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Binomial Theorem and Polynomial Expansion Precalculus, Quarter 2, Unit 2.1 Binomial Theorem and Polynomial Expansion Overview Number of instructional days: 10 (1 day = 45–60 minutes) Content to be learned Mathematical practices to be integrated • Expand binomials using Pascal’s triangle. Reason abstractly and quantitatively • Develop binomials using binomial theorem. • Connect the expanded form of the binomial to • Generate Pascal’s triangle. Pascal’s triangle. • Calculate a given term utilizing Pascal’s Attend to precision triangle. • Construct the coefficients with the binomial • Compute a given coefficient using binomial theorem calculations and correct Pascal rows. theorem. Look for and make use of structure • Produce the original binomial given a term or • Identify the degree, term, row, coefficient from an expanded binomial with leading coefficient the pattern of Pascal. of 1. Essential questions • How does Pascal’s triangle help us to expand • What are similarities and differences between binomials? using Pascal’s triangle versus Binomial theorem? • What does binomial theorem do to help us in binomial expansion? Hobbs Municipal Schools with process support from The Charles A. Dana Center at the University of Texas at Austin 17 Precalculus, Quarter 2, Unit 2.1 Binomial Theorem and Polynomial Expansion (10 days) Written Curriculum Common Core State Standards for Mathematical Content Arithmetic with Polynomials and Rational Expressions A-APR Use polynomial identities to solve problems. A-APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.1 1 The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. Common Core Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents— and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions Hobbs Municipal Schools with process support from The Charles A. Dana Center at the University of Texas at Austin 18 Precalculus, Quarter 2, Unit 2.1 Binomial Theorem and Polynomial Expansion (10 days) 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In grade 3 students multiplied within 100. In grade 4 students used multiplication to compare and fractions by whole numbers. In grade 5, students extended multiplication to include fractions with themselves. In grade 6 students multiply multi-digit decimals. In grade 7 students applied the distributive property. In algebra 1 students factored quadratics, and worked with laws of exponents. In algebra 2 students have graphed polynomial functions and students have worked with series and sigma notation for geometric series. Current Learning Students will be expanding factors of polynomial functions using Pascal’s triangle and binomial theorem. Future Learning Students will be using binomial expansion and modified Pascal triangles in Number theory and Calculus, Statistics with binomial distributions. Additional Findings Students struggle with creating the triangle, with understanding the relationship between the triangle entries and the coefficients of the polynomial functions. Students have trouble with evaluating sigma notation. Hobbs Municipal Schools with process support from The Charles A. Dana Center at the University of Texas at Austin 19 Precalculus, Quarter 2, Unit 2.1 Binomial Theorem and Polynomial Expansion (10 days) Hobbs Municipal Schools with process support from The Charles A. Dana Center at the University of Texas at Austin 20 Precalculus, Quarter 2, Unit 2.2 Performing Operations with Complex Numbers Overview Number of instructional days: 15 (1 day = 45–60 minutes) Content to be learned Mathematical practices to be integrated • Perform arithmetic operations on complex Construct viable arguments and critique the numbers. reasoning of others • Graph complex numbers on the complex plain. • Relate how to perform distance and midpoint • Graph polar coordinates on the polar coordinate calculations on the new complex number system. system from the Cartesian coordinate system. • Use complex conjugates to simplify complex Look for and make use of structure numbers. • Connect the rectangular and polar forms of the • Factor polynomial functions to find the complex number system by distance and complex roots. angular polar form. Look for and express regularity in repeated • Calculate the distance and midpoint on the complex plane. reasoning • Perform factoring to find complex roots from previous related material. Essential questions • How are the rectangular and polar forms of • How are distance and midpoint calculations of complex numbers related? complex numbers similar and different from distance and midpoint of Cartesian Why is the complex conjugate important in • coordinates? calculations of complex numbers? • What is magnitude and how does it relate to graphing in complex number systems. Hobbs Municipal Schools with process support from The Charles A. Dana Center at the University of Texas at Austin 21 Precalculus, Quarter 2, Unit 2.2 Performing Operations with Complex Numbers (15 days) Written Curriculum Common Core State Standards for Mathematical Content The Complex Number System N-CN Perform arithmetic operations with complex numbers. N-CN.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Represent complex numbers and their operations on the complex plane. N-CN.4 (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. N-CN.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus 2 and argument 120°. N-CN.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. Use complex numbers in polynomial identities and equations. N-CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). Common Core Standards for Mathematical Practice 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the
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