Grades 6–8AlgebraPlanned
Variables and algebraic expressions
Letters as variables standing for numbers, writing and evaluating algebraic expressions, combining like terms, using the distributive law, and the laws of exponents for whole-number powers.
math.algebra.variables-expressions
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 6
You will learn to
- Write algebraic expressions for situations and evaluate them for given values, including negative ones.
- Simplify expressions by combining like terms, and expand and factorise using the distributive law.
- Use the laws of exponents aᵐaⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ for whole-number exponents.
Definition
- A variable is a letter that stands for any number from a given set.
- Two expressions are equivalent if they have the same value for every value of their variables.
- For a whole number n ≥ 1, aⁿ is the product of n factors each equal to a; a⁰ = 1 for a ≠ 0.
Theorem
- a(b + c) = ab + ac for all numbers a, b and c.
- aᵐ · aⁿ = aᵐ⁺ⁿ and (ab)ⁿ = aⁿbⁿ.
- Testing an equivalence with a few values can refute it but cannot prove it.
Common misconceptions
- Writing (a + b)² = a² + b².
- Treating 2x and x² as the same.
- Believing x must be positive, or that different letters must have different values.
Leads to
Optional discoveries
- Rings, ideals and fields The rules for manipulating expressions are the laws of a commutative ring, so the same manipulations are valid wherever those laws hold, for example with remainders modulo n.
Learning pathways
Further reading
- I. M. Gelfand and A. Shen, Algebra (1993)