Math-I

Watch MATH-1 workshop videos online

• Number theory & operations (integers, fractions, ratios, percentages, exponents)

• Algebra (expressions, equations, inequalities, functions, linear & quadratic equations)

• Geometry & measurement (plane geometry, coordinate geometry, area, volume, transformations)

• Statistics & probability (data analysis, graphs, probability, counting principles)

Outline

Subtest I: Number and Quantity; Algebra  

Part I: Content Domains for Subject Matter Understanding and Skill  in Mathematics  

NUMBER AND QUANTITY (SMR Domain 1)  

Candidates demonstrate an understanding of number theory and a command of number sense as  outlined in the California Common Core Content Standards for Mathematics (Grade 6, Grade 7,  Grade 8, and High School). Candidates demonstrate a depth and breadth of conceptual knowledge to  ensure a rigorous view of number systems and their underlying structures. They prove and use  properties of natural numbers. They formulate conjectures about the natural numbers using inductive  reasoning and verify conjectures with proofs.  

0001 The Real and Complex Number Systems (SMR 1.1)  

a. Demonstrate knowledge of the properties of the real number system and of its subsets  

b. Perform operations and recognize equivalent expressions using various representations of  real numbers (e.g., fractions, decimals, exponents)  

c. Solve real-world and mathematical problems using numerical and algebraic expressions and  equations  

d. Apply proportional relationships to model and solve real-world and mathematical problems   

e. Reason quantitatively and use units to solve problems (i.e., dimensional analysis)  

f. Perform operations on complex numbers and represent complex numbers and their  operations on the complex plane  

0002 Number Theory (SMR 1.2)  

 a. Prove and use basic properties of natural numbers (e.g., properties of divisibility)   

b. Use the principle of mathematical induction to prove results in number theory   

c. Apply the Euclidean Algorithm 

 d. Apply the Fundamental Theorem of Arithmetic (e.g., find the greatest common factor and  the least common multiple; show that every fraction is equivalent to a unique fraction  where the numerator and denominator are relatively prime; prove that the square root of any  number, not a perfect square number, is irrational)  

ALGEBRA (SMR Domain 2)  

Candidates demonstrate an understanding of the foundations of algebra as outlined in the California  Common Core Content Standards for Mathematics (Grade 7, Grade 8, and High School). Candidates  demonstrate a depth and breadth of conceptual knowledge to ensure a rigorous view of algebra and its  underlying structures. They are skilled at symbolic reasoning and use algebraic skills and concepts to  model a variety of problem-solving situations. They understand the power of mathematical abstraction  and symbolism.  

0003 Algebraic Structures (SMR 2.1)  

 a. Demonstrate knowledge of why the real and complex numbers are each a field, and that  particular rings are not fields (e.g., integers, polynomial rings, matrix rings)  

 b. Apply basic properties of real and complex numbers in constructing mathematical  arguments (e.g., if a < b and c < 0, then ac > bc)  

 c. Demonstrate knowledge that the rational numbers and real numbers can be ordered and that  the complex numbers cannot be ordered, but that any polynomial equation with real  coefficients can be solved in the complex field  

 d. Identify and translate between equivalent forms of algebraic expressions and equations  using a variety of techniques (e.g., factoring, applying properties of operations)   

e. Justify the steps in manipulating algebraic expressions and solving algebraic equations and  inequalities  

 f. Represent situations and solve problems using algebraic equations and inequalities  

0004 Polynomial Equations and Inequalities (SMR 2.2)  

 a. Analyze and solve polynomial equations with real coefficients using:   the Fundamental Theorem of Algebra,  the Rational Root Theorem for polynomials with integer coefficients.  the Conjugate Root Theorem for polynomial equations with real coefficients,  the Binomial Theorem 

 b. Prove and use the Factor Theorem and the quadratic formula for real and complex quadratic  polynomials  

 c. Solve polynomial inequalities  

0005 Functions (SMR 2.3)  

 a. Analyze general properties of functions (i.e., domain and range, one-to-one, onto, inverses,  composition, and differences between relations and functions) and apply arithmetic  operations on functions  

 b. Analyze properties of linear functions (e.g., slope, intercepts) using a variety of  representations  

 c. Demonstrate knowledge of why graphs of linear inequalities are half planes and be able to  apply this fact  

 d. Analyze properties of polynomial, rational, radical, and absolute value functions in a  variety of ways (e.g., graphing, solving problems)  

 e. Analyze properties of exponential and logarithmic functions in a variety of ways (e.g.,  graphing, solving problems)  

 f. Model and solve problems using nonlinear functions  

0006 Linear Algebra (SMR 2.4)  

 a. Understand and apply the geometric interpretation and basic operations of vectors in two  and three dimensions, including their scalar multiples  

 b. Prove the basic properties of vectors (e.g., perpendicular vectors have zero dot product)  

 c. Understand and apply the basic properties and operations of matrices and determinants  (e.g., to determine the solvability of linear systems of equations)  

 d. Analyze the properties of proportional relationships, lines, linear equations, and their  graphs, and the connections between them  

 e. Model and solve problems using linear equations, pairs of simultaneous linear equations,  and their graphs  

Part II: Subject Matter Skills and Abilities  

Applicable to the Content Domains in Mathematics  

Candidates for Single Subject Teaching Credentials in mathematics use inductive and deductive  reasoning to develop, analyze, draw conclusions, and validate conjectures and arguments. As they  reason both abstractly and quantitatively, they use counterexamples, construct proofs using  contradictions, construct viable arguments, and critique the reasoning of others. They create multiple  representations of the same concept. They know the interconnections among mathematical ideas, use  appropriate tools strategically, and apply techniques and concepts from different domains and sub 

domains to model the same problem. They explain mathematical interconnections with other  disciplines. They are able to communicate their mathematical thinking clearly and coherently to  others, orally, graphically, and in writing. They attend to precision, including the use of precise  language and symbols.  

Candidates make sense of routine and complex problems, solving them by selecting from a variety of  strategies. They look for and make use of structure while demonstrating persistence and reflection in  their approaches. They analyze problems through pattern recognition, look for and express regularity  in repeated reasoning, and use analogies. They formulate and prove conjectures, and test conclusions  for reasonableness and accuracy. They use counterexamples to disprove conjectures.  

Candidates select and use different representational systems (e.g., coordinates, graphs). They  understand the usefulness of transformations and symmetry to help analyze and simplify problems.  They model with mathematics to analyze mathematical structures in real contexts. They use spatial  reasoning to model and solve problems that cross disciplines.