consensus map grade level - west jefferson hills … ii_honors_15162.docx · web viewcommon core:...

2015 - 2016

Honors Algebra II Honors Algebra 2 is a more extensive study of Algebra 2. This course will offer a more in-depth study of linear equations, quadratics, graphs and operations, and domain and range. Additional topics unique to Honors Algebra 2 will include:

Evaluating and Graphing Piecewise Functions including the Step Function,

Graphing rational functions (including those with oblique asymptotes), Evaluating the sum of a series and using summation notation, Evaluating and utilizing composite functions to prove inverse

relationships, Solving systems of equations and inequalities involving a linear and a

quadratic equation/inequality or two quadratic equations/inequalities.

Students will be expected to apply knowledge gained from this course to real-world problems. Students will use a graphing utility to illustrate concepts.

Course Information:Frequency and Duration: Daily for 40-45 minutes.Text: Pearson Algebra 2 Common Core (2012 )

Mathematics: Honors Algebra II v. 2015 - 2016

Content: General Transformations & Piecewise Functions

Duration: August/September (2 weeks)

Essential Question:

How do changes to the equation of a function affect the graph of a function?Do transformations on a graph always have the same effect regardless of the type of underlying function?

Skill:

Given the graph of a generic function and the equation of the transformed function, graph the transformation.

Given the graph of a function and its transformation, write a rule/equation for the transformed function.

Graph absolute value functions and their transformations. Give the domain and range of both f (x) and f '( x) in interval and inequality

notation. Define, evaluate, and graph piecewise functions (limit to linear equations

only). Show that the absolute value function can be defined as a piecewise

function. Evaluate and graph step functions including the greatest integer function.

Assessment:

Given the following graph of f ( x ), sketch the graph of f ( x )±3 ,

f ( x ±3 ) ,2 f ( x ) , 12f ( x ) ,−f ( x ) , f (−x). State the domain and range of both graphs.

Given the graph of f (x) and the graph of its transformation, identify the transformations that have taken place and then write the equation of the transformation in terms of f (x).


Graph f ( x )=|x+4|, f (x )=|x|−5 , f ( x )=−|x|, f ( x )=2∨x∨¿.

Given: f ( x )={ 2 x+6 for x≤−2−2 x−2 for−2<x ≤03 x−2 for x>0

, find f (−2 ) , f (0 ) , f (10)

Graph f ( x )={ 2 x+6 for x≤−2−2 x−2 for−2<x ≤03 x−2 for x>0

Graph f ( x )={3 for 0<x≤15 for 1<x≤27 for 2<x≤3

Graph f ( x )=[ x]

Resources: Pearson Algebra 2 Common Core: Chapter 2, Concept Byte pages 90 – 91, pages 99 – 113

Standards:

CC.2.2.HS.C.2 – Graph and analyze functions, and use their properties to make connections between the different representations.CC.2.2.HS.C.4 – Interpret the effects transformations have on functionsA2.1.3.2.1 A2.2.2.2.1

Common Core: Analyze functions using different representationsCommon Core: Graph piecewise defined functions including step functions

Vocabulary:

Absolute Value Function- any function in the form of f ( x )=a|x−h|+k; Dilation- stretch or compression; Greatest Integer Function- a function that represents the greatest integer less than or equal to x (f(x) = [x]);Parent function- the simplest form in a set of functions that form a family; Piecewise Function- a function that has different rules for different parts of its domain; Reflection- flipping the graph over the x-axis or y-axis; Step Function- a function that pairs every number in an interval with a single value; Transformation- dilation, reflection, or translation; Translation- vertical or horizontal shift

Comments:

f (x)f '( x)


Content: Quadratic Equations Duration: September/October (6 weeks)

Essential Question:

What are the advantages of a quadratic function in vertex form?What are the advantages of a quadratic function in standard form?How are the real solutions of a quadratic equation related to the graph of the related quadratic function?How is any quadratic function related to the parent function y=x2?

Skill:

Graph a quadratic function in standard and vertex form. Solve a quadratic equation by factoring including trinomials with a leading

coefficient that is not equal to 1 (trinomials, difference of squares, GCF, guess and check, key number).

Use the Quadratic Formula to solve a quadratic equation. Add, subtract, and multiply complex numbers. Use a complex conjugate divide complex numbers. Use the discriminant to find the number of x-intercepts/real

solutions/zeros/roots. Complete the square to solve quadratic equations or to convert from

standard to vertex form. Find complex solutions to a quadratic using the quadratic formula, and

simplify all solutions accordingly. Solve and graph systems of linear and quadratic equations/inequality or two

quadratic equations/inequalities

Assessment:

Graph y=x2 and any graph in the form of f ( x )=a ( x−h )2+k by manipulating a ,h ,∧k (i.e. transformation rules).

Graph quadratics in the form of y=a x2+bx+c by finding the coordinates of the vertex, equation of line of symmetry, and intercepts.

Convert y=2x2+20x+7 to vertex form. Solve: x2+5x−6=0 ,2 x2+10 x+12=0 ,3 x2+10 x+8=0 , x2−36=0 ,

3 ( x−4 )2=27 Solve by using the quadratic formula: −x2−3 x+11. How many real solutions does the following have: x2+7 x−15=0 ,

5 x2+6 x+4=0. Simplify the following: √−18, (4−3 i )+(−4+3 i ) , (5−3 i )−(−2+4 i) Multiply: 3 i (−5+2i ) ,(−1+2 i)(3−4 i)

Divide: 9+12 i3 i

, 2+3i1−4 i

Solve the following system by substitution and verify the solution by graphing: y=−x2−x+6 and y=x+3

Solve the following system by substitution and verify the solution by graphing: y=−x2−x+12 and y=x2+7 x+12

Solve the following system by substitution and verify the solution by graphing: y← x2−x+2 and y ≥x2−4


Resources Pearson Algebra 2 Common Core Chapter 4: pages 192 – 208, pages 216 – 255, & pages 258 - 264

Standards:

CC.2.3.HS.A.10 – Translate between the geometric description and the equation for a conic section.CC.2.2.HS.C.1 – Use the concept and notation of functions to interpret and apply them in terms of their context.CC.2.2.HS.C.2 – Graph and analyze functions, and use their properties to make connections between the different representations.CC.2.2.HS.C.3 – Write functions or sequences that model relationships between two quantities.CC.2.2.HS.C.5 – Construct and compare linear, quadratic, and exponential models to solve problems.CC.2.2.HS.C.6 – Interpret functions in terms of the situations they model.CC.2.2.HS.D.1 – Interpret the structure of expressions to represent a quantity in

terms of its context.CC.2.2.HS.D.2 – Write expressions in equivalent forms to solve problems.CC.2.2.HS.D.7 – Create and graph equations or inequalities to describe numbers or relationshipsCC.2.2.HS.D.8 – Apply inverse operations to solve equations or formulas for a given variableCC.2.2.HS.D.9 – Use reasoning to solve equations and justify the solution methodCC.2.2.HS.D.10 – Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.CC.2.1.HS.F.4 – Use units as a way to understand problems and to guide the solution of multi-step problems.CC.2.1.HS.F.6 – Extend the knowledge of arithmetic operations to the complex

numbers.CC.2.1.HS.F.7 Apply concepts of complex numbers in quadratic equations to

solve problems.A2.2.1.1.1 A2.2.1.1.3 A2.2.1.1.4 A2.1.3.2.2 A2.1.3.1.1

A2.1.1.1.1A2.1.1.2.2 A2.1.1.1.2 A2.1.1.2.1 A2.2.1.1.1 A2.2.1.1.3

A2.2.1.1.4

Common Core: Perform arithmetic operations with complex numbers.Common Core: Complete the square in a quadratic expression to reveal the

maximum or minimum value of the function it defines.Common Core: Solve a quadratic equation by taking square roots, completing

the square, factoring, or another appropriate method.National Standard: F.BF.3, 4a: Build new functions from existing functions

(Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types).

Vocabulary:

Complex Conjugate: number pairs of the form a+bi∧a−bi; Complex Number- the set of real numbers and imaginary numbers; Discriminant: b2 – 4ac; Imaginary Number- any number in the form of a+biwhere a∧bare realnumbers∧b≠0; Pure Imaginary Number- a number in the form of a+biwhere a=0∧b≠0; Quadratic- a secondary degree polynomial equation in the standard form of a x2+bx+c=0where a≠0; Real Number- a number in the form of a+bi where b=0

Comments:


Content: Polynomial Equations Duration: November (4 weeks)

Essential Question:

What does the degree tell you about its related polynomial function?How are factors, zeros, roots, and x-intercepts related?

Skill:

Create/Interpret a graph from a table of data. Given the graph of a polynomial function, state the intervals where the

function is increasing/decreasing in interval notation. Given the graph of a polynomial function, or using a graphing utility, locate

and describe the turning points (relative/local maxima, relative/local minima).

Given the equation of a polynomial function, classify it by its degree, describe its end-behavior, state the maximum/minimum number of turning points, and state the maximum/minimum number of x-intercepts.

Graph a polynomial function given its zeros. Graph a polynomial function written in factored form. State the multiplicity of each zero, and describe how the graph behaves at

each zero (bounce, cross). If given the root x=a, write the factor (x−a). Use the Conjugate Root Theorem. Use the Factor Theorem. Write the polynomial in standard form given the roots or zeros and their

multiplicity. Solve a polynomial equation using factoring techniques (including grouping,

GCF, sum/differences of cubes). Divide polynomials using long division and synthetic division. Use the Remainder Theorem. Determine the possible rational roots of a polynomial function using the

Rational Roots Theorem. Use Descartes Rule of Signs to determine the possible number of positive

and negative real roots of a polynomial. Graph a polynomial function that is not in factored form. Expand a binomial using Pascal’s Triangle and the Binomial Theorem.

Assessment:

Given the following graph, state the end-behavior, intervals of increasing/decreasing, turning points, and zeros.

Given the following polynomial, y=−2x4+8x3+14 x−1 :state the degree, maximum and minimum number of turning points, maximum and minimum number of x-intercepts, and end-behavior.

Given that a polynomial has the zeros x=3 , x=−1 , x=0 , x=3, write the polynomial in factored and then graph the polynomial.

Given that a polynomial has zeros at x=5 i , x=2, write the polynomial in factored form.


Graph the polynomial y= (x−3 ) (x+1 )2 ( x )3

Write the polynomial in factored form and then graph y=x3−2x2−15 x Determine if x−3 is a factor of x5−2 x3−x2+2 Solve y=x3+2 x2−3 x−6 Solve y=x3+64 Given the polynomial, y=3 x3+7 x2+6x−8, use Descartes Rule of Signs to

determine the maximum and minimum number of positive and negative real roots, use the Rational Roots Theorem to determine the possible real roots, use technology to find one real root, use long division or synthetic division to completely factor the polynomial and state all possible roots (both real and imaginary).

Use the Binomial Theorem to expand the following: ( x+1 )4 , (2x−3 )5 , (4−x )6

Resources: Pearson Algebra 2 Common Core: Chapter 5 (Sections 5.1 – 5.8): pages 280 – 337.

Standards:CC.2.2.HS.C.1 – Use the concept and notation of functions to interpret and apply them in terms of their context.CC.2.2.HS.C.6 – Interpret functions in terms of the situations they model.CC.2.2.HS.D.1 – Interpret the structure of expressions to represent quantities in terms of its contextCC.2.2.HS.D.2 – Write expressions in equivalent forms to solve problems.CC.2.2.HS.D.3 – Extend the knowledge of arithmetic operations and apply to polynomials.CC.2.2.HS.D.4 – Understand the relationship between zeros and factors of polynomials to make generalizations about functions and their graphs.CC.2.2.HS.D.5 – Use polynomial identities to solve problems.CC.2.2.HS.D.7 – Create and graph equations or inequalities to describe numbers

or relationships.CC.2.2.HS.D.9 – Use reasoning to solve equations and justify the solution methodCC.2.1.HS.F.4 – Use units as a way to understand problems and to guide the solution of multi-step problems.A2.2.1.1.1 A2.2.1.1.3 A2.2.1.1.4 A2.2.2.1.1 A2.1.2.2.1

A2.1.2.2.1A2.1.2.2.2 A2.2.2.1.4

Common Core: Graph functions expressed symbolically and show key features of the graph by hand in simple cases. Graph polynomial functions, identifying zeros and asymptotes and show end behavior.Common Core: Observe using graphs or tables that a quantity increasing exponentially eventually exceeds a graph increasing linearly.Common Core: Know and understand the relationship between zeros and

factors.

Common Core: Rewrite simple rational expressions in different forms: write a ( x )b ( x )


in the form q ( x )+ r ( x )b ( x )

using long division or other methods.

Vocabulary:

Binomial- the sum or difference of 2 monomials; Monomial- a real number, a variable, or the product of a real number and a variable; Polynomial- an expression with several terms; Root- synonym for x-intercept; Trinomial- the sum or difference of 3 monomials; Zero- synonym for x-intercept

Comments:


Content: Radical Functions and Rational Exponents

Duration: December (4 weeks)

Essential Question:

What must be true about a radical expression in order to simplify the nth root?When you square each side of an equation, is the resulting equation equivalent to the original?

Skill:

Simplify radical expressions. Multiply and divide radical expressions. Rationalize the denominator of a radical expression. Add and subtract radical expressions. Simplify expressions with rational exponents. Convert a radical expression to an expression including rational exponents

and vice versa. Combine radical expressions that have the same and different indexes. Solve radical equations and check for the existence of extraneous solutions. Graph radical functions (square root, cube root). Write the domain and range of a radical function.

Assessment:

Simplify radicals of the form √8∧ 3√54 ,√25 x2 . Perform the indicated operation. Write in simplest form:

4√x38√x2

, √3 ∙ 4√3

Simplify: 2√8+3√12. Simplify: 3+√6

2−√5.

Simplify: 21613 ,7

12 ∙7

12 ,5

14 ∙125

14

Write in radical form: 3 x12 ,¿, (−8 x √xy )

23

Write with rational exponents: √5 x3 y10 z Solve: 3+√2 x−3=8, 1−2√ x+2=9, √ x2+21 x=14 ,√3 x=√ x+6,

√3 x+2−√2x+7=0, √3 x+1−√x+1=2, 2 ( x+3 )23=8

Graph and write the domain and range of f ( x ) = √ x−1+3. Graph and write the domain and range of f ( x )= 3√x+3−1

Resources: Pearson Algebra 2 Common Core: Chapter 6 (Sections 6.1 – 6.5): pages 360 – 397.

Standards:CC.2.2.HS.D.2 – Write expressions in equivalent forms to solve problems.CC.2.2.HS.D.7 – Create and graph equations or inequalities to describe numbers or relationships


CC.2.2.HS.D.8 – Apply inverse operations to solve equations or formulas for a given variableCC.2.2.HS.D.9 – Use reasoning to solve equations and justify the solution method.CC.2.2.HS.D.10 – Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.CC.2.1.HS.F.1 – Apply and extend the properties of exponents to solve problems with rational exponents.A2.1.2.1.1 A.2.1.2.1.2 A.2.1.2.1.3 A.2.1.2.1.4 A2.1.3.2.2

A2.1.3.1.2

Vocabulary:

Extraneous solution- a solution that gives a false statement; Index- the root of a radical expression; Radicand- expression under a radical symbol; Rationalization- the process of removing a radical from the denominator

Comments:


Content: Inverse Relations and Functions

Duration: January (1 week)

Essential Question:

Is the inverse of a function always a function?How are a function and its inverse function related?

Skill:

Add, subtract, multiply, and divide functions. Compose functions. Given a table of values of a function, give the table of values that

corresponds to the inverse. Find the inverse of a relation or a function graphically. Find the inverse of a relation or a function algebraically. Use the horizontal line test to determine if a function’s inverse is a function. Given the equation of a quadratic function, find the inverse, and then sketch

the original function, its inverse, and restrict the domain accordingly. Show that the domain of a function is the range of its inverse and that the

range of a function is the domain of its inverse. Show that two functions are inverses of each other using composition.

Assessment:

If f ( x )=x+3 and g ( x )=x2−9, find, and h ( x )=x2+3x−7: find:

g+h ,h−g , fg , fh , fg, gf, f (g (x ) ) , f (g (3 ) ) , h (f ( x ) )

What is the inverse of relation s? Then graph s and its inverse.

Find the inverse of y=x2−1 then graph both relations. Find the inverse function of f ( x )=√x−2. Give the domain of the function

and its inverse. Graph both functions. Find the inverse and the domain and range of the function of its inverse.

Determine whether the inverse is a function: y= (7−x )2.

Determine if f ( x )=4 x+7 and g ( x )= x−74

are inverses.

Determine if f ( x )= 1x−1

and g ( x )=1x+1 are inverses.


Resources: Pearson Algebra 2 Common Core: Chapter 6 (Sections 6.6 and 6.7): pages 400 – 412.

Standards:

CC.2.2.HS.C.1 – Use the concept and notation of functions to interpret and apply them in terms of their context.CC.2.2.HS.C.3 – Write functions or sequences that model relationships between two quantities.CC.2.2.HS.C.4 – Interpret the effects transformations have on functions and find the inverses of functions.CC.2.2.HS.D.2 – Write expressions in equivalent forms to solve problems.A2.2.1.1.3 A2.2.1.1.3 A2.2.1.1.3

Common Core: Verify by composition that one function is the inverse of another.

Vocabulary:

Inverse Function- both the relation and its inverse are functions; Inverse Relation- if a relation pairs element a of its domain with element b of its range, the inverse relation “undoes” the relation and pairs b with a, and so if (a, b) is an ordered pair of a relation, then (b, a) is an ordered pair of its inverse; One-to-one function- a function in which each x value in the domain corresponds to exactly one y value in the range

Comments:


Content: Exponential Functions Duration: January (3 weeks)

Essential Question: How do you model a quantity that changes regularly over time?

Skill:

Given the equation of an exponential function, sketch the graph showing the correct horizontal asymptote.

Use transformation rules to graph exponential functions. Identify exponential growth and exponential decay given the equation or the

graph. Write exponential functions to model real-world exponential growth and

decay situations. Use Euler’s number to model real-world situations.

Assessment:

Graph the following: y=3x , y=3− x , y=3x+2 , y=3x−2 Without graphing, determine whether the function represents exponential

growth or decay: y=129 (1.63 ) x. Suppose you deposit $200 in a savings account that pays an annual rate of

4%. How much will be in the account after 3 years? If the annual rate of change is +70%, what is the growth factor? If the annual rate of change is -0.1%, what is the decay factor? The table shows the world population of the Iberian Lynx in 2003 and 2004.

If this trend continues and the population is decreasing exponentially, how many Iberian Lynx will there be in 2014?

Suppose you won a contest at the start of 5th grade that deposited $3000 in an account that pays 5% annual interest compounded continuously. How much will you have in the account 4 years later?

Resources: Pearson Algebra 2 Common Core: Chapter 7 (Sections 7.1 & 7.2): pages 434 – 450.

Standards:

CC.2.2.HS.C.1 – Use the concept and notation of functions to interpret and apply them in terms of their context.CC.2.2.HS.C.2 – Graph and analyze functions and use their properties to make connections between the different representations.CC.2.2.HS.C.3 – Write functions or sequences that model relationships between two quantities.CC.2.2.HS.C.4 – Interpret the effects transformations have on functions and find


the inverses of functions.CC.2.2.HS.C.5 –Construct and compare linear, quadratic, and exponential models to solve problems.CC.2.2.HS.C.6 – Interpret functions in terms of the situations they model.CC.2.2.HS.D.2 – Write expressions in equivalent forms to solve problems.

CC.2.2.HS.D.7 – Create and graph equations or inequalities to describe numbers or relationshipsCC.2.2.HS.D.10 – Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.CC.2.1.HS.F.4 – Use units as a way to understand problems and to guide the solution of multi-step problems.A2.2.1.1.1 A2.2.1.1.3 A2.1.1.4 A2.2.1.1.4 A2.2.1.1.1

A2.2.2.2.1A2.1.3.1.3 A2.1.3.1.4 A.2.1.2.1.4 A2.2.2.1.2 A2.1.3.1.3

A2.1.3.1.4

Vocabulary:

Exponential Function: a function in the form of y=abx where a≠0 , b>0 , b≠1, Exponential Growth: as x increases, y increases and when y=abx , b>1, Exponential Decay: as x increases, y decreases and approaches 0 and when y=abx ,0<b<1, Horizontal Asymptote: a horizontal line that the graph of a function approaches, for exponential functions, the graph will not cross the horizontal asymptote

Comments:


Content: Logarithmic Functions Duration: February (2 weeks)

Essential Question: How are exponential functions and logarithmic functions related?

Skill:

Write and evaluate logarithmic functions Graph logarithmic functions Define common logarithms and natural logarithms Condense logarithmic expressions into a single logarithm Expand a single logarithm Use the Change of Base Formula to evaluate logarithmic expressions

Assessment:

Write the following in logarithmic form: 102=100 ,81=34 , 827

=( 23 )3

Evaluate the following: log 832 , log5125 , log64132

Graph y=log2x , y=log2(x−3)

Condense into a single logarithm: 3 log2 x+12log2 y−log2 z

Expand: log x3√ yz5

Evaluate using the Change of Base Formula: log27

Resources: Pearson Algebra 2 Common Core: Chapter 7 (Sections 7.3 & 7.4): pages 451 - 468.

Standards: CC.2.2.HS.C.1 – Use the concept and notation of functions to interpret and apply them in terms of their context.CC.2.2.HS.C.2 – Graph and analyze functions and use their properties to make connections between the different representations.CC.2.2.HS.C.3 – Write functions or sequences that model relationships between two quantities.CC.2.2.HS.C.4 – Interpret the effects transformations have on functions and find the inverses of functions.CC.2.2.HS.C.6 – Interpret functions in terms of the situations they model.CC.2.2.HS.D.2 – Write expressions in equivalent forms to solve problems.CC.2.2.HS.D.7 – Create and graph equations or inequalities to describe numbers or relationshipsCC.2.2.HS.D.10 – Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.CC.2.1.HS.F.4 – Use units as a way to understand problems and to guide the


solution of multi-step problems.A2.2.1.1.1 A2.2.1.1.3 A2.1.1.4 A2.2.1.1.4 A2.2.1.1.1

A2.2.2.2.1A.2.1.2.1.4 A2.2.2.1.2 A2.1.3.1.3 A2.1.3.1.4

Vocabulary:

Common Logarithm- log base 10; Logarithm- log b x= y iff x=by; Logarithmic Function- the inverse of an exponential function; Natural Logarithm- log base e

Comments:


Content: Exponential and Logarithmic Equations

Duration: February (2 weeks)

Essential Question:

How do you solve equations involving variable exponents?How do you solve equations involving logarithms?

Skill:

Solve exponential equations that have the same base. Solve exponential equations by getting a common base. Solve exponential equations using logarithms including the natural log. Solve logarithmic equations using logarithmic properties and check for

extraneous solutions.

Assessment:

Solve for x: 3x+7=321 Solve for x: 24 x+1=16

Solve for x: ( 425 )x+1

=( 25 )12 x−11

Solve for x: 3x=7 Solve for x: 2x+1=54 x−3 Solve for x: e7x +1=15 Solve for x: log (3 x+1 )=2 Solve for x: log ( x−3 )+ log (x )=1 Solve for x: log3 (x2 )= log316 Solve for x: ln ( x+1 )− ln x=2 Solve for x: log ( x−3 )−log 2=log15

Resources: Pearson Algebra 2 Common Core: Chapter 7 (Sections 7.5 & 7.6): pages 469 - 485.

Standards:

CC.2.2.HS.D.2 – Write expressions in equivalent forms to solve problems.CC.2.2.HS.D.8 – Apply inverse operations to solve equations or formulas for a given variableCC.2.2.HS.D.9 – Use reasoning to solve equations and justify the solution methodCC.2.2.HS.D.10 – Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.A.2.1.2.1.4 A2.1.3.2.2 A2.1.3.1.3 A2.1.3.1.4

Vocabulary: No new vocabulary for this section.


Comments:


Content: Rational Expressions and Equations

Duration: March (4 weeks)

Essential Question:

Are a rational expression and its simplified from equivalent?How do rational functions model real world problems and their soltions?

Skill:

Simplify rational expressions and state any restrictions on the variable. Multiply and divide rational expressions. Add and subtract rational expressions. Solve rational equations. Graph rational functions by identifying zeros, holes, vertical asymptotes,

horizontal asymptotes, and oblique asymptotes.

Assessment: Simplify and state any restrictions on the variable: x

2−x−6x2+ x−2

Simplify and state any restrictions on the variable: x2+7 x+10x2−3x−10

Multiply: x2+x+6x−5

∙ x2−25

x2+4 x+3

Divide: x2

x2+2 x+1÷ 3 xx2−1

Add: 5 y+2x y2

+ 2x−44 xy

Add: xx−1+2 x−1x2−3 x+2

Subtract: x+2x2−2x

− x+22x−4

Simplify: x1x+ 1y

Simplify:

x−2x

+ 2x+1

3x−1

− 1x+1


Solve: 2x+3

= x−32

Solve: xx−3+xx+3

= 2x2−9

Solve: x−1x2+3 x+2

+ 2xx+2

= x−1x+1

Graph: f ( x )= x−3x+4

, f ( x )= x2

x2+1, f (x )=2 x

2−10 x+12x2−9

, f ( x )=2 x2−3 x+3x−2

showing all zeros and points of discontinuity.

Resources: Pearson Algebra 2 Common Core: Chapter 8: pages 495 - 548

Standards:

CC.2.2.HS.D.2 – Write expressions in equivalent forms to solve problems.CC.2.2.HS.D.6 – Extend the knowledge of rational functions to rewrite in

equivalent formsCC.2.2.HS.D.8 – Apply inverse operations to solve equations or formulas for a given variableCC.2.2.HS.D.10 – Represent, solve, and interpret equations/inequalities and

systems of equations/inequalities algebraically and graphically.A2.1.2.2.2 A2.1.3.2.2 A2.1.3.1.2

Common Core: Graph rational functions identifying zeros and asymptotes when suitable factorizations are available and showing end-behavior.

Vocabulary:

Simplest Form- the numerator and denominator are polynomials that have no common divisors; Rational Equation- contains at least one rational expression; Rational Expression- the quotient of two polynomial expressions

Comments:


Content: Sequences & Series Duration: April (2 weeks)

Essential Question:

How can you represent the terms of a sequence with an algebraic rule? Can a sequence have more than one rule?

Skill:

Recognize a mathematical pattern in a sequence. Use an explicit formula to find the nth term of a sequence. Use a recursive formula to find the nth term of a sequence. Write an explicit formula for a sequence. Write a recursive formula for a sequence. Find the common difference of an arithmetic sequence. Find the common ratio of a geometric sequence. Find the arithmetic and geometric mean between two numbers. Find the sum of a finite arithmetic series. Rewrite a series using summation notation. Find the sum of a series written in summation notation. Find the sum of a finite geometric series. Find the sum of an infinite geometric series. Determine whether a geometric series converges or diverges.

Assessment:

Are the following sequences arithmetic, geometric, or neither?o 4, 7, 10, 13, …o 1, 3, 6, 10, 15, 21, …

o12

, −14

, 18

, −116

, … Write an explicit and recursive formula for the following sequences

o 5, 8, 11, 14, 17, …o – 16, – 8, – 4, – 2, …o 49, 64, 81, 100, 121, …

Find the 32nd term of the sequence: 9, 4, – 1, – 6, – 11, … Find the 8th term of the sequence: – 3, 6, – 12, … Find the 100th term of the sequence: 1, 3, 6, 10, 15, 21, … Find the missing term of the geometric sequence: 12, _?_, 3, … Find the missing term of the arithmetic sequence: – 15, _?_, 1, … Find the first 5 terms in each sequence given the following formulas:

o an=2n−5o an=n

2+3o a1=1 and an=an−1+n

Find the sum of 1+4+7+…+31 Find the sum of −3+(−6 )+(−9 )+…+(−30) Write the series in summation notation: 1+4+7+…+31 Write the series in summation notation: −3+(−6 )+(−9 )+…+(−30) Find the sum of:

∑n=1

10

(3n−4)

Find the sum of:


∑n=1

9

2 (−1 )n

Find the sum of 3+6+12+24+…+3072 Find the sum of −5−10−20−40−…−2560 Determine whether the series converges or diverges: 1+2+4+… Determine whether the series converges or diverges: 1−1

2+ 14−…

Find the sum: 3+1+ 13+ 19+…

Find the sum: 1.1+0.11+0.011+… Find the sum:

∑n=1

∞

( 15 )n−1

Resources: Pearson Algebra 2 Common Core: Chapter 9: pages 561 – 601

Standards:

CC.2.2.HS.C.3 – Write functions or sequences that model relationships between two quantities.

A.2.1.1.1 A2.2.1.1.2

Common Core: Derive the formula for the sum of a finite geometric series, and use the formula to solve problems.

Vocabulary:

Arithmetic Mean- the average of two numbers (the arithmetic mean of x and y is x+ y

2); Arithmetic Sequence- a sequence that has a constant difference

between each term; Arithmetic Series- a series whose terms form an arithmetic sequence; Common Difference- the constant difference between terms of an arithmetic sequence; Common Ratio- the constant multiple(ratio) between two terms of a geometric sequence; Converge- if an infinite series gets closer and closer to a real number as the number of terms increases; Diverge- if an infinite series does not converge; Explicit Formula- a formula that expresses the nth term of a sequence in terms of n, where n is a positive integer; Finite Sequence- a sequence that has a first and last term; Finite Series- a series that has a first and last term; Geometric Mean- the square root of the product of two numbers (the geometric mean of x and y is √ xy); Geometric Sequence- a sequence that has a common ratio (multiple) between each term; Geometric Series- a series whose terms form a geometric series; Infinite Sequence- a sequence that continues without end; Infinite Series- a series that continues without end; Recursive Formula- gives the first term and defines the other terms by relating each term after the first term to the one before it; Sequence- An ordered list of numbers; Series- the sum of the terms of a sequence; Term- a number in a sequence

Comments:


Content: Probability and Samples Duration: April/May (3 weeks)

Essential Question:

What is the difference between a combination and a permutation?How can data be used to make predictions about future events?

Skill:

Use the Fundamental Counting Principal to determine the number of outcomes for a particular event.

Use a permutation to determine the number of ways that items can be arranged / selected.

Use a combination to determine the number of ways that items can be grouped.

Determine whether to use a combination or permutation to describe a situation.

Find the experimental and theoretical probability of an event. Find the geometric probability of an event. Use combination and permutations to determine the probability of an event. Determine the probability of multiple events occurring (and/or). Determine if two events are dependent or independent. Determine if two events are mutually exclusive. Use a contingency table to determine conditional probability. Use a tree diagram to determine conditional probability. Use probabilities to make fair decisions and analyze decisions. Identify sampling methods. Recognize bias in samples and surveys.

Assessment:

How many different sandwiches can be made if there is a choice between 3 types of bread, 4 types of meant, 3 types of cheese and 5 types of sauce? (Assuming you chose one bread, one meat, one cheese and one sauce)

How many different ways can 10 racers finish in 1st, 2nd and 3rd place? How many different teams of 6 players can be chosen from a group of 20

players? Evaluate 8 ! A baseball player got a hit 20 times out of his last 50 at bats, what is the

experimental probability that he will get a hit at his next at bat? What is the probability of rolling a 5 when one fair die is rolled? What is the probability of getting a sum of 5 when two fair die are rolled? What is the probability of being dealt exactly two 7’s in a 5-card hand from a

standard deck of 52 cards? A carnival game consists of throwing darts at a circular board as shown

below. What is the probability that a dart thrown will hit the shaded region?

A jar contains 30 red marble, 50 blue marbles, and 20 white marbles. What is P (red∨blue )?

At a party, there are 10 regular drinks and 4 diet drinks. There are also 6


bags of fat free chips and 10 bags of regular chips. If you pick a drink and a bag of chips at random, what is the probability of getting a diet drink and a fat free bag of chips?

If you draw two cards from a standard deck, what is the probability that you will draw a 2 or a face card?

If you draw two cards form a standard deck, what is the probability that you will draw a jack or a heart?

Suppose a jar contains 5 red jellybeans, 2 blue jellybeans and 8 green jellybeans. If one jellybean is chosen, put back in the jar then another is chosen, what is the probability of choosing a blue and a green jellybean?

Suppose a jar contains 5 red jelly beans, 2 blue jelly beans and 8 green jelly beans. If two jellybeans are chosen and not put back in the jar, what is the probability of choosing a blue and a green jellybean?

Suppose a jar contains 5 red jelly beans, 2 blue jelly beans and 8 green jelly beans. If two jellybeans are chosen and not put back in the jar, what is the probability of choosing two red jellybeans?

The table below shows students by age and ethnicity, if you pick a student at random what is the probability that they are female given that they are Spanish (P (female|spanish )?

You are off to soccer, and want to be the Goalkeeper, but that depends who is the Coach today: With Coach Sam, the probability that you are the goalkeeper is 0.5, but with Coach Alex, the probability that you are the goalkeeper is 0.3. Sam is Coach more often, about 6 out of every 10 games. So, what is the probability you will be a goalkeeper today? (Make a tree diagram to model the situation)

You and your three siblings need to decide which two of you are going to shovel the snow from the driveway. You each are assigned a number from 1 to 4 and then a spinner is used to determine the first person. Then that person decides the next person. Will this method result in a fair decision?

A supermarket wants to identify the percent of shoppers that use coupons, so a manager interviews every shopper in the makeup aisle. What type of sample does this represent and identify any bias.

A group of students is randomly selected from the school database; they are asked how much time they spend each week on household chores. What type of study method does this represent and can the results be used to make generalizations about the population?

Resources: Pearson Algebra 2 Common Core: Chapter 11 (Sections 11.1 – 11.5 & Section 11.8): pages 674 – 709 & pages 725 – 730

Standards:CC.2.4.HS.B.4 – Recognize and evaluate random processes underlying statistical experiments.CC.2.4.HS.B.5 – Make inferences and justify conclusions based on sample


surveys, experiments, and observational studies.CC.2.4.HS.B.6 – Use the concepts of independence and conditional probability to interpret data.CC.2.4.HS.B.7 – Apply the rules of probability to compute probabilities of compound d events in a uniform probability model.A2.2.3.2.3 A2.2.3.2.1 A2.2.3.2.2

Vocabulary:

Bias- a systematic error introduced by a sampling method; Combination: any unordered set of r objects from a set of n objects. The number of combinations of n objects taken r at a time is nCr¿

n !r ! (n−r )! ; Conditional Probability- the

probability that event B will occur, given that event A has already occurred P (A|B )= P (A∧B )

P (A ) ; Contingency table- a frequency table that contains data

from two different categories; Controlled Experiment- a sample is divided into two groups, one group receives “treatment” the other group (control group) does not receive treatment and then the two groups are compared; Convenience Sample- members of a population who are conveniently and ready available are chosen for a sample; Dependent Events- two events where the outcome of the first event does affect the probability of the second; Experimental Probability- the ratio of the number of times the event occurs to the total number of trials; Independent Events- two events where the outcome of the first event does not affect the probability of the second; Mutually Exclusive Events- When two events cannot happen at the same time (If two events are mutually exclusive then P (A∨B )=P ( A )+P (B ) and P (A∧B )=0); n factorial(n! )- for any positive integer n, n factorial is given by 1∗2∗3∗… (n−1 )∗n; Observational study- members of a sample are observed in a way that they are not affected by the study; Permutation- an arrangement of items in a particular order. For n items arranged r at a time nPr¿

n !(n−r )!

; Population- All the members of a set; Random

Sample- members of a population are selected at random to be part of a sample, each member has an equally likely chance of being selected; Sample- a part of a population; Sample Space- the set of all possible outcomes in an experiment; Self-Selected Sample- members of a population that volunteer are selected for a sample; Simulation- a model that imitates one or more events; Survey- every member of a sample is asked a set of questions; Systematic Sample- members of a population are ordered then selected at regular intervals for a sample; Theoretical Probability- If a sample space has n equally likely outcomes, and an event A occurs in m of theses outcomes, then P (A )=m

nComments:


Content: Regression Duration: May (2 weeks)

Essential Question:

How can real world data be model by an equation?How can you make predictions based on data?

Skill:

Model data with linear functions Model data with quadratic functions Model data with polynomial functions (cubic and quartic) Model data with exponential and logarithmic functions Determine the appropriate type of regression model for a set of data. Use regression equations to make predictions

Assessment:

Given a set of data, run a linear regression on the graphing calculator. Use the regression equation to make predictions.

Given a set of data, run a quadratic regression on the graphing calculator. Use the regression equation to make predictions.

Given a set of data, run a cubic and quartic regression on the graphing calculator. Use the regression equation to make predictions.

Given a set of data, run an exponential and logarithmic regression on the graphing calculator. Use the regression equation to make predictions.

Given a scatterplot and/or a set of data, determine the regression type that best models the data. Use the correlation coefficient (r-value) to confirm your prediction.

Resources:

Pearson Algebra 2 Common Core: Section 2.5 (pages 92 – 99), Section 4.3 (pages 209 – 214), Section 5.8 (pages 331 – 338), Section 7.2 (pages 445 – 450), Section 7.3 Concept Bite (pages 459 & 460)“Bungee Barbie” Lab“Challenger Space Shuttle (O-ring)” Lab

Standards:

CC.2.4.HS.B.2 – Summarize, represent, and interpret data on two categorical and quantitative variableCC.2.4.HS.B.3 – Analyze linear models to make interpretations based on the

data.CC.2.1.HS.F.3 – Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs and data displays.CC2.1.HS.F.5 – Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.A2.2.3.1.2 A2.2.3.1.1 A2.2.3.1.2 A2.2.3.1.1 A2.2.3.1.2


Vocabulary:

Correlation Coefficient (r-value)- a statistical measure of the degree to which changes to the value of one variable predict change to the value of another, correlation coefficients are expressed as values between +1 and -1; Regression Equation- an equation that represents the relationship between two quantitative variable; Scatterplot- the graph of plotted points that shows a relationship between two sets of data

Comments:

consensus map grade level - west jefferson hills … ii_honors_15162.docx · web viewcommon core:...

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