About this topic
Before you begin
Explain algebraic equivalence, simplify rational expressions and solve linear, simultaneous and quadratic conditions while retaining restrictions.
Learning goals
- Evaluate an algebraic expression by substitution
- Represent a situation with algebraic expressions
- Interpret algebraic notation and represented operations
- Simplify linear expressions by combining like terms
- Expand and factorise algebraic expressions
- Express and use the general term of a sequence
- Solve linear equations and check the solution
- Formulate, use and interpret a mathematical model
- Change the subject of a formula
- Multiply and divide algebraic fractions with domain restrictions
- Calculate and simplify algebraic fractions with restrictions
- Add and subtract algebraic fractions with domain restrictions
- Solve a first-degree equation by a specified expansion or denominator-clearing step, preserving original exclusions
- Solve a system of linear equations completely, including systems with no or infinitely many solutions
- Solve simultaneous equations by justified methods
- Interpret and draw graphs of functions
- Solve quadratic equations by suitable methods
- Apply the quadratic formula with signed coefficients, both branches and the complete real solution set
- Solve linear inequalities and show the solution set on a number line
- Solve inequalities and represent their solution sets
Syllabus statements covered
- Using letters to represent numbers
- Interpreting notations: ab as a × b; a/b as a ÷ b or a × (1/b); a² as a × a, a³ as a × a × a, a²b as a × a × b, …; 3y as y + y + y or 3 × y; 3(x + y) as 3 × (x + y); (3 + y)/5 as (3 + y) ÷ 5 or (1/5) × (3 + y)
- Evaluation of algebraic expressions and formulae
- Translation of simple real-world situations into algebraic expressions
- Recognising and representing patterns/relationships by finding an algebraic expression for the nth term
- Addition and subtraction of linear expressions
- Simplification of linear expressions such as −2(3x − 5) + 4x and 2x/3 − 3(x − 5)/2
- Use brackets and extract common factors
- Factorisation of linear expressions ax + bx + kay + kby
- Expansion of the product of algebraic expressions
- Changing the subject of a formula
- Finding the value of an unknown quantity in a given formula
- Use of (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², a² − b² = (a + b)(a − b)
- Factorisation of quadratic expressions ax² + bx + c
- Multiplication and division of simple algebraic fractions such as (3a/(4b²))(5ab/3); (3a/4) ÷ (9a²/10)
- Addition and subtraction of algebraic fractions with linear or quadratic denominator such as 1/(x − 2) + 2/(x − 3); 1/(x² − 9) + 2/(x − 3); 1/(x − 3) + 2/(x − 3)²
- Solving linear equations in one variable
- Solving simple fractional equations that reduce to linear equations, such as x/3 + (x − 2)/4 = 3 and 3/(x − 2) = 6
- Solving simultaneous linear equations in two variables by substitution and elimination or graphically
- Solving quadratic equations in one variable by factorisation, formula, completing the square for y = x² + px + q, or graphically
- Solving fractional equations that reduce to quadratics, such as 6/(x + 4) = x + 3 and 1/(x − 2) + 2/(x − 3) = 5
- Formulating equations to solve problems
- Solving linear inequalities in one variable and representing the solution on the number line
- Using letters to represent numbers
- Interpreting notations: ab as a × b; a/b as a ÷ b or a × (1/b); a² as a × a, a³ as a × a × a, a²b as a × a × b, …; 3y as y + y + y or 3 × y; 3(x + y) as 3 × (x + y); (3 + y)/5 as (3 + y) ÷ 5 or (1/5) × (3 + y)
- Evaluation of algebraic expressions and formulae
- Translation of simple real-world situations into algebraic expressions
- Recognising and representing patterns/relationships by finding an algebraic expression for the nth term
- Addition and subtraction of linear expressions
- Simplification of linear expressions such as −2(3x − 5) + 4x; 2x/3 − 3(x − 5)/2
- Use brackets and extract common factors
- Concept of equation
- Solving linear equations in one variable
- Solving simple fractional equations that can be reduced to linear equations such as x/3 + (x − 2)/4 = 3; 3/(x − 2) = 6
- Formulating a linear equation in one variable to solve problems
- Expansion of the product of algebraic expressions
- Changing the subject of a formula
- Finding the value of an unknown quantity in a given formula
- Use of:
- (a + b)² = a^2 + 2ab + b^2
- (a − b)² = a^2 − 2ab + b^2
- a^2 − b^2 = (a + b)(a − b)
- Factorisation of linear expressions of the form ax + bx + kay + kby
- Factorisation of quadratic expressions ax^2 + bx + c
- Multiplication and division of simple algebraic fractions such as (3a/(4b²))(5ab/3); (3a/4) ÷ (9a²/10)
- Addition and subtraction of algebraic fractions with linear or quadratic denominator such as 1/(x − 2) + 2/(x − 3); 1/(x² − 9) + 2/(x − 3); 1/(x − 3) + 2/(x − 3)²
- Concept of equation and inequality
- Solving simple inequalities in the form ax + b ≤ c and ax + b < c and representing the solutions on the number line
- Graphs of linear equations in two variables (ax + by = c)
- Solving simultaneous linear equations in two variables by:
- Substitution and elimination methods
- Graphical method
- Solving quadratic equations in one variable by factorisation
- Formulating a pair of linear equations in two variables to solve problems
- Solving quadratic equations in one variable by:
- Use of formula
- Completing the square for y = x^2 + px + q
- Graphical method
- Solving fractional equations that can be reduced to quadratic equations such as 6/(x + 4) = x + 3; 1/(x − 2) + 2/(x − 3) = 5
- Solving linear inequalities in one variable (including simultaneous inequalities) and representing the solution on the number line
- Formulating a quadratic equation in one variable to solve problems
Choose a lesson
Work through them in order.
- Not startedNotation, evaluation and algebraic modelsNextInterpret grouped algebraic notation and define a letter before substituting a value.
- Not startedExpressions and equivalenceNextDistinguish an expression from an equation and justify equivalent algebraic forms.
- Not startedRead a sequence ruleNextUse the given position in a stated rule without confusing the term value with its index.
- Not startedLinear expressions with integer coefficientsNextDistribute every sign through a bracket and combine only like terms.
- Not startedLinear expressions with fractional coefficientsNextUse common denominators to combine fractional linear coefficients without losing bracket signs.
- Not startedLinear equations with integer coefficientsNextApply the same reversible operation on both sides and check the answer in the original equation.
- Not startedEquations as balanced relationshipsNextPreserve equality through each solution step and test a candidate solution in the original equation.
- Not startedSequences and nth termsNextRelate a position to its term and distinguish a matching rule from evidence of a unique pattern.
- Not startedCommon factors and formula subjectsNextReverse distribution, change a formula's subject and find an unknown value from the given quantities.
- Not startedProducts and simple quadratic factorsNextDistribute two linear factors and reverse the product for monic quadratics.
- Not startedIdentities and general quadratic factorsNextDerive useful identities by distribution and verify every factorisation by expansion.
- Not startedFour-term common factorsNextGroup terms to expose a matching complete factor and verify it by expansion.
- Not startedAlgebraic fraction products and domainsNextCancel factors rather than terms and preserve exclusions when multiplying or dividing algebraic fractions.
- Not startedAlgebraic fraction sumsNextChoose a common denominator, retain excluded values and simplify only complete factors.
- Not startedFormulae and restrictionsNextRearrange formulae and simplify rational expressions while retaining nonzero-denominator restrictions.
- Not startedFractional linear equationsNextClear every denominator and check both the original domain and the solution.
- Not startedSimultaneous linear equationsNextConstruct two linear graphs, solve for their common pair and check graphical readings with algebra.
- Not startedZero products and quadratic equationsNextSet a factored quadratic equal to zero and retain every factor alternative.
- Not startedQuadratic formula and modelsNextIdentify coefficients, retain both possible roots and reject values only for a stated domain reason.
- Not startedCompleting the square and fractional quadraticsNextChoose an exact quadratic method and distinguish solving from rewriting a function.
- Not startedLinear inequalitiesNextReverse the order relation when multiplying or dividing by a negative number.
- Not startedSimultaneous linear inequalitiesNextIntersect conditions required together and retain the correct strict or inclusive endpoints.
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