Curriculum Documents by Quarter - Mathematics Grade 12

Unit of Study 1: Pre-Calculus Unit 1
Standards
Essential Questions

Learning Objectives

 represent situations that involve variable quantities with expressions, equations, inequalities, and matrices;
¨       use tables and graphs as tools to interpret expressions, equations, and inequalities;
¨       operate on expressions and matrices, and solve equations & inequalities;
¨       appreciate the power of mathematical abstraction and symbolism;
¨       model real-world phenomena with a variety of functions;
¨       represent and analyze relationships using tables, verbal rules, equations, and graphs;
¨       translate among tabular, symbolic, and graphical representations of functions;
¨       recognize that a variety of problem situations can be modeled by the same type of function;
¨       analyze the effects of parameter changes on the graphs of functions;
understand operations on, and the general properties and behavior of, classes of functions.

 §      What are different ways of representing numbers?

§      What is a number set?

§      How are mathematical operations related?

§      How do patterns and relationships help us understand mathematical situations?

§      How do we represent patterns and relationships in everyday life?

§      How can change be analyzed?

§      Which strategies and mathematical operations are used to solve problems?

§      How do we monitor and reflect on the process of mathematical problem solving?

§      How can mathematical thinking be organized and presented so that it can be shared with others?

§      How can we develop a mathematical language that is useful in the real world?

§      What are the relationships between mathematical concepts?

§      How can mathematics be used in other disciplines?

§      How is math used in the real world?

§      How can we use representations to model and interpret physical, social and mathematical phenomena?

Vocabulary

 

·     Rationalizing an Expression

·     Exponential vs. Rational Form

·     Factoring by Grouping

·     F.O.I.L.

·     Sum & Difference of Cubes

·     Domain

·     Range

·     Rational Expressions

·     Extraneous solution

·     Distance Formula

·     Midpoint Formula

·     x-axis, y-axis, and origin symmetry

·     increasing vs. decreasing vs. constant function

·     greatest integer function

·     even vs. odd function

·     Shifting vs. reflecting vs. stretching graphs

·     Composition of functions

·     Inverse function

·     Continuous function

·     Leading coefficient test

·     Intermediate Value Theorem

·     Long Division vs. Synthetic Division

·     Remainder Theorem

·     Factor Theorem

·     Rational Zero Test

·     Complex Number

·     Imaginary Number

·     Complex Conjugate

·     Horizontal vs. Vertical Asymptotes

Partial Fractions

Learning Outcomes

§     Compare and contrast the real number system and its various subsystems with regard to their structural characteristics

§     Develop conceptual understanding of the complex number system

§     Demonstrate facility with operations in the complex number system

§     Model real-world phenomena (e.g., compound interest or the flight of a ball) using functions and relations (e.g., linear, quadratic, sine and cosine, and exponential)

§     Represent and analyze relationships using written and verbal explanations, tables, equations, graphs and matrices and describe the connections among those representations

§     Analyze the effects of parameter changes on functions (e.g., linear, quadratic) using calculators and/or computers

§     Interpret algebraic equations and inequalities geometrically and describe geometric relationships algebraically

§     Perform mathematical operations on expressions and matrices, and solve equations and inequalities

§     Translate among tabular, symbolic and graphical representations of functions

§     Use the power of mathematical abstraction and algebraic symbolism to represent various situations

§     Determine maximum and minimum points of a graph and interpret results in problem situations

Understand operations on, and the general principles and behavior of, classes of functions (including logarithmic, exponential, and trigonometric functions)

Unit of Study 2: Pre-Calculus Unit 2

 

§      Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

§      Know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

§      Use the definition of logarithms to translate logarithms in any base.

§      Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.

§      Understand the notion of angle and how to measure it, expressing it in both degrees and radians.

§      Express sine and cosine as y- and x-coordinates of points on the unit circle and be able to graph the sine and cosine functions.

§      Understand the Pythagorean identity cos2(x) + sin2 (x) = 1 and relate it to the Pythagorean Theorem.

§      Prove a variety of trigonometric identities

§      Graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, & C in terms of amplitude, frequency, period, & phase shift.

§      Know the definition of the tangent, cotangent, secant, and cosecant functions and can graph them.

§      Know the definitions of the inverse trigonometric functions and graph the functions.

§      Compute, by hand, the values of the trigonometric and inverse trigonometric functions at various standard points.

§      Demonstrate an understanding of the addition & half-angle formulas for sines & cosines & use those formulas to prove and/or simplify other trigonometric identities.

§      Use trigonometry to determine unknown sides or angles in right triangles.

Use trigonometry in a variety of applications and word problems.

 

What are different ways of representing, organizing, and relating numbers to mathematical operations?

 

How do we understand and represent patterns, relationships, and change?

 

How do we collect, organize, display, and analyze data?

 

Which strategies and mathematical operations are used to solve problems and arguments, and how can we develop and improve them?

 

How can mathematical thinking be effectively communicated to different audiences?

 

How are mathematical concepts related to one another, other disciplines, and to the real world?

Vocabulary

 

·        Exponential Function

·        Logarithmic Function

·        Natural Logarithmic Function

·        Radian

·        Arc Length

·        Unit Circle

·        Sine, cosine, tangent, cotangent, secant, & cosecant

·        Period of a function

·        Hypotenuse, opposite, & adjacent side of a right triangle

·        Co-terminal Angle

·        Reference Angle

·        Amplitude

·        Phase shift

·        Inverse trigonometric functions

Learning Outcomes

§      Recognize, evaluate, and graph exponential functions.

§      Recognize, evaluate, and graph logarithmic functions

§      Use the change-of-base formula to rewrite and evaluate logarithmic expressions and

§      Use properties of logarithms to evaluate, rewrite, expand, or condense logarithmic expressions.

§      Solve exponential and logarithmic equations.

§      Use exponential growth models, exponential decay models, Gaussian models, logistic growth models, and logarithmic models to solve real-life problems.

§      Describe an angle and to convert between radian and degree measure.

§      Identify a unit circle and its relationship to real numbers.

§      Evaluate trigonometric functions of acute angles and how to use the fundamental trigonometric identities.

§      Evaluate trigonometric functions of any angle.

§      Sketch the graphs of sine and cosine functions and translations of these functions.

§      Sketch the graphs of other trigonometric functions.

§      Evaluate the inverse trigonometric functions and the compositions of trigonometric functions and inverse trigonometric functions.

§      Use trigonometric functions to solve real-life problems.

§      Use fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions.

§      Verify trigonometric identities.

§      Use standard algebraic techniques and inverse trigonometric functions to solve trigonometric equations.

§      Use sum and difference formulas to rewrite and evaluate trigonometric functions.

Use multiple–angle formulas, power-reducing formulas, half-angle formulas, and product-to-sum formulas to rewrite and evaluate trigonometric functions.

Unit of Study 3: Pre-Calculus Unit 3

 

Demonstrate an understanding of the addition & half-angle formulas for sines & cosines & use those formulas to prove and/or simplify other trigonometric identities.

Use trigonometry to determine unknown sides or angles in right triangles.

Use trigonometry in a variety of applications and word problems.

The learner will graph and use relations and functions to solve problems.

 

How are mathematical operations related?

How do patterns and relationships help us understand mathematical situations?

How do we represent patterns and relationships in everyday life?

Which strategies and mathematical operations are used to solve problems?

How do we monitor and reflect on the process of mathematical problem solving?

How can mathematical thinking be organized and presented so that it can be shared with others?

How can we develop a mathematical language that is useful in the real world?

What are the relationships between mathematical concepts?

How can mathematics be used in other disciplines?

How is math used in the real world?

How can we use representations to model and interpret physical, social and mathematical phenomena?

Vocabulary

 

·         Trigonometric Identity

·         Trigonometric Substitution

·         Verifying

·         Oblique Triangle

·         Law of Sines

·         Law of Cosines

·         Heron’s Area Formula

·         Vector v in the plane

·         Standard position

·         Zero vector

·         Unit vector

·         Direction angle

·         Orthogonal

·         Points of intersection

·         Break-even point

·         Method of elimination

·         Row-echelon form

·         Ordered triple

·         Row operations

·         Gaussian elimination

·         Non-square system of equations

·         Solution of a system of inequalities

·         Optimization

·         Linear programming

·         Objective function

·         Constraints

Feasible solutions

Learning Outcomes

  • Radian and Degree Measure
  • Trigonometric Functions: The Unit Circle
  • Right Triangle Trigonometry
  • Graphs of Sine and Cosine Functions
  • Graphs of Other Trigonometric Functions
  • Inverse Trigonometric Functions

Unit of Study 4:Pre-Calculus Unit 4

 

The learner will demonstrate an understanding of patterns, relationships, and fundamental algebraic concepts.

The learner will use relations and functions to solve problems.

The learner will create and use models of data for reporting and analysis.

 

How are mathematical operations related?

How do patterns and relationships help us understand mathematical situations?

How do we represent patterns and relationships in everyday life?

How can change be analyzed?

Which strategies and mathematical operations are used to solve problems?

How do we monitor and reflect on the process of mathematical problem solving?

How can mathematical thinking be organized and presented so that it can be shared with others?

How can we develop a mathematical language that is useful in the real world?

What are the relationships between mathematical concepts?

How can mathematics be used in other disciplines?

How is math used in the real world?

How can we use representations to model and interpret physical, social and mathematical phenomena?

Vocabulary

 

·         Terms of a sequence

·         Arithmetic sequence

·         Common difference

·         Geometric sequence

·         Common ratio

·         Infinite geometric series or geometric series

·         Mathematical induction

·         Foci

·         Vertices

·         Major axis

·         Center

·         Minor axis

·         Branches

·         Transverse axis

·         Conjugate axis

Learning Outcomes

  • Using Fundamental Identities
  • Verifying Trigonometric Identities
  • Solving Trigonometric Equations
  • Sum and Difference Formulas
  • Multiple-Angle and Product-Sum Formulas
  • Law of Sines
  • Law of Cosines
  • Vectors in the Plane
  • Vectors and Dot Products
  • Sequences and Summation Notation
  • Arithmetic Sequences
  • Geometric Sequences
  • Probability
  • Introduction to Conics: Parabolas
  • Ellipses
  • Hyperbolas
  • Rotation of Conics
  • Parametric Equations
  • Polar Coordinates
  • Graphs of Polar Equations
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