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Algebra · Recognition Drill
Algebra · 30 stems · 8s per card
Look at the stem. Classify it. Don't solve it. Each card gives you 5 seconds (8 in coach mode). After 30 stems you'll see your accuracy and median classification time. Aim: 80%+ accuracy in under 5 seconds per card.
Year-8 notation glossary — read this first
roots of P(x)
The values of x that make P(x) = 0. E.g., roots of x² − 5x + 6 are 2 and 3.
Σ (sigma) —
Add f(1) + f(2) + ⋯ + f(N).
⌊x⌋ (floor), ⌈x⌉ (ceiling)
⌊x⌋ rounds DOWN to an integer; ⌈x⌉ rounds UP.
i (imaginary unit)
A number with i² = −1. Then any 'complex number' is written a + bi.
|a + bi| (complex modulus)
Distance from 0 in the plane: √(a² + b²).
log_b(x)
The power you raise b to in order to get x. E.g., log₂ 8 = 3 because 2³ = 8.
The inverse function — undoes f. If f(2) = 5 then f⁻¹(5) = 2.
Technique deck — what the button labels mean
Vieta's formulas
Relate sum/product of roots to coefficients of a polynomial.
SFFT
Simon's Favourite Factoring Trick — add a constant to factor expressions like xy + ax + by.
Telescoping
Sums where consecutive terms cancel, leaving only the boundary terms.
Arithmetic sequence
Constant difference between consecutive terms.
Geometric sequence
Constant ratio between consecutive terms.
AM-GM
Arithmetic mean ≥ geometric mean — extremal trick for positive numbers.
Log laws
log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a.
Cycle of i
Powers of i repeat every 4: i, −1, −i, 1, …
De Moivre
(r cis θ)ⁿ = rⁿ cis(nθ) — for powers of complex numbers in polar form.
Complex modulus
|a + bi| = √(a² + b²) — the distance from the origin.
Roots of unity
Solutions of zⁿ = 1; n equally-spaced points on the unit circle.
Remainder theorem
The remainder of P(x) ÷ (x − a) equals P(a).
Factor theorem
If P(a) = 0 then (x − a) is a factor of P.
Polynomial interpolation (finite differences)
Reconstruct a low-degree polynomial from sample values; constant n-th differences mean degree ≤ n.
Word problem (rate equation)
Translate a story into algebraic equations (distance = rate × time, etc.).
Floor / ceiling
⌊x⌋ = round down to an integer; ⌈x⌉ = round up.
Functional iteration
Repeatedly apply a function: f(f(x)), f(f(f(x))), …
Absolute-value casework
Split into cases where each |·| expression is positive vs negative.
Difference of squares
a² − b² = (a − b)(a + b).
Substitution
Let y = some expression (often y = 2^x or y = √x) to collapse to a simpler equation.
Geometric mean (definition)
ⁿ√(x₁·x₂·…·xₙ) — the n-th root of the product.
Inverse function
f⁻¹(y) is the value of x for which f(x) = y.
Change of base
log_b(x) = log(x) / log(b) — convert any base to a known one.
Arithmetic series
Sum of terms in an arithmetic sequence: n × (first + last) / 2.
Quadratic formula
Solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a.
Symmetric identity
Identities like a² + b² = (a + b)² − 2ab that rewrite symmetric expressions.