Math

Queens Paideia’s uses Singapore Math, one of the most comprehensive and proven math curricula available, as a primary resource, and supplements it with manipulatives, math-based games, online programs, economic transactions, and accounting scenarios. Our total math program helps students develop an early and firm fluency in basic computational and problem-solving skills. Students learn to apply geometric concepts in spatial reasoning; concepts of probability in areas like decision making, prediction, sampling, uncertainty, interpretation of data, and statistical reasoning; and problem-solving strategies that require logical and algebraic reasoning skills, algorithms, and heuristics.

Our approach to mathematics education gives our students the practical tools they need to enhance their thinking and problem-solving skills, and to make sense of their world.

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Stage 1. Foundational Number Sense and Operations (approx. ages 5-7)

Stage 1 takes students from foundational numeracy to early development of the areas listed below, always with much support and guidance. Besides building their number skills through counting, ordering, and performing addition and subtraction operations up to 100, students practice interpreting, modeling, and verbally explaining their reasoning in simple mathematical applications, such as one- and two-step word problems involving whole numbers up to 100.

Where a student starts within Stage 1 depends on his/her preexisting skills upon entering QPS. We continuously evaluate our students to gauge their current capabilities and the most effective ways to address them.

Numeracy: Working with Whole Numbers to 1,000
  • Count/read/write numerals
  • Make, compare sets
  • Ordinal numbers
  • Place value
  • Number sentences
  • Greater than, less than, equal to
  • Early estimation
  • Patterns of numbers, shapes, symbols
Computation on Whole Numbers
  • Number bonds; part-whole concept
  • Addition/subtraction of numbers within 1,000
  • Commutative, associative, distributive properties
  • Rename in addition/subtraction to 100
  • Arrays to solve multiplication problems within 40
  • Multiply/divide by 2s, 3s, 4s, 5s, 10
  • Mental math strategies
  • Symbols for unknown numbers (algebraic thinking)
  • Identify a missing value (unknown or symbol) in a number sentence
Fractions
  • Halves and fourths
  • Fractions of a whole, with 1-12 denominators
  • Common denominators
  • Compare/order
Measurement
  • Currency (unit values, symbols, decimal notation, add/subtract to $10.00; measure value used in trade)
  • Time (clocks, calendars) with units of differing size (seconds, minutes, hours, days, weeks, months, years)
  • Duration of time intervals
  • Compare, measure length, weight, mass, capacity
  • Nonstandard and standard measurement units
  • Estimation
  • Generalizing/classifying
  • Read/draw simple bar graphs
Geometry
  • Area of shapes using unit squares
  • Common 2D and 3D objects
  • Extending patterns
Word Problems
  • 1- and 2-step problems involving whole numbers using all 4 operations Simple word problems involving fractions

Stage 2. Fluency with Number Sense and 5-Digit Operations (approx. ages 7-9 )

In Stage 2 students work toward fluency in interpreting place value and applying the basic arithmetic operations to whole numbers up to 10,000. They learn to represent and perform the basic operations on simple fractions, calculate area and perimeter for simple shapes, and explain in greater detail their reasoning in solving two-step word problems.

Where a student starts within Stage 2 depends on his/her preexisting skills upon entering QPS. We continuously evaluate our students to gauge their current capabilities and the most effective ways to address them.

Numeracy: Whole Numbers to 100,000
  • Count/read/write numerals, including standard and expanded notation
  • Place value
  • Number sentences
  • Estimation
  • Number patterns within 100,000
Computation on Whole Numbers
  • Addition/subtraction of numbers within 10,000
  • 0 and 1 in multiplication and division
  • Multiply/divide by 6s, 7s, 8s, 9s
  • Multiply/divide numbers within 10,000 by 1-digit and 2-digit numbers
  • Multiples and common multiples
  • Lowest common multiples of up to 3 numbers within 100
  • Commutative, associative, distributive properties
  • Mental math strategies
Fractions and Decimals
  • Compare/order fractions with different denominators
  • Equivalent fractions
  • Simplest form
  • Mixed numbers and improper fractions (conversions of)
  • Add/subtract like and unlike fractions
  • Multiply by whole numbers
  • Apply to money and time
  • Convert decimals to fractions; fractions to decimals
  • Compare/order decimals up to 3 decimal places
  • Add/subtract decimals up to 2 places
  • Multiply/divide decimals by 2-digit whole number
  • Estimation strategies
Measurement, Data Analysis, Probability
  • Tell time to the minute
  • Duration of time intervals
  • Convert between units of time
  • Estimate and measure length, weight, mass, capacity
  • Add/subtract/multiply/divide measurement units
  • Convert units within metric system using multiplication
  • Represent/compare data using bar graphs
  • Tables and bar graphs
Geometry
  • Area, perimeter, and length of unknown sides
  • Area, perimeter of composite figures of squares and rectangles
  • Volume of 3D figures via counting cubic units
  • Cubic centimeters, milliliters, liters
  • Identify, measure, construct acute, obtuse, right angles
  • Perpendicular and parallel lines
  • Right, equilateral, isosceles, scalene triangles
  • Line symmetry
Word Problems
  • 2-step problems with whole numbers, fractions, or decimals and all 4 operations Symbols for unknown numbers (algebraic thinking)

Stage 3. Fractions, Decimals, and Their Applications (approx. ages 9-11)

In Stage 3, students learn to express fractions, decimals, and percentages interchangeably. They perform arithmetic operations with rational numbers, demonstrate graphic representation of numeric concepts, and solve multi-step word problems involving ratios, proportions, rates, and percentages. They build geometry skills by finding angles through direct measurement and calculation of complimentary and supplementary angles and by using simple algebraic expressions to represent the formulae for finding the area and volume of complex shapes and solids.

Where a student starts within Stage 3 depends on his/her preexisting skills upon entering QPS. We continuously evaluate our students to gauge their current capabilities and the most effective ways to address them.

Numeracy: Understanding and Working with Whole Numbers to 100,000
  • Rounding numbers to nearest 10 or 100
  • Place value to 1,000,000
Computation on Whole Numbers
  • Estimate to verify reasonableness of addition and subtraction results
  • Multiply/divide numbers within 10,000 by 2-digit numbers
  • Commutative, associative, distributive properties
  • Order of operations
  • Mental math strategies
Fractions, Decimals, Percentage
  • Add/subtract like and unlike fractions
  • Add/subtract mixed numbers
  • Multiply by whole numbers
  • Divide fractions by whole numbers; whole number or fraction by a fraction
  • Order of operations
  • Apply to money and time
  • Convert decimals to fractions; fractions to decimals
  • Compare/order decimals up to 3 decimal places
  • Rounding
  • Add/subtract decimals up to 3 places
  • Multiply/divide by tends, hundreds, thousands
  • Multiply/divide by 2-digit whole number
  • Estimate to verify reasonableness of calculations
  • Decimal and fraction equivalents for percentage
  • Problems involving percentage of quantity
  • Percentage applied to sales, interest earned, percentage increase/decrease
Ratio/Average/Rate/Speed
  • Ratios to compare three quantities
  • Equivalent and simplified ratios
  • Changing ratios, proportions, rate
  • Multistep word problems involving ratio, speed, rate, average rate, percentage
Measurement, Data Analysis, Probability
  • Convert between units of time
  • Convert units within metric system using multiplication
  • Convert fractional measurements to different or compound units
  • Convert units involving decimals within measuring systems
  • Line graphs, pie charts
  • Collect, organize, display data
  • Average and other values
Geometry
  • Area of triangles
  • Find sides of rectangular prisms
  • Multistep problems involving volume and displacement
  • Circumference, radius, diameter of circles
  • Perimeter of compound 2D figures involving squares, triangles, half circles, circles
  • Classify 3D shapes
  • Attributes of triangles and quadrilaterals
  • Measure/construct angles
  • Vertical, adjacent, complementary, supplementary angles
  • Angle properties of intersecting lines, triangles, parallelograms, rhombuses, trapezoids
  • Geometric solids, nets of solids, solids of nets
Word Problems
  • Multi-step problems involving all 4 operations on whole numbers, fractions, decimals, percentages, ratios
Algebra
  • Bar diagrams to solve word problems involving whole numbers, fractions, decimals, percentage, ratio
  • Functional relationships
  • Simple algebraic expressions
  • Expressions to describe geometric quantities

Stage 4. Abstract Quantitative and Spatial Representations (approx. ages 11-13)

By the time students are in Stage 4, they are ready to advance their study of the number system. They learn the relationships between prime and composite numbers; rational and irrational numbers; prime factorization, exponents and roots; negative numbers; more advanced ways to model and numerically represent relationships described in word problems; and to use algebraic expressions to solve for unknown quantities. They also learn to use probability and statistics to predict outcomes, and basic techniques of analytic geometry for linear and quadratic functions.

Alongside their greater independence in approaching higher-level math, Students in Stage 4 review and become further grounded and fluent in the areas listed in Stage 3.

Where a student starts within Stage 4 depends on his/her preexisting skills upon entering QPS. We continuously evaluate our students to gauge their current capabilities and the most effective ways to address them.

Stage 5. Secondary School Mathematics (approx. ages 13-18)

QPS’s secondary mathematics curriculum presents mathematics as a formal discipline as well as a tool for making decisions and solving problems. The curriculum reflects and follows the New York State Board of Regents’ PARCC policies and guidance: Algebra I, Geometry, Algebra II (includes probability, statistical inference, analytic geometry, logarithms, and exponentials), Pre-Calculus, and Calculus. It exceeds most high school math programs in the following ways:

  • As students achieve specific mathematical skills and knowledge, they apply it to topics they are concurrently studying in science and social studies, with emphasis on the ways mathematics can be applied to situations, including ability to decide when and if to invoke mathematics, and if so which type.
  • Thinking skills, logic, deduction, inference, analytic skills, and the use of heuristics play a large role in Queens Paideia’s secondary math program.
  • Students learn about the function of mathematics as a model of the real world; its relationship to science; its pure or formal aspects; and its the history, status, and significance.
  • The curriculum is organized in terms of learning objectives, which are not indexed to age or grade level but rather to the student’s actual ability and stage of knowledge.


Progress through Queens Paideia’s mathematics curriculum is achievement driven, and assessment is continuous.


Algebra I
  • Solves problems involving analytic geometry, e.g., writing the equation of a line given the coordinates of a point and a slope.  
  • Adds, subtracts and multiplies polynomials.
  • Transforms polynomial expressions by factoring and completing the square.
  • Applies linear simultaneous equations in solving word problems
  • Manipulates radicals and exponents in linear equations
  • Does basic dimensional analysis involving units of physical quantities
  • Describes relationships between quantities by means of functions. 
Geometry
  • Applies triangle congruence and similarity criteria.
  • Uses graphical coordinates to obtain geometric results.
  • Applies the Pythagorean theorem in geometric calculation  
  • Calculates lengths, angles, areas, and volumes for  geometric shapes
  • Uses geometric representation and geometry software as a modeling tool.  
  • Applies physical and computational geometric modeling techniques.
  • Performs elementary Euclidean proofs
Algebra II
  • Applies factoring to rewrite polynomials
  • Solves equations involving logarithms and exponential functions,
  • Graphs functions involving logarithmic scales and exponential terms.
  • Determines whether a series is divergent or convergent, and identifies asymptotes.
  • Solves basic problems involving probability theory, distributions, and statistics
  • Solves quadratic equations and applies them to the representation of simple conic sections
Pre-Calculus
  • Defines rotations, reflections and dilations in the coordinate plane using 2 x 2 matrices (and translations by vector addition).
  • Uses matrices as tools for performing rotations and reflections in the coordinate plane.  
  • Uses equations for parabolas, hyperbolas, and ellipses.
  • Graphs and analyzes rational and exponential functions
  • Works with exponential and logarithmic functions.
  • Uses basic functions to describe relationships between two variables.
  • Uses software to generate graphs of geometric functions.
  • Performs basic trigonometric calculations using the Law of Sines, Law of Cosines, and other basic trigonometric formulas. .  
  • Applies probability and statistics to the process of inference and decision making.
Calculus
  • Expresses the meaning of the derivative in terms of a rate of change and local linear approximation, and uses derivatives to solve a variety of problems.       
  • Expresses the relationship between the derivative and the definite integral.       
  • Communicates the use of mathematics, both orally and in well-written sentences, to explain mathematical solutions to problems.
  • Determines the reasonableness of solutions, including sign, magnitude, relative accuracy, and units of measurement.