guidelines for standards-based instruction mathematics...secondary mathematics . grades 6 – 12 ....

Guidelines for Standards-Based Instruction

Secondary Mathematics

Grades 6 – 12

Los Angeles Unified School District Instructional Support Services, Secondary

Secondary Mathematics Branch Publication No. SC- XXXX.X (Revised October 2007)

NOT FOR DISTRIBUTION

Guidelines for Standards-Based Instruction

Secondary Mathematics Grades 6 - 12

Secondary Instructional Support Services Secondary Mathematics Branch Publication No. SC-863.14 (Revised July 2008)

Los Angeles Unified School District

Los Angeles Unified School District Secondary Mathematics Branch

Los Angeles Unified School District Instructional Support Services, Secondary Secondary Mathematics Branch 2007 All rights reserved. Permission is granted in advance for reproduction of this document by Los Angeles Unified School District employees. The content must remain unchanged and in its entirety as published by the District. To obtain permission to reproduce the information (text or charts) contained in this document for any commercial purpose, submit the specifics of your request in writing to the Los Angeles Unified School District, Secondary Mathematics Branch, 25th Floor, 333 S. Beaudry Ave, CA 90017, fax: (213) 241-8450. Printed in the United States of America. Publication No. SC-863.14 (Revised July 2008)

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Contents

Page(s)

Foreword…………………………………………………………………………………………………… iv

Acknowledgements…………………………………………………………………………………… v

List of Courses, Middle School…………………………………………………………………… 1

List of Courses, Senior High School…………………………………………………………… 2

Sequence of Math Courses…………………………………………………………….…………… 3

Middle School Courses (Grades 6 – 8)……………………………………………………….. 4

Senior High School Courses (Grades 9 – 12)………………………………..…………….. 22

Appendix ……………………………………………………………………………………………………

68

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Foreword

he Guidelines for Standards-Based Instruction in Secondary Mathematics, 2008 Edition, provides a directory of courses adopted by the Los Angeles Unified School District for students in grades 6-12. It is designed to communicate to stakeholders—students,

parents, school personnel and community representatives— the Mathematics content and skills students should master by the end of each grade level and course. It includes standards, descriptions, prerequisites, academic outcomes, course codes, syllabi, required assessments, and recommended instructional resources that meet the needs of diverse learners. As such, it is a comprehensive resource for implementation of and access to a rigorous standards-based secondary Mathematics curriculum that meets the District’s A through G graduation requirements and provides a gateway to multiple post-secondary options.

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In 1999, the California Department of Education adopted the Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve; this document, revised in 2005, established a curricular platform for instruction and assessment, state-wide. The subsequent Board adoption of content standards for Mathematics supported state and national efforts to improve student achievement. In 2001 the federal government reauthorized the No Child Left Behind Act and in 2003, the State Board of Education required that the class of 2006 to pass the California High School Exit Examination to receive a diploma. These efforts made revision of the guidelines for Mathematics essential. Consequently, the District’s secondary Mathematics program—the curricula, teaching and learning methodology, instructional resources, textbooks, assessments, and related resources—align to current accountabilities and to the academic demands of the 21st century. The Mathematics Guidelines for Standards-Based Instruction, 2008 reflects a philosophy of teaching and learning that is consistent with current research, best practices and national and state accountabilities. It also reflects the changing needs of students and society and supports what students need to know and be able to do to meet the challenges of the evolving global community of the 21st century. David L. Brewer III Jeanne F. Ramos

Director, Secondary Mathematics Superintendent

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v

Acknowledgements, 2008 Edition

he grade level scope and sequence of the courses in this 2008 edition of the Guidelines for Standards-Based Instruction in Mathematics are the result of the collective expertise of the LAUSD Secondary Mathematics Team

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The District extends its gratitude to the following: Kimberly Alafat, John Bailey, Moss Benmosche, Jane Berman, Marcia Bosma, Reginald Brookens, Alfred Chavez, Yee Chung, Felicia Clark, Francis Gesicki, Omer Hassan, Edwin Hayek, Firoza Kanji, Janice Lee, Suyen Moncada-Machado, Mary Olson, Caroline Piangerelli, Rodney Polte, Jeanne Ramos, Gary Scott, Cheryl Serge. Particular gratitude is extended to Peggy Rose, Philip Ogbuehi and Nigel Nisbet, who coordinated this initiative under the guidance of Jeanne Ramos, Director, Secondary Mathematics, and Nader Delnavaz, former Interim Director, Secondary Mathematics.


MATHEMATICS LIST OF COURSES

MIDDLE SCHOOL GRADES 6-8

Course Number Title Abbreviation Grade

Level Page(s)

CORE COURSES 310101 Mathematics 6A MATH 6A 6 310102 Mathematics 6B MATH 6B 6 6

310103 Mathematics 7A MATH 7A 7 310104 Mathematics 7B MATH 7B 7 9

310317 Algebra Readiness A ALGEBRA READ A 8 310318 Algebra Readiness B ALGEBRA READ B 8 12

310301 Algebra 1A ALGEBRA 1A 8-9 310302 Algebra 1B ALGEBRA 1B 8-9 16

INTERVENTION ELECTIVE COURSES 310231 Mathematics Tutorial Lab Middle School A MATH TU LB MS A 6-8 310232 Mathematics Tutorial Lab Middle School B MATH TU LB MS B 6-8 20

173101 ESL Mathematics A ESL MATH A 6-8 173102 ESL Mathematics B ESL MATH B 6-8 21

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MATHEMATICS LIST OF COURSES

SENIOR HIGH SCHOOL GRADES 9-12

Course Number Title Abbreviation Grade

Level Page(s)

CORE C REQUIREMENT COURSES 310301 Algebra 1A ALGEBRA 1A 8-12 310302 Algebra 1B ALGEBRA 1B 8-12 24

310401 Geometry A GEOMETRY A 9-12 310402 Geometry B GEOMETRY B 9-12 27

310303 Algebra 2A ALGEBRA 2A 10-12 310304 Algebra 2B ALGEBRA 2B 10-12 31

310601 Math Analysis A MATH ANALY A 10-12 310602 Math Analysis B MATH ANALY B 10-12 310505 Trigonometry/Math Analysis A TRG/MATH AN A 10-12 310506 Trigonometry/Math Analysis B TRG/MATH AN B 10-12

35

310503 Discrete Mathematics A DISCR MATH A 10-12 310504 Discrete Mathematics B DISCR MATH B 10-12 40

310607 Probability & Statistics A STAT & PROB A 10-12 310608 Probability & Statistics B STAT & PROB B 10-12 41

Advanced Placement Year-Long Courses 310701 AP Calculus A AP CALCULUS A 11-12

44 310702 or 310705

AP Calculus B AP Calculus B

AP CALCULUS B AP CALCULUS B

11-12

310706 AP Calculus C AP CALCULUS C 11-12 48

310609 AP Statistics A AP STATISTICS A 11-12 310610 AP Statistics B AP STATISTICS B 11-12 53

OTHER COURSES 310321 Advanced Applied Math A* ADV APP MATH A 10-12 310322 Advanced Applied Math B* ADV APP MATH B 10-12 58

INTERVENTION ELECTIVE COURSES 173101 ESL Mathematics A ESL MATH A 9-12 173102 ESL Mathematics B ESL MATH B 9-12 61

310209 Essential Standards in Mathematics ESS STAND MATH 10-12 62 310221 Mathematics Tutorial Lab A MATH TUT LAB A 9-12 310222 Mathematics Tutorial Lab B MATH TUT LAB B 9-12 67

*Note: Advanced Applied Math will meet LAUSD graduation requirements as part of two years of high school mathematics for students graduating through 2011 only.


Sequence of Secondary Mathematics Courses 6-12 Possible Pathways

Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12

Pathway A

Math 6 Math 7 Algebra Readiness Algebra 1

Algebra 2 Math Analysis

Pathway B Math 6 Math 7 Algebra 1 Geometry Math

Analysis

AP Calculus

or Discrete

Math or

Probability & Statistics

or AP Statistics

Pathway C Math 6 Algebra 1 Geometry Algebra 2

AP Calculus

or Discrete

Math or

Probability & Statistics

or AP Statistics

AP Calculus or

Discrete Math

or Probability & Statistics

or AP Statistics

CCAAHH SS EE EE

Analysis

Geometry

Algebra 2

Math

Meets the “C” requirement of A-G

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MIDDLE SCHOOL Mathematics

COURSES Grades 6-8

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Middle School Core Mathematics Courses

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Mathematics 6AB (Annual Course – Grade 6) Prerequisite: Mathematics 5AB 310101 Mathematics 6A 310102 Mathematics 6B Course Description The major purpose of this course is to serve as a vehicle by which students will master the four arithmetic operations with whole numbers, positive fractions, positive decimals, and positive and negative integers; and will accurately compute and solve problems. They will apply this knowledge to statistics and probability, and geometry. In this course, students will understand the concept of mean, median, and mode of data sets and how to calculate the range. They will analyze data and sampling processes for possible bias and misleading conclusions; they will use addition and multiplication of fractions routinely to calculate probabilities. Students will work with ratios and proportions. Students will continue their study of geometry, including complementary and supplementary angles, the sum of the angles in a triangle, the concept of the constant pi and its applications to the formulas for area and circumference of the circle. COURSE SYLLABUS

Unit 1

Recommended Focus Standards 6 NS 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. 6 NS 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions(e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction) Scope and Sequence As one of the most critical units, the number sense strand requires that students understand the position of the negative numbers and the geometric effect on the numbers on the number line when a number is subtracted from them. Interpreting and using ratios in different contexts will be essential for showing the relative size of two quantities.

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Unit 2

Recommended Focus Standards 6 NS 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to

show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).

6 NS 1.3 Use proportions to solve problems (e.g., determine the value of N if4

7 21

N= ; find the

length of a side of a polygon similar to a known polygon). Use cross multiplication as a method to solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

6 NS 1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.

6 NS 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.

6 AF 1.1 Write and solve one-step linear equations in one variable. 6 AF 2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of

another quantity. Scope and Sequence In this unit, the students will learn how to use, write and solve ratios and proportions. Equally important, students will develop an understanding of how to solve simple one-variable equations.

Unit 3 Recommended Focus Standards 6 SDAP 2.2 Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a Population. 6 SDAP 2.3 Analyze data displays and explain why the way in which the question was asked might

have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.

6 SDAP 2.4 Identify data that represent sampling errors and explain why the sample (and the display) might be biased. 6 SDAP 2.5 Identify claims based on statistical data and, in simple cases, evaluate the validity of the

claims. 6 SDAP 3.1 Represent all possible outcomes for compound events in an organized way (e.g., tables,

grids, tree diagrams) and express the theoretical probability of each outcome. 6 SDAP 3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages

between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1- P is the probability of an event not occurring.

6 SDAP 3.5 Understand the difference between independent and dependent events.

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Scope and Sequence Students will learn the concept of mean, median, and mode of data and how to calculate the range. In this unit, students will focus their attention on how to analyze data and sampling processes for possible bias nd misleading conclusions.

Unit 4

a

Recommended Focus Standards 6 MG 1.1 Understand the concept of a constant such as π ; know the formulas for the circumference

es and the sum of the angles angle to solve problems involving an unknown angle.

and area of a circle. 6 MG 2.2 Use the properties of complementary and supplementary angl of a triScope and Sequence In the final unit, students will learn how to use a constant such as π in formulas to calculate the circumference and area of a circle. Students will learn that the lengths of the sides of a polygon or the diameter of a circle are used to find the distance around the figure. Students will learn that the volumes of three-dimensional figures can often be found by dividing and combining them into figures whose volume re already known.

ills

s with whole numbers, positive fractions, positive decimals,

an, and mode of data sets and how to calculate the range

probabilities

equations Understand how to use a constant such as π

ds-based quizzes and tests

ative assessments Periodic Assessments

ctional Guide

• Supplemental materials and resources

a Representative Performance Outcomes and SkIn this course, students will know and be able to:

• Master the four arithmetic operationand positive and negative integers

• Understand the concept of mean, medi• Analyze data and sampling processes • Use addition and multiplication to calculate• Interpret and use ratios in different context • Understand how to solve simple one-variable•

Assessments will include: • Teacher designed standar• Projects and group tasks • Teacher designed form•

Texts/Materials • LAUSD Secondary Mathematics Instru• Textbook: District approved materials

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Mathematics 7AB (Annual Course – Grade 7) Prerequisite: Mathematics 6AB

10104 Mathematics 7B

OURSE DESCRIPTION

of tudents

ill know the Pythagorean Theorem and solve problems involving computing a missing side.

ia igh School Exit Exam (CAHSEE), it is essential that students become proficient in the key standards.

COURSE SYLLABUS Unit 1

310103 Mathematics 7A 3 C By the end of grade seven, students will be adept at manipulating numbers and equations and understandthe general principles at work. Students will gain a deeper understanding of rational numbers and their various forms of representation. They will increase their understanding of ratio and proportion and apply this knowledge to topics such as slopes of lines and the change in volume and surface area of basic three-dimensional figures when the scale is changed. Students will make conversions between different unitsmeasurement and compute percents of change and simple and compound interest. In addition, sw Since the seventh grade standards constitute the core content for the mathematics portion of the CalifornH

Recommended Focus Standards

Simplify numerical expressions by applying properties of rational numbers (e7 AF 1.3 .g., identity, , distributive, associative, commutative) and justify the process used. inverse

Scope and Sequence In this unit, students will have an opportunity to transition from prior mathematics through a review of topics from prior grade level standards. Students will then study algebraic expressions, equations and linear relationships.

Unit 2 Recommended Focus Standards

Add, Subtract, multiply, and divide rational numbers (integers, fractions7 NS 1.2 , and terminating

7 NS 1.5 repeating decimal and be

7 NS 1.7 counts, markups, commissions, and profit and compute

7 NS 2.5 number from zero on a number line; and determine the absolute value

numbers.

decimals) and take positive rational numbers to whole-number powers. Know that every rational number is either a terminating or a able to convert terminating decimals into reduced fractions. Solve problems that involve dissimple and compound interest.

7 NS 2.3 Multiply, divide, and simplify rational numbers by using exponent rules. Understand the meaning of the absolute value of a number; interpret the absolute value asthe distance of theof real

Scope and Sequence The focus of this unit is the in-depth study of the connections among properties, operations, and representation of rational numbers.

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Unit 3

Recommended Focus Standards 7NS 1.4 Differentiate between rational and irrational numbers.

Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutio

7 AF 4.1 ns in the context from which they arose, and

7 AF 4.2 istep problems involving rate, average speed, distance, and time or a direct

7 MG 3.3 , in

7 MG 3.4 means about the relationships between the sides and

of the two figures.

verify the reasonableness of the results. Solve multvariation.

Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments andsome situations, empirically verify the Pythagorean theorem by direct measurement.

Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence angles

Scope and Sequence Students will use the knowledge developed in the second unit to study linear relationships includgraphical representations and algebraic representations. Students will gain an understanding of congruency and the relationship of units of measurement and the use of ratios for conversions between measurement systems. N

ing

umber sense skills will be further utilized and developed through the study of the Pythagorean Theorem.

Unit 4 Recommended Focus Standards

Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is alwa

7 AF 3.3 ys the same and know that the ratio (“rise

7 AF 3.4 of e to the plot and

7 AF 4.2 step problems involving rate, average speed, distance, and time or a direct

7 MG 1.3 the solutions; and use

7 MG 3.6 ated in space (e.g., skew lines, the

7 SDAP 1.3 imum, the lower quartile, the , the upper quartile, and the maximum of a data set.

over run”) is called the slope of a graph. Plot the values of quantities whose ratios are always the same (e.g., cost to the number an item, feet to inches, circumference to diameter of a circle). Fit a linunderstand that the slope of the line equals the ratio of the quantities. Solve multivariation. Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units ofdimensional analysis to check the reasonableness of the answer.

Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangularsolids) and describe how two or more objects are relpossible ways three planes might intersect). Understand the meaning of, and be able to compute, the minmedian

Scope and Sequence The foci of this unit are: the representation and interpretation of data sets including quartiles; properties ofthree dimensional figures including surface area, volume and the effect of scale factor

s; and proportional

relationships and their representations including relating the slope of a line to a rate.

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REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS In this course, students will know and be able to:

• Simplify numerical and variable expressions by applying properties • Use the correct order of operations to evaluate expressions • Solve one step linear equations and inequalities in one variable and represent solutions

graphically • Use algebraic terminology correctly • Add, subtract, multiply, divide and simplify rational numbers • Calculate the absolute value of a sum or of a difference • Read, write, and compare rational numbers in scientific notation • Convert fractions, decimals, and percents from one form to another • Interpret absolute value as the distance of a number from zero on the number line • Know that every rational number is either a terminating or a repeating decimal • Solve problems involving discounts, mark-ups, commission, and profit • Compute simple and compound interest and calculate percentage of increase or of decrease • Solve one and two step linear equations and inequalities in one variable • Use multiple representations for linear relationships including tables and graphs • Solve problems involving rate, average speed, distance, and time • Identify congruent and similar figures and their corresponding parts • Determine scale factor, express it as a ratio, and determine how the scale factor affects area and

volume • Determine whether a triangle is right and calculate the missing side of a right triangle • Find area of squares and right triangles • Interpret a box and whisker plot, stem leaf plot, and scatter plot • Calculate volumes and surface areas • Graph linear functions by plotting points • Interpret a graph and its parts • Recognize that slope is a rate of change that is constant in a linear relationship • Use measures expressed as rates or products to solve problems

Assessments will include:

• Teacher designed standards-based quizzes and tests • Projects and group tasks • Teacher designed formative assessments • Periodic Assessments

Texts/Materials LAUSD Secondary Mathematics Instructional Guide

• Textbook: District approved materials • Supplemental materials and resources

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ALGEBRA READINESS AB (Intervention Course – Grade 8) Prerequisite: Mathematics 7AB 310317 Algebra Readiness A 310318 Algebra Readiness B COURSE DESCRIPTION Algebra Readiness is a one-year course designed to adequately prepare 8th grade students for Algebra. According to the Mathematics Framework for California Public Schools (2006 Revised Edition), it is imperative that students master pre-algebraic skills and concepts before they enroll in a course that meets or exceeds the rigor of the content standards for Algebra I adopted by the State board of Education The sixteen targeted standards for Algebra Readiness (thirteen from grade 7 and three from Algebra I) are grouped into the following topics: Operations on Rational Numbers, Equations and Functions, The Coordinate Plane, Graphing Proportional Relationships and Algebra (Introductory Examples). They are purposely limited in number to provide teachers the flexibility and time to rebuild the following foundational skills and concepts that may be missing from earlier grades: Whole Numbers, Operations on Whole Numbers, Rational Numbers, Operations on Rational Numbers, Symbolic Notation, Equations and Functions, and the Coordinate Plane. COURSE SYLLABUS

Unit 1

Recommended Focus Standards 7 NS 1.2 Add, subtract, multiply and divide rational numbers (integers, fractions and terminating

decimals) and take positive rational numbers to whole number powers. 7 NS 2.1 Understand negative whole-number exponents. Multiply and divide expressions

involving exponents with a common base. 7 AF 1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity,

inverse, distributive, associative and commutative) and justify the process used. 7 AF 2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-

number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.

Scope and Sequence Number Sense, Integers, Fractions, and Algebraic Thinking. Students work with whole numbers, integers and their operations. Next they study the properties of fractions.

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Unit 2 Recommended Focus Standards 7 NS 1.2 Add, subtract, multiply and divide rational numbers (integers, fractions and terminating

decimals) and take positive rational numbers to whole number powers. 7 NS 1.3 Convert fractions to decimals and percents and use these representations in estimations,

computations and applications. 7 NS 1.5 Know that every rational number is either a terminating or repeating decimal and be able

to convert terminating decimals into reduced fractions. 7 AF 4.2 Solve multistep problems involving rate, average speed, distance and time or a direct

variation. 7 MG 1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as

products (e.g., person-days) to solve problems; check the units of the solution; and use dimensional analysis to check the reasonableness of the answer.

Algebra I 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. [excluding fractional powers]

Scope and Sequence Fractions, Decimals, Ratios and Proportions. Students transition to operations on fractions and mixed numbers and then move to work on decimals and operations with them then study ratios, rates and proportions.

Unit 3 Recommended Focus Standards 7 NS 1.3 Convert fractions to decimals and percents and use these representations in estimations,

computations and applications. 7 NS 2.1 Understand negative whole-number exponents. Multiply and divide expressions

involving exponents with a common base. 7 AF 1.1 Use variables and appropriate operations to write an expression, an equation, an

inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).

7 AF 1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative and commutative) and justify the process used.

7 AF 4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.

7 AF 4.2 Solve multistep problems involving rate, average speed, distance and time or a direct variation.


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Scope and Sequence Percents and Algebraic Problem Solving. The unit opens with the study of percents, including percent increase and decrease. Students make the transition to algebraic problem solving including word problems including those involving average speed, distance, and time. They solve simple (one and two step) equations and inequalities.

Unit 4

Recommended Focus Standards 7 AF 3.3 Graph linear functions, noting that the vertical change (change in y- value) per unit of

horizontal change (change in x- value) is always the same and know that the ratio ("rise over run") is called the slope of a graph.

7 AF 3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities.

7 MG 3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.


Algebra I 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12. [excluding inequalities]

Algebra I 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. [excluding inequalities] Scope and Sequence Linear Functions and their Graphs, Linear Equations, and the Pythagorean Theorem. Students use the Cartesian coordinate system to graph points and lines and work with proportional relationships and linear functions using the slope to graph linear functions. Additionally students continue algebraic problem solving including multi-step problems.

Note: As students in this course will take the General Mathematics CST examination, in Unit 4 they study additional topics in Statistics, Data Analysis and Probability, Measurement and Geometry and the Real Number System.

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• Demonstrate the concept of place value in whole numbers. • Demonstrate fluency with operations on whole numbers. • Demonstrate fluency with representing fractions, mixed numbers, decimals and percentage. • Demonstrate fluency with operations on positive fractions • Demonstrate fluency with the use of symbols to express verbal information. • Demonstrate fluency in writing and solving simple linear equations. • Demonstrate fluency in plotting points, interpreting ordered pairs from a graph, and interpreting

lengths of horizontal and vertical line segments on a coordinate plane. • Demonstrate fluency in graphing and interpreting relationships of the form y mx =

• Use operations such as taking the opposite, finding the reciprocal, taking a root and the rules of exponents

• Simplify expressions before solving linear equations in one variable. • Solve multi-step problems, including word problems, involving linear equations in one variable

and provide justification for each step

ASSESSMENTS will include:


TEXTS/MATERIALS • LAUSD Secondary Mathematics Instructional Guide • Textbook: District approved materials • Supplemental materials and resources

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ALGEBRA 1 AB (Annual Course – Grade 8 or 9) Prerequisite: Mathematics 7AB 310301 Algebra 1A 310302 Algebra 1B COURSE DESCRIPTION The purpose of this course is to serve as the vehicle by which students make the transition from arithmetic to symbolic mathematical reasoning. It is an opportunity for students to extend and practice logical reasoning in the context of understanding, writing, solving, and graphing problems involving linear and quadratic equations (including systems of two linear equations in two unknowns). In this course, students are expected to demonstrate their ability to extend specific problems and conditions to general assertions about mathematical situations. Additionally, they are expected to justify steps in an algebraic procedure and check algebraic arguments for validity. COURSE SYLLABUS The following are recurring standards in the course: Algebra I 24.0 Students use and know simple aspects of a logical argument. Algebra I 25.0 Students use properties of the number system to judge the validity of results, to justify

each step of a procedure, and to prove or disprove statements.

Unit 1

Recommended Focus Standards Algebra I 2.0 Students understand and use such operations as taking the opposite, finding the

reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

Algebra I 4.0 Students simplify expressions before solving linear equations and inequalities in one variable.

Algebra I 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

Scope and Sequence This unit sets the stage for success in Algebra by providing time to review arithmetic (whole numbers, fractions, decimals, and percents) and proceeds on to cover foundational algebra skills necessary to solve equations. Subsequent to this review, students will proceed to solving equations in one variable (including equations with absolute value).

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Unit 2

Recommended Focus Standards Algebra I 6.0 Students graph a linear equation and compute the x- and y- intercepts. (e.g.,

graph 2 6 ). They are also able to sketch the region defined by linear inequalities (e.g., they sketch the region defined by

4x y+ =2 6 4x y+ < ).

Algebra I 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

Scope and Sequence The focus of this unit is graphing and deriving linear equations using a variety of techniques. The unit also addresses solving inequalities (including absolute value) in one variable.

Unit 3

Recommended Focus Standards Algebra I 9.0 Students solve a system of two linear equations in two variables algebraically and are

able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

Algebra I 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Algebra I 19.0 Students know the quadratic formula and are familiar with its proof by completing the

square. Algebra I 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to

solve quadratic equations. Algebra I 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Algebra I 23.0 Students apply quadratic equations to physical problems, such as the motion of an object

under the force of gravity. Scope and Sequence This unit includes two main foci. Linear relationships are concluded with solving systems of linear equations and inequalities. The students will then learn how to solve quadratic equations and how to interpret the graphs of quadratic functions.

Unit 4 Recommended Focus Standards Algebra I 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve

multistep problems, including word problems, by using these techniques. Algebra I 12.0 Students simplify fractions with polynomials in the numerator and denominator by

factoring both and reducing them to the lowest terms. Algebra I 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these

techniques. Algebra I 14.0 Students solve a quadratic equation by factoring or completing the square.

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Scope and Sequence It is important that students learn how to perform polynomial arithmetic (including factoring) and apply factoring as a technique to solve simple quadratics prior to the CST. After the CST, students will learn the arithmetic of rational expressions and will solve rational equations in one variable. REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS In this course, students will know and be able to:

• Interpret the meaning of variables and variable expressions • Solve linear equations (including proportions) in one variable • Apply the concept of absolute value to simple equations • Graph a linear function by plotting points, using intercepts, and using the slope and y-intercept • Sketch the region defined by a linear inequality • Derive the equation of a line when given a variety of parameters for that line • Solve inequalities in one variable (including those with absolute value) • Solve systems of linear equations by graphing, using substitution, and using elimination • Solve systems of linear inequalities by graphing • Solve quadratic equations by graphing, finding square roots, using the quadratic formula, and

factoring • Simplify and perform arithmetic operations on and with rational expressions • Solve rational equations • Solve application problems using the above techniques




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Middle School Elective Mathematics Courses

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Mathematics Tutorial Lab Middle School AB (Intervention Course Grade 6, 7, or 8) 310231 Mathematics Tutorial Lab Middle School A 310232 Mathematics Tutorial Lab Middle School B COURSE DESCRIPTION Mathematics tutorial lab is an elective mathematics course provided to students as a second course to support the core Mathematics class in grades 6, 7, or Algebra 1. The course is designed to enhance the student’s knowledge of prerequisite skills that are needed to access the grade level mathematics course. COURSE SYLLABUS Grade 6: The standards for the support class for Grade 6 are the set of Grade 5 intervention standards taken from the Mathematics Standards for the Mathematics Intervention Program on p 341 of the Mathematics Framework for California Public Schools. Namely: Grade 5 NS 1.2, 1.5, 2.0, 2.1, 2.5

AF 1.2, 1.3, 1.5 MG 1.1, 1.2, 1.3, 2.1, 2.2 SDAP 1.3, 1.4, 1.5 MR 1.0, 1.1, 1.2, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 3.0, 3.1, 3.2, 3.3

Grade 7: The standards for the support class for Grade 7 are the set of Grade 6 intervention standards taken from the Mathematics Standards for the Mathematics Intervention Program on p 341 of the Mathematics Framework for California Public Schools. Namely: Grade 6 NS 1.1, 1.2, 1.3, 1.4, 2.1, 2.3

AF 1.2, 2.1, 2.2, 2.3 MG 1.2, 1.3, 2.2 SDAP 3.3 MR 1.0, 1.1, 1.2,1.3, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 3.0, 3.1, 3.2, 3.3

Algebra I: The standards for the Algebra 1 support class are a subset of the sixteen targeted Mathematics Standards for the Algebra Readiness Program as laid out on page 340 of Appendix E of the Mathematics Framework for California Public Schools. Namely: Grade 7 NS 1.2, 1.3, 1.5, 2.1

AF 1.1, 1.3, 2.1, 3.3, 3.4, 4.1, 4.2 MG 1.3, 3.3 MR 1.0, 1.1, 1.2, 1.3, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 3.0, 3.1, 3.2, 3.3

ASSESSMENTS will include: • Teacher designed standards-based quizzes and tests • Projects and group tasks • Teacher designed formative assessments

TEXTS/MATERIALS • Textbook: District approved materials • Supplemental materials and resources

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ESL Mathematics (Grade 6, 7 or 8) 173101 ESL Mathematics A 173102 ESL Mathematics B COURSE DESCRIPTION ESL Mathematics is a one-year enabling course for newcomers enrolled in the Structured English Immersion Program. The course is designed to provide an introduction to key language and concepts in mathematics and to build a foundation for standards-based mathematics instruction taught in English. It may be offered under the following conditions:

• As a prerequisite for standards-based sheltered math courses taught using specially designed academic instruction in English (SDAIE).

• As an intervention for English Learners in need of basic language and conceptual development in

mathematics, to be offered during summer or intersession. COURSE SYLLABUS The standards should be taken from the Mathematics Standards for the Mathematics Intervention Program on page 340/341 of the Mathematics Framework for California Public Schools to be appropriate for each student’s needs.



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HIGH SCHOOL Mathematics

COURSES Grades 9-12

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High School Core Mathematics Courses

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ALGEBRA 1 AB (Annual Course – Grade 8 or 9) Prerequisite: Mathematics 7AB 310301 Algebra 1A 310302 Algebra 1B COURSE DESCRIPTION The purpose of this course is to serve as the vehicle by which students make the transition from arithmetic to symbolic mathematical reasoning. It is an opportunity for students to extend and practice logical reasoning in the context of understanding, writing, solving, and graphing problems involving linear and quadratic equations (including systems of two linear equations in two unknowns). In this course, students are expected to demonstrate their ability to extend specific problems and conditions to general assertions about mathematical situations. Additionally, they are expected to justify steps in an algebraic procedure and check algebraic arguments for validity. COURSE SYLLABUS The following are recurring standards in the course: Algebra I 24.0 Students use and know simple aspects of a logical argument. Algebra I 25.0 Students use properties of the number system to judge the validity of results, to justify

each step of a procedure, and to prove or disprove statements.

Unit 1

Recommended Focus Standards Algebra I 2.0 Students understand and use such operations as taking the opposite, finding the

reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

Algebra I 4.0 Students simplify expressions before solving linear equations and inequalities in one variable.

Algebra I 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

Scope and Sequence This unit sets the stage for success in Algebra by providing time to review arithmetic (whole numbers, fractions, decimals, and percents) and proceeds on to cover foundational algebra skills necessary to solve equations. Subsequent to this review, students will proceed to solving equations in one variable (including equations with absolute value).

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Unit 2

Recommended Focus Standards Algebra I 6.0 Students graph a linear equation and compute the x- and y- intercepts. (e.g.,

graph 2 ). They are also able to sketch the region defined by linear inequalities (e.g., they sketch the region defined by

6 4x y+ =2 6 4x y+ < ).

Algebra I 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

Scope and Sequence The focus of this unit is graphing and deriving linear equations using a variety of techniques. The unit also addresses solving inequalities (including absolute value) in one variable.

Unit 3 Recommended Focus Standards Algebra I 9.0 Students solve a system of two linear equations in two variables algebraically and are

able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

Algebra I 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Algebra I 19.0 Students know the quadratic formula and are familiar with its proof by completing the

square. Algebra I 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to

solve quadratic equations. Algebra I 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Algebra I 23.0 Students apply quadratic equations to physical problems, such as the motion of an object

under the force of gravity. Scope and Sequence This unit includes two main foci. Linear relationships are concluded with solving systems of linear equations and inequalities. The students will then learn how to solve quadratic equations and how to interpret the graphs of quadratic functions.

Unit 4

Recommended Focus Standards Algebra I 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve

multistep problems, including word problems, by using these techniques. Algebra I 12.0 Students simplify fractions with polynomials in the numerator and denominator by

factoring both and reducing them to the lowest terms. Algebra I 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these

techniques. Algebra I 14.0 Students solve a quadratic equation by factoring or completing the square.

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Scope and Sequence It is important that students learn how to perform polynomial arithmetic (including factoring) and apply factoring as a technique to solve simple quadratics prior to the CST. After the CST, students will learn the arithmetic of rational expressions and will solve rational equations in one variable. REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS In this course, students will know and be able to:

• Interpret the meaning of variables and variable expressions • Solve linear equations (including proportions) in one variable • Apply the concept of absolute value to simple equations • Graph a linear function by plotting points, using intercepts, and using the slope and y-intercept • Sketch the region defined by a linear inequality • Derive the equation of a line when given a variety of parameters for that line • Solve inequalities in one variable (including those with absolute value) • Solve systems of linear equations by graphing, using substitution, and using elimination • Solve systems of linear inequalities by graphing • Solve quadratic equations by graphing, finding square roots, using the quadratic formula, and

factoring • Simplify and perform arithmetic operations on and with rational expressions • Solve rational equations • Solve application problems using the above techniques




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GEOMETRY AB (Grade 8, 9 or 10) Prerequisite: Algebra 1AB 310401 Geometry A 310402 Geometry B COURSE DESCRIPTION The Geometry skills and concepts developed in this discipline are useful to all students. Aside from learning these skills and concepts, students will develop their ability to construct formal, logical arguments and proofs in geometric settings and problems. Although the curriculum is weighted heavily in favor of plane (synthetic) Euclidean geometry, there is room for placing special emphasis on coordinate geometry and its transformations. An important point to make to students concerning proofs is that while the written proofs presented in class should serve as models for exposition, they should in no way be a model of how proofs are discovered. The perfection of the finished product can easily mislead students into thinking that they must likewise arrive at their proofs with the same apparent ease. Teachers need to make clear to their students that the actual thought process is usually full of false starts and that there are many zigzags between promising leads and dead ends. Only trial and error can lead to a correct proof. This awareness of the nature of solving mathematical problems might lead to a de-emphasis of the rigid requirements on the writing of two-column proofs. Development of geometric intuition. The following geometric constructions are recommended to develop students’ geometric intuition. (In this context construction means “construction with straightedge and compass.”) It is understood that all of them will be proved at some time during the course of study. The constructions that students should be able to do are:

Bisecting an angle Constructing the perpendicular bisector of a line segment Constructing the perpendicular to a line from a point on the line and from a point not on the line Duplicating a given angle Constructing the parallel to a line through a point not on the line Constructing the circumcircle of a triangle Dividing a line segment into n equal parts Constructing the tangent to a circle from a point on the circle Constructing the tangents to a circle from a point not on the circle Locating the center of a given circle

Constructing a regular n-gon on a given circle for n = 3, 4, 5, 6

Use of technology. The availability of good computer software makes the accurate drawing of geometric figures far easier. Such software can enhance the experience of creating the constructions described previously. In addition, the ease of making accurate drawings encourages the formulation and exploration of geometric conjectures. If students do have access to such software, the potential for a more intense mathematical encounter is certainly there.

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COURSE SYLLABUS The following are recurring standards in each unit of the course:

Geometry 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.

Geometry 2.0 Students write geometric proofs, including proofs by contradiction. Geometry 3.0 Students construct and judge the validity of a logical argument and give counterexamples

to disprove a statement. Geometry 16.0 Students perform basic constructions with a straightedge and compass, such as angle

bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.

Geometry 17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.

Unit 1

Recommended Focus Standards Geometry 1.0 Students demonstrate understanding by identifying and giving examples of undefined

terms, axioms, theorems, and inductive and deductive reasoning. Geometry 2.0 Students write geometric proofs, including proofs by contradiction. Geometry 3.0 Students construct and judge the validity of a logical argument and give counterexamples

to disprove a statement. Geometry 7.0 Students prove and use theorems involving the properties of parallel lines cut by a

transversal, the properties of quadrilaterals, and the properties of circles. Geometry 12.0 Students find and use measures of sides and of interior and exterior angles of triangles

and polygons to classify figures and solve problems. Scope and Sequence This introductory unit helps students develop geometric sense by working through the foundations of geometric reasoning and developing geometric ideas connected to the study of polygons, angles and parallel lines. Students are provided with opportunities to perform constructions relating to these topics such as constructing the line parallel to a given line through a point off the line. Students are given opportunities to use reasoning (inductive and deductive), write proofs and disprove statements using logical arguments.

Unit 2

Recommended Focus Standards: Geometry 4.0 Students prove basic theorems involving congruence and similarity. Geometry 7.0 Students prove and use theorems involving the properties of parallel lines cut by a

transversal, the properties of quadrilaterals, and the properties of circles. Geometry 14.0 Students prove the Pythagorean theorem.

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Scope and Sequence The unit begins with the concepts of triangle congruence and similarity. Students then progress to study the properties of quadrilaterals. The unit concludes with the study of the Pythagorean Theorem, specifically its proof. Relevant constructions can be included throughout the unit, such as constructing the circumcircle of a triangle, and students should be given every opportunity to develop logical reasoning skills and mathematical proofs as they apply to each new topic of study. For example, using proof by contradiction to prove conjectures based on the triangle inequality theorem and using coordinate geometry to prove conjectures about triangle congruence or quadrilaterals.

Unit 3

Recommended Focus Standards: Geometry 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area,

volume, lateral area, and surface area of common geometric figures. Geometry 9.0 Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones,

and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.

Geometry 10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.

Geometry 18.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between

them. For example, sin( ) 2 2

tan( ) , and sin ( ) cos ( ) 1cos( )x

x xx x= + =

Geometry 19.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.

Scope and Sequence Students study special right triangles and trigonometric ratios. They then progress to a study of area, volume and surface area and investigate how changes in dimension affect perimeter, area and volume. Relevant constructions can be included throughout the unit. Students should be given every opportunity to develop logical reasoning skills and mathematical proofs as they apply to each new topic of study.

Unit 4

Recommended Focus Standards Geometry 7.0 Students prove and use theorems involving the properties of parallel lines cut by a

transversal, the properties of quadrilaterals, and the properties of circles. Geometry 21.0 Students prove and solve problems regarding relationships among chords, secants,

tangents, inscribed angles, and inscribed and circumscribed polygons of circles. Geometry 22.0 Students know the effect of rigid motions on figures in the coordinate plane and space,

including rotations, translations, and reflections. Scope and Sequence Students study the properties of circles and their relationships with lines and polygons. Students also study Transformations, i.e., Rigid motion in the coordinate plane. Relevant constructions can be included throughout the unit; for example, constructing the tangent to a circle from a point not on the circle, and students should be given every opportunity to develop logical reasoning skills and mathematical proofs as they apply to each new topic of study.

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• Identify and give examples of undefined terms, axioms, theorems, inductive and deductive reasoning

• Construct and judge the validity of a logical argument and give counterexamples to disprove a statement

• Write geometric proofs, including proofs by contradiction • Prove basic theorems involving congruence and similarity • Prove and use theorems involving the properties of parallel lines cut by a transversal, the

properties of quadrilaterals and the properties of circles • Find and use measures of sides and of interior and exterior angles of triangles and polygons to

classify figures and solve problems • Prove and solve problems regarding relationships among chords, secants, tangents, inscribed

angles and inscribed and circumscribed polygons of circles • Prove the Pythagorean Theorem • Derive and solve problems involving perimeter, circumference, area, volume, lateral area and

surface area of common geometric figures • Computer areas of polygons • Know the definitions of the basic trigonometric functions defined by the angles of a right triangle,

and the elementary relationships between them • Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an

angle and a length of a side • Prove theorems involving coordinate geometry • Know the effect of rigid motions on figures in the coordinate plane and space



TEXTS/MATERIALS

• LAUSD Secondary Mathematics Instructional Guide • Textbook: District approved materials • Supplemental materials and resources

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ALGEBRA 2AB (Grade 9, 10 or 11) Prerequisite: Algebra 1AB or Geometry AB 310303 Algebra 2A 310304 Algebra 2B COURSE DESCRIPTION Algebra II expands on the mathematical content of Algebra I and Geometry. There is no single unifying theme. Instead, many new concepts and techniques are introduced that will be basic to more advanced courses in mathematics and the sciences and useful in the workplace. In general terms the emphasis is on abstract thinking skills, the function concept, and the algebraic solution of problems in various content areas. Students who master Algebra II will gain experience with algebraic solutions of problems in various content areas, including the solution of systems of quadratic equations, logarithmic and exponential functions, the binomial theorem, and the complex number system.

COURSE SYLLABUS

Unit 1

Recommended Focus Standards

Algebra II 1.0 Students solve equations and inequalities involving absolute value. Algebra II 2.0 Students solve systems of linear equations and inequalities (in two or three variables) by

substitution, with graphs, or with matrices. Algebra II 3.0 Students are adept at operations on polynomials, including long division. Algebra II 4.0 Students factor polynomials representing the difference of squares, perfect square

trinomials, and the sum and difference of two cubes.

Scope and Sequence

This introductory unit sets the stage for success in Algebra II by providing a connection with the Algebra I concepts of graphing equations, solving systems of equations and inequalities, and working with polynomials. These concepts are expanded to include work with absolute value problems, work with three variables, specialized factoring and Polynomial long division. Concrete applications of simultaneous linear equations and linear programming to problems in daily life should be bought out, though there is no need to emphasize linear programming at this stage. While it would be inadvisable to advocate the use of graphing calculators all the time, such calculators are helpful for graphing regions in connection with linear programming once students are past the initial stage of learning.

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Unit 2

Recommended Focus Standards Algebra II 5.0 Students demonstrate knowledge of how real and complex numbers are related both

arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.

Algebra II 6.0 Students add, subtract, multiply, and divide complex numbers. Algebra II 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with

monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.

Algebra II 8.0 Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.

Algebra II 9.0 Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola

changes as a, b, and c vary in the equation 2

( )y a x b c= − + . Algebra II 10.0 Students graph quadratic functions and determine the maxima, minima, and zeros of the

function. Scope and Sequence This unit begins with previously learned concepts; rational expressions and parabolas. However these concepts are expanded upon to include quadratic (and other polynomial) denominators in the case of rational expressions and complex numbers in the case of quadratic equations (i.e. parabolas). From the beginning of the study of complex numbers, it is important to stress the geometric aspect; for example, the addition of two complex numbers can be shown in terms of a parallelogram. And the key difference between real and complex numbers should be pointed out: The complex numbers cannot be linearly ordered in the same way as real numbers are (the real line).

Unit 3

Recommended Focus Standards Algebra II 11.0 Students prove simple laws of logarithms. 11.1 Students understand the inverse relationship between exponents and logarithms and

use this relationship to solve problems involving logarithms and exponents. 11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.

Algebra II 12.0 Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

Algebra II 18.0 Students use fundamental counting principles to compute combinations and permutations.

Algebra II 19.0 Students use combinations and permutations to compute probabilities.

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Algebra II 20.0 Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.

Scope and Sequence The “big ideas” in this unit are exponential and logarithmic functions, combinations and permutations and probability and statistics. While the probability and statistics standards are not listed as Algebra II standards they comprise 8% of the questions on the Algebra II CST examination.

Unit 4

Recommended Focus Standards Algebra II 15.0 Students determine whether a specific algebraic statement involving rational expressions,

radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.

Algebra II 23.0 Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.

Scope and Sequence Standard 15.0 allows teachers to review several topics; rational expressions, radical expressions and logarithmic and exponential functions through the lens of logical reasoning; i.e., is a given statement sometimes true, always true or never true. Other topics covered include; arithmetic and geometric series, conic sections, functional concepts and mathematical induction.

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• Solve equations and inequalities involving absolute value • Solve systems of linear equations and inequalities in two and three variables • Demonstrate knowledge of how real and complex numbers are related both arithmetically and

graphically • Add, subtract, multiply and divide complex numbers • Be adept at operations on polynomials, including long division • Factor polynomials representing the difference of squares, perfect square trinomials, and the sum

and difference of two cubes • Add, subtract, multiply, divide, reduce and evaluate rational expressions with monomial and

polynomial denominators • Solve and graph quadratic equations • Demonstrate and explain the effect that changing a coefficient has on the graph of quadratic

functions • Graph quadratic functions and determine the maxima, minima, and zeros of the function • Prove simple laws of logarithms • Know the laws of fractional exponents and understand exponential functions • Determine the truth of specific algebraic statements involving rational expressions, radical

expressions, or logarithmic or exponential functions • Derive summation formulae for arithmetic series and for both finite and infinite geometric series. • Know the binomial theorem and use it to expand binomial expressions • Use fundamental counting principles to compute combinations and permutations • Use combinations and permutations to compute probabilities


• Teacher designed standards-based quizzes and tests • Projects and group tasks • Teacher designed formative assessments


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Math Analysis AB or Trigonometry/Math Analysis AB (Grade 10, 11 or 12) Prerequisite: Algebra 2AB 310601 Math Analysis A 310602 Math Analysis B 310505 Trigonometry/Math Analysis A 310506 Trigonometry/Math Analysis B COURSE DESCRIPTION Mathematical Analysis AB is generally taught as Trigonometry during one semester and Mathematical Analysis/Pre-Calculus in the other. COURSE SYLLABUS Trigonometry

Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric functions studied are defined geometrically rather than in terms of algebraic equations, but one of the goals of this course is to acquaint students with a more algebraic viewpoint toward these functions.

Students should have a clear understanding that the definition of the trigonometric functions is made possible by the notion of similarity between triangles.

A basic difficulty confronting students is one of superabundance: There are six trigonometric functions and seemingly an infinite number of identities relating to them. The situation is actually very simple, however. Sine and cosine are by far the most important of the six functions. Students must be thoroughly familiar with their basic properties, including their graphs and the fact that they give the coordinates of every point on the unit circle (Standard 2.0). Moreover, three identities stand out above all others: sin2 x + cos2 x = 1 and the addition formulas of sine and cosine:

Trig 3.0 Students know the identity cos2(x) + sin2(x) = 1: 3.1. Students prove that this identity is equivalent to the Pythagorean theorem (i.e.,

students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity).

3.2. Students prove other trigonometric identities and simplify others by using the identity cos2(x) + sin2(x) = 1. For example, students use this identity to prove that sec2(x) = tan2(x) + 1.

Trig 10.0 Students demonstrate an understanding of the addition formulas for sines and cosines and

their proofs and can use those formulas to prove and/or simplify other trigonometric identities.

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Students should know the proofs of these addition formulas. An acceptable approach is to use the fact that the distance between two points on the unit circle depends only on the angle between them. Thus, suppose that angles a and b satisfy 0 < a < b, and let A and B be points on the unit circle making angles a and b with the positive x-axis. Then A = (cos a, sin a), B = (cos b, sin b), and the distance d(A, B) from A to B satisfies the equation:

d(A, B)2 = (cos b − cos a)2 + (sin b − sin a)2.

On the other hand, the angle from A to B is (b − a), so that the distance from the point C = (cos(b − a), sin(b − a)) to (1, 0) is also d(A, B) because the angle from C to (1, 0) is (b − a) as well. Thus:

d(A, B)2 = (cos(b − a) − 1)2 + sin2(b − a).

Equating the two gives the formula:

cos(b − a) = cos a cos b + sin a sin b.

From this formula both the sine and cosine addition formulas follow easily.

Students should also know the special cases of these addition formulas in the form of half-angle and double-angle formulas of sine and cosine (Standard 11.0). These are important in advanced courses, such as calculus. Moreover, the addition formulas make possible the rewriting of trigonometric sums of the form A sin(x) + B cos(x) as C sin(x + D) for suitably chosen constants C and D, thereby showing that such a sum is basically a displaced sine function. This fact should be made known to students because it is important in the study of wave motions in physics and engineering. Students should have a moderate amount of practice in deriving trigonometric identities, but identity proving is no longer a central topic.

Of the remaining four trigonometric functions, students should make a special effort to get to know

tangent, its domain of definition( ,2 2

)π π− , and its graph (Standard 5.0). The tangent function

naturally arises because of the standard:

Trig 7.0 Students know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.

Because trigonometric functions arose historically from computational needs in astronomy, their practical applications should be stressed (Standard 19.0). Among the most important are:

Trig 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems.

Trig 14.0 Students determine the area of a triangle, given one angle and the two adjacent sides. These formulas have innumerable practical consequences.

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Complex numbers can be expressed in polar forms with the help of trigonometric functions (Standard 17.0). The geometric interpretations of the multiplication and division of complex numbers in terms of the angle and modulus should be emphasized, especially for complex numbers on the unit circle. Mention should be made of the connection between the nth roots of 1 and the vertices of a regular n-gon inscribed in the unit circle: Trig 18.0 Students know DeMoivre’s theorem and can give nth roots of a complex number given in

polar form. Mathematical Analysis

This discipline combines many of the trigonometric, geometric, and algebraic techniques needed to prepare students for the study of calculus and other advanced courses. It also brings a measure of closure to some topics first brought up in earlier courses, such as Algebra II. The functional viewpoint is emphasized in this course. Mathematical Induction

The eight standards are fairly self-explanatory. However, some comments on four of them may be of value. The first is mathematical induction:

MA 3.0 Students can give proofs of various formulas by using the technique of mathematical induction.

This basic technique was barely hinted at in Algebra II; but at this level, to understand why the technique works, students should be able to use the technique fluently and to learn enough about the natural numbers. They should also see examples of why the step to get the induction started and the induction step itself are both necessary. Among the applications of the technique, students should be able to prove by induction the binomial theorem and the formulas for the sum of squares and cubes of the first n integers.

Roots of Polynomials

Roots of polynomials were not studied in depth in Algebra II, and the key theorem about them was not mentioned:

MA 4.0 Students know the statement of, and can apply, the fundamental theorem of algebra.

This theorem should not be proved here because the most natural proof requires mathematical techniques well beyond this level. However, there are “elementary” proofs that can be made accessible to some of the students. In a sense this theorem justifies the introduction of complex numbers. An application that should be mentioned and proved on the basis of the fundamental theorem of algebra is that for polynomials with real coefficients, complex roots come in conjugate pairs. Consequently, all polynomials with real coefficients can be written as the product of real quadratic polynomials. The quadratic formula should be reviewed from the standpoint of this theorem.

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Conic Sections

The third area is conic sections (see Standard 5.0). Students learn not only the geometry of conic sections in detail (e.g., major and minor axes, asymptotes, and foci) but also the equivalence of the algebraic and geometric definitions (the latter refers to the definitions of the ellipse and hyperbola in terms of distances to the foci and the definition of the parabola in terms of distances to the focus and directrix). A knowledge of conic sections is important not only in mathematics but also in classical physics.

Limits

Finally, students are introduced to limits:

MA 8.0 Students are familiar with the notion of the limit of a sequence and the limit of a function as the independent variable approaches a number or infinity. They determine whether certain sequences converge or diverge.

This standard is an introduction to calculus. The discussion should be intuitive and buttressed by much numerical data. The calculator is useful in helping students explore convergence and divergence and guess the limit of sequences. If desired, the precise definition of limit can be carefully explained; and students may even be made to memorize it, but it should not be emphasized. For example, students can be

taught to prove why for linear functions ( ), lim ( ) ( )f x f x f a ax a

= ∀→

, but it is more likely a ritual of

manipulating ε’s and δ’s in a special situation than a real understanding of the concept. The time can probably be better spent on other proofs (e.g., mathematical induction).

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REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS In this course, students will:

• Know the identity 2 2

cos ( ) sin ( ) 1x x+ =

• Demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use these formulas to prove and/or simplify other trigonometric identities

• Know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line

• Know the law of sines and the law of cosines and apply those laws to solve problems • Determine the area of a triangle, given one angle and two adjacent sides • Know DeMoivre’s theorem and can give nth roots of a complex number given in polar form • Give proofs of various formulas by using the technique of mathematical induction • Know the statement of, and can apply, the fundamental theorem of algebra • Be familiar with the notion of the limit if a sequence and the limit of a function as the

independent variable approaches a number or infinity. They determine whether certain sequences converge or diverge




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Discrete Mathematics AB (Grade 11 or 12) Prerequisite: Math Analysis AB 310503 Discrete Mathematics A 310504 Discrete Mathematics B COURSE DESCRIPTION Discrete mathematics is centered around elementary logic, methods of proof, set theory, basic counting, mathematical induction, recursion, matrices, optimization techniques and their applications in computer science. INTENDED LEARNING OUTCOMES

• Utilize elementary truth tables, operations, and logical equivalences to prove compound and conditional statements.

• Understand and exploit the contrapositive, converse, and inverse forms of a conditional statement.

• Utilize the universal and existential quantifiers to construct valid propositional statements. • Form valid arguments with compound and quantified statements using the modus ponens and

modus tollens principles. • Understand the difference between and requirements of Necessary and Sufficient conditions. • Correctly parse and understand multiply quantified statements • Utilize the method of direct proof, proof by contradiction, and proof by contraposition. • Understand the technique of counter-examples and how it relates to mathematical ideas. • Properly set up and carry out mathematical induction arguments. • Use proper notation for set theory. • Find the union, intersection, difference, complement, and product of two sets. • Prove basic set identities. • Use matrices to solve systems of linear equations. • Understand and use permutations and combinations to solve problems in counting and

probability. • Apply the inclusion-exclusion principle to count the number of elements in a union of sets. • Understand recursion and recursively defined sequences. • Solve basic recurrence relations by iteration.




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Probability & Statistics AB (Grade 10, 11 or 12) Prerequisite: Algebra 2AB 310607 Probability & Statistics A 310608 Probability & Statistics B COURSE DESCRIPTION This discipline is an introduction to the study of probability, interpretation of data, and fundamental statistical problem solving. Mastery of this academic content will provide students with a solid foundation in probability and facility in processing statistical information. COURSE SYLLABUS Some of the topics addressed in this course review material found in the standards for the earlier grades and reflect that this content should not disappear from the curriculum. These topics include the material with respect to the common concepts of mean, median, and mode and to the various display methods in common use, as stated in these standards:

P&S 6.0: Students know the definitions of the mean, median, and mode of a distribution of data and can compute each in particular situations.

P&S 8.0: Students organize and describe distributions of data by using a number of different

methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

In the early grades students also receive an introduction to probability at a basic level. The next topic will expand on this base so that students can find probabilities for multiple discrete events in various combinations and sequences. The standards in Algebra II related to permutations and combinations and the fundamental counting principles are also reflective of the content in these standards:

P&S 1.0: Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces.

P&S 2.0: Students know the definition of conditional probability and use it to solve for

probabilities in finite sample spaces. P&S 3.0: Students demonstrate an understanding of the notion of discrete random variables by

using them to solve for the probabilities of outcomes, such as the probability of the occurrence of five heads in 14 coin tosses.

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The most substantial new material in this discipline is found in Standard 4.0:

P&S 4.0: Students are familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families.

Instruction typically flows from the counting principles for discrete binomial variables to the rules for elaborating probabilities in binomial distributions. The fact that these probabilities are simply the terms in a binomial expansion provides a strong link to Algebra II and the binomial theorem. From this base, basic probability topics can be expanded into the treatment of these standard distributions. In the binomial case students should now be able to define the probability for a range of possible outcomes for a set of events based on a single-event probability and thus to develop better understanding of probability and density functions. The normal distribution, which is the limiting form of a binomial distribution, is typically introduced next. Students are not to be expected to integrate this distribution, but they can answer probability questions based on it by referring to tabled values. Students need to know that the mean and the standard deviation are parameters for this distribution. Therefore, it is important to understand variance, based on averaged squared deviation, as an index of variability and its importance in normal distributions, as stated in these standards:

P&S 5.0: Students determine the mean and the standard deviation of a normally distributed random variable.

P&S 7.0: Students compute the variance and the standard deviation of a distribution of data.

Standard 4.0 also includes exponential distributions with applications, for example, in lifetime of service and radioactive decay problems. Including this distribution acquaints students with probability calculations for other types of processes. Here, students learn that the distribution is defined by a scale parameter, and they learn simple probability computations based on this parameter.

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REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS In this course, students will:

• Know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces

• Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces

• Demonstrate an understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of five heads in 14 coin tosses

• Be familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families

• Determine the mean and standard deviation of a normally distributed random variable • Know the definitions of the mean, median and mode of a distribution of data and can compute

each in particular situations • Students compute the variance and the standard deviation of a distribution of data • Organize and describe distributions of data by using a number of different methods, including

frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots




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AP Calculus AB (Grade 11 or 12) Prerequisite: Math Analysis AB Before studying calculus, all students should complete four years of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise-defined functions. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on) and know the values of the trigonometric functions at the numbers 0, , , , ,

6 4 3 2π π π π and their multiples.

310701 AP Calculus A 310702 or 310705

AP Calculus B

COURSE DESCRIPTION Calculus AB is primarily concerned with developing the students’ understanding of the concepts of calculus and providing experience with its methods and applications. The courses emphasize a multi-representational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The connections among these representations also are important. COURSE SYLLABUS I. Functions, Graphs, and Limits

Analysis of graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

Limits of functions (including one-sided limits) • An intuitive understanding of the limiting process • Calculating limits using algebra • Estimating limits from graphs or tables of data

Asymptotic and unbounded behavior • Understanding asymptotes in terms of graphical behavior • Describing asymptotic behavior in terms of limits involving infinity • Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth) Continuity as a property of functions • An intuitive understanding of continuity (The function values can be made as close as desired by taking

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sufficiently close values of the domain). • Understanding continuity in terms of limits • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem) II. Derivatives

Concept of the derivative • Derivative presented graphically, numerically, and analytically • Derivative interpreted as an instantaneous rate of change • Derivative defined as the limit of the difference quotient • Relationship between differentiability and continuity

Derivative at a point • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation • Instantaneous rate of change as the limit of average rate of change • Approximate rate of change from graphs and tables of values

Derivative as a function • Corresponding characteristics of graphs of ƒ and 'f • Relationship between the increasing and decreasing behavior of ƒ and the sign of 'f • The Mean Value Theorem and its geometric interpretation • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

Second derivatives • Corresponding characteristics of the graphs of ƒ, 'f , and ''f • Relationship between the concavity of ƒ and the sign of ''f • Points of inflection as places where concavity changes

Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity • Optimization, both absolute (global) and relative (local) extrema • Modeling rates of change, including related rates problems • Use of implicit differentiation to find the derivative of an inverse function • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations

Computation of derivatives • Knowledge of derivatives of basic functions, including power, exponential,

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logarithmic, trigonometric, and inverse trigonometric functions • Derivative rules for sums, products, and quotients of functions • Chain rule and implicit differentiation III. Integrals

Interpretations and properties of definite integrals • Definite integral as a limit of Riemann sums • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

' ( ) ( ) ( )b

f x dx f b f aa∫ = −

• Basic properties of definite integrals (examples include additivity and linearity).

Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and accumulated change from a rate of change.

Fundamental Theorem of Calculus • Use of the Fundamental Theorem to evaluate definite integrals • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined

Techniques of antidifferentiation • Antiderivatives following directly from derivatives of basic functions • Antiderivatives by substitution of variables (including change of limits for definite integrals)

Applications of antidifferentiation • Finding specific antiderivatives using initial conditions, including applications to motion along a line • Solving separable differential equations and using them in modeling (including the study of the equation y’ = ky and exponential growth)

Numerical approximations to definite integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values

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• Work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations

• Understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems

• Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems

• Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus

• Communicate mathematics and explain solutions to problems both verbally and in written sentences

• Model a written description of a physical situation with a function, a differential equation, or an integral

• Use technology to help solve problems, experiment, interpret results, and support conclusions • Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of

measurement • Develop an appreciation of calculus as a coherent body of knowledge and as a human

accomplishment


• Teacher designed standards-based quizzes and tests • Projects and group tasks • Teacher designed formative assessments • College Board AP Calculus AB Examination

TEXTS/MATERIALS


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AP Calculus BC (Grade 11 or 12) Prerequisite: Math Analysis AB Before studying calculus, all students should complete four years of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise-defined functions. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on) and know the values of the trigonometric functions at the numbers 0, , , , ,

6 4 3 2π π π π and their multiples.

310702 or 310705

AP Calculus B

310706 AP Calculus C COURSE DESCRIPTION Calculus BC is primarily concerned with developing the students’ understanding of the concepts of calculus and providing experience with its methods and applications. The courses emphasize a multi-representational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The connections among these representations also are important. Calculus BC is an extension rather than an enhancement of Calculus AB; common topics require a similar depth of understanding. COURSE SYLLABUS Note: The course syllabus for Calculus BC includes all Calculus AB topics. Entirely new topics are marked with an asterisk (*), whereas new bullets within existing topics are marked with a plus sign (+). I. Functions, Graphs, and Limits

Analysis of graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

Limits of functions (including one-sided limits) • An intuitive understanding of the limiting process • Calculating limits using algebra • Estimating limits from graphs or tables of data Asymptotic and unbounded behavior • Understanding asymptotes in terms of graphical behavior • Describing asymptotic behavior in terms of limits involving infinity

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• Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth)

Continuity as a property of functions • An intuitive understanding of continuity (The function values can be made as close as desired by taking sufficiently close values of the domain). • Understanding continuity in terms of limits • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)

*Parametric, polar, and vector functions The analysis of planar curves includes those given in parametric form, polar form, and vector form. II. Derivatives

Concept of the derivative • Derivative presented graphically, numerically, and analytically • Derivative interpreted as an instantaneous rate of change • Derivative defined as the limit of the difference quotient • Relationship between differentiability and continuity

Derivative at a point • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation • Instantaneous rate of change as the limit of average rate of change • Approximate rate of change from graphs and tables of values

Derivative as a function • Corresponding characteristics of graphs of ƒ and 'f • Relationship between the increasing and decreasing behavior of ƒ and the sign of 'f • The Mean Value Theorem and its geometric interpretation • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

Second derivatives • Corresponding characteristics of the graphs of ƒ, 'f , and ''f • Relationship between the concavity of ƒ and the sign of ''f • Points of inflection as places where concavity changes

Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity +Analysis of planar curves given in parametric form, polar form, vector form, including velocity and acceleration

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• Optimization, both absolute (global) and relative (local) extrema • Modeling rates of change, including related rates problems • Use of implicit differentiation to find the derivative of an inverse function • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations +Numerical solution of differential equations using Euler’s method +L’Hopital’s Rule, including its use in determining limits and convergence of improper integrals and series.

Computation of derivatives • Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions • Derivative rules for sums, products, and quotients of functions • Chain rule and implicit differentiation +Derivatives of parametric, polar and vector functions

III. Integrals

Interpretations and properties of definite integrals • Definite integral as a limit of Riemann sums • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

' ( ) ( ) ( )b

f x dx f b f aa∫ = −

• Basic properties of definite integrals (examples include additivity and linearity)

*Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve (including a curve given in parametric form).

Fundamental Theorem of Calculus • Use of the Fundamental Theorem to evaluate definite integrals • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined

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Techniques of antidifferentiation • Antiderivatives following directly from derivatives of basic functions +Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (non-repeating linear factors only) +Improper integrals (as limits of definite integrals)

Applications of antidifferentiation • Finding specific antiderivatives using initial conditions, including applications to motion along a line • Solving separable differential equations and using them in modeling (including the study of the equation y’ = ky and exponential growth) +Solving logistic differential equations and using them in modeling

Numerical approximations to definite integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values *IV. Polynomial Approximations and Series

*Concept of series A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.

*Series of constants +Motivating examples, including decimal expansion +Geometric series with applications +The harmonic series +Alternating series with error bound +Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series +The ratio test for convergence and divergence +Comparing series to test for convergence or divergence

*Taylor series +Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve) +Maclaurin series and the general Taylor series centered at x = a

+Maclaurin series for the functions 1

, sin , cos , and 1

xe x x

x−

+Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series +Functions defined by power series +Radius and interval of convergence of power series +Lagrange error bound for Taylor polynomials

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• Work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations

• Understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems

• Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems

• Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus

• Communicate mathematics and explain solutions to problems both verbally and in written sentences

• Model a written description of a physical situation with a function, a differential equation, or an integral

• Use technology to help solve problems, experiment, interpret results, and support conclusions. • Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of

measurement • Develop an appreciation of calculus as a coherent body of knowledge and as a human

accomplishment


• Teacher designed standards-based quizzes and tests • Projects and group tasks • Teacher designed formative assessments • College Board AP Calculus BC Examination

TEXTS/MATERIALS


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AP Statistics AB (Grade 11 or 12) Prerequisite: Algebra 2AB 310609 AP Statistics A 310610 AP Statistics B COURSE DESCRIPTION The purpose of the AP course in statistics is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: 1. Exploring Data: Describing patterns and departures from patterns 2. Sampling and Experimentation: Planning and conducting a study 3. Anticipating Patterns: Exploring random phenomena using probability and simulation 4. Statistical Inference: Estimating population parameters and testing hypotheses COURSE SYLLABUS I. Exploring Data: Describing patterns and departures from patterns Exploratory analysis of data makes use of graphical and numerical techniques to study patterns and departures from patterns. Emphasis should be placed on interpreting information from graphical and numerical displays and summaries. A. Constructing and interpreting graphical displays of distributions of univariate data (dotplot, stemplot, histogram, cumulative frequency plot) 1. Center and spread 2. Clusters and gaps 3. Outliers and other unusual features 4. Shape B. Summarizing distributions of univariate data 1. Measuring center: median, mean 2. Measuring spread: range, interquartile range, standard deviation 3. Measuring position: quartiles, percentiles, standardized scores (z-scores) 4. Using boxplots 5. The effect of changing units on summary measures C. Comparing distributions of univariate data (dotplots, back-to-back stemplots, parallel boxplots) 1. Comparing center and spread: within group, between group variation 2. Comparing clusters and gaps 3. Comparing outliers and other unusual features 4. Comparing shapes

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D. Exploring bivariate data 1. Analyzing patterns in scatterplots 2. Correlation and linearity 3. Least-squares regression line 4. Residual plots, outliers, and influential points 5. Transformations to achieve linearity: logarithmic and power transformations E. Exploring categorical data 1. Frequency tables and bar charts 2. Marginal and joint frequencies for two-way tables 3. Conditional relative frequencies and association 4. Comparing distributions using bar charts II. Sampling and Experimentation: Planning and conducting a study Data must be collected according to a well-developed plan if valid information on a conjecture is to be obtained. This plan includes clarifying the question and deciding upon a method of data collection and analysis. A. Overview of methods of data collection 1. Census 2. Sample survey 3. Experiment 4. Observational study B. Planning and conducting surveys 1. Characteristics of a well-designed and well-conducted survey 2. Populations, samples, and random selection 3. Sources of bias in sampling and surveys 4. Sampling methods, including simple random sampling, stratified random sampling, and cluster sampling C. Planning and conducting experiments 1. Characteristics of a well-designed and well-conducted experiment 2. Treatments, control groups, experimental units, random assignments, and replication 3. Sources of bias and confounding, including placebo effect and blinding 4. Completely randomized design 5. Randomized block design, including matched pairs design D. Generalizability of results and types of conclusions that can be drawn from observational studies, experiments, and surveys

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III. Anticipating Patterns: Exploring random phenomena using probability and simulation Probability is the tool used for anticipating what the distribution of data should look like under a given model. A. Probability 1. Interpreting probability, including long-run relative frequency interpretation 2. “Law of Large Numbers” concept 3. Addition rule, multiplication rule, conditional probability, and independence 4. Discrete random variables and their probability distributions, including binomial and geometric 5. Simulation of random behavior and probability distributions 6. Mean (expected value) and standard deviation of a random variable, and linear transformation of a random variable B. Combining independent random variables 1. Notion of independence versus dependence 2. Mean and standard deviation for sums and differences of independent random variables C. The normal distribution 1. Properties of the normal distribution 2. Using tables of the normal distribution 3. The normal distribution as a model for measurements D. Sampling distributions 1. Sampling distribution of a sample proportion 2. Sampling distribution of a sample mean 3. Central Limit Theorem 4. Sampling distribution of a difference between two independent sample proportions 5. Sampling distribution of a difference between two independent sample means 6. Simulation of sampling distributions 7. t-distribution 8. Chi-square distribution IV. Statistical Inference: Estimating population parameters and testing hypotheses Statistical inference guides the selection of appropriate models. A. Estimation (point estimators and confidence intervals) 1. Estimating population parameters and margins of error 2. Properties of point estimators, including unbiasedness and variability 3. Logic of confidence intervals, meaning of confidence level and confidence intervals, and properties of confidence intervals 4. Large sample confidence interval for a proportion 5. Large sample confidence interval for a difference between two proportions 6. Confidence interval for a mean 7. Confidence interval for a difference between two means (unpaired and paired)

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8. Confidence interval for the slope of a least-squares regression line B. Tests of significance 1. Logic of significance testing, null and alternative hypotheses; p-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of power 2. Large sample test for a proportion 3. Large sample test for a difference between two proportions 4. Test for a mean 5. Test for a difference between two means (unpaired and paired) 6. Chi-square test for goodness of fit, homogeneity of proportions, and independence (one- and two-way tables) 7. Test for the slope of a least-squares regression line REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS In this course, students will:

• Solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions and understand the simplifications that arise with independent events

• Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces

• Demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of five heads in 14 coin tosses

• Understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable

• Know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable

• Know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable

• Be familiar with the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families

• Determine the mean and standard deviation of a normally distributed random variable • Know the central limit theorem and can use it to obtain approximations for probabilities in

problems of finite sample spaces in which the probabilities are distributed binomially • Know the definitions of the mean, median and mode of a distribution of data and can compute

each of them in particular situations • Compute the variance and the standard deviation of a distribution of data • Find the line of best fit to a given distribution of data by using least squares regression

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• Know what the correlation coefficient of two variables means and are familiar with the coefficient’s properties

• Organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots

• Be familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic

• Know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution

• Determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error

• Determine the P-value for a statistic for a simple random sample from a normal distribution • Be familiar with the chi-square distribution and chi-square test and understand their uses


• Teacher designed standards-based quizzes and tests • Projects and group tasks • Teacher designed formative assessments • College Board AP Statistics Examination

TEXTS/MATERIALS


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Advanced Applied Math AB (Grade 11 or 12) Prerequisite: Algebra 1AB 310321 Advanced Applied Math A 310322 Advanced Applied Math B COURSE DESCRIPTION This course is designed for students whose past performance in mathematics places them in jeopardy of not meeting the high school graduation mathematics requirement if they take Geometry. Advanced Applied Mathematics will meet the district graduation requirements for students graduating through 2011, however, Advanced Applied Mathematics does NOT satisfy the “C” requirement of A-G. Schools may enroll students in the Advanced Applied Mathematics course based upon the decision of the parent, counselor, mathematics department chairperson, and the site administrator under the guidelines set by the Local District Superintendent. COURSE SYLLABUS The course is to be taught using the Geometry course materials with an emphasis on those standards that stress application. REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS Below is a modified set of Geometry standards that reflect the emphasis on application: Geometry 1.0 Students demonstrate understanding by identifying and giving examples of undefined

terms, axioms, theorems, and inductive and deductive reasoning.

Geometry 7.0 (Modified) Students use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.

Geometry 8.0 Students know, derive and solve problems involving the perimeter, circumference, area,

volume, lateral area, and surface area of common geometric figures. Geometry 9.0 (Modified) Students compute the volumes and surface areas of prisms, pyramids,

cylinders, cones, and spheres.

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Geometry 10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms and trapezoids.

Geometry 11.0 Students determine how changes in dimension affect the perimeter, area and volume of

common geometric figures and solids. Geometry 12.0 Students find and use measures of sides and of interior and exterior angles of triangles

and polygons to classify figures and solve problems. Geometry 13.0 (Modified) Students solve problems concerning the relationships between angles in

polygons by using properties of complementary, supplementary, vertical, and exterior angles.

Geometry 15.0 Students use the Pythagorean Theorem to determine distance and find missing lengths of

sides of right triangles. Geometry 17.0 (Modified) Students solve problems in coordinate geometry, using the midpoint of a line

segment, the distance formula, and various forms of equations of lines. Geometry 18.0 (Modified) Students know the definitions of the basic trigonometric functions. Geometry 19.0 Students use trigonometric functions to solve for an unknown length of a side of a right

triangle, given an angle and a length of a side. Geometry 20.0 Students know and are able to use angle and side relationships in problems with special

right triangles, such as . 30 , 60 , and 90 triangles and 45 , 45 and 90 triangles° ° ° ° ° °

Geometry 21.0 (Modified) Students solve problems regarding relationships among chords, secants,

tangents, inscribed angles, and inscribed and circumscribed polygons of circles. Geometry 22.0 (Modified) Students know the effect of rigid motions on figures in the coordinate plane,

including rotations, translations, and reflections.




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High School Elective Mathematics Courses

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ESL Mathematics (Grade 9 - 12) 173101 ESL Mathematics A 173102 ESL Mathematics B COURSE DESCRIPTION ESL Mathematics is a one-year enabling course for newcomers enrolled in the Structured English Immersion Program. The course is designed to provide an introduction to key language and concepts in mathematics and to build a foundation for standards-based mathematics instruction taught in English. It may be offered under the following conditions:

• As a prerequisite for standards-based sheltered math courses taught using specially designed academic instruction in English (SDAIE).

• As an intervention for English Learners in need of basic language and conceptual development in

mathematics, to be offered during summer or intersession. COURSE SYLLABUS The standards should be taken from the Mathematics Standards for the Mathematics Intervention Program on pages 340/341 of the Mathematics Framework for California Public Schools so as to be appropriate for the student’s needs.



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Essential Standards in Mathematics (Grade 10, 11 or 12) 310209 Essential Standards in Mathematics COURSE DESCRIPTION This one semester course is designed as a preparation for the CAHSEE with instruction (review) in pre-algebraic and introductory algebra concepts and skills that are not yet understood well enough by the student to earn a passing score. It is intended for either 11th or 12th grade students who did not pass the CAHSEE or 10th grade students whose past performance in mathematics places them in jeopardy of not passing. The course will focus on meeting the California content standards in mathematics in some selected standards from Algebra 1 and strands from Grades 6 and 7 including Number Sense; Algebra and Function; Measurement and Geometry; Statistics, Data Analysis and Probability; and Mathematical Reasoning. COURSE SYLLABUS The course covers essential standards from middle school mathematics and a selection of Algebra 1 standards. Algebra topics include much of the concepts in the first part of Algebra 1A course: understanding, writing, solving, and graphing linear equations and inequalities, including systems of linear equations in two unknowns, and the solving of problems utilizing algebraic techniques. REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS Below is the set of CAHSEE standards with strikethroughs to indicate portions of standards that are not covered: 6 SDAP 1.0 Students compute and analyze statistical measurements for data sets: 1.1 Compute the range, mean, median, and mode of data sets.

6SDAP 2.0 Students use data samples of a population and describe the characteristics and limitations of the samples:

2.5 Identify claims based on statistical data and, in simple cases, evaluate the validity of the claims

6 SDAP 3.0 Students determine theoretical and experimental probabilities and use these to make

predictions about events 3.1 Represent all possible outcomes for compound events in an organized way (e.g., tables,

grids, tree diagrams) and express the theoretical probability of each outcome. 3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages

between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1-P is the probability of an event not occurring.

3.5 Understand the difference between independent and dependent events.

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7 NS 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms:

1.1 Read, write, and compare rational numbers in scientific notation (positive and negative

powers of 10) with approximate numbers using scientific notation. 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating

decimals) and take positive rational numbers to whole-number powers. 1.3 Convert fractions to decimals and percents and use these representations in estimations,

computations, and applications. 1.4 Differentiate between rational and irrational numbers. 1.5 Know that every rational number is either a terminating or repeating decimal and be able

to convert terminating decimals into reduced fractions. 1.6 Calculate the percentage of increases and decreases of a quantity. 1.7 Solve problems that involve discounts, markups, commissions, and profit and compute

simple and compound interest.

7 NS 2.0 Students use exponents, powers, and roots and use exponents in working with fractions: 2.1 Understand negative whole-number exponents. Multiply and divide expressions involving

exponents with a common base. 2.2 Add and subtract fractions by using factoring to find common denominators. 2.3 Multiply, divide, and simplify rational numbers by using exponent rules. 2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect

square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why.

2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.

7AF 1.0 Students express quantitative relationships by using algebraic terminology, expressions,

equations, inequalities, and graphs: 1.1 Use variables and appropriate operations to write an expression, an equation, an

inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).

1.2 Use the correct order of operations to evaluate algebraic expressions such as . 23(2 5)x + 1.5 Represent quantitative relationships graphically and interpret the meaning of a specific

part of a graph in the situation represented by the graph.

7AF 2.0 Students interpret and evaluate expressions involving integer powers and simple roots: 2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-

number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.

2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent.

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7 AF 3.0 Students graph and interpret linear and some nonlinear functions: 3.1 Graph functions of the form y = nx2 and y = nx3 and use in solving problems. 3.3 Graph linear functions, noting that the vertical change (change in y- value) per unit of

horizontal change (change in x- value) is always the same and know that the ratio ("rise over run") is called the slope of a graph.

3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities.

7 AF 4.0 Students solve simple linear equations and inequalities over the rational numbers:

4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.

4.2 Solve multi step problems involving rate, average speed, distance, and time or a direct variation.

7 MG 1.0 Students choose appropriate units of measure and use ratios to convert within and between

measurement systems to solve problems: 1.1 Compare weights, capacities, geometric measures, times, and temperatures within and

between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters).

1.2 Construct and read drawings and models made to scale. 1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products

(e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.

7 MG 2.0 Students compute the perimeter, area, and volume of common geometric objects and use

the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale:

2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

2.2 Estimate and compute the area of more complex or irregular two-and three-dimensional figures by breaking the figures down into more basic geometric objects.

2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.

2.4 Relate the changes in measurement with a change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 144 square inches or [1 ft2] = [144 in2], 1 cubic inch is approximately 16.38 cubic centimeters or [1 in3] = [16.38 cm3]).

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7 MG 3.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures:

3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.

3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.

3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.

7 SDAP 1.0 Students collect, organize, and represent data sets that have one or more variables and

identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet software program:

1.1 Know various forms of display for data sets, including a stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data.

1.2 Represent two numerical variables on a scatter plot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level).

7 MR 1.0 Students make decisions about how to approach problems: 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant

information, identifying missing information, sequencing and prioritizing information, and observing patterns.

1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.

7 MR 2.0 Students use strategies, skills, and concepts in finding solutions:

2.1 Use estimation to verify the reasonableness of calculated results. 2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning

and arithmetic and algebraic techniques. 2.4 Make and test conjectures by using both inductive and deductive reasoning.

7 MR 3.0 Students determine a solution is complete and move beyond a particular problem by

generalizing to other situations: 3.3 Develop generalizations of the results obtained and the strategies used and apply them to

new problem situations.

Algebra 1, 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

Algebra 1, 3.0 Students solve equations and inequalities involving absolute values. Algebra 1, 4.0 Students simplify expressions before solving linear equations and inequalities in one

variable, such as 3(2x-5) + 4(x-2) = 12.

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Algebra 1, 5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

Algebra 1, 6.0 Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).

Algebra 1, 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

Algebra 1, 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.

Algebra 1, 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

Algebra 1, 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques.

Algebra 1, 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.




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Mathematics Tutorial Lab AB (Intervention Course Grades 9 - 12) 310221 Mathematics Tutorial Lab A 310222 Mathematics Tutorial Lab B COURSE DESCRIPTION Mathematics tutorial lab is an elective mathematics course provided to students as a second course to support the core Mathematics class in Algebra 1, Geometry, or Algebra 2. The course is designed to enhance the student’s knowledge of prerequisite skills that are needed to access the grade level mathematics course. COURSE SYLLABUS Algebra I: The standards for the Algebra 1 support class are a subset of the sixteen targeted Mathematics Standards for the Algebra Readiness Program as laid out on page 340 of Appendix E of the Mathematics Framework for California Public Schools. Namely: Grade 7 NS 1.2, 1.3, 1.5, 2.1

AF 1.1, 1.3, 2.1, 3.3, 3.4, 4.1, 4.2 MG 1.3, 3.3 MR 1.0, 1.1, 1.2, 1.3, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 3.0, 3.1, 3.2, 3.3

Geometry: The standards for the Geometry support class are taken from the Measurement and Geometry standard strands in grades 6 and 7, and a subset of the Algebra 1 standards as defined on pages 64, 65, 74–76, and 80–84 of the Mathematics Framework for California Public Schools. Namely: Grade 6 Grade 7 Algebra I

MG 1.1, 1.2, 1.3, 2.1, 2.2, 2.3 MG 1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4, 3.5 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 19.0, 20.0, 21.0, 22.0, 24.0

Algebra 2: The standards for the Algebra 2 support class are taken from the Algebra 1 and Geometry standards as defined on pages 80–84, and 85–89 of the Mathematics Framework for California Public Schools. Namely: Algebra 1 Geometry

2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0 8.0, 9.0, 10.0, 11.0, 15.0, 17.0, 18.0, 19.0, 20.0



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Appendix

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The A-G Curriculum and Mathematics

alifornia’s optional A-G curriculum, aligned with the state’s public university entrance requirements for the California State University (CSU) and University of California (UC) systems, is not just for college-bound students. According to a report from

Education Trust-West (2004), completing the A-G requirements “makes for a more meaningful high school experience by providing the challenges that encourage high schoolers to learn more and to live up to high expectations.” The A-G curriculum is not just preparation for college and work, the report contents; it is preparation for life.

C On June 14, 2005, the Board of Education approved a Resolution to create educational equity through the implementation of the A-G course sequence as a part of the high school graduation requirement for the class of 2012. The required A-G courses comprise about 65 percent of LAUSD graduation unit total of 230. If a student enrolls in the more rigorous recommended sequence of A-G courses, he/she will have completed approximately 78 percent of LAUSD graduation requirements.

A-G SEQUENCE OF COURSES A History/Social Studies 2 years required 20 semester units

B English 4 years required 40 semester units of core requirement courses

C Mathematics 3 years required 4 years recommended

30 semester units 40 semester units

D Laboratory Science 2 years required 3 years recommended


E Languages other than English - All years of same language

2 years required 3 years recommended


F Visual and Performing Arts 1 year required of year-long course

10 semester units

G Electives - Interdisciplinary 2 semesters required 10 semester units

Minimum Required Units 150 semester units 65% of LAUSD graduation total

Recommended Units 180 semester units 78% of LAUSD graduation total

LAUSD Graduation Unit Total including Electives

230 semester units

References Education Trust-West. (2004). “The A-G Curriculum: College-Prep? Work-Prep? Life Prep. Understanding and Implementing a Rigorous Core Curriculum for All.” http://www.edtrustwest.org.

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http://www.edtrustwest.org/


Designing Advanced and Honors Courses

tudents need opportunities to take advanced and enriched mathematics courses in middle and high school when they demonstrate higher levels of proficiency, effort, and achievement. Research shows that coursework in advanced and honors-level classes should be differentiated, or specially designed for students whose achievement is significantly above that of their peers. In the

Mathematics Framework for California Public Schools, these courses provide students with opportunities necessary to reach their fullest potential. Differentiation in a core mathematics class includes curriculum, instruction, and assessment that are enriched along four dimensions: acceleration/pacing, depth, complexity, and novelty (CDE, 1994). Differentiation in advanced and honors-level courses implies that students will be working on concepts that are more cognitively demanding than those addressed in core courses, and students will be engaged in both collaborative and independent study that exceeds grade-level standards and builds students’ independence with difficult reading, writing, listening, and speaking tasks.

S

• Acceleration/Pacing provides arrangements for students to move more rapidly through a

curricular sequence. An accelerated curriculum would include challenging and appropriate opportunities above and beyond the usual grade-level content: special projects, seminars, independent study, alternate assessments, and flexible grouping.

• Depth allows students who demonstrate an extraordinary knowledge, skills, or interest in a topic or task to pursue it in greater detail and to a greater level of understanding. Depth refers to approaching or studying something from the concrete to the abstract, from the familiar to the unfamiliar, and from the known to the unknown. An in-depth study would often include a significant amount of outside, independent research guided by essential questions that lead to advanced insight and comprehension.

• Complexity involves making relationships between and among ideas, connecting other concepts, and layering—a why/how interdisciplinary approach that connects and bridges to other disciplines, always enhancing the meanings of ideas. Students working individually or together on relatively complex ideas and relationships should be particularly encouraged to examine their own thinking.

• Novelty differs primarily from the other forms of differentiation because it is primarily student-initiated. Differentiating the curriculum through increasing depth and complexity should always begin with the students’ response to the topics, issues, ideas, and tasks presented. Providing advanced learning opportunities through novelty depends entirely on the students’ perceptions and responses, their inquiry and exploration using personalized and nontraditional approaches to finding the irony, paradoxes, metaphors, and other sophisticated symbolic processes within and across content areas. Teachers should encourage students to develop original interpretations, reinterpretations, or new implications among or within disciplines.

The University of California grants special “honors” designation and extra credit in students’ grade point average computation only to those level courses that meet specific criteria. (See High School Honors Level Mathematics Courses) References California Department of Education. (1999, revised 2005) Mathematics Framework for California Public

Schools: Kindergarten Through Grade 12. California Department of Education and California Association for the Gifted. (1994). Differentiating the

Core Curriculum and Instruction to Provide Advanced Learning Opportunities: A Position Paper.

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High School Honors Level Mathematics Courses

he University of California grants special "honors" designation and extra credit in students' grade point average computation only to those high school honors level courses that meet the following criteria. The University strongly encourages that such courses be available to all sectors of the school population. T

• AP Courses. Advanced Placement (AP) courses in the "a-g" subjects which are designed to prepare students for an Advanced Placement Examination of the College Board are automatically granted honors status, even if they are offered at the 10th grade level (e.g., newly developed courses/exams in Human Geography and World History). For more information about AP, go to the College Board's web site at www.collegeboard.org/ap/.

• International Baccalaureate. Designated International Baccalaureate (IB) courses offered by schools participating in the IB program are automatically granted honors status. For a list of IB courses that are granted honors status, search for the "International Baccalaureate" program list on the Doorways course list web site at https://doorways.ucop.edu/list. For more information about IB programs, go to www.ibo.org.

• College Courses. College courses in the "a-g" subjects that are transferable to the University of California. To determine whether a course is transferable, go to www.assist.org.

• Other Honors Courses. Other honors courses (that are not AP, IB, or college courses) specifically designed by the high school are acceptable if they are in the disciplines of history, English, advanced mathematics, laboratory science, languages other than English, and advanced visual and performing arts and have distinctive features which set them apart from regular high school courses in the same discipline areas. These courses should be seen as comparable in terms of workload and emphasis to AP, IB, or introductory college courses in the subject. Acceptable honors level courses are specialized, advanced, collegiate-level courses offered at the 11th and 12th grade levels. Please refer to the notes below for special requirements for the certification of these honors courses.

NOTE

The only non-AP, UC-approved Honors mathematics courses are Honors Mathematical Analysis AB and Honors Trigonometry/Math Analysis AB

Reference University of California Office of the President, 2008 Guide to “a-g” Requirements and Instructions for Updating Your School’s “a-g” Course List. LAUSD Reference Guide, REF-1469.3, January 23, 2008

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http://www.collegeboard.org/ap/

https://doorways.ucop.edu/list

http://www.ibo.org/

http://www.assist.org/


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GUIDELINES FOR STANDARDS-BASED INSTRUCTION

References: Mathematics Framework for California Public Schools, Kindergarten through Grade twelve.

Adopted by the California State Board of Education, March 2005 Published by the California Department of Education Sacramento, 2006

Secondary Mathematics Instructional Guide 2007-2008

Los Angeles Unified School District

Calculus, Calculus AB, Calculus BC Course Description May 2006, May 2007

AP College Board

Statistics Course Description May 2007, May 2008

AP College Board

Prepared by the LAUSD Secondary Mathematics Unit, July 2008

guidelines for standards-based instruction mathematics...secondary mathematics . grades 6 – 12 ....

Documents