Notation And Manipulation
Overview
Algebra is arithmetic with the numbers left out on purpose. A letter stands for a number you do not yet know, or for any number at all, and once you can write the four basic arithmetic processes with letters instead of digits you can say something true about every case at once. That is why this topic sits at the front of the Algebra section: everything after it, from solving equations to differentiating a curve, is manipulation of the objects defined here.
You will use letters to express generalised numbers, substitute into formulae and transform them, tell an expression from an equation from an identity, collect like terms and expand brackets, factorise by taking out common factors and by recognising the difference of two squares, apply the index laws, and handle algebraic fractions. On the Extension Tier you will transform complex formulae where the subject appears twice, expand products of two or three binomials, factorise quadratics with a leading coefficient, work with fractional powers and algebraic denominators, and construct proofs.
Objectives
- [Core] use letters to express generalised numbers and express basic arithmetic processes algebraically
- [Core] substitute numbers for words and letters in formulae and transform simple formulae
- [Extension] transform complex formulae including when the subject appears twice
- [Core] understand and use the concepts of expressions, equations, formulae, identities, inequalities, terms and factors
- [Core] collecting like terms and expanding brackets up to expanding products of two linear expressions
- [Extension] expanding products of two or three binomials
- [Core] taking out common factors, factorising quadratic expressions of the form x² + bx + c; including the difference of two squares
- [Extension] factorising quadratic expressions of the form ax² + bx + c; including the difference of two squares
- [Core] index laws for multiplication and division using integer powers
- [Extension] including fractional powers
- [Core] manipulation of rational expressions: use of + - × ÷ for algebraic fractions with denominators being numeric
- [Extension] linear or quadratic algebraic expressions
- [Core] argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments
- [Extension] to include proofs
Mind map
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Lesson Note
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Lesson Evaluation
Congratulations on completing the lesson on Notation And Manipulation. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- Circle the expression that is equivalent to x^3 + 6x. A. x(x + 6) B. x^2(x + 6) C. x(x^2 + 6) D. x(x^2 + 6x) Answer: C
- You are given that p = m + 5. Which one of the following is true? A. m = p + 5 B. m + p = 5 C. m = 5 - p D. m = p - 5 Answer: D
- Factorise fully 6x^3 - 24x. A. 6x(x^2 - 4) B. 6x(x + 2)(x - 2) C. 6(x^3 - 4x) D. 3x(2x^2 - 8) Answer: B
- Simplify 5c^3 x 4c^7. A. 9c^10 B. 20c^10 C. 20c^21 D. 9c^21 Answer: B
- Which of these is an identity? A. 2x + 1 = 9 B. 3x - 1 > 8 C. A = pi r^2 D. (x + 3)^2 = x^2 + 6x + 9 Answer: D
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Work through these questions in the app
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