Advanced Counting I
Advanced Counting I is the third course offered in Classic Math School’s Enrichment Program, and is intended for students from 2nd to 5th grade. Prerequisites include knowing the multiplication table by heart, plus a certain level of fluency with whole numbers. The scope and depth of the course may vary significantly depending on the student's’ level and duration of the course.
This interactive 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 I course may include:
- Continued development of number sense and number fluency with whole numbers: reviewing all four operations on whole numbers, using proper names for operands and the results of the operations (i.e., ‘'addend’, ‘sum’, ‘factor’, ‘product’, etc).
- Mastering multiplication of multidigit numbers using a standard algorithm, as well as strategies based on place value, the properties of operations and area models. Developing a strong habit of estimating the product with desired accuracy, using appropriate rounding and mental multiplication by the powers of 10.
- Introducing the concepts of factors and multiples. Finding all factor pairs for whole numbers up to 100. Finding common factors for two or more numbers, and then their greatest common factor (GCF). Gaining familiarity with factors and multiples by listing them for different numbers and looking for the patterns. Discovering prime and composite numbers. Introducing some rules of divisibility.
- Introducing division with a remainder. Dividing a multidigit number by a one digit divisor. Performing this division mentally by breaking down a dividend into a sum of easy multiples of a divisor and a remainder (if any), and then using distributive property.
- Introducing long division algorithm, and validating it by applying the distributive property for division over addition to the dividend in the expanded notation. Writing a numerical equation relating dividend, divisor, quotient, and remainder for any division problem. Discovering restrictions on remainders by considering different division scenarios, and then making general conjectures.
- Practicing long division with multi-digit dividends and 2-3 digit divisors, always using estimation to verify the reasonableness of the quotient, and sometimes using multiplication to check the precision of the answer.
- Solving and making up multistep word problems involving all four operations on whole numbers. Exploring different considerations for division with remainders in the context of a word problem.
- Introducing the counting principle.
- Introducing the concept of average for a set of whole numbers. Exploring the properties and meaning of the average. Solving word problems involving finding averages of whole numbers and finding the missing element of a set with the given average.
- Introducing the identity property for addition and multiplication. Revisiting the concept of order of operations. Changing the order of operations using grouping symbols and/or properties. Revisiting the Commutative and Associative properties for addition and multiplication. Introducing the Distributive property of multiplication (division) over addition (subtraction), emphasizing that it should be used with special discretion. Illustrating how all those properties could be used for mental math.
- Extending the concept of fraction. Introducing the concept of proper and improper fractions. Introducing mixed numbers. Converting improper fractions to mixed numbers and vice versa.
- Comparing fractions and mixed numbers; plotting them on the number line. Developing fluency in recognizing, generating, and visualizing equivalent fractions. Recognizing the process of simplifying a fraction as the process of ‘removing’ the GCF of its numerator and denominator. Demonstrating that the process of ‘removing’ their common factors one by one will inevitably lead to the same result, but will take longer. Realizing that while the list of equivalent fractions is endless, the simplest form for any fraction is unique.
- Adding and subtracting fractions and mixed numbers with common denominators, and denominators that are multiples/factors of each other. Decomposing proper and improper fractions into a sum of unit fractions. Understanding mixed numbers as a sum of a whole number and a proper fraction. While recalling that fraction bar is also a division sign, recognizing that a mixed number/improper fraction is a different notation for dividend-divisor-quotient-remainder relationship.
- Extending previous understanding of two whole numbers multiplication to multiplication of a whole number by a fraction. Recognizing that multiplying a whole number by a certain fraction means finding that certain fractional part of the whole number. Using pictorial models, extending that understanding to multiplication of two fractions as finding that certain fractional part of a given fraction. Demonstrating cross-cancelling steps as a shortcut in multiplication process and encouraging its use as a routine.
- Introducing decimals up to hundreds, relating them to fractions. Integrating decimals into the place value system, and considering whole numbers as decimals. Reading and writing decimals.
- Comparing decimals. Plotting decimals on the number line. Adding and subtracting decimals. Multiplying decimals by a whole number. Rounding decimals to the nearest whole number.
- Converting decimals to fractions (proper, improper, mixed numbers). Comparing simple fractions and decimals by finding their relative positions on the number line.
- Extending 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 measurement and between the systems.
- Solving and making up money management and time management problems involving fractions, decimals and unit conversions.
- Understanding basic geometric terms (point, line, segment, ray, angle) and their proper notation. Identifying and classifying different angles (acute, right, obtuse), and measuring them with a protractor to the accuracy of a whole degree. Constructing angles of a given whole degree measures with a protractor.
- Identifying and classifying polygons (triangles, quadrilaterals, pentagons, hexagons, etc.) and three dimensional solids: prisms and pyramids. Introducing circles and related three dimensional solids: spheres, cylinders and cones.
- Identifying and classifying triangles (equilateral, isosceles, scalene, acute, right, obtuse), and quadrilaterals (square, rectangle, parallelogram, rhombus, kite, trapezoid).
- Continuing on perimeter and area concepts. Introducing the concept of volume. Exploring simple exponential notation in regards to area of a square and volume of a cube.
- Continuing on two-dimensional coordinate system. Plotting points with given fractional coordinates and finding coordinates of given points in the first quadrant. Translations, reflections, and rotations of figures.
- Continuing on the basics of Set Theory. Considering Venn Diagram modeling for three sets. Solving a variety of logic problems, including Venn diagram problems.