Unit 1:  Expanding the number system

 

Instructional Days:

 6

 

Lessons

 

 

ELO’s

Know and be able to…

·         Simplify expressions using rules of exponents

·         Graph exponential functions and recognize them as growth or decay

·         Perform operations (add, subtract, multiply) on:

o    Radicals (including simplifying/dividing)

o   Polynomials

·         Understand the complex number system and different classifications of numbers

·         Represent expressions in radical and rational exponent form

 

Topic:  Rational and Irrational Numbers

1

-Review the types of numbers (whole, integers, rational, irrational)

-Review simplifying radicals        

 

 

1

Explore sums, differences, products and (quotients) of irrational numbers

 

Topic:  Exponents and Exponential Functions

 

Review rules of exponents

 

 

1

-Use properties of exponents with rational exponents

-Be able to go back and forth between radical and rational exponent form

 

 

1

-Use properties of exponents to interpret and transform exponential functions and classify them as either growth or decay (review graphing exponential functions as needed)

 

Topic:  Polynomial Functions

1

-Define terms and coefficients of polynomial functions

-Add, subtract and multiply polynomial expressions

 


 

 

Unit:  Analyzing and Graphing Functions

 

 

15 days

 

Lessons

 

 

 

ELO’s

Know and be able to…

·         Analyze graphs of various functions and relations:  domain and range, x and y intercepts, determine if the graph is a function, determine the type of symmetry, determine if the function is increasing/decreasing on an interval

·         Graph lines in slope intercept form and standard form

·         Find the average rate of change using a graph or x-values

·         Find the Inverse of a function from a graph

·         Identify a function as linear, quadratic, exponential, absolute value, square root or cube root

·         Graph functions using transformations

·         Apply linear, quadratic, and exponential models

·         Compare and contrast the family of functions using tables, equations and graphs

 

 

Topic:  The Family of Functions

3

Introduce the family of functions:  square root, cube root, cubic, absolute value, piecewise, step, quadratic, linear, exponential

 

 

3

Compare and Contrast functions in terms of:

·         Where the function is increasing/decreasing

·         Domain and range (include real life examples)

·         Maximums and minimums

·         Positive and negative

·         x and y intercepts

·         symmetry (even vs. odd function)

 

 

2

Transform functions (not step or piecewise) and identify the effects of a, h and k on the graph of each

 

Topic:  Inverses of functions

2

Find the inverse of a function (use graphs to explore and reflect over line y = x)

 


 

Topic:  Exploring functions in various forms (tables, graphs, words, etc.)

2

Expand graphs to include tables, verbal descriptions, and use the various representations to compare and contrast functions

 

 

1

Observe using a table and graph that an exponential function will eventually exceed any other function

 

Topic:  Average rate of change

2

Find the average rate of change from an equation or a table; estimate the average rate of change from a graph, and compare average rates of change from different functions

 

 

 

 

 


 

Unit :  Quadratics and Systems of Equations

20 days

 

ELO’s:

Know and be able to:

·         Factor polynomials

o   Greatest Common Factors

o   Factor by Grouping

o   Factor Trinomials (guess and check)

·         Solve a quadratic equation  by factoring, using square roots, and the quadratic formula.

·         Graph quadratic functions in both vertex form and standard form

·         Write a quadratic equation given the zeros of the function

·         Write a quadratic equation in vertex form, given that it was in standard form

 

Topic:  Graphing Quadratics

2

-Graph quadratics using vertex form, standard form and intercept form

 

Topic:  Factoring Quadratics

3

-Review GCF

-Factor quadratics – emphasize intercepts vs. factors

-Solve quadratics by factoring (using terms such as roots, zeros, solutions)

 

Topic:  Solving Quadratics

2

-Solve quadratics by taking square roots (including those with complex solutions)

-Introduce and expand on imaginary numbers

 

 

2

-Complete the square

-Use to solve quadratics and write in vertex form

-find the maximum/minimum of a quadratic(applied)

 

 

1

-Solve quadratics using the quadratic formula

 

 

1

-Explain the Fundamental Theorem of Algebra

-Write quadratics as products of linear factors (real and complex)

 

Topic:  Quadratic Inequalities

1

-Solve quadratic inequalities in one variable

 

 

1

-Create equations and inequalities in one variable and use them to solve problems

 


 

Topic:  Systems of Equations

1

-Create equations in two or more variables to represent relations between quantities; graph equations in coordinate plane with labels and scales

 

 

1

-Review systems of equations

 

 

1

-Solve simple systems of linear and quadratic equations

 

 

1

-Rearrange formulas (with squares) to solve for a specific variable; interpret complicated expressions by viewing one or more of the parts as a single entity

 


 

Unit:  Probability

7 days

 

ELO’s

Know and be able to…  

 

·         Define and apply vocabulary– theoretical and experimental probability of a random event, sample space,  independent/dependent events, joint/disjoint events, conditional probability, union (OR) intersection (AND), complement of an event, permutation, combination

·         Understand probability notation:  P(A), P(A or B), P(A and B), P(Ac),  P(B|A)

·         Calculate probabilities of joint and disjoint events using the addition rule

 

Topic:  Probability Vocabulary

1

-Describe events of a probability experiment:  sample space, union, intersection, complements

-Draw Venn diagrams to show relationships between sets within a sample space

 

Topic:  Independent and Dependent Events

1

-Understand the difference between independent and dependent events

-Compute probabilities of independent and dependent events

 

 

Topic:  Conditional Probability

1

-Understand and compute conditional probability (from Venn diagrams, two way tables and tree diagrams) and use it to determine independence

Interpret conditional probabilities and independence in context

 

 

Topic:  Two way tables

1

-Use two way tables to calculate probabilities (joint, conditional, unions, etc.)

 

Topic:  Compound Probabilities

2

-Apply the Additional Rule for joint and disjoint events

 

 

 

-Apply the Multiplication Rule for independent and dependent events (honors)

 

Topic:  Counting

1

-Use permutations and combinations to determine outcomes and compute probabilities

 

Unit:  Geometry

12 days

 

ELO’s

Know and be able to…  

·         Define and apply angle relationships in parallel and intersecting lines

·         Define and apply angle relationships in triangles and isosceles triangles

·         Define and apply segment relationships in triangles (medians, midsegments and perpendicular bisectors)

·         Define and apply side, angle and diagonal relationships in parallelograms and rectangles

·         Find missing sides and angles of similar figures

·         Prove that two figures are similar

 

Topic:  Angles

1

-Review:  types of angles and angle pairs

-Prove vertical angles are congruent

 

Topic:  Transversals of Parallel Lines

1

-Prove theorems about transversals of parallel lines and angles pairs (AIA, AEA, CA, CIA)

 

 

 

-Prove that the perpendicular bisector of a line segment is exactly equidistant from the segment’s endpoint

 

Topic:  Angles of a triangle

1

-Prove the triangle sum theorem

-Prove that the base angles of an isosceles triangle are congruent

 

Topic:  Congruent Triangles

2

-Prove that triangles are congruent using SSS, SAS, ASA, AAS

 

Topic:  Parallelograms

1

 

-Prove properties of parallelograms:

·         Opposite sides are congruent

·         Opposite angles are congruent

·         Diagonals bisect each other

·         Rectangles are parallelograms with congruent diagonals

 

Topic:  Segments of Triangles

1

-Prove the midsegment of a triangle is parallel to the third side and half the length

-Show that the medians of a triangle meet at a single point; relationship between median and centroid

 


 

Topic:  Similar Triangles

1

-Define similar triangles

- Find missing sides of similar triangles

 

 

1

-Prove triangles are similar

 

 

1

-Prove a line parallel to a one side of a triangle divides the other parts proportionally

 

 

1

-Find the point on a line segment between two given points that partitions the segment in a given ratio

 

 

1

-Prove a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves the line passing through the center unchanged

-The dilation of a line segment is longer or shorter than the ratio given by the scale factor

 


 

Unit:  Trigonometry

 

 

ELO’s

Know and be able to…  

·         Solve right triangles with the Pythagorean Theorem

·         Know the definitions of the 6 trig functions: sin, cos, tan, csc, sec, cot

·         Find missing sides and angles of right triangles

·         Solve application problems involving right triangles

 

Topic:  Pythagorean Theorem

1

-Prove Pythagorean Theorem using similar triangles

-Use Pythagorean Theorem to solve for missing sides of a triangle

 

Topic:  Trigonometric ratios and their applications

2

-Define the trig functions

-Prove the Pythagorean Identity and use it to find the sin, cos, tan of an angle

-Explain and use the relationship between the sine and cosine of complementary angles

 

 

2

-Use trig ratios and Pythagorean Theorem to solve application problems

 


 

Unit:  Circles

 

 

ELO’s

Know and be able to…  

·         Define/identify the parts of a circle and explain how they relate to each other

·         Calculate the measure of central angles, inscribed angles, minor arcs, major arcs

·         Calculate the length of an arc and the area of a sector

·         Graph a circle in the coordinate plane

·          Knowing information about the circle, either write an equation in standard form

·         Work with constructions involving inscribed and circumscribed figures

 

Topic:  Circle Vocabulary

1

-Prove all circles are similar

-Describe and describe relationships between radii, chords, and angles

 

 

1

-Explain the differences between central angles, inscribed angles and circumscribed angles

 

 

1

-Explain the relationship between the inscribed angle and the diameter

-Explain the relationship between a line tangent to a circle and the radius to the point of tangency

-Construct a tangent line from a point outside a given circle to the circle

 

Topic:  inscribed/circumscribed triangles and quadrilaterals

1

-Construct the inscribed and circumscribed circles of a triangle

-Prove properties of angles for a quadrilateral inscribed in a circle

 

Topic:  Circles in the Coordinate Plane

2

-Graph circles

-Use coordinate Geometry to prove relationships and theorems about circles (Review distance/midpoint formula as needed)

 

Topic:  Radian Measure, arc length and sector area

2

-Use the concept of similarity to understand that arc length intercepted by the central angle is proportional to the radius

-Develop the definition of radians as a unit of measure by relating to arc length

-Derive the formula for area of a sector

 

Unit:  Understanding Volume

4 days

 

ELO’s

Know and be able to…  

·         Calculate the volume of various sold figures

·         Explain how the formulas for volume are derived

 

 

2

Provide an informal argument for the formulas for circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone

 

 

2

Use volume formulas for cylinders, pyramids and cones to solve problems