Algebraic Methods

Scheme of work: Year 13 A-Level: Pure 2: Algebraic Methods

Prerequisite Knowledge

  1. Algebraic Manipulation: You should be comfortable with manipulating algebraic expressions, including simplification, factoring, expanding, and solving equations involving quadratic and linear terms.
  2. Equations and Inequalities: A strong grasp of solving various types of equations (quadratic, linear, and polynomial) and inequalities, as well as understanding their solutions.
  3. Differentiation and Integration: Basics of differentiation (power rule, product rule, chain rule) and integration (basic integration rules) along with their applications.
  4. Proof Techniques: Basic familiarity with direct proofs, proof by contradiction, and proof by induction at an introductory level.

Success Criteria

  • Proof by contradiction:
    • Students should be able to identify the hypothesis, the negation of the hypothesis, and the contradiction in a proof by contradiction.
    • Students should be able to construct a proof by contradiction for a given statement.
  • Multiplying and dividing with algebraic fractions:
    • Students should be able to multiply and divide algebraic fractions using the distributive property and the concept of a common denominator.
    • Students should be able to simplify algebraic fractions using factoring and cancellation.
  • Writing an algebraic fraction in partial fraction form:
    • Students should be able to identify the denominators of a given algebraic fraction that can be factored.
    • Students should be able to write an algebraic fraction in partial fraction form using the method of partial fractions.
  • Working with partial fractions with repeated linear factors:
    • Students should be able to identify the repeated linear factors in the denominator of a given algebraic fraction.
    • Students should be able to manipulate algebraic fractions with repeated linear factors using the method of partial fractions.
  • Calculating with improper algebraic fractions:
    • Students should be able to convert improper algebraic fractions into mixed numbers and vice versa.
    • Students should be able to add, subtract, multiply, and divide improper algebraic fractions.

Key Concepts

  • Proof by contradiction:
    • The hypothesis of a statement is assumed to be true.
    • The negation of the hypothesis is then shown to lead to a contradiction.
    • This contradiction shows that the hypothesis must be false, and therefore the statement must be true.
  • Multiplying and dividing with algebraic fractions:
    • Algebraic fractions can be multiplied and divided using the distributive property and the concept of a common denominator.
    • Algebraic fractions can also be simplified using factoring and cancellation.
  • Writing an algebraic fraction in partial fraction form:
    • An algebraic fraction can be written in partial fraction form by expressing it as a sum of simpler fractions, each of which has a linear numerator and a denominator that is the product of linear factors.
  • Working with partial fractions with repeated linear factors:
    • When the denominator of an algebraic fraction has repeated linear factors, the fraction can be written in partial fraction form using the method of partial fractions with repeated linear factors.
  • Calculating with improper algebraic fractions:
    • Improper algebraic fractions can be converted into mixed numbers and vice versa.
    • Improper algebraic fractions can be added, subtracted, multiplied, and divided using the same rules as proper algebraic fractions.

Common Misconceptions

  • Proof by contradiction:
    • Students may misunderstand the concept of a contradiction. A contradiction is not just any false statement. It is a statement that is the opposite of a true statement.
    • Students may also make mistakes when constructing proofs by contradiction. For example, they may forget to assume the negation of the hypothesis, or they may not show that the negation of the hypothesis leads to a contradiction.
  • Multiplying and dividing with algebraic fractions:
    • Students may make mistakes when multiplying and dividing algebraic fractions. For example, they may forget to distribute the numerator and denominator, or they may not cancel common factors.
    • Students may also have difficulty simplifying algebraic fractions. For example, they may not be familiar with the rules of factoring and cancellation.
  • Writing an algebraic fraction in partial fraction form:
    • Students may misunderstand the concept of a partial fraction. A partial fraction is not just a fraction with a linear numerator and a quadratic denominator. It is a way of writing an algebraic fraction as a sum of simpler fractions, each of which has a linear numerator and a denominator that is the product of linear factors.
    • Students may also make mistakes when writing algebraic fractions in partial fraction form. For example, they may not identify all the factors in the denominator, or they may not find the correct coefficients for the numerators.
  • Working with partial fractions with repeated linear factors:
    • Students may not be familiar with the method of partial fractions with repeated linear factors. This method is used to write algebraic fractions with repeated linear factors in partial fraction form.
    • Students may also make mistakes when working with partial fractions with repeated linear factors. For example, they may not find the correct denominators for the fractions, or they may not find the correct coefficients for the numerators.
  • Calculating with improper algebraic fractions:
    • Students may not be familiar with the concept of an improper algebraic fraction. An improper algebraic fraction is an algebraic fraction where the degree of the numerator is greater than or equal to the degree of the denominator.
    • Students may also make mistakes when calculating with improper algebraic fractions. For example, they may forget to convert the improper fraction into a mixed number, or they may not perform the arithmetic correctly.

Algebraic Methods Lessons

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