# Elementary Mathematics

# Overview

Students who study math gain the ability to better form and deploy complex knowledge — in general. Whether its solving a difficult engineering problem or solving a thorny personal problem, one needs to be able to: identify and separate different operative factors, organize these factors, bring different elements of knowledge to bear on the problem in a systematic way, and pressure-test one’s thinking for mistakes, problems, and weaknesses. These are the skills that our elementary programs impart when teaching mathematics.

And so the key outcomes of our math education are two-fold:

*A deep understanding of the way numbers work and relate to each other, as well as the ability to manipulate them quickly and easily**The acquisition of cognitive powers and habits that can apply to all other areas of one’s life.*

Our students, in mathematics, gain:

*Explicit competence in foundational domains of mathematics (see below for a content list), along with an intuitive number sense.**The ability to build new knowledge from previous knowledge (by rigorous mathematical inference).**The ability to organize knowledge into an integrated, hierarchical structure (e.g. logical chains).**The ability to deploy highly abstract knowledge in solving particular problems (e.g. word problems, heuristics, models).**The ability to participate in discussion and debate with others using clear, step-wise reasoning and truths (such as measurements and prior axioms).*

# Program Elements

- Hands-On
Our math curriculum begins with hands-on materials that foster an intuition of quantity, place value, and geometry. By manipulating these scientifically designed examples of abstract ideas, students build a library of experiences that develop into mental models of mathematical principles. A set of developmentally and mathematically refined hands-on Montessori learning materials, along with unique learning materials developed by our pedagogy team, forms the backbone of the math curriculum.

Each learning material not only concretizes but isolates an aspect of mathematics. The cleanliness and objectivity of math is manifest in the learning materials, and reinforced in the more abstract presentations. Definitions are clear and concise, statements are correct and precise, and problems are solved by laying bare every step. The precision and isolation in the materials allows students to practice one skill or concept at a time, repeated as often as needed for mastery.

- Concrete to Abstract
- Intentionally Sequenced
- Mastery Based

# Lower Elementary Curriculum

### Numbers and Operations

**Great Stories***The Origin Story of Our Number System*

**The Decimal System***Composing NumbersPlace Value to the Millions*

**Intro to Math Operations***AdditionSubtractionMultiplicationDivision*

**Memorization of Math Facts**

*Strategies for Mental Math*

Fact Fluency

Fact Fluency

**Intro to Multi-Digit Multiplication***Concept of Pace Value in Long MultiplicationGeometric and Cross Multiplication*

**Laws of Multiplication***Commutative LawDistributive Law*

**Long Division***Concept of Place Value in Long DivisionDistributive Division*

**Divisibility***Divisibility by 2Divisibility by 5Divisibility by 25*

**Multiples and Factors***Multiples through 100Least Common MultipleFactors*

**Squares and Cubes of Numbers***Concept of Squares and CubesNotation of Squares and CubesOperations using Bead MaterialBuilding the Decanomial*

**Intro to Fractions***Concept and Notation of FractionsConnection Between Fractions and DivisionFraction Equivalence and ComparisonOperations with Like Denominator*

**Intro to Decimal Fractions***Concept and Notation of Decimal FractionsDecimal HierarchiesDecimal Fraction Comparison*

**Word Problems**

*Strategies for Understanding*

Applied to All Math Topics

Applied to All Math Topics

**Money***Counting CurrencyConnection to DecimalsAdding and Subtracting Money*

**Temperature***Concept of Measuring TemperatureReading Weather Reports, Tracking*

**Data and Graphing**

*Concept of a Graph*

Collecting Data

Pictographs

Bar Graphs

Collecting Data

Pictographs

Bar Graphs

### Geometry

**Great Stories***The Harpedonaptae and Measuring the Land of Egypt*

*Geometric Concepts**Plane FiguresGeometric ConstructionsDesigning with InsetsPoint to Solid Symmetry*

**Lines***Parts of a LineTypes of LinesRelationships Between Two Lines*

**Angles***Parts of an AngleTypes of AnglesPairs of Angles*

**Polygons***Parts of a PolygonPolygon ClassificationRegular and Irregular Properties of a Triangle, Quadrilaterals*

**The Circle**

*Parts of a Circle*

Relationship of Triangles and Circles

Relationship of Triangles and Circles

**Geometrical Relationships**

*Congruence*

Similarity

Equivalence

Similarity

Equivalence

**Equivalence Study**

*Triangle and Rectangle*

Rhombus and Rectangle

Trapezoid and Rectangle

Regular Decagon

Triangle with Same Base and Height

Rhombus and Rectangle

Trapezoid and Rectangle

Regular Decagon

Triangle with Same Base and Height

**Transformations***Concepts of ReflectionRotation and TranslationTessellations*

**Length***Concept of Measuring by UnitMetric and US Customary Systems*

**Perimeter and Area**

*Concepts of Perimeter and Area*

Relationship of Area and Multiplication

Deriving the Area Formula for the Rectangle and Parallelogram

Relationship of Area and Multiplication

Deriving the Area Formula for the Rectangle and Parallelogram

**Solids**, **Surface Area, and Volume***Parts of a Solid Solid ClassificationThree Important DimensionsConcept of a NetEquivalence with Liquid Volume*

**Volume and Weight Measurement***Concept of Measuring Liquid VolumeMeasuring Spoons, Cups, Graduated CylindersConcept of Measuring Weight*

# Upper Elementary Curriculum

### Numbers and Operations

**Multi-Digit Multiplication**

*Accurate Computation on Paper*

Strategies for Mental Math Estimation

Strategies for Mental Math Estimation

**Multi-Digit Division***Group Division: Stamp GameComputation with and without RemaindersStrategies for Mental Math Estimation*

**Divisibility***Divisibility by 2, 5, and 25Divisibility by 4 and 8 Divisibility by 3, 6, and 9Divisibility by 11 and 7*

**Multiples and Factors***Least Common MultipleFactors and Prime FactorsGreatest Common Factor*

**Negative Integers***Comparing Negative IntegersNegative Integer Operations*

*Non-Decimal Base Systems**Historical BasisCounting and OperationsConversionsBase Systems in Time and Angles*

**Fractions***Fractions in Lowest TermsReciprocalsFraction Operations with Unlike DenominatorsMixed Number Operations*

**Decimal Fractions***Calculation with MaterialsCentesimal Frame Conversion of Fractions to DecimalsFraction and Decimal Conversion on Paper*

**Percents***Concept and DefinitionConversions of Fractions, Decimals and PercentsPercents in Area, Scale, and Finance*

*Ratio**Concept and Connection to FractionsRatios in Unit Rates, Figuring Simple Interest, and Scale Drawings*

### Geometry

**Geometric Concepts***Constructions Using a Straight Edge and CompassGeometry in Art (Perspective, Symmetry, Transformation)Geometry in Engineering (Bridges, Arches)*

**Advanced Study of Geometric Figures***Construction and NotationAngle Measurement and EstimationSum of the Degrees of Interior Angles of a PolygonCircumference and Pi*

**Congruence, Similarity, Equivalence***Side Lengths and Angle Measures of Congruent FiguresSide Lengths of Similar FiguresSimilar Figures and Indirect Measurement*

**Theorems and Proofs***Pythagorean Theorem Statement and ConceptEuclid's ProofFormula for the Length of the HypotenuseConverse of the Pythagorean Theorem: Is it a Right Triangle?*

**Transformations***Reflection, Rotation, TranslationTransformations on a Coordinate Plane*

**Area***Relationship Between Perimeter and AreaApply Fractions and Decimals to Figuring AreaArea of a Triangle, Trapezoid, Rhombus Area of Compound Figures*

**Solids, Surface Area, and Volume***Platonic Solids, PolyhedraSurface Area of Cubes, Prisms, Pyramids, and CylindersVolume of Cubes, Prisms, Pyramids, and Cylinders*

### Applied Mathematics

**Research and Timelines***Timeline and Research of the History of Human Mathematical DiscoveryBuilding Models of Great Geometric and Mathematical Concepts*

**Accounting***Operations with MoneyCash LedgersBudgeting*

**Measurement**

*Customary and Metric Estimation*

Customary and Metric Conversion

Customary and Metric Conversion

*Length, Weight, Volume and Temperature*

*Applications to Cooking, Science Labs, and Building*

**Data and Statistics***MeanMedianModeLine PlotsFrequency ChartsStem-and-Leaf PlotsBox Plots*

### Algebra

**Square and Cube Root**

*Concept of a Root with Bead Materials*

Concept of a Root with Peg Boards

Concept of a Root with Peg Boards

**Exponents***Operations with PowersNegative ExponentsExponential Notation*

**Numerical Expressions***Parenthesis in Numerical ExpressionsExpressions with Mixed Operations*

**Squaring and Cubing***PolynomialsSquaring and Cubing SumsBinomials and TrinomialsOrders of Magnitude*

**Equations***Concept of BalanceInput-Output TablesSolving One-Step Equations with Whole Numbers and DecimalsTables and Graphs of Two-Variable Equations*