Key to Algebra
(two semester course)
Key to Algebra is the fifth course offered in the sequence offered by Classic Math School’s Enrichment program, and is intended for students from 4th to 6th grade. The course prerequisites include fluency with all operations on non negative rational numbers in any form ( i.e., 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 Key to Algebra course is designed to be taught in a small group setting using a variety of instructional methods and materials. The main idea of the course can be described by the slogan 'Welcome to Algebra or have you already been there?’. Hands-on activities, math games and puzzles form an integral part of each class session to enhance student's understanding and retention.
The topics and problems that are studied in Key to Algebra course may include:
- Understanding the point and need of elementary Algebra as a convenient tool for summarizing the very well familiar concepts (e.g. properties of arithmetic operations, family facts, parity, divisibility, etc.).
- Refining the concept of divisibility. Developing formulas for even and odd numbers, as well as for any whole number divisible by another whole number with any given remainder. Reiterating the concepts of factors and multiples, GCF and LCM.
- Refining the concept of prime numbers, relatively prime numbers, and prime factorization with exponential notation. Exploring Goldbach's conjecture.
- Using algebraic notation for all the arithmetic properties and developing fluency in naming and applying them in different settings.
- Reintroducing the concept of the order of operations as a convention rather than a math law. Scrutinizing PEMDAS and other common misbeliefs associated with order of operations. Incorporating exponentiation into the hierarchy of operations.
- Introducing simple radicals as an application of the concept of inverse operations, emphasizing the essential restrictions.
- Overviewing the Fractions/Decimals/Percents Concepts using algebraic notation.
- Introducing negative integral exponents. Introducing scientific notation and expanded notation for decimals. Developing thorough understanding of decimals’ place value using expanded notation.
- Mastering the concept of percent. Understanding fractions, decimals, and percents as different notations for the same number (with some exceptions to be considered), developing fluency in conversion process.
- Solving multistep sophisticated word problems involving fractions, decimals and percents by any means (pictures, tape diagrams, systematic trial and error, etc.), but algebraic means. Further in this text those problems will be referred as ‘Problems Challenging without Algebra’.
- Building on the concept of integers. Introducing the concept of opposites and absolute value, illustrating them with the number line.
- Adding and subtracting integers. Additive inverses.
- Multiplying and dividing integers. Building on understanding of inverse operations and implied restrictions. Introducing reciprocals.
- Introducing the concept of variables, algebraic expressions and terms.
- Evaluating algebraic expressions versus evaluating arithmetic expressions. Introducing the concept of equivalent algebraic expressions.
- Multiplying Terms. Considering like terms and unlike terms. Combining like terms.
- Reconsidering Commutative, Associative and Distributive Principles of Algebra as a natural extension of the similar principles/properties of Arithmetic. Using Distributive Principle as the only legal way of overriding the Order of Operations (opening parenthesis).
- Simplifying algebraic expressions. Scrutinizing this simplification as a multistep process of creating a chain of equivalent algebraic expressions. Prioritizing simplification over evaluation.
- Introducing the concept of Ratio. Introducing Rational numbers as a result of dividing integers. Comparing rational numbers, graphing rational numbers on the number line. Complex calculations with rational numbers.
- Revisiting the concept of Rate, Time, and Distance. Exploring the formula relating them with unit analysis and applying it to a variety of word problems, including 'Word Problems Challenging Without Algebra’.
- Introducing the goal and process of solving equations.
- Balancing linear equations by different means using different models. Introducing equivalent equations.
- Using equations to solve problems. Reconsidering previously introduced “Word Problems Challenging without Algebra” to appreciate Algebra.
- Introducing proportion as a special type of equation, denoting two equal ratios. Encouraging shortcuts in solving proportions. Reconsidering percents as proportions.
- Introducing Relations and Functions. Assigning functions/relations by ordered pairs, graphs, tables, formulas. Introducing the concept of Domain and Range of function/relation.