Algebra II and PreAP Algebra II Scope and Sequence with …...NMSI’s Laying the Foundation Lesson:...

Algebra II Unit 1 Page 1

Algebra II and PreAP Algebra II Scope and Sequence with NMSI’s Laying the Foundation lessons and Quality Core Modules

Unit 1: Functions, Equations, Absolute Value and Inequalities 15 days of instruction plus assessment time- 3.5 weeks

Teacher Note: This unit is designed as a review and extension of material covered in the PreAP Algebra I and PreAP Geometry Course. Discussion with previous teachers about how much of the curriculum was actually covered and concepts that were not well received by students is vital to planning the time for this unit. The teacher needs to go to the Quality Core website and download Quality Core Alg II Unit 2 Linear Equations and Inequalities for use in this unit and go to the NMSI website and download all of the lessons for this unit. A graphing calculator is needed for many of the lessons in this unit. NMSI’s Laying the Foundation Lesson: Literal Equations (1 day) Teacher Note: For #1 the students need access to formulas. You could give them the Quality Core reference sheet so that the formulas are not holding them back from what we really want them to do and that is to be able to manipulate equations to solve for specific variables. This would be a great 1st or 2nd day activity. ** Checkpoint Unit 1 1,2,3 can be used with this lesson.

AL COS Common Core Standard

Common

Core

Algebra II 23

Rearrange formulas to highlight a quantity of interest, using the same reasoning as

in solving equations.

A-CED4

NMSI’s Laying the Foundation lesson: Introducing Interval Notation (.5 days) Teacher Note: Provides a review of inequalities and their graphs. Students should understand the concepts of and “increasing”, “decreasing”, and “constant” function. Introduces the concept of + and – infinity. This lesson takes students from the inequality way of writing an interval to the interval notation and ties the graph on the number line to both. Answers to Quality core tests samples are both in inequality notation and interval notation. The committee believes that this is a standard that should have been left in Algebra II and is a great review of Algebra I.


Common

Core

Old Algebra II

22

For a function that models a relationship between two quantities, interpret key

features of graphs and tables in terms of the quantities, and sketch graphs showing

key features given a verbal description of the relationship. Key features include

intercepts; intervals where the function is increasing, decreasing, positive, or

negative; relative maximums and minimums; symmetries; end behavior; and

periodicity.

F-IF4


NMSI’s Laying the Foundation lesson: Transformations of Functions Exploration (2.5 days) Teacher Note: Have students complete the first page for homework the night before. Students should be familiar with the following function notations: f(x) + c, f(x) – c, f(-x), -f(x); and be able to do powers, square roots, and absolute value. Suggest students work in groups on part 2. Graphing Calculators are needed for this lesson and students will need graph paper. This may take a little longer if graphing calculators have not been used before. ** Checkpoint Unit 1 5,6,7,8 can be used with this lesson.


Common Core

Algebra II 34

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)

for specific values of k (both positive and negative); find the value of k given the

graphs. Experiment with cases and illustrate an explanation of the effects on the

graph using technology. Include recognizing even and odd functions from their

graphs and algebraic expressions for them.

F-BF3

NMSI’s Laying the Foundation lesson: Even/Odd functions (1 day) Teacher Note: You want to talk with your Algebra I teacher and see if they did this lesson. If not, this is a great review of Algebra I to start the year. The teacher instructions suggest to use your information from Transformations of Functions Exploration to help you make connections. Graphing Calculators are needed for this lesson. The students

must understand how to define an even/odd function ( ) ( ) ( ) ( )f x f x and f x f x and connect that with the

points on the graph. The committee believes that this is a standard that should have been left in Algebra II and is a great review of Algebra I. ** Checkpoint Unit 1 9 can be used with this lesson.


Common

Core

Old Algebra II

22






periodicity.

F-IF4

NMSI’s Laying the Foundation lesson: Transforming Domain and Range (1 day) Teacher Note: The activity asks students to transform the absolute value function and then to discuss how the domain and range are affected. Problems 6-9 will be discovery for students. ** Checkpoint Unit 1 13,17,18,19 can be used with this lesson.


Common

Core

Algebra II 34






F-BF3

Algebra II 29

Relate the domain of a function to its graph and, where applicable, to the

quantitative relationship it describes.

F-IF5


NMSI’s Laying the Foundation lesson: Applying Piecewise Functions (1 day) Teacher Note: This lesson makes connections between distance-time graphs and speed graphs. The graphs are linear piecewise graphs. ** Checkpoint Unit 1 10,11,12,14,15,16,22,23 and Illustrative Mathematics Pizza Promotion can be used with this lesson.


Common

Core

Algebra II 30

Graph functions expressed symbolically and show key features of the graph, by

hand in simple cases and using technology for more complicated cases (piecewise

functions)

F-IF7b

Algebra II 31

Write a function defined by an expression in different but equivalent forms to

reveal and explain different properties of the function.

F-IF8

Algebra II 29



F-IF5

Old Algebra II

24

Calculate and interpret the average rate of change of a function (presented

symbolically or as a table) over a specified interval. Estimate the rate of change

from a graph.

F-IF6

NMSI’s Laying the Foundation lesson: Walking Piecewise Graphs (1 day) Teacher Note: Another option for LTF lessons if you have the CBR. Students walk piecewise graphs.


Common

Core

Algebra II 30


hand in simple cases and using technology for more complicated cases

F-IF7

Algebra II 30a

Graph square root, cube root, and piecewise-defined functions, including step functions and

absolute value functions.

F-IF7b

Old Algebra II

24



from a graph.

F-IF6

Quality Core Alg II Unit 2 Linear Equations and Inequalities (C-2 – C-7) (.5 day) Solving Inequalities Matching Game

Teacher Note: This should be a review of how to solve different types of inequalities. This activity has all of the steps of solving an inequality and students must match and give justification for each step.


Common

Core

Algebra II 20

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

Include equations arising from linear and quadratic functions, and simple rational

and exponential functions.

A-CED1


Quality Core Alg II Unit 2 Linear Equations and Inequalities (D-3-D-8, E-4- E-8) (3 days) Absolute Value and Inequalities

Teacher Note: These activities show students how to solve different types of absolute equations and inequalities.


Common

Core

Algebra II 21

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]

A-CED2

NMSI’s Laying the Foundation lesson: Exploring Inequalities (.5 days) Teacher Note: Lesson links the solution of an inequality in 2 variables to an inequality in one variable. Students should have previous experience graphing linear, quadratic, and absolute value functions. ** Checkpoint Unit 1 #24 can be used with this lesson.


Common

Core

Algebra II 27

Explain why the x-coordinates of the points where the graphs of the equations y =

f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the

solutions approximately, e.g., using technology to graph the functions, make tables

of values, or find successive approximations. Include cases where f(x) and/or g(x)

are linear, polynomial, rational, absolute value, exponential, and logarithmic

functions.

A-REI11

NMSI’s Laying the Foundation lesson: Systems of Linear Inequalities (1 day) Teacher Note: Prepare the model of the solid using cardstock prior to the lesson. Students should be familiar with graphing linear inequalities, finding areas and perimeters of polygons, and volumes of solids. The reference sheet could be used for the formulas of the different shapes. ** Checkpoint Unit 1 25,26,27,28 can be used with this lesson.


Common

Core

Algebra II 27






functions.

A-REI11

Algebra II 22

Represent constraints by equations or inequalities, and by systems of equations

and/or inequalities, and interpret solutions as viable or nonviable options in a

modeling context.

A-CED3

Quality Core Alg II Unit 2 Linear Equations and Inequalities (J-7 and L-2) (2days)

Linear Programming Teacher Note: This should have been covered in Algebra I and should be a quick review. Talk with Algebra I teacher to determine how much time will need to be spent on this. This could be covered on the EOC test for Algebra II.


Common

Core

Algebra II 21

Create equations in two or more variables to represent relationships between

quantities; graph equations on coordinate axes with labels and scales.

A-CED2

Algebra II Represent constraints by equations or inequalities, and by systems of equations A-CED3


22 and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

Algebra II 27






functions.

A-REI11

Teacher led Instruction (1 day) Teacher Note: Use illuminations- Egg Launch contest (Quadratic).


Common

Core

Algebra II 32

Compare properties of two functions each represented in a different way

(algebraically, graphically, numerically in tables, or by verbal descriptions).

Example: Given a graph of one quadratic function and an algebraic expression for

another, say which has the larger maximum.

F-IF9

Algebra II and PreAP Algebra II Unit 2 Page 1


Unit 2: Arithmetic Series and Matrices 9 days of instruction plus assessment time-2 weeks

Teacher Note: Teachers will need to go to Quality Core Website and download from Algebra II Unit 1 The purpose and Predictability of Patterns and Unit 3 What is a Matrix- Really Qulaity Core Alg II Unit 1 The Purpose and Predictability of Patterns- (D-6 – F-10) (3 days) Arithmetic Sequences Series

Teacher Notes: Students should have been introduced to arithmetic sequences in Algebra I. Communicate with the Algebra I teacher to understand depth of knowledge. It is very important to make the connection between linear functions that have constant slope and arithmetic sequences that have common differences.

1 1

( 1)

na a d n

y b m x

The comparison of these two should be key to what you are doing. It is not a new formula, it is just

a line moved to the right one because you cannot start with the 0 term, so you move the line over to the right one (x-1) to start with the 1st term.


Common

Core

Algebra I 38

Construct linear and exponential functions, including arithmetic and sequences,

given a graph, a description of a relationship, or two input-output pairs (include

reading these from a table).

F-LE2 Partial

Algebra I 35

Write arithmetic sequences both recursively and with an explicit formula, use them

to model situations, and translate between the two forms. F-BF2 Partial


Quality Core Alg II Unit 3 What is a Matrix- Really? (Entire Unit) (6 days) Teacher Notes: Students have probably not been introduced to matrices until now. Adding, Subtracting and multiplying by a scalar is simple and could be done in one day (B-5 – D-6). Multiplying matrices will take two days (E-1 – F-7) one day for practice and another for application. This unit does a good job with application. Solving Matrices Equations requires the discussion of the identity matrix and determinant. This unit goes into great detail about students discovering the pattern for the inverse matrix by using the identity matrix. There may not be time for this and you may need a direct approach to finding determinants and inverse matrices. You might want to skip H and K if you are worried about time. The key to this concept is the application of a matrix to solve a problem. This is where the technology comes in to save on time. Students need to know how to solve a 3 variable equation by hand, but do not need to spend a great deal of time on this. The connection of how to solve a 3 variable equation using matrices is very important. Quality Core put the solving 3 variable equations in the graphing calculator part of their test. There is not enough graphing calculator practice problems and I would find another source, textbook or google it, to obtain some more. There is no mention of row reduction or row echelon form in either CCSS or QC. (3 or 4 days)


Common

Core

Algebra II 7

Use matrices to represent and manipulate data, e.g., to represent payoffs or

incidence relationships in a network.

N-VM6

Algebra II 8

Multiply matrices by scalars to produce new matrices, e.g., as when all of the

payoffs in a game are doubled.

N-VM7

Algebra II 9

Add, subtract, and multiply matrices of appropriate dimensions. N-VM8

Algebra II 10

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

N-VM9

Algebra II 11

Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

N-VM10

This is not specifically designated in Common Core. They suggest the use of technology.

PreCalc Represent a system of linear equations as a single matrix equation in a vector variable.

A-REI8

Algebra II 26

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

A-REI9



Unit 3: Quadratic and Root Functions 17 days of instruction plus Checkpoint time- 3.5 weeks

NMSI’s Laying the Foundation lesson: Transformations of Conic Sections (1 day) Teacher Note: Graphing and translating 2 2 2, ,x x and x r . The students should have graphed They should graph

2y x in Algebra 1 and 2 2y x r in Geometry. This lesson should be a good review of these and introduce y x as the

function that is made when you reflect 2y x about the line y x . Only Responsible for the parabolas, square root and

circles (would be okay if did not do v-vii on circles). Hyperbolas are taken care of in PreCalc.


Common

Core

Precalc 28

Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles,

and degenerate conics, from second-degree equations. Example: Graph x2 – 6x +

y2 – 12y + 41 = 0 or y2 – 4x + 2y + 5 = 0.

a. Formulate equations of conic sections from their determining characteristics.

AL Standard

?

Old Algebra II

22






periodicity.

F-IF4

Algebra II 29



F-IF5

Algebra II 30


hand in simple cases and using technology for more complicated cases.*

a. Graph square root, cube root, and piecewise-defined functions, including step

functions and absolute value functions.

F-IF7

F-IF7b

Algebra II 31

Write a function that describes a relationship between two quantities.* [F-BF1]

a. Combine standard function types using arithmetic operations.

F-BF1

F-BF1b

Algebra II 34






F-BF3


NMSI’s Laying the Foundation lesson: Composition of Functions Graphically (.5 day) Teacher Note: This is a neat lesson to teach composition of functions. ** Checkpoints Unit 1 #4 can be used with this lesson.


Common

Core

This is a foundational lesson.

NMSI’s Laying the Foundation lesson: Composition of Functions ( .5 day) Teacher Note: Introduction of the ( )f g x symbol for composition and begins to compose functions that are and are not inverses. #2 deals with composing 2y x and y x .

**Checkpoints Unit 3 #3 can be used with this lesson.

AL COS

Common Core Standard

Common

Core

Algebra II 35a

Solve an equation of the form f(x) = c for a simple function f that has an inverse,

and write an expression for the inverse.

F-BF4a

NMSI’s Laying the Foundation lesson: Composition of Functions Exploration (1 day) Teacher Note: Introduces 1f notation for inverse and drives home the fact that the composition of inverse functions

results in y x .


Common

Core

Algebra II 35

Solve an equation of the form f(x) = c for a simple function f that has an inverse,

and write an expression for the inverse.

F-BF4a

NMSI’s Laying the Foundation lesson: Solving equations Graphically-Is there a solution? (1 day) Teacher Note: This lesson asks students to find solutions of linear, absolute value, and quadratic functions. The radical function is then introduced and the idea of extraneous answers is discussed **Checkpoint Unit 3 #6 and Illustrative Mathematics How does the Solution change? can be used with this lesson


Common

Core

Algebra II 24

Solve simple rational and radical equations in one variable, and give examples

showing how extraneous solutions may arise. A-REI2

Algebra II 27






functions.

A-REI11

Old Algebra II

22






periodicity.

F-IF4


Algebra II 29


quantitative relationship it describes. F-IF5

Algebra II 30





F-IF7

F-IF7b

Algebra II 33

Write a function that describes a relationship between two quantities.*


F-BF1

F-BF1b

Algebra II 34






F-BF3

Algebra II 28

Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. a. Formulate equations of conic sections from their determining characteristics.

AL Standard

Teacher Led Discussion: Complex Roots and Quadratics (3 days) Teacher Note: There is not a specific lesson that discusses complex roots. Your textbook should do a good job of explaining the complex roots and how the Quadratic Formula’s discriminant helps us decide what type of roots a quadratic has. This might also be a good place to graph quadratic inequalities. ** Checkpoint Unit 3 5,7 can be used with this lesson.


Common

Core

Algebra II 1

Know there is a complex number i such that 2 1i , and every complex number has the form a + bi with a and b real.

N-CN1

Algebra II 2

Use the relation 2 1i and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

N-CN2

Algebra II 3

Find the conjugate of a complex number; use conjugates to find moduli and

quotients of complex numbers.

N-NC3

Algebra II 4

Solve quadratic equations with real coefficients that have complex solutions. N-CN7

Algebra II 5

Extend polynomial identities to the complex numbers. Example: Rewrite 2 4 as 2 2x x i x i

N-CN8

Algebra II 6

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

N-CN9

Algebra II 7

Use the structure of an expression to identify ways to rewrite it.

Example: See 2 2

4 4 2 2asx y x y , thus recognizing it as a difference of squares

that can be factored as 2 2 2 2x y x y .

A-SSE2

Algebra II 27




of values, or find successive approximations.

A-REI11

Old Algebra II




F-IF4


22 intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and

periodicity. Algebra II

29


quantitative relationship it describes. F-IF5

Algebra II 30





F-IF7

F-IF7b

Algebra II 33



F-BF1

F-BF1b

Algebra II 34






F-BF3

Algebra II 28


AL Standard

NMSI’s Laying the Foundation lesson: Quadratic Functions- Adaptation of AP Calculus 1997 AB 2 (1 or 2 days) Teacher Note: Students are asked to graph 2y x and a transformation and discuss the different aspects of the new

graph. The students then find average rate of change, instantaneous rate of change and area under the curve using rectangles. ** Checkpoint Unit 3 2,8,22 and Illustrative Mathematics Building a general quadratic function can be used with this lesson.


Common

Core

Algebra II 20




A-CED1

Algebra II 21



A-CED2

Old Algebra II

22






periodicity.*

F-IF4

Algebra II 29


quantitative relationship it describes.*

F-IF5

Algebra II 30





F-IF7

F-IF7b

Algebra II Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) F-BF3


34 for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the



NMSI’s Laying the Foundation lesson: Accumulation with a Quadratic Function (1 day) Teacher Note: Table of data about rate of water flow is input into calculator. Use regression capabilities to find quadratic fit and use function to estimate another data point. Draw rectangles and estimate area under curve to estimate total amount of water after a certain amount of time to help family make a decision. #7-11 is an extension of this by using the sum/sequence capabilities of calculator to get the area more precise. ** Checkpoint Unit 3 23 can be used with this lesson.


Common

Core

Algebra II 20




A-CED1

Algebra II 21



A-CED2

Algebra II 33



F-BF1

F-BF1b

NMSI’s Laying the Foundation lesson: Takin’Care of Business (1 day) Teacher Note: This lesson is a revenue, cost, profit problem that will use a quadratic model.


Common

Core

Algebra II 31



F-IF8

Algebra II 28a

20. Create graphs of conic sections, including parabolas, hyperbolas, ellipses,

circles, and degenerate conics, from second-degree equations.

a. Formulate equations of conic sections from their determining characteristics.

Old Algebra II

22






periodicity.*

F-IF4

NMSI’s Laying the Foundation lesson: Quadratic Optimization (1.5 days) Teacher Note: This is similar to the Algebra I Lesson by the same title. This is a great time to pull in the symmetry of parabolas and mention that if they know the roots, the vertex must be in the middle of the roots! This would allow them to not even need a graphing calculator to get the maxima. ** Checkpoint Unit 3 1,4,10,11,12 can be used with this lesson.


Common

Core

Algebra II 20




A-CED1


Algebra II 21



A-CED2

Algebra II 23

Rearrange formulas to highlight a quantity of interest, using the same reasoning as

in solving equations.

A-CED4

Algebra II 28


Old Algebra II

22






periodicity.*

F-IF4

Algebra II 29



F-IF5

Algebra II 30





F-IF7

F-IF7b

Algebra II 33



F-BF1

F-BF1b

Teacher Led Discussion--Radical equations (5 days)

Teacher Note: There is not a specific NMSI’s Laying the Foundation lesson that discusses Radical Equations. Your textbook should do a good job of having these type of problems. G.1.b, G.1.c, G.1.d, G.1.e should have been covered in Algebra I and Geometry. A quick review of those and then a day of G.1.f and G.1.g and this should be covered. ** Checkpoint Unit 3 13,14,15,16,17,19,21 and Illustrative Mathematics 1. Checking a calculation of a decimal exponent and 2. Rational or Irrational? can be used with this lesson.



Unit 4: Polynomials and Complex Roots 9 days of instruction plus assessment time- 2 weeks

NMSI’s Laying the Foundation lesson: Graphical Transformations (1 day) Teacher Note: The students are given a function that looks like a polynomial. They are asked to transform the graph and how the domain, maximum, x-intercept and y-intercept change. The students are then asked to find the average rate of change and area bounded by the graph. ** Checkpoint- Illustrative Mathematics Interpreting the Graph


Common

Core

Old Algebra II

22






periodicity.*

F-IF4

Algebra II 29



F-IF5

Old Algebra II

24



from a graph.*

F-IF6

Algebra II 34






F-BF3

NMSI’s Laying the Foundation lesson: Investigating Functions (2 days) Teacher Note: Connects the characteristics of polynomial functions with transformations. Focuses on how the roots,

maximum and minimum are affected by transformations. Introduces students to ( )f x and f x ** Checkpoint Unit 4 #1 can be used with this lesson.


Common

Core

Algebra II 17

Identify zeros of polynomials when suitable factorizations are available, and use the

zeros to construct a rough graph of the function defined by the polynomial.

A-APR3

Old Algebra II

22



key features given a verbal description of the relationship.

F-IF4

Algebra II 30b

Graph polynomial functions, identifying zeros when suitable factorizations are

available, and showing end behavior. F-IF7c

Algebra II 34




F-BF3




Teacher Led Discussion --Polynomials end behavior and roots (4 days) Teacher Note: This part of the course is where students discover how polynomials act due to the highest power and the leading coefficient. Long or synthetic division is used to break down polynomials in order to find factors/roots in order to make a rough sketch of the polynomial. Use the rational root theorem to help you decide what numbers might be roots to use in the synthetic/long division. Look at graphs of polynomials and if there are not as many roots as the highest degree, then there must be pairs of imaginary roots. The discussion of double roots will probably be a part of this discussion. The connection about how many maxima and minimum and the highest power is also an outcome of this discussion. Textbooks should handle this nicely. There will be a polynomial discovery lesson for the graphing calculator in the dropbox. ** Checkpoint Illustrative Mathematics Computations With Complex Numbers can be used with this lesson.


Common

Core

Algebra II 1

Know there is a complex number i such that i2 = –1, and every complex number

has the form a + bi with a and b real.

N-CN1

Algebra II 6

Know the Fundamental Theorem of Algebra; show that it is true for quadratic

polynomials.

N-CN9

Algebra II 15

Understand that polynomials form a system analogous to the integers; namely, they

are closed under the operations of addition, subtraction, and multiplication; add,

subtract, and multiply polynomials.

A-APR1

Algebra II 16

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a,

the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a

factor of p(x).

A-APR2

Algebra II 17



A-APR3

Algebra II 18

Prove polynomial identities and use them to describe numerical relationships.

Example: The polynomial identity 22 2 2 2 2 2( ) ( – ) 2x y x y xy can be used

to generate Pythagorean triples.

A-APR4

Algebra II 21



A-CED2

Algebra II 30



F-IF7

Algebra II 31



F-IF8

Algebra II 33



F-BF1

F-BF1b


NMSI’s Laying the Foundation lesson: Adaptation of AP Calculus 1997 AB1 (2 days) Teacher Note: Graphing Calculator is required. Students must graph functions, calculate values, using the table, and finding zeros of a function. This is a velocity and position function problem. Average rate of change and particle motion are key concepts of this lesson. Students explore motion on a horizontal line utilizing both graphical and analytical skills. **Checkpoint Unit 4 #2 can be used with this lesson.


Common

Core

Algebra II 12

Interpret expressions that represent a quantity in terms of its context.*

a. Interpret parts of an expression such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a

single entity.

A-SSE1

A-SSE1a

A-SSE1b

Algebra II 17



A-APR3

Algebra II 27






functions.*

A-REI11

Old Algebra II

22



key features given a verbal description of the relationship.

F-IF4

Old Algebra II

24



from a graph.*

F-IF6

Algebra II 30b

Graph polynomial functions, identifying zeros when suitable factorizations are

available, and showing end behavior. F-IF7c



Unit 5: Exponential and Logarithmic Functions 15 days of instruction plus Checkpoint time- 3.5 weeks

Teacher Note: The Quality Core Algebra II Unit 1 must be downloaded from the Quality Core website. Additional use of textbook and other teacher sources are needed for this unit.

NMSI’s Laying the Foundation lesson: And So They Grow (2 days) Teacher Note: Students will use dice and a graphing calculator to simulate the growth of a population. Since we cannot physically measure how a population increases and decreases, the students will use a simulation to model the situation. Students should have some knowledge of the behavior of exponential functions and know how to use a graphing calculator to create an exponential regression function. This is a great reintroduction to exponential growth. ** Checkpoint Illustrative Mathematics Extending the Definitions of Exponents can be used with this lesson.


Common

Core

Algebra II 20




A-CED1

Algebra II 21



A-CED2

Old Algebra II

22



key features given a verbal description of the relationship. Key features include:



periodicity.★

F-IF4

Old Algebra II

24



from a graph.★

F-IF6

Teacher Led Discussion --Graphing Exponential and Logarithmic functions (1 Day) Teacher Note: This would be a good time to make the connection between the graphs of exponential graphs that they

have already seen and xy e . They may have never seen e before, so you might want to explain that it is a special base

and the approximate value is 2.718281828… The formal explanation for e should take place in Precalculus, but you

could do it here if you choose. Include a discussion of translations of xy e and its symmetry and asymptotic behavior.

Introduce logs as the inverse “undo” function of exponential graphs and show graphs. Introduce y= ln x as the inverse of xy e .

**Checkpoint Unit 5 1,2,3,4,5,7,8,9,13 and Illustrative Mathematics Comparing Exponential can be used

with this lesson.


Common

Core

Old Algebra II

22




F-IF4




periodicity.★

Algebra II 34




graph using technology.

[F-BF3]

Algebra II 36

For exponential models, express as a logarithm the solution to abct = d where a, c,

and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using

technology.

F-LE4

Teacher Led Discussion -Exponential and Log Laws (2 Days) Teacher Note: The students have been exposed to the exponential laws, but with variables as bases. You will need to review these with bases that are numbers and the base of e. Introduce the log laws as the inverse or “going backwards” of the exponential laws. Use your textbook to practice these. ** Checkpoints Unit 5 6,10,11,12 can be used with this lesson.


Common

Core

Algebra II 36



technology.

F-LE4

NMSI’s Laying the Foundation lesson: Solving Systems of Exponential, Logarithmic, and Linear Equations (1 day) Teacher Note: Students will solve systems of equations using properties of exponents and logarithms. Students should be familiar with properties of exponents and logarithms, solving exponential, logarithmic and quadratic equations, solving systems of equations, and graphing functions then finding the intersection of the functions. Graphing calculator required. #1 and #2 can be solved very similarly to solving linear systems after a few laws are applied. #3 and #4 requires substitution and factoring, then applying log laws to solve. #5 requires rewriting in terms of ln and then solving. The concept of ln both sides in order to solve exponential is designated for PreCalculus, but would be very helpful for this exercise. ** Checkpoints Unit 5 #14 can be used with this lesson.


Common

Core

Algebra II 13

Use the structure of an expression to identify ways to rewrite it.

A-SSE2

Algebra II 27






functions.★

A-REI11

Algebra II 29



F-IF5

Algebra II Graph functions expressed symbolically and show key features of the graph, by F-IF7


30 hand in simple cases and using technology for more complicated cases.★

Algebra II 33

Combine standard function types using arithmetic operations. F-BF1b

Algebra II 36



technology.

F-LE4

NMSI’s Laying the Foundation lesson: Exponential and Natural Log Functions (2 days) Teacher Note: This lesson serves as a good review of many topics covered with exponential and natural logarithmic functions, such as sketching graphs using transformations, using properties of exponents and logarithms, solving both types of equations, growth and decay problems, and linearization of data.


Common

Core

Algebra II 22

Represent constraints by equations or inequalities, and by systems of equations

and/or inequalities, and interpret solutions as viable or nonviable options in a

modeling context.

A-CED3

Algebra II 30


hand in simple cases and using technology for more complicated cases.★

c. Graph exponential and logarithmic functions, showing intercepts and end

behavior, and trigonometric functions, showing period, midline, and amplitude.

F-IF7

F-IF7e

NMSI’s Laying the Foundation lesson: Motion Problems Using Exponential and Natural Logarithmic Functions (1 day) Teacher Note: Students will apply exponential and logarithmic functions to the concepts of motion of an object along a horizontal line. Students should be able to factor polynomial expressions and be familiar with natural logarithms and exponential functions. Students should be able to solve both equations and inequalities involving both. This lesson is designed to be completed without a graphing calculator. Be sure to check out the two web sites listed under material and resources.


Common

Core

Algebra II 13

Use the structure of an expression to identify ways to rewrite it. A-SSE2

Old Algebra II

22






periodicity.★

F-IF4

Algebra II 36



technology.

F-LE4


NMSI’s Laying the Foundation lesson: Curing the Sniffles (1 day) Teacher Note: Students will determine the amount of medicine that remains in the body at any given time. A simulation will be used to model the situation. Students should have some experience with recursively defined sequences. This requires that students use the sequential mode on their graphing calculator. ** Checkpoints Unit 5 #16 can be used with this lesson.


Common

Core

Old Algebra II

22



key features given a verbal description of the relationship

F-IF4

Algebra I 35

Write arithmetic and geometric sequences both recursively and with an explicit

formula, use them to model situations, and translate between the two forms.

F-BF2

Quality Core Algebra II Unit 1 Patterns (G-10 – J-4) (3 days) Teacher Note: The students should have been introduced to geometric series in Algebra I, but you may need to remind/reteach this concept. The pages listed have vocabulary and activities starting with the introduction to a geometric sequence to summing of a finite and infinite geometric series. You will need to choose what you need of this

unit depending on the knowledge of your students.


Common

Core

Algebra II 14

Derive the formula for the sum of a finite geometric series (when the common ratio

is not 1), and use the formula to solve problems

A-SSE4

Algebra I #38

Construct exponential functions and geometric sequences, given a graph, a

description of a relationship, or two input-output pairs (include reading these from

a table).

F-LE2

Algebra I #34

Write arithmetic and geometric sequences both recursively and with an explicit

formula, use them to model situations, and translate between the two forms.

F-BF2



Unit 6: Rational Functions 8 days of instruction plus assessment time- 2 weeks

NMSI’s Laying the Foundation lesson: Rational Functions-Short Run Behavior (1 day) Teacher Note: Students will investigate the behavior of rational functions near the vertical asymptotes. Graphing calculator required.

AL COS

Common Core Standard

Common

Core

Old Algebra II

22






periodicity.*

F-IF4

Algebra II 29



F-IF5

Old Algebra II

24



from a graph.*

F-IF6

Algebra II 31



F-IF8

Algebra II 12

Interpret expressions that represent a quantity in terms of its context.*



single entity.

A-SSE1

A-SSE1a

A-SSE1b

NMSI’s Laying the Foundation lesson: Rational Functions-Long Run Behavior (1 day) Teacher Note: Students will find the end behavior asymptote of a rational function. Graphing calculator required.


Common

Core

Old Algebra II

22






periodicity.*

F-IF4

Algebra II 29



F-IF5

Old Algebra II

24



from a graph.*

F-IF6

Algebra II Write a function defined by an expression in different but equivalent forms to F-IF8


31 reveal and explain different properties of the function. Algebra II

12 Interpret expressions that represent a quantity in terms of its context.*



single entity.

A-SSE1

A-SSE1a

A-SSE1b

NMSI’s Laying the Foundation lesson: Rational Functions-Transformations of Rational Functions (1 day) Teacher Note: Students will apply transformations to the graphs of rational functions, describe the transformations, and graph the transformed functions. They will need to do long division in order to accomplish writing the rational functions as a sum of two parts. Students need graph paper for this lesson. ** Checkpoint Unit 6 3,5,6 can be used with this lesson.


Common

Core

Algebra II 19

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form

q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of

r(x) less than the degree of b(x), using inspection, long division, or for the more

complicated examples, a computer algebra system.

A-APR6

Algebra I 11

(+) Understand that rational expressions form a system analogous to the rational

numbers, closed under addition, subtraction, multiplication, and division by a

nonzero rational expression; add, subtract, multiply, and divide rational

expressions.

A-APR7

Algebra II 34






F-BF3

NMSI’s Laying the Foundation lesson: Rational Function Exploration (1 day) Teacher Note: Students will investigate end behavior asymptotes of rational functions. Students should be able to graph using a table of values, graph a function, and use function notation. Graphing calculator required.


Common

Core

Algebra II 34






F-BF3

Algebra II 20




A-CED1

Illuminations lesson: Light It Up illuminations.nctm.org AL COS Common Core Standard

Common

Core

Algebra II 33



F-BF1

F-BF1b


NMSI’s Laying the Foundation lesson: Optimization with Rational Functions (1 day) Teacher Note: Students will write a rational function for a real life situation. They will find the minimum value of the rational function using a table, graph and graphing calculator. Students will need to know how to use the trace function on the graphing calculator.


Common

Core

Algebra II 20




A-CED1

Algebra II 21



A-CED2

Algebra II 24

Solve simple rational and radical equations in one variable, and give examples

showing how extraneous solutions may arise.

A-REI2

NMSI’s Laying the Foundation lesson: Piecewise and Rational Functions (2 days) Teacher Note: Students will use an AP Calculus, free response exam question to review previously learned material and experience new material such as limits as the independent variable approaches infinity. Students should be familiar with equations of piecewise functions, graphing rational functions, solving systems of quadratic equations and inequalities, transformations, finding domain and finding area under a curve using triangles, rectangles and trapezoids. Graphing calculator required.


Common

Core

Algebra II 27






functions.*

A-REI11



Unit 7: Probability and Statistics 6 days of instructional days plus assessment time- 2 weeks

NMSI’s Laying the Foundation lesson: Probability Rules! (1.5 day) Teacher Note: This lesson introduces students to the probability notation and emphasizes the addition and multiplication rule. Rather than encourage students to memorize these rules, you might want to encourage them to make a table and compute the probabilities the way they did in Movie Probability. If the students have not used Movie Probability, this could be given as homework and then lead into Probability Rules. A teacher might consider not doing #1 and 2 until after doing #3. The Addition Rule can be taught easily with 3e and discuss that you counted some of the students twice, so it had to be subtracted. The Multiplication Rule is a manipulation of the conditional probability learned in Movie Probability and found in #3c and can be used in that way instead of being memorized. For #4 and #5 make a table of values and then answer the questions.


Common

Core

Algebra II 43

Describe events as subsets of a sample space (the set of outcomes), using

characteristics (or categories) of the outcomes, or as unions, intersections, or

complements of other events (“or,” “and,” “not”).

S-CP1

Algebra II 44

Understand the conditional probability of A given B as P(A and B)/P(B), and

interpret independence of A and B as saying that the conditional probability of A

given B is the same as the probability of A, and the conditional probability of B

given A is the same as the probability of B.

S-CP3 Partial

Algebra II 46

Recognize and explain the concepts of conditional probability and independence in

everyday language and everyday situations. S-CP5

Algebra II 47

Find the conditional probability of A given B as the fraction of B’s outcomes that

also belong to A, and interpret the answer in terms of the model. S-CP6

Algebra II 48

Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the

answer in terms of the model.

S-CP7

Algebra II 49

Apply the General Multiplication Rule in a uniform probability model, P(A and B)

= P(A)P(B|A)=P(B)P(A|B), and interpret the answer in terms of the model.

S-CP8

NMSI’s Laying the Foundation lesson: Calculate Probabilities with Tree Diagrams (1 day) Teacher Note: Tree Diagrams are used to find probabilities. Students make sample spaces and use them to produce probabilities. Students are asked to analyze the probability of a medical test and make a decision about treatment. Teachers may need to emphasize that this is another way to organize data in order to find probabilities.


Common

Core

Algebra II 42

Analyze decisions and strategies using probability concepts (e.g., product testing,

medical testing, pulling a hockey goalie at the end of a game).

S-MD7

Algebra II 43




S-CP1

Algebra II 44




S-CP3 Partial



Algebra II 46



Algebra II 47



Algebra II 48

Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the

answer in terms of the model.

S-CP7

Algebra II 49



S-CP8

NMSI’s Laying the Foundation lesson: Independence (1.5 days) Teacher Note: Students investigate conditional probabilities from a two-way table and to use these probabilities to determine if two events are independent. Mathematical notation used in probability is introduced. Students make decisions about data by analyzing the conditional probabilities associated with the information about the survivors of the Titanic.


Common

Core

Algebra II 42



S-MD7

Algebra II 43




S-CP1

Algebra II 44





S-CP3

Algebra II 45

Construct and interpret two-way frequency tables of data when two categories are

associated with each object being classified. Use the two-way table as a sample

space to decide if events are independent and to approximate conditional

probabilities.

S-CP4

Algebra II 46



Algebra II 47



Algebra II 49



S-CP8


NMSI’s Laying the Foundation lesson: Probability Using Sample Spaces, Permutations, and Combinations (1.5 days) Teacher Note: Students are introduced to permutations and combinations as a way to do the basic counting principle for a large sample space. Students use the combinations and permutations to determine how many favorable outcomes are possible and to compute the probability of a chance event. Students use a two-way frequency table to get information in this lesson. If students have not been exposed to the NMSI’s Laying the Foundation lesson Family Fun (Middle Grades), they may need to work through this before doing this lesson. ** Checkpoint Unit 7 #1 can be used as a culmination for this unit.


Common

Core

Algebra II 41

(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random

number generator). S-MD6

Algebra II 42



S-MD7

Algebra II 43




S-CP1

Algebra II 45

Construct and interpret two-way frequency tables of data when two categories are

associated with each object being classified. Use the two-way table as a sample

space to decide if events are independent and to approximate conditional

probabilities.

S-CP4

Algebra II 50

Use permutations and combinations to compute probabilities of compound events and solve problems.

S-CP9



Unit 8: Trigonometric Functions 8 days of instruction plus assessment time-2 weeks

NMSI’s Laying the Foundation lesson: A Transformation Story (.5 day) Teacher Note: This unit is used here to remind students about transformation of functions. This could be used as a homework that kicks off the next day class discussion.


Common

Core

Algebra II 34






F-BF3

Old Algebra II

22






periodicity.

F-IF4

NMSI’s Laying the Foundation lessons: Introduction to Trigonometric Ratios with Special Right Triangles (1 day) Teacher Note: Students will develop the unit circle. Students should be familiar with special right triangles and six trig ratios. Arc length formula can be used to introduce radians in this lesson if desired. Good for developing the unit circle. The beginning of this may have been introduced in Geometry. There will be additional resources in the dropbox to support this concept.


Common Core

Geo 19 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. H.1.a. Apply properties of 45°-45°-90° and 30°-60°-90° triangles to determine lengths of sides of triangles

G.SRT.6

Algebra II 37

Understand radian measure of an angle as the length of the arc on the unit circle

subtended by the angle.

F-TF1

Algebra II 38

Explain how the unit circle in the coordinate plane enables the extension of

trigonometric functions to all real numbers, interpreted as radian measures of angles

traversed counterclockwise around the unit circle.

F-TF2

Algebra II 39

Define the six trigonometric functions using ratios of the sides of a right triangle,

coordinates on the unit circle, and the reciprocal of other functions. AL

Standard


Teacher Led Discussion-Graphing sin x and cos x (3 to 4 days) Teacher Note: These are concepts that concentrate on the teaching of the simple graphs of sin x and cosx. There are not LTF lessons that really address the following standards. There will be some resources in the dropbox to support this concept.


Common

Core

Algebra II 40

Choose trigonometric functions to model periodic phenomena with specified

amplitude, frequency, and midline.*

F-TF5

Old Algebra II

22






periodicity.*

F-IF4

Algebra II 30



c. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F-IF7

F-IF7e

Algebra II 33



F-BF1

F-BF1b

Algebra II and PreAP Algebra II Scope and Sequence with …...NMSI’s Laying the Foundation Lesson:...

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