Algebra II Suggested Big Idea Unit 1: Algebra and Functions Content Emphasis Cluster Interpret the structure of expressions Understand the relationship between zeros and factors of polynomials Rewrite rational expressions Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Represent and solve equations and inequalities graphically Analyze functions using different representations Build a function that models a relationship between two quantities Mathematical Practices MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning. Common Assessment End of Unit Assessment Graduate Competency Prepared graduates understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations Prepared graduates are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency Prepared graduates make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data CCSS Priority Standards Cross-Content Connections Writing Focus Language/ Vocabulary Misconceptions A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A.APR.2 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). Literacy Connections RST.6-8.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 6-8 texts and topics. Writing Connection WHST.6-8.2 Write informative/expl anatory texts, including the narration of historical events, scientific procedures/ experiments, or technical processes. a. Introduce a topic clearly, previewing what is to follow; Academic Vocabulary- Operations, Derive, Sequence, Series, Modeling, Identify, Prove Technical Vocabulary- Domain, Range, End Behavior, Standard Form, Factored Form, A.SSE.2-3 Some students may believe that factoring and completing the square are isolated techniques within a unit of quadratic equations. Teachers should help students to see the value of these skills in the context of solving higher degree equations and examining different families of functions. Students may think that the minimum (the vertex) of the graph of y = (x + 5)2 is shifted to the right of the minimum (the vertex) of the graph y = x2 due to the addition sign. Students should explore examples both analytically and graphically to overcome this misconception. Some students may believe that the minimum of the graph of a quadratic function always occur at the y-intercept.
Transcript
Algebra II
Suggested Big Idea Unit 1: Algebra and Functions
Content Emphasis Cluster Interpret the structure of expressions
Understand the relationship between zeros and factors of polynomials
Rewrite rational expressions
Understand solving equations as a process of reasoning and explain the reasoning
Solve equations and inequalities in one variable
Represent and solve equations and inequalities graphically
Analyze functions using different representations
Build a function that models a relationship between two quantities
Mathematical Practices MP.1. Make sense of problems and persevere in solving them.
MP.2. Reason abstractly and quantitatively.
MP.3. Construct viable arguments and critique the reasoning of others.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
Common Assessment End of Unit Assessment
Graduate Competency Prepared graduates understand that equivalence is a foundation of mathematics represented in numbers,
shapes, measures, expressions, and equations
Prepared graduates are fluent with basic numerical and symbolic facts and algorithms, and are able to select
and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding
of their efficiency, precision, and transparency
Prepared graduates make sound predictions and generalizations based on patterns and relationships that arise
from numbers, shapes, symbols, and data
CCSS Priority Standards Cross-Content
Connections Writing Focus Language/
Vocabulary
Misconceptions
A.SSE.2 Use the structure of an expression to identify ways to
rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus
recognizing it as a difference of squares that can be factored as
(x2 – y2)(x2 + y2).
A.SSE.3 Choose and produce an equivalent form of an expression
to reveal and explain properties of the quantity represented by the
expression. a. Factor a quadratic expression to reveal the zeros of
the function it defines. b. Complete the square in a quadratic
expression to reveal the maximum or minimum value of the
function it defines. c. Use the properties of exponents to
transform expressions for exponential functions. For example the
expression 1.15t can be rewritten as (1.151/12)12t ≈1.01212t to
reveal the approximate equivalent monthly interest rate if the
annual rate is 15%.
A.APR.2 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).
Literacy
Connections
RST.6-8.4
Determine the
meaning of
symbols, key
terms, and other
domain-specific
words and
phrases as they
are used in a
specific
scientific or
technical
context relevant
to grades 6-8
texts and topics.
Writing
Connection
WHST.6-8.2
Write
informative/expl
anatory texts,
including the
narration of
historical
events, scientific
procedures/
experiments, or
technical
processes.
a. Introduce a
topic clearly,
previewing what
is to follow;
Academic
Vocabulary-
Operations,
Derive,
Sequence,
Series,
Modeling,
Identify,
Prove
Technical
Vocabulary-
Domain, Range,
End Behavior,
Standard
Form,
Factored
Form,
A.SSE.2-3
Some students may believe that factoring and
completing the square are isolated techniques
within a unit of quadratic equations.
Teachers should help students to see the
value of these skills in the context of solving
higher degree equations and examining
different families of functions.
Students may think that the minimum (the
vertex) of the graph of y = (x + 5)2 is shifted
to the right of the minimum (the vertex) of
the graph y = x2 due to the addition sign.
Students should explore examples both
analytically and graphically to overcome this
misconception.
Some students may believe that the minimum
of the graph of a quadratic function always
occur at the y-intercept.
Algebra II
A.APR.3 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.APR.6 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.REI.1 Explain each step in solving a simple equation as
following from the equality of numbers asserted at the previous
step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
A.REI.2 Solve simple rational and radical equations in one
variable, and give examples showing how extraneous solutions
may arise.
A.REI.4 Solve quadratic equations in one variable. a. Use the
method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the
same solutions. Derive the quadratic formula from this form. b.
Solve quadratic equations by inspection (e.g., for x2 = 49), taking
square roots, completing the square, the quadratic formula and
factoring, as appropriate to the initial form of the equation.
Recognize when the quadratic formula gives complex solutions
and write them as a ± bi for real numbers a and b.
A.REI.11 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
F.IF.4 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.IF.7 Graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and using
technology for more complicated cases. a. Graph linear and
quadratic functions and show intercepts, maxima, and minima. b.
Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions. c. Graph
polynomial functions, identifying zeros when suitable
factorizations are available, and showing end behavior. d. (+)
Graph rational functions, identifying zeros and asymptotes when
suitable factorizations are available, and showing end behavior. e.
Graph exponential and logarithmic functions, showing intercepts
and end behavior, and trigonometric functions, showing period,
midline, and amplitude.
F.IF.8 Write a function defined by an expression in different but
equivalent forms to reveal and explain different properties of the
function. a. Use the process of factoring and completing the
square in a quadratic function to show zeros, extreme values, and
symmetry of the graph, and interpret these in terms of a context.
b. Use the properties of exponents to interpret expressions for
exponential functions. For example, identify percent rate of
change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t,
y = (1.2)t/10, and classify them as representing exponential
growth or decay.
F.BF.1 Write a function that describes a relationship between two
quantities. a. Determine an explicit expression, a recursive
process, or steps for calculation from a context. b. Combine
standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by
adding a constant function to a decaying exponential, and relate
these functions to the model. c. (+) Compose functions. For
example, if T(y) is the temperature in the atmosphere as a
function of height, and h(t) is the height of a weather balloon as a
function of time, then T(h(t)) is the temperature at the location of
the weather balloon as a function of time.
F.BF.2 Write arithmetic and geometric sequences both recursively
and with an explicit formula, use them to model situations, and
translate between the two forms.
f. Provide a
concluding
statement or
section that
follows from
and supports the
information or
explanation
presented.
WHST.6-8.4
Produce clear and
coherent writing
in which the
development,
organization,
and style are
appropriate to
task, purpose,
and audience.
flexibly from
a range of
strategies.
F.IF.7
Students may believe that each family of
functions (e.g., quadratic, square root, etc.) is
independent of the others, so they may not
recognize commonalities among all functions
and their graphs.
Students may also believe that skills such as
factoring a trinomial or completing the
square are isolated within a unit on
polynomials, and that they will come to
understand the usefulness of these skills in
the context of examining characteristics of
functions.
Additionally, student may believe that the
process of rewriting equations into various
forms is simply an algebra symbol
manipulation exercise, rather than serving a
purpose of allowing different features of the
function to be exhibited.
F.BF.1-2
Students may believe that the best (or only)
way to generalize a table of data is by using a
recursive formula.
Students naturally tend to look “down” a table
to find the pattern but need to realize that
finding the 100th term requires knowing the
99th term unless an explicit formula is
developed.
Students may also believe that arithmetic and
geometric sequences are the same. Students
need experiences with both types of
sequences to be able to recognize the
difference and more readily develop
formulas to describe them.
Algebra II
Unit 1
POLY WANT A NOMIAL?
Length of Unit
Part 1 – 3 Weeks Part 2 – 4 Weeks Part 3 – 3Weeks
TOTAL - 10 WEEKS
Concepts Function Notation, Operations on functions, Domain and Range, Polynomial Operations, Maximum, Minimum, Vertex, Intercepts, End Behavior, The
Remainder Theorem Complex Numbers, Conjugates, Quadratics, Zeros, Factoring, Completing the Square, Quadratic Formula, Writing polys given roots, Inverses of Quadratics, Radicals, Radical Equations, Systems of Equations.
Standards of Mathematical Practices
MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning.
Content Standards
*Priority Standards*
A.APR.2 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.APR.3 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.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
A.APR.6 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.CED.1 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.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A.REI.4 Solve quadratic equations in one variable.
Algebra II
A.REI.4.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
A.REI.4.b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
A.REI.11
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.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
A.SSE.3.a Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Factor a quadratic expression to real the zeros of the function
A.SSE.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
F.BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
F.BF.1.b Combine standard function types using arithmetic operations.
F.BF.1.c (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
F.BF.4 Find inverse functions.
F.BF.4.a 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. For example, f(x) =2x3 or f(x) = (x+1)/(x–1) for x = 1.
F.BF.4.c Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
F.IF.4
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.
Algebra II
F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F.IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
N.CN.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.CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.
N.Q .2 Define appropriate quantities for the purpose of descriptive modeling.
Inquiry Questions
Why do different types of equations require different types of solution processes? What is the square root of -1? What are the implications of having a solution to this problem?
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…
Topic 1 – Graphing Polynomials (Graphing Calculator) (A.APR.3, F.IF.4, F.IF.9) The students will be able to identify and interpret key features of graphs; use given key features to construct a graph. Topic 2 – Operations on Polynomials (A.APR.6, F.BF.1.b, F.BF.1.c) The students will be able to combine functions and polynomials using arithmetic operations. Topic 3- Solving Polynomials by factoring and Simple Inverses (Algebraically with integer roots) (A.APR.2, A.REI.1, A.SSE.3.a, F.IF.8, F.BF.4, F.BF.4.a) The students will be able to rewrite expressions in different but equivalent forms to find roots of polynomials; find the inverse of any function graphically or
by using a table. The students will be able to find some inverses algebraically. Topic 4 – Solving Radical Equations (A.REI.2) Students will be able to solve radical equations and will be able to determine if there are extraneous solutions.
Algebra II
Topic 5 – Complex Numbers (N.CN.1, N.CN.2, N.CN.7) Students will be able to preformed arithmetic operations (add, sub, and multiply) on complex number; recognize when a polynomial has a complex solutions. Topic 6 – Completing the Square (CTS) (A.SSE.3.b, A.REI.4.a, A.REI.4.b) Students will be able to use completing the square to rewrite a quadratic function in vertex form and will understand how to derive the quadratic formula
using completing the square; find the inverse of quadratic trinomials using CTS. Topic 7 – Quadratic Formula (A.REI.4, A.REI.4.b) Students will be able to derive the formula; solve using the quadratic formula. Topic 8 – Solving Polynomials (with any roots) (N.CN.7) Students will be able to solve any polynomial. Topic 9 – Arithmetic Sequences and Finite Arithmetic Series (F.BF.1.a, F.BF.2, F.IF.3) Students will be able to recognize arithmetic sequences and series and write functions to model them. (Algebra 1 spends a lot of time on this topic.) Topic 10 – Systems of Equations (A.REI.11, A.REI.7) Students will be able to determine a solution to a system of equations no matter what type of equations compose the system.
Domain, Range, End Behavior, Standard Form, Factored Form, Minimum, Maximum, Variable, Y-intercept, Zero, Root, Odd, Even, Degree, Binomial, Trinomial, Inverse, Arithmetic, System or Equations.
Resources
Lessons
Optional Resource- EngageNY Module 1- https://www.engageny.org/resource/algebra-ii-module-1 Proposed sequence: Topic 1 – Graphing Polynomials (Graphing Calculator) Identify and interpret key features of graphs (intercepts, maximum, minimum, zeros, degree, end behavior, odd or even from the graph
only); use given key features to construct a graph.
EngageNY Module 1 -Lesson 14 EngageNY Module 1 -Lesson 15 Illustrative Mathematics: “Influenza epidemic” http://www.illustrativemathematics.org/illustrations/637 Illustrative Mathematics: “Warming and Cooling” http://www.illustrativemathematics.org/illustrations/639 Illustrative Mathematics: “How is the weather?” http://www.illustrativemathematics.org/illustrations/649 Illustrative Mathematics: “Telling a Story with Graphs” http://www.illustrativemathematics.org/illustrations/650 Topic 2 – Operations on Polynomials Combine functions and polynomials using arithmetic operations of addition, subtraction, multiplication, division (long and synthetic), and
composition. EngageNY Module 1 -Lesson 5 EngageNY Module 1 -Lesson 18 - 20 “Manipulating Polynomials” – Mathematics Assessment Project This lesson unit is intended to help you assess how well students are able to manipulate and calculate with polynomials. In particular, it
aims to identify and help students who have difficulties in switching between visual and algebraic representations of polynomial expressions and performing arithmetic operations on algebraic representations of polynomials, factorizing and expanding appropriately when it helps to make the operations easier.
http://map.mathshell.org/materials/lessons.php?taskid=437#task437 Illustrative Mathematics: “A Sum of Functions” http://www.illustrativemathematics.org/illustrations/230 Topic 3- Solving Polynomials by factoring and Simple Inverses (Algebraically with integer roots) *Spend most time on this: Factor by grouping to find roots of polynomials (quadratics and above); find the inverse of any function
graphically or by using a table. The students will be able to find some inverses algebraically. EngageNY Module 1 -Lesson 13 Illustrative Mathematics: “The Missing Coefficient” http://www.illustrativemathematics.org/illustrations/592 Illustrative Mathematics: “Zeroes and factorization of a quadratic polynomial I” http://www.illustrativemathematics.org/illustrations/787 Illustrative Mathematics: “Zeroes and factorization of a quadratic polynomial II” http://www.illustrativemathematics.org/illustrations/789 Illustrative Mathematics: “Zeroes and factorization of a general polynomial” http://www.illustrativemathematics.org/illustrations/788 Illustrative Mathematics: “Zeroes and factorization of a non polynomial function” http://www.illustrativemathematics.org/illustrations/796
Optional: Unit 1 Part 1 Assessment (Calculator and Non Calculator ) 2015-16 D6 Alg II Common Assessment Unit 1 Part 1 (on Schoolcity-paper version or online) Given to students in September Topic 4 – Solving Radical Equations Solve radical equations and will be able to determine if there are extraneous solutions. EngageNY Module 1 -Lesson 28 and 29 “Building and Solving Equations 2” – Mathematics Assessment Project This lesson unit is intended to help you assess how well students are able to create and solve linear and non-linear equations. In particular,
the lesson will help identify and help students who have the following difficulties: Solving equations where the unknown appears once or more than once; Solving equations in more than one way. http://map.mathshell.org/materials/lessons.php?taskid=554#task554
Illustrative Mathematics: “Same solutions?” http://www.illustrativemathematics.org/illustrations/613 Illustrative Mathematics: “How does the solution change?” http://www.illustrativemathematics.org/illustrations/614 Topic 5 – Complex Numbers Preformed arithmetic operations (add, sub, and multiply) on complex number; recognize when a polynomial has a complex solutions. EngageNY Module 1 -Lesson 36-40 Illustrative Mathematics: “Complex number patterns” http://www.illustrativemathematics.org/illustrations/722 “Manipulating Radicals” – Mathematics Assessment Project This lesson unit is intended to help you assess how well students are able to: •Use the properties of exponents, including rational
exponents, and manipulate algebraic statements involving radicals. Discriminate between equations and identities. In this lesson there is also an opportunity to consider the role of the imaginary number, but this is optional. http://map.mathshell.org/materials/lessons.php?taskid=547#task547
Topic 6 – Completing the Square (CTS) Use completing the square to rewrite a quadratic function in vertex form and will understand how to derive the quadratic formula using
completing the square. USE ALGEBRA TILES TO INTRODUCE! Find the inverse of quadratic trinomials using CTS. EngageNY Module 1 -Lesson 12 Illustrative Mathematics: “Increasing or Decreasing?, Variation 2” http://www.illustrativemathematics.org/illustrations/167 Illustrative Mathematics: “Ice Cream” http://www.illustrativemathematics.org/illustrations/551 Illustrative Mathematics: “Profit of a Company” http://www.illustrativemathematics.org/illustrations/434 Illustrative Mathematics: “Profit of a Company, assessment variation” http://www.illustrativemathematics.org/illustrations/1344 Illustrative Mathematics: “Forms of exponential expressions” http://www.illustrativemathematics.org/illustrations/1305
Illustrative Mathematics: “Braking Distance” http://www.illustrativemathematics.org/illustrations/586 Topic 8 – Solving Polynomials (with any roots) Solve any polynomial; build polynomials given roots. EngageNY Module 1 -Lesson 11
Illustrative Mathematics: “Braking Distance” http://www.illustrativemathematics.org/illustrations/586 Optional: Unit 1 Part 2 Assessment Non Calculator 2015-16 D6 Alg II Common Assessment Unit 1 Part 2 (on Schoolcity-paper version or online) Given to students in early October Topic 9 – Arithmetic Sequences and Finite Arithmetic Series (F.BF.1.a, F.BF.2, F.IF.3) Arithmetic sequences and series and write functions to model them. (Algebra 1 spends a lot of time on this topic.) Summation and Series
is the new topics and the focus of Alg 2. Illustrative Mathematics: “The Skeleton Tower” http://www.illustrativemathematics.org/illustrations/75 Illustrative Mathematics: “The Summer Intern” http://www.illustrativemathematics.org/illustrations/72 Illustrative Mathematics: “Kimi and Jordan” http://www.illustrativemathematics.org/illustrations/241 Topic 10 – Systems of Equations (A.REI.11, A.REI.7) Find a solution(s) to a system of equations no matter what type of equations compose the system. This should include, linear, quadratic, circle, rational, exponential, etc. graphically. Algebraically solve linear and quadratic systems. EngageNY Module 1 -Lesson 30 - 32 Illustrative Mathematics: “Two Squares are Equal” http://www.illustrativemathematics.org/illustrations/618 Illustrative Mathematics: “Introduction to Polynomials – College Fund” http://www.illustrativemathematics.org/illustrations/1551 “Optimization Problems: Boomerangs” – Mathematics Assessment Project This lesson unit is intended to help you assess how well students are able to: •Interpret a situation and represent the constraints and variables mathematically. •Select appropriate mathematical methods to use. •Explore the effects of systematically varying the constraints. •Interpret and evaluate the data generated and identify the optimum case, checking it for confirmation. •Communicate their reasoning clearly.
http://map.mathshell.org/materials/lessons.php?taskid=207#task207 Illustrative Mathematics: “The Circle and The Line” http://www.illustrativemathematics.org/illustrations/223 Illustrative Mathematics: “A Linear and Quadratic System” http://www.illustrativemathematics.org/illustrations/576 Illustrative Mathematics: “What functions do two graph points determine?” http://www.illustrativemathematics.org/illustrations/376 Optional: Unit 1 Part 3 Assessment Non Calculator 2015-16 D6 Alg II Common Assessment Unit 1 Part 3 (on Schoolcity-paper version or online) Given to students by the end of October
Core Instructional Task
Operations of Polynomials Operation of Functions Quadratic Functions Operations of Complex Numbers Radical Functions System of Equations
Technology Graphing Calculator
Materials
Algebra II – Section 2-1 Algebra II – Section 6-4 Algebra II – Section 6-5 Algebra II – Section 6-5 Algebra II – Section 5-3 (Just by grouping) Algebra II – Section 6-3 Algebra II – Section 6-6 Algebra II – Section 6-7 Algebra II – Section 6-8 Algebra II – Section 5-3 GROUPING ONLY
Algebra II- Section 5-5 MUST USE ALGEBRA TILES Algebra II – Section 5-6 DERIVE THE QUADRATIC FORMULA Algebra II – Section 5-4 (complex numbers with 5-6) Algebra II – Section 7-4 Algebra II – Section 7-6 Algebra II – Section 7-7 (Just Equations) Algebra II – Section 11-6
Algebra II – Section 11-1 Algebra II – Section 11-2Domain Representation (Illuminations) Manipulating Polynomials (Shell Lesson) Representing Polynomials (Shell) Engage New York: Module 1 – Topics A, B and D (New resource) https://www.engageny.org/resource/algebra-ii-module-1 High School Flip Book -PDF (New resource) http://community.ksde.org/LinkClick.aspx?fileticket=VNhMSEbSycI%3d&tabid=5646&mid=13290
Performance Task(s) (Assessments)
2015-16 D6 Alg II Common Assessment Unit 1 Part 1 (on Schoolcity-paper version or online) Given to students in September 2015-16 D6 Alg II Common Assessment Unit 1 Part 2 (on Schoolcity-paper version or online) Given to students in early October 2015-16 D6 Alg II Common Assessment Unit 1 Part 3 (on Schoolcity-paper version or online) Given to students by the end of October
Scanned into School City or students take the assessment online
Should be in addition to individually developed site based extended response exam that will not be entered into School City and teacher
created formative assessments
Misconceptions Students might confuse irrational numbers with non-real or complex numbers; remind students about the relationships
Students might think that taking square roots, factoring, and completing the square are isolated techniques used only for quadratics. Help students understand the value of these skills in the context of solving higher degree equations and examining different families of functions.
Students also might believe expressions cannot be factored because they do not fit a recognizable form. They need help manipulating terms until structures become evident.
Students might try to combine terms that are not like or change the degree of variables when combining terms.
Students often forget to distribute to all terms when multiplying and fail to use the property of exponents correctly when distributing.
Students tend to forget the negative solution for quadratic equations not involving the use of the quadratic formula. For example, x2 = 9 has 3 as a solution but also -3.
Ensure students understand the idea of rate of change as it applies to polynomial functions and that it applies to real-world situations; it is not just an abstract idea.
Students tend to look down tables to find patterns but must realize that finding the 100th term requires the 99th term unless an explicit formula is developed.
Students might not understand what it means to find sums of series. For example, if asked to find the sum of the first 17 terms of a series, they might only find the 17th term.
Instructional Notes
Instructional Notes Polynomial Unit 1 part 1:
End behaviors
o Limit notation
Domain and Range
o Written in interval notation. Example: [-4, 0) or (- ∞ , ∞)
o Restrictions
Factoring
o Emphasize grouping strategy
Compositions
Interconnectedness of factors, roots, zeros
Remainders of polynomial division stated as rational expressions
Instructional Notes Polynomial Unit 1 part 2:
Solve quadratics by:
o Factoring
o Completing the square
o Quadratic formula
Characteristics of functions and their inverses
o Algebraic relationship
o Relationship of ordered pairs
Algebra II
o Graphical symmetry with respect to the line y=x
Patterns for powers of I, addition, subtraction and multiplication. Stress a + bi (Help pre-calc with vectors on the complex plane)
Instructional Notes Polynomial Unit 1 part 3:
Arithmetic Sequences are taught to mastery in Alg 1 in regards to formulas. Series and summation is the new part for Alg 2.
Systems should be taught using multiple types of functions solving algebraically and graphically.
Suggested Big Idea Unit 2: Algebra and Functions
Content Emphasis Cluster Understand solving equations as a process of reasoning and explain the reasoning
Analyze functions using different representations
Build a function that models a relationship between two quantities
Interpret functions that arise in applications in terms of the context
Mathematical Practices MP.1. Make sense of problems and persevere in solving them.
MP.2. Reason abstractly and quantitatively.
MP.3. Construct viable arguments and critique the reasoning of others.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
Common Assessment End of Unit Assessment
Graduate Competency Prepared graduates are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate
(mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and
transparency
Prepared graduates make sound predictions and generalizations based on patterns and relationships that arise from numbers,
MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning.
Content Standards
*Priority Standards*
A.CED.1 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.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
F.BF.1.b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
F.BF.4 Find inverse functions.
F.BF.4.a 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. For example, f(x) =2x3 or f(x) = (x+1)/(x–1) for x = 1.
F.BF.4.c (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.IF.4
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.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F.IF.7.d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
Algebra II
F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
N.Q .2 Define appropriate quantities for the purpose of descriptive modeling.
Inquiry Questions
Where will you find discontinuities in the real world? What is a real world example of a rational function? How are the models of rational and radical equations related? Can graphs of rational and radical functions be transformed in the same way as quadratic and linear functions?
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…
Topic 1 – Rational functions Graphically (A.REI.2, A.SSE.3, F.BF.3, F.BF.4.c, F.IF.4, F.IF.7, F.IF.7.d) Find intercepts, discontinuities (holes and vertical asymptotes), horizontal asymptotes, end behavior and domain and range. The students will be able to graph rational functions, identifying zeros, asymptotes, points of discontinuity, and will be able to describe end behavior in limit
notation.
Topic 2 – Rational functions Algebraically (A.REI.2, A.SSE.3, F.BF.4, F.BF.4.a, F.IF.8) The students will be able to rewrite rational functions so they are able to graph rational functions, identifying zeros, asymptotes, points of discontinuity, and
will be able to describe end behavior in limit notation. Topic 3- Operations on Rational Expressions (F.BF.1.b) The students will be able to combine rational functions using the four basic operations and can graph their simplifications; find inverses of simple rational
equations.
Topic 4 – Solving Rational Equations (A.CED.1, N.Q.2) The students will be able to solve rational equations and determine if their solutions are extraneous.
Critical Language:
Academic Vocabulary Cross discipline language-
Rewrite, Simplify Equality, Solve, Definition, Meaning, Extending, Least Common Denominator
Technical Vocabulary Discipline-specific language
Vertical Asymptote, Rational Expression, Horizontal Asymptote, Undefined, Point of Discontinuity, End Behavior, Common Denominator, Rational Equation, Limit, Continuity, Complex Fraction, Extraneous Solution, Rational Function,
Algebra II
Resources
Lessons
Optional Resource: EngageNY Module 1 https://www.engageny.org/resource/algebra-ii-module-1 and Module 2 https://www.engageny.org/resource/algebra-ii-module-3
Proposed sequence: Topic 1 – Rational functions Graphically Find intercepts, discontinuities (holes and vertical asymptotes), horizontal asymptotes, end behavior and domain and range. The students will be able to graph rational functions, identifying zeros, asymptotes, points of discontinuity, and will be able to describe
end behavior in limit notation.
EngageNY Module 1 -Lesson 22
Illustrative Mathematics: “Graphing Rational Functions” http://www.illustrativemathematics.org/illustrations/1694 Topic 2 – Rational functions Algebraically Simplify rational functions and identify discontinuities and classify as holes and/or vertical asymptotes; determine equations for
horizontal asymptotes using limit notation. Topic 3- Operations on Rational Expressions
Add, subtract, multiply and divide rational expressions; inverses to simple rational equations (Examples 𝑦 = 3
𝑥+2 𝑎𝑛𝑑 𝑦 =
𝑥+2
𝑥−5 ) paying
attention to domain and range of both the function and its inverse EngageNY Module 1 -Lesson 24 and 25 Topic 4 – Solving Rational Equations Solve rational equations in one variable algebraically and graphically (as a system of equations). EngageNY Module 1 -Lesson 26 and 28
Illustrative Mathematics: “Harvesting the Fields” http://www.illustrativemathematics.org/illustrations/83 Optional: Unit 2 Rational Functions Non – Calculator
Algebra II – Extend in Section 8-3 Algebra II – Section 8-1 (Example 1) Algebra II – Section 8-1
Algebra II – Section 8-6 (Equations only) Algebra II – Section 8-6 Extend Engage New York: Module 1 – Topics C
https://www.engageny.org/resource/algebra-ii-module-1 High School Flip Book -PDF (New resource) http://community.ksde.org/LinkClick.aspx?fileticket=VNhMSEbSycI%3d&tabid=5646&mid=13290
Performance Task(s) (Assessments)
2015-16 D6 Alg II Common Assessment Unit 2 (Schoolcity) Given to students by December break Scanned into School City or students take the assessment online
Should be in addition to individually developed site based extended response exam that will not be entered into School City and teacher
created formative assessments
Misconceptions
Students might be tempted to cancel terms, rather than factors of, in numerators and denominators of fractions. Emphasize the whole numerator is divided by the whole denominator. Be careful using the word “cancel.” Rather, talk about dividing both numerators and denominators by a common factor.
Students might believe all solutions to rational equations are viable, without recognizing instances in which extraneous solutions are generated and must be eliminated.
Notes Be sure to discuss effective domain, algebraically finding the horizontal asymptotes, clearing out the denominator to solve equations.
suitable factorizations are available, and showing end
behavior. e. Graph exponential and logarithmic
functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and
amplitude.
F.IF.8 Write a function defined by an expression in
different but equivalent forms to reveal and explain
different properties of the function. a. Use the process
of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry
of the graph, and interpret these in terms of a context.
b. Use the properties of exponents to interpret
expressions for exponential functions. For example,
identify percent rate of change in functions such as y =
(1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and
classify them as representing exponential growth or
decay.
N.RN.1 Explain how the definition of the meaning of
rational exponents follows from extending the
properties of integer exponents to those values,
allowing for a notation for radicals in terms of rational
exponents. For example, we define 51/3 to be the cube
root of 5 because we want (51/3)3 = 5(1/3)3 to hold,
so (51/3)3 must equal 5.
N.RN.2 Rewrite expressions involving radicals and
rational exponents using the properties of exponents.
manipulation exercise, rather than serving a
purpose of allowing different features of the
function to be exhibited.
N.RN.1-2
Students sometimes misunderstand the
meaning of exponential operations, the way
powers and roots relate to one another, and
the order in which they should be
performed. Attention to the base is very
important.
position of a negative sign of a term with a
rational exponent can mean that the rational
exponent should be either applied first to the
base, 81, and then the opposite of the result
is taken,
answer of 4 -81 will be not real if the
denominator of the exponent is even. If the
root is odd, the answer will be a negative
number.
Students should be able to make use of
estimation when incorrectly using
multiplication instead of exponentiation.
Students may believe that the fractional
exponent in the expression 36 means the
same as a factor of 1/3 in multiplication
by the exponent.
Algebra II
Unit 3
LOG JAMS
Length of Unit 5 WEEKS
Concepts Geometric Sequences and Series, Exponential Functions, Growth, Decay, Finance, Properties of Exponents, Properties of Logarithms, Logarithmic Functions, Logarithms, Inverse, End Behavior.
Standards of Mathematical Practices
MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning.
Content Standards
*Priority Standards*
A.CED.1 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.SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A.SSE.3.c
Use the properties of exponents to transform expressions for exponential functions. For example the expression
1.15t can be rewritten as (1+.𝟏𝟓
𝟏𝟐)12t ≈ 1.012512t to reveal the approximate equivalent monthly interest rate if the
annual rate is 15%.
A.SSE.4 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. For example, calculate mortgage payments.
F.BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
F.BF.1.b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
F.BF.4 Find inverse functions.
F.BF.4.a 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. For example, f(x) =2x3 or f(x) = (x+1)/(x–1) for x = 1.
F.BF.4.c (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Algebra II
F.IF.4
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.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F.IF.8.b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F.LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b I s2, 10, or e; evaluate the logarithm using technology.
F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
N.RN.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3 = 𝟓𝟏/𝟑𝟑 to hold, so (51/3)3 must equal 5.
N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Inquiry Questions
What financial phenomena can be modeled with exponential and logarithmic functions?
Algebra II
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…
Topic 1 – Geometric Sequences and Series (A.SSE.4, F.BF.1.a, F.BF.2, F.IF.3) Students will be able to write equations of explicit and recursive geometric sequences and series and use those to solve problems (Algebra 1 does this to
mastery). Converge and Diverge are the new topics for Algebra 2. Topic 2 – Properties of Exponents (A.SSE.3, F.IF.8.b, N.RN.1, N.RN.2) Students will be able to use and apply properties of exponents including integer and rational exponents. Make the connection to radicals and rational
exponents. Topic 3- Exponential Growth and Decay (A.CED.1, A.REI.11, A.SSE.1.b, A.SSE.3.c, F.BF.1.b, F.BF.3, F.IF.4, F.IF.7, F.IF.7.e, F.IF.9, F.LE.2, F.LE.5) Students will be able to model exponential growth or decay algebraically, graphically, or by using a table and understand properties of exponential functions.
Compound interest (annual, monthly, daily, etc.) and natural growth/decay. Topic 4 – Logarithms as Inverse Functions (A.CED.1, A.REI.11, F.BF.3, F.BF.4, F.BF.4.c, F.IF.4, F.IF.7, F.IF.7.e, F.IF.9) Students will be able to model derive logarithmic functions as inverses of exponential functions algebraically, graphically, or by using a table and understand
properties of logarithmic functions including transformations, and end behavior. Topic 5 – Properties of Logarithms (A.SSE.3) Students will be able to use and apply properties of logarithms to expand and condense expressions. Topic 6 – Solving Logarithmic and Exponential equations (A.CED.1, A.REI.11, A.SSE.3, F.BF.4, F.BF.4.a, F.BF.4.c, F.LE.4) Students will be able to solve logarithmic and exponential equations algebraically and graphically.
Nonlinear, average rate of change, parameters, explicit, recursive, relative maximum, relative minimum, symmetry, logarithms, exponential functions, growth, properties of exponents, geometric series, inverse functions, intercepts, end behavior, geometric sequence, explicit, recursive, discrete, continuous, common ratio, domain, range, horizontal asymptote
Algebra II
Resources
Lessons
Optional Resource: EngageNY Module 2- https://www.engageny.org/resource/algebra-ii-module-2 Proposed sequence: Topic 1 – Geometric Sequences and Series Find nth term of a geometric sequence; describe a geometric sequence as a function defined on the set of positive integers using a graph,
table and algebra; recursive formula (All done in Alg 1) New for Alg 2: sum of a finite geometric series; sum of an infinite geometric series; Use as an intro to exponential functions.
Illustrative Mathematics: “Triangle Series” http://www.illustrativemathematics.org/illustrations/442 Illustrative Mathematics: “Course of Antibiotics” http://www.illustrativemathematics.org/illustrations/805 Illustrative Mathematics: “Cantor Set” http://www.illustrativemathematics.org/illustrations/929 Illustrative Mathematics: “A Lifetime of Savings” http://www.illustrativemathematics.org/illustrations/1283 Illustrative Mathematics: “Susita’s Account” http://www.illustrativemathematics.org/illustrations/218 Topic 2 – Properties of Exponents Product rule, Quotient rule, Power rule, Negative exponents, zero exponents (review from Algebra 1); Rational exponents; writing as a
radical or an exponent; Properties of exponents using rational exponents Illustrative Mathematics: “Forms of exponential expressions” http://www.illustrativemathematics.org/illustrations/1305 Illustrative Mathematics: “Extending the Definitions of Exponents http://www.illustrativemathematics.org/illustrations/385
Topic 3- Exponential Growth and Decay General form of an exponential function (y = abx); y-intercept, rate of growth/decay; transformations of exponential functions; Alternate
form of exponential growth/decay (A = A0(1 +/- r)t); derive compound interest formula and formula for compounding continuously A = Pert– emphasis on why, not plug and chug
llustrative Mathematics: “Buying a car” http://www.illustrativemathematics.org/illustrations/582 Illustrative Mathematics: “Sum of angles in a polygon” http://www.illustrativemathematics.org/illustrations/1124 Illustrative Mathematics: “Doubling Your Money” http://www.illustrativemathematics.org/illustrations/214 Illustrative Mathematics: “Bacteria Populations” http://www.illustrativemathematics.org/illustrations/370 Illustrative Mathematics: “Algae Blooms” http://www.illustrativemathematics.org/illustrations/570 Illustrative Mathematics: “Snail Invasion” http://www.illustrativemathematics.org/illustrations/638 Topic 4 – Logarithms as Inverse Functions Derive the logarithmic function as an inverse using a table and graph; emphasize the inverse relationship between logarithmic functions
and exponential functions; domain and range, intercepts and end behavior; include natural logarithms and Euler’s number
Illustrative Mathematics: “Carbon 14 dating in practice II” http://www.illustrativemathematics.org/illustrations/760 Illustrative Mathematics: “Accuracy of Carbon 14 Dating II” http://www.illustrativemathematics.org/illustrations/784 Illustrative Mathematics: “Graphene” http://www.illustrativemathematics.org/illustrations/1569 Topic 5 – Properties of Logarithms Exponential vs. Logarithmic form; sum, difference and power rules; change of base Topic 6 – Solving Logarithmic and Exponential equations Solve equations that contain logarithms; solve exponential equations that require logarithms Optional: Unit 3 Assessment Calculator (This assessment is all calculator)
Core Instructional Task
Exponential an logarithmic functions
Technology Graphing Calculator
Materials
Algebra II – Section 11-3 Algebra II – Section 11-4 Algebra II – Section 11-5 Algebra II – Explore in Section 11-6 Kuta Algebra II
Algebra II – Section 9-1 Algebra II –Section 6-1 Algebra II – Section 9-2 (not solving) Algebra II – Section 9-2 Algebra II – Section 9-3
Algebra II – Section 9-4 Algebra II – Section 9-5 Algebra II – Section 9-6 Engage New York: Module 2 (New resource) https://www.engageny.org/resource/algebra-ii-module-2 High School Flip Book -PDF (New resource) http://community.ksde.org/LinkClick.aspx?fileticket=VNhMSEbSycI%3d&tabid=5646&mid=13290
Performance Task(s) (Assessments)
2015-16 D6 Alg II Common Assessment Unit 3 Given to students in February Scanned into School City or students take the assessment online
Should be in addition to individually developed site based extended response exam that will not be entered into School City and teacher
created formative assessments
Misconceptions
Students might multiply base and exponent, rather than raising the base to the power, when working with exponential equations.
Students might believe all solutions to radical equations are viable, without recognizing instances in which extraneous solutions are generated and must be eliminated.
Students often think a “+” next to x is a shift to the right due to the addition sign, which indicates positive change. Students should explore examples both analytically and graphically to overcome this notion.
Ensure students understand the idea of rate of change as it applies to exponential functions and that it applies to real-world situations; it is not just an abstract idea.
Students might misinterpret the percent change in a problem such as y = (1.01)12t as 1% (see F-IF.8b).
Students might believe all functions have inverses and need to see counterexamples.
F.IF.4 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.IF.7 Graph functions expressed
symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph
Literacy Connections
RST.6-8.4
Determine the meaning of
symbols, key terms, and
other domain-specific
words and phrases as
they are used in a
specific scientific or
technical context
relevant to grades 6-8
texts and topics.
RST.6-8.5
Analyze the structure an
author uses to organize
a text, including how
the major sections
contribute to the whole
and to an understanding
of the topic.
RST.6-8.7
Integrate quantitative or
technical information
expressed in words in a
text with a version of
that information
expressed visually (e.g.,
in a flowchart, diagram,
model, graph, or table).
Writing Connection
WHST.6-8.2
Write
informative/explanatory
texts, including the
narration of historical
events, scientific
procedures/
experiments, or
technical processes.
p. Introduce a topic
clearly, previewing
what is to follow;
organize ideas,
concepts, and
information into
broader categories as
appropriate to
achieving purpose;
include formatting (e.g.,
headings), graphics
(e.g., charts, tables),
and multimedia when
useful to aiding
comprehension.
q. Develop the topic with
relevant, well-chosen
facts, definitions,
concrete details,
quotations, or other
Academic Vocabulary-
Solve, combine,
recognize, compare,
calculate, construct,
define, interpret,
increase, decrease,
intersection, solution,
positive, negative,
input,
output, explain, prove,
graph, key features,
interpret, angles,
model,
counterclockwise,
clockwise
Technical Vocabulary-
Nonlinear, average rate
of change,
parameters,
periodicity, unit
circle, coordinate
plane, trigonometric
functions, periodic
phenomena, radian
measure, subtend,
amplitude, frequency,
midline, period,
Pythagorean identity,
sine, cosine, tangent,
Students may believe that it is reasonable to
input any x-value into a function, so they
will need to examine multiple situations in
which there are various limitations to the
domains.
Students may also believe that the slope of a
linear function is merely a number used to
sketch the graph of the line. In reality,
slopes have real-world meaning, and the
idea of a rate of change is fundamental to
understanding major concepts from
geometry to calculus.
Students may believe that each family of
functions (e.g., quadratic, square root, etc.)
is independent of the others, so they may not
recognize commonalities among all
functions and their graphs.
Students may also believe that skills such as
factoring a trinomial or completing the
square are isolated within a unit on
polynomials, and that they will come to
understand the usefulness of these skills in
the context of examining characteristics of
functions.
Additionally, student may believe that the
process of rewriting equations into various
forms is simply an algebra symbol
manipulation exercise, rather than serving a
Algebra II
rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Concepts Nonlinear, Unit Circle, Trigonometric Functions, Model, Periodic Phenomena, Transformations, Systems of Equations
Standards of Mathematical Practices
MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning.
Content Standards
*Priority Standards*
G.SRT.8 Use trigonometric rations and the Pythagorean Theorem to solve right triangles in applied problems.
F.BF.3 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.TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for 𝜋
3,
𝝅
𝟒𝒂𝒏𝒅
𝝅
𝟔 .
F.IF.4
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.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F.TF.2 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.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
F.TF.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios.
Algebra II
Inquiry Questions
What phenomena can be modeled with trigonometric functions? How does the change in altitude affect the angle of elevation/depression? How do the table, graph, and function notation of a trigonometric function compare to polynomial and rational functions? How does the periodicity in the unit circle correspond to the periodicity in graphs of models of periodic phenomena? Why can the same class of functions model diverse types of situations (e.g., sales, manufacturing, temperature, amusement park rides)?
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…
Topic 1 – Right Triangle Trigonometry (G.SRT.8) Students will be able to apply the trigonometric ratios as side of a triangle to solve right triangle problems. Angle of elevation and depression application
problems.
Topic 2 – Special Right Triangles (F.TF.3) Students will be able to identify and determine missing side lengths (or angles) of a given special right triangle.
Topic 3a- The Unit Circle (F.TF.2) Students will be able to use special right triangles to determine coordinates on a circle of radius r to begin understanding the three basic trigonometric
functions in terms of coordinates instead of sides of a triangle.
Topic 3b – Radian Measure (F.TF.1) Students will be able to understand the necessity of radian measure and discover the relationship between radian and degree measurement. Students will
also use both measurement systems to find areas of sectors, arc length.
Topic 4 – Graphs of the three basic trigonometric functions (F.IF.7, F.IF.7.e, F.BF.3, F.IF.4, F.IF.5) Students will be able to view the three basic trigonometric functions as circular functions, unwrapping each function to graph it. Students will also use
vocabulary such as ‘period’ and ‘amplitude’ to discuss transformations of the three trigonometric graphs.
Topic 5 – The Six Trigonometric Functions (F.TF.2) Students will be able to identify and know the relationship between the three basic trigonometric functions and their respective reciprocals.
Students will be able to solve simple trigonometric equations as systems of equations. Example: 𝑆𝑖𝑛 𝜃 =√3
2
*Students will not be required to understand the inverse trigonometric functions with the exception of right triangle trigonometry Example: 𝑆𝑒𝑐 𝜃 = √3 Topic 7 – Pythagorean Identities (F.TF.8) Students will be able to derive and use the Pythagorean Identities.
Nonlinear, periodicity, unit circle, coordinate plane, trigonometric functions, periodic phenomena, radian measure, amplitude, frequency, midline, period, Pythagorean identity, sine, cosine, tangent, cosecant, secant, cotangent, arc length, area of sector, quadrant
Resources
Lessons
Optional Resources: Engage NY Module 2 https://www.engageny.org/resource/algebra-ii-module-2 Proposed sequence: Topic 1 – Right Triangle Trigonometry Apply the trigonometric ratios as side of a triangle to solve right triangle problems. Angle of elevation and depression application
problems. Should be all review.
Illustrative Mathematics: “Shortest line segment from a point P to a line L” http://www.illustrativemathematics.org/illustrations/962 Topic 2 – Special Right Triangles Find missing side lengths of a given special right triangles using the rules they find by using a hypotenuse of 1. Should be review but is
necessary to derive the unit circle. Topic 3a- The Unit Circle Use special right triangles to determine coordinates on a circle of radius r to begin understanding the three basic trigonometric
functions in terms of coordinates instead of sides of a triangle. Topic 3b – Radian Measure Understand the necessity of radian measure and discover the relationship between radian and degree measurement. Students will
also use both measurement systems to find areas of sectors, arc length. “Sectors of Circles” – Mathematics Assessment Project This lesson unit is intended to help you assess how well students are able to solve problems involving area and arc length of a sector of a circle using radians. It assumes familiarity with radians and should not be treated as an introduction to the topic. This lesson is intended to help you identify and assist students who have difficulties in: •Computing perimeters, areas, and arc lengths of sectors using formulas. •Finding the relationships between arc lengths, and areas of sectors after scaling..
Illustrative Mathematics: “Orbiting Satellite” http://www.illustrativemathematics.org/illustrations/1639 . Topic 4 – Graphs of the three basic trigonometric functions Only the three basic trigonometric functions as circular functions, unwrapping each function to graph it. Students will also use
vocabulary such as ‘period’ and ‘amplitude’ to discuss transformations of the three trigonometric graphs. “Ferris Wheel” – Mathematics Assessment Project This lesson unit is intended to help you assess how well students are able to: •Model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions. •Interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person
above the ground and t is the elapsed time. http://map.mathshell.org/materials/lessons.php?taskid=427#task427 Illustrative Mathematics: “As the Wheel Turns” http://www.illustrativemathematics.org/illustrations/595 Illustrative Mathematics: “Foxes and Rabbits 2” http://www.illustrativemathematics.org/illustrations/816 Illustrative Mathematics: “Foxes and Rabbits 3” http://www.illustrativemathematics.org/illustrations/817 Topic 5 – The Six Trigonometric Functions Identify and know the relationship between the three basic trigonometric functions and their respective reciprocals. Writing the six
trig functions given one of the ratios. Topic 6 – Solving Trigonometric Equations*
Students will be able to solve simple trigonometric equations as systems of equations. Example: 𝑆𝑖𝑛 𝜃 =√3
2 as sin is in y1 and
√3
2 is in
y2 and find intersections. *Students will not be required to understand the inverse trigonometric functions with the exception of right triangle trigonometry
Example: 𝑆𝑒𝑐 𝜃 = √3 Topic 7 – Pythagorean Identities *If time permits* Derive and use the Pythagorean Identities. Optional: Unit 4 Trigonometry Assessment (Calculator and Non Calculator)
Core Instructional Task Sinusoidal functions
Technology Graphing Calculators
Materials
Algebra II – Section 14-3 Algebra II- Section 14-1
Algebra II- Section 14-2 Engage New York: Module 2 (New resource) https://www.engageny.org/resource/algebra-ii-module-2 High School Flip Book -PDF (New resource) http://community.ksde.org/LinkClick.aspx?fileticket=VNhMSEbSycI%3d&tabid=5646&mid=13290
Performance Task(s) (Assessments)
2015-16 D6 Alg II Common Assessment Unit 4 (Schoolcity) Given to students the end of March Scanned into School City or students take the assessment online
Should be in addition to individually developed site based extended response exam that will not be entered into School City and teacher
created formative assessments
Misconceptions
Students often think a “+” next to x is a shift to the right due to the addition sign, which indicates positive change. Students should explore examples both analytically and graphically to overcome this notion.
Students might believe there is no need for radians if one already knows how to use degrees. Show a rationale for how radians are unique in terms of finding function values in trigonometry since the radius of the unit circle is 1.
Students might believe all angles having the same reference values have identical sine, cosine, and tangent values, forgetting that certain trigonometric functions carry certain signs (+ or -) in different quadrants.
Students might believe all trigonometric functions have a range of 1 to -1. Students must see examples showing how coefficients can change the range and appearance of graphs.
Students need help seeing the connection between the Pythagorean Theorem and the study of trigonometry.
Showing students the relationship between the sine and cosine values for a particular angle (that the sum of the squares of these values always equals 1) is a unique way to view trigonometry through the lens of geometry.
Ensure students understand the idea of rate of change as it applies to trigonometric functions and that it applies to real-world situations; it is not just an abstract idea.
Instructional Notes
Be sure to develop a conceptual understanding of the six trig functions. NOT a memorization.
Use unit circle to graph 3 trig functions.
Spend time teaching trig vocab to students.
Focus on radians instead of degrees.
Trig equations (𝑆𝑖𝑛 𝜃 =√3
2) without a calculator and with a calculator by graphing.
S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
S.IC.4 Use data from a sample survey to
estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
S.IC.5 Use data from a randomized
experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
Observational Studies, Statistical Results, Simulation, Normal Distribution
Standards of Mathematical Practices
MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically.
Content Standards (Priority Standards)
F.BF.D Build a function that models a relationship between two quantities
S.CP.1 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.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
S.CP.3 Understand the conditional probability of P(A|B) = P(A ∩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.CP.4
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. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
S.CP.6 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.CP.7 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.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
Algebra II
S.IC.2
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
S.IC.6 Evaluate reports based on data.
S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.2
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
7.SP.3
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
Algebra II
Key Knowledge and Skills (Procedural Skill and Application)
My students will be able to (Do)…
Topic A - Probability (S-IC.2, S-CP.1, S-CP.2, S-CP.3, S-CP.4, S-CP.5, S-CP.6, S-CP.7)
Students determine the sample space for a chance experiment.
Given a description of a chance experiment and an event, students identify the subset of outcomes from the sample space corresponding to the
complement of an event.
Given a description of a chance experiment and two events, students identify the subset of outcomes from the sample space corresponding to the
union or intersection of two events.
Students calculate the probability of events defined in terms of unions, intersections, and complements for a simple chance experiment with equally
likely outcomes.
Students calculate probabilities given a two-way table of data.
Students calculate conditional probabilities given a two-way data table or using a hypothetical 1000 two-way table.
Students interpret probabilities, including conditional probabilities, in context.
Students use a hypothetical 1000 two-way table to calculate probabilities of events.
Students use two-way tables (data tables or hypothetical 1000 two-way tables) to determine if two events are independent.
Students represent events by shading appropriate regions in a Venn diagram.
Given a chance experiment with equally likely outcomes, students calculate counts and probabilities by adding or subtracting given counts or
probabilities.
Students interpret probabilities in context.
Students use the complement rule to calculate the probability of the complement of an event and the multiplication rule for independent events to
calculate the probability of the intersection of two independent events.
Students recognize that two events 𝐴 and 𝐵 are independent if and only if 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴)𝑃(𝐵) and interpret independence of two events 𝐴 and
𝐵 as meaning that the conditional probability of 𝐴 given 𝐵 is equal to 𝑃(𝐴).
Students use the formula for conditional probability to calculate conditional probabilities and interpret probabilities in context.
Students use the addition rule to calculate the probability of a union of two events.
Algebra II
Topic B – Modeling Data Distributions (S-ID.4)
Students describe data distributions in terms of shape, center, and variability.
Students use the mean and standard deviation to describe center and variability for a data distribution that is approximately symmetric.
Students draw a smooth curve that could be used as a model for given data distribution.
Students recognize when it is reasonable and when it is not reasonable to use a normal curve as a model for a given data distribution.
Students calculate 𝑧-scores.
Students use technology and tables to estimate the area under a normal curve.
Students interpret probabilities in context.
Students use tables and technology to estimate the area under a normal curve.
Students interpret probabilities in context.
When appropriate, students select an appropriate normal distribution to serve as a model for a given data distribution.
Topic C – Drawing Conclusions Using Data from a Sample (S-IC.1, S-IC.3, S-IC.4, S-IC.6)
Students distinguish between observational studies, surveys, and experiments.
Students explain why random selection is an important consideration in observational studies and surveys and why random assignment is an
important consideration in experiments.
Students recognize when it is reasonable to generalize the results of an observation study or survey to some larger population and when it is
reasonable to reach a cause-and-effect conclusion about the relationship between two variables.
Students differentiate between a population and a sample.
Students differentiate between a population characteristic and a sample statistic.
Students recognize statistical questions that are answered by estimating a population mean or a population proportion.
Students understand the term sampling variability in the context of estimating a population proportion.
Students understand that the standard deviation of the sampling distribution of the sample proportion offers insight into the accuracy of the sample
proportion as an estimate of the population proportion.
Students use data from a random sample to estimate a population proportion.
Students calculate and interpret margin of error in context.
Students know the relationship between sample size and margin of error in the context of estimating a population proportion.
Students understand that the standard deviation of the sampling distribution of the sample mean conveys information about the anticipated accuracy
of the sample mean as an estimate of the population mean.
Students know the relationship between sample size and margin of error in the context of estimating a population mean.
Students interpret margin of error from reports that appear in newspapers and other media.
Students critique and evaluate statements in published reports that involve estimating a population proportion or a population mean.
Algebra II
Topic D – Drawing Conclusions Using Data from an Experiment (S-IC.3, S-IC.5, S-IC.6)
Given a description of a statistical experiment, students identify the response variable and the treatments.
Students recognize the different purposes of random selection and of random assignment.
Students recognize the importance of random assignment in statistical experiments.
Students understand that when one group is randomly divided into two groups, the two groups’ means differ just by chance (a consequence of the
random division).
Students understand that when one group is randomly divided into two groups, the distribution of the difference in the two groups’ means can be
described in terms of shape, center, and spread.
Given data from a statistical experiment with two treatments, students create a randomization distribution.
Students use a randomization distribution to determine if there is a significant difference between two treatments.
Students carry out a statistical experiment to compare two treatments.
Students critique and evaluate statements in published reports that involve determining if there is a significant difference between two treatments in
Complement of an Event, Conditional Probability, Experiment, Hypothetical 1000 Table, Independent Events, Intersection of Two Events, Lurking Variable, Margin of Error, Normal Distribution, Observational Study, Random Assignment, Random Selection, Sample Survey, Treatment, Union of Two Events.
Resources
Lessons
Eureka Math Algebra II Module 4 Topic A - Probability (S-IC.2, S-CP.1, S-CP.2, S-CP.3, S-CP.4, S-CP.5, S-CP.6, S-CP.7) Lesson 1: Chance Experiments, Sample Spaces, and Events Lesson 2: Calculating Probabilities of Events Using Two-Way Tables Lessons 3–4: Calculating Conditional Probabilities and Evaluating Independence Using Two-Way Tables Lesson 5: Events and Venn Diagrams Lessons 6–7: Probability Rules
Algebra II
Topic B – Modeling Data Distributions (S-ID.4) Lesson 8: Distributions—Center, Shape, and Spread Lesson 9: Using a Curve to Model a Data Distribution Lessons 10–11: Normal Distributions Topic C – Drawing Conclusions Using Data from a Sample (S-IC.1, S-IC.3, S-IC.4, S-IC.6) Lesson 12: Types of Statistical Studies Lesson 13: Using Sample Data to Estimate a Population Characteristic Lessons 14–15: Sampling Variability in the Sample Proportion Lessons 16–17: Margin of Error When Estimating a Population Proportion Lessons 18–19: Sampling Variability in the Sample Mean Lessons 20–21: Margin of Error When Estimating a Population Mean Lesson 22: Evaluating Reports Based on Data from a Sample Topic D – Drawing Conclusions Using Data from an Experiment (S-IC.3, S-IC.5, S-IC.6) Lesson 23: Experiments and the Role of Random Assignment Lesson 24: Differences Due to Random Assignment Alone Lessons 25–27: Ruling Out Chance Lessons 28–29: Drawing a Conclusion from an Experiment Lesson 30: Evaluating Reports Based on Data from an Experiment
Suggested Tools and Representations
Graphing calculator or graphing software, Random number tables, Random number software, Normal distribution, Two-way frequency tables, Spreadsheets
Performance Task(s) (Assessments)
2015-16 D6 Alg II Common Assessment Unit 5 (Schoolcity) Given to students in May
Scanned into School City or students take the assessment online
Should be in addition to individually developed site based extended response exam that will not be entered into School City and
teacher created formative assessments
Misconceptions
Students might believe inferences from samples to populations can be done only in experiments. They need to realize that inferences can be done in samplings and observational studies if data are collected through random process.
Students might believe population parameters and sample statistics are the same. For example, they may think there is no difference between the population mean, which is a constant, and the sample mean, which is a variable.
Students might believe that making decisions is simply comparing the value of one observation of a sample statistic to the value of a population parameter, not realizing the distribution of the sample statistic needs to be created.
Students might believe all bell-shaped curves are normal distributions. A bell-shaped curve is normal when 68% of the distribution is within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations.
