Math Path

Study library · 8 curated guides

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Curated high-yield

ALG · High-yield

See structure before you compute

Algebra items often hide an equivalent form. Pause to factor, complete the square, or rewrite radicals before grinding arithmetic.

  • Ask: can I factor, conjugate, or substitute?
  • Track domain after every rewrite (especially radicals/rationals).
  • Zero product property only after the equation is literally a product = 0.

Exam tip: If two answer choices are algebraically equivalent, you mis-simplified — back up.

ALG · High-yield

Complex numbers on Praxis

Know i-cycle powers, conjugates, and that nonreal roots of real polynomials come in conjugate pairs.

  • i¹=i, i²=−1, i³=−i, i⁴=1, then repeat.
  • Multiply by conjugate to simplify quotients.
  • Discriminant < 0 ⇒ complex conjugate roots for real coefficients.

Exam tip: Reduce exponents of i mod 4 before multiplying big products.

FUNC · High-yield

Four representations or bust

Functions questions move among graphs, tables, equations, and stories. Practice translating the same feature all four ways.

  • Intercepts, asymptotes, monotonicity, and end behavior should be namable from any representation.
  • Transformations: outside = vertical; inside = horizontal (and opposite sign).
  • Average rate of change is the slope of the secant — always write the formula first.

Exam tip: When stuck, sketch a quick graph even if the item is algebraic.

FUNC · High-yield

Calculus meaning over machinery

5165 emphasizes derivative as rate and integral as accumulation more than heavy symbolic drills.

  • f′ > 0 increasing; f′ < 0 decreasing; f′=0 critical.
  • f′′ sign ↔ concavity; inflection where concavity changes.
  • Net change = integral of rate; total distance needs |rate| thinking.

Exam tip: Read the units in worded calculus items — they often reveal whether you need f, f′, or ∫f.

GEO · High-yield

Congruence vs similarity checklist

Keep a one-page cheat sheet of SSS/SAS/ASA/AAS vs AA similarity, and remember SSA is dangerous.

  • Congruence criteria need corresponding parts carefully matched.
  • Similar ⇒ proportional sides; congruent ⇒ equal sides.
  • Isometries (translate/rotate/reflect) preserve distance; dilations do not.

Exam tip: On diagrams, mark given ticks/arcs before hunting a criterion.

STAT · High-yield

Stats is a reading test too

Many misses are language: bias, confounding, conditional probability wording, and correlation vs causation.

  • Describe distributions with shape, center, spread, outliers.
  • P(A|B) means restrict to B first — use a table.
  • Experiments assign treatments; observational studies do not.

Exam tip: If a choice claims 'proves causes,' be suspicious unless it was a well-designed experiment.

ALG · High-yield

Tasks of Teaching Mathematics

About 25% of Praxis items embed content in teaching moves: diagnose errors, choose representations, evaluate explanations.

  • Name the misconception precisely (not just 'student is wrong').
  • Pick the representation that makes the structure visible.
  • Next instructional step should address the misconception, not pile on new content.

Exam tip: Ask: what would a strong teacher say or show next?

FUNC · High-yield

NJ CE path reminder

Content study is half the battle — you still need 30 math credits (12 advanced), Praxis 5165 ≥ 159, and the CE application steps.

  • This site builds Praxis fluency; college courses create transcript eligibility.
  • Physiology & hygiene can be NJ's online test if you lack bio/health credits.
  • After CE + job offer comes Provisional Teaching Program coursework.

Exam tip: Keep a running list of completed math courses with levels (100/200 vs 300+).