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 Number and Quantity ● Unpacked Content
For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
What is the purpose of this document?
To increase student achievement by ensuring educators understand specifically what the new standards mean a student must know, understand and be
able to do.
What is in the document?
Descriptions of what each standard means a student will know, understand and be able to do. The “unpacking” of the standards done in this document is
an effort to answer a simple question “What does this standard mean that a student must know and be able to do?” and to ensure the description is
helpful, specific and comprehensive for educators.
How do I send feedback?
We intend the explanations and examples in this document to be helpful and specific. That said, we believe that as this document is used, teachers and
educators will find ways in which the unpacking can be improved and made ever more useful. Please send feedback to us at [email protected]
and we will use your input to refine our unpacking of the standards. Thank You!
Just want the standards alone?
You can find the standards alone at www.corestandards.org.
The Real Number System
N-RN
Common Core Cluster
Extend the properties of exponents to rational exponents.
Common Core Standard
Unpacking
What does this standard mean that a student will know and be able to do?
N-RN.1 Explain how the definition of
the meaning of rational exponents
follows from extending the properties
of integer exponents to those values,
allowing for a notation for radicals in
terms of rational exponents. For
example, we define 51/3to be the cube
root of 5 because we want (51/3)3=
5(1/3)3to hold, so (51/3)3must equal 5.
N-RN.1 Understand that the denominator of the rational exponent is the root index and the numerator is the
exponent of the radicand. For example, 51/2 = 5 Assessment Prototype
levels must be identified
for this document.
What is the pattern for the exponents? They are reduced by a factor of
N-RN.1 In order to understand the meaning of rational exponents, students can initially investigate them by
considering a pattern such as:
€
each time. What is the pattern of the
simplified values? Each successive value is the square root of the previous value. If we continue this pattern, then
.
Once the meaning of a rational exponent (with a numerator of 1) is established, students can verify that the
properties of integer exponents hold for rational exponents as well. For example,
since
since
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 2 (Level I)
Ex. Use an example to show why
holds true for expressions involving rational exponents like or .
Ex: Medical scientists model the amount of active insulin, y, left in the bloodstream after x minutes using the formula (knowing that 10 units of insulin were administered): What is the amount of active insulin left after 20 seconds?
N-RN.1 Extend the properties of exponents to justify that (51/2 )2 = 5
(Level I)
Ex: Using what you know about properties of exponents, simplify:
Ex: Using the connection between rational exponents and radicals, determine the value of x: 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 radicals as
expressions using rational exponents. At the Math 1 level, focus on fractional exponents with a numerator of 1.
(Level I)
Ex. Simplify the following.
a.
b.
Ex. Simplify the following.
a.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 3 b.
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?
The Real Number System
N-RN
Common Core Cluster
Use properties of rational and irrational numbers.
Common Core Standard
N-RN.3 Explain why the sum or
product of two rational numbers is
rational; that the sum of a rational
number and an irrational number is
irrational; and that the product of a
nonzero rational number and an
irrational number is irrational.
Unpacking
What does this standard mean that a student will know and be able to do?
N-RN.3 Know and justify that when adding or multiplying two rational numbers the result is a rational number
(Level III)
Ex: What set(s) of numbers in the Real Number System is the sum of and an element of? What set(s) of numbers
in the Real Number System is the product of and an element of? Explain how you know whether the solution
would be rational or irrational without computing.
N-RN.3 Know and justify that when adding a rational number and an irrational number the result is irrational.
(Level III)
Ex: What set(s) of numbers in the Real Number System is the sum of and
an element of? Explain how you
know whether the solution would be rational or irrational without computing.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 4 Ex: Can two unique irrational numbers have a rational number as their sum? Justify your response.
N.RN.3 Know and justify that when multiplying a nonzero rational number and an irrational number the result is
irrational.
Ex: What set(s) of numbers in the Real Number System is the product of and
an element of? Explain how
you know whether the solution would be rational or irrational without computing.
Ex: Can two unique irrational numbers have a rational number as their product? Justify your answer.
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 Interpret units in the context of the problem. 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 When solving a multi-step problem, use units to evaluate the appropriateness of the solution.
(Level I) Ex. You are mixing concrete for a school project and you have calculated according to the directions that you need 8 gallons of water for your mix. Your bucket is not calibrated, so you do not know how much it holds. On the other hand, you have just finished a 2 liter bottle of soda. If you use the bottle to measure your water, how many times will you need to fill it? Conversion factor 1 gallon = 3.785 liters. Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 5 and
to find the number of bottles needed. Ex: You are sitting at a stop light and the light turns green. Knowing the acceleration a in m/s2 and the time t in s,
and the fact that s=at, what is the speed, s, and in what unit should it be expressed?
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.
Ex. The Columbus High School store sells bottled drinks before and after school. During the first few weeks of
school, the manager set a price of $1.25 per bottle, and daily sales averaged 90 bottles per day. The manager then
increased the price to $1.75 per bottle, and sales decreased to an average of 70 bottles per day. What is the rate of
change in average daily sales as the price per bottle increases from $1.25 to $1.75? What units would you use to
describe this rate of change? Why?
Ex. When finding the area of a circle using the formula
the radius?
a.
square feet
b.
inches
c.
cubic yards
d.
pounds
, which unit of measure would be appropriate for
Ex. What is the unit of measure when calculating perimeter, area, and volume? Why?
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.
(Level I/II)
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. Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 6 Ex. What scale would be appropriate when creating a histogram using the following data that describes the level
of lead in the blood of children (in micrograms per deciliter) who were exposed to lead from their parents’
workplace? Why?
10, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 23, 24, 25, 27, 31, 34, 34, 35, 35, 36, 37, 38, 39, 39, 41, 43, 44, 45, 48, 49,
62, 73
N-Q.2 Define appropriate quantities
for the purpose of descriptive
modeling.
N-Q.2 Determine and interpret appropriate quantities when using descriptive modeling. 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.
(Level I)
Ex. What quantities could you use to describe the best city in North Carolina?
Ex. What quantities could you use to describe the effectiveness of a basketball player?
Ex. If you were opening your own restaurant, what criteria must be considered for your restaurant to be
successful? Now that you have defined the variables, which ones will provide income and which ones will be
expenses?
Ex. What quantities could you use to describe a safe bungee jump apparatus? If you were to build an in-classroom
bungee jump apparatus, what units would be best to use for your measurements? Explain. If you were to build a
real bungee jump apparatus, what units would it be best to use for your measurements? Explain. How can you
relate the model from the classroom to the real life bungee jump?
Ex. Elizabeth is working on a crime report for the city in which she lives. After the interview process, she found
some people thought that building more police stations, which means living closer to a police station, would result
in less crime. Others she interviewed thought that the closeness to the police station was not an important factor in
a neighborhood’s crime level. Elizabeth wanted to research these claims further, so she called her local police
stations to get data on crimes within the past year, focusing specifically on robberies and how far away each
occured from the police station. Below is a table displaying the data she gathered.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 7 N-Q.3 Choose a level of accuracy
appropriate to limitations on
measurement when reporting
quantities.
Number of blocks
Number of crimes
from police station
per block
0-5
1.4
6-10
1.5
Greater than 10
1.7
1. Analyze this information determining the relationship that exists between the closeness to the police
station and the number of crimes per block? Justify your reasoning.
2. Write a statement describing the relationship you found, how you found it, and why your explanation
makes sense.
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.
(Level II)
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?
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 8 Ex. What would an appropriate level of accuracy be when measuring the length of a beach, from sand to water?
Explain your reasoning.
The Complex Number System
N-CN
Common Core Cluster
Perform arithmetic operations with complex numbers.
Common Core Standard
N-CN.1 Know there is a complex
number i such that i2= –1, and every
complex number has the form a + bi
with a and b real.
Unpacking
What does this standard mean that a student will know and be able to do?
N-CN.1 Every number is a complex number or the form a + bi, where a and b are elements of the Real Numbers
and bi is an element of the Pure Imaginary Numbers. Students should know the sets and subsets of the Complex
Number System. The identity, i= − 1 , is not only used to identify non-real solutions for particular functions, but
2
is also be used to find the identity, i = –1, which is used to simplify expressions.
(Level III)
Ex. Is it possible to simplify
2
N-CN.2 Use the relation i = –1 and
the commutative, associative, and
distributive properties to add,
subtract, and multiply complex
numbers.
N-CN.3 (+) Find the conjugate of a
complex number; use conjugates to
find moduli and quotients of complex
using the Real Number System? Justify your reasoning.
N.CN.2 When adding, subtracting, or multiplying Complex Numbers and i 2 remains in the expression, use the
identity i 2 = −1 and the commutative, associative, and distributive properties to simplify the expression further.
(Level III)
Ex. The voltage E, current I, and resistance R in an electrical circuit are related by Ohm’s Law: E=IR.
Respectively, these quantities are measured in volts, amperes, and ohms, respectively. Find the voltage in an
electrical circuit with current (2 + 4i) amperes and resistance (5-4i) ohms.
N-CN.3 Given a complex number, when multiplying the number by it’s conjugate, the product is a Real Number.
Therefore, this means that complex conjugates are complex numbers that have the same Real part, and their
Imaginary parts are only different in sign, as in opposite signs.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 9 numbers.
Ex. The voltage E, current I, and resistance R in an electrical circuit are related by Ohm’s Law: E=IR.
Respectively, these quantities are measured in volts, amperes, and ohms, respectively. If an electrical circuit has
(3 − 4i) amperes, find the necessary resistance value to produce a non-Complex voltage.
N-CN.3 Find the magnitude(length), modulus(length) or absolute value(length), of the vector representation of a
complex number.
The Complex Number System
N-CN
Common Core Cluster
Represent complex numbers and their operations on the complex plane.
Common Core Standard
Unpacking
What does this standard mean that a student will know and be able to do?
N-CN.4 (+) Represent complex
numbers on the complex plane in
rectangular and polar form (including
real and imaginary numbers), and
explain why the rectangular and polar
forms of a given complex number
represent the same number.
N-CN.4 Transform complex numbers in a complex plane from rectangular to polar form and vise versa,
N-CN.5 (+) Represent addition,
subtraction, multiplication, and
conjugation of complex numbers
geometrically on the complex plane;
use properties of this representation
for computation. For example,
(1 – √3i)3= 8 because (1 – √3i) has
modulus 2 and argument 120°.
N-CN.5 Geometrically show addition, subtraction, and multiplication of complex numbers on the complex
coordinate plane.
N-CN.4 Know and explain why both forms, rectangular and polar, represent the same number.
N-CN.5 Geometrically show that the conjugate of complex numbers in a complex plane is the reflection across the
x-axis.
N-CN.5 Evaluate the power of a complex number, in rectangular form, using the polar form of that complex
number.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 10 N-CN.6 (+) Calculate the distance
between numbers in the complex
plane as the modulus of the
difference, and the midpoint of a
segment as the average of the
numbers at its endpoints.
N-CN.6 Find the distance between complex numbers using the modulus of the difference, and then find the
midpoint using the average of the numbers at the endpoints of a segment in the complex plane.
The Complex Number System
N-CN
Common Core Cluster
Use complex numbers in polynomial identities and equations.
Common Core Standard
N-CN.7 Solve quadratic equations
with real coefficients that have
complex solutions.
Unpacking
What does this standard mean that a student will know and be able to do?
N-CN.7 Extend strategies for solving quadratics such as; taking the square root and applying the quadratic
formula, to find solutions of the form, a + bi, for quadratic equations. This extension is made when the identity
i= − 1 is used to simplify radicals having negative numbers under the radical.
(Level III)
Ex. Describe how and explain why the solution(s) to a quadratic equation may be:
a. One Real solution b. Two Complex solutions c. Two Real solutions Ex. Your class is given the quadratic equation ax2 + bx + c = 0, knowing that one solution is 2 + 3i. The class
came up with several different possibilities for the second solution; (3 + 2i), (2 – 3i), (-2 – 3i), and (-2 + 3i). Are
either of these proposed solutions correct? Why, or why not?
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 11 N-CN.8 (+) Extend polynomial
identities to the complex numbers.
For example, rewrite x2+ 4 as
(x + 2i)(x – 2i).
N-CN.8 Use polynomial identities to write equivalent expressions in the form of complex numbers
N-CN.9 (+) Know the Fundamental
Theorem of Algebra; show that it is
true for quadratic polynomials.
N-CN.9 Understand The Fundamental Theorem of Algebra, which says that the number of complex solutions to a
polynomial equation is the same as the degree of the polynomial. Show that this is true for any quadratic
polynomial.
(Level III)
Ex. If a function has a degree of 5, how many solutions would that function have? Why?
Ex. Solve the general form of a quadratic equation for x. What do the solutions show in relation to the
Fundamental Theorem of Algebra for quadratic equations? Vector and Matrix Quantities
N-VM
Common Core Cluster
Represent and model with vector quantities.
Common Core Standard
N-VM.1 (+) Recognize vector
quantities as having both magnitude
and direction. Represent vector
quantities by directed line segments,
and use appropriate symbols for
vectors and their magnitudes (e.g., v,
|v|, ||v||, v).
Unpacking
What does this standard mean that a student will know and be able to do?
N-VM.1 Know that a vector is a directed line segment representing magnitude and direction.
N-VM.1 Use the appropriate symbol representation for vectors and their magnitude.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 12 N-VM.2 (+) Find the components of a
vector by subtracting the coordinates
of an initial point from the
coordinates of a terminal point.
N-VM.2 Find the component form of a vector by subtracting the coordinates of an initial point from the coordinates
of a terminal point, therefore placing the initial point of the vector at the origin.
N-VM.3 (+) Solve problems
involving velocity and other
quantities that can be represented by
vectors.
N-VM.3 Solve problems such as velocity and other quantities that can be represented using vectors.
Vector and Matrix Quantities
N-VM
Common Core Cluster
Perform operations on vectors.
Common Core Standard
N-VM.4 (+) Add and subtract
vectors.
a. Add vectors end-to-end,
component-wise, and by the
parallelogram rule. Understand
that the magnitude of a sum of
two vectors is typically not the
sum of the magnitudes.
b. Given two vectors in magnitude
and direction form, determine the
magnitude and direction of their
sum.
c. Understand vector subtraction v –
w as v + (–w), where –w is the
additive inverse of w, with the
same magnitude as w and
pointing in the opposite direction.
Unpacking
What does this standard mean that a student will know and be able to do?
N-VM.4a Know how to add vectors head to tail, using the horizontal and vertical components, and by finding the
diagonal formed by the parallelogram.
N-VM.4b Understand that the magnitude of a sum of two vectors is not the sum of the magnitudes unless the
vectors have the same heading or direction.
N-VM.4c Know how to subtract vectors and that vector subtraction is defined much like subtraction of real
numbers, in that v – w is the same as v + (–w), where –w is the additive inverse of w. The opposite of w, -w, has
the same magnitude, but the direction of the angle differs by 180°.
N-VM.4c Represent vector subtraction on a graph by connecting the vectors head to tail in the correct order and
using the components of those vectors to find the difference.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 13 Represent vector subtraction
graphically by connecting the tips
in the appropriate order, and
perform vector subtraction
component-wise.
N-VM.5 (+) Multiply a vector by a
scalar.
a. Represent scalar multiplication
graphically by scaling vectors and
possibly reversing their direction;
perform scalar multiplication
component-wise, e.g., as c(vx, vy)
= (cvx, cvy).
b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
N-VM.5a Represent scalar multiplication of vectors on a graph by increasing or decreasing the magnitude of the
vector by the factor of the given scalar. If the scalar is less than zero, the new vector’s direction is opposite the
original vector’s direction.
N-VM.5a Represent scalar multiplication of vectors using the component form, such as c(vx, vy) = (cvx, cvy).
N-VM.5b Find the magnitude of a scalar multiple, cv, is the magnitude of v multiplied by the factor of the |c|.
Know when c > 0, the direction is the same, and when c < 0, then the direction of the vector is opposite the
direction of the original vector.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 14 Vector and Matrix Quantities
N-VM
Common Core Cluster
Perform operations on matrices and use matrices in applications.
Common Core Standard
Unpacking
What does this standard mean that a student will know and be able to do?
N-VM.6 (+) Use matrices to represent
and manipulate data, e.g., to represent
payoffs or incidence relationships in a
network.
N-VM.7 (+) Multiply matrices by
scalars to produce new matrices, e.g.,
as when all of the payoffs in a game
are doubled.
N-VM.8 (+) Add, subtract, and
multiply matrices of appropriate
dimensions.
N-VM.9 (+) Understand that, unlike
multiplication of numbers, matrix
multiplication for square matrices is
not a commutative operation, but still
satisfies the associative and
distributive properties.
N-VM.6 Represent and manipulate data using matrices, e.g., to organize merchandise, keep total sales, costs, and
using graph theory and adjacency matrices to make predictions.
N-VM.10 (+) Understand that the
zero and identity matrices play a role
in matrix addition and multiplication
similar to the role of 0 and 1 in the
real numbers. The determinant of a
square matrix is nonzero if and only if
the matrix has a multiplicative
inverse.
N-VM.10 Identify a zero matrix and understand that it behaves in matrix addition, subtraction, and multiplication,
much like 0 in the real numbers system.
N-VM.7 Multiply matrices by a scalar, e.g., when the inventory of jeans for July is twice that for January.
N-VM.8 Know that the dimensions of a matrix are based on the number of rows and columns.
N-VM.8 Add, subtract, and multiply matrices of appropriate dimensions.
N-VM.9 Understand that matrix multiplication is not commutative, AB ≠ BA, however it is associative and
satisfies the distributive properties.
N-VM.10 Identify an identity matrix for a square matrix and understand that it behaves in matrix multiplication
much like the number 1 in the real number system.
N-VM.10 Find the determinant of a square matrix, and know that it is a nonzero value if the matrix has an inverse.
N-VM.10 Know that if a matrix has an inverse, then the determinant of a square matrix is a nonzero value.
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 15 N-VM.11 (+) Multiply a vector
(regarded as a matrix with one
column) by a matrix of suitable
dimensions to produce another vector.
Work with matrices as
transformations of vectors.
N-VM.11 To translate the vector AB , where A(1,3) and B(4,9), 2 units to the right and 5 units up, perform the
following matrix multiplication.
N-VM.12 (+) Work with 2 × 2
matrices as a transformations of the
plane, and interpret the absolute
€ value
of the determinant in terms of area.
N-VM.12 Given the coordinates of the vertices of a parallelogram in the coordinate plane, find the vector
representation for two adjacent sides with the same initial point. Write the components of the vectors in a 2x2
matrix and find the determinant of the 2x2 matrix. The absolute value of the determinant is the area of the
parallelogram. (This is called the dot product of the two vectors.)
"1 0 2 %"1 4 % " 3 6 %
$
'$
' $
'
$0 1 5 '$ 3 9 ' = $€8 14 '
$#0 0 1 '&#$1 1 '& $#1 1 '&
Numbers and Quantity Theme Unpacked Content Current as of April 10, 2011 16