Download Math II Unpacking Document

North Carolina High School Mathematics Math II Unpacking Document The Real Number System N-­‐RN Common Core Cluster Extend the properties of exponents to rational exponents. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? N-­‐RN.2 Rewrite expressions involving N.RN.2 Students should be able to rewrite expressions involving rational exponents as expressions involving radicals and rational exponents using radicals and simplify those expressions. the properties of exponents. Ex: Use what you know about the properties of exponents to simplify the following expressions: !
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= ! = 𝑏!
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Ex. The expression 8! also can be written as (8! )2 or as 8! ! . Write each expression in a form that involves radicals. Simplify your answers to confirm that they are equivalent. N.RN.2 Students should be able to rewrite expressions involving radicals as expressions using rational exponents and use the properties of exponents to simplify those expressions. !
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Ex. Given that 81! = 81! = ( 81)! , which form would be easiest for you to calculate WITHOUT using a calculator? Why? Ex. Complete the table below. Equivalent Forms Exponential Form Radical Form Whole Number or Decimal !
64! !
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9 16 !
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512 !
36 NC DEPARTMENT OF PUBLIC INSTRUCTION 1 North Carolina High School Mathematics Math II Unpacking Document !
Ex. When evaluating 9 , Kyle entered 9^3/2 into his calculator. Karen entered √(9)^3. Caleb entered 9^1.5 They did not all get the same results. Which entry method(s) give the correct result? How could the incorrect entry method(s) be modified to get the correct result? Quantities N.Q Common Core Cluster Reason quantitatively and use units to solve problems. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? N.Q.1 Use units as a way to understand problems and to guide the solution of multi-­‐step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N.Q.1 Use units as a tool to help solve multi-­‐step problems. For example, students should use the units assigned to quantities in a problem to help identify which variable they correspond to in a formula. Students should also analyze units to determine which operations to use when solving a problem. For example, given the speed in mph and time traveled in hours, what is the distance traveled? From looking at the units, we can determine that we must multiply miles per hour times hours to get an answer expressed in miles: !"
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ℎ𝑟 = 𝑚𝑖 Ex: You are sitting at a stoplight when the light turns green. If you know the acceleration (a) in m/s2 and the time (t) in s, and that s=at, what unit would the speed be expressed? (Note that knowledge of the distance formula is not required to determine the need to multiply in this case.) N.Q.1 Based on the type of quantities represented by variables in a formula, choose the appropriate units to express the solution and interpret the meaning of the units in the context of the relationships described. !
Ex: The unit of measure for force, F, is called a Newton. We know that 𝐹 = 𝑚 ∙ , where m is the mass !
measured in kg, t is the time measured in seconds, and v is the velocity (speed). What are the units of Newtons (in terms of m, kg and s)? NC DEPARTMENT OF PUBLIC INSTRUCTION 2 North Carolina High School Mathematics Math II Unpacking Document N.Q.1 When given a graph or data display, read and interpret the scale and origin. When creating a graph or data display, choose a scale that is appropriate for viewing the features of a graph or data display. Understand that using larger intervals on the scale effectively “zooms out” from the graph and choosing smaller intervals “zooms in.” Ex. The Columbus High School store sells bottled drinks before and after school. During the first few weeks of school, the manager set a price of $1.25 per bottle, and daily sales averaged 90 bottles per day. The manager then increased the price to $1.75 per bottle, and sales decreased to an average of 70 bottles per day. a. What is the rate of change in average daily sales as the price per bottle increases from $1.25 to $1.75? What units would you use to describe this rate of change? Describe the change. b. Assume that sales are a function of price, and create a graph of the relationship. Explain your reasoning as to how you chose the scale and set the intervals for the graph. c. What rule did you use to graph the relationship? d. Use your graph or rule to find the amount of daily sales if the price changes to $2.25? $0.90? Ex. A group of students organized a local concert to raise awareness for The American Diabetes Foundation. They have several expenses for promoting and operating the concert and will be making money through selling tickets. Their profit can be modeled by the formula P = x(4,000 – 250x) – 7500. Sketch the following profit model. Explain the scale and the properties of the graph. N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. N.Q.2 Define the appropriate quantities to describe the characteristics of interest for a population. For example, if you want to describe how dangerous the roads are, you may choose to report the number of accidents per year on a particular stretch of interstate. Generally speaking, it would not be appropriate to report the number of exits on that stretch of interstate to describe the level of danger. Ex. Needed NC DEPARTMENT OF PUBLIC INSTRUCTION 3 North Carolina High School Mathematics Math II Unpacking Document N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. N.Q.3 Understand that the tool and context used determines the level of accuracy that is reported for a measurement. For example, when using a ruler, you can only legitimately report accuracy to the nearest division. If I use a ruler that has centimeter divisions to measure the length of my pencil, I can only report its length to the nearest centimeter. When doing calculations involving measurements, the level of accuracy of the final answer should correspond to the level of accuracy reported in the measurements. Ex. The following sketch shows a surveyor’s notes that will be used to calculate the height of a mountain. (Level II) 48° 2,600 ft a. Use the information given in the sketch to calculate the height of the mountain. b. What is an appropriate level of accuracy to use when reporting your answer to part a? Explain. NC DEPARTMENT OF PUBLIC INSTRUCTION 4 North Carolina High School Mathematics Math II Unpacking Document Seeing Structure in Expressions A.SSE Common Core Cluster Interpret the structure of expressions Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? 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 or 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. Note: At this level include polynomial expressions. A.SSE.1a. Students manipulate the terms, factors, and coefficients in difficult expressions to explain the meaning of the individual parts of the expression. Use them to make sense of the multiple factors and terms 5
of the expression. For example, the expression $10, 000 (1.055) represents the amount of money I have in an account. My account has a starting value of $10,000 with a 5.5% interest rate every 5 years, where 10,000 and (1+.055) are factors, and the $10,000 does not depend on the amount the account is increased by. Another example: A ball is thrown vertically by an astronaut on Planet X that has a lighter gravity than earth. The expression −𝑥 ! + 2𝑥 + 3 represents the height of the ball 𝑥 seconds after it was thrown. The expression −(𝑥 − 3)(𝑥 + 1) represents the equivalent factored form. The zeros of the function 𝑦 = −𝑥 ! + 2𝑥 + 3 would then be 𝑥 = 3 and 𝑥 = −1. The zero of 3 can be interpreted as the number of seconds it took for the ball to hit the ground (where the height, 𝑦, is equal to zero). The -­‐1 does not have meaning in the context of this problem since time cannot have a negative value. Ex. The expression −4.9𝑡 ! + 17𝑡 + 0.6 describes the height in meters of a basketball t seconds after it has been thrown vertically in the air. Interpret the terms and coefficients of the expression in the context of this situation. A.SSE.1b Students group together parts of an expression to reveal underlying structure. Ex. What information related to symmetry is revealed by rewriting the quadratic formula as 𝑥 =
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? NC DEPARTMENT OF PUBLIC INSTRUCTION 5 North Carolina High School Mathematics Math II Unpacking Document A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). A.SSE.2 Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression. Ex. The expression 4000𝑝 − 250𝑝 ! represents the income at a concert, where p is the price per ticket. Rewrite this expression in another form to reveal the expression that represents the number of people in attendance based on the price charged. Seeing Structure in Expressions A.SSE Common Core Cluster Write expressions in equivalent forms to solve problems Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15! can be rewritten A.SSE.3c Use properties of exponents to write an equivalent form of an exponential function to reveal and explain specific information about the rate of growth or decay. Ex. The equation 𝑦 = 14,000(0.8) ! represents the value of an automobile x years after purchase. Find the yearly and the monthly rate of depreciation of the car. (Level II) !
as (1.15!" )!"! ≈ (1.012)!"! to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. NC DEPARTMENT OF PUBLIC INSTRUCTION 6 North Carolina High School Mathematics Math II Unpacking Document Arithmetic With Polynomials and Rational Expressions A.APR Common Core Cluster Perform arithmetic operations on polynomials Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? 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. Note: At this level, add and subtract any polynomial and extend multiplication to as many as three linear expressions. A.APR.1 The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or product is also a polynomial. Polynomials are not closed under division because in some cases the result is a rational expression rather than a polynomial. A.APR.1 Add, subtract, and multiply polynomials. Ex. Simplify. a. 4𝑏 𝑐𝑏 − 𝑧𝑑 b. 4𝑥 ! − 3𝑦 ! + 5𝑥𝑦 − 8𝑥𝑦 + 3𝑦 ! c. 𝑥 + 4 𝑥 − 2 3𝑥 + 5 Ex. A small online company estimates that the cost, in dollars, of producing x units of one of their products is given by the expression 0.001𝑥 ! + 2𝑥 + 550. The revenue from selling x units is given by the expression 8.50𝑥. Write a polynomial expression to represent the profit generated by selling x units of the product. What would be the profit from sales of 2000 units? Arithmetic With Polynomials and Rational Expressions A.APR Common Core Cluster Understand the relationship between zeros and factors of polynomials Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? A.APR.3 Identify zeros of polynomials when suitable A.APR.3 Find the zeros of a polynomial when the polynomial is factored. Then use the zeros to sketch the graph. NC DEPARTMENT OF PUBLIC INSTRUCTION 7 North Carolina High School Mathematics Math II Unpacking Document factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Note: At this level, limit to quadratic expressions. Ex. Given the function 𝑦 = 2𝑥 ! + 6𝑥 – 3, list the zeros of the function and sketch the graph. (Level II) Ex. Sketch the graph of the function 𝑓 𝑥 = 𝑥 + 5 ! . What is the multiplicity of the zeros of this function? How does the multiplicity relate to the graph of the function? (Level II) Creating Equations A.CED Common Core Cluster Create equations that describe numbers or relationships Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? 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. Note: At this level extend to quadratic and inverse variation (the simplest rational) functions and use common logs to solve exponential equations. A.CED.1 From contextual situations, write equations and inequalities in one variable and use them to solve problems. Include one-­‐variable equations that arise from functions by the selection of a particular target y-­‐
value. For example, in the radioactive decay problem below, 25 would be substituted for y in the equation 𝑦 = 100
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, which results in the one-­‐variable equation 25 = 100
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. Note, the resulting equation can be solved in Level I using a table or graph. See A-­‐REI.11. Ex. A pool can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the pool is full, it can be drained by pipe C in 4 hours. If the pool is initially empty and all three pipes are open, write an equation to find how long it takes to fill the pool. NC DEPARTMENT OF PUBLIC INSTRUCTION 8 North Carolina High School Mathematics Math II Unpacking Document Ex. A cardboard box company has been contracted to manufacture open-­‐top rectangular storage boxes for small hardware parts. The company has 30 cm x 16 cm cardboard sheets. They plan to cut a square from each corner of the sheet and bend up the sides to form the box. If the company wants to make boxes with the largest possible volume, what should be the dimensions of the square to be cut out? What are the dimensions of the box? What is the maximum volume? x cm x cm 16 cm 30 NC DEPARTMENT OF PUBLIC INSTRUCTION 9 North Carolina High School Mathematics Math II Unpacking Document A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Note: At this level extend to simple trigonometric equations that involve right triangle trigonometry. A.CED.2 Given a contextual situation, write equations in two variables that represent the relationship that exists between the quantities. Also graph the equation with appropriate labels and scales. Make sure students are exposed to a variety of equations arising from the functions they have studied. Ex. The intensity of light radiating from a point source varies inversely as the square of the distance from the source. Write an equation to model the relationship between these quantities given a fixed energy output. 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. Note: Extend to linear-­‐quadratic and linear-­‐inverse variation (simplest rational) systems of equations. A.CED.3 When given a problem situation involving limits or restrictions, represent the situation symbolically using an equation or inequality. Interpret the solution(s) in the context of the problem. When given a real world situation involving multiple restrictions, develop a system of equations and/or inequalities that models the situation. In the case of linear programming, use the Objective Equation and the Corner Principle to determine the solution to the problem. With regard to systems of inequalities: In Math I, students begin to lay the groundwork of concepts needed for linear programming and optimization. Students should be able to write and graph a system of inequalities based on a modeling context. They should choose solution points from the intersection, or feasible, region and interpret them in the context of the problem, determining whether or not they make sense. For example, consider the calculator company situation below. Given the graph of the constraints where x is the number of graphing calculators produced and y is the number of scientific calculators produced, students can select and interpret points that fall in the feasible region. The point (20, 80) would indicate that 20 graphing calculators and 80 scientific calculators are produced. In this instance, 1200 circuits and 100 plastic cases would be needed, well within the given amounts. Students can explore other points, seeking to find which combination(s) use all of the given materials. Students should also NC DEPARTMENT OF PUBLIC INSTRUCTION 10 North Carolina High School Mathematics Math II Unpacking Document explore points outside the feasible region, realizing that they do not meet the constraints. Points such as (21.5, 126.2) should also be discussed as non-­‐viable solutions in this context. In Math II, students continue to work with key concepts underlying linear programming and optimization. Students explore the concept of optimization by selecting points from the feasible region and finding the optimal solution for a given situation by trial and error. For example, in the calculator company problem, students are given the amount of profit made on each type of calculator. Students select viable solutions from the feasible region and calculate the amount of profit earned. Through a process of trial and error, they determine the solution that maximizes profit. In Math III, students fully develop the concept of linear programming and optimization. The concept of an Objective Function is introduced. For a set of linear inequalities, they use the Corner Principle to determine the optimum solution from the feasible region. Ex. Imagine that you are a production manager at a calculator company. Your company makes two types of calculators, a scientific calculator and a graphing calculator. Each model uses the same plastic case and the same circuits. However, the graphing calculator requires 20 circuits and the scientific calculator requires only 10. The company has 240 plastic cases and 3200 circuits in stock. Write the system of inequalities that satisfy this situation. How many different combinations of calculators can be made to satisfy this criteria? 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. Note: At this level, extend to compound variation relationships. A.CED.4 Solve multi-­‐variable formulas or literal equations, for a specific variable. Explicitly connect this to the process of solving equations using inverse operations. Ex. The formula 𝐹𝑉 = 𝑃𝑀𝑇
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is used to calculate the future value of an annuity, where FV = future value, PMT = amount of periodic payment, n = number of compounding periods, and i = interest rate. Solve the formula for PMT. (Level II) Ex. The “condition of average” is the insurance term used when calculating a payout against a claim where the policy undervalues the sum insured. In the event of a partial loss, the amount paid against a claim will be in the same proportion as the value of the underinsurance. The formula used is !"# !"#$%&'
𝑃𝑎𝑦𝑜𝑢𝑡 = 𝐶𝑙𝑎𝑖𝑚×
. Solve the formula for the Current Value. (Level II) !"##!"# !"#$%
NC DEPARTMENT OF PUBLIC INSTRUCTION 11 North Carolina High School Mathematics Math II Unpacking Document Reasoning with Equations and Inequalities A.REI Common Core Cluster Understand solving equations as a process of reasoning and explain the reasoning Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? 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. Note: At this level, limit to factorable quadratics. A.REI.1 Relate the concept of equality to the concrete representation of the balance of two equal quantities. Properties of equality are ways of transforming equations while still maintaining equality/balance. Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process with mathematical properties. Ex. Needed A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Note: At this level limit to inverse variation. A.REI.2 Solve simple rational and radical equations in one variable and provide examples of how extraneous solutions arise. Ex. Solve the following equations for 𝑥 . a.
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NC DEPARTMENT OF PUBLIC INSTRUCTION 12 North Carolina High School Mathematics Math II Unpacking Document Reasoning with Equations and Inequalities A.REI Common Core Cluster Solve equations and inequalities in one variable Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? A.REI.4 Solve Equations in One Variable b. Solve quadratic equations by inspection (e.g., for x2 = 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. Note: At this level, limit to solving quadratic equations by inspection, taking square roots, quadratic formula, and factoring when the lead coefficient is one. Writing complex solutions is not expected; however, recognizing when the formula generates non-­‐real solutions is expected. A.REI.4b Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square. Ex. Find the solution to the following quadratic equations: a. 𝑥 ! – 7𝑥 − 18 = 0 b. 𝑥 ! = 81 c. 𝑥 ! − 10𝑥 + 5 = 0 A.REI.4b Use the quadratic formula to solve any quadratic equation, recognizing the formula always produces solutions. Write the solutions in the form a ± bi , where a and b are real numbers. Students should understand that the solutions are always complex numbers of the form a ± bi . Real solutions are produced when b = 0, and pure imaginary solutions are found when a = 0. The value of the discriminant b 2 − 4ac determines how many and what type of solutions the quadratic equation has. Ex. Ryan used the quadratic formula to solve an equation and his result was =
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. a. Write the quadratic equation Ryan started with. b. Simplify the expression to find the solutions. c. What are the x-­‐intercepts of the graph of the corresponding quadratic function? Reasoning with Equations and Inequalities A.REI NC DEPARTMENT OF PUBLIC INSTRUCTION 13 North Carolina High School Mathematics Math II Unpacking Document Common Core Cluster Solve systems of equations Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. A.REI.7 Solve a system containing a linear equation and a quadratic equation in two variables (conic sections possible) graphically and symbolically. Add context or analysis Ex. Solve the following system graphically and symbolically: x2 + y2 = 1 y = x Reasoning with Equations and Inequalities A.REI Common Core Cluster Represent and solve equations and inequalities graphically Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A.REI.10 The solutions to equations in two variables can be shown in a coordinate plane where every ordered pair that appears on the graph of the equation is a solution. Understand that all points on the graph of a two-­‐variable equation are solutions because when substituted into the equation, they make the equation true. Ex. Which of the following points lie on the graph of the equation −5𝑥 + 2𝑦 = 20? a. (4, 0) b. (0, 10) c. (-­‐1, 7.5) d. (2.3, 5) How many solutions does this equation have? Justify your reasoning. NC DEPARTMENT OF PUBLIC INSTRUCTION 14 North Carolina High School Mathematics Math II Unpacking Document 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, exponential, and logarithmic ★
functions. A.REI.11 Construct an argument to demonstrate understanding that the solution to every equation can be found by treating each side of the equation as separate functions that are set equal to each other, f(x) = g(x). Allow y1=f (x) and y2= g(x) and find their intersection(s). The x-­‐coordinate of the point of intersection is the value at which these two functions are equivalent, therefore the solution(s) to the original equation. Students should understand that this can be treated as a system of equations and should also include the use of technology to justify their argument using graphs, tables of values, or successive approximations. Ex. John and Jerry both have jobs working at the town carnival. They have different employers, so their daily wages are calculated differently. John’s earnings are represented by the equation, p(x) = 2x and Jerry’s by g(x) = 10 + 0.25x. a. What does the variable x represent? b. If they begin work next Monday, Michelle told them that Friday would be the only day they made the same amount of money. Is she correct in her assumption? Explain your reasoning. c. When will Jerry earn more money than John? When will John earn more money than Jerry? During what day will their earnings be the same? Justify your conclusions. Interpreting Functions F.IF Common Core Cluster Understand the concept of a function and use function notation. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F.IF.2 Students should continue to use function notation throughout high school mathematics. F.IF.2 Students should be comfortable finding output given input (i.e. f(3) = ?) and finding inputs given outputs (f(x) = 10). Example: If h(x)= -­‐16t2 + 5t +7 models the path of an object projected inro the air. a. Find h(5) and explain the meaning of the solution. b. Find t, when h(t) = 0. Explain your reasoning. Notes – All courses are expected to be able evaluate a function given a graphical display. At the course I level, evaluating and creating functions should be limited to linear and exponential. Course II students, should extend functions to include quadratic, simple power, and inverse variation functions. NC DEPARTMENT OF PUBLIC INSTRUCTION 15 North Carolina High School Mathematics Math II Unpacking Document Interpreting Functions F.IF Common Core Cluster Interpret functions that arise in applications in terms of the context. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? 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.* F.IF.4 This standard should be taught alongside the specific function your class is studying. Students should be able to move fluidly between graphs, tables, and words and understand the interplay between the different representations. When given a table or graph of a function that models a real-­‐life situation, explain the meaning of the characteristics of the graph in the context of the problem. At the course one level, the focus is on linear, exponentials, and quadratics o Linear – x/y-­‐intercepts and slope as increasing/decreasing at a constant rate. o Exponential-­‐ y-­‐intercept and increasing at an increasing rate or decreasing at a decreasing rate. o Quadratics – x-­‐intercepts/zeroes, y-­‐intercepts, vertex, the effects of the coefficient of x2 on the concavity of the graph, symmetry of a parabola. At the course two level, the focus is on power functions, and inverse functions. o Power functions – the effects of a positive/negative coefficient, the effects of the exponent on end behavior. o Inverse functions – understanding the effects on the graph of having a variable in the denominator (asymptotes). At the course three level, in addition to the previous course work, students should focus on polynomial and trigonometric functions. o Polynomials – emphasis should be on the commonalities of quadratics and power functions. o Trigonometric functions – intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Include amplitude, frequency, and midline (F-­‐TF 5). Note – This standard should be seen as related to F.IF.7 with the key difference being students can interpret from a graph or sketch graphs from a verbal description of key features. NC DEPARTMENT OF PUBLIC INSTRUCTION 16 North Carolina High School Mathematics Math II Unpacking Document Ex. Insert picture of a quadratic function modeling a real-­‐life situation. a. What are the x-­‐intercepts and y-­‐intercepts and explain their meaning in the context of the problem. b. Identify any maximums or minimums and explain their meaning in the context of the problem. c. Describe the intervals of increase and decrease and explain them in the context of the problem. F-­‐IF.4 When given a verbal description of the relationship between two quantities, sketch a graph of the relationship, showing key features. Ex. Needed NC DEPARTMENT OF PUBLIC INSTRUCTION 17 North Carolina High School Mathematics Math II Unpacking Document 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 factory, then the positive integers would be an appropriate domain for the function.* F.IF.5 Given a function and context, determine the practical domain of the function as input values that make sense to the constraints of the problem context. Ex. A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h(t) = – 16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet. a. What is the practical domain for t in this context? Why? b. What is the height of the rocket two seconds after it was launched? c. What is the maximum value of the function and what does it mean in context? d. When is the rocket 100 feet above the ground? e. When is the rocket 250 feet above the ground? f. Why are there two answers to part e but only one practical answer for part d? g. What are the intercepts of this function? What do they mean in the context of this problem? h. What are the intervals of increase and decrease on the practical domain? What do they mean in the context of the problem? Interpreting Functions F.IF Common Core Cluster Analyze functions using different representations. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. Graph square root, cube root, and piecewise-­‐defined functions, including step functions and absolute value functions. e. Graph exponential and logarithmic F.IF.7 Part a., b., and c. are learned by students sequentially in Courses I – III. Part e. is carried through Courses I – III with a focus on exponential in Course I and moving towards logarithms in Courses II and III. Note: This standard should be seen as related to F.IF.4 with the key difference being students can create graphs from the symbolic function in this standard. Ex. The all start kicker kicks a field goal. The path of the ball is modeled by f ( x ) = −4.9t 2 + 20t . Find the maximum and minimum realistic values for the path of the ball and describe what each means in the context of this problem. NC DEPARTMENT OF PUBLIC INSTRUCTION 18 North Carolina High School Mathematics Math II Unpacking Document functions, showing intercepts and end behavior, and graph trigonometric functions, showing period, midline, and amplitude. NC DEPARTMENT OF PUBLIC INSTRUCTION 19 North Carolina High School Mathematics Math II Unpacking Document Ex. Income taxes are calculated at a rate dependent on the range of the taxable income. The table below shows income tax rates up to 100,000. Taxable Income (dollars) Income Tax $0-­‐$10,000 10% of income $10,001-­‐$45,550 15% of income over 10,000 plus $1,000 $45,551-­‐$100,000 25% of income over 45,552 plus $6832.50 (a) Last year, Ravi earned $15,000 washing cars. How much should he pay in personal income taxes? (b) Jordan earned $50,000 managing a car washing business. How much does he owe in taxes? (c) INSERT FUNCTION! Explain the correspondences between this piecewise function and the tax table. (d) Graph this function on a set of axis labeling key features. Ex. Graph f ( x ) = 2 x+1 , showing its intercepts and describe its end behavior. Ex. Graph f(x) = x2 + 6x + 5 and f(x) = x2 + 6x – 16 showing intercepts and maximums or minimums, and describe the key differences between the graphs. NC DEPARTMENT OF PUBLIC INSTRUCTION 20 North Carolina High School Mathematics Math II Unpacking Document Interpreting Functions F.IF Common Core Cluster Analyze functions using different representations. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F.IF.8 a is carried throughout Courses I -­‐ III . In Course I and II, students will be limited to factoring quadratic functions. In Course III, students will be able to complete the square. Students should take a quadratic function and manipulate it in a different form (standard to factored) so that they can show and explain special properties of the function such as; zeros, extreme values, and symmetry. An exemplar lesson plan of F.IF.8 and F.IF.9 is available from the Shell Center for Mathematics Education titled L20: Forming Quadratics at http://map.mathshell.org/materials/lessons.php?taskid=224 Ex. Coyote was chasing road runner, seeing no easy escape, Road Runner jumped off a cliff towering above the roaring river below. Molly mathematician was observing the chase and obtained a digital picture of this fall. Using her mathematical knowledge, Molly modeled the Road Runner’s fall with the following quadratic functions: h(t) = -­‐16t2 + 32t + 48 h(t) = -­‐16(t+ 1)(t – 3) a. How can Molly have two equations? b. Which of the rules would be most helpful in answering each of these questions? Explain. i.
What is the maximum height the Road Runner reaches and when will it occur? ii.
When would the Road Runner splash into the river? iii.
At what height was the Road Runner when he jumped off the cliff? Ex. Stephanie is considering moving to Carmel or Lunsford. Carmel’s population curve in thousands of people in year x can be shown by the equation y = 200(1.015)x. Lunsford’s population curve in thousands of people in year x can be shown by the equation y = 60(0.998)x. a. What is the rate of change in Carmel? And what is the rate of change in Lunsford? Which population model represents decay and which one represents growth. Justify your answer. F.IF.9 Compare properties of two F.IF.9 This standard includes comparing two different functions in two different forms and comparing one NC DEPARTMENT OF PUBLIC INSTRUCTION 21 North Carolina High School Mathematics Math II Unpacking Document 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. function in two different forms. An exemplar lesson plan of F.IF.8 and F.IF.9 is available from the Shell Center for Mathematics Education titled L20: Forming Quadratics at http://map.mathshell.org/materials/lessons.php?taskid=224 Ex: A herd of horses at Corolla Beach was first counted at 100 heads. Repopulation efforts have yielded a net growth of 16% yearly of the existing horse population. Simultaneously, biologists have recorded the sea turtle population growth in the following table: Year 0 1 2 5 10 Number of Sea Turtles 75 90 108 Which population is growing at a faster rate? Explain your reasoning. Ex: Compare and contrast 6x + 2y = 12 and y = 6 – 3x. 186 464 NC DEPARTMENT OF PUBLIC INSTRUCTION 22 North Carolina High School Mathematics Math II Unpacking Document Building Functions F.BF Common Core Cluster Build a function that models a relationship between two quantities. Unpacking What does this standard mean that a student will know and be able to do? Common Core Standard F.BF.1 Write a function that Ex: Ten bacteria are placed in a test tube and each one splits in two after one minute. After 1 minute, the describes a relationship between two resulting 10 bacteria each split in two, creating 20 bacteria. This process continues for one hour until test quantities.* tube is filled up. a. Determine an explicit expression, a. How many bacteria are in the test tube after 5 minutes? 15 minutes? a recursive process, or steps for b. Describe how you can take any current number of bacteria to find the number of bacteria at the next calculation from a context. minute (this is writing a NOW -­‐ NEXT rule). b. Combine standard function types c. Write an “N = …” rule that gives the number of bacteria after each minute. using arithmetic operations. For d. How many bacteria are in the test tube after one hour? example, build a function that e. For further research, Dr. Bland removes 5 bacteria after each minute from the original test tube to models the temperature of a start a new cell culture. How does this affect your rule in parts b. and c.? cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. NC DEPARTMENT OF PUBLIC INSTRUCTION 23 North Carolina High School Mathematics Math II Unpacking Document Building Functions F.BF Common Core Cluster Build new functions from existing functions. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? 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. F.BF.3 Identify the effect of transformations to functions in multiple modalities. Student should be fluent in representations of functions as equations, tables, graphs, and descriptions and understand the interplay among these representations. Fill in all missing components to the below table. Description of Original Function Multiply the original Change Output function output by 5 x f(x) f(x)+5 -­‐3 9 -­‐2 4 -­‐1 1 5 0 0 1 1 2 4 9 3 9 (a) Graph and label each of the function outputs with the corresponding x-­‐values on the same set of axis in three different colors. (b) Carefully explain the relationship between the original function and both transformed functions. Given the function f(x)=-­‐.5x2+x2+9.5x-­‐10 ,g(x)=f(x-­‐3), h(x)=f(3x) (a) Using technology plot the above functions in different colors. Analyze g(x) and h(x) in comparison to f(x). (b) For each of the new functions (g(x) and h(x)) compare the local minimum, local maximum, and zeros to the original function f(x). (c) Complete a table of values for f(x), h(x), and g(x) for x=-­‐5,-­‐4,-­‐3,-­‐2,-­‐1,0,1,2,3,4,5 NC DEPARTMENT OF PUBLIC INSTRUCTION 24 North Carolina High School Mathematics Math II Unpacking Document (d) Explain correspondences between the function representations, graphs created in part (a), verbal descriptions. NC DEPARTMENT OF PUBLIC INSTRUCTION 25 North Carolina High School Mathematics Math II Unpacking Document F.BF.3 Given the graphs of the original function and a transformation, determine the value of (k). Below are graphs representing the height of a two Farris Wheels at the county fair. (a) Explain how the speed is different on the blue Ferris Wheel from that of the red Ferris Wheel. How does this impact the positions of the local minimums and maximums. (b) Write the rule for the red graph assuming that the blue graph is represented by f(x) Note: The intent of this question is NOT to develop a sophisticated trigonometric equation. It is acceptable to answer part (b) in terms of f(x). NC DEPARTMENT OF PUBLIC INSTRUCTION 26 North Carolina High School Mathematics Math II Unpacking Document F.BF.3 Recognize even and odd functions from their graphs and equations. Ex. (a) Graph f(x)=3x4+x2+1 and construct a table of values from -­‐5 to 5. (b) Graph g(x)=x3-­‐4x and construct a table of values from -­‐5 to 5. (c) The graph of f(x) is said to be an even function. Using the table and graph of f(x) develop a definition for even function. (d) The graph of g(x) is said to be an odd function. Using the table and graph of f(x) develop a definition for odd function. (e) In learning teams create two additional even functions and two additional odd functions. Defend your created functions and definitions to your classmates. (f) Are all functions either even or odd? Justify your conjecture. Congruence G.CO Common Core Cluster Experiment with transformations in the plane. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.2 Describe and compare geometric and algebraic transformations on a set of points as inputs to produce another set of points as outputs, to include translations and horizontal and vertical stretching. Also compare and contrast the angle measures, side lengths, and perimeter of the pre-­‐image and image after the transformation occurs. Ex. Using Interactive Geometry Software perform the following dilations (x,y)→(4x,4y) (x,y)→(x,4y) (x,y)→(4x, y) on the triangle defined by the points (1,1) (6,3) (2,13). NC DEPARTMENT OF PUBLIC INSTRUCTION 27 North Carolina High School Mathematics Math II Unpacking Document G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.3 Describe the rotations and reflections of a rectangle, parallelogram, trapezoid, or regular polygon that maps each figure onto itself, beginning and ending with the same geometric shape. E. Given rotations and reflections and combinations of rotations and reflections, illustrate each with a diagram. Where a combination is not possible, give examples to illustrate why? (co-­‐ordinate arguments). G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.4 Students should understand that when a figure is reflected about a line, the line of reflection perpendicularly bisects the segment that joins the pre-­‐image point to its corresponding image point. When figures are rotated, the points travel in a circular path over some specified angle of rotation. When figures are translated, the segments of the pre-­‐image are parallel to the corresponding segments of the image. Ex. Needed G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.CO.5 Given different figures, identify the sequence of transformations that transforms one figure onto the other figure. Ex. Using Interactive Geometry Software or graph paper, perform the following transformations on the triangle ABC with coordinates A(4,5), B(8,7) and C(7,9). First, reflect the triangle over the line y=x. Then rotate the figure 180° about the origin. Finally, translate the figure up 4 units and to the left 2 units. Write the algebraic rule in the form (x,y)→(x’,y’) that represents these composite transformations. NC DEPARTMENT OF PUBLIC INSTRUCTION 28 North Carolina High School Mathematics Math II Unpacking Document Congruence G.CO Common Core Cluster Understand congruence in terms of rigid motions. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.CO.6 Use geometric descriptions of
rigid motions to transform figures and to
predict the effect of a given rigid motion
on a given figure; given two figures, use
the definition of congruence in terms of
rigid motions to decide if they are
congruent G.CO.6 Use descriptions of rigid motions to move figures in a coordinate plane, and predict the effects rigid
motion has on figures in the coordinate plane.
G.CO.6 Use this fact knowing rigid transformations preserve size and shape or distance and angle measure, to
connect the idea of congruency and to develop the definition of congruent.
Ex. Consider parallelogram ABCD with coordinates A(2,-2), B(4,4), C(12,4) and D(10,-2). Perform the following
transformations. Make predictions about how the lengths, perimeter, area and angle measures will change under
each transformation.
a. A reflection over the x-axis.
b. A rotation of 270° about the origin.
c. A dilation of scale factor 3 about the origin.
d. A translation to the right 5 and down 3.
Verify your predictions. Compare and contrast which transformations preserved the size and/or shape with those
that did not preserve size and/or shape. Generalize, how could you determine if a transformation would maintain
congruency from the pre-image to the image? G.CO.7 Use the definition of
congruence in terms of rigid motions to
show that two triangles are congruent if
and only if corresponding pairs of sides
and corresponding pairs of angles are
congruent. G.CO.7 Use the definition of congruence, based on rigid motion, to show two triangles are congruent if and only if
their corresponding sides and corresponding angles are congruent.
This standard connects with GSRT.3. Students should connect that two triangles are congruent if and only if they
are similar with a scale factor of one. Building on their definition of similarity, this means that a rigid
transformation will preserve the angle measures and the sides will change (or not change) by a scale factor of one.
Ex. Using Interactive Geometry Software or graph paper graph the following triangle M(-1,1), N(-4,2) and P(-3,5)
and perform the following transformations. Verify that the pre-image and the image are congruent. Justify your
answer.
NC DEPARTMENT OF PUBLIC INSTRUCTION 29 North Carolina High School Mathematics Math II Unpacking Document a. (x,y)→(x+2, y-6)
b. (x,y)→(-x,y)
c. (x,y)→(-y,x)
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.8 Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence criteria; ASA, SSS, SAS, AAS, and HL. (Do NOT Prove) Students should connect that these triangle congruence criteria are special cases of the similarity criteria in GSRT.3. ASA and AAS are modified versions of the AA criteria for similarity. Students should note that the “S” in ASA and AAS has to be present to include the scale factor of one, which is necessary to show that it is a rigid transformation. Students should also investigate why SSA and AAA are not useful for determining whether triangles are congruent. Ex. Andy and Javier are designing triangular gardens for their yards. Andy and Javier want to determine if their gardens that they build will be congruent by looking at the measures of the boards they will use for the boarders, and the angles measures of the vertices. Andy and Javier use the following combinations to build their gardens. Will these combinations create gardens that enclose the same area? If so, how do you know? a. Each garden has length measurements of 12ft, 32ft and 28ft. b. Both of the gardens have angle measure of 110°, 25° and 45°. c. One side of the garden is 20ft another side is 30ft and the angle between those two boards is 40°. d. One side of the garden is 20ft and the angles on each side of that board are 60° and 80°. e. Two sides measure 16ft and 18ft and the non-­‐included angle of the garden measures 30°. NC DEPARTMENT OF PUBLIC INSTRUCTION 30 North Carolina High School Mathematics Math II Unpacking Document Congruence G.CO Common Core Cluster Prove geometric theorems. Using logic and deductive reasoning, algebraic and geometric properties, definitions, and proven theorems to draw conclusions about given geometric situations. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.CO.10 Using any method you choose, construct the medians of a triangle. Each median is divided up by the centroid. Investigate the relationships of the distances of these segments. Can you create a deductive argument to justify why these relationships are true? Can you prove why the medians all meet at one point for all triangles? Extension: using coordinate geometry, how can you calculate the coordinate of the centroid? Can you provide an algebraic argument for why this works for any triangle? Using Interactive Geometry Software or tracing paper, investigate the relationships of sides and angles when you connect the midpoints of the sides of a triangle. Using coordinates can you justify why the segment that connects the midpoints of two of the sides is parallel to the opposite side. If you have not done so already, can you generalize your argument and show that it works for all cases? Using coordinates justify that the segment that connects the midpoints of two of the sides is half the length of the opposite side. If you have not done so already, can you generalize your argument and show that it works for all cases? NC DEPARTMENT OF PUBLIC INSTRUCTION 31 North Carolina High School Mathematics Math II Unpacking Document Congruence G.CO Common Core Cluster Make geometric constructions Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.CO.13 Ex. Using a compass and straightedge or Interactive Geometry Software, construct an equilateral triangle so that each vertex of the equilateral triangle is on the circle. Construct an argument to show that your construction method will produce an equilateral triangle. If your method and/or argument are different than another students’, critique his/her argument deciding whether it makes sense, looking for flaws in their reasoning. Repeat this process for a square inscribed in a circle and a regular hexagon inscribed in a circle. Similarity, Right Triangles, and Trigonometry G.SRT Common Core Cluster Understand similarity in terms of similarity transformations. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor. a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G.SRT.1a Given a center of a dilation and a scale factor, verify using coordinates that when dilating a figure in a coordinate plane, a segment of the pre-­‐image that does not pass through the center of the dilation, is parallel to it’s image when the dilation is preformed. However, a segment that passes through the center does not change. NC DEPARTMENT OF PUBLIC INSTRUCTION 32 North Carolina High School Mathematics Math II Unpacking Document 1
y
x
2
4
6
−1
−2
−3
Rule for the transformation is (x,y)→(⅓x,⅓y). Comment on the slopes and lengths of the segments that make up the triangles. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor G.SRT.1b Given a center and a scale factor, verify using coordinates, that when performing dilations of the pre-­‐image, the segment which becomes the image, is longer or shorter based on the ratio given by the scale factor. y
x
2
4
6
8
−2
−4
Rule for the transformation is (x,y)→(½x,½y). Comment on the slopes and lengths of the segments that make up the triangles. Similarity, Right Triangles, and Trigonometry G.SRT Common Core Cluster Define trigonometric ratios and solve problems involving right triangles. NC DEPARTMENT OF PUBLIC INSTRUCTION 33 North Carolina High School Mathematics Math II Unpacking Document Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.6 Using corresponding angles of similar right triangles, show that the relationships of the side ratios are the same, which leads to the definition of trigonometric ratios for acute angles. There are three points on a line that goes through the origin, (5,12) (10,24) (15,36). Sketch this graph. How y
compare? Why does this make sense? Call the distance from the origin to each point (“r”). x
x
y
Find the r for each point. Find the ratios of and . r
r
do the ratios of Students should use their knowledge of dilations and similarity to justify why these triangles are congruent. It is not expected at this stage that they will know the Triangle Similarity Theorems (comes in course 3). NC DEPARTMENT OF PUBLIC INSTRUCTION 34 North Carolina High School Mathematics Math II Unpacking Document G.SRT.7 Explain and use the G.SRT.7 Show that the sine of an angle is the same as the cosine of the angles’s complement. relationship between the sine and Draw yourself a right triangle XYZ with X as the right angle. cosine of complementary angles. (i) If m Y = 65, what is m Z? How do you know Z once you know Y? Is this always true? (ii) What is the link between the sine of an angle and the ‘co’sine of the complementary angle? (iii) Suppose that M and N are the acute angles of a right triangle. (a) Use a diagram to explain/show why sin M = cos N. (b) Why can the equation in (a) be re-­‐written as sin M = cos (90 – M)? What would the equivalent equation be for cos N? G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* G.SRT.8 Apply both trigonometric ratios and Pythagorean Theorem to solve application problems involving right triangles. a. An onlooker stands at the top of a cliff 119 meters above the water’s surface. With a clinometer, she spots two ships due west. The angle of depression to each of the sailboats is 11° and 16°. Calculate the distance between the two sailboats. b. What is the distance from the onlooker’s eyes to each of the sailboats? What is the difference of those distances? Explain why or why not this difference is not the same as the distance between the two sailboats. Similarity, Right Triangles, and Trigonometry G.SRT Common Core Cluster Apply trigonometry to general triangles. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.SRT.9 (+) Derive the formula A=½ ab sin (C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.9 For a triangle that is not a right triangle, draw an auxiliary line from a vertex, perpendicular to the opposite side and derive the formula, A=½ ab sin (C), for the area of a triangle, using the fact that the height of the triangle is, h=a sin(C). Ex. Needed NC DEPARTMENT OF PUBLIC INSTRUCTION 35 North Carolina High School Mathematics Math II Unpacking Document G.SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-­‐
right triangles (e.g., surveying problems, resultant forces). Ex. A surveyor standing at point C is measuring the length of a property boundary between two points located at A and B. Explain what measurements he is able to collect using his transit. Create a plan for this surveyor to find the length of the boundary between A and B. How does the surveyor use the law of sines and/or cosines in this problem? Will your process develop a reliable answer? Why or why not? Expressing Geometric Properties with Equations G.GPE Common Core Cluster Translate between the geometric description and the equation for a conic section. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.GPE.1 Use the Pythagorean Theorem to derive the equation of a circle, given the center and the radius. G.GPE.1 Given an equation of a circle, complete the square to find the center and radius of a circle. Ex. Given a coordinate and a distance from that coordinate develop a rule that shows the locus of points that is that given distance from the given point (based on the Pythagorean theorem). Expressing Geometric Properties with Equations G.GPE Common Core Cluster Use coordinates to prove simple geometric theorems algebraically. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? NC DEPARTMENT OF PUBLIC INSTRUCTION 36 North Carolina High School Mathematics Math II Unpacking Document G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.6 Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Ex. If you are given the midpoint of a segment and one endpoint. Find the other endpoint. a. midpoint: (6, 2) endpoint: (1, 3)G-­‐GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.! b. midpoint: (-­‐1, -­‐2) endpoint: (3.5, -­‐7) Ex. If Jennifer and Jane are best friends. They placed a map of their town on a coordinate grid and found the point at which each of their house lies. If Jennifer’s house lies at (9, 7) and Jane’s house is at (15, 9) and they wanted to meet in the middle, what are the coordinates of the place they should meet NC DEPARTMENT OF PUBLIC INSTRUCTION 37 North Carolina High School Mathematics Math II Unpacking Document Geometric Measurement and Dimension G.GMD Common Core Cluster Visualize relationships between two-­‐dimensional and three-­‐dimensional objects. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.GMD.4 Identify the shapes of two-­‐
dimensional cross-­‐sections of three-­‐
dimensional objects, and identify three-­‐
dimensional objects generated by rotations of two-­‐dimensional objects. G.GMD.4 Given a three-­‐ dimensional object, identify the shape made when the object is cut into cross-­‐sections. G.GMD.4 When rotating a two-­‐ dimensional figure, such as a square, know the three-­‐dimensional figure that is generated, such as a cylinder. Understand that a cross section of a solid is an intersection of a plane (two-­‐
dimensional) and a solid (three-­‐dimensional). Ex. Needed NC DEPARTMENT OF PUBLIC INSTRUCTION 38 North Carolina High School Mathematics Math II Unpacking Document Modeling with Geometry G.MG Common Core Cluster Apply geometric concepts in modeling situations. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a soda can or the paper towel roll as a cylinder).* Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep and extends for 5 feet. Then for 15 feet (horizontally) there is a constant slope downwards to the 10 foot-­‐deep end. (a) Sketch the pool and indicate all measures on the sketch. (b) How much water is needed to fill the pool to the top? To a level 6 inches below the top? (c) One gallon of pool paint covers approximately 75 sq feet of surface. How many gallons of paint are needed to paint the inside walls of the pool? If the pool paint comes in 5-­‐gallon cans, how many cans are needed? (d) How much material is needed to make a rectangular pool cover that extends 2 feet beyond the pool on all sides? (e) How many 6-­‐inch square ceramic tiles are needed to tile the top 18 inches of the inside faces of the pool? If the lowest line of tiles is to be in a contrasting color, how many of each tile are needed? 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).* G.MG.2 Use the concept of density when referring to situations involving area and volume models, such as persons per square mile. Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep and extends for 5 feet. Then for 15 feet (horizontally) there is a constant slope downwards to the 10 foot-­‐
deep end. (a) Sketch the pool and indicate all measures on the sketch. (b) How much water is needed to fill the pool to the top? To a level 6 inches below the top? (c) One gallon of pool paint covers approximately 75 sq feet of surface. How many gallons of paint are needed to paint the inside walls of the pool? If the pool paint comes in 5-­‐gallon cans, how many cans are needed? (d) How much material is needed to make a rectangular pool cover that extends 2 feet beyond the pool on all sides? (e) How many 6-­‐inch square ceramic tiles are needed to tile the top 18 inches of the inside faces of the pool? If the lowest line of tiles is to be in a contrasting color, how many of each tile are needed? NC DEPARTMENT OF PUBLIC INSTRUCTION 39 North Carolina High School Mathematics Math II Unpacking Document G.MG.3 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).* G.MG.3 Solve design problems by designing an object or structure that satisfies certain constraints, such as minimizing cost or working with a grid system based on ratios (i.e., The enlargement of a picture using a grid and ratios and proportions) Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep and extends for 5 feet. Then for 15 feet (horizontally) there is a constant slope downwards to the 10 foot-­‐
deep end. (a) Sketch the pool and indicate all measures on the sketch. (b) How much water is needed to fill the pool to the top? To a level 6 inches below the top? (c) One gallon of pool paint covers approximately 75 sq feet of surface. How many gallons of paint are needed to paint the inside walls of the pool? If the pool paint comes in 5-­‐gallon cans, how many cans are needed? (d) How much material is needed to make a rectangular pool cover that extends 2 feet beyond the pool on all sides? (e) How many 6-­‐inch square ceramic tiles are needed to tile the top 18 inches of the inside faces of the pool? If the lowest line of tiles is to be in a contrasting color, how many of each tile are needed? Making Inferences and Justifying Conclusions S.IC Common Core Cluster Understand and evaluate random processes underlying statistical experiments. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? S.IC.2 Decide if a specified model is consistent with results from a given data-­‐generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? S.IC.2 Use data-­‐generating processes such as simulations to evaluate the validity of a statistical model. Ex. Jack rolls a 6 sided die 15 times and gets the following results: 4, 6, 1, 3, 6, 6, 2, 5, 6, 5, 4, 1, 6, 3, 2 Based on these results, is Jack rolling a fair die? Justify your answer using a simulation. (Level II) NC DEPARTMENT OF PUBLIC INSTRUCTION 40 North Carolina High School Mathematics Math II Unpacking Document Making Inferences and Justifying Conclusions S.IC Common Core Cluster Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? S.IC.6 Evaluate reports based on data. S.IC.6 Evaluate reports based on data on multiple aspects (e.g. experimental design, controlling for lurking variables, representativeness of samples, choice of summary statistics, etc.) Ex. Find a statistical report based on an experiment and evaluate the experimental design. (Level III) Conditional Probability and the Rules of Probability S.CP Common Core Cluster Understand independence and conditional probability and use them to interpret data. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S.CP.1 Define the sample space for a given situation. Ex. What is the sample space for rolling a die? (Level II) Ex. What is the sample space for randomly selecting one letter from the word MATHEMATICS? (Level II) S.CP.1 Describe an event in terms of categories or characteristics of the outcomes in the sample space. Ex. Describe different subsets of outcomes for rolling a die using a single category or characteristic. (Level II) S.CP.1 Describe an event as the union, intersection, or complement of other events. Ex. Describe the following subset of outcomes for choosing one card from a standard deck of cards as the NC DEPARTMENT OF PUBLIC INSTRUCTION 41 North Carolina High School Mathematics Math II Unpacking Document intersection of two events: {queen of hearts, queen of diamonds}. (Level II) S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S.CP.2 Understand that two events A and B are independent if and only if 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) ∙ 𝑃(𝐵). S.CP.2 Determine whether two events are independent using the Multiplication Rule (stated above). Ex. For the situation of drawing a card from a standard deck of cards, consider the two events of “draw a diamond” and “draw an ace.” Determine if these two events are independent. (Level II) Ex. Create and prove two events are independent from drawing a card from a standard deck. S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.3 Understand that the conditional probability of event A given event B has already happened is given by the formula: S.CP.4 Construct and interpret two-­‐
way frequency tables of data when two categories are associated with each object being classified. Use the two-­‐way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from S.CP.4 Create a two-­‐way frequency table from a set of data on two categorical variables. Ex. Make a two-­‐way frequency table for the following set of data. Use the following age groups: 3-­‐5, 6-­‐8, 9-­‐
11, 12-­‐14, 15-­‐17. (Level II) Youth Soccer League Gender Age Gender Age Gender Age Gender Age Gender Age M 4 F 7 M 17 M 5 F 10 M 7 M 7 M 16 M 9 M 6 F 8 F 15 F 14 F 13 F 4 F 6 M 13 M 14 M 15 M 5 M 4 M 12 F 12 M 17 M 9 F 10 M 15 F 8 M 12 M 10 𝑃(𝐴 𝐵) =
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S.CP.3 Understand that two events A and B are independent if and only if 𝑃 (𝐴 𝐵) = 𝑃(𝐴) and 𝑃 (𝐵 𝐴) = 𝑃 𝐵 . In other words, the fact that one of the events happened does not change the probability of the other event happening. S.CP.3 Prove that two events A and B are independent by showing that 𝑃 (𝐴 𝐵) = 𝑃(𝐴) and 𝑃 (𝐵 𝐴) = 𝑃 𝐵 . Ex. For the situation of drawing a card from a standard deck of cards, consider the two events of “draw a spade” and “draw a king.” Prove that these two events are independent. (Level II) Ex. Create and prove two events are dependent from drawing a card from a standard deck. NC DEPARTMENT OF PUBLIC INSTRUCTION 42 North Carolina High School Mathematics Math II Unpacking Document your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. F 11 F 16 M 13 F 13 F 15 S.CP.4 Determine if two categorical variables are independent by analyzing a two-­‐way table of data collected on the two variables. NC DEPARTMENT OF PUBLIC INSTRUCTION 43 North Carolina High School Mathematics Math II Unpacking Document S.CP.4 Calculate conditional probabilities based on two categorical variables and interpret in context. Ex. Use the frequency table to answer the following questions. (Level II) Youth Soccer League Age Group Gender 3-­‐5 6-­‐8 9-­‐11 12-­‐14 15-­‐17 Total years old years old years old years old years old Male 4 3 3 5 5 20 Female 1 4 3 4 3 15 Total 5 7 6 9 8 35 a) Given that a league member is female, how likely is she to be 9-­‐11 years old? b) What is the probability that a league member is aged 9-­‐11? c) Given that a league member is aged 9-­‐11, what is the probability that a member of this league is a female? d) What is the probability that a league member is female? e) Are the events “9-­‐11 years old” and “female” independent? Justify your answer. S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S.CP.5 Given an everyday situation describing two events, use the context to construct an argument as to whether the events are independent or dependent. Ex. Felix is a good chess player and a good math student. Do you think that the events “being good at playing chess” and “being a good math student” are independent or dependent? Justify your answer. (Level II) Ex. Juanita flipped a coin 10 times and got the following results: T, H, T, T, H, H, H, H, H, H. Her math partner Harold thinks that the next flip is going to result in tails because there have been so many heads in a row. Do you agree? Explain why or why not. (Level II) NC DEPARTMENT OF PUBLIC INSTRUCTION 44 North Carolina High School Mathematics Math II Unpacking Document Conditional Probability and the Rules of Probability S.CP Common Core Cluster Use the rules of probability to compute probabilities of compound events in a uniform probability model. Common Core Standard Unpacking What does this standard mean that a student will know and be able to do? S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.6 Understand that when finding the conditional probability of A given B, the sample space is reduced to the possible outcomes for event B. Therefore, the probability of event A happening is the fraction of event B’s outcomes that also belong to A. Before event B happens, the sample space can be visualized as below: Both Outcomes in A + B Once B occurs, the situation can be viewed as follows: So, Sample Space 𝑃(𝐴 𝐵) =
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NC DEPARTMENT OF PUBLIC INSTRUCTION 45 North Carolina High School Mathematics Math II Unpacking Document S.CP.6 Understand that drawing without replacement produces situations involving conditional probability. S.CP.6 Calculate conditional probabilities using the definition. Interpret the probability in context. Ex. Peter has a bag of marbles. In the bag are 4 white marbles, 2 blue marbles, and 6 green marbles. Peter randomly draws one marble, sets it aside, and then randomly draws another marble. What is the probability of Peter drawing out two green marbles? S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. S.CP.7 Understand that two events A and B are mutually exclusive if and only if 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 0. In other words, mutually exclusive events cannot occur at the same time. S.CP.7 Determine whether two events are disjoint (mutually exclusive). Ex. Given the situation of rolling a six-­‐sided die, determine whether the following pairs of events are disjoint: a. rolling an odd number; rolling a five b. rolling a six; rolling a prime number c. rolling an even number; rolling a three d. rolling a number less than 4; rolling a two S.CP.7 Given events A and B, calculate 𝑃 (𝐴 𝑜𝑟 𝐵) using the Addition Rule. Ex. Given the situation of drawing a card from a standard deck of cards, calculate the probability of the following: a. drawing a red card or a king b. drawing a ten or a spade c. drawing a four or a queen d. drawing a black jack or a club e. drawing a red queen or a spade S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. S.CP.8 Calculate probabilities using the General Multiplication Rule. Interpret in context. Ex. Needed NC DEPARTMENT OF PUBLIC INSTRUCTION 46 North Carolina High School Mathematics Math II Unpacking Document S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. S.CP.9 Identify situations as appropriate for use of a permutation or combination to calculate probabilities. Use permutations and combinations in conjunction with other probability methods to calculate probabilities of compound events and solve problems. Ex: Medical scientists model the amount of active insulin, 𝑦, left in the bloodstream after 𝑥 minutes using the formula (knowing that 10 units of insulin were administered): 𝑦 = 10 ∙ 0.95 ! What is the amount of active insulin left after 20 seconds? NC DEPARTMENT OF PUBLIC INSTRUCTION 47