Advanced Counting II
(two-semester course)

Advanced Counting II is a fourth course offered in Classic Math School’s Enrichment Program, and is intended for students from 3nd to 6th grade. Prerequisites include confidence in all four operations on whole numbers plus certain familiarity with fractions and decimals. The scope and depth of the course may vary significantly depending on the actual students’ level and duration of the course.

Interactive Advanced Counting II course is designed to be taught in a small group setting using a variety of instructional methods and materials. Hands-on activities, math games and puzzles form an integral part of each class session to enhance students’ understanding and retention.

The topics and problems that are studied in Advanced Counting course may include:

  1. Finalizing fluency with multi-digit whole number addition, subtraction, multiplication and division. Extending the concept of the Order of operations by introducing more grouping symbols and exponentiation.  Demonstrating how their use  changing the order of operations using grouping symbols and/or properties. Revisiting  commutative and associative properties for addition and multiplication, distributive property of multiplication (division) over addition (subtraction). Considering grouping symbols and/or properties as convenient tools for overriding the common order of operations for the purposes of mental math.
  2. Introducing the concept of divisibility, applying rules of divisibility to greater numbers. Reiterating the concept of prime numbers. Introducing exponential notation and its use in prime factorization. Extending the concepts of factors and multiples to Greatest Common Factor (GCF) and Least Common Multiple (LCM). Introducing relatively prime numbers.
  3. Building on the concept of fractions. Developing fluency in recognizing and generating equivalent fractions, simplifying fractions, and converting improper fractions into mixed numbers and backwards. Comparing fractions mentally by different methods, including comparing each fraction with a half first and then using transitivity.
  4. Introducing addition/subtraction of fractions and mixed numbers with different denominators. Relating LCD of fractions to LCM of their denominators. Encouraging mixed numbers subtraction without changing them to improper fractions to avoid extra calculations.
  5.  Reviewing multiplication of fractions and mixed numbers to present their division as the inverse of multiplication with all that it implies. Introducing reciprocals.
  6. Solving and making up word problems that involve all four operations on fractions. Finding a certain part of a given whole and finding a whole from a given part.
  7. Building on the concept of probability. Probability of independent events.
  8. Extending the concept of decimals. Considering different decimal notations for the same numbers, recognizing whole numbers as decimals. Reading and writing decimals up to millionths, integrating decimals into the place value system. Comparing decimals, placing them on the number lines with appropriate scales. Using place value understanding, rounding decimal to tenths, hundredths, thousandths, ten thousandths, or hundred thousandths positions.
  9. Reviewing multiplication of a decimal by a whole number, expecting the product to be less than the whole number, if the decimal is less than one. Introducing multiplication of decimal by decimal, predicting when the product will exceed one/both factors, estimating products.
  10. Converting common fractions to decimals by making equivalent ones with denominators that are multiples of ten, analyzing in which cases it is impossible. While recalling that fraction bar is also a division sign, converting fractions/mixed numbers into decimals through long division process. By introducing the non-terminating repeating (periodic) decimals, encouraging a conjecture about fraction’s certain features that will guarantee it conversion to a decimal of a certain type. Recognizing that this conversion is equivalent to a long division of two whole numbers (numerator and denominator) running into a decimal form of a quotient rather than a whole number quotient with a remainder.
  11. Introducing long division of decimals. Mastering all four operations on decimals, making a special case of multiplying and dividing decimals by a power of ten to be performed mentally.
  12.  Solving and making up multistep word problems that involve all four operations on fractions and decimals.
  13.  Introducing the concept of Percents. Relating percents to decimals and fractions.
  14.  Simplifying arithmetic expressions involving fractions, decimals  and a variety of grouping symbols (parentheses, brackets, braces and fraction bars). Understanding the need of conversion between fractional and decimal notations to perform calculations. Considering relative advantages and disadvantages of each notation, making the optimum choice for every problem.
  15. Distinguishing between arithmetic expressions and equations. Introducing scale modeling for equations.
  16.  The concept of measurement. Identifying appropriate units (metric and imperial) of measuring length, area, volume, weight/mass, time, and money. Carrying out simple unit conversion within a system of measurements and between the systems.
  17.  Solving and making up money “management” and time “management” problems involving fractions, decimals, percents and unit conversions.
  18.  Reviewing basic geometric terms (point, line, segment, ray, angle) and their proper notation. Identifying and classifying different angles (acute, right, obtuse, straight).  Building on identification and classification of polygons (triangles, quadrilaterals, pentagons, hexagons, etc.).
  19.  Continuing on identification and classification of triangles (equilateral, isosceles, scalene, acute, right, obtuse) and quadrilaterals (square, rectangle, rhombus, parallelogram, kite, trapezoid). Exploring the relationship between the lengths of the sides in a triangle and its interior angle measures.
  20. Applying Venn diagram modeling to identify set relationship between different types of triangles and between different types of quadrilaterals.
  21.  Introducing the concepts of variable and formula through the simple perimeter and area formulas. Considering formulas as a convenient device for repetitive calculations. Derivation and application of area formulas for the square, rectangle, triangle, parallelogram, and trapezoid based on area addition postulate.
  22. Reviewing the concept of congruency. Introducing the concept of similarity and its relationship to congruency. Exploring scale factors.
  23.  Introduction to integers.  Comparing integers using the number line. Using all four quadrants of  two-dimensional coordinate system to plot points with the given coordinates and to find coordinates of the given points.
  24.  Finding, describing, and extending patterns of numbers, symbols, and geometric figures.
  25.  Introducing inductive and deductive reasoning. Solving sophisticated logic problems of different nature.