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ALGEBRA LAB
Learn it by playing it
Four short missions on live algebra stages. Grab a dial, watch the picture move, and figure out the rule. When a mission glows green, read why it worked.
1234
Mission 1 of 4
Build 5, 8, 11, 14, ...
Two dials grow a sequence: the step d and the offset b. Make the bars read 5, 8, 11, 14.
In Tn = d*n + b, which letter is the constant difference between terms?
Your gaps are 1, but the target climbs by 3. Set d first.
Study notes
Expressions and substitution C2.1 / E2.1
A letter in algebra stands for a number - either an unknown to find or a variable that can change.
Substitute carefully: put brackets around negative values, e.g. if x = −3 then x² = (−3)² = 9, not −9.
Follow the normal order of operations when evaluating: brackets, indices, multiply/divide, add/subtract.
An expression has no equals sign (2x + 5); an equation states two expressions are equal and can be solved.
A formula expresses one quantity in terms of others, e.g. v = u + at; each letter has a meaning in context.
Expanding and factorising C2.2 / E2.2
Collect like terms only: 2a + 3b + 5a − 9b = 7a − 6b. Terms with different letters or powers stay separate.
Expand a single bracket by multiplying every term inside: 3x(2x − 4y) = 6x² − 12xy.
Expand double brackets term by term (sometimes called FOIL): (2x + 1)(x − 4) = 2x² − 8x + x − 4 = 2x² − 7x − 4.
Factorise fully by taking out the highest common factor of every term: 9x² + 15xy = 3x(3x + 5y).
Check by expanding back: a correct factorisation reproduces the original expression exactly.
Algebraic fractions E2.3 (Extended only)
Simplify algebraic fractions by factorising numerator and denominator, then cancelling common factors: (x² − 9)/(x + 3) = x − 3 for x ≠ −3.
Never cancel across added terms: (x + 3)/3 does not simplify to x; only common FACTORS of the whole numerator and denominator cancel.
Add or subtract algebraic fractions using a common denominator, exactly as with number fractions.
Multiply by multiplying tops and bottoms; divide by inverting the second fraction and multiplying.
Note the values a denominator excludes: a fraction is undefined wherever its denominator is zero.
Indices C2.4 / E2.4
Rules of indices for a non-zero base: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ.
For a non-zero base a, a⁰ = 1; a⁻ⁿ = 1/aⁿ flips the power to a reciprocal.
Fractional powers are roots: a^(1/n) = ⁿ√a (for a ≥ 0 when n is even), and a^(m/n) = (ⁿ√a)ᵐ.
Solve index equations by writing both sides with the same base: 2ˣ = 32 = 2⁵ gives x = 5.
Simplify expressions term by term: 12a⁵ ÷ 3a⁻² = 4a⁵⁻⁽⁻²⁾ = 4a⁷; (5x³)² = 25x⁶.
Equations and changing the subject C2.5 / E2.5
Solve a linear equation by doing the same thing to both sides: 3x + 4 = 10 gives 3x = 6, so x = 2.
With brackets or unknowns on both sides, expand and collect first: 5 − 2x = 3(x + 7) becomes 5 − 2x = 3x + 21, so −16 = 5x and x = −16/5.
Simultaneous linear equations: eliminate one unknown by adding or subtracting scaled equations, then substitute back to find the other.
Construct the equations from the words first when a problem is given in context - define the letters you use.
Change the subject by reversing the operations in reverse order; the new subject must end up alone on one side.
Inequalities C2.6 / E2.6
Solve inequalities like equations, but multiplying or dividing by a NEGATIVE number reverses the sign: −2x > 6 gives x < −3.
On a number line use an open circle for strict bounds (<, >) and a closed (filled) circle for inclusive bounds (≤, ≥).
A combined inequality like −3 ≤ x < 1 keeps both conditions at once; read it as x is at least −3 and less than 1.
Always state the solution set the question asks for - sometimes integer values only.
Sequences and the nth term C2.7 / E2.7
Continue a sequence by spotting the term-to-term rule: 1, 3, 6, 10, 15, ... increases by one more each time (next: 21, 28).
A linear sequence has a constant difference d; its nth term is dn + b, found by matching the first term.
A simple quadratic sequence has constant SECOND differences; its nth term starts with (d/2)n² where d is the second difference.
Recognise special sequences: squares (1, 4, 9, ...), cubes, triangular numbers and powers of 2.
Subscript notation Tn means the nth term of sequence T; use it to write rules like Tn = 3n + 2.
Proportion E2.8 (Extended only)
Direct proportion: y = kx for a constant k; doubling x doubles y. Find k from one pair of values first.
Inverse proportion: y = k/x for a non-zero x; doubling x halves y.
Powers also appear: y proportional to x² means y = kx², and inverse square means y = k/x².
Method: write the proportion as an equation with k, substitute known values to find k, then use the equation for new values.
Graphs in practical situations C2.9 / E2.9
The gradient of a straight-line graph is a rate of change: on a distance-time graph it is speed.
A horizontal section of a distance-time graph means stopped; a steeper section means faster.
On a conversion graph, read both directions and check the units on each axis.
Draw graphs from data with a sensible scale, labelled axes and a smooth fit through the points where appropriate.
Graphs of functions and solving graphically C2.10 / E2.10
Build a table of values before plotting; choose x values that show the interesting behaviour (roots, turning points, asymptotes).
Know the shapes: y = ax + b is a straight line; y = ±x² + ax + b is a parabola; y = a/x (x ≠ 0) is a reciprocal curve in two branches with the axes as asymptotes.
Solve an equation graphically by reading the x-coordinates where the graph meets the x-axis (roots) or meets another graph (intersections).
The intersection of a line and a curve gives the solutions of their simultaneous equations - read both coordinates.
Sketching linear and quadratic curves C2.11 / E2.11
A linear sketch needs just the gradient and the y-intercept (or any two points).
A quadratic sketch needs the direction (up for a positive x² coefficient, down for negative), the roots and the y-intercept.
Quadratics are symmetric about the vertical line through their turning point; use that to place the curve.
Extended: cubics, reciprocals and exponentials have characteristic shapes worth recognising on sight.
Functions and gradient of a curve E2.12-E2.13 (Extended only)
Function notation f(x) names a rule: f(x) = 2x + 1 means double the input and add 1; fg(x) means apply g first, then f.
The inverse f⁻¹(x) reverses the rule; it exists when the function is one-to-one.
The gradient of a curve at a point is the gradient of the tangent there - estimated from a drawn tangent as change in y over change in x.
Differentiation gives the gradient function directly: if y = xⁿ then dy/dx = nxⁿ⁻¹.
EduTribe study notes - original material aligned to the Cambridge IGCSE Mathematics 0580 syllabus (2025-2027). Automated checks completed; not yet reviewed by an educator.