GOVERNOR LIVINGSTON HIGH SCHOOL MATHEMATICS …

BERKELEY HEIGHTS PUBLIC SCHOOLS BERKELEY HEIGHTS, NEW JERSEY

GOVERNOR LIVINGSTON HIGH SCHOOL MATHEMATICS DEPARTMENT

ALGEBRA 1/PROJECT CONNECT ALGEBRA 1 Course#MAY0910

Curriculum Guide

Date: September 2014 Updated: December 2019

Dr. Melissa Varley, Superintendent Mr. Scott McKinney, Assistant Superintendent Mr. Andrew Ziobro, District Supervisor

Developed by: Nicole Manganelli Andrea Sperling Michelle Vieria

Carol Chory William Ayres

Steven Hess

This curriculum may be modified through varying techniques, strategies, and materials, as per an individual student’s

Individualized Educational Plan (IEP)

Approved by the Berkeley Heights Board of Education at the regular meeting held on 12/5/19 .

VISION STATEMENT Algebra 1 continues to build on the knowledge from grades 6 - 8 using expressions, equations and inequalities. An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.

Berkeley Heights Public Schools 1

MISSION STATEMENT

Many of the concepts presented in Algebra I are progressions of the concepts that were started in grades 6 through 8; the content presented in this course is intended to extend and deepen the previous understandings. Unit 1 begins with setting the stage for work with expressions and equations through understanding quantities and the relationships between them. The work in unit 2 will build on the grade 8 concepts for linear and exponential relationships. Success in unit 2 will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. The standards included in unit 3 blend the conceptual understandings of expressions and equations with procedural fluency and problem solving. The students will not encounter solutions of quadratic equations that are complex. The standards presented in unit 4 involve functions and extending the concepts of integer exponents to concepts of rational exponents. These understandings will be applied to other types of equations in future courses. Unit 5 will build on previous work with descriptive statistics. Linear models will be used to assess how a model fits data. National and state standards are aligned and integrated throughout the curriculum.


COURSE PROFICIENCIES COURSE OBJECTIVES

Seeing Structure in Expressions

● Interpret the structure of expressions ● Write expressions in equivalent forms to solve problems

Arithmetic with Polynomials and Rational Functions ● Perform arithmetic operations on polynomials ● Understand the relationship between zeros and factors of polynomials ● Use polynomial identities to solve problems ● Rewrite rational functions

Creating Equations ● Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities ● Understand solving equations as a process of reasoning and explain the reasoning ● Solve equations and inequalities in one variable ● Solve systems of equations ● Represent and solve equations and inequalities graphically

Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.


STUDENT PROFICIENCIES

Some clusters that are not major emphases in themselves are designed to support and strengthen areas of major emphasis, while other clusters that may not connect tightly or explicitly to the major work of the grade would fairly be called additional. To say that some things have greater emphasis is not to say that anything in the standards can safely be neglected in instruction. Neglecting material will leave gaps in student skill and understanding and may leave students unprepared for the challenges of a later grade. Major Content Supporting Content Additional Content (Identified by PARCC Model Content Frameworks).

Bold type indicates grade level fluency requirements. (Identified by PARCC Model Content Frameworks).

The student will know or be able to:

UNIT 1: Relationships Between Quantities and Reasoning with Equations

# STUDENT LEARNING OBJECTIVES CORRESPONDING CCSS/NJCCCS

A1u1-1 Solve multi-step problems that can be represented algebraically with accurate and appropriately defined units, scales, and models (such as graphs, tables, and data displays).

N.Q.1 N.Q.2 N.Q.3

A1u1-2 Interpret terms, factors, coefficients, and expressions (including complex linear and exponential expressions) in terms of context.

A.SSE.1

A1u1-3 Solve linear equations and inequalities in one variable (including literal equations). Justify each step in the process and solution.

A.CED.4 A.REI.3

A1u1-4 Create linear equations and inequalities in one variable and use them to solve problems. Justify each step in the process and the solution.

A.CED.1 A.REI.1 A.REI.3

A1u1-5 Create linear equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.CED.2

A1u1-6

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.1


UNIT 2: Linear Relationships

# STUDENT LEARNING OBJECTIVES CORRESPONDING CCSS

A1u2-1 Solve systems of linear equations in two variables graphically and algebraically. Include solutions that have been found by replacing one equation by the sum of that equation and a multiple of the other.

A.REI.5 A.REI.6

A1u2-2 Find approximate solutions of linear equations by making a table of values, using technology to graph and successive approximations.

A.REI.11

A1u2-3 Graph equations, inequalities, and systems of inequalities in two variables and explain that the solution to an equation is all points along the curve, the solution to a system of linear functions is the point of intersection, and the solution to a system of inequalities is the intersection of the corresponding half-planes; describe constraints with linear equations and inequalities and systems of equations and/or inequalities to determine if solutions are viable or nonviable. ★

A.REI.10 A.REI.11

A.REI.12 A.CED.3

A1u2-4 Explain and interpret the definition of functions including domain and range and how they are related; correctly use function notation in a context and evaluate functions for inputs and their corresponding outputs.

F.IF.1 F.IF.2

A1u2-5 Write a function for a geometric sequence defined recursively, whose domain is a subset of the integers.

F.IF.3

A1u2-6 Graph functions by hand (in simple cases) and with technology (in complex cases) to describe linear relationships between two quantities and identify, describe, and compare domain and other key features in one or multiple representations. ★

F.IF.5 F.IF.7a F.IF.9

A1u2-7 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

F.IF.9


UNIT 3: Expressions and Equations

# STUDENT LEARNING OBJECTIVES CORRESPONDING CCSS

A1u3-1 Interpret parts of expressions in terms of context including those that represent square and cube roots; use the structure of an expression to identify ways to rewrite it. ★

A.SSE.1, A.SSE.2

A1u3-2

Rewrite expressions using factoring, completing the square and properties of exponents to produce equivalent forms that highlight particular properties such as the zeros or the maximum or minimum value of the function. ★

A.SSE.3

A1u3-3 Perform addition, subtraction and multiplication with polynomials and relate it to arithmetic operations with integers.

A.APR.1

A1u3-4

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, simple rational and exponential functions and highlighting a quantity of interest in a formula.

A.CED.1, A.CED.4

A1u3-5 Create linear and quadratic equations that represent a relationship between two or more variables. Graph equations on the coordinate axes with labels and scale.

A.CED.2,

A1u3-6 Derive the quadratic formula by completing the square and recognize when there are no real solutions.

A.REI.4

A1u3-7 Solve quadratic equations in one variable using a variety of methods [including inspection (e.g. x2 = 81), factoring, completing the square, and the quadratic formula].

A.REI.4


UNIT 4: Functions and Modeling

# STUDENT LEARNING OBJECTIVES CORRESPONDIN

G CCSS

A1u4-1 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.3

A1u4-2

Use the properties of rational and irrational numbers to explain why the sum or product of two rational numbers is rational; the sum of a rational number and an irrational number is irrational; and the product of a nonzero rational number and an irrational number is irrational.

N.RN.3

A1u4-3

Sketch the graph of a function that models a relationship between two quantities (expressed symbolically or from a verbal description) showing key features ( including intercepts, minimums/maximums, domain, and rate of change) by hand in simple cases and using technology in more complicated cases and relate the domain of the function to its graph. ★

F.IF.4 F.IF.5 F.1F.7

A1u4-4

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.

F.IF.9

A1u4-5 Calculate (over a specified period if presented symbolically or as a table) or estimate (if presented graphically) and interpret the average rate of change of a function. ★

F.IF.6

A1u4-6

Write functions in different but equivalent forms by translating quadratic expressions using methods such as factoring and completing the square to show zeros, extreme values, and symmetry of the graph; interpret these in terms of a context.

F.IF.8a

A1u4-7

Write a function that describes a linear, quadratic or exponential relationship (e.g. growth/decay and arithmetic and geometric sequences) from graphs, tables, or a written description of the relationship, recursively and with an explicit formula, and describe how quantities increase over equal intervals relate these functions to the model. ★

F.BF.1

F.LE.1 F.LE.2

A1u4-8 Identify the effects of transformations [ f(x) + k, k f(x), f(kx), and f(x + k)] on a function and find the value of k given the graphs.

F.BF.3

A1u4-9

Compare (using graphs and tables) linear, quadratic, and exponential models to determine that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function; include interpret parameters in terms of a context.

F.LE.3 F.LE.5


UNIT 5: Descriptive Statistics

# STUDENT LEARNING OBJECTIVES CORRESPONDING CCSS/NJCCCS

A1u5-1

Represent data on the real number line (i.e. dot plots, histograms, and box plots) and use statistics, appropriate to the shape of the data distribution, to interpret and compare center and spread in the context of the data (account for effects of outliers)

S.ID.1 S.ID.2

S.ID.3

A1u5-2 Summarize and interpret categorical data for two categories in two-way frequency tables; recognize associations and trends in the data.

S.ID.5

A1u5-3

Represent and describe data for two variables on a scatter plot, fit a function to the data, analyze residuals (in order to informally assess fit), and use the function to solve problems.

Uses a given function or choose a function suggested by the context. Emphasize linear and exponential models.

S.ID.6a

A1u5-4 Interpret the slope and intercept of a linear model in the context of the data; compute (using technology) and interpret the correlation coefficient of a linear fit

S.ID.7 S.ID.8

A1u5-5 Distinguish between correlation and causation. S.ID.9


METHODS OF EVALUATION

1. Projects/ reports/ presentations 2. Laboratory activities 3. Simulated activities 4. Classroom participation and cooperation 5. Performance on worksheets, quizzes and unit tests 6. Final exam, projects and/or reports


MODIFICATIONS & ACCOMMODATIONS

Modifications and Accommodations for Special Education students, students with 504s, English Language Learners and Gifted and Talented students may include but are not limited to the following:

Special Education ● Individualized Education Plans (IEPs) ● Exemplars of varied performance levels ● Multimedia presentations ● Sheltered instruction ● Consultation with ESL teachers ● Manipulatives ● Tiered/Scaffolded Lessons ● Mnemonic devices ● Visual aids ● Modeling ● Guided note-taking ● Study Guides ● Modified homework ● Differentiated pre-typed class notes and example problems ● Use of the special education teacher to re-instruct in flexible small groups for the struggling

learner ● Manipulatives ● Flipped Instruction ● Word banks ● Reduced choice on assessments ● Preferential seating ● Choice activities ● Modified time requirements ● Modified notes ● Modified lesson, assessment and study guide format ● Provide an enriched curriculum and activities ● Independent projects ● Contracts/behavior support plans ● Open-ended responses ● Project-based learning ● Group activities


● Guided Notes ● Functional learning incorporated into each lesson ● Exploration Activities ● Assessment read aloud ● Small group assessments ● Organizational Support ● Oral questioning assessments to supplement written response ● Pre-writing Structural Supports for extended writing tasks ● Ongoing teacher feedback as part of the writing process ● Interactive Study Guides ● Multi-sensory approach to instruction ● Written and spoken step-by-step directions ● Content-focused assessment (not grading for spelling/grammar) ● Graphic organizers ● Non-verbal cues to begin task/remain on task/refocus ● Individual monitoring for understanding/reinforced instruction ● Printed copies of class readings for application of Active Reading Strategies

Gifted & Talented

● Provide one-to-one teacher support ● Curriculum Compacting ● Advanced problems to extend the critical thinking skills of the advanced learner ● Supplemental reading material for independent study ● Elevated questioning techniques using Webb’s Depth of Knowledge matrix ● Curriculum Compacting ● Flexible grouping ● Tiered assignments ● Topic selection by interest ● Manipulatives ● Tiered Lessons ● Flipped Instruction ● Multimedia Presentations ● Open-ended responses ● Project-based learning ● Group activities ● Guided Notes ● Conclusions and analysis of exploratory activities ● Career based learning incorporated into each lesson ● Exploration Activities


● Student choice

ELLs ● Exemplars of varied performance levels ● Multimedia presentations ● Sheltered instruction ● Consultation with ESL teachers ● Manipulatives ● Tiered/Scaffolded Lessons ● Mnemonic devices ● Visual aids ● Modeling ● Guided note-taking ● Study Guides ● Modified homework ● Differentiated pre-typed class notes and example problems ● Individualized instruction plans ● Manipulatives ● Flipped Instruction ● Words banks ● Reduced choice on assessments ● Preferential seating ● Choice activities ● Modified time requirements ● Modified notes ● Modify lesson, assessment and study guide format ● Provide an enriched curriculum and activities ● Contracts/management plans ● Open-ended responses ● Project-based learning ● Group activities ● Guided Notes ● Exploration Activities ● Assessment read aloud ● Small group assessments ● Oral questioning assessments to supplement written response ● Pre-writing Structural Supports for extended writing tasks ● Ongoing teacher feedback as part of the writing process ● Interactive Study Guides


● Multi-sensory approach to instruction ● Written and spoken step-by-step directions ● Graphic organizers ● Non-verbal cues to begin task/remain on task/refocus ● Individual monitoring for understanding/reinforced instruction ● Printed copies of class readings for application of Active Reading Strategies

504s

● Exemplars of varied performance levels ● Multimedia presentations ● Sheltered instruction ● Tiered/Scaffolded Lessons ● Mnemonic devices ● Visual aids ● Modeling ● Guided note-taking ● Study Guides ● Differentiated pre-typed class notes and example problems ● Manipulatives ● Words banks ● Reduced choice on assessments ● Preferential seating ● Modified time requirements ● Modified notes ● Modify lesson, assessment and study guide format ● Modified homework ● Independent projects ● Contracts/management plans ● Open-ended responses ● Project-based learning ● Group activities ● Guided Notes ● Exploration Activities ● Assessment read aloud ● Small group assessments ● Organizational Support ● Oral questioning assessments to supplement written response ● Pre-writing Structural Supports for extended writing tasks ● Ongoing teacher feedback as part of the writing process ● Interactive Study Guides ● Multi-sensory approach to instruction ● Written and spoken step-by-step directions ● Content-focused assessment (not grading for spelling/grammar) ● Graphic organizers


● Non-verbal cues to begin task/remain on task/refocus ● Individual monitoring for understanding/reinforced instruction ● Printed copies of class readings for application of Active Reading Strategies

Students at Risk of Failure

● Exemplars of varied performance levels ● Multimedia presentations ● Tiered/Scaffolded Lessons ● Modeling ● Guided note-taking ● Study Guides ● Differentiated pre-typed class notes and example problems ● Individualized instruction plans ● Words banks ● Reduced choice on assessments ● Preferential seating ● Choice activities ● Modified time requirements ● Modified notes ● Modified lesson, assessment and study guide format ● Modified homework ● Provide an enriched curriculum and activities ● Contracts/management plans ● Open-ended responses ● Project-based learning ● Group activities ● Guided Notes ● Exploration Activities ● Assessment read aloud ● Small group assessments ● Oral questioning assessments to supplement written response ● Pre-writing Structural Supports for extended writing tasks ● Ongoing teacher feedback as part of the writing process ● Interactive Study Guides ● Multi-sensory approach to instruction ● Written and spoken step-by-step directions ● Graphic organizers ● Non-verbal cues to begin task/remain on task/refocus ● Individual monitoring for understanding/reinforced instruction ● Printed copies of class readings for application of Active Reading Strategies


SCOPE AND SEQUENCE COURSE OUTLINE/STUDENT OBJECTIVE

Unit 1: Relationships Between Quantities and Reasoning with Equations (6-8 weeks) Technology:8.1.12.A1,3 21st Century:CRP1,2,4,8,11 Cross Curricular:HS-PS2-2;NJSLSA.R8 The student will be able to:

Reason quantitatively and use units to solve problems.

N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.

N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Interpret the structure of expressions. * A.SSE.1: focus on linear, quadratic, and an introduction to exponential expressions.

A.SSE.1 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. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

Create equations that describe numbers or relationships. ** A.CED.1 is limited to quadratic equations. *** A.CED.4 excludes cases that require extraction of roots or inverse functions.

A.CED.1** Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions.

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.CED.4*** Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Understand solving equations as a process of reasoning and explain the reasoning.

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.

Solve equations and A.REI.3 Solve linear equations and inequalities in one variable, including


inequalities in one variable.

equations with coefficients represented by letters.

Unit 2: Linear Relationships (6-8 weeks) Technology:8.1.12.A1,3 21st Century:CRP1,2,4,8,11 Cross Curricular:HS-PS2-2;NJSLSA.R8 The student will be able to:

Solve systems of equations.

A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Represent and solve equations and inequalities graphically.

A.REI.10

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

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.REI.12

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Create equations that describe numbers or relationships

A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Understand the concept of a function and use function notation.

F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

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.


Interpret functions that arise in applications in terms of the context.

F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★

Analyze functions using different representations.

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 functions.

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.

Unit 3: Expressions and Equations (7-9 weeks) Technology:8.1.12.A1,3 21st Century:CRP1,2,4,8,11 Cross Curricular:HS-PS2-2;NJSLSA.R8 The student will be able to:

Interpret the structure of expressions. * A.SSE.1: focus on linear, quadratic, and an introduction to exponential expressions. ** A.SSE.2: focus on polynomial expressions.

A.SSE.1 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. For example, interpret: P(1+r)n as the product of P and a factor not depending on P.

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).

Write expressions in equivalent forms to solve problems. *** A.SSE.3 is limited to real numbers.

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%.

Perform arithmetic operations on polynomials.

A.APR.1 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.


Create equations that describe numbers or relationships.

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.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Solve equations and inequalities in one variable.

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.

Unit 4: Functions and Modeling (8-10 weeks) Technology:8.1.12.A1,3 21st Century:CRP1,2,4,8,11 Cross Curricular:HS-PS2-2;NJSLSA.R8;HS-PS3-1 The student will be able to:

Understand the relationship between zeros and factors of polynomials.

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.

Extend the properties of exponents to rational exponents.

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.

N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a non-zero rational number and an irrational number is irrational.

Interpret functions that arise in

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


applications in terms of the context. * F.IF.4 and F.IF.5 are limited to linear and quadratic functions.

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.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★

F.IF.6 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. ★

Analyze functions using different representations.

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.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F.IF.8a 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.

** F.IF.9 is limited to linear and quadratic functions.

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.

Build a function that models a relationship between two quantities. *** F.BF.1 is limited to linear and quadratic functions.

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.

Build new functions from existing functions.

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.

Construct and compare linear, quadratic, and

F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.


exponential models and solve problems.

a. Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative another.

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.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Interpret expressions for functions in terms of the situation they model.

F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.

Unit 5: Descriptive Statistics (5-6 weeks) Technology:8.1.12.A1,3 21st Century:CRP1,2,4,8,11 Cross Curricular:NJSLSA.R8; 6.1.4.B.10 I The student will be able to:

Summarize, represent, and interpret data on a single count or measurement variable.

S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

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.

Summarize, represent, and interpret on two categorical and quantitative variables.

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.


S.ID.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a.Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

Interpret linear models. S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

S.ID.9 Distinguish between correlation and causation.


RESOURCES/ACTIVITIES GUIDE

Selected Opportunities for Connections to Mathematical Practices

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. A1u1-4. Create abstract algebraic models of real-world problems then transform them. A1u2- 4. Determine the relationship between domain and range of a function and explain the connection to the inputs and outputs. A1u3-1. Analyze expressions by examining their parts in context. A1u4-3. Create a graph from a verbal description or symbolic representation reasoning about the relationship between two quantities. A1u5-5. Examine several related events to determine if the relationship is correlation or causation. Ex. Watching TV and getting a low score on a test 3. Construct viable arguments and critique the reasoning of others. A1u3-6. Examine the steps in the derivation of the quadratic formula to determine if they are logical and ask questions to improve the clarity. 4. Model with mathematics. 5. Use appropriate tools strategically. A1u4-3. Use graphing technology when graphing complicated functions and to identify the key features of the graph. A1u5-5. Compute the correlation coefficient for a linear model using technology. 6. Attend to precision. A1u1-6. Use precise language when giving descriptions. A1u5-3. Create data displays and give oral or written descriptions, in context, using explicit language. 7. Look for and make use of structure. A1u3-2. Look for structural similarities when writing equivalent expressions. 8. Look for and express regularity in repeated reasoning. A1u1-3. Write general formulas after working with specific examples. A1u2-5. Describing the regularity in the way terms cancel can lead to a general formula for a geometric sequence. A1u4-5 Continually evaluate the results of calculating the rate of change between two quantities over a period of time. All of the content presented in this course has connections to the standards for mathematical practices.


SUGGESTED MATERIALS

1. Texas Instruments TI-84+ graphing calculator. 2. Lesson and help websites:

http://www.khanacademy.org/ http://www.coolmath.com/prealgebra

http://www.webmath.com/index4.html http://www.thatquiz.org/

http://www.ck-12.org/ https://www.desmos.com/calculator

3. Manipulatives. (Algebra tiles, geometric figures, tangrams, etc.) 4. Internet support from publisher: http://connected.mcgraw-hill.com/ provides

interactive text online with activities and videos Resources for Students https://im.kendallhunt.com/HS/index.html Carter, John A., Gilbert J. Cuevas, Roger Day, Carol Malloy, Berchie Holliday, and Beatrice

Moore Luchin. Algebra 1. Common Core Edition ed. Columbus, OH: McGraw-Hill Companies, 2014. Print.

Glencoe Math Hands-On Activities.

https://www.khanacademy.org/math/algebra

Texas Instruments TI-84+ graphing calculator www.mathspace.co Resources for Teachers Carter, John A., Gilbert J. Cuevas, Roger Day, Carol Malloy, Berchie Holliday, and Beatrice

Moore Luchin. Algebra 1. Common Core Edition ed. Columbus, OH: McGraw-Hill Companies, 2014. Print.

Glencoe Math Teacher Center

(Check, eHelp eSolutions, Tools, Tutor, Vocab, Watch, Worksheets, Presentations)

Manipulatives

CK-12.ORG


http://www.khanacademy.org/

http://www.khanacademy.org/

http://www.coolmath.com/prealgebra

http://www.coolmath.com/prealgebra

http://www.webmath.com/index4.html

http://www.webmath.com/index4.html

http://www.thatquiz.org/

http://www.thatquiz.org/

http://www.ck-12.org/

http://www.ck-12.org/

https://www.desmos.com/calculator

http://connected.mcgraw-hill.com/

https://im.kendallhunt.com/HS/index.html

Documents

GOVERNOR LIVINGSTON HIGH SCHOOL MATHEMATICS …