Table of Contents
 Introduction
 Numeration
 Introduction
 Quantities in the Decimal System
 Numerals (Symbols)
 Union of Quantities and Numerals (Symbols)
 Additional Exercises in Numeration
 The Hundred Board
 The Seguin Boards
 Introduction to Operations Using the Change Game
 Golden Bead Chains
 Hierarchical Material
 Introduction to Other Mathematical Material
 Introduction to Memorization
 Introduction
 Addition
 Stamp Game Addition
 Memorization
 Strip Board (including Doubles)
 Games and Explorations using Bead Bars
 Addition Charts and Combination Card Exercises
 Passage from Chart I to Chart II
 Passage from Chart II to Chart III (the Whole Chart)
 Passage from Chart III to Chart IV (the Half Chart)
 Passage from Chart IV to Chart V (the Simplified Tables)
 The Bingo Game for Addition (using Chart VI)
 Bead Frame Addition
 More Memorization Exercises
 Further Explorations using Bead Bars
 Special Cases
 0 – Calculating the Sum
 1 – Calculating the Second Addend
 2 – Calculating the First Addend
 3 – Inverse of Case Zero
 4 – Inverse of 1st Case, Calculating the Second Addend
 5 – Inverse of 2nd Case, Calculating the First Addend
 6 – Calculating the First and Second Addends
 Collective Activity:
 Individual Work:
 Word Problems
 Multiplication
 Stamp Game Multiplication
 Memorization Exercise
 Multiplication Boards (Bead Boards)
 Introduction and List of Materials:
 Initial Presentation:
 Multiplication Booklets
 Combination Cards
 Skip Counting (Linear Counting)
 List of Materials
 Initial Presentation
 Short Chain:
 Long Chain:
 Games of Comparison
 Bead Bar Multiplication
 Multiplication Charts and Combination Cards
 Passage from Chart I to Chart II
 Passage from Chart II to Chart III (the Whole Chart)
 Passage from Chart III to Chart IV (the Half Chart)
 The Bingo Game of Multiplication (using Chart V)
 Exercise
 Exercise
 Exercise
 Group Games:
 Multiplication By 10, 100, 1000
 Presentation:
 By ten
 By one hundred
 By one thousand
 Multiplication Boards (Bead Boards)
 Checkerboard – Geometrical Analysis of Multiplication
 Introduction to the Checkerboard
 Presentation:
 Games:
 Multiplication with the Checkerboard
 1st level
 2nd levelSmall Multiplication
 3rd levelPartial Products (this passage can be skipped)
 4th levelMental Carrying Over
 Multiplication and Drawing
 Introduction to the Checkerboard
 Bead Frame Multiplication
 Small Bead Frame
 Multiplication By 10, 100, 1000
 Multiplication with a OneDigit Multiplier
 Large Bead FrameMultipliers of 2 or More Digits
 The Whole Product
 Partial Products
 Horizontal Golden Bead Frame
 The Whole Product
 Partial Products
 Carrying Mentally
 Small Bead Frame
 More Memorization Exercises
 The Snake Game
 Various Ways of Constructing a Product
 Small Multiplication
 Inverse Products
 Construction of a Square
 Multiplication of a Sum
 Analysis of the SquaresBinomial
 Analysis of the SquaresTrinomial
 Passage From One Square to a Succeeding Square
 Passage From One Square to a NonSuccessive Square
 Skip Count Chains – Further Exploration
 List of Materials
 Construction of Geometric Figures
 Decanomial (a polynomial having ten terms) & The Construction of Chart I
 List of Materials
 Vertical Presentation
 Horizontal Presentation
 Angular Presentation
 Commuted Decanomial
 Part One
 Part TwoBuilding the Tower
 Variation on the 1st Method
 2nd Method
 Part ThreeDecomposition of the Tower
 Skip Counting ChainsFurther Explorations
 Decanomial (a polynomial having ten terms) & The Construction of Chart I
 Numeric Decanomial
 Special Cases
 0 Calculate the product
 1 Calculate the Multiplier
 2 Calculate the Multiplicand
 3 Inverse of Case Zero – Calculate the Product
 4 Inverse of The First Case – Calculate the Multiplier
 5 Inverse of the Second Case – Calculate the Multiplicand
 6 Calculate the Multiplier and the Multiplicand
 Activity: The Bank Game
 Word Problems
 Subtraction
 Stamp Game Subtraction
 Memorization Exercises
 Strip Board
 Introduction and List of Materials
 Initial Presentation
 Subtraction Booklets
 Combination Cards
 Decomposition of a Number
 Decomposition of a Number
with Zero as the Subtrahend
 The Snake Game
 Subtraction Charts and Combination Card Exercises
 Passage From Chart I to Chart II
 The Bingo Game of Subtraction (using Chart III)
 Exercise:
 Exercise:
 Exercise:
 Group Games
 Strip Board
 Bead Frame Subtraction
 Static Subtraction
 Dynamic Subtraction
 Calculating the Difference
a little more Abstractly Presentation:
 Games:
 More Memorization Exercises (Special Cases)
 0 Calculating the Difference
 1 Calculating the Subtrahend
 2 Calculating the Minuend
 3 Inverse of Case Zero – Calculating the Difference
 4 Inverse of the 1st Case Calculating the Subtrahend
 5 Inverse of the 2nd Case Calculating the Minuend
 6 Calculating the Minuend and the Subtrahend
 Collective Activity:
 Individual Activity:
 Word Problems
 Division
 Stamp Game Division – Single Digit
 Memorization Exercises
 Division Boards (Bead Boards)
 Introduction and List of Materials
 Initial Presentation
 Preparation for Presentation
 Presentation
 Division Booklets
 Division Charts and Combination Cards
 Chart I
 Bingo Game (Chart II)
 Exercise A:
 Exercise B.
 Exercise C.
 Group Game 1.
 Group Game 2.
 Division Boards (Bead Boards)
 Division By More Than One Digit
 Double Digit Division
 Intro. to Double Digit Division (on The Change Game)
 Double Digit Division on The Stamp Game
 Triple Digit Division
 Intro. to Triple Digit Division (on The Change Game)
 Triple Digit Division on The Stamp Game
 Double Digit Division
 Division Involving Zeros
 Division with Zero in the Dividend
 Division with Zero in the Divisor
 Group Division
 Single Digit Group Division
 Double Digit Group Division
 More Memorization Exercises
 Special Cases
 0 Calculate the quotient
 1 Calculate the Divisor
 2 Calculate the Dividend
 3 Inverse of Case Zero: Calculate the Quotient
 4 Inverse of The First Case: Calculate the Divisor
 5 Inverse of the Second Case: Calculate the Dividend
 6 Calculate the Divisor and the Dividend
 Search for Quotients
 Prime Numbers
 Special Cases
 Division With Racks & Tubes
 Introduction
 Materials: racks and tubes:
 Small Division (1digit divisor, 4digit dividend)
 1st level
 2nd level
 3rd Level – Group Division
 Big Division (2 or more digits in the divisor)
 1st level:
 2nd level:
 3rd levelGroup Division with a two digit divisor
 Long Division
 Division with Zero in the
Divisor
 Word Problems
 From Combination Cards
 Distributive vs. Group Division
 Divisibility
 1. Introduction
 2. Div. by 2
 3. Div. by 4
 4. Div. by 5
 5. Div. by 25
 6. Div. by 9
 Synthesis of Multiples & Divisibility
 Fractions
 Introduction
 Understanding What Fractions Are
 Understanding the Concept of Fractions
 Introducing Fractions Using the Fraction Skittles
 Introducing Fractions Using the Fraction Circles
 Putting Fractions in Order from a Whole to a Tenth
 Matching Fractions from a Whole to a Tenth
 Understanding the Parts of Fractions
 Writing Fractions and Finding Equivalencies
 Understanding Written Fractions
 Matching Fraction Tickets to Fraction Circles
 Constructing Fraction Circles to Match Fraction Tickets
 Finding Fraction Equivalencies
 Adding and Subtracting Fractions
 Adding Fractions with the Same Denominator That Add Up to One or Less
 Adding Fractions with the Same Denominator That Add Up to More Than One
 Introduction to Adding Fractions With Different Denominators
 Subtracting Fractions with the Same Denominator
 Introduction to Subtracting Fractions with a Different Denominator
 Multiplying and Dividing Fractions
 Multiplying Fractions By Whole Numbers
 Dividing Fractions By Whole Numbers
 Dividing Fractions By Whole Numbers when Equivalent Fractions Must Be Made First
 Geometry

 Chapter One – Sensorial Exploration of Shapes
 Introduction to Geometry
 The Geometry Cabinet
 Introduction
 Presentation tray
 Triangles
 Rectangles
 Regular Polygons
 Circles
 Other figures
 Constructive Triangles – First Series
 First box
 Second box – Construction of Quadrilaterals
 Second box – Heads and Tails
 Trapezoid
 Third Box
 Chapter Two – Knowledge of Plane Figures and Details
 Introduction
 Fundamental Concepts
 Lines
 Types of lines
 Parts of a straight line
 Positions of a straight line
 Horizontal line – curved or straight?
 Straight line in a horizontal plane
 Two straight lines lying in a plane – coplanar lines
 Divergent and Convergent Lines
 Oblique and perpendicular lines
 Two straight lines crossed by a transversal
 Two parallel straight lines crossed by a transversal
 Angles
 Types of Angles and the Parts of an Angle
 Study of angles
 Vertical or Opposite Angles
 Measurement of Angles
 Presentation of the Montessori protractor
 Use of the Protractor
 Operations with Angles
 Addition
 Subtraction
 Multiplication
 Division – Bisecting an Angle
 Exercises in measurement and drawing of angles
 Operations with Angles
 Convex and reflex angles
 Research in the environment
 A new definition of an angle
 Plane Figures – Difference Between Closed Curve Figures and Polygons
 Classification Exercises
 Classification of closed curve regions and polygons
 A sensorial classification of convex and concave (reentrant)
 Closed curve regions/polygons – convex/concave
 Classification Exercises
 Polygons – Triangles
 Triangles – Classification
 Triangles – Parts of the triangle
 Special nomenclature of the rightangled triangle
 Study of Quadrilaterals
 Classification
 Quadrilaterals – Types of trapezoids
 Quadrilaterals – Classification according to set theory
 Parts of the Quadrilateral
 Polygons With More Than 4 Sides
 Parts of Polygons
 Comparative Examination of Polygons
 Study of the apothem
 The polygon in relation to the first circle
 The polygon in relation to the 2nd circle
 The polygon in relation to the 1st and 2nd circles simultaneously
 From Irregular to Regular Polygons
 Construction of Polygons
 Circle – Level One
 Circle and its parts
 Relationship between a circumference and a straight line
 Relationship between two circumferences
 Relationship between a circumference and a straight line
 Relationship between two circumferences
 Chapter Three – Congruency, Similarity and Equivalence
 Chapter One – Sensorial Exploration of Shapes

 Introduction to the Material – Squares
 Presentation of the Material – Triangles
 Presentation of the Concepts
 Similarity
 Equivalence
 Concepts in Action – Constructive Triangles – Second Series
 Triangular box
 Large hexagonal box
 Smaller hexagonal box
 Relationship between T1 and T2
 Difference and ratios between similar figures
 Sensorial demonstration of the difference between H1 and H2
 Equivalence between two rhombi
 Equivalence between the trapezoid and T2
 Ratio between circumscribing and inscribed figures
