. Word problems involving division

Racks and Tubes Purpose: Long division with multi-digit divisors

Manipulatives:

• Racks and Tubes Division Board
• Small green skittles
• Number cards
• Recording sheets

Demonstrations (4):

1. Setup for multi-digit division
2. Complete division process
3. Division with zeros in quotient
4. Advanced division problems

2. Fractions

2.1 Fraction Circles

Purpose: Visualization of fraction relationships

Manipulatives:

• Fraction Circles (insets divided into halves through tenths)
• Fraction labels

Demonstrations (8):

1. Review of fraction concepts
2. Equivalent fractions using circles
3. Comparing fractions with the same denominator
4. Comparing fractions with different denominators
5. Addition of fractions with like denominators
6. Subtraction of fractions with like denominators
7. Finding common denominators
8. Mixed numbers and improper fractions

2.2 Fraction Insets

Purpose: Deeper understanding of fraction relationships

Manipulatives:

• Metal fraction insets
• Control charts

Demonstrations (6):

1. Representation of fractions
2. Finding equivalent fractions
3. Simplifying fractions
4. Finding common denominators
5. Addition with unlike denominators
6. Subtraction with unlike denominators

2.3 Fraction Operations Materials

Addition and Subtraction Strip Board Purpose: Operations with fractions

Manipulatives:

• Fraction Addition/Subtraction Board
• Fraction strips
• Recording sheets

Demonstrations (4):

1. Addition with like denominators
2. Addition with unlike denominators
3. Subtraction with like denominators
4. Subtraction with unlike denominators

Multiplication and Division Board Purpose: Operations with fractions

Manipulatives:

• Fraction Multiplication Board
• Fraction Division Board
• Fraction pieces
• Recording sheets

Demonstrations (8):

1. Multiplication of fraction by whole number
2. Multiplication of fraction by fraction
3. Product of mixed numbers
4. Simplifying products
5. Division of fractions using board
6. Reciprocals
7. Division with mixed numbers
8. Converting between improper fractions and mixed numbers

3. Decimal System

3.1 Decimal Board

Purpose: Understanding decimals and place value

Manipulatives:

• Decimal Board
• Decimal number cards
• Colored beads

Demonstrations (9):

1. Introduction to decimal notation
2. Reading and writing decimals
3. Converting between fractions and decimals
4. Decimal place value
5. Comparing decimals
6. Addition of decimals
7. Subtraction of decimals
8. Multiplication of decimals
9. Division of decimals

3.2 Decimal Operations

Decimal Checkerboard Purpose: Multiplication with decimals

Manipulatives:

• Decimal Checkerboard
• Decimal beads
• Recording sheets

Demonstrations (5):

1. Setting up decimal multiplication
2. Multiplying with one decimal place
3. Multiplying with multiple decimal places
4. Placing the decimal point in the product
5. Word problems involving decimal multiplication

4. Powers and Roots

4.1 Squaring and Cubing

Bead Chains Purpose: Concrete experience with squares and cubes

Manipulatives:

• Bead Chains (short and long)
• Square and cube number cards
• Recording materials

Demonstrations (8):

1. Review of short bead chains (1-10)
2. Skip counting with chains
3. Long bead chains (1-10)
4. Relationships between numbers and their squares
5. Relationships between numbers and their cubes
6. Recording squares and cubes
7. Finding square roots with chains
8. Finding cube roots with chains

Squaring and Cubing Materials Purpose: Geometric representation of powers

Manipulatives:

• Wooden squares and cubes
• Power of 2 material
• Power of 3 material

Demonstrations (6):

1. Building squares geometrically
2. Building cubes geometrically
3. Numerical pattern of squares
4. Numerical pattern of cubes
5. Binomial expansions
6. Trinomial expansions

4.2 Pythagoras

Pythagorean Theorem Materials Purpose: Discover relationships in right triangles

Manipulatives:

• Pythagorean Boards
• Colored squares
• Metal right triangles

Demonstrations (4):

1. Concrete proof of Pythagorean Theorem
2. Finding hypotenuse using theorem
3. Finding sides using theorem
4. Applications in real-world problems

5. Measurement

5.1 Linear Measurement

Purpose: Understanding standard units and conversions

Manipulatives:

• Measurement chains
• Rulers (metric and customary)
• Conversion charts

Demonstrations (5):

1. Metric system relationships
2. Customary system relationships
3. Conversion between systems
4. Perimeter calculation
5. Problem-solving with linear measurement

5.2 Area Measurement

Purpose: Understanding area concepts

Manipulatives:

• Area material with squares and rectangles
• Unit squares
• Recording materials

Demonstrations (6):

1. Concept of area as square units
2. Area of rectangle and square
3. Area of triangles
4. Area of parallelograms
5. Area of irregular shapes
6. Real-world area problems

5.3 Volume Measurement

Purpose: Understanding three-dimensional measurement

Manipulatives:

• Volume cubes
• Prisms and cylinders
• Graduated containers

Demonstrations (5):

1. Concept of volume as cubic units
2. Volume of cube and rectangular prism
3. Volume of cylinder
4. Relationship between capacity and volume
5. Real-world volume problems

6. Data and Probability

6.1 Data Analysis

Purpose: Organizing and interpreting data

Manipulatives:

• Graph paper
• Colored pencils
• Data cards

Demonstrations (6):

1. Creating tally charts
2. Bar graphs construction and interpretation
3. Line graphs construction and interpretation
4. Pie charts construction and interpretation
5. Finding mean, median, mode, and range
6. Drawing conclusions from data

6.2 Probability

Purpose: Understanding chance and likelihood

Manipulatives:

• Colored cubes
• Spinners
• Dice
• Probability boards

Demonstrations (5):

1. Probability as a fraction
2. Experimental vs. theoretical probability
3. Sample spaces
4. Compound events
5. Fair and unfair games

7. Geometry

7.1 Constructive Triangles

Purpose: Exploring properties of triangles

Manipulatives:

• Constructive triangle boxes
• Recording materials

Demonstrations (7):

1. Review of triangle types
2. Equivalence of triangles
3. Congruence
4. Constructing similar triangles
5. Relationship between triangles and quadrilaterals
6. Area of triangles
7. Pythagorean applications

7.2 Geometric Solids

Purpose: Exploring 3D shapes

Manipulatives:

• Geometric solids
• Nets of solids
• Bases and height rods

Demonstrations (6):

1. Properties of 3D shapes
2. Surface area calculation
3. Volume calculation
4. Relationship between 2D and 3D shapes
5. Nets and development of solids
6. Classification of polyhedra

7.3 Geometric Constructions

Purpose: Precision drawing of geometric figures

Manipulatives:

• Compass
• Straightedge
• Protractor
• Construction paper

Demonstrations (8):

1. Bisecting lines
2. Bisecting angles
3. Constructing perpendicular lines
4. Constructing parallel lines
5. Constructing regular polygons
6. Circle constructions
7. Inscribed and circumscribed figures
8. Applications of constructions

8. Pre-Algebra and Algebraic Thinking

8.1 Binomial and Trinomial Cubes

Purpose: Pattern recognition and algebraic thinking

Manipulatives:

• Binomial Cube
• Trinomial Cube

Demonstrations (4):

1. Building the binomial cube
2. Algebraic representation (a+b)³
3. Building the trinomial cube
4. Algebraic representation (a+b+c)³

8.2 Algebraic Pegboard

Purpose: Concrete representation of algebraic concepts

Manipulatives:

• Pegboard
• Colored pegs
• Equation cards

Demonstrations (6):

1. Representing variables
2. Building linear equations
3. Solving for unknowns
4. Representing inequalities
5. Systems of equations
6. Patterns and functions

8.3 Equation-Solving Materials

Purpose: Balance approach to equations

Manipulatives:

• Equation trays
• Positive/negative counters
• Variable cards

Demonstrations (5):

1. Concept of equivalence
2. Solving one-step equations
3. Solving two-step equations
4. Equations with variables on both sides
5. Word problems with unknowns

9. Number Theory

9.1 Multiples and Factors

Purpose: Understanding number relationships

Manipulatives:

• Pegboard
• Colored pegs
• Number cards

Demonstrations (7):

1. Finding multiples
2. Finding factors
3. Greatest common factor
4. Least common multiple
5. Prime factorization
6. Prime and composite numbers
7. Applications in fraction operations

9.2 Divisibility

Purpose: Understanding divisibility rules

Manipulatives:

• Divisibility charts
• Number cards
• Recording materials

Demonstrations (5):

1. Divisibility rules for 2, 5, 10
2. Divisibility rules for 3, 9
3. Divisibility rules for 4, 6, 8
4. Divisibility rules for 7, 11
5. Applications in factoring and fractions

10. Advanced Topics

10.1 Ratio and Proportion

Purpose: Understanding relationships between quantities

Manipulatives:

• Proportion boards
• Ratio cards
• Color-coded counters

Demonstrations (6):

1. Concept of ratio
2. Writing ratios in different forms
3. Equivalent ratios
4. Setting up proportions
5. Solving proportions
6. Real-world applications

10.2 Percentages

Purpose: Understanding parts of 100

Manipulatives:

• Percentage board
• Percentage cards
• Hundred squares

Demonstrations (7):

1. Concept of percentage
2. Converting between fractions and percentages
3. Converting between decimals and percentages
4. Finding the percentage of a number
5. Finding what percentage one number is of another
6. Finding the whole from a percentage
7. Real-world percentage problems

10.3 Negative Numbers

Purpose: Extending the number system

Manipulatives:

• Number line
• Positive/negative counters
• Temperature thermometer model

Demonstrations (6):

1. Concept of negative numbers
2. Addition with negative numbers
3. Subtraction with negative numbers
4. Multiplication with negative numbers
5. Division with negative numbers
6. Real-world applications of negative numbers

Integration Activities

Throughout the 9-12 curriculum, mathematics is integrated with other subject areas through:

1. Research Projects - Using mathematical concepts in cultural studies
2. Going Out - Real-world application of mathematical concepts
3. Great Lessons - Connection of mathematics to the universe and human development
4. Cosmic Education - Mathematics as a language of the universe

Assessment Approaches

1. Observation - Teacher notes on work with materials
2. Portfolios - Collection of student work
3. Journals - Student reflection on mathematical thinking
4. Presentations - Student explanations of mathematical concepts
5. Projects - Application of multiple concepts
6. Self-assessment - Student tracking of progress

Progression Throughout Upper Elementary

9-Year-Olds (4th Grade Equivalent)

• Solidify operations with whole numbers
• Begin fraction operations
• Explore decimal concepts
• Basic geometric concepts
• Introduction to data analysis

10-Year-Olds (5th Grade Equivalent)

• Master fraction and decimal operations
• Explore ratio and proportion
• Deepen geometric understanding
• Begin algebraic thinking
• Develop problem-solving strategies

11-12-Year-Olds (6th Grade Equivalent)

• Negative numbers and integers
• Pre-algebraic concepts
• Advanced geometry
• Percentage applications
• Statistical thinking

Summary of Manipulatives by Category

1. Numeration & Operations: Golden Beads, Stamp Game, Bead Frames, Checkerboard, Test Tube Division, Racks and Tubes
2. Fractions: Fraction Circles, Fraction Insets, Addition/Subtraction Strip Board, Multiplication/Division Board
3. Decimals: Decimal Board, Decimal Checkerboard
4. Powers & Roots: Bead Chains, Wooden Squares and Cubes, Pythagorean Boards
5. Measurement: Measurement Chains, Area Materials, Volume Cubes
6. Data & Probability: Graph Materials, Probability Tools
7. Geometry: Constructive Triangles, Geometric Solids, Construction Tools
8. Algebraic Thinking: Binomial/Trinomial Cubes, Algebraic Pegboard, Equation Materials
9. Number Theory: Pegboards, Divisibility Charts
10. Advanced Topics: Proportion Boards, Percentage Materials, Negative Number Line

Total number of unique manipulative systems: 33 Total number of demonstrations across all systems: 189