Canyons School District Secondary III Scope and Sequence. CANYONS SCHOOL DISTRICT SECONDARY III and III H SCOPE AND SEQUENCE

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1 CANYONS SCHOOL DISTRICT SECONDARY III and III H SCOPE AND SEQUENCE

2 Unit 1: Inferences and Conclusions from Data Regular: 5 6 weeks Honors: 5 weeks Honors Advanced: 3 weeks S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Concepts and Skills to Master I can understand that the shape of a normal distribution is symmetric, single- peaked, and bell shaped. I can distinguish between examples and non- examples of approximately normally distributed data. I can show that I know that any normal distribution can be described by its mean and standard deviation. I can understand how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). I can show that I know that 1, 2, and 3 standard deviations refer to 68%, 95%, or 99.7% of the population, respectively. I can use technology or tables to estimate areas under the curve of a normal distribution (Limit area calculations to 1, 2, and 3 standard deviations, and use approximations for other distances from the mean). Curriculum Supports Walch Unit 1 Lesson 1: Normal Distributions Walch Unit 1 Lesson 1: Standard Normal Calculations Walch Unit 1 Lesson 1: Assessing Normality ACT test scores are approximately normally distributed. One year the scores had a mean of 21 and a standard deviation of 5.2. Sketch and label the distribution, labeling 1, 2, and 2 standard deviations. What proportion of ACT scores are less than 25.2? What is the interval that contains 95% of scores? 2

3 S.IC.1 Understand that statistics allows inferences to be made about population parameters based on a random sample from that population. Concepts and Skills to Master I can understand the importance of randomness in Do students like school lunch? Explain why selecting the obtaining a representative sample from a population. first 10 people in the lunch line is not a representative I can use randomly collected data to make an inference sample of the opinions of students. Describe a process for about a population. selecting a representative sample. Walch Unit 1 Lesson 2: Differences Between Populations and Samples S.IC.2 Decide if a specified model is consistent with results from a given data- generating process, e.g., using simulations. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of five tails in a row cause you to question the model? Concepts and Skills to Master I can design a model to simulate random outcomes using dice, coins, cards, or technology. I can evaluate the results of a simulation to determine if the model is consistent with the results of the simulation. I can use the Law of Large Numbers to understand the relationships between theoretical and experimental probability. Walch Unit 1 Lesson 2: Simple Random Sampling Walch Unit 1 Lesson 2: Other Methods of Random Sampling The local newspaper randomly selects 10 students for an interview about the school dress code. Nine of the students are boys. Does the number of boys selected cause you to question to selection process? Justify your answer. Sample Task (DOK 4) Create a model for obtaining a sample and defend why the model will generate consistent results. 3

4 Recognize the purposes of and differences among sample surveys, experiments, and observational studies. S.IC.3 Concepts and Skills to Master I can distinguish between and recognize the purposes and limitations of sample surveys, experiments, and observational studies. I can describe how randomization should occur in surveys, experiments, and observational studies. A research question is Can people distinguish between a name brand lemonade and a generic brand lemonade? Out of sample surveys, experiments, and observations studies, which design would provide results that would best represent the population? Why? Sample Task (DOK 4) Design a study. Describe the appropriate use of randomization for the study. Walch Unit 1 Lesson 3: Identifying Surveys, Experiments, and Observational Studies Walch Unit 1 Lesson 3: Designing Surveys, Experiments, and Observational Studies S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Concepts and Skills to Master I can perform a simulation to model data. A recent school poll showed that 47% of respondents I can understand that the margin of error refers to the favored Allison Duncan for student body president, while expected range of variation in a poll if it were to be 51% favored Mindy Robison, with a margin of error of 4% conducted multiple times under the same procedures. for each poll. Can a winner be determined from the poll? I can use sample survey results to interpret margins of error Explain that estimate a population mean of proportion. I can understand that the margin of error is greater when the population has more variability. Walch Unit 1 Lesson 4: Estimating Sample Proportions Walch Unit 1 Lesson 4: The Binomial Distribution Walch Unit 1 Lesson 4: Estimating Sample Means Walch Unit 1 Lesson 4: Estimating with Confidence 4

5 S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. Concepts and Skills to Master I can analyze data from a randomized experiment or Is heart rate affected by sitting vs. standing? Measure heart simulation that compares two treatments. rates for each, and find the difference between the rates for I can determine whether the results of two treatments are each student. Plot the differences. Are the differences significantly different. positive? Negative? Centered over zero? Determine ifthe I can understand when differences in parameters are differences are significant. significant. Walch Unit 1 Lesson 5: Evaluating Treatments Walch Unit 1 Lesson 5: Designing and Simulating Treatments S.IC.6 Evaluate reports based on data. Concepts and Skills to Master I can evaluate the validity of research designs, analyses, and conclusions in published reports. Make a list of criteria you would use to evaluate a statistical report. Walch Unit 1 Lesson 5: Reading Reports Sample Task (DOK 4) Research and write a statistical report on a topic of your choosing. Evaluate the reports of others in the class for statistical validity. 5

6 Unit 2A: Polynomial Relationships Regular: 5-6 weeks Honors: 6 weeks Honors Advanced: 4 weeks A.SSE.1 Interpret expressions that represent a quantity in terms of its context a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one of more of their parts as a single entity. For example, interpret P(1 + r)! as the product of P and a factor not depending on P. Concepts and Skills to Master I can identify the parts of an expression as terms, factors, coefficients, exponents, quotients, divisors, dividends, remainders, and constants. I can explain the meaning of the parts of an expression as they relate to the entire expression and the context of the problem. I can identify when a rational expression would be defined and the practical domain of a problem situation. Walch Unit 2A Lesson 1: 1a Structure of Expressions Walch Unit 2A Lesson 2: 1a HONORS: The Binomial Theorem Walch Unit 2A Lesson 2: 1b Polynomial Identities Walch Unit 2A Lesson 2: 1b HONORS: Complex Polynomial Identities Walch Unit 2A Lesson 2: 1b HONORS: The Binomial Theorem A.APR.1 Given that the volume of a box is x! + 4x! + 5x + 2 with a height of x + 1, what are the other dimensions? 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. I can fluently add, subtract, and multiply two or more polynomials. Simplify: (x! + 5x! + 3x 2) + (x! 3x! + 7x 1) x! + 5x! + 3x 2 (x! 3x! + 7x 1) (x! + 5x! + 3x 2)(x! 3x! + 7x 1) 6

7 Walch Unit 2 Lesson 1: Adding and Subtracting Polynomials Walch Unit 2 Lesson 1: Multiplying POlynomials A.SSE.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. I can solve rational equations in one variable. I can solve radical equations in one variable. I can detect the presence of extraneous roots and explain conditions that give rise to them. Walch Unit 2A Lesson 2: Polynomial Identities Walch Unit 2A Lesson 2: HONORS: Complex Polynomial Identities Walch Unit 2A Lesson 2: HONORS: The Binomial Theorem A.APR.4 Solve and eliminate extraneous solutions if they arise.!! +! =!!!!!!!!!!!!!! Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x! + y! )! = (x! y! )! + (2xy)! can be used to generate Pythagorean triples. I can prove polynomial identities that expand or factor polynomials. Walch Unit 2A Lesson 2: Polynomial Identities Walch Unit 2A Lesson 2: HONORS: Complex Polynomial Identities Walch Unit 2A Lesson 2: HONORS: The Binomial Theorem Prove: x! + y! = (x + y)(x! xy + y! ) 7

8 HONORS: N.CN.8(+) Extend polynomial identities to the complex numbers. For example, rewrite x! + 4 as (x + 2i)(x 2i). I can use polynomial identities to rewrite polynomial expressions using complex numbers. I can factor polynomial expressions over the set of complex numbers. Walch Unit 2A Lesson 2: HONORS: Complex Polynomial Identities HONORS: A.APR.5(+) Determine linear factors of x! 8 over the complex number system. Assume that 1 + i is a zero of the polynomial f with real coefficients. Justify that x! 2x + 5 is a factor. Know and apply the Binomial Theorem for the expansion of (x + y) n in power 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. I can fluently expand binomials by hand, recognizing Pascal s Triangle as a tool of efficiency. I can expand binomials of the form (ax + by) n using Pascal s Triangle. Walch Unit 2A Lesson 2: HONORS: The Binomial Theorem Expand (x + 2) 5. Explain how using Pascal s Triangle can be used to expand (x + y) n. 8

9 F.IF.7c Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. I can graph simple polynomials functions by hand and identify key features of the graph. I can graph complex polynomial functions using technology and identify key features of the graph. I can identify the domain and range of a polynomial function. Walch Unit 2A Lesson 3: Describing End Behavior and Turns Walch Unit 2A Lesson 3: HONORS: Finding Zeros A.APR.2 Graph the function: f x = 4x! 8x! 19x! + 23x 6 Provide an appropriate viewing window where key features are visible. Identify the extrema, domain, and range. Determine the intervals over which the function is increasing, or decreasing. Approximate the value of the roots, and give the multiplicity of each. Describe the end behavior. Know and apply the Remainder Theorem: For a polynomial (x) and a number a, the remainder on the division by x a is p(a), so p(a) = 0 if and only if (a x) is a factor of p(x). I can recognize that if p(a) = 0 then (x a) is a factor of p(x). I can recognize that if (x a) is a factor of p(x) then p(a) = 0. I can use the Remainder Theorem to determine factors of polynomials. Walch Unit 2A Lesson 3: The Remainder Theorem Using the Remainder Theorem, decide whether (x 5) and (x + 2) are factors of the polynomial f x = 2x! 5x! 28x

10 A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. I can use the Remainder Theorem to draw a rough graph of a polynomial. I can recognize that repeated factors indicate multiplicity of roots and graph polynomials with repeated factors. Identify the factors of the polynomial graphed below: Walch Unit 2A Lesson 3: HONORS: Finding Zeros Walch Unit 2A Lesson 3: The Rational Root Theorem HONORS: N.CN.9(+) I can show that polynomials with degree n have at most n roots over the real number system. I can show that polynomials with degree n have exactly n roots over the complex number system. Given a fourth degree polynomial, how could you have: Zero real roots? One real root? Two real roots? Three real roots? Four real roots? Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Walch Unit 2A Lesson 3: HONORS: Finding Zeros Sketch a polynomial function of degree 5 with the following characteristics: One real root Three real roots Five real roots Why can the polynomial function of degree 5 not have exactly two distinct roots? 10

11 A.REI.11 Explain why the x- coordinates of the points where the graph 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, exponential, and logarithmic functions. I can approximate solutions to systems of two equations using graphing technology. I can approximate solutions to systems of two equations using tables of values. I can 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). I can express that when f(x) = g(x), the two equations have the same solution(s). Walch Unit 2A Lesson 4: Solving Systems of Equations Graphically A.SSE.4 Graph the following functions in the same viewing window: f x = 2! g x = 10! h x = e! Determine their common point of intersection and explain what that point represents in terms of the functions. Derive the formula for the sum of a geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. I can derive the formula for the sum of a geometric series. I can find the sum of a geometric series in context. Walch Unit 2A Lesson 5: Geometric Sequences Walch Unit 2A Lesson 5: Sum of a Finite Geometric Series Walch Unit 2A Lesson 5: Sum of a Infinite Geometric Series You are starting to save in a fixed rate savings account that earns 6% interest annually but is compounded monthly. Each month you deposit $100. Write a formula for the total money saved after n months. Write a formula for the total money earned after n months. 11

12 Unit 2B: Rational and Radical Relationships Regular: 4 weeks Honors: 4 weeks Honors Advanced: 3 weeks A.SSE.1: Interpret expressions that represent a quantity in terms of its context a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one of more of their parts as a single entity. For example, interpret P(1 + r)! as the product of P and a factor not depending on P. I can identify the parts of an expression as terms, factors, coefficients, exponents, quotients, divisors, dividends, remainders, and constants. I can explain the meaning of the parts of an expression as they relate to the entire expression and the context of the problem. I can identify when a rational expression would be defined and the practical domain of a problem situation. Sample Task (DOK 4) Find a radical, rational, or logarithmic function that models natural phenomena, and explain the role of the various parts of the expression. Walch Unit 2B Lesson 1: Structures of Rational Expressions A.SSE.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. I can solve rational equations in one variable. Rewrite!!! as a product of two rational expressions.!!! I can solve radical equations in one variable. I can detect the presence of extraneous roots and explain conditions that give rise to them. Walch Unit 2B Lesson 1: Structures of Rational Expressions Walch Unit 2B Lesson 1: HONORS: Adding and Subtracting Rational Expressions 12

13 Walch Unit 2B Lesson 1: HONORS: Multiplying Rational Expressions Walch Unit 2B Lesson 1: HONORS: Dividing Rational Expressions A.APR.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspections, long division, or, for the more complicated examples, a computer algebra system. I can divide polynomials and recognize when the divisor is a factor and when you will have non- zero remainders. I can use long division to rewrite a rational expression in the form q(x) + r(x)/b(x). I can use a computer algebra system to divide complicated polynomials. Walch Unit 2B Lesson 1: Rewriting Rational Expressions Walch Unit 2B Lesson 1: HONORS: Dividing Rational Expressions HONORS: A.APR.7(+) Perform each operation: Understand that rational expression form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. I can add, subtract, multiply, and divide rational expressions. I can demonstrate that rational expressions are closed under addition, subtraction, multiplication, and non- zero division. Walch Unit 2B Lesson 1: HONORS: Adding and Subtracting Rational Expressions Walch Unit 2B Lesson 1: HONORS: Multiplying Rational Expressions Walch Unit 2B Lesson 1: HONORS: Dividing Rational Expressions Compare and contrast the operations addition, subtraction, division, multiplication, and division on whole numbers to the same operations performed on polynomials. For example:! +! and! +!!!!!!!!! 13

14 A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. I can solve rational equations in one variable. I can solve radical equations in one variable. I can detect the presence of extraneous roots and explain conditions that give rise to them. Write a radical equation with a true solution at x = 5 and extraneous solutions at x = - 1 and x = - 2. Walch Unit 2B Lesson 2: Solving Rational Equations Walch Unit 2B Lesson 2: Solving Radical Equations A.REI.11 Explain why the x- coordinates of the points where the graph 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, exponential, and logarithmic functions. I can approximate solutions to systems of two equations using graphing technology. I can approximate solutions to systems of two equations using tables of values. I can 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). I can express that when f(x) = g(x), the two equations have the same solution(s). Walch Unit 2B Lesson 2: Solving Systems of Equations How many liters of a 70% alcohol solution must be added to 50 L of a 40% alcohol solution to product a 50% alcohol solution? 14

15 F.TF.1 Unit 3: Trigonometry of General Triangles and Trigonometric Functions Regular: 6 7 weeks Honors: 7 weeks Honors Advanced: 5 weeks Understand that radian measures of an angle as the length of the arc on the unit circle subtended by the angle. I can define a radian as the length of the arc on the unit circle subtended by the angle. I can locate radian measures on the unit circle. Walch Unit 3 Lesson 1: Radians F.TF.2 I can find the measure of an angle, given a coordinate on the unit circle and given an angle. I can find the corresponding coordinates on the unit circle. I can explain how the use of positive and negative rotations can be used to obtain the domain of the sine and cosine functions. I can use co- terminal reference angles to find values for trigonometric functions. I can recognize that the coordinate of any point on the unit circle may be defined as (cos θ, sin θ). I can use the angles and corresponding coordinates on the unit circle to create the graphs of the sine and cosine functions. 15 Draw the angle whose measure is 2 radians. Explain who the unit circle in the coordinate plan enables the extension of the trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Use the unit circle to find cos (- 7π/3) Walch Unit 3 Lesson 1: The Unit Circle Walch Unit 3 Lesson 1: Special Angles in the Unit Circle

16 Walch Unit 3 Lesson 1: Evaluating Trigonometric Functions HONORS: Derive the formula A = ½ ab sin (C) for the area of a triangle by drawing an auxiliary line from a vertex G.SRT.9(+) perpendicular to the opposite side. I can derive the formula A = ½ ab sin (C) for the area of a triangle The two sides of a triangle are 4 cm and 5 cm, and the by drawing an auxiliary line from a vertex perpendicular to the included angle is 32 degrees. Find the area of the opposite side. triangle. Walch Unit 3 Lesson 2: HONORS: Proving the Law of Sines HONORS: Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.10(+) I can prove the Law of Sines. Write a paragraph proof of the Law of Sines. I can prove the Law of Cosines. I can use the Law of Sines and the Law of Cosines to solve problems. Walch Unit 3 Lesson 2: HONORS: Proving the Law of Sines Walch Unit 3 Lesson 2: HONORS: Proving the Law of Cosines Walch Unit 3 Lesson 2: HONORS: Applying the Laws of Sines and Cosines F.TF.5: I can graph trigonometric parent functions. I can identify the amplitude, frequency, period, and midline given either an equation or a graph of a trigonometric function. I can use trigonometric functions to model real- world problems. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. The Ferris Wheel at Lagoon has a diameter of 21.8 m rotating on a platform 3m above the ground. If it completes one revolution in 40 seconds, sketch a graph of height vs. time, extending the graph for more than one revolution. Walch Unit 3 Lesson 3: Periodic Phenomena and Amplitude, Frequency, and Midline Walch Unit 3 Lesson 3: Using Trigonometric Functions to Model Periodic Phenomena 16

17 Unit 4A: Mathematical Modeling of Inverse, Logarithmic, and Trigonometric Functions Regular: 6 7 weeks Honors: 7 weeks Honors Advanced: 5 weeks F.BF.4a Find inverse functions. a. 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. For example, f x = 2x! or f x = x + 1 x 1 for x 1. Explain to your classmates the clues you use to determine whether or not a function has an inverse. I can determine whether or not a given function has an inverse, and find the inverse if its exists (including rational, radical, trigonometric, and exponential functions) For the following functions, find the inverse if it exists: Walch Unit 4a Lesson 1: Determining Inverses of Quadratic Functions Walch Unit 4a Lesson 1: Determine Inverses of Other Functions Walch Unit 4a Lesson 2: Logarithmic Functions as Inverses HONORS: Find inverse functions. F.BF.4c(+) c. Read values of an inverse function from a graph or a table, given that the function has an inverse. When provided with a table of values for a function, I can write the table for the inverse function. When provided with a set of ordered pairs for a function, I can Sketch the graph of an inverse function given a table, set of ordered pairs, or graph of the original function. 17

18 write the set of ordered pairs for the inverse function. When provided with a graph of a function, I can sketch the graph of the inverse function. Walch Unit 4a Lesson 1: HONORS: Finding Inverses of Functions in Various Forms HONORS: Find inverse functions F.BF.4d(+) I can restrict the domain and find the inverse of a function. I can sketch the inverted relation and restrict the domain in such a manner as to create a function. I can provide the table of an inverted relation and restrict the domain in such a manner as to create a function. d. Produce an invertible function from a non- invertible function by restricting the domain. What is the inverse of the function f x = x! 1 where x 0? Walch Unit 4a Lesson 1: HONORS: Restricting the Domain to Find Inverses HONORS: Understand the inverse relationship between exponents and logarithms and use this relationship to solve F.BF.5(+) problems involving logarithms and exponents I can understand the inverse relationship between exponents and Find the inverse of f x = 3! logarithms and use this relationship to solve problems involving logarithms and exponents. Walch Unit 5 Lesson 2: Defining Rotations, Reflections, and Translations F.LE.4 For exponential models, express as a logarithm the solution to ab!" = d where a, c, and d are numbers and the base b is 2, 10 or e; evaluate the logarithm using technology. I can understand the relationship between the properties of Using technology, find the value of log! 4. Using exponents and the properties of logarithms. properties of exponents, explain the result. I can convert exponential equations to logarithmic equations. I can convert logarithmic equations to exponential equations. I can use technology to evaluate logarithms. 18

19 19 Walch Unit 4a Lesson 2: Logarithmic Functions as Inverses Walch Unit 4a Lesson 2: Common Logarithms Walch Unit 4a Lesson 2: Natural Logarithms F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. I can write an equivalent form of a function defined by an Write the function f x =!!!!!! in factored form to expression for functions given in : F.IF.7e as well as!!!!!! identify any asymptotes and points of discontinuity. simple rational functions. o I can identify zeros, transformations, points of discontinuity, and asymptotes when suitable factorizations are available. o I can use properties of logarithms to write equivalent forms. I can transition between equivalent forms to identify desired key features. Walch Unit 4a Lesson 2: Common Logarithms Walch Unit 4a Lesson 2: Natural Logarithms F.IF.7e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. I can graph exponential, logarithmic functions, and trigonometric functions by hand given an equation. I can use technology to graph exponential, logarithmic, and trigonometric functions for more complicated cases. I can find and interpret key features of exponential, logarithmic, and trigonometric functions. Walch Unit 4a Lesson 2: Graphing Logarithmic Functions Graph a sine, cosine, and tangent function that all have the same period.

20 Walch Unit 4a Lesson 3: Graphing the Sine Function Walch Unit 4a Lesson 3: Graphing the Cosine Function 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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. I can distinguish between logarithmic and exponential equations based on equations, tables, and verbal descriptions. I can identify key features such as intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior. I can use key features of logarithmic functions to sketch a graph. I can interpret key features in terms of context. Walch Unit 4a Lesson 2: Interpreting Logarithmic Models F.IF.5 We can use a logarithmic regression to forecast the population of Hawaii: ln(x). Graph the function. Identify and interpret the intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior. Relate the domain of a function to its graph, and where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person- hours it takes to assemble n engines in a factor, then the positive integers would be an appropriate domain for the function. I can identify appropriate values for the domain of a function based on context. I can identify the domain of a function from the graph. Walch Unit 4a Lesson 2: Interpreting Logarithmic Models The ph scale measures how basic or acidic a substance is. The scale ranges from 0 14 with 0 being the most acidic and 14 being the most basic. In chemistry, the ph solution is defined by ph = - log(h+), where H+ is the hydrogen ion concentration of the solution in moles per liter. Find all possible hydrogen ion concentration for solutions with ph values that lie between 2 and 5. Justify your answer graphically. 20

21 F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. I can calculate the rate of change over a given interval for logarithmic function within a context. I can calculate the rate of change when presented as an equation or table. I can estimate the rate of change from a graph. The table below represents the population growth of Hawaii since Years represents the number of years from Calculate the average rate of change from 1995 to Population of Hawaii People Years (thousands) Since 1995 Walch Unit 4a Lesson 2: Interpreting Logarithmic Models 1, , , , ,

22 Secondary Strand III Unit 4B: Mathematical Modeling and Choosing a Model Regular: 7 weeks Honors: 7 weeks Honors Advanced: 4 weeks 22 A.CED.1 Create equation 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. I can create simple rational, square root, cube root, A man is blowing up a balloon. His lung capacity is 6 liters of air. polynomial, trigonometric, and logarithmic equations in one If he blows 5 times into the balloon, assuming the balloon is variable, and use them to solve problems. perfectly spherical, what would be your prediction for the radius I can create simple rational, square root, cube root, of the balloon? Justify your answer mathematically. polynomial, trigonometric, and logarithmic inequalities in one variable, and use them to solve problems. I can understand the meaning of solutions, including extraneous, in reference to context. Walch Unit 4B Lesson 1: Creating Equations in One Variable A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. I can identify constraints such as domain, range, asymptotes, and points of discontinuity when given a context involving You are going on a river- rafting trip. The trip will cost at least $1,000 and you can take up to 10 people. You want to keep the equations, inequalities, and systems. average cost under $200 per person. Write a system of I can interpret solutions as viable or non- viable based on the constraints. inequalities and graph the solution set representing the average cost per person based on the number of people you invite. Give the domain and range of the solution set. What are the constraints on the domains of the data? How many people will you invite and why?

23 Walch Unit 4B Lesson 1: Creating Constraints A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. I can solve rational, square root, cube root, polynomial, and logarithmic formula for a quantity of interest. Walch Unit 4B Lesson 1: Rearranging Formulas F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, 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 effect on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. I perform translation, reflections, and dilations on any function with and without technology. I can describe the effect of a transformation on a function. I can identify functions that are even, odd, or neither. Walch Unit 4B Lesson 2: Transformations of Parent Graphs Walch Unit 4B Lesson 2: Recognizing Odd and Even Functions F.BF.1b Describe the graphical relationship between the two functions: Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Given a radical, rational, exponential, polynomial, or The potential energy of an object is given by PE = mgh. If the object 23

24 trigonometric context, I can find an explicit algebraic expression or series of steps to model the context with mathematical representations. I can combine radical, rational, exponential, polynomial, or trigonometric functions using addition, subtraction, or multiplication. Walch Unit 4B Lesson 2: Combining Functions HONORS: F.BF.1c(+) is thrown and its height h is a function of time such that h t = 0.6t! + 3t, what is the potential energy of the object in terms of time? Write a function that describes a relationship between two quantities. c. Compose functions. For example, if T(y) is the temperature of the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. I can apply function operations to find the composite of two functions. Walch Unit 4B Lesson 2: HONORS: Composition of Functions HONORS: F.BF.4b(+) Find inverse functions b. Verify by composition that one function is the inverse of another. I can verify that two functions are inverses of each other by taking the composition of the two functions. Walch Unit 4B Lesson 2: Verifying Function Inverses by composition The manufacturer of a computer is offering two discounts on last year s model. The first discount is a $200 rebate, and the second discount is 20% off the regular price. Find the composite functions that describe how a customer could receive both discounts. Which composite function will provide the customer with the largest discount? Justify your reasoning. Given two functions, find f(f!! x ) f!! (f x ) 24

25 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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. I can distinguish between rational and radical equations based on equations, tables, and verbal descriptions. I can identify key features such as intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior. I can use key features of a rational, square root, cube root, polynomial, and trigonometric function to sketch a graph. I can interpret key features in terms of context. Walch Unit 4B Lesson 3: Reading and Identifying Key Features of Real- World Situation Graphs Walch Unit 4B Lesson 4: Linear, Exponential, and Quadratic Functions Walch Unit 4B Lesson 4: Piecewise, Step, and Absolute Value Functions Walch Unit 4B Lesson 4: Square Roots and Cube Root Functions Draw a graphical representation predicting the temperatures for the next seven days. Identify and interpret key features. 25

26 F.IF.5 Relate the domain of a function to its graph, and where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person- hours it takes to assemble n engines in a factor, then the positive integers would be an appropriate domain for the function I can identify appropriate values for the domain of a function based on context. I can identify the domain of a function from the graph. State the domain for the following graph in both set and interval notation: Walch Unit 4B Lesson 3: Reading and Identifying Key Features of Real- World Graphs Walch Unit 4B Lesson 4: Linear, Exponential, and Quadratic Functions Walch Unit 4B Lesson 4: Piecewise, Step, and Absolute Value Functions Walch Unit 4B Lesson 4: Square Root and Cube Root Functions 26

27 F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. I can calculate the rate of change over a given interval for rational, square root, cube root, polynomial, logarithmic, and trigonometric functions within a context. I can calculate the rate of change when presented as an equation or table. I can estimate the rate of change from a graph. Given the graph below: 1. Identify all the intervals where the average rate of change is negative. 2. Identify two intervals where one has a greater average rate of change than the other. 3. Find two points on the graph where the average rate of change is zero. Walch Unit 4B Lesson 3: Reading and Identifying Key Features of Real- World Situation Graphs Walch Unit 4B Lesson 3: Calculating Average Rates of Change Walch Unit 4B Lesson 3: Comparing Functions 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. I can compare properties of two functions, where one is represented algebraically, graphically, numerically in tables, or by verbal description and the other is modeled using a different representation. 27

28 Walch Unit 4B Lesson 3: Comparing 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. I can write and graph equations to represent a rational, square root, cube root, polynomial, trigonometric and logarithmic relationships. Walch Unit 4B Lesson 4: Linear, Exponential, and Quadratic Functions F.BF.3 You are buying a refridgerator. Refridgerator 1 costs $550, the average cost per year in electricity is $92, and is expected to last 10 years. Refridgerator 2 costs $1200, electricity costs $50 per year, and it is expected to last 20 years. Write and graph rational equations representing the average cost of each refrigerator per year. After how many years will the average cost per year be the same? Which refrigerator would you buy and why? Identify the effect on the graph of replacing f(x) by f(x) + k, 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 effect on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. I perform translation, reflections, and dilations on any function with and without technology. I can describe the effect of a transformation on a function. I can identify functions that are even, odd, or neither. Walch Unit 4B Lesson 4: Linear, Exponential, and Quadratic Functions Walch Unit 4B Lesson 4: Piecewise, Step, and Absolute Value Functions Walch Unit 4B Lesson 4: Square Root and Cube Root Functions Describe the graphical relationship between the two functions: 28

29 F.IF.7 29 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise- defined functions, including step functions and absolute value functions. I can graph graph functions by hand given an equation. I can use technology to graph more complicated cases. I can find and interpret key features of functions. Walch Unit 4B Lesson 4: Piecewise, Step, and Absolute Value Functions Walch Unit 4B Lesson 4: Square Root and Cube Root Functions G.GMD.4 Some professions charge fees based upon the length of time: Up to 6 minutes cost $50 Over 6 minutes to 15 minutes costs $80 Over 15 minutes costs $80 plus $5 per minute above 15 minutes Graph the function and identify key features. Identify the shapes of two- dimensional cross- sections of three- dimensional objects, and identify three- dimensional objected generated by rotations of two- dimensional objects. I can identify the two- dimensional shapes created from the cross- sections of three- dimensional objects. I can rotate two- dimensional objects and identify the three- dimensional objects created by the rotation. Walch Unit 4B Lesson 5: Two- Dimensional Cross Sections of Three- Dimensional Shapes G.MG.1 Demonstrate how you could slice an octahedron to create a triangle, a square, a rhombus that is not a square. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder) I can use geometric shapes to deconstruct objects or situations. I use cross- sections to deconstruct three- dimensional objects. I can use measures of appropriate two- and three- dimensional shapes to estimate the measures of complex objects taking into account any overlap that may occur. Estimate the area of the shaded region and the non- shaded region, given that the radius of the wheel is 10 inches.

30 Walch Unit 4B Lesson 5: Two- Dimensional Cross Sections of Three- Dimensional Shapes G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g. persons per square mile, BTUs per cubic foot). I can understand density as a ratio. I can differentiate between area and volume densities, their units, and situations in which they are appropriate (e.g., area density is ideal for measuring population density spread out over land, and the concentration of oxygen in the air is best measured with volume density). Walch Unit 4B Lesson 5: Density G.MG.3 The current population of New York City is 3.8 million. The area of new York City if 300 square miles. Calculate the population density of New York City. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). I can design solutions to problem through geometric modeling. Sample Task (DOK 4) Maximize the number of parking spaces in a given complex- shaped parking lot. Work with given constraints such as parking stall size, area needed between sections of stalls, etc Justify your work Walch Unit 4B Lesson 5: Design 30