Build formula fluency, then train four habits - plug‑in, back‑solve, draw‑and‑label, units‑first. Run timed, adaptive sets, tag errors by type, and reteach fast. Use the assets below to operationalize your class or center.
Who this page is for?
ACT teachers, tutors, department leads, and test‑prep business owners.
Copy is classroom‑ready and cohort‑friendly. Use it for lesson plans, homework, and remediation.
Domains: algebra, functions, geometry, trigonometry, statistics, probability.
Most questions are single‑ or two‑step; a few multi‑step items add traps. Calculator is allowed.
Program targets for a 4–6 week cycle:
Timing accuracy ≥ 80% on mixed sets.
Formula recall ≥ 90% (3‑min sprint checks).
Repeat‑error rate ≤ 10% (using the error log).
Two full timed tests in the final week.
Planning help for your calendar and parent comms:
ACT Test Dates & Deadlines 2025-26
Translate → Inventory → Connect → Compute.
Underline the ask, list givens with units, choose a method, compute cleanly.
Core methods to teach and reuse across topics:
Plug‑in: Pick easy numbers for abstract algebra.
Back‑solve: Substitute choices into equations; stop at first fit
Draw‑and‑label: Quick diagram for geometry and word problems.
Units‑first: Track units at every step; cancel like factors.
Share the printable with students and use 3‑minute “formula sprints” each class.
m = (y₂ − y₁) / (x₂ − x₁)
y = m x + b
y − y₁ = m(x − x₁)
Midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2)
√((x₂ − x₁)² + (y₂ − y₁)²)
Parallels: slopes equal; Perpendicular: m₁·m₂ = −1
x = [−b ± √(b² − 4ac)] / (2a)
Discriminant: b² − 4ac
Vertex x = −b/(2a)
Zeros from a(x − r₁)(x − r₂)
(x − h)² + (y − k)² = r²
a² + b² = c²
45‑45‑90 (1:1:√2), 30‑60‑90 (1:√3:2)
A = ½ b h
A = b h
A = ½(b₁ + b₂)h
A = l w; A = s²
A = π r²; C = 2π r
A_sector = (θ/360)π r²; Arc = (θ/360)·2π r
V = B h; cylinder V = π r² h
V = ⅓ B h; cone V = ⅓ π r² h
Sphere V = ⁴⁄₃ π r³; SA = 4π r²
Exponents: a^m a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n
Logs: log(ab) = log a + log b; log(a^k) = k·log a
Mean/Median/Mode/Range
P = favorable/total; independent: P(A∩B) = P(A)P(B)
nPr = n!/(n−r)!; nCr = n!/[r!(n−r)!]
aₙ = a₁ + (n−1)d; Sₙ = (n/2)(a₁ + aₙ)
aₙ = a₁ r^(n−1); Sₙ = a₁(1 − r^n)/(1 − r)
180° = π rad
sin = opp/hyp; cos = adj/hyp; tan = opp/adj
Coaching cues:
Teach “name → trigger → 1 example.”
Test recall weekly; reteach any miss in the first five.
Pair each formula with one “wrong-path” example so students learn the trap.
Algebra & Equations
Systems: Prefer elimination; align coefficients to reduce steps.
Quadratics: Connect factors ↔ zeros ↔ vertex; choose the fastest view.
Example: If 2x + 3 = 13 → x = 5. If the ask is 2x − 12x − 1, answer 99 directly—don’t stop at x.
Functions & Graphs
Lines: Identify slope and intercept quickly.
Transformations: y = f(x ± a) ± b → shifts left/right, up/down; a outside scales.
Example: y = 2f(x − 3) + 4 ⇒ right 3, stretch by 2, up 4.
Geometry
Similar triangles: Set side ratios; one‑step solve.
Angles: Parallel lines create equal or supplementary angles.
Example: Scale factor 2 doubles all sides; area scales by 4.
Trigonometry
Defaults: SOH‑CAH‑TOA.
Special triangles: 30°‑60°‑90° and 45°‑45°‑90° for instant side recovery.
Example: In 30°‑60°‑90° with hypotenuse 10, short = 5, long = 5√3.
Statistics & Probability
Tables/trees: Keep totals straight; avoid double counts.
Complements: “At least one” = 1 − P(none).
Example: Two independent events 0.3 and 0.5 → both 0.15; at least one 1 − (0.7 × 0.5) = 0.65.
One minute per item on average; some take 20–30 seconds.
Teach: skip early, star time‑eaters, return later.
No penalty for wrong answers — bubble everything and guess one letter at the end.
Structure & policies: confirm using ACT Mathematics Test overview and ACT Calculator Policy
2‑Week Sprint (daily 45–60 min)
Day 1–2: Formula sprints + algebra core
Day 3–4: Geometry/Trig labs + 40‑Q timed sets.
Day 5: Mixed 60‑Q under real timing.
Day 6: Error‑type reteach (formula/setup/arithmetic/timing)
Day 7–8: Functions + stats/probability.
Day 9: Hard‑only set (plug‑in/back‑solve).
Day 10: Full test.
Day 11: Deep review + redo starred.
Day 12–14: Speed sets + final full test.
4-Week Standard (5 days/week)
Week 1: Formulas + algebra; 30‑Q checkpoint.
Week 2: Geometry + trig; 30‑Q checkpoint.
Week 3: Functions + stats/probability; 60‑Q test
Week 4: Two full tests + targeted reteach based on error logs.
Formula Sheet (PDF) - one page with 31 formulas.
Error Log (Google Doc template) - track cause of miss + fix.
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