The Lovett School Upper School Mathematics Curriculum
The Lovett School Vision for Learning Lovett offers experiences that inspire our students to love learning. We encourage them to think critically, communicate effectively, engage creatively, and collaborate purposefully. We provide the opportunities and resources that help our students develop independence and self-direction and extend their learning beyond the walls of the classroom as they grow intellectually, emotionally, physically, aesthetically, morally, and spiritually.
200 - Algebra I Advanced Course Description Grade: 9 Group: I Units: 1.0 In Algebra I Advanced, students will learn to reason symbolically and apply those skills to solve a variety of problems. The key content involves writing, solving, and graphing linear and quadratic equations. Additional topics of study include data analysis, systems of equations, inequalities, exponents, polynomials, radicals, ratios, proportions, and absolute value. Note: Students who enroll in Algebra I Advanced in Upper School must pass that class in the Upper School to receive graduation credit and move into Geometry. Essential Questions 1. How does algebra differ from basic math and arithmetic? 2. How is algebra like a language? 3. How do we use linear functions and quadratic equations to model real-life situations? 4. What other subjects use algebraic processes and techniques? 5. How does algebra improve the critical thinking process? Assessment 1. Classwork 2. Homework 3. Quizzes 4. Unit tests 5. Semester exams 6. Project Skills Benchmarks A student will be able to: 1. Simplify and evaluate algebraic expressions using the order of operations. 2. Solve linear algebraic equations. 3. Add, subtract, and multiply polynomials. 4. Evaluate integer exponents. 5. Factor polynomials. 6. Solve equations with rational expressions. 7. Graph linear equations. 8. Graph inequalities on a number line and a Cartesian plane. 9. Graph and/or solve a system of equations or inequalities. 10. Add, subtract, multiply, divide, and simplify radical expressions. Solve radical equations. 11. Solve a quadratic equation using the quadratic formula or factoring. 12. Translate word problems into algebraic equations or inequalities and solve.
Units 1. Introduction to Critical Thinking and Problem Solving 2. Data Analysis 3. Equations, Inequalities, and their Graphs 4. Linear Equations and their Graphs 5. Systems of Equations and their Graphs 6. Absolute Value Expressions and Equations 7. Exponents and Polynomials 8. Quadratic Equations and their Graphs 9. Rational Expressions with a specific emphasis on Factoring 10. Radical Expressions and Equations Textbooks and Resources 1. Prentice Hall Classics Algebra 1, by Smith, Charles, Dossey, and Bittinger. 2. TI-84 Calculator 3. Handouts Updated May 2016
210 - Geometry Course Description Grades: 9-10 Group: I Units: 1.0 Prerequisite: Algebra I Geometry focuses on important geometric facts, proofs using deductive reasoning, the integration of algebra and geometry, and applications of geometry. During the year, the student will learn the properties of parallel lines, circles, triangles, parallelograms, and other polygons. These properties will be used to study coordinate geometry, congruence, similarity, right triangle trigonometry, perimeter, area, surface area, and volume. Throughout the course, the student will use and further develop skills learned in algebra. Essential Questions 1. What is the role of algebra in geometry? 2. How does geometry differ from algebra? 3. What is the role of proofs in the development of a deductive system? 4. How does geometry enhance reasoning abilities/logical thinking? 5. How do we use geometry to model real-life situations? Assessment 1. Homework 2. Classwork 3. Unit quizzes and/or tests 4. Semester exams 5. Projects, usually done in collaboration with another student 6. Example Assessment - The Unit 3 Test measures students' ability to differentiate given information about angles and segments, including congruence marks and basic angle relationships (vertical angles, supplementary, complementary, etc.). As the first assessment with proofs, there is a strong emphasis on making a claim and providing evidence to support the claim. Students analyze diagrams and must think critically regarding the postulates of congruence; is there enough information given to conclude the two triangles are congruent? Why or why not? The assessment incorporates this type of deductive reasoning, along with applying algebraic concepts to solve conclusions. If a student deduces and makes a statement regarding congruence, he or she then uses algebraic expressions to solve for missing measures. Lastly, vocabulary provided throughout the semester in a geometric context (midpoint, distance, segment congruence) are applied on an x-y coordinate plane to incorporate the mathematical problem solving into familiar geometric terms. The combination of the geometric concepts and algebraic process ensure the course standards are met while retaining essential practices from algebra.
Skills Benchmarks A student will be able to: 1. Construct a two-column proof involving parallel lines and congruent triangles. 2. Solve right triangles using the Pythagorean Theorem, the relationships of sides in special right triangles, and the trig ratios. 3. Apply the theorems of tangents, arcs, chords, and angles of a circle. 4. Find the perimeter and area of triangles, quadrilaterals, regular polygons, and circles. 5. Find the surface area and volume of prisms, pyramids, cylinders, cones, and spheres. 6. Use the algebra of geometry. a. Ratio and proportions b. Distance formula c. Midpoint formula d. Slopes of lines e. Equations of lines and circles 7. Model real-world applications using geometry. Units 1. 2. 3. 4. 5. 6. 7. 8. 9.
Points, Lines, Planes, and Angles Parallel Lines and Planes Congruent Triangles Quadrilaterals Similar Polygons Right Triangles Circles Areas of Plane Figures Areas and Volumes of Solids
Textbooks and Resources 1. Jurgensen, Ray C., Richard G. Brown, and John W. Jurgensen, Geometry, McDougal Littell 2. TI-84 Graphing Calculator 3. Geometer’s Sketchpad software 4. Geometers Updated May 2016
215 - Honors Geometry Course Description Grades: 9-10 Group: I Units: 1.0 Prerequisite: Algebra I AND teacher recommendation The honors level of geometry covers all topics included in the regular course but at an accelerated pace. Many concepts are extended and developed more completely; there is a greater emphasis on proof which involves critically analyzing and solving non-routine problems that require creative strategies. The pace of the course affords the opportunity to explore additional topics, such as non-Euclidean geometries which will allow for the continued development of problem-solving, collaboration, and critical thinking skills. Outside readings, projects, and/or independent research may be used to enrich this advanced study. Throughout the course, students will use and further develop skills learned in algebra. Essential Questions 1. What is the role of algebra in geometry? 2. How does geometry differ from algebra? 3. What is the role of proofs in the development of a deductive system? 4. How does geometry enhance reasoning abilities/logical thinking? 5. How do we use geometry to model real-life situations? Assessment 1. Homework 2. In-class group work 3. Unit quizzes and/or tests 4. Projects completed collaboratively with a partner 5. Semester exams 6. Example project: At the end of the fall semester, students will have the opportunity to collaboratively research certain mathematical phenomenon of their choice. Students will demonstrate the understanding and application of the chosen topic by presenting their findings to the class. Skills Benchmarks A student will be able to: 1. Construct a two-column proof involving parallel lines, congruent triangles and other polygons, similarity, and relationships within a circle. 2. Solve right triangles using the Pythagorean Theorem, the relationships of sides in special right triangles, and the trig ratios. 3. Apply the theorems of tangents, arcs, chords, and angles of a circle. 4. Find the perimeter and area of triangles, quadrilaterals, regular polygons, and circles. 5. Find the area and volume of prisms, pyramids, cylinders, cones, and spheres.
6. Use the algebra of geometry. a. Ratio and proportions b. Distance formula c. Midpoint formula d. Slopes of lines e. Equations of lines and circles 7. Model real-world applications using geometry. 8. Apply transformations (translations, rotations, reflections, and dilations) to geometric figures, if time permits. Units 1. Points, Lines, Planes, and Angles 2. Deductive Reasoning 3. Parallel Lines and Planes 4. Congruent Triangles 5. Quadrilaterals 6. Inequalities in Geometry 7. Similar Polygons 8. Right Triangles 9. Circles 10. Areas of Plane Figures 11. Areas and Volumes of Solids 12. Coordinate Geometry 13. Transformations, if time permits Textbooks and Resources 1. Jurgensen, Ray C., Richard G. Brown, and John W. Jurgensen, Geometry, McDougal Littell 2. TI-84 Graphing Calculator Updated September 2018
220 - Algebra II Course Description Grades: 9-11 Group: I Units: 1.0 Prerequisite: 210 - Geometry or 215 - Geometry Honors Algebra II begins with a review of basic algebra topics but quickly moves to more advanced material, including relations, functions (linear, quadratic, radical, rational, exponential, and logarithmic), complex numbers, and matrices. Emphasis is placed not only on the acquisition of skills but also on problem-solving, applications of algebra, and written and verbal communication of concepts. Students engage in both individual and collaborative problemsolving, applying the techniques learned to non-routine problems in a variety of disciplines, such as economics, physical science, and financial math. A heavy emphasis is also placed on using the TI-84 Graphing Calculator to help solve problems.
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Essential Questions How do we use functions to model real-life situations? How do graphs of functions add a visual perspective to a mathematical problem? How does the use of technology enhance student understanding? How do the skills learned in algebra provide a foundation for the study of more advanced mathematics? How important is the language of mathematics? Why are functions important? What are some basic "toolkit" functions? Assessment Homework Quizzes Tests Semester exams Assessment Example: Following the unit on Quadratic Functions and Complex Numbers, students have the opportunity to work in small groups to investigate quadratic modeling. The goal of this lab is to investigate the relationships that a projectile has relative to time. Each group will video record a ball thrown in the air. Students will then use the application, LoggerPro, to track the motion of a ball, creating a scatter plot of time and height. Students need to make decisions based on the skills covered during this unit to come up with a quadratic curve of best fit. Communication and Collaboration is also key as students will need to divide tasks and verify findings along the way.
Skills Benchmarks A student will be able to: 1. Solve linear equations and inequalities.
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Graph and write equations of linear functions. Solve and/or graph a system of linear equations or inequalities. Factor a polynomial using common factors and/or products of binomials. Solve polynomial equations over the set of complex numbers. Model real-world scenarios using a variety of functions, including linear, quadratic, and exponential. 7. Solve rational equations and identify extraneous solutions. 8. Simplify expressions containing rational exponents. 9. Solve radical equations and identify extraneous solutions. 10. Find all six trigonometric values from all angles on the Unit Circle. 11. Solve oblique triangles using the Laws of Sines and Cosines. 12. Solve problems involving probability and statistics. Units 1. Equations and Inequalities 2. Linear Relations and Functions 3. Systems of Linear Equations & Inequalities 4. Matrices 5. Quadratic Functions & Complex Numbers 6. Polynomial Expressions and Functions 7. Inverses and Radical Functions 8. Exponential and Logarithmic Functions 9. Rational Functions 10. Statistics and Probability 11. Trigonometric Functions Textbooks and Resources 1. Glencoe Algebra 2 (Common Core State Standards Edition), Columbus, Ohio; McGraw-Hill, 2012. 2. TI-84 Graphing Calculator Updated June 2018
225 - Honors Algebra II Course Description Grades: 9-11 Group: I Units: 1.0 Prerequisite: 210 - Geometry or 215 - Geometry Honors AND teacher recommendation Honors Algebra II is fast-paced and challenging, designed to go well beyond the topics of Algebra II. This accelerated study of algebra will include a thorough investigation of exponential functions, logarithmic functions, conic sections, trigonometry, vectors, and an introduction to descriptive statistics. Collaboration, problem solving, and critical thinking are emphasized in the real world applications of the study of trigonometric functions. Embedded through each unit of study will be opportunities for students to collaborate on the topics studied in class. Essential Questions 1. How do we use functions to model real-life situations? 2. How do graphs of functions add a visual perspective to a mathematical problem? 3. How does the use of technology enhance student understanding? 4. How do the skills learned in algebra provide a foundation for the study of more advanced mathematics? 5. How important is the language of mathematics? 6. Why are functions important? 7. What are some basic "toolkit" functions? Assessment 1. Homework, homework solutions will be discussed in a collaborative large group discussion. 2. Class work will provide opportunities for students to communicate effectively and engage creatively their understanding of the mathematical topics. 3. Quizzes 4. Tests include non-traditional, challenging problems that target students’ ability to think critically and problem solve. On the trigonometry unit test, students encounter real-world trigonometric applications, taking data (e.g. tidal ebb and flow, sunrise and sunset times) to create a trigonometric model to predict future events. 5. Application problem sets 6. Project 7. Semester exams Skills Benchmarks A student will be able to: 1. Solve linear equations and inequalities. 2. Graph and write equations of linear functions. 3. Solve and/or graph a system of linear equations or inequalities.
4. Use matrices and determinants to solve two and three-variable systems. 5. Factor a polynomial using common factors and/or products of binomials. 6. Solve quadratic equations and inequalities over the set of complex numbers. 7. Solve rational equations and identify extraneous solutions. 8. Simplify expressions containing rational exponents. 9. Solve radical equations and identify extraneous solutions. 10. Sketch, from their equations, the graphs of circles, ellipses, hyperbolas, and parabolas. 11. Solve triangle problems using right triangle trigonometric ratios, the Law of Cosines, and the Law of Sines. 12. Either sketch the graph of a sinusoidal function or create the equation of the trigonometric function based on the graph. 13. With the aid of the technology, model real-world applications using the appropriate function. 14. Identify a vector and find its component form, magnitude, direction, and unit vector. 15. Add, subtract, or find the angle in between multiple vectors. 16. Sketch a component vector. 17. Identify a geometric or arithmetic sequence and determine the formula to describe each particular sequence. 18. Calculate the sum of an arithmetic, geometric, and convergent infinite series. Units 1. 2. 3. 4. 5. 6. 7. 8. 9.
Equations, Inequalities, and Linear Functions and Relations Linear Functions Systems of Equations and Inequalities Quadratic Functions and Complex Numbers Higher Degree Functions Inverses and Irrational Algebraic Functions Exponential and Logarithmic Functions Rational Algebraic Functions and Relations Quadratic Relations: Conics Sections, studying the properties and transformations of parabolas, circles, ellipses, and hyperbolas, systems of quadratic relations. 10. Trigonometric Functions, an introduction of the six trigonometric functions, construction of the unit circle, using points on a unit circle to graph trigonometric functions, and applications of trigonometric functions. 11. Vectors, properties of vectors, operations with vectors, the definition of the dot product, and determining the angle in between two vectors. 12. Sequences and Series, comparing and contrasting arithmetic and geometric sequences and series, finding the sum of geometric and arithmetic series, finding the sum of an infinite convergent geometric series.
Textbooks and Resources 1. Glencoe, Algebra 2 (Common Core Edition), Menlo Park California: McGraw-Hill Companies, Inc., 2010. 2. TI-84 Graphing Calculator
3. Geometer’s Sketchpad v 4.07, mathematical exploration program. 4. Desmos an online graphing utility, https://www.desmos.com/ Updated May 2016
230 - College Algebra and Trigonometry Course Description Grades: 11-12 Group: I Units: 1.0 Prerequisite: 220 - Algebra II and teacher recommendation This course has the threefold focus of strengthening conceptual foundations, improving manipulative skills, and providing knowledge of relevant applications. The content includes a study of the real number system, functions and graphs, polynomial functions, rational functions, exponential and logarithmic functions, conic sections, and applications of trigonometry. After the start of the school year, students will be permitted to move into College Algebra only when the course change is initiated by the teacher. Essential Questions 1. What algebraic skills are necessary to complete a course in college algebra and trigonometry, and how will these skills be used in the study of more advanced mathematics? 2. How do we differentiate among the various subsets of the complex number system? 3. What is the connection between the equations and graphs of functions? 4. What role does technology play in the study of algebra and trigonometry? 5. How do the basic rules for transforming graphs of functions apply throughout the course? 6. What must students learn to converse mathematically? Assessment 1. Homework 2. Classwork 3. Quizzes 4. Tests 5. Semester exams 6. Projects 7. Participation Skills Benchmarks A student will be able to: 1. Simplify expressions containing rational exponents. 2. Simplify rational and radical expressions. 3. Factor polynomials using various methods. 4. Graph and write equations of linear functions. 5. Solve linear equations and inequalities. 6. Solve absolute equations and inequalities. 7. Solve polynomial equations and inequalities.
8. Solve rational equations and inequalities. 9. Solve radical equations. 10. Solve exponential and logarithmic equations. 11. Solve right triangles using trigonometric ratios. 12. Solve oblique triangles using the Law of Sines and the Law of Cosines. 13. Sketch graphs and write equations for each of the four conic sections. Units 1. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction to Graphs and Graphers Basic Concepts of Algebra Graphs, Functions, and Models Functions and Equations: Zeros and Solutions Polynomial Functions Rational Functions Exponential and Logarithmic Functions Trigonometry Conic Sections
Textbooks and Resources 1. Beecher, et. al. Algebra and Trigonometry. 3rd ed. Addison Wesley 2. TI-84 Graphing Calculator Updated May 2016
240 - Precalculus Course Description Grades: 10-12 Group: I Units: 1.0 Prerequisite: Year-long average of 80 or higher in 220 - Algebra II OR 230 - College Algebra and Trigonometry and teacher recommendation Precalculus mathematics encompasses concepts that develop out of topics from Algebra II. The content begins with a review of linear and quadratic functions. It continues with an emphasis on polynomial and rational functions, logarithmic and exponential functions, sequences and series, trigonometric functions, applications of trigonometry, vectors, conic sections, and an introduction to limits. Upon successful completion of Precalculus, students should be prepared for a course in Calculus. Essential Questions 1. Why do we use different sets of numbers in mathematics? 2. How do we use functions to model real-life situations? 3. How do graphs of functions add a visual perspective to a mathematical problem? 4. How do basic rules for transforming functions apply throughout Precalculus? 5. How does the use of technology enhance student understanding? 6. How do the skills learned in Precalculus provide a foundation for the study of more advanced mathematics? Assessment 1. Homework 2. Classwork 3. Quizzes 4. Tests 5. Projects - during the Trigonometry unit, students will have to exercise their critical thinking and problem solving abilities to approach a real-world trigonometric application such as taking tidal or temperature data to create a periodic mathematical model to predict future events. 6. Cumulative semester exams 7. Example: The Chapter 6 Assessment focuses on the application of right triangle rules and properties, emphasizing angles of depression and elevation, and real-world scenarios of height, distance, diagonals, and the relationship between objects separated by a distance. In the 2017-2018 school year, students created a triangle word problem of their own on the test to engage their creativity and anchor the concepts to something tangible and meaningful. In using right triangle problem in real-world contexts, students think critically about the answers found for problems: does this distance or angle make sense given the other information? Does something about this not make sense given the information provided? A key component of this application is taking an abstract concept
like the Law of Sines or Law of Cosines and assigning meaning to the values given and found. Skills Benchmarks A student will be able to: 1. Solve equations and inequalities using a variety of techniques, including simplification, factoring, the quadratic formula, sign diagrams, logs, and trig identities. 2. Apply the Leading Coefficient Test, Descartes Rule of Signs, and information about xand y-intercepts to graph and analyze polynomial functions, including using synthetic and long division to find the zeros of a function. 3. Work with polynomial, rational, radical, exponential, logarithmic, and sinusoidal functions as follows: a. Identifying domain and range b. Locating intercepts c. Sketching a graph d. Applying transformations e. Modeling real-world applications using the appropriate function 4. Solve problems involving arithmetic and geometric sequences and series. 5. Memorize and apply the Unit Circle, including simplifying trigonometric identities and solving trigonometric equations. 6. Solve triangle problems using right triangle ratios, the Law of Cosines, and the Law of Sines. 7. Graph the sinusoidal and rational trig functions and the inverse functions. 8. Write equations of the four conic sections and sketch the resulting graphs, identifying significant characteristics. 9. Calculate the limit of a difference quotient. 10. Calculate the limit of a difference quotient. Units 1. 2. 3. 4. 5. 6. 7. 8. 9.
Functions & Their Graphs Polynomial & Rational Functions Exponential & Logarithmic Functions Sequences, Series, & Probability Trigonometric Functions Analytic Trigonometry Additional Topics in Trigonometry Conic Sections Limits and an introduction to Calculus
Textbooks and Resources 1. Larson, Ron. Precalculus with Limits: A Graphing Approach, Cengage Learning, 2016 th
(7 ed.) 2. TI-84 Graphing Calculator
Updated June 2018
245 - Honors Precalculus Course Description Grades: 10-12 Group: I Units: 1.0 Prerequisite: 225 - Algebra II Honors AND teacher recommendation Honors Precalculus covers the same topics as general Precalculus but in much greater depth. It includes a continuation of the study of vectors begun in Honors Algebra II, an exploration of polar and parametric equations, an investigation of limits, and an introduction to the derivative. Throughout the course, an emphasis is placed on critical thinking and problem solving. A student wishing to make the transition from Algebra II to Precalculus Honors must do preparatory work during the summer and demonstrate mastery of specific topics through satisfactory completion of a placement exam taken prior to the beginning of the school year. Essential Questions 1. Why do we use different sets of numbers in mathematics? 2. Why do we use functions to model real-life situations? 3. How do graphs of functions add a visual perspective to a mathematical problem? 4. How does the use of technology enhance student understanding? 5. How do the skills learned in Precalculus provide a foundation for the study of more advanced mathematics? 6. How do basic rules for transforming functions apply throughout Precalculus? 7. How important is the language of mathematics? Assessment 1. Homework 2. Classwork 3. Quizzes 4. Tests 5. Projects 6. Semester exams 7. Examples Assessment - As an end-of-semester culminating assessment, students collaboratively and independently solve higher-level modeling problems using the functions learned that semester, such as modeling the flight of a rocket or determining the required length of a tunnel under a mountain. Students must be able to apply the proper function to the proper situation (Benchmark 2e), draw pictures of proposed model functions (Benchmarks 3 and 5), identify boundary conditions on models (Benchmarks 1, 2a and 2b), and communicate their results in context. Skills Benchmarks A student will be able to: 1. Solve equations and inequalities using a variety of techniques, including simplification, factoring, the quadratic formula, sign diagrams, logarithms, and trigonometric identities.
2. Work with polynomial, rational, radical, exponential, logarithmic, and sinusoidal functions as follows: a. Identifying domain and range b. Locating intercepts c. Sketching a graph d. Applying transformations e. Modeling real-world applications using the appropriate function 3. Graph the sinusoidal and rational trigonometric functions and the inverse functions. 4. Solve triangle problems using right triangle ratios, the Law of Cosines, and the Law of Sines. 5. Write equations of the four conic sections and sketch the resulting graphs, identifying significant points. 6. Work with two-dimensional vector quantities in the following ways: a. Represent them as magnitude and direction (in a variety of direction systems including standard rotation and compass quadrant) b. Represent them as horizontal and vertical components. c. Compute and interpret the dot product. 7. Find the value of the limit of a function graphically and algebraically, including end behaviors of functions. 8. Compute and interpret the derivatives of simple polynomial functions and their products and quotients. Units 1. Equations and Inequalities 1. Functions and Graphs 2. Polynomial and Rational Functions 3. Inverse, Exponential and Logarithmic Functions 4. Sequences, Series, and Probability 5. The Trigonometric Functions 6. Analytic Trigonometry 7. Topics from Analytic Geometry 8. Systems of Equations and Inequalities 9. Parametric and Polar Equations 10. Topics from Calculus Textbooks and Resources 1. Larson, Ron. Precalculus with Limits: A Graphing Approach, Cengage Learning, 2016 th
(7 ed.) 2. TI-84 Graphing Calculator Updated August 2018
250 - Calculus Course Description Grades: 11-12 Group: I Units: 1.0 Prerequisite: 240 - Precalculus or 245 - Precalculus Honors Calculus provides an introduction to the mathematics of differential and integral calculus. Limits and derivations are introduced theoretically and technically, but the emphasis is on problem solving and applications. The indefinite integral, separable differential equations, and the definite integral are studied and applications of the definite integral are emphasized. Students are expected to understand the mechanics of the course as well as critically analyze and solve problems that require creative strategies. Students are expected to communicate their results both verbally and written in a complete and correct fashion. Collaborating together in a group project at the end of spring semester, they will tie together what they have learned throughout the year by collecting raw data and using that data to answer a series of calculus-based questions. Essential Questions 1. How is calculus different from the mathematics a student has studied previously? 2. How are the ideas of the tangent to a curve and area under a curve related to collected data? 3. How is it that the concept of a limit arises throughout the study of calculus? 4. What is the relationship between differentiation and integration? Assessment 1. Homework 2. Quizzes 3. Tests 4. Project - students will be required to exercise and develop purposeful collaboration and effective communication to mathematically model the volume of a vase. Students will also be required to think critically to approach an analysis of the arc length and surface area of the vase. Students will use problem solving strategies to creatively reach conjectures to determine differences between the actual volume and arc length of the vase versus the theoretical volume and arc length of the vase. As final submittal in addition to the calculations and work for the analysis of the vase, a reflection of the student’s work will be submitted. 5. Semester exams Skills Benchmarks 1. Students should recognize calculus as a valuable tool, with the concept of "limit" at its core.
2. Students should have a clear understanding of functions and should be comfortable working with functions represented in various ways: graphs, charts, analytical, or verbal. 3. Students should understand the meaning of the derivative as a rate of change. 4. Students should understand that a definite integral is a limit of Riemann sums and that it is an accumulation of a rate of change. 5. Students should understand that derivatives and integrals are inverses as stated in the Fundamental Theorem. 6. Students should be able to model a written description of a physical situation with a function, differential equation, or an integral. 7. Students should be able to analyze, process, and interpret real-world data with various features of the graphing calculator. Units 1. Differentiation – Limits, Continuity, Rates of Change, Exponential Functions, Logarithmic Functions 2. Applications of Differentiation – Extrema of Functions, Graph Sketching, Business and Economics Applications, Implicit Differentiation, Related Rates 3. Integration – Area under a Curve, Average Value, Exponential Functions, Logarithmic Functions 4. Applications of Integration – Growth and Decay Models, Probability Density Function, Volumes of Solids of Revolution, Volumes by Cross-Sections, Arc length, Surface Area, Differential Equations 5. Supplemental enrichment to higher level mathematics if time permits – Simplifying, solving rational equations and expressions, expanding binomial expressions, factoring, graphing piecewise functions Textbooks and Resources 1. Bittinger, Ellenbogen, and Surgent, Calculus and Its Applications, Addison-Wesley, Tenth Edition 2. TI-84 Graphing Calculator 3. Calculus in Motion Software 4. Logger Pro Software Updated May 2016
252 - Statistics Course Description Grades: 11-12 Group: I Units: 0.5 Offered: Fall/Spring Prerequisite: 220 - Algebra II Statistics is the science and art of collecting, analyzing, and drawing conclusions from data. In today’s increasingly data-centric world, making and defending data-driven decisions is an essential hallmark of quantitative literacy. In this course, students will learn and apply methods for quality data collection, summarizing and analyzing data, and drawing valid inferences from data, all with the aid of appropriate computational tools. Students will develop a critical eye for the application of statistics in the media and an appreciation for the power of statistical thinking in a variety of fields. Note: This course cannot fulfill the mathematics graduation requirement of one unit beyond Algebra II. This course is intended for seniors for whom a semester of mathematics study is appropriate and other students seeking an elective option in mathematics. 252 - Statistics is not a prerequisite for 265 - AP Statistics. Students enrolled in 265 - AP Statistics may not enroll in 252 - Statistics. Essential Questions 1. How can mathematical and statistical reasoning about data be used to assess risks and evaluate conclusions and claims? 2. How can graphical displays of data amplify our abilities to determine patterns within and associations between the variables under consideration? 3. How can we design studies and experiments from both a qualitative and quantitative standpoint to maximize our ability to draw valid conclusions from them? 4. How can we use the language of probability to describe the inferences we make from statistical estimates? Assessment Daily homework throughout the unit 1. One quiz and test per major unit of study 2. Final research project: a study designed to answer a student-generated research question and draw an inference with statistical justification; a summary paper, approximately 8 pages in length; and a presentation of the research to the class or an authentic audience (e.g., a project on student homework data presented to academic administrators) 3. Example assessment: Students will design an experiment to assess the goodness of paper airplane designs in terms of air flight distance. Students must collaborate to determine a statistically valid way to test paper airplanes across construction material,
testing conditions, and constraints on measurement (student abilities to throw, available measuring equipment) [Benchmark 4]. Students will use simulation to draw an appropriate statistical inference [Benchmark 6]. In this process, they will amplify their ability to think critically about experimental design and communicate effectively using the language and methods of statistics. Skills Benchmarks 1. Students will be able to differentiate between categorical data and quantitative data, as well as differentiate between continuous and discrete quantitative data. 2. Students will be able to describe data with respect to what is typical (shape, variability) as well as what is not (outliers). 3. Students will be able to generate appropriate graphical displays for data and draw conclusions within and across the represented groups. 4. Students will be able to differentiate between types of data collection measures (surveys, experiments, observational studies) and select the most appropriate one. 5. Students will be able to recognize potential bias in the design of a study. 6. Students will be able to compute the probabilities and conditional probabilities of events and combinations of events within a sample space. 7. Students will be able to describe the ideal sampling distribution of a statistic (sample mean, sample proportion) based upon the sample size and true parameter values for a population under consideration. 8. Students will make and critically evaluate inferences from a point estimate of an unknown population parameter. Units 1. Exploring Univariate Data a. Types of data b. Graphical displays of univariate data c. Measures of center and spread d. Comparison of subgroups across a variable 2. Design of Studies a. Design of surveys and types of bias b. Simple random sampling c. Observational studies versus experiments d. The role of randomization and valid conclusions from experiments 3. Foundations of Statistical Inference a. Independent events and conditional probability b. Normal distributions and associated probabilities c. The notion of a sampling distribution 4. Techniques of Statistical Inference a. Language and interpretation of confidence intervals, including point estimates and margins of error b. Language of significance tests (null and alternative hypothesis, one-tailed vs. two-tailed, test statistic)
5. Technological Tools for Statistics [as time permits] a. Data organization and processing b. Data visualization 6. Exploring Bivariate Data [as time permits] a. Numerical and graphical summaries of bivariate data b. Quantitative and qualitative justification of linear association (correlation coefficient, residual analysis) c. Other association models (quadratic, exponential) Textbooks and Resources 1. Main course textbook: Statistics and Probability Through Applications, Third Edition, Daren Starnes and Josh Tabor, 2017, Bedford, Freeman, and Worth Publishers. 2. TI-84 graphing calculator 3. Online statistical applets on publisher site Last updated January 2019
255 - Advanced Placement Calculus AB Course Description Grades: 11-12 Group: I Units: 1.0 Prerequisite: 240 - Precalculus or 245 - Precalculus Honors AND teacher recommendation Fee: $94 AP Exam Fee Advanced Placement Calculus AB provides a detailed introduction to the mathematics of differential and integral calculus. It covers the theory and applications of limits, derivatives, and the indefinite and definite integral. Students are expected to understand both theory and application--mastering the mechanics of the course and critically analyzing and solving nonroutine problems that require creative strategies. Moreover, communicating results with written justifications is essential. The course follows the AP Calculus syllabus as described by the College Board, and all students take the AP exam at the end of the second semester. Essential Questions 1. How is calculus different from the mathematics a student has studied previously? 2. How are the concepts of the area under a curve and the tangent to a curve related to each other? 3. How does the concept of a limit arise throughout the study of calculus? 4. What is the relationship between differentiation and integration? 5. What was the inspiration for creating this “mathematics of motion”? Assessment 1. Homework 2. AP practice sets, usually completed in collaborative groups 3. Quizzes 4. Tests 5. Fall semester exam 6. Final assessment and mock AP exam in the spring 7. Spring Advanced Placement exam 8. Example: After teaching Part 2 of the Fundamental Theorem, students are divided into groups of three or four and assigned a free response problem from the 2005 AP Exam. This problem describes a scenario where sand on the beach is washed away by the tide and replenished by a pump simultaneously with both rates modeled by trig functions. In order to calculate the amount of sand on the beach at a given time and the time at which the sand is a minimum, students must create the “accumulation function” using the narrative description of this function. This is a “calculator active” problem, requiring a high level of critical thinking and, for most students, a lengthy period of collaborative discussion. Certain calculator skills are necessary, but students must also have a clear understanding of the First Derivative Test and be able to execute a creative application of the Extreme Value Theorem. All results require a written justification.
Skills Benchmarks 1. Students should recognize calculus as a valuable tool, with the concept of "limit" at its core. 2. Students should have a clear understanding of functions and should be comfortable working with functions represented in various ways: graphs, charts, analytical, or verbal. 3. Students should understand the meaning of the derivative as a rate of change. 4. Students should understand that a definite integral is a limit of Riemann sums and that it is an accumulation of a rate of change. 5. Students should understand that derivatives and integrals are inverses as stated in the Fundamental Theorem. 6. Students should be able to model a written description of a physical situation with a function, differential equation, or an integral. 7. Students should be able to analyze, process, and interpret real-world data with various features of the graphing calculator Units 1. Limits and Continuity 2. The Derivative and Techniques of Differentiation / the Derivative as a Function 3. Applications of Differentiation---Extrema of Functions, The Mean Value Theorem, Related Rates, Particle Motion, Optimization 4. Integration and Integration Techniques 5. Logarithmic and Exponential Functions 6. Differential Equations and Slope Fields 7. Applications of Integration--Area under a Curve, Average Value, Accumulation Functions, Particle Motion, Volumes of Solids of Revolution, Volumes by Cross-Sections 8. Other Topics: Inverse trig derivatives and integrals and L’Hopital’s Rule Textbooks and Resources 1. Larson, Ron and Bruce H. Edwards, Calculus of a Single Variable, Brooks/Cole, Ninth Edition 2. TI-84 Graphing Calculator 3. AP Free Response sets from previous years 4. AP Multiple Choice sets from previous years 5. Calculus in Motion Software 6. Desmos Software Updated July 2018
260 - Advanced Placement Calculus BC Course Description Grades: 11-12 Group: I Units: 1.0 Prerequisite: 245 - Precalculus Honors AND teacher recommendation Fee: $94 AP Exam Fee AP Calculus BC is a full-year course designed to develop a students' understanding of the concepts of calculus and to provide experience with its methods and applications. AP Calculus BC includes all of the AB material plus: additional techniques of differentiation and integration; parametric, polar, and vector functions; and infinite sequences and series. The content in the course has many applications to physics and other mathematical sciences and are methods used daily in the engineering world. The course follows the AP Calculus BC syllabus as described by the College Board, and all students take the AP exam at the end of the second semester. Essential Questions 1. How is calculus different from the mathematics a student has studied previously? 2. How are the ideas of the area under a curve and the tangent to a curve related to each other? 3. How is it that the concept of a limit arises throughout the study of calculus? Assessment 1. Homework 2. AP practice sets 3. Tests 4. Semester exam (fall only) 5. Spring Advanced Placement exam 6. AP Free Response practice sets are given throughout the year to help students revisit previous topics and require them to think creatively about how to approach any given problem. These sets are comprised of two problems from past year's AP Free Response exams. They are multi-part and each part requires the uses of concepts from different units. Throughout these problems, students are required to write a clear justification of their mathematical process and conclusion. Skills Benchmarks 1. Students should recognize calculus as a valuable tool, with the concept of "limit" at its core. 2. Students should have a clear understanding of functions and should be comfortable working with functions represented in various ways: graphs, charts, analytical, or verbal. 3. Students should understand the meaning of the derivative as a rate of change.
4. Students should understand that a definite integral is a limit of Riemann sums and that it is an accumulation of a rate of change. 5. Students should understand that derivatives and integrals are inverses as stated in the Fundamental Theorem. 6. Students should be able to model a written description of a physical situation with a function, differential equation, or an integral. 7. Students should be able to analyze, process, and interpret real-world data with various features of the graphing calculator Units 1. 2. 3. 4. 5. 6. 7. 8.
Limits, Continuity, and Definition of Derivative Calculating Derivatives Applications of the Derivative Introduction to Area and Integrals Calculating Integrals Applications of the Integral Parametrics, Polars, and Vectors Sequences and Series
Textbooks and Resources 1. Larson, Ron and Bruce H. Edwards, Calculus of a Single Variable, Brooks/Cole, Ninth Edition 2. TI-84 Graphing Calculator 3. AP Free Response sets from prior years 4. AP Multiple Choice sets from prior years Updated July 2016
265 - Advanced Placement Statistics Course Description Grade: 11-12 Group: I Units: 1.0 Prerequisite: Application and departmental recommendation required; minimum requirements include fall semester average of 80 or higher in Precalculus or 85 or higher in Algebra II Corequisite: 217 - Precalculus or higher-level math Fee: $94 AP Exam Fee AP Statistics is designed to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. It is built around four main topics: exploring data, planning a study, probability as it relates to distributions of data, and inferential reasoning. Small group learning activities are a major element of the course. In these groups, students are encouraged to communicate effectively using the language of statistics, to search for mutual understanding, and to creatively solve problems they have not seen before. The course follows the Statistics AP syllabus as described by the College Board and all students take the AP examination at the end of the second semester. Essential Questions 1. How does the analysis of data lead to a study of patterns and departures from patterns? 2. Why is a well-developed plan critical to the collection of data? 3. How is probability used to anticipate the distribution of data under a given model? 4. How does statistical inference guide the selection of appropriate models? Assessment 1. Homework 2. Group work 3. Quizzes 4. Tests 5. Chapter-specific projects 6. Previous AP multiple choice and free response questions 7. Semester exam (fall semester only) 8. Spring culminating project 9. Spring Advanced Placement exam Skills Benchmarks 1. Students should be able to detect important characteristics such as shape, location, variability, and unusual values when examining distributions of data. 2. Students should be able to develop a careful plan to collect data, a plan that allows for the appropriate type of analysis. 3. Students should be able to use probability distributions to describe data. 4. Students should be able to use statistical inference to guide the selection of appropriate models used to draw conclusions from data.
Units 1. Exploring Data 2. Modeling Distributions of Data 3. Describing Relationships 4. Designing Studies 5. Probability: What Are the Chances 6. Random Variables 7. Sampling Distributions 8. Estimating with Confidence 9. Testing a Claim 10. Comparing Two Populations or Groups 11. Inference for Distributions of Categorical Data 12. More about Regression Textbooks and Resources 1. Starnes, Daren S., Daniel S. Yates, and David S. Moore, The Practice of Statistics, New York: W. H. Freeman and Company, 6th Edition 2. TI-84 Graphing Calculator 3. Free response and multiple choice problems from previous AP exams 4. Various online applets Updated August 2018
270 - Honors Multivariable Calculus (Post-AP) Course Description Grade: 12 Group: I Units: 0.5 Offered: Fall only Prerequisite: 250 - Calculus or higher level mathematics AND teacher recommendation Many of the topics of this course extend concepts of single-variable calculus to functions with more than one independent variable. The course begins with a review of polar and parametric equations, followed by a thorough study of vectors and the geometry of space and surfaces in space. Other topics include differentiation and integration of vector-valued functions; rates of change and extrema of functions of several variables; multiple integration; and important theorems of vector calculus. In every case, related applications are emphasized. Students regularly engage in both individual and collaborative problem-solving, applying the techniques learned to non-routine problems in a variety of disciplines. Students employ multiple representations of problems with the aid of appropriate technology to visualize and conceptualize higher-level abstractions. Additional emphasis is placed on communicating an understanding of concepts and mathematical models, not just following set processes. Essential Questions 1. Much like in single-variable calculus, where we zoom in on a point on the graph of a differentiable function and the graph becomes indistinguishable from the graph of its tangent line; what would it look like to zoom in on a point on a surface of a graph of a function of two variables? How can we approximate this new function? 2. What is the distinction between a vector-valued function r and a real-valued function f, g, or h? Why is a vector-valued function useful? 3. How can we evaluate the same volume in different ways using multiple integrals? Why is it important to understand several ways of evaluating the same problem? 4. What is the distinction between a vector-valued function and a vector field? What are some properties of vector fields that can be used to model real-world processes? Assessment 1. Weekly problem sets 2. Unit tests 3. Cumulative semester exam The weekly problem sets in Honors Multivariable Calculus are higher-level problems that require students to apply current techniques to new problems, evaluate multiple approaches to the same problem, and create mathematical proofs of novel conjectures. Students work collaboratively on these assessments without teacher input, discovering the best method for approaching a problem and discussing the quality of their own solutions. For example, students
will compare two methods for determining the shortest distance from a point to a plane: vector projections (Benchmark 1) and simultaneous equations (Benchmark 2). Skills Benchmarks A student will be able to: 1. Use vectors in space to represent lines and planes, and also to represent quantities such as force and velocity. 2. Use three-dimensional coordinate systems (rectangular, cylindrical, and spherical) to represent points, planes, spheres, and cylindrical and quadric surfaces. 3. Use and interpret vector-valued functions representing planes and surfaces in space and study the motion of an object along a given curve or surface. 4. Sketch a function of more than one independent variable and extend the idea of the derivative to finding partial derivatives. 5. Find and interpret directional derivatives and the gradient of a surface. 6. Solve optimization problems involving functions of several variables. 7. Use iterated integrals, double integrals, and triple integrals to find areas and volumes in space. 8. Write and evaluate a line integral. 9. Write and evaluate a surface integral. (if time permits) Units 1. 2. 3. 4.
Parametric Equations and Vectors in a Plane Surfaces and Paths in Space Multivariable Functions and Differential Calculus Multivariable Integral and Vector Calculus
Textbooks and Resources 1. Text: Larson, Ron and Bruce H. Edwards, Calculus AP Edition, Chapters 11-15, Brooks/Cole, 9th ed. 2. TI-84 Graphing Calculator 3. TI nSpire CAS Graphing Calculator (in class) Updated May 2016
275 - Honors Linear Algebra (Post-AP) Course Description Grade: 12 Group: I Units: 0.5 Offered: Spring only Prerequisite: 270 - Honors Multivariable Calculus (fall semester) AND teacher recommendation (previous year) Linear Algebra is the branch of mathematics that explores linear systems, matrix representations of linear systems, vectors, and vector spaces. Other topics will include determinants, linear transformations, eigenvalues/eigenvectors, and related applications. Students will use computer software, such as MATLAB, to further explore these applications through programming. Students regularly engage in both individual and collaborative problemsolving, applying the techniques learned to non-routine problems in a variety of disciplines. Additional emphasis is placed on communicating understanding of concepts and mathematical models, not just following set processes. Essential Questions 1. When a matrix is converted to row-reduced echelon form, how has the nature of the matrix changed? How can we interpret the result of row-reducing a matrix? 2. How do terms such as subspace, span, and basis facilitate our understanding of a given set of vectors? 3. What exactly is an eigenvector of a matrix? How is the idea of the “eigenspace� of a given matrix useful? 4. When can a matrix be diagonalized, and why is a diagonal matrix useful? 5. What additional properties do orthogonal vectors and matrices possess, and how can these properties be applied? Assessments 1. Weekly problem sets 2. Unit tests 3. Final research and coding project - Each student decides on a natural, social, or technological system that they wish to model using the matrix techniques learned in the class. Students code an interface in MATLAB or similar software that allows the user to compare different aspects of the model. The final research project in Honors Linear Algebra is a capstone experience that combines original research - potentially involving navigating an unfamiliar discipline or integrating an unusual set of constraints - with the mathematical literacy developed throughout the student’s Lovett career. The student gets a sense not only of why matrix models are so prevalent in a variety of disciplines, but a chance to draw their own conclusions about a model and its
implications (Benchmarks 4 and 7). Students gain experience in modeling software and using its facilities to create interactive, impactful presentations of results (Benchmark 8). Skills Benchmarks A student will be able to: 1. Solve linear systems using direct methods and interpret their solution(s). 2. Use matrix operations and compute inverses and determinants of n x n matrices by a variety of methods. 3. Understand the terms “subspace,” “span,” and “basis” as they relate to a set of vectors in any dimension of Rn. 4. Find eigenvalues and eigenvectors of a given matrix and interpret what they mean. 5. Determine whether a set of vectors is orthogonal, and whether such sets are orthonormal. 6. Express rotation and reflection transformations as orthogonal matrices and apply these transformations to vectors in R2 and R3. 7. Use the tools of linear algebra in real world applications. 8. Do basic programming in the context of a mathematical software package, such as MATLAB. Units 1. 2. 3. 4.
Matrices and Systems of Linear Equations Eigenvalues and Eigenvectors Orthogonality and Distance Coding in MATLAB
Textbooks and Resources 1. Poole, David , Linear Algebra: A Modern Introduction, Brooks/Cole, 3rd ed. 2. TI-84 Graphing Calculator 3. TI nSpire CAS Graphing Calculator (in class) 4. MATLAB, The MathWorks, Inc. Updated May 2016