Alg 2 / Geom Yr 1

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In this first part of the two-year Integrated Algebra 2/Geometry course, emphasis is placed on discovery and developing intuition for algebraic and geometric properties. Many new topics are introduced this year, with a focus on finding patterns and making conjectures through inductive reasoning, rather than through formal proof. Algebra is employed throughout the course to solve geometric problems, work with sequences, and investigate exponential and power functions. Heavy emphasis is placed on the connection between algebra and geometry through the study of transformation of 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 1: Inductive reasoning, sequences, and linear equations
  • What is inductive reasoning?
  • What are sequences, and more specifically, what are arithmetic sequences?
  • How can we graph linear equations?
  • What is direct variation?
  • How can we solve absolute value equalities?
  •  Inductive reasoning
  • Sequences
  • Arithmetic Sequences
  • Linear Equations
  • Direct variation
  • Absolute value equalities

 

  • Using inductive reasoning to find patterns and draw conclusions
  • Finding the next term in a sequence
  • Finding the explicit formula and graphing arithmetic sequences
  • Writing linear equations in the three forms and graphing them
  • Setting up and solving problems involving direct variation
  • Solving absolute value equalities
Unit 2: Functions
  • What is a function and what can we do with it?
  • What is the domain and range of a function?
  • What are some special parent functions?
  • How do we translate functions?
  • Functions
  • Domain and range
  • Linear, quadratic, cubic, and absolute value parent functions
  • Translation of functions
  • Finding domain, range, and intercepts of functions
  • Graphing linear, quadratic, cubic, and absolute value parent functions
  • Translating functions

 

Unit 3: Geometric Building Blocks
  • What makes a good definition?
  • What are some essential definitions in geometry that we will return to all year?
  • How do we solve basic problems of coordinate geometry?

 

  • Basic geometric terms
  • Midpoint formula
  • Parallel and perpendicular slopes
  • Triangle coordinate geometry
  • Creating a clear and concise definition
  • Becoming familiar with basic geometric terms
  • Using the midpoint formula
  • Finding equations for lines that are parallel/perpendicular to other lines
  • Finding equations for the median, perpendicular bisector, and altitude of a triangle
Unit 4: Radicals, Reflections, Special Angles, and Constructions

  • How can we work with expressions involving radicals or factoring?
  • How can we reflect functions?
  • What special pairs of angles are formed by intersecting and parallel lines?
  • How can we perform basic constructions using compass and straightedge, patty paper, and Geometers Sketchpad
  • Radicals
  • Factoring
  • The square root parent function
  • Reflecting functions
  • Special angles, including linear pairs, vertical pairs, and angles formed by parallel lines intersected by a transversal
  • Basic constructions using straightedge and compass, patty paper, and Geometers Sketchpad
  • Simplifying radicals and solving equations involving them
  • Factoring expressions
  • Graphing the square root parent function
  • Reflecting functions
  • Recognizing special angles and solving problems involving them
  • Performing basic constructions, including duplication and bisection of segments and angles, and creating parallel and perpendicular lines
Unit 5: Quadratic Functions
  • How can we solve quadratic equations analytically and graphically?
  • What are the three different forms of quadratic equations?
  • What are imaginary numbers and what are their basic properties?
 
  • Solving quadratic equations
  • Graphing quadratic equations
  • Expressing quadratic equations in the three forms
  • Basic properties of imaginary numbers
  • Solving quadratic equations by factoring, completing the square, graphing, and the quadratic formula
  • Expressing quadratic functions in standard, vertex, and factored forms
  • Using the discriminant to find the number of roots of a quadratic equation
  • Defining an imaginary number
  • Adding and multipling imaginary numbers and finding their absolute value and complex conjugate
Unit 6: Triangles
  • How do we know that a set of lengths forms an acute, obtuse, right, or nonexistent triangle?
  •  What are special types of right triangles and their properties?
  • What is trigonometry and how can we use it to solve problems?
  • How do we know when two triangles are similar, and how can we use similarity to solve for missing information?
  • The distance formula
  • The Pythagorean Theorem and its converse
  • 30-60-90 and 45-45-90 triangles
  • Sine, cosine, and tangent
  • Ratios and proportions
  • Similarity
  • Using the Pythagorean Theorem and its converse
  • Determining whether a triangle exists, and if so, what type it is, by its side lengths
  • Using the special properties of 30-60-90 and 45-45-90 triangles to solve problems
  • Discovering interior and exterior angle properties of triangles.
  • Using sine, cosine, and tangent to solve right triangle problems.
  • Using the SSS, SAS, and AA similarity conjectures to solve triangle problems.
Unit 7: Power, Exponential, and Inverse Functions
  • What are properties of inverse, exponential, power functions, and how do we graph them?
  • What is a geometric sequence?
  • How can we solve equations involving rational exponents?
  • What are the points of concurrency in a triangle?
  • Power, exponential, and inverse functions
  • Inverse variation
  • Geometric sequences
  • Points of concurrency in a triangle
  • Solving equations involving rational exponents and roots.
  • Graphing inverse and exponential functions and finding asymptotes.
  • Solving problems involving inverse variation.
  • Finding the explicit formula for a geometric sequence.
  • Finding points of concurrency algebraically and through constructions
Unit 8: Polygons
  • What are the interior and exterior angle relationships of a polygon?
  •  What are the special types of quadrilaterals and their properties?
  • How can we create polgons using constructions?
  • Interior and exterior angle sum formulas for polygons
  • Special types of quadrilaterals
  • Constructions involving polynomials
  • Using the interior and exterior angle relationships of a polygon to solve problems
  • Exploring the properties of special quadrilaterals, and using their properties to construct them.
Unit 9: Circles
  • What are special properties and formulas involving circles?
  • How can we dilate a function vertically?
  • Defining circle terms including tangent, chord, secant, inscribed, and central angles, and solving basic problems involving them
  • Circumference and arc length conjectures
  • Circle coordinate geometry
  • Vertical dilations
  • Discovering special properties about tangents, central angles, and inscribed angles of  a circle, and using them to solve problems
  • Using circumference and arc length formulas
  • Dilating functions vertically
Unit 10: Area
  • How can we find the area of polygons and circles?
  • How can we dilate a function horizontally?
  • Area formulas for polygons and circles
  • Dilating a function horizontally
  • Using the area formulas for circles, rectangles, parallelograms, triangles, trapezoids, and regular polygons
  • Dilating functions horizontally
Unit 11: Solids
  • How do we find the volume and surface area of prisms, cylinders, pyramids, and cones?
  • Volume and surface area of prisms, cylinders, pyramids, and cones
  • Discovering the volume and surface area formulas for prisms, cylinders, pyramids, and cones, and using them to solve problems