Praxis (Praxis Core and Praxis Subject Assessments) Praxis Subject Assessment: Mathematics: Content Knowledge (5165) Syllabus

Praxis Subject Assessment: Mathematics: Content Knowledge (5165) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Praxis Subject Assessment: Mathematics: Content Knowledge (5165) in Praxis (Praxis Core and Praxis Subject Assessments), not a summary of it.

1. 1

   Number and Quantity

   3 topics

   • Real and Complex Number Systems
     • Properties of real numbers
     • Operations with complex numbers
   • Vectors and Matrices
     • Vector operations
     • Matrix algebra and determinants
   • Quantitative Reasoning and Units
2. 2

   Algebra and Functions

   4 topics

   • Expressions, Equations, and Inequalities
     • Polynomial and rational expressions
     • Solving systems and inequalities
   • Function Families
     • Linear, quadratic, and polynomial functions
     • Exponential and logarithmic functions
     • Trigonometric functions
   • Modeling with Functions
   • Sequences and Series
3. 3

   Calculus and Discrete Mathematics

   4 topics

   • Limits and Continuity
   • Derivatives and Applications
     • Rules of differentiation
     • Optimization and related rates
   • Integrals and Applications
     • Definite and indefinite integrals
     • Area and accumulation
   • Discrete Mathematics
     • Combinatorics and counting
     • Logic and set theory
4. 4

   Geometry, Probability, and Statistics

   4 topics

   • Euclidean and Coordinate Geometry
     • Congruence and similarity proofs
     • Circles and conic sections
   • Transformations and Trigonometry
   • Probability
     • Conditional probability and independence
     • Probability distributions
   • Statistics and Data Analysis
     • Sampling and experimental design
     • Inference and interpretation

Praxis Subject Assessment: Mathematics: Content Knowledge (5165) flashcards for Praxis (Praxis Core and Praxis Subject Assessments)

20 of 52 cards from the Praxis Subject Assessment: Mathematics: Content Knowledge (5165) deck — real questions with worked answers.

1. What are the standard subsets of the real number system, from most restrictive to least?

   Natural numbers (counting), whole numbers (naturals + 0), integers (whole + negatives), rationals (ratios of integers), and irrationals; rationals and irrationals together form the real numbers.

2. How do you define i (the imaginary unit), and what are the first four powers of i?

   i = √(-1). Powers cycle with period 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1.

3. How do you find the modulus (absolute value) of a complex number a + bi?

   |a + bi| = √(a² + b²), the distance from the origin to the point (a, b) in the complex plane.

4. What is the complex conjugate of a + bi, and what is the product of a complex number and its conjugate?

   The conjugate is a − bi. Their product is (a + bi)(a − bi) = a² + b², a real, nonnegative number.

5. How do you add two vectors and multiply a vector by a scalar (component form)?

   Add componentwise: ⟨a,b⟩ + ⟨c,d⟩ = ⟨a+c, b+d⟩. Scalar multiply each component: k⟨a,b⟩ = ⟨ka, kb⟩.

6. How is the dot product of vectors ⟨a,b⟩ and ⟨c,d⟩ computed, and what does it equal in terms of angle?

   Dot product = ac + bd. It also equals |u||v|cos θ, where θ is the angle between them; a dot product of 0 means the vectors are perpendicular.

7. When can two matrices A (m×n) and B (p×q) be multiplied, and what is the size of the product?

   They can be multiplied only if n = p (inner dimensions match). The product AB is m×q.

8. How do you compute the determinant of a 2×2 matrix [[a,b],[c,d]]?

   Determinant = ad − bc.

9. What is the inverse of a 2×2 matrix [[a,b],[c,d]], and when does it fail to exist?

   Inverse = (1/(ad−bc)) [[d,−b],[−c,a]]. It does not exist when the determinant ad−bc = 0 (singular matrix).

10. In dimensional analysis, how do you convert 60 miles per hour to feet per second?

    Multiply by conversion factors: 60 mi/hr × 5280 ft/mi × 1 hr/3600 s = 88 ft/s.

11. What is the difference between accuracy and precision in measurement?

    Accuracy is how close a measurement is to the true value; precision is how close repeated measurements are to each other (consistency/reproducibility).

12. How do you determine the number of significant figures appropriate when multiplying measured quantities?

    The result should have the same number of significant figures as the factor with the fewest significant figures.

13. What does the discriminant b² − 4ac reveal about the roots of a quadratic ax² + bx + c = 0?

    Positive: two distinct real roots; zero: one repeated real root; negative: two complex conjugate roots.

14. State the quadratic formula for solving ax² + bx + c = 0.

    x = (−b ± √(b² − 4ac)) / (2a).

15. How do you solve a system of two linear equations by elimination?

    Scale one or both equations so a variable's coefficients are opposites, add the equations to eliminate that variable, solve for the remaining variable, then back-substitute.

16. What rule governs the inequality sign when you multiply or divide both sides by a negative number?

    The inequality sign must be reversed (flipped).

17. How do you factor the difference of two squares a² − b²?

    a² − b² = (a + b)(a − b).

18. What are the key characteristics (domain, range, shape) of the parent linear function f(x) = x?

    Domain and range are all real numbers; the graph is a straight line through the origin with slope 1; it is increasing everywhere.

19. What are the domain and range of the parent square root function f(x) = √x?

    Domain: x ≥ 0; range: y ≥ 0; the graph starts at the origin and increases, concave down.

20. What are the domain and range of the exponential parent function f(x) = bˣ (b > 1)?

    Domain: all real numbers; range: y > 0; horizontal asymptote y = 0; increasing with y-intercept (0, 1).

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