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Algebra and equations

Build and connect the representations and methods in algebra and equations.

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  • 19 lessons
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Before you begin

Build and connect the representations and methods in algebra and equations.

Learning goals
  • Interpret algebraic notation and represented operations
  • Evaluate an algebraic expression by substitution
  • Represent a situation with algebraic expressions
  • Express and use the general term of a sequence
  • Simplify linear expressions by combining like terms
  • Solve linear equations and check the solution
  • Formulate, use and interpret a mathematical model
  • Expand and factorise algebraic expressions
  • 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 simultaneous equations by justified methods
  • Interpret and draw graphs of functions
  • Solve quadratic equations by suitable methods
  • 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 simple inequalities ax ≤ b, ax ≥ b, ax < b, ax > b where a and b are integers
  • 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 with integral coefficients such as 2(x − 3y); 4x − 2(3x − 5); 3(x − y) − (2y + x) − y
  • concepts of equation
  • solving linear equations with integral coefficients in one variable
  • formulating a linear equation in one variable to solve problems
  • simplification of linear expressions with fractional coefficients such as 2x/3 − 3(x − 5)/2
  • expansion of the product of two linear expressions
  • use brackets and extract common factors
  • 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 quadratic expressions ax^2 + bx + c
  • multiplication and division of simple algebraic fractions such as (3a/(4b²))(5ab/3); (3a/4) ÷ (9a²/10)
  • solving linear equations in one variable (including fractional coefficients)
  • concept and properties of inequality
  • solving simple inequalities in the form ax ≤ b, ax ≥ b , ax < b and ax > b, where a and b are integers.
  • solving simple fractional equations that can be reduced to linear equations such as x/3 + (x − 2)/4 = 3; 3/(x − 2) = 6
  • graphs of linear equations in two variables(ax + by = c)
  • solving simultaneous linear equations in two variables by: • substitution and elimination methods • graphical method
  • formulating a linear equation in one variable or a pair of linear equations in two variables to solve problems
  • factorisation of linear expressions of the form 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
  • 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 quadratic equations in one variable by: • factorisation • 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
  • formulating a quadratic equation in one variable to solve problems

Lessons

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  1. Notation, evaluation and algebraic modelsG2 Mathematics 2027 · Normal (Academic) Secondary 1 Mathematics 2020Interpret grouped algebraic notation and define a letter before substituting a value.
  2. Read a sequence ruleG2 Mathematics 2027 · Normal (Academic) Secondary 1 Mathematics 2020Use the given position in a stated rule without confusing the term value with its index.
  3. Linear expressions with integer coefficientsG2 Mathematics 2027 · Normal (Academic) Secondary 1 Mathematics 2020Distribute every sign through a bracket and combine only like terms.
  4. Linear expressions with fractional coefficientsG2 Mathematics 2027 · Normal (Academic) Secondary 2 Mathematics 2021Use common denominators to combine fractional linear coefficients without losing bracket signs.
  5. Linear equations with integer coefficientsG2 Mathematics 2027 · Normal (Academic) Secondary 1 Mathematics 2020Apply the same reversible operation on both sides and check the answer in the original equation.
  6. Equations as balanced relationshipsG2 Mathematics 2027 · Normal (Academic) Secondary 2 Mathematics 2021Preserve equality through each solution step and test a candidate solution in the original equation.
  7. Sequences and nth termsG2 Mathematics 2027 · Normal (Academic) Secondary 1 Mathematics 2020Relate a position to its term and distinguish a matching rule from evidence of a unique pattern.
  8. Common factors and formula subjectsG2 Mathematics 2027 · Normal (Academic) Secondary 3–4 Mathematics 2022Reverse distribution, change a formula's subject and find an unknown value from the given quantities.
  9. Products and simple quadratic factorsG2 Mathematics 2027 · Normal (Academic) Secondary 2 Mathematics 2021Distribute two linear factors and reverse the product for monic quadratics.
  10. Identities and general quadratic factorsG2 Mathematics 2027 · Normal (Academic) Secondary 2 Mathematics 2021Derive useful identities by distribution and verify every factorisation by expansion.
  11. Four-term common factorsG2 Mathematics 2027 · Normal (Academic) Secondary 3–4 Mathematics 2022Group terms to expose a matching complete factor and verify it by expansion.
  12. Algebraic fraction products and domainsG2 Mathematics 2027 · Normal (Academic) Secondary 2 Mathematics 2021Cancel factors rather than terms and preserve exclusions when multiplying or dividing algebraic fractions.
  13. Algebraic fraction sumsG2 Mathematics 2027 · Normal (Academic) Secondary 3–4 Mathematics 2022Choose a common denominator, retain excluded values and simplify only complete factors.
  14. Fractional linear equationsG2 Mathematics 2027 · Normal (Academic) Secondary 2 Mathematics 2021Clear every denominator and check both the original domain and the solution.
  15. Simultaneous linear equationsG2 Mathematics 2027 · Normal (Academic) Secondary 2 Mathematics 2021Construct two linear graphs, solve for their common pair and check graphical readings with algebra.
  16. Zero products and quadratic equationsG2 Mathematics 2027 · Normal (Academic) Secondary 3–4 Mathematics 2022Set a factored quadratic equal to zero and retain every factor alternative.
  17. Quadratic formula and modelsG2 Mathematics 2027 · Normal (Academic) Secondary 3–4 Mathematics 2022Identify coefficients, retain both possible roots and reject values only for a stated domain reason.
  18. Completing the square and fractional quadraticsG2 Mathematics 2027 · Normal (Academic) Secondary 3–4 Mathematics 2022Choose an exact quadratic method and distinguish solving from rewriting a function.
  19. Linear inequalitiesG2 Mathematics 2027 · Normal (Academic) Secondary 2 Mathematics 2021Reverse the order relation when multiplying or dividing by a negative number.

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