Big Ideas Chapter 6: Exponential Functions and Sequences

Transcription

1 Big Ideas Chapter 6: Exponential Functions and Sequences We are in the middle of the year, having finished work with linear equations. The work that follows this chapter involves polynomials and work with quadratics. This chapter introduces students to exponential functions and sequences. Students will revisit exponential functions and learn about logarithmic functions in Algebra 2. The properties of exponents presented in the first lesson should be a review for students. Many of the problems involve numeric expressions, although there are algebraic expressions as well. The same properties of exponents are applied to radicals and rational exponents in the next lesson. The next three lessons are about exponential functions and the attributes of exponential growth and decay functions. Exponential equations are solved using the properties of exponents from the beginning of the chapter and graphically with a graphing calculator. At the end of the chapter, geometric sequences are introduced with the connection made to exponential functions. This is similar to the lesson on arithmetic sequences and the connection to linear functions earlier in the book. The last lesson on recursively defined sequences looks at both arithmetic and geometric sequences, writing them as recursive rules. Mathematical Progressions: Before: Middle School Work done prior to the Algebra 1 course, students have had experiences in writing and evaluating numerical expressions with whole number exponents. They have used properties of exponents, Evaluated square roots and cube roots, as well as used scientific notation and performed operations with numbers in scientific notation. The have spent much time with the use order of operations to evaluate expressions. Current: Algebra I Evaluate and simplify expressions with exponents, including rational exponents; find nth roots. Identify, evaluate, and graph exponential functions; use, identify, interpret, and rewrite exponential growth and decay functions; solve real-life problems. Solve exponential equations. Identify, extend, and graph geometric sequences; write geometric sequences as functions. Write terms and rules of recursively defined sequences; translate between recursive and explicit rules. After: Geometry Students will continue with these concepts in Geometry as they will spend much time with exponents in the calculations of finding area, volume, and the use of the Pythagorean Theorem. The concept of simplifying radicals appear throughout many standards and concepts throughout Geometry.

2 Chapter 6: Exponential Functions and Sequences Unit Planner TIME: 4 weeks UNIT NARRATIVE: In this chapter, students will continue to learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. Textbook Correlations: Additional Resources Big Ideas: Chapter 6 Lesson Tutorials, Game Closet, Student Journal, Skills Review Handbook, Dynamic Classroom, Interactive Lessons, Lesson Planning Tool, Puzzle Time, Practice A and B, Enrichment ESSENTIAL QUESTIONS: ACADEMIC VOCABULARY: Order of operations, Square roots, 1. How can you write general rules involving properties of exponents? Arithmetic sequences, power, exponent, base, scientific notation, 2. How can you write and evaluate an nth root of a number? n th root of a number, radical, index of a radical, exponential function, 3. What are some of the characteristics of the graph of an exponential function? independent variable, dependent variable, parent function, 4. What are some of the characteristics of exponential growth and exponential exponential growth, exponential growth function, exponential decay, decay functions? exponential decay function, compound interest, zero exponent, 5. How can you solve an exponential equation graphically? negative exponents, Product of Powers Property, Quotient of Powers 6. How can you use a geometric sequence to describe a pattern? Property, Power of a Power Property, Power of a Product Property, 7. How can you define a sequence recursively? Power of a Quotient Property, exponential equation, geometric sequence, common ratio, arithmetic sequence, common difference, explicit rule, recursive rule,

3 CLUSTER HEADING & STANDARDS: Create equations that describe numbers or relationships. A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand solving equations as a process of reasoning and explain the reasoning. A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Represent and solve equations and inequalities graphically. A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, and exponential functions. Understand the concept of a function and use function notation. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Interpret functions that arise in applications in terms of the context. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. MATHEMATICAL PRACTICE: All mathematical practice standards are addressed in every unit. MP 1 Make sense of problems and persevere when solving them. MP 2 Reason quantitatively and abstractly. MP 3 Construct viable arguments and critique the reasoning of others. MP 4 Model with mathematics. MP 5 Use appropriate tools strategically. MP 6 Attend to precision. MP 7 Look for and make use of structure. MP 8 Look for and express regularity in repeating reasoning. Supporting and Additional Standards Extend the properties of exponents to rational exponents. N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

4 Construct and compare linear, quadratic, and exponential models and solve problems. F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). Write expressions in equivalent forms to solve problems. A.SSE.3 Use the properties of exponents to transform expressions for exponential functions. Extend the properties of exponents to rational exponents. N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. Analyze functions using different representations. F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F.IF.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) ᵗ, y = (0.97) ᵗ, y = (1.01)12ᵗ, y = (1.2) ᵗ/10, and classify them as representing exponential growth or decay. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Build a function that models a relationship between two quantities. F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.

5 F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.* Build new functions from existing functions. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Learning Outcomes: Using the properties of exponents, rewrite a radical expression as an expression with a rational exponent. Using the properties of exponents, rewrite an expression with a rational exponent as a radical expression. Explain the properties of operations of rational exponents as an extension of the properties of integer exponents. Explain how radical notation, rational exponents, and properties of integer exponents relate to one another. Relate the domain of a function to its graph and to the quantitative relationship it describes, where applicable. Explain why a domain is appropriate for a given situation. Determine the differences between simple and complicated linear, exponential and quadratic functions and know when the use of technology is appropriate. Graph exponential functions, by hand in simple cases or using technology for more complicated cases, and show intercepts and end behavior. Write a function that describes a relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context. Use given and constructed arithmetic and geometric sequences, expressed both recursively and with explicit formulas, to model real-life situations. Distinguish between situations that can be modeled with linear functions and exponential functions to solve mathematical and real-world problems. Prove that linear functions grow by equal differences over equal intervals. Prove that exponential functions grow by equal factors over equal intervals. Determine when a graph, a description of a relationship, or two input output pairs (include reading these from a table) represents a linear or exponential function in order to solve problems. Compare tables and graphs of exponential and other polynomial functions to observe that a quantity, increasing exponentially, exceeds all others to solve mathematical and real world problems. Interpret the parameters in a linear or exponential function in terms of a context. End of the Unit Assessment: AC Driven DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Tie this concept into the use of Unit 1 and 2 to allow students to conceptually understand. Allow a graphic organizer to be used throughout unit with applied theorems and properties as they are developmentally discovered. Have students verbally explain the concept prior to constructing or sketching the function. Virtual Technology of real-world situations. Use of Rational expressions Multiple step with multiple simplifications. Multi standard processes Talk Moves Presentations ELD Literacy Standards Graphic organizers Highlighting : cloze activities SIOP strategies Real-world visuals Group collaboration Number talks Small group instruction One on one peer support Smaller size quantities Number Talks

6 Big Ideas Chapter 7: Polynomial Equations and Factoring Before: 8 th Grade A study of algebraic expressions generalizes ordinary arithmetic. This aspect of algebra revolves around the properties of operations, and it rewards looking at the structure of an expression. Learning about algebraic expressions provides an excellent opportunity for students to generalize all of their learning about rational numbers. For example, in kindergarten students learn that adding means putting together. They also learn that 3 apples can be added to 5 oranges to give 8 fruits, but that if 3 paper clips were added to 5 nails, then no sensible quantity will result. Thus, kindergarten students learn that only quantities with the same unit can be added. It is exactly this learning that is required when students start to add and subtract linear expressions (7.EE.1). For example, if students add 3a + 3b + 5a, it will make sense from Kindergarten learning that they can put 3a and 5a together to give 8a and that the result is 8a+ 3b. The work of kindergarten can be extended to the adding and subtracting of monomials. Unfortunately, this link with previous learning is rarely invoked. Instead, when it comes to adding and subtracting monomials students are usually taught to combine or collect like terms. Although we all know that when we ask students to combine or collect like terms we are asking them to add or subtract monomials, our students do not. Instead of connecting with previous learning, learning to combine like terms creates the opportunity for mistakes and misconceptions. For example, many students imagine that you cannot multiply 3a by 5b because 3a and 3b are not like terms. We recommend that students are not taught to combine or collect like terms but instead are taught to add or subtract monomials. Students perform arithmetic operations on polynomials, and understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Students use the structure of an expression to identify ways to rewrite it. Students also solve quadratic equations. During: Algebra Students perform arithmetic operations on polynomials, and understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Students use the structure of an expression to identify ways to rewrite it. Students also solve quadratic equations. The study of algebraic expressions is at the core of high school mathematics. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances. Reading an expression with comprehension involves analysis of its underlying structure. In courses beyond middle school, students interpret the structure of expressions and write expressions in equivalent forms to solve problems. In high school courses students perform arithmetic operations on polynomials, and rewrite rational expressions. After: Geometry and Algebra 2 Students will continue to factor and apply the use of Polynomials through the entire course of Algebra. This unit leads to graph and analyze the graphs of polynomial functions, including transformations. Add, subtract, multiply, divide, and factor polynomials, including cubic polynomials. Find solutions of polynomial equations and zeroes of polynomial functions. Use the Fundamental Theorem of Algebra. Write polynomial functions.

7 Polynomial Equations and Factoring This is a long chapter about polynomial equations and factoring. It is positioned here in the book in preparation for upcoming work with quadratics. In the first few lessons, the vocabulary and representation of polynomials is introduced, along with operations with polynomials. Operations of addition, subtraction, and multiplication are presented. The remainder of the chapter is on solving polynomial equations, which can be done when the polynomial is written in factored form. Students will use the Zero-Product Property to solve polynomial equations in factored form. Students will learn a series of techniques for factoring polynomials, aided by visual explorations using algebra tiles. Chapter 7: Polynomial Equations and Factoring TIME: 4 weeks UNIT NARRATIVE: In this unit, students draw on their foundation of the analogies between polynomial arithmetic and base ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi digit integers, and division of polynomials with long division of integers. Students identify zeroes of polynomials, including complex zeroes of quadratic polynomials, and make connections between zeroes of polynomials and solutions of polynomial equations. The role of factoring, as both an aid to the algebra and to the graphing of polynomials, is explored. Students continue to build upon the reasoning process of solving equations as they solve polynomial, rational, and radical equations, as well as linear and non linear systems of equations. This unit culminates with the fundamental theorem of algebra as the ultimate result in factoring. Connections to applications in prime numbers in encryption theory, Pythagorean triples, and modeling problems are pursued. An additional theme of this module is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Students use appropriate tools to analyze the key features of a graph or table of a polynomial function and relate those features back to the two quantities in the problem that the function is modeling. Textbook Correlations: Additional Resources Big Ideas: Chapter 7 Lesson Tutorials, Game Closet, Student Journal, Skills Review Handbook, Dynamic Classroom, Interactive Lessons, Lesson Planning Tool, Puzzle Time, Practice A and B, Enrichment ESSENTIAL QUESTIONS: ACADEMIC VOCABULARY: Commutative Property of Addition, 1. How can you add and subtract polynomials? Distributive Property, Greatest common factor, monomial, degree of a 2. How can you multiply two polynomials? monomial, polynomial, binomial, trinomial, degree of a polynomial, 3. What are the patterns in the special products? standard form, leading coefficient, polynomial, binomial, trinomial, 4. How can you solve a polynomial equation? factored form, Zero-Product Property, roots, repeated roots 5. How can you use algebra tiles to factor the trinomial x2 + bx + c into the product of two binomials? 6. How can you use algebra tiles to factor the trinomial ax2 + bx + c into the product of two binomials?

8 CLUSTER HEADING & STANDARDS: Perform arithmetic operations on polynomials. A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Understand the relationship between zeroes and factors of polynomials. A.APR.3 Identify zeroes of polynomials when suitable factorizations are available, and use the zeroes to construct a rough graph of the function defined by the polynomial. MATHEMATICAL PRACTICE: All mathematical practice standards are addressed in every unit. MP 1 Make sense of problems and persevere when solving them. MP 2 Reason quantitatively and abstractly. MP 3 Construct viable arguments and critique the reasoning of others. MP 4 Model with mathematics. MP 5 Use appropriate tools strategically. MP 6 Attend to precision. MP 7 Look for and make use of structure. MP 8 Look for and express regularity in repeating reasoning Solve equations and inequalities in one variable. A.REI.4b Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Write expressions in equivalent forms to solve problems. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 - y 4 as (x 2 ) 2 - (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 - y 2 )(x 2 + y 2 ). A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Learning Outcomes: Explain the connection between the factored form of a quadratic expression and the zeroes of the function it defines. Explain the properties of the quantity represented by the quadratic expression. Factor a quadratic expression to produce an equivalent form of the original expression. Identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are closed under the operations of addition, subtraction, and multiplication. Apply arithmetic operations of addition, subtraction, and multiplication to polynomials. Prove polynomial identities. Use polynomial identities to describe numerical relationships. Use all available types of functions to create such equations, including root functions, but constrain to simple cases. Create equations and inequalities in one variable to model real world situations. Create equations (linear and exponential) and inequalities in one variable and use them to solve problems. Solve quadratic equations by inspection taking square roots, completing the square, the quadratic formula and factoring. Use the process of factoring and completing the square in a quadratic function to show zeroes, extreme values, and

9 symmetry of the graph, and interpret these in terms of a context. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. End of the Unit Assessment: AC Driven DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Virtual Technology of real-world ELD Literacy Standards situations. Graphic organizers Use of Rational expressions Highlighting : cloze activities Multiple step with multiple SIOP strategies simplifications. Real-world visuals Multi standard processes Group collaboration Talk Moves Number talks Presentations Visual aids: Allow students to take real world situations. Scaffolding of basic concepts regarding prime factorization. Begin with basic monomials and progress. Revisit 7 th, 8 th, Common Core Standards in regard to factors and primes. Allow conceptual understanding using the area model of multiplication. Develop lessons with various polynomial processes prior to complex numbers. Small group/ 1x1 instruction Smaller size quantities, number talks Visual aids: Allow students to take real world surveys. Scaffolding of basic concepts regarding prime factorization. Real-world situations where students can work problems in real context. Allow the use of the area model for conceptual understanding. Develop lessons with various polynomial processes prior to complex numbers. Graphic organizers.