## Elementary Mathematics

## Overview

The study of mathematics offers gifts far beyond numeracy and calculation. It allows children the opportunity to develop and exercise their reasoning mind. It teaches them how to evaluate situations, test hypotheses, employ problem solving strategies, derive conclusions and articulate them precisely. These are critical skills that children will need in their future careers, in their personal lives and in every aspect of a world increasingly characterized by rapid change.

The reason we study mathematics is two-fold:

- First, we want students to understand how numbers work and relate to each other, as well as the ability manipulate them quickly and easily.
- The second reason is that mathematics is cognitive training that develops precise mental habits one can apply to all areas of life.

A student with a well rounded education in mathematics will gain:

- Competence in everyday math concepts, like quantity, and an intuitive number sense.
- Ability to build on prior knowledge
- The ability to organize and integrate knowledge
- The ability to think laterally. Applying solutions from one problem to another.
- The ability to think about problem solving in a methodical way, considering all the steps required to complete work

## The Montessori Method of Mathematics

Our math curriculum begins with hands-on materials that help the children concretely experience quantity, place value, and geometry. By manipulating these materials students create mental models of abstract mathematical principles.

They aren’t shown examples of abstract ideas, they create abstract ideas from examples. They aren’t verbally told a “two plus two equals four” and then shown props that demonstrate this idea. Children work with beads and physically combine groups of two and discover, that no matter what, two items plus two items will always make a group of four.

Children learn by *doing*.

The Montessori classroom is filled with beautiful and enticing materials. These materials are unique to Montessori education, and they are used in a unique way.

In a conventional classroom, a teacher might use materials to lecture to students. The teacher might hold up colored blocks and explain how they fit together, and then tell the students the formula for calculating a certain quantity about the blocks.

In a Montessori classroom, the materials are used to help students discover the formulas themselves. The materials are designed to be self-correcting, so that students can learn by trial and error. This allows students to develop a deep understanding of mathematical concepts.

After much practice with the materials, children develop an abstract mental model of the mathematical concepts. At this point, they no longer need the materials to help them solve problems. Only then do we show them the algorithm (provided, they didn’t find it themselves already!)

Subjects are ordered based on how easy it is for the child to understand not how easy it is to lecture on. This seems like it should be the norm, but in conventional schools, it’s not.

For example in conventional classrooms geometry is taught in middle school or high school, because the abstract mathematical representations of geometric concepts is difficult to teach without a grounding in higher mathematics.

But a very young child can distinguish the difference between a circle, triangle and square without the need of abstract representation.

With the hands-on materials we are able to sequence the curriculum to match a child’s developmental abilities. The approach is understandable at every level to the child and always firmly rooted in reality.

The lessons are ordered in such a way that only one new difficulty is introduced at at time. These lessons are usually very short and given to students who are all ready for that new challenge. Thus, children learn each step precisely and thoroughly before being asked to add a new level of complexity.

The hands-on materials are designed to be enjoyable, so that repetition- a requirement for mastery- is fun. Practice becomes play.

Math lessons are often presented to individuals or pairs of students. When students need more time or support, they have as many lessons and as much time as they need. Students who excel are able to move forward without waiting for the group to “catch up” to them.

Time is variable, but mastery is fixed.

## Lower Elementary Curriculum

### Arithmetic

**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 Numbers***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

### Arithmetic

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

**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*

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