Precalculus, Quarter 3, Unit 3.1 Polar and Rectangular ... · math models for wave behavior and planetary motion ... the area under a curve and the volume of a curve using partitioning,

Cumberland, Lincoln, and Woonsocket Public Schools C-27 in collaboration with the Charles A. Dana Center at the University of Texas at Austin

Precalculus, Quarter 3, Unit 3.1

Polar and Rectangular Coordinates

Overview Number of instructional days: 7 (1 day = 45 minutes)

Content to be learned Mathematical practices to be integrated • Make a table of values to find values of r for

given values of θ.

• Plot points (r, θ).

• Graph equations in polar coordinates.

• Convert between rectangular and polar coordinates.

• Represent complex numbers in polar form.

• Find the nth root of complex numbers.

Make sense of problems and perervere in solving them.

• Work between different representations.

• Use technology to solve problems.

Model with mathematics.

• Relate what has been learned in mathematics to every day life.

• Interpret results in the context of a problem.

Look for and make use of structure.

• Identify patterns and structures.

Attend to precision.

• Use labels of axes and units of measure correctly.

Essential questions • What examples can be found in nature that

display the characteristics of a polar graph? In what ways do they accomplish this?

• What advantages do polar coordinates have over rectangular coordinates?

• In the equation of the form !bar sin= or !bar cos= , what effect does changing the

values of a and b have on the graph?

• What impact does a polar equation have on the symmetry of a graph?

• What is the connection between the real numbers, complex numbers, and numbers represented in polar form?

Precalculus, Quarter 3, Unit 3.1 Polar and Rectangular Coordinates (7 days) Final, July 2011

C-28 Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

Written Curriculum

Grade Span Expectations

M(N&O)–AM –4 Accurately solves problems and demonstrates understanding of complex numbers by interpreting them geometrically and by computing with them ( e, g., adding, multiplying, dividing, finding the nth root, or by finding conjugates). Understands complex numbers as an extension of the real numbers (e.g. arising in solutions of polynomial equations). Manipulates complex numbers using rectangular and polar coordinates. Knows the fundamental theorem of algebra and knows that non-constant polynomials always factor into linear factors over the complex numbers. (Local)

Clarifying the Standards

Prior Learning

In Algebra 2, students learned the operations using complex numbers. They also used complex numbers as solutions to polynomial equations.

Current Learning

Students manipulate complex numbers using rectangular and polar coordinates.

Future Learning

The study of the rectangular and polar coordinate systems will lead to the development of parametric equations, which is useful for studying motion in calculus. The derivatives of parametric, polar, and rectangular equations will be emphasized in calculus, with emphasis on related rates and real-world applications.

Additional Research Findings

Science for All Americans indicates that rectangular, polar, and parametric equations are the patterns in math models for wave behavior and planetary motion (pp. 52–56, 124, and 145–151).

Principles and Standards for School Mathematics indicates that polar coordinates are used to analyze geometric situations. The polar coordinate representation can be simpler and may be more useful for solving certain problems (pp. 308, 314).



Sequences and Series


Content to be learned Mathematical practices to be integrated • Identify arithmetic and geometric sequences.

• Find the nth term of arithmetic and geometric sequences.

• Generalize to find a specific term of an arithmetic or geometric sequence.

• Compute partial sums of infinite arithmetic and geometric sequences.

• Determine when and if a geometric series converges.

• Find the sum of an infinite geometric series.

• Connect arithmetic and geometric sequences to linear and exponential functions, respectively.

Look for and express regularity in repeated reasoning.

• Look for general methods, patterns, repeated calculations, and shortcuts.

• Look for patterns to find generalizations.


• Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively.

• Calculate and compute accurately (including technology).

Model with mathematics.

• Relate what has been learned in mathematics in everyday life.

Essential questions • What algebraic concept relates to the common

difference of an arithmetic sequence?

• What algebraic concept relates to the common ratio of a geometric sequence?

• Why is the initial condition in a recursive definition important?

• What is the connection among a geometric sequence, its partial sum, and an exponential function?

• What are the similarities and differences between an arithmetic and a geometric sequence defined explicitly or recursively?

• What real-world applications are modeled by arithmetic and geometric series?

• What is the primary distinction between a sequence and a series?

• What is the connection among an arithmetic sequence, its partial sum, and a linear function?

Precalculus, Quarter 3, Unit 3.2 Sequences and Series (10 days) Final, July 2011


Written Curriculum


M(F&A)–12–1 Identifies arithmetic and geometric sequences and finds the nth term; then uses the generalization to find a specific term. (Local)

M(F&A)–AM-1 Identifies and computes partial sums of infinite arithmetic and geometric sequences, determines when an infinite geometric series converges, and finds its sum. Connects arithmetic and geometric sequences to linear and exponential functions, respectively. (Local)


Prior Learning

In kindergarten, students identified patterns to find the next one, two, or three elements. In grade 1, patterns were represented in models, tables, or sequences. Student also found missing elements in a numerical pattern. In grade 4, students studied nonlinear patterns and wrote rules in words or symbols to find the next case. In grade 5, patterns were studied in problem situations. In grade 6, students wrote patterns in expression or equation form, using words or symbols to express the generalization of a linear relationship. In grade 7, students extended this writing of patterns to nonlinear relationships. In grade 8, students continued their study of these concepts in more depth. In grade 9 and 10, students solved problems involving patterns, and in grade 11, they studied first, second, and third differences.

Current Learning

Students identify arithmetic and geometric sequences and find the nth term. Students determine when an infinite geometric series converges and finds its sum. Students connect concepts to linear and exponential functions. Students compute sums and partial sums of arithmetic and geometric series.

Future Learning

The study of sequences and series ultimately leads to an analysis of limits, integral calculus, engineering calculus, and applications of higher calculus. Students will compute limits for various functions, the integral as a net accumulator, and Taylor series for various functions. In addition, students will calculate the area under a curve and the volume of a curve using partitioning, slicing, and cross-sections.


According to Beyond Numeracy, infinite series and their applications constitute an important area of mathematical analysis. Used informally by mathematicians long before they were completely understood, series appeal to our intuitions about numbers and infinity. Annuities provide a practical application of infinite geometric series. Geometric series also arise in determining the quantity of medicine in the blood of a person on a long-term regimen of the medicine. Finding derivatives, integrals, solving differential equations, and working with complex and imaginary numbers are simplified and represented by power series (pp. 221–224).



Permutations, Combinations, Probability

Overview Number of instructional days: 13 (1 day = 45–50 minutes)

Content to be learned Mathematical practices to be integrated • Solve real-world problems using permutations

and combinations.

• Calculate permutations and combinations using factorial notation.

• Compute the theoretical and experimental probabilities for a sample spaces containing equally and non-equally likely outcomes.

• Calculate and compare the odds and probability of an event.

• State the differences between independent and dependent events using lists, tables, and Venn diagrams.

• Determine the union, intersection, and the complement of sets.

• Determine the complement and the negation of a single set.

• Solve problems involving conditional probability.

• Develop Pascal’s triangle and the binomial theorem with reference to (a + b)n

Make sense of problems and preserve in solving them.

• Think about simpler problems to help solve more complex problems involving permutations, combinations, and probability.

Reason abstractly and quantitatively.

• Use abstract reasoning in solving problems involving permutations, combinations, and probability, and check answers to ensure they are quantitatively sound.

• Represent mathematically what is read regarding a permutation, combination, or probability situation and do something with it.

Essential questions • What are the similarities and differences

between permutations and combinations?

• In what real-world situations would you encounter conditional probability?

• How does one determine whether events are independent or dependent?

• What are the similarities and differences between odds and probability?

• What careers use probability?

• What is the relationship between theoretical and experimental probability?

Precalculus, Quarter 3, Unit 3.3 Permutations, Combinations, and Probability (13 days) Final, July 2011


Written Curriculum


M(DSP)-12-4 Uses counting techniques to solve problems in context involving combination or permutations using a variety of strategies (e.g., nCr, nPr, or n!); and finds unions, intersections, and complements of sets. (Local)

M(DSP)-12-5 For a probability event in which the sample space may or may not contain equally likely outcomes, predicts the theoretical probability of an event and tests the prediction through experiments and simulations; compares and contrasts theoretical and experimental probabilities; finds the odds of an event and understands the relationship between probability and odds. (Local)

M(DSP)-AM-5 Solves probability problems (e.g., by applying concepts of counting, random variables, independence/dependence of events, and conditional probability). (Local)


Prior Learning

In grade 1, students worked with concepts of equally likely, more likely, and less likely. In grade 2, the terms certain and impossible, counting techniques, and tree diagrams were introduced. In grade 3 combinations, simple permutations, and sample space are introduced. In grade 4, theoretical probability of an event was expressed as a fraction. In grade 5, students worked with experimental versus theoretical probability. Grade 6 introduced the fundamental counting principle and the concept of a “fair” game. In grade 7, students were introduced to more advanced permutations and compare/contrast experimental versus theoretical probability. In grades 9 and 10, the concepts of permutations, combinations, fundamental counting principle, and experimental probability were expanded.

Current Learning

Students continue learning the concepts of permutations, combinations, and the similarities and differences between them. Students continue working with theoretical and experimental probability. New concepts are introduced, including odds of an event, independent/dependent events, and conditional probability.

Future Learning

Concepts of probability are used in the field of economics, health, politics, actuarial science (e.g. insurance), gambling, and statistical studies.




Research information on this content can be found in the following references: • Principles and Standards for School Mathematics, (pp. 324 and 331–333); • Beyond Numeracy, “Multiplication Principle,” (pp. 150-153), “Probability” (pp. 187–191); • Benchmarks for Science Literacy, (p. 129); and • A Research Companion to Principles and Standards for School Mathematics, (pp. 216–225).

This research indicates that students bring to class beliefs and conceptions about chance that are often incorrect, and it identifies numerous teaching opportunities to assess students’ thinking about chance and to change their beliefs. The research also states that high school students should learn to identify mutually exclusive, joint, and conditional events by drawing on their knowledge of combinations, permutations, and counting to compute the probabilities associated with such events.

Notes About Resources and Materials





Vectors


Content to be learned Mathematical practices to be integrated • Apply the properties of numbers and the field

properties of sets of numbers to solve problems involving vectors.

• Interpret vectors algebraically and geometrically.

• Perform vector addition and scalar multiplication in the plane, both algebraically and geometrically, using arithmetic properties.

Use appropriate tools strategically.

• Use various tools, including technology, to help solve problems.

• Use technology to visualize results.


• Use precise mathematical vocabulary and clear and accurate definitions and symbols to communicate efficiently and effectively.

Essential questions • What are the similarities and differences

between the magnitude of a number and the vector representation for the number?

• How does absolute value impact the vector representation of a number?

• In what situations is the magnitude of a number used in place of the vector of that same number, and vice versa?

Precalculus, Quarter 3, Unit 3.4 Vectors (10 days) Final, July 2011


Written Curriculum


M(N&O)–AM–8 Applies properties to add and multiply numerical matrices with attention to the arithmetic properties of these operations. Algebraically and geometrically interpret vectors, vector addition, and scalar multiplication in the plane, with attention to arithmetic properties. Knows and uses the principle of mathematical induction. (Local)


Prior Learning

In grade 1, the properties of numbers (odd, even, composition, decomposition) and field properties (commutative, identity, +) were introduced to solve problems and simplify computations. In grade 2, associative addition was introduced. In grade 3, the multiplication property of zero for single-digit whole numbers was taught. The field properties of identity multiplication and the commutative multiplication of single-digit whole numbers were taught. In grade 4, the multiplication property of zero, remainders, and field properties (commutative, associative, and identity multiplication) were added. In grade 5, divisibility rules of numbers and field properties of distributive were introduced. In grade 6, prime factorization, multiplicative property of one, and additive inverse began. In grade 7, students demonstrated a conceptual understanding of field properties. In grade 8, prime factorization and the field properties of numbers included exponents. At this time, addition and subtraction was represented in a nontraditional way. (Example: a ∆ b = a+b-1. Is ∆ commutative?)

In grades 9 and 10 (algebra 1), comparing and contrasting properties of numbers, number systems, and subsets of number systems was studied. In grades 11 and 12 (algebra 2), subsets of numbers were studied in depth under the field properties of numbers (Example: Are the set of numbers closed under the property of addition or multiplication?).

Current Learning

Current learning includes applying the properties of numbers and the field properties of sets of numbers to include:

• Addition and scalar multiplication of vectors

• Algebraic and geometric interpretation of vectors

• Use of arithmetic properties to vector analysis in the plane

Future Learning

Vector analysis will introduce the student to parametric equations, graphing relations, and the application of derivatives to motion. Applications of the derivative will include optimization, and related rates will incorporate vector analysis.




According to Beyond Numeracy, vectors are used with quantities that require two or more dimensions to be specified. They play a primary role in linear algebra as well as in many other areas of applied mathematics (pp. 136–140).

Notes About Resources and Materials



Documents

Precalculus, Quarter 3, Unit 3.1 Polar and Rectangular ... · math models for wave behavior and planetary motion ... the area under a curve and the volume of a curve using partitioning,