Working with Algebraic Fractions

Students learn how to simplify and perform addition, subtraction, multiplication and division with algebraic fractions.  As learning progresses they combine these skills to solve equations with algebraic fractions using the quadratic formula.  This unit takes place in Term 2 of Year 11 and follows on from solving quadratic equations.

Prerequisite Knowledge
• Solve linear equations in one unknown algebraically(including those with the unknown on both sides of the equation)
• Apply the four operations, including formal written methods, simple fractions (proper and improper)
• Calculate exactly with fractions
• Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax2 + bx + c

Success Criteria
• Simplify and manipulate algebraic fractions by:
• collecting like terms
• multiplying a single term over a bracket
• taking out common factors
• expanding products of two or more binomials
• simplifying expressions involving sums, products and powers, including the laws of indices
• Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula

Key Concepts
• Students need to apply the same numerical techniques with algebraic fractions as they have done with numerical ones.
• Like numerical fractions algebraic fractions need to have a common denominator when performing addition or subtraction.
• Simplifying algebraic fractions involves factorising the expression into either one or more brackets.
• Multiply the fractions through by a common denominator to cancel out the division when solving fractions.

Common Misconceptions
• Students who understand the need for common denominators when adding or subtracting fractions are often let down by their poor algebraic skills. Particularly when multiplying out by a negative.
• When attempting to simplify fractions students tend to cancel down incorrectly thus losing marks for final accuracy.
• Students can forget to use the difference of two squares when finding common denominators.
• Students struggle with factorising quadratics when the coefficient of x2 is greater than one.
• It is common for students to try and solve for the unknown when they have only been asked to simplify.

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