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 Standard
N-RN.2 Rewrite expressions
involving radicals and rational
exponents using the properties of
exponents.
Unpacking
What does this standard mean that a student will know and be able to do?
N-RN.2 Students should be able to use the properties of exponents to rewrite expressions involving rational exponents as
expressions using radicals.
2
1
1
Ex. The expression 8 3 can be written as ( 8 3 )2 or as (82 ) 3 .
Write these expressions in radical form. How would you confirm that these forms are equivalent?
3
4
Considering that 81  ( 4 81)3 , which form would be easier to simplify without a calculator? Why?
3
Ex. When calculating ( 9)3 , Kyle entered 9^3/2 into his calculator. Karen entered it as 9 . 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
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Quantities
N-Q
Common Core Cluster
Reason quantitatively and use units to solve problems.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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:
(
mi
)( hr )  mi
. Note that knowledge of the distance formula is not required to determine the
hr
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 selling
tickets. Their profit can be modeled by the formula P = x(4,000 – 250x) – 7500. Graph the profit model using an
appropriate 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.
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
2
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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. For
example, when using a ruler, you can only legitimately report accuracy to the nearest division. Further, 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. Also, when using a calculator, measures can be given to the nearest desired decimal place of a
particular 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 measuring
instrument. 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
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
3
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Common Core Cluster
Interpret the structure of expressions.
Common Core Standard
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 )n as the product of P
and a factor not depending on
P.
Unpacking
What does this standard mean that a student will know and be able to do?
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 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. For example, consider the
2
expression 4000 p  250 p that represents income from a concert where p is the price per ticket. The equivalent
factored form, p (4000  250 p) , 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).
b
b2  4ac
?

2a
2a
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.
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
4
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Seeing Structure in Expressions
A-SSE
Common Core Cluster
Write expressions in equivalent forms to solve problems.
Common Core Standard
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 the
approximate equivalent monthly
interest rate if the annual rate is
15%.
Unpacking
What does this standard mean that a student will know and be able to do?
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.
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
5
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Arithmetic with Polynomials and Rational Expressions
A-APR
Common Core Cluster
Perform arithmetic operations on polynomials.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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.
Common Core Standard
A-APR.3 Identify zeros of
polynomials when suitable
factorizations are available, and use
the zeros to construct a rough graph
of the function defined by the
polynomial.
Unpacking
What does this standard mean that a student will know and be able to do?
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?
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
6
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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.
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.
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.
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.
(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 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. What do you think the student had in mind when using the numbers 20 and 56?
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 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.
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
7
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
combinations of different foods.
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.
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 H =
kA (T1 - T2 )
, solve for T2.
L
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
8
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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-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
a simple equation as following from
the equality of numbers asserted at the solution, construct a convincing argument that justifies each step in the solution process with mathematical properties.
previous step, starting from the
assumption that the original equation
has a solution. Construct a viable
argument to justify a solution method.
A-REI.2 Solve simple rational and
radical equations in one variable, and
give examples showing how
extraneous solutions may arise.
A-REI.2 Solve simple rational and radical equations in one variable and provide examples of how extraneous
solutions arise. Add context.
Ex. Solve 5  ( x  4)  2 for x.
Ex. Mary solved x 
Why or why not?
Ex. Solve
2  x for x and got x = -2, and x = 1. Are all the values she found solutions to the equation?
3
x
3

 for x.
x 3 x 3 2
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
9
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Reasoning with Equations and Inequalities
A-REI
Common Core Cluster
Solve equations and equalities in one variable.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
A-REI.4b Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and
completing 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 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,
b2  4ac , 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 x 
8  (8)2  4(1)(2)
was his result.
2(1)
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?
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
10
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Reasoning with Equations and Inequalities
A-REI
Common Core Cluster
Solve systems of equations.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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.
Common Core Standard
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).
Unpacking
What does this standard mean that a student will know and be able to do?
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.
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
11
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
A-REI.11 Explain why the xcoordinates 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 twovariable system y = f(x) and y = g(x). When solving by graphing, the x-value(s) of the intersection point(s) of
y = 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 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.
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
12
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Interpreting Functions
F-IF
Common Core Cluster
Understand the concept of a function and use function notation.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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.
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
13
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Interpreting Functions
F-IF
Common Core Cluster
Interpret functions that arise in applications in terms of the context.
Common Core Standard
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.

Unpacking
What does this standard mean that a student will know and be able to do?
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 the
characteristics 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 180 (  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.
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
14
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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, 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
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.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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.
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.
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
15
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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.9 Compare properties of two
functions each represented in a
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 at
http://map.mathshell.org/materials/lessons.php?taskid=224
At this level, completing the square is still not expected.
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
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
16
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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.
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:
Forming Quadratics at
http://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.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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.)
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
17
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Building Functions
F-BF
Common Core Cluster
Build a function that models a relationship between two quantities.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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’s
graph, 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.
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
18
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Congruence
C-GO
Common Core Cluster
Experiment with transformation 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. 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 and
reflections that carry it onto itself.
G-CO.4 Develop definitions of
rotations, reflections, and translations
in terms of angles, circles,
perpendicular lines, parallel lines, and
line segments.
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.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. 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 Refer to 8.G where students have already experimented with the properties of rigid transformations and
dilations. Students should understand and be able to explain that when a figure is reflected about a line, the
segment that joins the preimage point to its corresponding image point is perpendicularly bisected by the line of
reflection. 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 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 this composite transformation.
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
19
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Congruence
G-CO
Common Core Cluster
Understand congruence in terms of rigid motions.
Common Core Standard
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.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.
Unpacking
What does this standard mean that a student will know and be able to do?
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.
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 preimage to the image?.
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.
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)
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
20
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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.
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.
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°.
Students can make sense of this problem by drawing diagrams of important features and relationships and using
concrete objects or pictures to help conceptualize and solve this problem.
Congruence
G-CO
Common Core Cluster
Prove geometric theorems.
Common Core Standard
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
Unpacking
What does this standard mean that a student will know and be able to do?
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?
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
21
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
sides of a triangle is parallel to the
third side and half the length; the
medians of a triangle meet at a point.
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?
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.
Common Core Standard
G-CO.13 Construct an equilateral
triangle, a square, and a regular
hexagon inscribed in a circle.
Unpacking
What does this standard mean that a student will know and be able to do?
G-CO.13 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 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 circumscribe
the hexagonal nuts that she is using. Each side of the regular hexagonal nut is 8mm. Construct a diagram that
shows what size drill she should use to countersink the nut.
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
22
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Similarity, Right Triangles, and Trigonometry
G-SRT
Common Core Cluster
Understand similarity in terms of similarity transformations.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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 its
image 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? How
could 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,
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
23
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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.
Similarity, Right Triangles, and Trigonometry
G-SRT
Common Core Cluster
Define trigonometric ratios and solve problems involving right triangles.
Common Core Standard
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.7 Explain and use the
relationship between the sine and
cosine of complementary angles.
Unpacking
What does this standard mean that a student will know and be able to do?
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.
Ex. There are three points on a line that goes through the origin, (5, 12) (10, 24) (15, 36). Sketch this graph. How dothe
y
compare? Why does this make sense? Call the distance from the origin to each point (“r”). Find the r for
x
y
x
each point. Find the ratios of and .
r
r
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).
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?
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
24
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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?
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
25
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Similarity, Right Triangles, and Trigonometry
G-SRT
Common Core Cluster
Apply trigonometry to general triangles.
Common Core Standard
G-SRT.9 (+) Derive the formula
A
1
ab sin(C ) for the area of a
2
triangle by drawing an auxiliary line
from a vertex perpendicular to the
opposite side.
Unpacking
What does this standard mean that a student will know and be able to do?
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 
h  a sin(C ) .
1
ab sin(C ) , for the area of a triangle, using the fact that the height of the triangle is,
2
Instructional note: The focus is on deriving the formula, not using it.
Show that for each of the following cases, A 
1
ab  cos C
2
Before students derive this problem, pose the following problem to help them see the usefulness of this shortcut.
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
26
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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).
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?
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
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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 that
given 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 DH 4 units, can you write a rule that
represents the relationship of the x value, the y value and the radius? Why is this relationship true? As point H
rotates around the circle, does this relationship stay true?
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
27
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
In the diagram below, circle D translated 4 units to the right to create circle E. Why is the equation of this new
circle (x − 4)2 + y2 = 42? Why is the equation for circle I x2 + (y − 4)2 = 42? Using similar reasoning, could you
write 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
center at (h, k) and a radius of r?
5 ? Can you generalize this equation for a circle with a
Common Core Cluster
Use coordinates to prove simple geometric theorems algebraically.
Common Core Standard
G-GPE.6 Find the point on a directed
line segment between two given
points that partitions the segment in a
given ratio.
Unpacking
What does this standard mean that a student will know and be able to do?
G-GPE.6 Given two points on a line, find the point that divides the segment into an equal number of parts. If
finding the mid-point, it is always halfway between the two endpoints. The x-coordinate of the mid-point will be
the mean of the x-coordinates of the endpoints and the y-coordinate will be the mean of the y-coordinates of the
endpoints.
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?
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
28
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Geometric Measurement and Dimension
G-GMD
Common Core Cluster
Visualize relationships between two-dimensional and three-dimensional objects.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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).
Modeling with Geometry
G-MG
Common Core Cluster
Apply geometric concepts in modeling situations.
Common Core Standard
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). 
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 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.
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
29
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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 a
cylinder). 
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). 
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 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?
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?
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
30
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
Interpreting Categorical and Quantitative Data
S-ID
Common Core Cluster
Understand and evaluate random processes underlying statistical experiments.
Common Core Standard
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?
Unpacking
What does this standard mean that a student will know and be able to do?
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.
Common Core Standard
S-IC.6 Evaluate reports based on
data.
Unpacking
What does this standard mean that a student will know and be able to do?
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.)
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
31
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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
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”).
Unpacking
What does this standard mean that a student will know and be able to do?
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.
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.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
Ex. Describe the following subset of outcomes for choosing one card from a standard deck of cards as the
intersection of two events: {queen of hearts, queen of diamonds}.
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.
Ex. Create and prove two events are independent from drawing a card from a standard deck.
S-CP.3 Understand that the conditional probability of event A given event B has already happened is given by the
formula:
P( A and B)
P( A | B) 
P( B)
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
32
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
of A, and the conditional probability
of B given A is the same as the
probability of B.
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 twoway 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
your 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 on
the 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.
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
33
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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.
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.
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
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.
Unpacking
What does this standard mean that a student will know and be able to do?
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.
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
34
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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.
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.
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 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
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
35
Formatted by WS/FCS July, 2013
Unpacking Document for Math II Standards (Created from NCDPI Conceptual Category/Theme Unpackings)
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.9 (+)Use permutations and
combinations to compute probabilities
of compound events and solve
problems.
S-CP.8 (+) Calculate probabilities using the General Multiplication Rule. Interpret in context.
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.
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
36
Formatted by WS/FCS July, 2013