### OVERVIEW

The “Story of Numbers” helps children understand the power of mathematics and motivates them to continue exploring numbers. Progression through the Montessori math curriculum is not strictly linear. Instead, Maria Montessori envisioned elementary math as a three-tiered progression. The first tier consists of the numbers to ten, place value, and the four operations. The second tier is dedicated to the memorization of math facts. The third tier is where the children study hierarchy, that is, how the numbers in the decimal system are related and grouped. The children explore different concepts of math simultaneously. Children frequently ask for the biggest problems possible. They also enjoy writing their own BIG problems. The younger children practice using the materials representing whole numbers, fractions and decimals, and through repeated experiences with them, they “discover” algorithms or concepts by themselves or under the guidance of the teacher. Montessori places great emphasis on the study of geometry, and all the math materials have a geometric aspect. Children in the lower elementary classrooms study lines, angles, and plane figures, as well as linear and cubic measurement. In the upper elementary the children use boxes of cubes and prisms, which they previously manipulated in the primary classroom, to cube a binomial or trinomial. Through their studies, the students are able to discover abstract concepts of algebra, using materials that once were a part of their sensorial experiences only. The upper elementary children also take great delight in further study of different systems of numeration, both those used by ancient civilizations, and other possible systems, such as base two or base twelve.

### CURRICULUM

#### The History of Mathematics

• Key lessons on the history of mathematics (e.g., the invention of zero)
• Work with different systems (Babylonian, Roman, Mayan, etc.)
• Comparison of number systems (Number bases vs. place-value)

#### Numeration and Concepts of Numbers

• Writing and reading large numbers – expanded notation
• Rounding to the nearest ten, hundred, thousand, etc.
• Estimation
• Properties of numbers: commutative, associative, distributive
• Number sentences (order of operations, use of parentheses)
• Ratio and proportion
• Arithmetic mean and median
• Statistics and probability
• Integers (positive and negative numbers)
• Representation of pairs of numbers on a coordinate plane
• Numeration and operations in other number bases

#### Whole Number Operations

• Review of facts for all operations
• Review of dynamic subtraction, especially with zeros
• Mental multiplication of factors with zeros
• Multiple-digit multiplication and division problems
• Cross-multiplication
• All four operations with integers

#### Fractions, Decimals, and Percent

• Addition and subtraction with fractions and mixed numbers (like and unlike denominators)
• Multiplication and division with fractions and mixed numbers
• Changing fractions to decimals
• Changing decimals to fractions
• Decimals in expanded notation
• Skip counting, comparing, and rounding decimals
• Addition and subtraction of decimals
• Multiplication and division of decimals
• Concept of percent
• Comparison and equivalence of fractions, decimals, and percent
• Finding the percent of a number

#### Multiples and Factors

• Factor trees and prime factorization
• Using primes to find the LCM and GCF
• Divisibility

#### Powers of Numbers

• Squaring a polynomial (geometric and algebraic representations)
• Finding the numerical value of the square of a polynomial
• Finding square roots, with materials and abstractly
• Cubing a binomial
• Cubing a trinomial
• Powers of ten
• Powers of other numbers
• Expanded notation, including with exponents

#### Measurement

• English and metric units of measurement (length, weight, liquid capacity)
• Equivalences within a system (e.g. converting inches to yards)
• Introduction to very small and very large measurements, scientific notation

#### Geometry

• History of geometry (contributions by various people and cultures)
• Geometric design: tessellations, 3-D constructions, origami, scale drawing, symmetry, computer applications, plane figures
• Review of triangles, quadrilaterals and their parts
• Translation, rotation and reflection
• The circle: its parts and relationship to other figures
• The theorem of Pythagoras
• Area of triangles, rhombi, trapezoids, regular polygons, circles
• Review of solids and their parts
• Surface area of polyhedrons
• Volume of cubes, prisms, pyramids
• Volume of cylinders, cones, spheres

#### Algebra

• Concepts (variables and constants, expressions, introduction to functions, equations, etc.)
• Computations (order of operations)

#### Problem-solving

• Techniques of problem solving
• Problems using whole numbers, fractions, decimals, percents, and integer
• Problems involving traveling: velocity, distance, and time
• Money problems (purchasing, figuring tax, interest, tip, check-writing)
• Geometry problems (angles, perimeter, area and volume)
• Other practical applications of math (weather, sports, nutrition. etc.)
• Interpretation and construction of tables and graphs (line, bar, circle)
• Use of calculators and computers to record and relay data.