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1 Mathematics Curriculum Guide Precalculus Page 1 of 12

2 Paramount Unified School District High School Math Curriculum Guides In , PUSD will continue to implement the Standards by providing rigorous math instruction that promotes inquiry based learning, rich math discussions, problem solving and critical thinking. What s Continuing? The Scope and Sequence document sequences instruction for the year for each grade level. Content is organized into instructional units. For each unit of instruction, the scope and sequence cites the standards addressed and the suggested timeframe. What s New? The order of some course units may be different from last year. Adjustments have been made as well so most course content is addressed before SBAC testing. Curriculum Guides The Unit Overview is the big picture of what students will learn throughout the unit. Standards are unpacked to determine specific learning outcomes. Students demonstrate their learning of the standards through Knowledge, Skills and Understandings. Knowledge refers to factual knowledge students must acquire to show mastery, Skills are proficiencies students should demonstrate independently to meet the standard and Understandings are big ideas students come to understand by learning the content. In the Instructional Sequence, the corresponding learning outcomes for each lesson are cited (Knowledge, Skills and Understandings). Focus questions are included to guide instruction while specific textbook lessons and instructional resources are cited to address specific learning targets (e.g., LearnZillion, Smarter Balanced). Units will be posted on the District website as they are completed. Because the Standards require students to demonstrate mastery differently than they did in the past, it is critical for teachers to spend time examining the unit overview to better understand the expectations of the standards and the student learning outcomes. Skills cited in the overview have been revised to align more closely to the standards and to the Smarter Balanced Achievement Level Descriptors. Revisions may be made to the instructional sequence to reflect teacher feedback. Inquiry questions have been included in the units to assist teachers in developing inquiry based lessons that promote problem solving, questioning and rich mathematical discussions. Examples of SBAC type questions are cited in the pacing. Page 2 of 12

3 Instruction Assessments What s Continuing? Unit assessments will continue to reflect a variety of question types including (Part 1) selected response questions, (Part 2) short constructed response and extended constructed response items. Part 2 of the assessments will require students to show their work and explain their thinking. By providing opportunities to demonstrate mathematical reasoning, students apply higherlevel thinking such as analysis, synthesis and evaluation. Unit assessments will be administered at the end of each unit and assessment dates will vary by course and with careful thought to the end of the semester. To review sample items of the Smarter Balanced Assessment which will be implemented in , go to and performance tasks/. For the first three weeks of school, all 9 12 students will receive instruction in an Introduction to a Classroom. The concepts and skills included in this Introduction are to be incorporated into and taught within the context of the Unit 1 content that is taught during this time. The purpose of this unit is to: introduce students to the routines and procedures of the mathematics classroom introduce the skills necessary for building a classroom community review Standards for Mathematical Practice review the proper use of Thinking Maps as tools for discussion and constructing arguments set individual and classroom goals that align to course expectations and SMPs All math classrooms should incorporate these daily practices: thought provoking questioning techniques that develop the habits of mind of the Standards for Mathematical Practice Talk Moves to facilitate discussion and math discourse opportunities to work independently and collaboratively to problem solve inquiry lessons that provide opportunities to explore and investigate concepts opportunities to explain thinking and justify reasoning both orally and in writing What s New? This year, targeted courses will pilot online unit assessments in math to prepare students for Smarter Balanced testing. Inquiry based instruction should be evident in all math classrooms. In an inquiry based math classroom, students persevere to problem solve and think critically as they uncover new concepts. Teachers support this productive struggle by asking guided questions that are planned in advance and scaffolded. This approach is different from a direct instruction where teachers specifically tell students the steps to follow to solve a problem. An inquiry lesson planning template was piloted last year with lead teachers. This year it will be shared with all schools during professional development. In order to support instruction through inquiry, the expectations for posting a daily objective have changed. With inquiry based instruction, students should not be told what they are going to learn as this takes away the opportunity to learn through discovery. Instead, teachers will post a daily Focus Question from the instructional sequence in the curriculum guide. This question guides student thinking as they explore skills and concepts and discover new learning. Teachers can also post Essential Questions from the unit overview to revisit throughout the unit. Page 3 of 12

4 Paramount Unified School District Educational Services Introduction to the Classroom Grades 9 12 Transfer Goals 1) Interpret and persevere in solving complex math problems using strategic thinking and expressing answers with a degree of precision appropriate for the problem context. 2) Express appropriate math reasoning by constructing viable arguments, critiquing the reasoning of others, attending to precision when making mathematical statements, and using tools to support reasoning. 3) Apply mathematical knowledge to analyze and model math relationships in the context of a situation in order to make decisions, draw conclusions, and solve problems. Unit Focus The purpose of this unit is to: introduce students to the routines and procedures of the classroom introduce the skills necessary for building a classroom community (e.g., participate in whole group, small group, and partner conversations using Talk Moves; respond to questions and prompts through academic discussions and writing) review all Standards for Mathematical Practice (especially SMPs #1 3) review the proper use of Thinking Maps as tools for discussion and constructing arguments set individual and classroom goals that align to grade level expectations and SMPs Understandings Students will understand that Mathematical learning is supported with structured routines and procedures Participating in academic conversations requires purposeful listening, speaking and writing in math which leads to greater understanding and shared reasoning Mathematicians don t give up and try different strategies to solve problems (SMP #1) Mathematicians can show their reasoning using both numbers and words (SMP #2) Mathematicians construct arguments and critique the reasoning of others (SMP #3) Thinking Maps are a useful tool for organizing thinking Setting goals is important to monitor both individual and class progress Knowledge Students will know Agreed upon classroom rules and procedures All 8 Talk Moves Linguistic patterns used for partaking in academic discussions The Standards for Mathematical Practices (SMPs) Thinking Maps Individual and classroom goals Essential Questions Students will consider. By following classroom rules and procedures, how does this support my learning? How can I effectively communicate my ideas orally and in writing to construct a viable argument about what I am learning and thinking? What strategies can I use to persevere in problem solving? How can I respectfully critique the reasoning of others? How can Thinking Maps be used during math to help me organize and explain my thinking? By setting goals, how will this help me to progress in meeting my grade level expectations and develop the habits of mind of a mathematician? Skills Students will be skilled at and able to Follow classroom rules and procedures Participate in academic conversations using Talk Moves, appropriate body language and a variety of active participation strategies Respond to questions and prompts orally and in writing using academic language Use a variety of strategies to persevere in solving problems (e.g., numbers, words) Construct viable arguments and critique the reasoning of others Use Thinking Maps to represent mathematical thinking Set individual and classroom goals Suggested Timeframe: Within first 3 weeks Instructional Activities Collaboratively establish classroom rules and procedures (behavior, retrieve materials, independent work, group work) Review each Talk Move individually model first, check for understanding and use in context to promote shared reasoning Establish appropriate behaviors for listening and speaking (e.g., eye contact, lean in, strong voice, speak clearly, linguistic patterns, etc.) Review expectations for working collaboratively (everyone contributes, listen to each other, add on to classmates thinking, disagree with and critique others respectfully) Establish individual and classroom goals for the year, quarter, etc. that align to grade level expectations and the SMPs Use Thinking Maps for the activities cited above (e.g., Circle Map to identify practices for each of the SMPs, Multi flow Map to show the effects of effective collaboration, etc.) Resources Talk Moves posters SMP posters Problem solving response frames Thinking Maps posters Goal setting documents Page 4 of 12

5 Paramount Unified School District Educational Services Topic 1: Functions and Graphs Mathematics Topic 1 Overview Course: Precalculus Transfer Goals 1) Interpret and persevere in solving complex math problems using strategic thinking and expressing answers with a degree of precision appropriate for the problem context. 2) Express appropriate math reasoning by constructing viable arguments, critiquing the reasoning of others, attending to precision when making mathematical statements, and using tools to support reasoning. 3) Apply mathematical knowledge to analyze and model math relationships in the context of a situation in order to make decisions, draw conclusions, and solve problems. Understandings Standard s ALGEB RA A CED 1 A CED 2 A REI 10, 11 F IF 1 F IF 2 F IF 4 F IF 5 F IF 6 F IF 7, F IF 8 F IF 9 F BF 1 F BF 3 F BF 4 A SSE 3 Quantities N Q 1 Students will understand that Functions are use usually given in terms of equations involving x and y. (F IF 1) Functions are used to determine the function s domain and its range. (F IF 1, 3) Ordered pair describes the order of the coordinate and makes a difference in a point s location. (A REI 10) A parent function is the simplest form of a set of functions that forma a family. Each function in the family is a transformation of the parent function. (F IF 7) A function involving more than one transformation can be graphed in the following order: horizontal shifting, stretching or shrinking, reflecting and vertical shifting. (F BF 3) Transformations help to understand the relationship between graphs that have similarities but are not the same. (F BF 3) Functions and graphs form the basis for understanding mathematics and applications. (A CED 1, 2) The ratio of the vertical change to the horizontal change is constant. That constant ratio describes the slope of the line. (F IF 6) Linear functions can be graphed using the slope and the y intercept. Linear equations in standard form can be graphed using the x and y intercepts. (A REI 10) Operations of functions are performed same as for real numbers. One difference is that the domain of each function must be considered. (F BF 1) Numerical, algebraic, and graphical models provide different methods to visualize, analyze, and understand data. (A CED 1, 2) Knowledge Students will know Vocabulary: Zeros, intercepts, domain, range, function, interval notation, vertical line test, increasing, decreasing, minimum, maximum, regression, vertical and horizontal shift and stretch, reflection, composition, inverse functions, horizontal line test, one to one function, verbal models Properties of the twelve basic functions. The graph of f has x intercepts at the real zeros, intervals of increasing, decreasing, and relative maximum and minimum. A function involving more than one transformation can be graphed in the following order: horizontal shifting, stretching or shrinking, reflecting and vertical shifting. Transformations help to understand the relationship between graphs that have similarities but are not the same. If a function does not model a verbal condition, its domain is the largest set of real numbers for which the value of f(x) is a real number, excluding real numbers that cause division by zero and real numbers that result in an event root of a negative numbers. Most functions can be created by combining or modifying other functions. Essential Questions Students will keep considering Suggested Timeframe: 4 weeks, 20 days Assessment Dates: Sept , 2015 How do represent functions graphically and interpret information given by the graphs? How are domains affected when functions are combined arithmetically or composed? How can inverse functions be determined, verified, and graphed? How can verbal models are used to obtain functions from verbal descriptions? Skills Students will be skilled at and able to do the following Investigate end behavior. Identify different forms of linear equations. Know the different forms of a linear function. Graph linear equations from a given form. Identify graphs of the twelve basic functions. Analyze functions both algebraically and graphically for various properties including domain, range, continuity, intervals of increasing and decreasing, local extrema, symmetry, and end behavior. Identify algebraically and graphically representations of translations, reflections, stretches, and shrinks of functions. Perform transformations of graph, which include vertical and horizontal shifts, reflections, vertical stretching and shrinking, and horizontal stretching and shrinking. Perform operations that apply to all types of functions and build new functions from existing functions. Consider the domain of functions as the largest set of real numbers, excluding real numbers that cause division by zero and real numbers that result in an event root of a negative numbers. Find the inverse functions by restricting the domain using algebraic method, using tables, and using graphs. Verify inverse functions by composing the functions in both orders. Identify appropriate basic functions with which to model real world problems and be able to produce specific functions to model data, formulas, graphs, and verbal descriptions. Page 5 of 12

6 Paramount Unified School District Educational Services Topic 1: Functions and Graphs Essential Questions: How do represent functions graphically and interpret information given by the graphs? How are domains affected when functions are combined arithmetically or composed? How can inverse functions be determined, verified, and graphed? How can verbal models are used to obtain functions from verbal descriptions? Lesson/ Time Activity 3 days Review concepts of Algebra (Sections P.1,P.5, P.9) Focus Questions for Lessons Focus Question How do you recognize subsets of the real numbers and identify their properties to simplify algebraic expressions? Is a rule that can help to determine the factors of an expression? Standards: A CED 1, 2, F IF 1, 2, 4, 5, 6, 7, 8, 9, A REI 11, F BF 1, 3, 4, N Q 1 Suggested Timeframe: 4 weeks, 20 days Start Date: August 19, 2015 Assessment Dates: September 15 16, 2015 Understandings Knowledge Skills Resources The set of real numbers contains rational and irrational numbers Absolute value expresses distance Mathematical modeling is the process of finding an equation to describe realworld phenomena Rules of factorization Domain of an inequality Vocabulary: absolute value, mathematical modeling, setbuilder notation, intersection of sets, union of sets, rea; numbers, factoring, domain, inequalities, interval notation. Simplify algebraic expressions Identify intersection and union of sets Use the properties of real numbers Factoring expressions Solving linear, compound and absolute value inequalities Use interval notation to determine the domain of an inequality Mathematics Topic 1 Pacing Course: Precalculus P.1: # , P.5: # , P.9: # days Lesson 1.4: Linear Functions and Slope A REI 10 Lesson 1.5: More on Slope F IF 6, 7, 8, 9 Focus Question. How do you write and graph linear equations? Explain how to derive the slopeintercept form of line s equation, y = mx +b from the point intercept form, y y 1 = m(x x 1 ) The ratio of the vertical change to the horizontal change is constant. That constant ratio describes the slope of the line. Linear functions can be graphed using the slope and the y intercept. Linear equations in standard form can be graphed using the x and y intercepts. Vocabulary: slope, scatter plot, intercept form, point slope form, horizontal and vertical lines, parallel and perpendicular lines, rate of change. Parallel lines have the same slope. Perpendicular lines slopes are opposite reciprocals. Identify different forms of linear equations. Know the different forms of a linear function. Graph linear equations from a given form. 1.4: #87 99, : #27 38, : #25 34, 87 88, 118, : #5 12, day Topic 1 Opening Activity Introduction to the Performance Task (see attached) Practices Instruction in the Standards for Mathematical Practices Use of Talk Moves Note taking Use of Manipulatives Use of Technology Use of Real world Scenarios Project based Learning Thinking Maps Page 6 of 12

7 Time Lesson/ Activity Focus Questions for Lessons Understandings Knowledge Skills Additional Resources 1 day Lesson 1.1: Graphs and graphing utilities F IF 1 Focus Question. How do you graph functions and interpret information given by the graphs? Are single numbers the only way to represent intercepts? Can ordered pairs also be used? Functions are use usually given in terms of equations involving x and y Functions are used to determine the function s domain, range and intercepts. Ordered pair describes the order of the coordinates and makes a difference in a point s location. Vocabulary: x and y axis, quadrants, ordered pair, x and y intercept, origin, equation in two variables. Ordered pairs satisfy solutions of equations. Plot points and graph equations in the rectangular system. Use a graph to determine intercepts. Interpret information given by graphs. Use a graphing Utility 1.1: #55 65, 67 74, : # days Lessons 1.2: Basics of Functions and Their Graphs F IF 5, 7, 9 Lesson 1.3: More on Functions and Their Graphs Focus Question How do you graph relations and identify their properties? Pg. 185 Chose from A relation is identified as a function if it passes the vertical line test and its domain values appear only once. Dots, open dots, or arrows on the left and right sides of a graph give information about a function. The x coordinate and not the y coordinate describe the interval where the functions increase, decrease, or are constant. Vocabulary: domain, range, function, vertical line test, intercepts, intervals notation, open interval, interval of increasing, decreasing, or constant, symmetry, local maximum and minimum. F(x) notation describes the value of the function at x. Identify the graphs of twelve basic functions. Obtain information about a function from its graph and visualize that lines will or will not extend on both sides of the line depending on a dot, open dot or arrow. Analyze functions both algebraically and graphically for various properties including domain and range, intervals of increasing and decreasing, local and absolute extrema, symmetry, asymptotes and end behavior. 1.2: #99 116, : #79 82, , : #1 10, : #33 36, 37 42, Practices Instruction in the Standards for Mathematical Practices Use of Talk Moves Note taking Use of Manipulatives Use of Technology Use of Real world Scenarios Project based Learning Thinking Maps Page 7 of 12

8 Time Lesson/ Activity Focus Questions for Lessons Understandings Knowledge Skills Additional Resources 2 days Lesson 1.6: Transformation s of functions 2 days F IF 7 F BF 3 Lesson 1.7: Combinations of functions: Composite Functions F BF 1 Focus Question How transformations could help you to understand the relationship between graphs that have similarities but are not the same? Pg. 229 # 128 Focus Question How are domains affected when functions are combined arithmetically or composed? Describe the values of x that must be excluded from the domain of (f g)(x) A parent function is the simplest form of a set of functions that forma a family. Each function in the family is a transformation of the parent function. A function involving more than one transformation can be graphed in the following order: horizontal shifting, stretching or shrinking, reflecting and vertical shifting. Transformations help to understand the relationship between graphs that have similarities but are not the same. Operations of functions are performed same are for real numbers. One difference is that the domain of each functions must be considered If a function does not model a verbal condition, its domain is the largest set of real numbers for which the value of f(x) is a real number, excluding real numbers that cause division by zero and real numbers that result in an event root of a negative numbers. Vocabulary: common functions, vertical and horizontal shift, reflections, vertical and horizontal stretching and shrinking, Graphing procedures Transformation of a familiar graph makes graphing easier. Vocabulary: inequality symbols and related operations, solid and dashed lines, region of solutions The composition of two functions involves replacing the variable of one function with the expression equal to the other function. Identify algebraically and graphically representations of translations, reflections, stretches, and shrinks of functions. Perform transformations of graph, which include vertical and horizontal shifts, reflections, vertical stretching and shrinking, and horizontal stretching and shrinking. Perform operations that apply to all types of functions and build new functions from existing functions. Consider the domain of functions as the largest set of real numbers, excluding real numbers that cause division by zero and real numbers that result in an event root of a negative numbers. 1.6: # , : #17 24, 53 66, : #97 107, : p #a d, p #8 12, 25 26, 51 66, day Lesson 1.8: Inverse Functions F BF 4 Focus Question How can a function s inverse be determined, verify, and graphed? Pg. 256 # 89, 90 The inverse of a may be or not be a function. The range of the relation is the domain of the inverse and domain of the relation is the range of the inverse. If f and f 1 are functions and if either maps a to b, then the other maps b to a. Vocabulary: inverse relation, inverse function, one to one function, horizontal line test, The inverse of a function will only be a function if the function is a one toone. Find the inverse functions by restricting the domain using algebraic method, using tables, and using graphs. Verify inverse functions by composing the functions in both orders. 1.8: #65 75, : #1 10, 11 28, Instruction in the Standards for Mathematical Practices Use of Talk Moves Note taking Practices Use of Manipulatives Use of Technology Use of Real world Scenarios Project based Learning Thinking Maps Page 8 of 12

9 Time Lesson/ Activity Focus Questions for Lessons Understandings Knowledge Skills Additional Resources 1 day Lesson 1.10: Modeling with Functions A CED 1,2 Focus Question How can you construct functions from verbal descriptions and from formulas? Pg. 281 # 68 Data can be modeled from real world situations with equations, formulas, or graphs. These tools can be used to draw conclusions about the situations. Real world situations can be analyzed and solved by functions.. Vocabulary: verbal models When developing functions that model situations, it is helpful to draw a diagram Identify appropriate basic functions with which to model real world problems and be able to produce specific functions to model data, formulas, graphs, and verbal descriptions. This whole lesson has cc problems. 1.10: #21 22, days 1 day Review & Assess Chapter 1 Concepts & Skills Use Textbook Resources and/or Teacher Created Items for Assessment Topic 1 Performance Task Have students work collaboratively to reflect on the Performance Task 2 days Unit 1 Assessment (Created and provided by PUSD) Instruction in the Standards for Mathematical Practices Use of Talk Moves Note taking Practices Use of Manipulatives Use of Technology Use of Real world Scenarios Project based Learning Thinking Maps Page 9 of 12

10 Topic 1 Performance Task Page 1 Precalculus Name: Date: Per: Page 10 of 12

11 Topic 1 Performance Task Page 2 Page 11 of 12

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