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Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)

The Real Number System N-RN

Common Core Cluster

Extend the properties of exponents to rational exponents.

Common Core StandardUnpackingWhat does this standard mean that a student will know and be able to do?

N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

N-RN.2 Students should be able to use the properties of exponents to rewrite expressions involving rational exponents as expressions using radicals.

Ex. The expression can be written as ( )2 or as . Write these expressions in radical form. How would you confirm that these forms are equivalent?

Considering that , which form would be easier to simplify without a calculator? Why?

Ex. When calculating , Kyle entered 9^3/2 into his calculator. Karen entered it as . They came out with different results. Which was right and how could the other student modify their input?

Modeling is best interpreted not as a collection of isolated topics, but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol ().

Math II NCDPI Unpacked Content 1 Formatted by WS/FCS July, 2013


Quantities N-Q

Common Core Cluster

Reason quantitatively and use units to solve problems.


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. 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 mph times hours to get

an answer expressed in miles: . 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 variables and interpret the meaning of the units in the context of the relationships that the formula describes.

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 values for the tick marks on the scale effectively “zooms out” from the graph and choosing smaller values “zooms in.” Understand that the viewing window does not necessarily show the x- or y-axis, but the apparent axes are parallel to the x- and y-axes. Hence, the intersection of the apparent axes in the viewing window may not be the origin. Also be aware that apparent intercepts may not correspond to the actual x- or y-intercepts of the graph of a function.

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 sellingtickets. Their profit can be modeled by the formula P = x(4,000 – 250x) – 7500. Graph the profit model using anappropriate scale and explain your reasoning.




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.

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N-Q.3 Understand that the tool used determines the level of accuracy that can be reported for a measurement. Forexample, when using a ruler, you can only legitimately report accuracy to the nearest division. Further, if I use aruler that has centimeter divisions to measure the length of my pencil, I can only report its length to the nearestcentimeter. Also, when using a calculator, measures can be given to the nearest desired decimal place of aparticular unit.

Ex. What is the accuracy of a ruler with 16 divisions per inch?

Ex. What would an appropriate level of accuracy be when studying the number of Facebook users?

Ex. Vivian, John’s mother is a chemist, and she brought home a very delicate and responsive measuringinstrument. Her children enjoyed learning how to use the device by measuring the weight of pennies one at a time.Here is a list from lightest to heaviest (weights are in milligrams).

2480 2484 2487 2491 2493 2495 2496 2498 2501 2503 2506 2507 2511 2515 2516

1. Given the information above, what do you think is the best estimate of the weight of a penny in units? Explain your reasoning. 2. Vivian and John’s Aunt, Maria, said she had a penny that was counterfeit. They decided to weigh the penny and discovered the penny weighed 2541 milligrams. John said that because the penny weighed more than all of their other measurements, it must be counterfeit. Vivian does not believe it is counterfeit based only on one weight measurement; she believes if they weigh the penny again it might be closer to the weight of the other pennies. Vivian also had a thought that if they measured the weights of more pennies, that Aunt Maria’s penny might not seem so strange. Why might this be so?

Ex. What would an appropriate level of accuracy be when measuring the length of a beach, from sand to water?Explain your reasoning.




Seeing Structure in Expressions A-SSE

Common Core Cluster

Interpret the structure of expressions.


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

as the product of P and a factor not depending on P.

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 of the expression. For example, consider the expression 10,000(1.055)5. This expression can be viewed as the product of 10,000 and 1.055 raised to the 5th power. 10,000 could represent the initial amount of money I have invested in an account. The exponent tells me that I have invested this amount of money for 5 years. The base of 1.055 can be rewritten as (1 + 0.055), revealing the growth rate of 5.5% per year.

Ex. The expression −4.9t2 + 17t + 0.6 describes the height in meters of a basketball t seconds after it has beenthrown 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. For example, consider the

expression that represents income from a concert where p is the price per ticket. The equivalent

factored form, , shows that the income can be interpreted as the price times the number of people in attendance based on the price charged. At this level, include polynomial expressions.

Ex. What information related to symmetry is revealed by rewriting the quadratic formula as x = ?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 4000p2 − 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.


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 1.15t can be rewritten as (1.151/12)12t ≈(1.012)12t to reveal theapproximate equivalent monthly interest rate if the annual rate is 15%.

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 = 14000(0.8)x represents the value of an automobile x years after purchase. Find the yearly and the monthly rate of depreciation of the car.




Arithmetic with Polynomials and Rational Expressions A-APR

Common Core Cluster

Perform arithmetic operations on polynomials.


A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

A-APR.1 The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or product is also a polynomial.

A-APR.1 Add, subtract, and multiply polynomials. At this level, add and subtract any polynomial and extend multiplication to as many as three linear expressions.

Common Core Cluster

Understand the relationship between zeros and factors of polynomials.


A-APR.3 Identify zeros ofpolynomials when suitablefactorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

A-APR.3 Find the zeros of a polynomial when the polynomial is factored. Then use the zeros to sketch the graph. At this level, limit to quadratic expressions.

Ex. Given the function y = 2x2 + 6x – 3, list the zeros of the function and sketch the graph.

Ex. Sketch the graph of the function f(x) = (x + 5)2. What is the multiplicity of the zeros of this function? How

does the multiplicity relate to the graph of the function?




Creating Equations A-CED

Common Core Cluster

Create equations that describe numbers or relationships.


A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.1 From contextual situations, write equations and inequalities in one variable and use them to solve problems. Include linear and exponential functions. At this level, extend to quadratic and inverse variation (the simplest rational) functions and use common logs to solve exponential equations.

(Level II)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 bypipe C in 4 hours. If the pool is initially empty and all three pipes are open, write an equation to find how long ittakes to fill the pool. What do you think the student had in mind when using the numbers 20 and 56?

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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. At this level, extend to simple trigonometric equations.

Ex. The intensity of light radiating from a point source varies inversely as the square of the distance from thesource. 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

A-CED.3 Use constraints which are situations that are restricted to develop equations and inequalities and systems of equations or inequalities. Describe the solutions in context and explain why any particular one would be the optimal solution. 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.




combinations of different foods.Ex. Imagine that you are a production manager at a calculator company. Your company makes two types ofcalculators, a scientific calculator and a graphing calculator.

a. 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. Graph the system of inequalities that represents these constraints.

b. The profit on a scientific calculator is $8.00, while the profit on a graphing calculator is $16.00. Write an equation that describes the company’s profit from calculator sales.c. How many of each type of calculator should the company produce to maximize profit using the stock on

hand?

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.

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. At this level, extend to compound variation relationships.

(Level II)

Ex. If , solve for T2.




Reasoning with Equations and Inequalities A-REI

Common Core Cluster

Understand solving equations as a process of reasoning and explain the reasoning.


A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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.

A-REI.2 Solve simple rational andradical equations in one variable, and give examples showing howextraneous solutions may arise.

A-REI.2 Solve simple rational and radical equations in one variable and provide examples of how extraneoussolutions arise. Add context.

Ex. Solve for x.

Ex. Mary solved for x and got x = -2, and x = 1. Are all the values she found solutions to the equation?Why or why not?

Ex. Solve for x.





Common Core Cluster

Solve equations and equalities in one variable.


A-REI.4 Solve quadratic 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.

A-REI.4b Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, andcompleting the square. Add context or analysis. At this level, limit solving quadratic equations by inspection, taking square roots, quadratic formula, and factoring when lead coefficient is one. Writing complex solutions is not expected; however, recognizing when the formula gives complex solutions is expected.

Ex. Find the solution to the following quadratic equations: a. x2 – 7x – 18 = 0 b. x2 = 81 c. x2 – 10x + 5 = 0

A-REI.4b Use the quadratic formula to solve any quadratic equation, recognizing the formula always producessolutions. 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 areproduced when b = 0, and pure imaginary solutions are found when a = 0. The value of the discriminant,

, determines how many and what type of solutions the quadratic equation has.

(Level II)

Ex. Ryan used the quadratic formula to solve an equation and was his result. 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 this quadratic function?





Common Core Cluster

Solve systems of equations.


A-REI.7 Solve a simple systemconsisting 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 circlex2 + 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

Common Core Cluster

Represent and solve equations and inequalities graphically.


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 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. At this level, extend to quadratics.




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. At this level, extend to quadratic functions.

A-REI.11 Understand that solving a one-variable equation of the form f(x) = g(x) is the same as solving the two-variable system y = f(x) and y = g(x). When solving by graphing, the x-value(s) of the intersection point(s) ofy = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear and exponential functions. Use technology, entering f(x) in y1 and g(x) in y2, graphing the equations to find their point of equality. At this level, extend to quadratic functions.

Ex. How do you find the solution to an equation graphically?

A-REI.11 Solve graphically, finding approximate solutions using technology. At this level, extend to quadratic functions.

A-REI.11 Solve by making tables for each side of the equation. Use the results from substituting previous values of x to decide whether to try a larger or smaller value of x to find where the two sides are equal. The x-value that makes the two sides equal is the solution to the equation. At this level, extend to quadratic functions.

(Level 1/II)Ex. John and Jerry both have jobs working at the town carnival. They have different employers, so their dailywages are calculated differently. John’s earnings are represented by the equation, p(x) = 2x and Jerry’s byg(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 sameamount 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 whatday 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.


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, understanding f(input) = output, f(x)=y.

F-IF.2 Students should be comfortable finding output given input (i.e. f(3) = ?) and finding inputs given outputs (f(x) = 10) and describe their meanings in the context in which they are used. At this level, extend to quadratic, simple power and inverse variation functions.

Ex.

a. Solve h(5) and explain the meaning of the solution. b. Solve h(t) = 0 and explain the meaning of the solution.

Notes – All courses are expected to be able evaluate a function given a graphical display.At the course one level, evaluating functions should be limited to linear and exponential.Course II students should extend functions to include quadratic, simple power, and inverse variation functions.




Interpreting Functions F-IF

Common Core Cluster

Interpret functions that arise in applications in terms of the context.


F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. 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 revisited with every function your class is studying. Students should be able to move fluidly between graphs, tables, words, and symbols and understand the connections between the different representations. When given a table and/or graph of a function that models a real-life situation, explain how the table relates to the graph and vise versa. The characteristics described should include rate of change, intercepts, maximums/minimums, symmetries, and intervals of increase and/or decrease. Also explain the meaning of thecharacteristics of the graph and table in the context of the problem. At this level, limit to simple trigonometric functions

(sine, cosine, and tangent in standard position) with angle measures of ( radians) or less. Periodicity not addressed.

At the course one level, the focus is on linear, exponentials, and quadratics• Linear – x/y-intercepts and slope as increasing/decreasing at a constant rate.• Exponential- y-intercept and increasing at an increasing rate or decreasing at a decreasing rate.• Quadratics – x-intercepts/zeroes, y-intercepts, vertex, intervals of increase/decrease, the effects of the coefficient of x2 on the concavity of the graph, symmetry of a parabola.

At the course two level, students should extend the previous course work with function types to focus on power functions, and inverse functions.• Power functions – the effects of a positive/negative coefficient, the effects of the exponent on end behavior, the rate of increase or decrease on certain intervals,• Inverse functions – understanding the effects on the graph of having a variable in the denominator (asymptotes).

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.

F-IF.5 Relate the domain of a function to its graph and, where

F- IF.5 Given a function, determine its domain. Describe the connections between the domain and the graph of the function. Know that the domain taken out of context is a theoretical domain and that the practical domain of a function is




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.

found based on a contextual situation given, and is the input values that make sense to the constraints of the problem context. In context, students will identify the domain, stating any restrictions and why they are restrictions. At this level, extend to quadratic, right triangle trigonometric and inverse variation functions.

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 theoretical domain for the function? How do you know this? b. What is the practical domain for t in this context? Explain. c. What is the height of the rocket two seconds after it was launched? d. What is the maximum value of the function and what does it mean in context? e. When is the rocket 100 feet above the ground? f. When is the rocket 250 feet above the ground? g. Why are there two answers to part e but only one practical answer for part d? h. What are the intercepts of this function? What do they mean in the context of this problem? i. What are the intervals of increase and decrease on the practical domain? What do they mean in the context of the problem?

Common Core Cluster

Analyze functions using different representations.


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.

b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

F-IF.7 Part a., b., and c. are learned by students sequentially in Courses I – III. Part e. is carried throughCourses I – III with a focus on exponential in Course I and moving towards logarithms in Courses II and III.At this level for part e, extend to simple trigonometric functions (sine, cosine, and tangent in standard position).

This standard should be seen as related to F-IF.4 with the key difference being students can create graphs, by hand and using technology, from the symbolic function in this standard.

Ex. Income taxes are calculated at a rate dependent on the range of a person’s taxable income. The table below shows income tax rates up to 100,000.




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. Explain the connections between this piecewise function and the tax table. d. Graph this function and label any characteristic features that are key to the function.

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and graph trigonometric functions, showing period, midline, and amplitude.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 Students should take a function and manipulate it in a different form so that they can show and explain special properties of the function such as; zeros, extreme values, and symmetries.

F-IF.8a This standard is developed throughout Courses I - III. In Course I and II, students will factor quadratic functions and in Course III, students will complete the square. Students should manipulate a quadratic function to identify its different forms (standard, factored, and vertex) so that they can show and explain special properties of the function such as; zeros, extreme values, and symmetry. Students should be able to distinguish when a particular form is more revealing of special properties given the context of the situation. At this level, completing the square is still not expected.

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 athttp://map.mathshell.org/materials/lessons.php?taskid=224At this level, completing the square is still not expected.

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

F-IF.9 This standard compares different functions when they have different representations. For example, one may be represented using a graph and the other an equation. Students should compare the properties of two functions represented by verbal descriptions, tables, graphs, and equations. At this level, extend to quadratic, simple power, and inverse variation functions.

An exemplar lesson plan of F-IF.8 and F-IF.9 is available from the Shell Center for Mathematics Education titled L20:



http://map.mathshell.org/materials/lessons.php?taskid=224


function and an algebraic expression for another, say which has the larger maximum.

Forming Quadratics athttp://map.mathshell.org/materials/lessons.php?taskid=224

Building Functions F-BF

Common Core Cluster

Build a function that models a relationship between two quantities.


F-BF.1 Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

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.

F-BF.1a Recognize when a relationship exists between two quantities and write a function to describe them. Use steps, the recursive process, to make the calculations from context in order to write the explicit expression that represents the relationship. Continue to allow information recursive notation through this level.

F-BF.1b Use addition, subtraction, multiplication, and division to combine functions to create other functions. When used in context, interpret the effect the combination of the functions had on the given situation. (Students should take standard function types such as constant, linear and exponential functions and add, subtract, multiply and divide them. Also explain how the function is affected and how it relates to the model.)

Building Functions F-BF

Common Core ClusterModeling is best interpreted not as a collection of isolated topics, but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol ().


http://map.mathshell.org/materials/lessons.php?taskid=224


Build a function that models a relationship between two quantities.


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 transformations have on functions in multiple modalities or ways. Student should be fluent with representations of functions as equations, tables, graphs, and descriptions. They should also understand how each representation of a function is related to the others. For example, the equation of the line is related to it’sgraph, table, and when in context, the problem being solved. Know that when adding a constant, k, to a function, it moves the graph of the function vertically. If k is positive, it translates the graph up, and if k is negative, it translates the graph down.

If k is either added or subtracted from the x-value, it translates the graph of the function horizontally. If we add k, the graph shifts left and if we subtract k, the graph shifts right. The expression (x + k) shifts the graphs k units to the left because when x + k = 0, x = -k.

Use the calculator to explore the effects these values have when applied to a function and explain why the values affect the function the way it does. The calculator visually displays the function and its translation making it simple for every student to describe and understand translations. At this level, extend to quadratic functions and kf(x).

(Level II)Ex. Given the function f(x) = -5x2 + 2x - 10 , g(x) = f(x-3), and 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), what do you find interesting when comparing the local minimum, local maximum, and zeros to the original function f(x)? c. Using technology, complete a table of values for f(x), h(x), and g(x) when the domain is the set of Integers over the interval [-5, 5]. Explain the relationship between the symbolic representation, the table of values, and the graphs created in part (a) for each of the functions.

Congruence C-GO

Common Core Cluster




Experiment with transformation in the plane.


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. 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). Compare and contrast the following from each dilation: angle measure, side length, perimeter.

G-CO.3 Given a rectangle,parallelogram, trapezoid, or regular polygon, describe the rotations andreflections that carry it onto itself.

G-CO.3 Describe the rotations and reflections of a rectangle, parallelogram, trapezoid, or regular polygon thatmaps each figure onto itself, beginning and ending with the same geometric shape. Given combinations of rotations and reflections, illustrate each combination with a diagram. Where a combination is not possible, give examples to illustrate why that will not work (coordinate arguments).

G-CO.4 Develop definitions ofrotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO.4 Refer to 8.G where students have already experimented with the properties of rigid transformations anddilations. Students should understand and be able to explain that when a figure is reflected about a line, thesegment that joins the preimage point to its corresponding image point is perpendicularly bisected by the line ofreflection. 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 preimage are parallel to the corresponding segments of the image.

G-CO.5 Given a geometric figure anda 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 Using Interactive Geometry Software or graph paper, perform the following transformations on thetriangle 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 this composite transformation.

Congruence G-CO




Common Core Cluster

Understand congruence in terms of rigid motions.


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 ofcongruence 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 rigidmotion 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, toconnect the idea of congruency and to develop the definition of congruent.

Consider parallelogram ABCD with coordinates A(2,-2), B(4,4), C(12,4) and D(10,-2). Perform the followingtransformations. Make predictions about how the lengths, perimeter, area and angle measures will change undereach 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 thosethat did not preserve size and/or shape. Generalize, how could you determine if a transformation would maintaincongruency from the preimage to the image?.

G-CO.7 Use the definition ofcongruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides andcorresponding pairs of angles arecongruent.

G-CO.7 Use the definition of congruence, based on rigid motion, to show two triangles are congruent if and only iftheir corresponding sides and corresponding angles are congruent.

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 preimage and the image are congruent. Justify your answer. 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 congruencecriteria; ASA, SSS, SAS, AAS, and HL.

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 ASAand AAS has to be present to include the scale factor of one, which is necessary to show that it is a rigidtransformation. Students should also investigate why SSA and AAA are not useful for determining whethertriangles are congruent.

Andy and Javier are designing triangular gardens for their yards. Andy and Javier want to determine if theirgardens 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°.

Students can make sense of this problem by drawing diagrams of important features and relationships and usingconcrete objects or pictures to help conceptualize and solve this problem.

Congruence G-CO

Common Core Cluster

Prove geometric theorems.


G-CO.10 Prove theorems abouttriangles. 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 youconnect the midpoints of the sides of a triangle. Using coordinates can you justify why the segment that connectsthe midpoints of two of the sides is parallel to the opposite side. If you have not done so already, can yougeneralize 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?At this level, include measures of interior angles of a triangle sum to 180 º and the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length.

Congruence G-CO

Common Core Cluster

Make geometric constructions.


G-CO.13 Construct an equilateraltriangle, a square, and a regularhexagon inscribed in a circle.

G-CO.13 Using a compass and straightedge or Interactive Geometry Software, construct an equilateral triangle sothat each vertex of the equilateral triangle is on the circle. Construct an argument to show that your constructionmethod will produce an equilateral triangle. If your method and/or argument are different than another student’s,understand his/her argument and decide whether it makes sense and look for any flaws in their reasoning.

Repeat this process for a square inscribed in a circle and a regular hexagon inscribed in a circle.

Donna is building a swing set. She wants to countersink the nuts by drilling a hole that will perfectly circumscribethe hexagonal nuts that she is using. Each side of the regular hexagonal nut is 8mm. Construct a diagram thatshows what size drill she should use to countersink the nut.

Similarity, Right Triangles, and Trigonometry G-SRT

Common Core Cluster




Understand similarity in terms of similarity transformations.


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. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT.1a Given a center of a dilation and a scale factor, verify using coordinates that when dilating a figure in acoordinate plane, a segment of the pre-image that does not pass through the center of the dilation, is parallel to itsimage when the dilation is preformed. However, a segment that passes through the center does not change.

Rule for the transformation is (x, y)→(⅓ x, ⅓y).Comment on the slopes and lengths of the segments that make up the triangles. What patterns do you notice? Howcould you be confident that this pattern would be true for all cases?

Rule for the transformation is (x, y)→(½x, ½y).Comment on the slopes and lengths of the segments that make up the triangles.

G-SRT.1b Given a center and a scale factor, verify using coordinates, that when performing dilations of the preimage, the segment which becomes the image, is longer or shorter based on the ratio given by the scale factor.What patterns do you notice? How could you be confident that this pattern would be true for all cases?

In 8.G-3 and 8.G-4 students have already had experience with dilations and other transformations.Note proportional reasoning in 6.RP and 7.RP.





Common Core Cluster

Define trigonometric ratios and solve problems involving right triangles.


G-SRT.6 Understand that bysimilarity, 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 thesame, which leads to the definition of trigonometric ratios for acute angles.

Ex. There are three points on a line that goes through the origin, (5, 12) (10, 24) (15, 36). Sketch this graph. How dothe

ratios of compare? Why does this make sense? Call the distance from the origin to each point (“r”). Find the r for

each point. Find the ratios of and .

Students should use their knowledge of dilations and similarity to justify why these triangles are congruent. It is notexpected at this stage that they will know the Triangle Similarity Theorems (comes in course 3).

G-SRT.7 Explain and use therelationship between the sine andcosine of complementary angles.

G-SRT.7 Show that the sine of an angle is the same as the cosine of the angle’s complement.(Level II)Draw a right triangle XYZ with <X as the right angle.(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? Why or why not?


Common Core Cluster

Apply trigonometry to general triangles.Modeling is best interpreted not as a collection of isolated topics, but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol ().




G-SRT.9 (+) Derive the formula

for the area of atriangle 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, , for the area of a triangle, using the fact that the height of the triangle is,

.Instructional note: The focus is on deriving the formula, not using it.

Show that for each of the following cases,

Before students derive this problem, pose the following problem to help them see the usefulness of this shortcut.

G-SRT.11 (+) Understand andapply the Law of Sines and the Law of Cosines to find unknownmeasurements in right and non-right

G-SRT.11 (+) 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?




triangles (e.g., surveyingproblems, resultant forces).

Expressing Geometric Properties with Equations G-GPE

Common Core Cluster

Translate between the geometric description and the equation for a conic section.


G-GPE.1 Derive the equation of acircle 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. At this level, derive the equation of the circle using the Pythagorean Theorem.

G-GPE.1 Given an equation of a circle, complete the square to find the center and radius of a circle.Given a coordinate and a distance from that coordinate develop a rule that shows the locus of points that is thatgiven distance from the given point (based on the Pythagorean theorem).

If the coordinate of point H in the diagram below is (x, y) and the length of 4 units, can you write a rule thatrepresents the relationship of the x value, the y value and the radius? Why is this relationship true? As point Hrotates around the circle, does this relationship stay true?

In the diagram below, circle D translated 4 units to the right to create circle E. Why is the equation of this newcircle (x − 4)2 + y2 = 42? Why is the equation for circle I x2 + (y − 4)2 = 42? Using similar reasoning, could youwrite an equation for a circle with the center at (-4, 0) and a radius of 4? Center of (0, -4) and radius of 4? What is




the equation of a circle with center at (-8, 11) and a radius of ? Can you generalize this equation for a circle with a center at (h, k) and a radius of r?

Common Core Cluster

Use coordinates to prove simple geometric theorems algebraically.


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 on a line, find the point that divides the segment into an equal number of parts. Iffinding the mid-point, it is always halfway between the two endpoints. The x-coordinate of the mid-point will bethe mean of the x-coordinates of the endpoints and the y-coordinate will be the mean of the y-coordinates of theendpoints.

Ex. If general points N at (a, b) and P at (c, d) are given, why are the coordinates of point Q (a, d)? Can you find thecoordinates of point M?

Geometric Measurement and Dimension G-GMD

Common Core Cluster




Visualize relationships between two-dimensional and three-dimensional objects.


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 isgenerated, 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).

Modeling with Geometry G-MG

Common Core Cluster

Apply geometric concepts in modeling situations.


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 acylinder).

G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a soda canor 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 andextends 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 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as acylinder).

G-MG.2 Use the concept of density when referring to situations involving area and volume models, such aspersons per square mile.

Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep andextends 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.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 gridsystems 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 andratios and proportions)

Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep andextends 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?

Interpreting Categorical and Quantitative Data S-ID

Common Core Cluster

Understand and evaluate random processes underlying statistical experiments.





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.

Common Core Cluster

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.


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 lurkingvariables, representativeness of samples, choice of summary statistics, etc.)

Conditional Probability and the Rules of Probability S-CP

Common Core Cluster

Understand independence and conditional probability and use them to interpret data.





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?Ex. What is the sample space for randomly selecting one letter from the word MATHEMATICS?

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.

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 theintersection of two events: {queen of hearts, queen of diamonds}.

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 thischaracterization to determine if they are independent.

S-CP.2 Understand that two events A and B are independent if and only if .

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.

Ex. Create and prove two events are independent from drawing a card from a standard deck.S-CP.3 Understand the conditionalprobability of A given B asP(A and B)/P(B), and interpretindependence of A and B as sayingthat 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.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.




Ex. Create and prove two events are dependent from drawing a card from a standard deck.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 fromyour school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

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.

S-CP.4 Determine if two categorical variables are independent by analyzing a two-way table of data collected onthe two variables.

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.

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 theconcepts of conditional probabilityand independence in everydaylanguage and everyday situations. For example, compare the chance of having lung cancer if you are a

S-CP.5 Given an everyday situation describing two events, use the context to construct an argument as to whetherthe 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 playingchess” and “being a good math student” are independent or dependent? Justify your answer.

Ex. Juanita flipped a coin 10 times and got the following results: T, H, T, T, H, H, H, H, H, H. Her math partnerModeling is best interpreted not as a collection of isolated topics, but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol ().



smoker with the chance of being asmoker if you have lung cancer.

Harold thinks that the next flip is going to result in tails because there have been so many heads in a row. Do youagree? Explain why or why not.

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.


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.

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. Peterrandomly draws one marble, sets it aside, and then randomly draws another marble. What is the probability ofPeter 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 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 generalMultiplication Rule in a uniformprobability model, P(A and B) =P(A)P(B|A) = P(B)P(A|B), andinterpret the answer in terms of themodel.

S-CP.8 (+) Calculate probabilities using the General Multiplication Rule. Interpret in context.

S-CP.9 (+)Use permutations andcombinations 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. Usepermutations and combinations in conjunction with other probability methods to calculate probabilities ofcompound events and solve problems.



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