# Alg 2 / Geom Yr 2

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In this second part of the two-year Integrated Algebra 2/Geometry course, emphasis is placed on mathematical justification of algebraic and geometric properties. Many of the topics introduced in the previous year are revisited and expanded, with a focus on proving relationships and general principles deductively. Step-by-step, coordinate, paragraph, and indirect proofs are used. Algebra is employed throughout the course to solve geometric problems, work with series, and investigate exponential, logarithmic, and rational functions.

Assessments will include group and individual projects, daily homework assignments, quizzes, and tests. Resources used throughout the course include textbooks, graphing calculators, Geometer's Sketchpad, Moodle, Wiki, and addtional websites.

The accelerated version will cover the same material, as well as explore additional advanced topics. The pace of the accelerated course will also move considerably faster, as the class meets four times per week rather than five.

## Units

Unit Essential Questions Content Skills and Processes
Unit 0: Introduction to Functions
• What is a function?
• How can we represent functions?
• How can we combine functions
• Introduce five families of functions ( linear, quadratic, exponential, logarithmic, rational) through data collection activity
• Domain, range, notation, transformations, zeroes, intercepts
• Composition of functions
• Inverse functions
• One to one functions
• Use a function's graph or equation to model numerical data
• Find the composition of functions
• Find the inverse of a function
Unit 1: Exponential Functions
• What is exponential growth?
• What is an exponential function and how does it behave?
• How can we use a recursive definition to express a sequence of numbers?
• Recursive definitions for sequences
• Real number exponents and rules for using exponents
• Exponential Functions
• Applications of exponential functions
• Solving exponential equations
• Given a sequence, write its recursive definition
• Given a recursive definition for a sequence, write the sequence
• Prove laws of exponents for rational number exponents
• Use exponential models to solve problems involving exponential growth and decay
• Use a calculator to find exponential models and solve exponential equations
Unit 2: Logarithmic Functions
• How can we solve an exponential equation with a variable exponent?
• What is a logarithm?
• What are the properties of a logarithmic function?
• What are properties of a logarithm and how do we know they are valid?
• How can we use logarithms to solve applied problems?
• Definition of a logarithm
• Logarithmic functions
• Properties of logarithms
• Applications of logarithms
• Solving equations involving logarithms
• Evaluate logarithms without a calculator
• Graph logarithmic functions and their transformations
• Use logarithm properties to simplify and evaluate expressions
• Use a calculator to solve problems involving exponential and logarithmic growth
• Prove properties of logarithms
Unit 3: Deductive Reasoning
• How can we determine mathematical truth?
• What is deductive reasoning?
• What properties of lines can we prove true using deductive reasoning?
• What is a coordinate proof?
• How can we write a deductive proof in paragraph form?
• Vertical angles
• Complements of congruent angles
• Supplements of congruent angles
• Parallel and perpendicular lines
• Alternate interior and alternate exterior angles
• Same side interior angles
• Triangle angle sum theorem and its corollaries
• Exterior angle of a triangle
• Write step by step deductive proofs
• Write coordinate proofs
• Write paragraph proofs
Unit 4: Concurrency and Systems
• What special segments in triangles are concurrent?
• How can we use algebraic methods to prove that lines are concurrent?
• How can we extend our understanding of solving systems of equations to three dimensions?
• How can we apply our knowledge of systems to solve optimization problems?
• Triangle concurrency problems and proofs
• Systems of linear and non-linear equations
• Intersection of two planes
• Absolute value inequalities
• Systems of inequalities
• Linear programming
• Solve systems of linear and non-linear equations
• Prove given sets of lines are concurrent
• Solve linear programming problems
• Use technology to visualize and explore the intersection of planes in space
Unit 5: Triangle Congruence
• What are the minimum conditions necessary to determine whether triangles are congruent?
• How can we use deductive reasoning to prove properties of equilateral and isosceles triangles?
• What is an indirect proof?
• If side lengths in triangles are unequal, how do the corresponding angles compare, and vice versa?
• Postulates and Theorems about Triangle Congruence
• Corresponding Parts of Congruent Triangles
• Isosceles and Equilateral Triangle Properties
• Indirect Proof
• Triangle Inequalities in one and two triangles
• Write deductive proofs to prove triangles congruent
• Use congruent triangles to determine congruence of segments and angles
• Write indirect proofs
• Compare angles and sides in triangles that are not congruent

• How can we categorize quadrilaterals?
• What properties of quadrilaterals can be proven?
• How can coordinate proofs be used to prove quadrilateral properties?
• What minimum conditions are needed to prove that a quadrilateral must be a special type of quadrilateral?
• Definitions of special quadrilaterals
• Deductive proofs of quadrilateral properties
• Coordinate proofs of quadrilateral properties
• Proofs that a quadrilateral must be a special type, given a specific set of properties
• Geometric probabliltiy
• Write a deductive proof
• Write a coordinate proof
Unit 7: Quadratic Functions
• How can we find the minimum or maximum of a function?
• What is the relationship of important points on a graph to the equation of the graph?
• Relationship of length to area (linear -> quadratic).
• Review forms of a quadratic: standard, vertex, factored
• Meaning of roots and zeroes - relationship to the graph of the function
• Relationship of sum and product of roots.

• Completing the square (method of solution as well as rewriting standard into vertex form).
• Proof: Proving quadratic formula by two different means.
• Graphing a quadratic function - plotting roots as well as how to plot when roots are complex.
Unit 8: Parametric Equations
• What are alternate forms of functions?
• How is it possible to rewrite a non-function (such as a circle) as a function (or set of functions)?
• In what ways can we write and use functions that are dependent on another variable, such as time?
• Definition of parametric equations
• Applications of parametric equations
• Eliminating the parameter to write as an x-y equation, paying attention to restrictions on the domain.
• Plotting parametric curves given a table of data.
• Plotting parametric curves (domain and direction necessary) from the equations.
•  Using the calculator to plot and solve problems with parametric equations
Unit 9: Similarity
• Why do scale models work?
• Why is similarity so important in the real world?
• How do we know when figures are similar?
• AA Postulate
• SSS and SAS Similarity Theorems
• Right triangle similarities (altitude drawn from the right angle to the hypotenuse).
• Geometric mean
• Ratios between similar figures: lengths, areas, and volumes
• Proof: Similarity thoeorms -> Angle bisector theorem, side-splitter theroem
• Solving proportions between similar figures.
• Apply similarity to fractal problems.
Unit 10: Right Triangle Trigonometry
• How can we name functions?
• What are qualities of the trigonometric functions that other functions we have studied do not have?
• How can we use sine, cosine, and tangent to solve real world problems?
• Sine, cosine, and tangent of a right triangle (SOHCAHTOA).
• Values for sine, cosine, and tangent can be related to the similarity of right triangles with congruent angles.
• Graphs of sine and cosine from 0 to 90 degrees.
• Sine, cosine, and tangent of special angles.
• Sine and cosine as parametric functions.
• Inverse sine, inverse cosine, and inverse tangent problems.
• Projectile motion problems (optional).
• Solving for sides and angles of right triangles.
• Using sine and cosine with parametric equations to solve wind and water problems.
• Apply sine and cosine to application problems.
Unit 11: Rational Functions
• When we find the variable in the denominator, what are the fundamental differences with other functions that we should be aware of?
• How are rational functions related to the parent function of 1/x?
• Why are there restrictions placed on rational functions?
• What does end behavior mean?
• Definition of rational function: r(x) = p(x)/q(x), where p and q are polynomials.
• Review operations on rational expressions (addition, subtraction, multiplication, and division).
• Simplifying complex fractions.
• Recognize and find the domain, range, and zeroes of a rational function.
• Recognize the difference between the vertical and horizontal (if present) asymptotes, and be able to find them.
• Long division of polynomials.
• Find the slant asymptote of a rational function when it has one.
• Simplifying rational expressions.
• Graphing Rational Functions (domain, range, asymptotes, zeroes, end behavior).
• Long division of polynomials
• Transformations on 1/x
Unit 12: Series
• How is it possible to find the sum of an infinite number of terms?
• What are the advantages of knowing the sum of an infinite series if we can't actually know all the terms?
• What does infinity really mean if there is a way to add up an infinite number of terms to get a finite answer?
• Geometric connection to series (Fractals)
• Definition of series.
• Sigma notation - beginning value, a_n term, number of terms in series
• Formula for partial sum of an arithmetic series
• Formula for partial sum of a geometric series
• Formula for sum of an infinite geometric series - restrictions?
• Writing series in sigma notation.
• Find the a_n term in simple series.
• Prove the formulas for the sums of a partial arithmetic series, partial geometric series, and an inifinite geometric series.
• Solving problems using the series formulas.
Unit 13: Circles and Spheres
• What figures are always similar to each other?
• How are circles and Pythagorean Theorem related?
• Why are we still using triangles with circles?
• What does a circle become in three dimensions? What are the similarities and differences between that object and a circle?
• Definition of a circle.
• Representation of a circle on a coordinate plane - equation.
• Coordinate proofs for: angle inscribed in a semicircle is right, line through center of circle and midpoint of chord is perpendicular to the chord, distances between congruent chords and the center of the circle are equal.
• When can't we use coordinate proofs?
• Intersection of circles and lines.
• Using similarity theorems to prove circle theorems.
• Comparing circles to spheres.
• Surface area and volume of a sphere.
• Generate equation of a circle using distance formula.
• Write equation of a circle given center and radius.
• Write equation of a circle given a center and a point on the radius.
• Proving circle theorems with and without coordinate proofs.
• Solving systems of circles and lines.
• Calculate volume and surface area of spheres.