NUMBER THEORY (6 Multiple Choice Items, 1 Constructed Response Item)
Natural Numbers
a. Prove and use basic properties of natural numbers (e.g., properties of divisibility)
Lesson 92. Divisibility Concept: https://www.youtube.com/watch?v=dIfpZzX7bKo,
https://www.youtube.com/watch?v=DZKIzksdbpQ, https://www.youtube.com/watch?v=Df9h5t64NlQ
Lesson 93. Proving divisibility theorems: https://www.youtube.com/watch?v=0vIj582-rbQ
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b. Use the Principle of Mathematical Induction to prove results in number theory
Lesson 94. Introduction: https://www.youtube.com/watch?v=ruBnYcLzVlU
Lesson 95. Examples: https://www.youtube.com/watch?v=wblW_M_HVQ8,
https://www.youtube.com/watch?v=IFqna5F0kW8, https://www.youtube.com/watch?v=twA6vZgX_U4
Lesson 96. Divisibility and Induction: https://www.youtube.com/watch?v=99t7DkeLqWE,
https://www.youtube.com/watch?v=_OgoNmxBopQ, https://www.youtube.com/watch?v=3_0wkumMaN4
CSET Math Questions: https://www.youtube.com/watch?v=Zl91f5yuf-0,
https://www.youtube.com/watch?v=048Kf2ht8SE&list=PLS0TM4AjRKMbYou37x7I4eMTTmI_eSagw&index=13
c. Know and apply the Euclidean Algorithm
Lesson 97. Introduction: https://www.youtube.com/watch?v=fwuj4yzoX1o,
https://www.youtube.com/watch?v=JUzYl1TYMcU, https://www.youtube.com/watch?v=p5gn2hj51hs
CSET Math Question: https://www.youtube.com/watch?v=-qhLorxxI5Q
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)
Lesson 98. Introduction: https://www.youtube.com/watch?v=Ws42X_u1MR8,
https://www.youtube.com/watch?v=uzvopM4PZLw,
https://www.youtube.com/watch?v=33N2dkhIF4w
Lesson 99. Proof of Irrationationality of Certain Square Roots: https://www.youtube.com/watch?v=mX91_3GQqLY,
https://www.youtube.com/watch?v=176lZt90Xvs
NN1. Assessment: Practice Problems (1.1: 22-27, 30; 1.2: 23, 24, 27-30), Induction (all)
ADDITIONAL CSET MATH SUBTEST I REVIEW PROBLEMS - MUST DO!
ALGEBRA AND FUNCTIONS
Algebra Practice Questions: Math Guru
More Practice Precalculus Exams (Focus on Functions, Polynomial Equations and Inequalities, and Algebra): Skip those problems that are not covered by the CSET I exam
LINEAR ALGEBRA
Matrix Problems Set A Set B (Matrix Tutorial has several sections with multiple choice items)
ALGEBRAIC STRUCTURES AND NUMBER THEORY
Divisibility Proof Exercises: Hildebrand's Problems
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Great resource on Abstract Algebra for Beginners with good problems: B. A. Sethuraman's A Gentle Introduction to Abstract Algebra
INTEGRATED REVIEW
Released CSET Math Subtest I Items
Free CSET Math Subtest I Practice Tests: Version 1 Version 2
CSET 1 REVIEW
ALGEBRA (24 Multiple Choice Items, 3 Constructed Response Items)
Preliminaries: Basic Algebra (Not a part of CSET I specs)
Lesson 1. Definition of a Rational Number: https://www.youtube.com/watch?v=6QKcW22IYe0, https://www.youtube.com/watch?v=X6t1e77fNU4
Lesson 2. Definition of a Complex Number: https://www.youtube.com/watch?v=uaI_4KVtp2k,
https://www.youtube.com/watch?v=zE3ZV8-FJ98, https://www.youtube.com/watch?v=cTqZYEtTQhA,
https://www.youtube.com/watch?v=H_50Y36AK4k, https://www.youtube.com/watch?v=JoLq9NaLOvI, https://www.youtube.com/watch?v=KgYl6YIaVU4,
Functions Part I
a. Analyze and prove general properties of functions (i.e., domain and range, one-to-one, onto, inverses, composition, and differences between relations and functions)
a.1 Functions and Relations
Lesson 3. Functions and Relations: https://www.youtube.com/watch?v=NlhQqmLN51M
Lesson 4. Functions and Relations: https://www.youtube.com/watch?v=Uz0MtFlLD-k
Lesson 5. Function Notation: https://www.youtube.com/watch?v=kagLx6MW9JY
a.2 Domain and Range
Lesson 6. Interval Notation: https://www.youtube.com/watch?v=sq2SlrkZeI8
Lesson 7. Domain and Range Graphically: https://www.youtube.com/watch?v=I0f9O7Y2xI4
Lesson 8. Domain and Range Graphically: https://www.youtube.com/watch?v=QAxZEelInJc
Lesson 9. Domain and Range Algebraically: https://www.youtube.com/watch?v=sbvSBH2Mo20
Lesson 10. Finding the Domain and Range of a Function Algebraically: https://www.youtube.com/watch?v=hZEGZMb4uzQ
Required Functions to Know: Linear, Quadratic, and Odd/Even Functions
Lesson 11. Linear Functions: Khan Academy Module on LFs
Lesson 12. Quadratic Functions: Khan Academy Module on QFs
Lesson 13. Graphs of Odd and Even Functions: Definitions, Symmetry
Assessment: Practice Problems Lines #s 1-8 excluding #7, Quadratic Equations #5, Quadratic Equations #s9 and 10
a.3 One-to-One Functions
Lesson 14. Definition: https://www.youtube.com/watch?v=1NPvkYLjjws
Lesson 15. Determining Whether a Function is 1-1: https://www.youtube.com/watch?v=9b8Xd77f-qk
a.4 Onto Functions
Lesson 16. Onto Functions: https://www.youtube.com/watch?v=xKNX8BUWR0g
Lesson 17. Onto Functions: https://www.youtube.com/watch?v=KiKJt4gywYE
a.4 Inverse Functions
Lesson 18. Definition of an Inverse of a Function: https://www.youtube.com/watch?v=JPPitlVKjWQ
Lesson 19. Finding the Inverse of a Function: https://www.youtube.com/watch?v=q739Wcf0ZUg
Lesson 20. Checking the Inverse: https://www.youtube.com/watch?v=nSmFzOpxhbY
Lesson 21. Finding the Inverse of a Function: https://www.youtube.com/watch?v=Bq7Jd7vXC08
a.5 Operations Involving Functions
Lesson 22. Basic Operations: https://www.youtube.com/watch?v=5934UqzXvKA
Lesson 23. Composition of Functions: https://www.youtube.com/watch?v=PCHkzAxiJwU
Lesson 24. All Operations of Functions: https://www.youtube.com/watch?v=X5Qyf3UxIrU
F1. Assessment: Practice Problems Domain and Range, Quadratic Functions, Inverse Functions, All Functions,
Operations Involving Functions #6, Inverse Functions #2, More Functions
Polynomial Equations and Inequalities
a. Know why graphs of linear inequalities are half planes and be able to apply this fact (e.g., linear programming)
Lesson 25. Graphing Lines: https://www.youtube.com/watch?v=2UrcUfBizyw
Lesson 26. Graphing Linear Inequalities: https://www.youtube.com/watch?v=lxTQrsUip9g
Lesson 27. Solving Linear Programming Problems Introduction: https://www.youtube.com/watch?v=6M6jKr4rMoI
Lesson 28. Solving Linear Programming Problems: https://www.youtube.com/watch?v=2ACJ9ewUC6U
Lesson 29. Solving Linear Programming Problems: https://www.youtube.com/watch?v=hccsOVEqizU
PEI1. Assessment: Practice Problems Review 1 #7, Review 2 #5, Review 3 #7
b. Prove and use the following:
b.1 The Rational Root Theorem for polynomials with integer coefficients
Lesson 30. Rational Roots Theorem: https://www.youtube.com/watch?v=YMyv9-9VXw4
Lesson 31. Rational Roots Theorem: https://www.youtube.com/watch?v=jERVeo3Dtkw
b.2 The Factor Theorem
Lesson 32. Factor and Remainder Theorems: https://www.youtube.com/watch?v=_IPqCaspZOs
Lesson 33. Factor Theorem Problems: https://www.youtube.com/watch?v=6G3iAgpK4kA
CSET Math Questions: https://www.youtube.com/watch?v=6G3iAgpK4kA&list=PL6AC21728CD1693DC
https://www.youtube.com/watch?v=yseY3IIGVus
b.3 The Conjugate Roots Theorem for polynomial equations with real coefficients
Lesson 34. Conjugate Pairs: https://www.youtube.com/watch?v=ZQjq-o9ZBek
Lesson 35. Polynomials that Have Conjugate Pairs: https://www.youtube.com/watch?v=PQr0yVq5ysc
b.4 The Quadratic Formula for real and complex quadratic polynomials
Lesson 36. Quadratic Formula with Real Roots: https://www.youtube.com/watch?v=i7idZfS8t8w
Lesson 37. Quadratic Formula with Complex Roots: https://www.youtube.com/watch?v=dnjK4DPqh0k
CSET Math Questions: https://www.youtube.com/watch?v=ciFe5sRdRzI, https://www.youtube.com/watch?v=0vkqaOOfn1A,
https://www.youtube.com/watch?v=LWOK7IoHgd4&list=PL06D3A067F97DA04F
b.5 The Binomial Theorem
Lesson 38. Binomial Theorem Using Pascal's Triangle: https://www.youtube.com/watch?v=NLQmQGA4a3M
Lesson 39. Binomial Theorem: https://www.youtube.com/watch?v=iPwrDWQ7hPc
PEI2. Assessment: Practice Problems (p. 8)
c. Analyze and solve polynomial equations with real coefficients using the Fundamental Theorem of Algebra
Lesson 40. Fundamental Theorem of Algebra: https://www.youtube.com/watch?v=d8-LO6FCna0
Lesson 41. Fundamental Theorem of Algebra: https://www.youtube.com/watch?v=b4VHq7NDsVo
Functions Part II
b. Analyze properties of polynomial, rational, radical, and absolute value functions in a variety of ways (e.g., graphing, solving problems)
b.1 Properties of Graphs of Polynomial Functions
Lesson 42. End Behavior of Graphs of Polynomial Functions: https://www.youtube.com/watch?v=PSGL95nQBy8
Lesson 43. Graphs of Polynomial Functions with Repeated Zeros: https://www.youtube.com/watch?v=honyxJPl0-0
CSET Math Question: https://www.youtube.com/watch?v=4vh8sTztdz0&index=12&list=PLS0TM4AjRKMbYou37x7I4eMTTmI_eSagw
b.2 Properties of Graphs of Rational Functions
Lesson 44. Graphs of Rational Functions Introduction: https://www.youtube.com/watch?v=-DUMVzvu7-g
Lesson 45. Graphs of Rational Functions: https://www.youtube.com/watch?v=hWjMovgqvi4
Lesson 46. Graphs of Rational Functions (Harder): https://www.youtube.com/watch?v=IbsLYHzKoH0
b.3 Properties of Graphs of Radical Functions
Lesson 47. Graphs of Radical Functions and Transformations: https://www.youtube.com/watch?v=CnH5cs_SPpM
Lesson 48. Graphs of Radical Functions: https://www.youtube.com/watch?v=Uteoh0PtWwI
b.4 Properties of Graphs of Absolute Value Functions
Lesson 49. Graphs of Absolute Value Functions Introduction: https://www.youtube.com/watch?v=uSCfgSK7J-k
Lesson 50. Graphs of Absolute Value Function Using a Piecewise Approach: https://www.youtube.com/watch?v=71SfBO-B4dE
Lesson 51. Graphs of Absolute Value Functions: https://www.youtube.com/watch?v=1OtzfP8fCNU
F2. Assessment: Practice Problems Functions 1, Functions 2, Functions 3, Functions 4, Functions 5 (#s 1-5), Functions 6, Functions 7, Functions 8, Functions 9
c. Analyze properties of exponential and logarithmic functions in a variety of ways (e.g., graphing, solving problems)
c.1 Properties of Graphs of Exponential Functions
Lesson 52. Graphs of Exponential Functions: https://www.youtube.com/watch?v=8ZNpM7aSkuA
Lesson 53. Finding the Equation, Given the Graph of an Exponential Function: https://www.youtube.com/watch?v=vmFiraM8qTw
c.2 Properties of Graphs of Logarithmic Functions
Lesson 54. Definition of a Logarithm: https://www.youtube.com/watch?v=mQTWzLpCcW0
Lesson 55. Graphs of Logarithmic Function: https://www.youtube.com/watch?v=jS_52xEjT_k
Lesson 56. Graphs of Logarithmic Functions: https://www.youtube.com/watch?v=qTLezcX3jNo
Lesson 57. Comparing Graphs of Exponential and Logarithmic Functions: https://www.youtube.com/watch?v=w1A2ZYmfGco
Lesson 58. Domain and Range of Logarithmic Functions: https://www.youtube.com/watch?v=qTLezcX3jNo
CSET Math Question: https://www.youtube.com/watch?v=eTRlmWj1Xkc
Assesssment: Exponential and Logarithmic Problems: Set A Set B Set C
CSET Math Questions: https://www.youtube.com/watch?v=eTRlmWj1Xkc,
https://www.youtube.com/watch?v=eu7Ufwuhvf8&list=PLS0TM4AjRKMbYou37x7I4eMTTmI_eSagw
Preliminaries: Writing Proofs (Not a part of CSET I specs)
Lesson 59. Proofs Involving Odd and Even Numbers: https://www.youtube.com/watch?v=Ei3lFjDvXD0
Lesson 60. Direct Proofs: https://www.youtube.com/watch?v=W2v6FG7u4HY
Lesson 61. More Direct Proofs: https://www.youtube.com/watch?v=qlpFIx2eVR8
Lesson 62. Indirect Proofs: https://www.youtube.com/watch?v=IZ2MJO1HtMY
Lesson 63. More Indirect Proofs: https://www.youtube.com/watch?v=JlvGxZndAG4
Algebraic Structures
a. Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings)
Lesson 64. Axioms for the Real Numbers: http://www.math.uchicago.edu/~boller/HarrisMathCamp/axioms.pdf
Lesson 65. Groups with Examples: https://www.youtube.com/watch?v=WwndchnEDS4
Lesson 66. Rings and Fields with Examples: https://www.youtube.com/watch?v=syHBApgJhsA
Lesson 67. Number Systems and Fields: http://www.math.uchicago.edu/~boller/HarrisMathCamp/systems.pdf
Lesson 68. More Rings and Fields: http://mathrefresher.blogspot.com/2006/05/fields-and-rings.html
A1. Assessment: Practice Problems (3.1 #32)
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b. Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a < b and c < 0, then ac > bc)
Lesson 69. Basic Proof Techniques: http://www.math.psu.edu/wysocki/M403/Notes403_2.pdf
Lesson 70. Properties of Real Numbers: http://www.cengage.com/resource_uploads/downloads/1133963021_375161.pdf
Lesson 71. Properties of Real Numbers: http://www.math.psu.edu/wysocki/M403/Notes403_2.pdf
Lesson 72. Properties of Real Numbers: http://www.cs.sunysb.edu/~pfodor/courses/CSE215/L08_1-Number.pdf
c. Know 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
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Lesson 73. Axioms for Ordered Fields: http://homepages.math.uic.edu/~kauffman/axioms1.pdf
Lesson 74. More Ordered Fields: https://faculty.etsu.edu/gardnerr/4217/notes/1-2.pdf
Lesson 75. More Ordered Fields: http://www.uobabylon.edu.iq/eprints/paper_12_27257_140.pdf
Lesson 76. Real Numbers and Fields: http://www.math.vanderbilt.edu/~schectex/courses/thereals/
Lesson 77. Real Numbers and Fields: http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch02.pdf
Preliminaries: Right Triangle Trigonometry and the Pythagorean Theorem (Not a part of CSET I specs)
Lesson 78. Pythagorean Theorem: https://www.youtube.com/watch?v=AA6RfgP-AHU
Lesson 79. Pythagorean Theorem Problems: https://www.youtube.com/watch?v=-vXr5gwBxVg
Lesson 80. Basic Right Triangle Trigonometry: https://www.youtube.com/watch?v=Jsiy4TxgIME
Lesson 81. More Right Triangle Trigonometry: https://www.youtube.com/watch?v=G-T_6hCdMQc
Linear Algebra
a. Understand and apply the geometric interpretation and basic operations of vectors in two and three dimensions, including their scalar multiples and scalar (dot) and cross products
Lesson 82. Vectors in Two Dimensions: https://www.youtube.com/playlist?list=PLtvClhypCw9lfliZAg-rJx_ah_qNLHl6i
Lesson 83. Vectors in Three Dimensions: https://www.youtube.com/watch?v=cQVTMmGurCQ,
https://www.youtube.com/watch?v=AXQXuPnOYjo, https://www.youtube.com/watch?v=AXQXuPnOYjo
Lesson 84. Dot Product: https://www.youtube.com/watch?v=KDHuWxy53uM, https://www.youtube.com/watch?v=SpEnXMzDZAI
Lesson 85. Dot Product vs Cross Product: https://www.youtube.com/watch?v=E34CftP455k
CSET Math Question: https://www.youtube.com/watch?v=z6w6o55PJok,https://www.youtube.com/watch?v=2tAmJLEz9gs
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b. Prove the basic properties of vectors (e.g., perpendicular vectors have zero dot product)
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Lesson 86. Proofs of Basic Properties: https://www.youtube.com/watch?v=rVQ3G9epCjw,
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c. Understand and apply the basic properties and operations of matrices and determinants (e.g., to determine the solvability of linear systems of equations)
Lesson 87. Introduction: https://www.youtube.com/watch?v=ZR9nVXWTfcY,
https://www.youtube.com/watch?v=rBkWTQ_D3wA (complex numbers in matrix form)
Lesson 88. More on Operations: https://www.youtube.com/watch?v=5o8GRIzfYP8,
https://www.youtube.com/watch?v=iWnhjBl5xOs
Lesson 89. Determinants: https://www.youtube.com/watch?v=umXgFjqPfsk
Lesson 90. Inverse of a Square Matrix: https://www.youtube.com/watch?v=kZwItfmFtyk,
https://www.youtube.com/watch?v=iUQR0enP7RQ
Lesson 91. Solving Linear Systems of Equations Using Matrices: https://www.youtube.com/watch?v=9LYVi-n-6Jw,
https://www.youtube.com/watch?v=PTN8rganLWo,
https://www.youtube.com/watch?v=eXhf18KlYdg, https://www.youtube.com/watch?v=Re1F4d24Fxc
CSET Math Questions: https://www.youtube.com/watch?v=vZHgXtX9O9A&list=PLS0TM4AjRKMbYou37x7I4eMTTmI_eSagw&index=7,
https://www.youtube.com/watch?v=017nmnFFcVg, https://www.youtube.com/watch?v=8L-VRz9JTk0