keystone exams: algebra ii · keystone exams: algebra ii module 1—number systems and non-linear...

Keystone Exams: Algebra IIAssessment Anchors and Eligible Content

with Sample Questions and Glossary

Pennsylvania Department of Education

www.education.state.pa.us

April 2014

Pennsylvania Department of Education—Assessment Anchors and Eligible Content

PENNSYLVANIA DEPARTMENT OF EDUCATION

General Introduction to the Keystone Exam Assessment Anchors

Introduction

Since the introduction of the Keystone Exams, the Pennsylvania Department of Education (PDE) has been working to create a set of tools designed to help educators improve instructional practices and better understand the Keystone Exams. The Assessment Anchors, as defined by the Eligible Content, are one of the many tools the Department believes will better align curriculum, instruction, and assessment practices throughout the Commonwealth. Without this alignment, it will not be possible to significantly improve student achievement across the Commonwealth.

How were Keystone Exam Assessment Anchors developed?

Prior to the development of the Assessment Anchors, multiple groups of PA educators convened to create a set of standards for each of the Keystone Exams. Enhanced Standards, derived from a review of existing standards, focused on what students need to know and be able to do in order to be college and career ready. (Note: Since that time, PA Core Standards have replaced the Enhanced Standards and reflect the college- and career-ready focus.) Additionally, the Assessment Anchors and Eligible Content statements were created by other groups of educators charged with the task of clarifying the standards assessed on the Keystone Exams. The Assessment Anchors, as defined by the Eligible Content, have been designed to hold together, or anchor, the state assessment system and the curriculum/instructional practices in schools.

Assessment Anchors, as defined by the Eligible Content, were created with the following design parameters: Clear: The Assessment Anchors are easy to read and are user friendly; they clearly detail which

standards are assessed on the Keystone Exams.

Focused: The Assessment Anchors identify a core set of standards that can be reasonably assessed on a large-scale assessment; this will keep educators from having to guess which standards are critical.

Rigorous: The Assessment Anchors support the rigor of the state standards by assessing higher-order and reasoning skills.

Manageable: The Assessment Anchors define the standards in a way that can be easily incorporated into a course to prepare students for success.

How can teachers, administrators, schools, and districts use these Assessment Anchors?

The Assessment Anchors, as defined by the Eligible Content, can help focus teaching and learning because they are clear, manageable, and closely aligned with the Keystone Exams. Teachers and administrators will be better informed about which standards will be assessed. The Assessment Anchors and Eligible Content should be used along with the Standards and the Curriculum Framework of the Standards Aligned System (SAS) to build curriculum, design lessons, and support student achievement.

The Assessment Anchors and Eligible Content are designed to enable educators to determine when they feel students are prepared to be successful in the Keystone Exams. An evaluation of current course offerings, through the lens of what is assessed on those particular Keystone Exams, may provide an opportunity for an alignment to ensure student preparedness.


How are the Assessment Anchors organized?

The Assessment Anchors, as defined by the Eligible Content, are organized into cohesive blueprints, each structured with a common labeling system that can be read like an outline. This framework is organized first by module, then by Assessment Anchor, followed by Anchor Descriptor, and then finally, at the greatest level of detail, by an Eligible Content statement. The common format of this outline is followed across the Keystone Exams.

Here is a description of each level in the labeling system for the Keystone Exams: Module: The Assessment Anchors are organized into two thematic modules for each of the

Keystone Exams. The module title appears at the top of each page. The module level is important because the Keystone Exams are built using a module format, with each of the Keystone Exams divided into two equal-size test modules. Each module is made up of two or more Assessment Anchors.

Assessment Anchor: The Assessment Anchor appears in the shaded bar across the top of each Assessment Anchor table. The Assessment Anchors represent categories of subject matter that anchor the content of the Keystone Exams. Each Assessment Anchor is part of a module and has one or more Anchor Descriptors unified under it.

Anchor Descriptor: Below each Assessment Anchor is a specific Anchor Descriptor. The Anchor Descriptor level provides further details that delineate the scope of content covered by the Assessment Anchor. Each Anchor Descriptor is part of an Assessment Anchor and has one or more Eligible Content statements unified under it.

Eligible Content: The column to the right of the Anchor Descriptor contains the Eligible Content statements. The Eligible Content is the most specific description of the content that is assessed on the Keystone Exams. This level is considered the assessment limit and helps educators identify the range of the content covered on the Keystone Exams.

PA Core Standard: In the column to the right of each Eligible Content statement is a code representing one or more PA Core Standards that correlate to the Eligible Content statement. Some Eligible Content statements include annotations that indicate certain clarifications about the scope of an Eligible Content.

“e.g.” (“for example”)—sample approach, but not a limit to the Eligible Content

“Note”—content exclusions or definable range of the Eligible Content

How do the K–12 Pennsylvania Core Standards affect this document?

Assessment Anchor and Eligible Content statements are aligned to the PA Core Standards; thus, the former enhanced standards are no longer necessary. Within this document, all standard references reflect the PA Core Standards.

Standards Aligned System—www.pdesas.org

Pennsylvania Department of Education—www.education.state.pa.us


Keystone Exams: Algebra II

Logarithmic Properties

xy

logax = y ↔ x = a y log x = y ↔ x = 10 y In x = y ↔ x = ey

loga(x · y ) = loga x + loga y

loga xp = p · loga x

loga = loga x − loga y

Quadratic Functions

General Formula: f (x) = ax2 + bx + c

Standard (Vertex) Form: f (x) = a (x − h )2 + k

Factored Form: f (x ) = a(x − x1)(x − x2)

Quadratic Formula: x =

when ax2 + bx + c = 0 and a Þ 0

2a

ˉb ± b2 − 4ac

Compound Interest Equations

Annual: A = P (1 + r )t

Periodic: A = P (1 + )nt

Continuous: A = Pert

A = account total after t years

P = principal amount

r = annual rate of interest

t = time (years)

n = number of periods interest is compounded per year

n

r

Permutation: nPr =

Combination: nCr =

(n − r )!

n !

r !(n − r )!

n !

Data Analysis

Exponential Properties

1a

am

an

am · an = am + n (am)n = am · n

= am − n a ¯1 =

Powers of the Imaginary Unit

i = ¯1 i2 = ¯1

i3 = ¯ i i4 = 1

A = lw

l

w

V = lwh

lw

h

Shapes

FORMULA SHEET

Formulas that you may need to work questions in this document are found below.You may use calculator π or the number 3.14.



MODULE 1—Number Systems and Non-Linear Expressions & Equations

ASSESSMENT ANCHOR

A2.1.1 Operations with Complex Numbers

Anchor Descriptor Eligible ContentPA Core

Standards

A2.1.1.1 Represent and/or use imaginary numbers in equivalent forms (e.g., square roots and exponents).

A2.1.1.1.1 Simplify/write square roots in terms of i (e.g., Ï

}

-24 = 2i Ï}

6 ).CC.2.1.HS.F.6

A2.1.1.1.2 Simplify/evaluate expressions involving powers of i (e.g., i6 + i3 = –1 – i).

Sample Exam Questions

Standard A2.1.1.1.1

The expression Ï} x is equivalent to 14i Ï

}

3 . What is the value of x?

A. –588

B. –588i

C. 588

D. 588i

Standard A2.1.1.1.2

An expression is shown below.

x5 + 6x3 + 8x

Which value of x makes the expression equal to 0?

A. –2i2

B. –2i

C. 4i

D. 4i2




ASSESSMENT ANCHOR



Standards

A2.1.1.2 Apply the order of operations in computation and in problem-solving situations.

A2.1.1.2.1 Add and subtract complex numbers (e.g., (7 – 3i ) – (2 + i ) = 5 – 4i ).

CC.2.1.HS.F.6

A2.1.1.2.2 Multiply and divide complex numbers (e.g., (7 – 3i )(2 + i ) = 17 + i ).


Standard A2.1.1.2.1

An equation with real numbers a, b, c, and d is shown below.

(4i + ab) – (6i + cd) = –2i

Which relationship must be true?

A. ab = –cd

B. ab = cd

C. a = c

D. (a – c) = (b – d)

Standard A2.1.1.2.2

An equation is shown below.

(a + bi )(4 – 2i ) = 40

What is the value of b?

A. 2

B. 4

C. 10

D. 20




Standard A2.1.1

Lily is practicing multiplying complex numbers using the complex number (2 + i ).

To determine the value of (2 + i )2, Lily performs the following operations:

step 1: (2 + i )2 = 4 + i2

step 2: 4 + i2 = 4 + (–1)

step 3: 4 + (–1) = 3

Lily made an error.

A. Explain Lily’s error and correct the step which contains her error.

Lily says that (2 + i )n is a complex number for every positive integer value of n.

B. Explain how you know that Lily is correct.

Continued on next page.

ASSESSMENT ANCHOR






Continued. Please refer to the previous page for task explanation.

Lily is continuing to explore different ways in which complex numbers can be multiplied so the answer is not a complex number. Lily multiplies (2 + i ) and (a + bi ), where a and b are real numbers, and finds that her answer is not a complex number.

C. Write an equation that expresses the relationship between a and b.

equation:

D. Explain why the expression (c + di )2 is always a complex number for nonzero, real values of c and d.




Standard A2.1.1

To find the roots of a quadratic equation, ax2 + bx + c, where a, b, and c are real numbers, Jan uses the quadratic formula.

Jan finds that a quadratic equation has 2 distinct roots, but neither are real numbers.

A. Write an inequality using the variables a, b, and c that must always be true for Jan’s quadratic equation.

inequality:

The expression 3 + Ï}

–4 is a solution of the quadratic equation x2 – 6x +13 = 0.

B. What is 3 + Ï}

–4 written as a complex number?

3 + Ï}

–4 =






C. What is (5 + 2i )2 expressed as a complex number? Use the form a + bi, where a and b are real numbers.

(5 + 2i )2 =

D. What is a possible solution to the equation 5 = Ï}}

(a – bi )(a + bi ) when a and b are whole numbers greater than zero?

a = b =




ASSESSMENT ANCHOR

A2.1.2 Non-Linear Expressions


Standards

A2.1.2.1 Use exponents, roots, and/or absolute values to represent equivalent forms or to solve problems.

A2.1.2.1.1 Use exponential expressions to represent rational numbers.

CC.2.1.HS.F.1

CC.2.2.HS.D.2

A2.1.2.1.2 Simplify/evaluate expressions involving positive and negative exponents and/or roots (may contain all types of real numbers—exponents should not exceed power of 10).

A2.1.2.1.3 Simplify/evaluate expressions involving multiplying with exponents (e.g., x6 • x7 = x13), powers of powers (e.g., (x6)7 = x42), and powers of products (e.g., (2x2)3 = 8x6). Note: Limit to rational exponents.

A2.1.2.1.4 Simplify or evaluate expressions involving logarithms and exponents (e.g., log28 = 3 or log42 = 1 }

2 ).


Standard A2.1.2.1.1

Which expression is equivalent to 1 ___ 25

?

A. 4 • 10–2

B. 25 • 10–2

C. 4 • 10–1

D. 25 • 10–1

Standard A2.1.2.1.2


8 √___

x-8x + 5 √___

x-5x

Which value of x makes the expression equal to 1 } 2

?

A. 0

B. 1

C. 2

D. 4




Standard A2.1.2.1.3


x10n(x5 • (x–5 )n) = x–10

Which is the value of n?

A. –3

B. –1

C. – 1 } 2

D. 2

Standard A2.1.2.1.4


log 4 √

___ x

16 ___

y4

What is the value of the expression when log x = 8 and log y = 1?

A. 7

B. 15

C. 16

D. 31




Standard A2.1.2.2.1


6x2 – 19x + 10

Which is a factor of the expression?

A. 2x + 2

B. 2x + 5

C. 3x – 2

D. 3x – 5

Standard A2.1.2.2.2


3x2 – 4x – 15 ____________ 2x2 – 5x – 3

; x Þ – 1 } 2 , 3

Which expression is equivalent to the one shown?

A. 3x + 5 ______ 2x + 1

B. 3x – 5 } 2x – 1

C. x2 + x – 12

D. 5x2 – 9x – 18

ASSESSMENT ANCHOR



Standards

A2.1.2.2 Simplify expressions involving polynomials.

A2.1.2.2.1 Factor algebraic expressions, including difference of squares and trinomials.Note: Trinomials limited to the form ax2+bx+c where a is not equal to 0.

CC.2.2.HS.D.1

CC.2.2.HS.D.2

CC.2.2.HS.D.3

CC.2.2.HS.D.4

CC.2.2.HS.D.5

A2.1.2.2.2 Simplify rational algebraic expressions.





Standard A2.1.2

The expression (10d)3/2 is used to find how many times more energy is released by an earthquake of greater magnitude than by an earthquake of lesser magnitude, where d is the difference in magnitudes.

A. How many times more energy is given off by an earthquake with magnitude 5.2 than by an earthquake with magnitude 3.2?

times more energy:

B. What is the difference (d) when 100 times more energy is released by an earthquake of greater magnitude than by an earthquake of lesser magnitude?

d =

ASSESSMENT ANCHOR








C. What is an equivalent exponential expression to (10d)3/2 with a base of 1,000?

equivalent exponential expression:

D. Explain why ( Ï}

10 )3d is equivalent to (10d)3/2.




Standard A2.1.2

Beatriz is simplifying exponential and radical expressions.

A. What rational number is the result of simplifying 3 √_____

16-3/2 ?

rational number:

The exponential expression 52x + 3 can be simplified to the form a(bx) where a and b are integers.

B. What are the values of a and b?

a =

b =






The variable c represents a whole number between 1 and 100. The values of the expressions c1/2 and c2/3 are both whole numbers for only one value of c.

C. What whole number does c represent?

c =




ASSESSMENT ANCHOR

A2.1.3 Non-Linear Equations


Standards

A2.1.3.1 Write and/or solve non-linear equations using various methods.

A2.1.3.1.1 Write and/or solve quadratic equations (including factoring and using the Quadratic Formula).

CC.2.2.HS.C.2

CC.2.2.HS.C.3

CC.2.2.HS.C.4

CC.2.2.HS.C.5

CC.2.2.HS.C.6

CC.2.2.HS.D.5

CC.2.2.HS.D.6

CC.2.2.HS.D.7

CC.2.2.HS.D.8

CC.2.2.HS.D.9

CC.2.2.HS.D.10

A2.1.3.1.2 Solve equations involving rational and/or radical expressions (e.g., 10/(x + 3) + 12/(x – 2) = 1 or Ï}

x2 + 21x = 14).A2.1.3.1.3 Write and/or solve a simple exponential or

logarithmic equation (including common and natural logarithms).

A2.1.3.1.4 Write, solve, and/or apply linear or exponential growth or decay (including problem situations).


Standard A2.1.3.1.1

The equation x2 + bx + c = 0 has exactly 1 real solution when b and c are real numbers. Which equation describes b in terms of c?

A. b = c2

B. b = Ï}

c

C. b = 2c

D. b = 2 Ï}

c

Standard A2.1.3.1.2


64x4 + 8x3 _________

8x3 = x2 + 8

What is the solution set of the equation?

A. {–7, –1}

B. {–7, 1}

C. {–1, 7}

D. {1, 7}




Standard A2.1.3.1.3


35x = 92x – 1

Which equation has the same solution?

A. 3x = 10x – 5

B. 5x = 4x – 2

C. 8x = 11x – 1

D. 15x = 18x – 9

Standard A2.1.3.1.4

A patient is given a 100-milligram dosage of a drug that decays exponentially, with a half-life of 6 hours. Which equation could be used to find the milligrams of drug remaining (y) after x hours?

A. y = 100(6)0.5x

B. y = 100(x)0.5/6

C. y = 100(0.5)x/6

D. y = 100(0.5x)1/6




ASSESSMENT ANCHOR



Standards

A2.1.3.2 Describe and/or determine change.

A2.1.3.2.1 Determine how a change in one variable relates to a change in a second variable (e.g., y = 4/x; if x doubles, what happens to y?).

CC.2.2.HS.C.2

CC.2.2.HS.C.3

CC.2.2.HS.C.4

CC.2.2.HS.D.7

CC.2.2.HS.D.8

CC.2.2.HS.D.9 A2.1.3.2.2 Use algebraic processes to solve a

formula for a given variable (e.g., solve d = rt for r).


Standard A2.1.3.2.1

A moving object’s kinetic energy (Ek ) is dependent on the mass of the object (m) and the object’s velocity (v), as shown in the equation below.

Ek = 1 __ 2 mv2

How does the value of Ek change when the value of m is unchanged and the value of v is multiplied by 2?

A. The value of Ek is squared.

B. The value of Ek is multiplied by 2.

C. The value of Ek is multiplied by 4.

D. The value of Ek is multiplied by 8.

Standard A2.1.3.2.2

Physicists use the formula shown below to determine total energy (E) of a body by usingmomentum (p), mass (m), and the speed of light (c).

E = Ï}}

(pc)2 + (mc2)2

A physicist knows the speed of light, the mass of the body, and the total energy used by the body. Which formula could be used to determine the momentum of the body?

A. p = Ï}

E2 – m2c4 } c

B. p = Ï}

E – mc2 } c

C. p = E __ c – mc

D. p = E – mc




Standard A2.1.3

Michaela is solving rational equations.

A. What is the solution set of the equation x2 – 7x + 12 ___________ x2 + x – 12

= 3? Show or explain all your work.

B. The only solution of the equation x2 + bx – 18 }} –2x + 4

= 4 is x = –17. What is the value of b?

b =

ASSESSMENT ANCHOR








Michaela solved for x in the rational equation as shown below.

x2 + 2x – 15 ___________

x – 3 = 3

(x – 3) • x2 + 2x – 15 ___________

x – 3 = 3 • (x – 3)

x2 + 2x – 15 = 3x – 9

x2 – x – 6 = 0

(x – 3)(x + 2) = 0

x – 3 = 0 or x + 2 = 0

x = {3, –2}

The solution set for x is incorrect.

C. Explain Michaela’s error.




Standard A2.1.3

A fully-charged lithium-ion computer battery loses 20% of its permanent capacity each year of storage.

A. Write an exponential equation showing the capacity (c) remaining in a fully-charged lithium-ion computer battery after y years.

c =

B. What capacity is remaining in a fully-charged lithium-ion battery after 1.5 years?

capacity:






C. Solve the exponential equation from part A for y.

y =

D. For how many years will a fully-charged lithium-ion computer battery have been stored when it has lost exactly half of its capacity?

years:



MODULE 2—Functions and Data Analysis

ASSESSMENT ANCHOR

A2.2.1 Patterns, Relations, and Functions


Standards

A2.2.1.1 Analyze and/or use patterns or numbers.

A2.2.1.1.1 Analyze a set of data for the existence of a pattern, and represent the pattern with a rule algebraically and/or graphically.

CC.2.1.HS.F.7

CC.2.2.HS.C.1

CC.2.2.HS.C.2

CC.2.2.HS.C.3

CC.2.2.HS.C.5

CC.2.2.HS.C.6

CC.2.3.HS.A.10

CC.2.4.HS.B.2

A2.2.1.1.2 Identify and/or extend a pattern as either an arithmetic or geometric sequence (e.g., given a geometric sequence, fi nd the 20th term).

A2.2.1.1.3 Determine the domain, range, or inverse of a relation.

A2.2.1.1.4 Identify and/or determine the characteristics of an exponential, quadratic, or polynomial function (e.g., intervals of increase/decrease, intercepts, zeros, and asymptotes).


Standard A2.2.1.1.1

Terms 1 through 5 of a pattern are listed below.

4 3 4 7 12

The pattern continues. Which expression could be used to determine the nth term in the pattern?

A. 2n + 2

B. Zn – 2Z + 3

C. n2 – 4n + 7

D. n3 – 5n2 + 7n + 1

Standard A2.2.1.1.2

Terms 1 through 5 of a sequence are shown below.

8 } 81

4 } 27

2 } 9 1 }

3 1 }

2

What is the 10th term in the sequence?

A. 81 } 64

B. 81 } 32

C. 243 } 64

D. 243 ____ 32




Standard A2.2.1.1.3

What is the inverse of y = ln (x – 15) + 3?

A. y = ex – 3 + 15

B. y = ex + 12

C. y = ex + 12 + 15

D. y = ex – 15 + 3

Standard A2.2.1.1.4

When is f(x) = x2 – x – 12 increasing?

A. x > 1 } 2

B. x < 1 } 2

C. x > –3

D. x < 4




Standard A2.2.1

The path of a roller coaster after it has reached the top of the first hill follows a polynomial function, as shown in the graph below.

300

200

100

0 100 200 300 400 500

f(x)

x

Hei

ght (

in fe

et)

Distance (in feet)

Path of a Roller Coaster

A. Over what interval is f(x) increasing?

≤ x ≤


ASSESSMENT ANCHOR

A2.2.1 Patterns, Relations, and Functions






B. At what value of x is there a minimum of f(x) over the interval 0 ≤ x ≤ 300?

value of x:

C. At what value of x is there a zero of f(x)?

value of x:






D. Explain why f(x) cannot be a quadratic function.




Standard A2.2.1

The function below describes the graph of a quadratic function where c is a positive real number.

y = x2 – c

A. What are the x-intercept(s) of the graph of the quadratic function?

x-intercept(s):

B. What is a y-value, in terms of c, which cannot be in the range of the quadratic function?

y-value:






C. What is the domain of the inverse of the quadratic function?

domain of the inverse of the quadratic function:




ASSESSMENT ANCHOR

A2.2.2 Applications of Functions


Standards

A2.2.2.1 Create, interpret, and/or use polynomial, exponential, and/or logarithmic functions and their equations, graphs, or tables.

A2.2.2.1.1 Create, interpret, and/or use the equation, graph, or table of a polynomial function (including quadratics).

CC.2.1.HS.F.3

CC.2.1.HS.F.4

CC.2.2.HS.C.3

CC.2.2.HS.C.4

CC.2.2.HS.C.5

CC.2.2.HS.C.6

CC.2.2.HS.D.7

CC.2.3.HS.A.10

A2.2.2.1.2 Create, interpret, and/or use the equation, graph, or table of an exponential or logarithmic function (including common and natural logarithms).

A2.2.2.1.3 Determine, use, and/or interpret minimum and maximum values over a specifi ed interval of a graph of a polynomial, exponential, or logarithmic function.

A2.2.2.1.4 Translate a polynomial, exponential, or logarithmic function from one representation of a function to another (graph, table, and equation).


Standard A2.2.2.1.1

The table below represents a quadratic function.

f(x)x

5 3

4 0

1 3

0 8

1 15

Which describes a complete list where the zeros of f(x) occur?

A. x = 8 and x = 4

B. x = 4 and x = 2

C. x = –8 and x = –4

D. x = –4 and x = –2




Standard A2.2.2.1.2

A logarithmic function is graphed below.

x

f(x)

21 3 4–2–3–4

–4–3–2

1234

–1–1

What is the value of f(8)?

A. 3

B. 4

C. 16

D. 256

Standard A2.2.2.1.3

A function of x is shown below.

y = –3(x – 2)(x + 4)

What is the maximum value of the function over the interval –3 ≤ x ≤ 2?

A. 0

B. 15

C. 24

D. 27

Standard A2.2.2.1.4

A function of x is graphed below.

x

y

2 4 6 8–2–6–8

–4–6–8

–2

2468

–4

Which equation best describes the graph?

A. y = x2 + 5

B. y = (x – 2)2 + 1

C. y = (x + 2)2 + 1

D. y = (x + 2)(x – 1)




ASSESSMENT ANCHOR



Standards

A2.2.2.2 Describe and/or determine families of functions.

A2.2.2.2.1 Identify or describe the effect of changing parameters within a family of functions (e.g., y = x2 and y = x2 + 3, or y = x2 and y = 3x2).

CC.2.2.HS.C.4

CC.2.2.HS.C.5

CC.2.2.HS.C.6

Sample Exam Question

Standard A2.2.2.2.1

The graph of the equation y = 3x2 has its vertex at the coordinate point (0, 0). What coordinate point describes the vertex of the graph of the equation y = 3x2 – 3?

A. (0, –3)

B. (0, 3)

C. (–3, 0)

D. (3, 0)




Standard A2.2.2

An exponential function of the form f(x) = a • bx + c is represented by the pairs of values shown in the table below.

f(x)x

2 1,215

1 135

0 15

153

2527

A. Determine the exponential function that contains the 5 points shown in the table.

f(x) =


ASSESSMENT ANCHOR







B. What is the minimum value of f(x) over the interval –5 ≤ x ≤ 5?

minimum:

C. Describe the difference in the graph of the exponential function g(x) = a • bx + 2 + c and the graph of the exponential function f(x) = a • bx + c when a, b, and c remain unchanged.






D. What is the value of g(0)?

g(0) =




Standard A2.2.2

The number of meals a restaurant serves is a function of the price of each meal. The restaurant found it will serve 72 meals when it charges a price of $7.00 per meal. It will serve 52 meals when it charges a price of $12.00 per meal. The relationship between the number of meals served and the price of each meal is linear.

A. Write a linear function that represents the relationship between the number of meals served (f(x) ) and the price of each meal (x).

f(x) =






The table below shows profits for the same restaurant based on the price of each meal.

Restaurant Profits

Price (in dollars) Profit (in dollars)

5 80

7 216

16 432

22 216

The relationship between the profit and the price of each meal can be represented by a quadratic function.

B. Write a quadratic function that represents the relationship between the restaurant’s profit (g(x) ) and the price of each meal (x).

g(x) =

C. Based on the quadratic function from part B, what is a price per meal where the profit will be exactly $0.00?

price: $

D. Based on the quadratic function from part B, what is the maximum profit, in dollars?

maximum profit: $




ASSESSMENT ANCHOR

A2.2.3 Data Analysis


Standards

A2.2.3.1 Analyze and/or interpret data on a scatter plot and/or use a scatter plot to make predictions.

A2.2.3.1.1 Draw, identify, fi nd, interpret, and/or write an equation for a regression model (lines and curves of best fi t) for a scatter plot.

CC.2.1.HS.F.3

CC.2.1.HS.F.5

CC.2.4.HS.B.2

CC.2.4.HS.B.3A2.2.3.1.2 Make predictions using the equations or

graphs of regression models (lines and curves of best fi t) of scatter plots.


Standard A2.2.3.1.1

Nehla recorded the volume in decibels for different settings on her amplifier as shown in the scatter plot below.

Amplifier Volume

Setting

Dec

ibel

s

14012010080604020

10 2 3 4 65 7 8 9 10 11x

y

Which equation best describes the curve of best fit?

A. y = x } 9

B. y = 30x

C. y = 30x2

D. y = 30 Ï} x

Standard A2.2.3.1.2

Students conducted an experiment dropping balls and measuring how high the balls bounced. They recorded their results in the scatter plot shown below.

Ball Bounce

Drop Height (cm)

Bou

nce

Hei

ght (

cm) 52

504846444240

50 52 54 56 6058 62x

y

Based on the line of best fit shown on the scatter plot, which is most likely the height of the bounce of a ball that was dropped from a height of 30 centimeters (cm)?

A. 20 cm

B. 24 cm

C. 30 cm

D. 38 cm




ASSESSMENT ANCHOR



Standards

A2.2.3.2 Apply probability to practical situations.

A2.2.3.2.1 Use combinations, permutations, and the fundamental counting principle to solve problems involving probability.

CC.2.4.HS.B.4

CC.2.4.HS.B.5

CC.2.4.HS.B.6

CC.2.4.HS.B.7A2.2.3.2.2 Use odds to fi nd probability and/or use

probability to fi nd odds.A2.2.3.2.3 Use probability for independent,

dependent, or compound events to predict outcomes.


Standard A2.2.3.2.1

A 5-character key code is randomly generated by a computer using the 26 letters of the alphabet and the 10 digits 0–9. What is the probability that the 5 characters in a key code, listed as they are randomly generated, will spell the word "GREAT"?

A. 1 }} 60,466,176

B. 1 }} 45,239,040

C. 5 }} 60,466,176

D. 5 }} 45,239,040

Standard A2.2.3.2.2

The probability an airplane will be delayed is 5 } 23

.

What are the odds in favor of the airplane being delayed?

A. 5:23

B. 5:18

C. 18:23

D. 18:5




Standard A2.2.3.2.3

A candy store owner gives 2 sample jelly beans to each customer. The owner randomly selects the 2 samples from a large container of jelly beans. The list below shows the percent of each fl avor of jelly bean in the container.

• 25% of the jelly beans are strawberry • 40% of the jelly beans are green apple • 20% of the jelly beans are grape • 15% of the jelly beans are lemon

Last month 550 people each received 2 sample jelly beans. Based on the information above, which is most likely the number of people who received 1 lemon-fl avored jelly bean and 1 green-apple-fl avored jelly bean?

A. 10

B. 33

C. 66

D. 151




Standard A2.2.3

A naturalist introduced a new species of fish to a lake. He started by putting 20 fish of the new species into the lake. The naturalist then recorded the total number of fish of the new species in the lake on the same day for each of the next three years. His data is listed below.

Fish of the New Species

YearNumber of

Fish

0 20

1 96

2 158

3 204

The exponential curve of best fit for this data can be expressed in the form y = –380 (b)x + c, where x represents the year and y represents the number of fish. The base b and constant c are real numbers.

A. What is the equation of the exponential curve of best fit for the data?

equation:


ASSESSMENT ANCHOR







B. Based on your equation of the curve of best fit from part A, how many fish of the new species will be in the lake when the naturalist records his data in year 6?

fish of the new species:

C. Based on your equation of the curve of best fit from part A, what is the first year in which there will be at least 350 fish of the new species in the lake?

year:






The naturalist wants to use the data to predict how many fish of the new species will be in the lake in 50 years.

D. What is one possible reason the correct equation for the curve of best fit may not provide an accurate prediction?




Standard A2.2.3

The cast of a play consists of 7 males and 5 females. There is 1 male lead role and 1 female lead role.

A. In how many different ways can the two lead roles be cast?

different ways:

Of the 7 males and 5 females, there are two sets of siblings. One set consists of a brother and sister, and the other set consists of a brother and his 2 sisters. The lead roles are cast randomly.

B. What is the probability that a pair of siblings are cast in the lead roles?

probability:






There are 8 speaking roles in the play. Of the 8 speaking roles, 5 are for males and 3 are for females. After the 8 speaking roles in the play are cast, the remaining cast members will make up the chorus.

C. What is the probability that the set of siblings consisting of a brother and his 2 sisters are all in the chorus?

probability:

The director decides to add n more females to the cast.

D. Write an expression that represents the probability that the set of siblings that consists of a brother and his 2 sisters are all in the chorus.

expression:


KEYSTONE ALGEBRA II ASSESSMENT ANCHORS

KEY TO SAMPLE MULTIPLE-CHOICE ITEMS

Algebra II

Eligible Content Key

A2.1.1.1.1 A

A2.1.1.1.2 B

A2.1.1.2.1 B

A2.1.1.2.2 B


A2.1.2.1.1 A

A2.1.2.1.2 C

A2.1.2.1.3 A

A2.1.2.1.4 D

A2.1.2.2.1 C

A2.1.2.2.2 A


A2.1.3.1.1 D

A2.1.3.1.2 D

A2.1.3.1.3 B

A2.1.3.1.4 C

A2.1.3.2.1 C

A2.1.3.2.2 A


A2.2.1.1.1 C

A2.2.1.1.2 C

A2.2.1.1.3 A

A2.2.1.1.4 A


A2.2.2.1.1 D

A2.2.2.1.2 A

A2.2.2.1.3 D

A2.2.2.1.4 C

A2.2.2.2.1 A


A2.2.3.1.1 D

A2.2.3.1.2 B

A2.2.3.2.1 A

A2.2.3.2.2 B

A2.2.3.2.3 C

Ke

yst

on

e E

xa

ms:

Alg

eb

ra

Glo

ssar

y t

o t

he

A

sses

smen

t A

nch

or

& E

ligi

ble

Co

nte

nt

The

Key

sto

ne

Glo

ssar

y in

clu

des

te

rms

and

de

fin

itio

ns

asso

ciat

ed

wit

h t

he

Ke

ysto

ne

Ass

essm

en

t A

nch

ors

an

d

Elig

ible

Co

nte

nt.

Th

e te

rms

and

de

fin

itio

ns

incl

ud

ed

in

th

e g

loss

ary

are

in

ten

ded

to

ass

ist

Pe

nn

sylv

ania

ed

uca

tors

in

bet

ter

un

der

stan

din

g th

e K

eys

ton

e A

sse

ssm

en

t A

nch

ors

an

d E

ligib

le C

on

ten

t. T

he

glo

ssar

y d

oes

n

ot

def

ine

all

po

ssib

le t

erm

s in

clu

de

d o

n a

n a

ctu

al K

eys

ton

e E

xam

, an

d i

t is

no

t in

ten

ded

to

def

ine

term

s fo

r u

se in

cla

ssro

om

inst

ruct

ion

fo

r a

par

ticu

lar

grad

e le

vel o

r co

urs

e.

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

ww

w.e

du

cati

on

.sta

te.p

a.u

s A

pri

l 20

14

http://www.education.state.pa.us/

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

A

pri

l 20

14

Ab

so

lute

Va

lue

A

nu

mb

er’s d

ista

nce

fro

m z

ero

on

th

e n

um

be

r lin

e.

It is w

ritt

en

|a

| a

nd

is r

ead

“th

e a

bso

lute

va

lue

of

a.”

It

resu

lts in

a n

um

be

r gre

ate

r th

an

or

equ

al to

ze

ro (

e.g

., |4

| =

4 a

nd

|–4

| =

4).

Exa

mp

le o

f a

bso

lute

va

lue

s o

f –4

and

4 o

n a

nu

mb

er

line

:

Ad

dit

ive

In

ve

rse

T

he

op

po

site

of

a n

um

be

r (i.e

., fo

r an

y n

um

be

r a

, th

e a

dd

itiv

e in

ve

rse

is –

a).

An

y n

um

be

r an

d its

a

dd

itiv

e in

ve

rse w

ill h

ave

a s

um

of

ze

ro (

e.g

., –

4 is th

e a

dd

itiv

e inve

rse o

f 4

sin

ce

4 +

–4 =

0;

like

wis

e,

the

ad

ditiv

e in

ve

rse o

f –4

is 4

sin

ce

–4

+ 4

= 0

).

Ari

thm

eti

c S

eq

uen

ce

A

n o

rde

red

lis

t of

nu

mb

ers

tha

t in

cre

ase

s o

r d

ecre

ase

s a

t a

co

nsta

nt

rate

(i.e.,

th

e d

iffe

ren

ce

be

twe

en

n

um

be

rs r

em

ain

s t

he

sa

me

). E

xa

mp

le:

1,

7, 13

, 19

, …

is a

n a

rith

me

tic s

equ

en

ce

as it h

as a

co

nsta

nt

diffe

ren

ce

of

+6

(i.e., 6

is a

dde

d o

ve

r an

d o

ver)

.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 3

A

pri

l 20

14

As

ym

pto

te

A s

traig

ht

line

to

wh

ich

th

e c

urv

e o

f a

gra

ph

co

me

s c

lose

r an

d c

loser.

The

dis

tan

ce

betw

ee

n t

he

cu

rve

an

d th

e a

sym

pto

te a

ppro

ach

es z

ero

as t

he

y t

en

d to

infin

ity.

The

asym

pto

te is d

en

ote

d b

y a

da

she

d

line

on a

gra

ph.

The m

ost

co

mm

on

asym

pto

tes a

re h

orizo

nta

l an

d v

ert

ical. E

xa

mp

le o

f a

ho

rizo

nta

l a

sym

pto

te:

Ba

r G

rap

h

A g

rap

h th

at

sh

ow

s a

se

t of fr

equ

en

cie

s u

sin

g b

ars

of

equ

al w

idth

, bu

t he

igh

ts t

ha

t a

re p

rop

ort

iona

l to

th

e f

reque

ncie

s. It

is u

se

d to

su

mm

ari

ze

dis

cre

te d

ata

. E

xa

mp

le o

f a

ba

r gra

ph

:

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 4

A

pri

l 20

14

Bin

om

ial

A p

oly

no

mia

l w

ith

tw

o u

nlik

e te

rms (

e.g

., 3

x +

4y o

r a

3 –

4b

2).

Ea

ch

te

rm is a

mo

no

mia

l, a

nd

the

mo

no

mia

ls a

re jo

ine

d b

y a

n a

dd

itio

n s

ym

bo

l (+

) or

a s

ub

tractio

n s

ym

bo

l (–

). I

t is

co

nsid

ere

d a

n

alg

eb

raic

exp

ressio

n.

Bo

x-a

nd

-Wh

iske

r P

lot

A g

rap

hic

me

tho

d fo

r sh

ow

ing a

su

mm

ary

an

d d

istr

ibu

tio

n o

f da

ta u

sin

g m

ed

ian

, qu

art

iles, a

nd

extr

em

es (

i.e

., m

inim

um

an

d m

axim

um

) of

data

. T

his

sh

ow

s h

ow

fa

r ap

art

an

d h

ow

eve

nly

da

ta is

dis

trib

ute

d.

It is h

elp

ful w

he

n a

vis

ua

l is

ne

ed

ed

to

se

e if

a d

istr

ibu

tion

is s

ke

we

d o

r if th

ere

are

an

y

ou

tlie

rs.

Exa

mp

le o

f a b

ox-a

nd

-wh

isker

plo

t:

Cir

cle

Gra

ph

(o

r P

ie C

hart

) A

circula

r dia

gra

m u

sin

g d

iffe

ren

t-siz

ed

se

cto

rs o

f a c

ircle

wh

ose

an

gle

s a

t th

e c

en

ter

are

pro

po

rtio

na

l to

the

fre

qu

en

cy.

Se

cto

rs c

an

be

vis

ua

lly c

om

pa

red

to

sho

w in

form

atio

n (

e.g

., s

tatistica

l da

ta).

Se

cto

rs

rese

mb

le s

lice

s o

f a

pie

. E

xa

mp

le o

f a

circle

gra

ph

:

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 5

A

pri

l 20

14

Co

eff

icie

nt

The

nu

mb

er,

usu

ally

a c

on

sta

nt,

th

at

is m

ultip

lied

by a

va

ria

ble

in a

te

rm (

e.g

., 3

5 is t

he

co

eff

icie

nt of

35

x2y);

the

ab

sen

ce

of a

co

eff

icie

nt is

th

e s

am

e a

s a

1 b

ein

g p

rese

nt (e

.g.,

x is t

he s

am

e a

s 1

x).

Co

mb

ina

tio

n

An

un

ord

ere

d a

rra

nge

me

nt,

lis

tin

g o

r sele

ction

of

ob

jects

(e.g

., t

wo

-le

tte

r com

bin

ation

s o

f th

e th

ree

le

tte

rs X

, Y

, a

nd Z

wo

uld

be

XY

, X

Z,

an

d Y

Z; X

Y is t

he

sa

me

as Y

X a

nd

is n

ot co

un

ted

as a

diffe

ren

t co

mb

ina

tio

n).

A c

om

bin

ation

is s

imila

r to

, b

ut n

ot

the

sa

me

as,

a p

erm

uta

tio

n.

Co

mm

on

Lo

ga

rith

m

A loga

rith

m w

ith

ba

se

10

. It is w

ritt

en

log x

. T

he

co

mm

on

lo

ga

rith

m is t

he

po

we

r o

f 10

ne

cessa

ry t

o

equ

al a

giv

en

nu

mb

er

(i.e

., log x

= y

is e

qu

ivale

nt

to 1

0y =

x).

Co

mp

lex

Nu

mb

er

The

su

m o

r diffe

ren

ce

of

a r

ea

l nu

mb

er

an

d a

n im

agin

ary

nu

mb

er.

It

is w

ritt

en

in th

e fo

rm a

+ b

i,

wh

ere

a a

nd

b a

re r

ea

l n

um

be

rs a

nd i

is th

e im

agin

ary

un

it (

i.e

., i

=

1

). T

he

a is c

alle

d t

he r

ea

l p

art

,

an

d th

e b

i is

ca

lled

th

e im

agin

ary

pa

rt.

Co

mp

osit

e N

um

be

r A

ny n

atu

ral n

um

be

r w

ith

mo

re t

ha

n t

wo

fa

cto

rs (

e.g

., 6

is a

co

mp

osite

nu

mb

er

sin

ce

it h

as fo

ur

facto

rs: 1

, 2,

3,

and

6).

A c

om

po

site

nu

mb

er

is n

ot a

prim

e n

um

be

r.

Co

mp

ou

nd

(o

r C

om

bin

ed

) E

ve

nt

An

eve

nt th

at

is m

ad

e u

p o

f tw

o o

r m

ore

sim

ple

even

ts,

su

ch

as t

he

flip

pin

g o

f tw

o o

r m

ore

co

ins.

Co

mp

ou

nd

In

eq

ua

lity

W

hen t

wo

or

mo

re ine

qu

alit

ies a

re ta

ke

n t

oge

the

r a

nd w

ritt

en

with

th

e ine

qu

alit

ies c

on

ne

cte

d b

y t

he

w

ord

s a

nd

or

or

(e.g

., x

> 6

and

x <

12

, w

hic

h c

an a

lso b

e w

ritt

en

as 6

< x

< 1

2).

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 6

A

pri

l 20

14

Co

ns

tan

t A

te

rm o

r exp

ressio

n w

ith

no v

aria

ble

in it. It

ha

s t

he

sa

me

va

lue a

ll th

e t

ime

.

Co

ord

ina

te P

lan

e

A p

lane

fo

rme

d b

y p

erp

end

icula

r nu

mb

er

line

s.

The

ho

rizo

nta

l nu

mb

er

line

is t

he

x-a

xis

, a

nd

the

ve

rtic

al n

um

be

r lin

e is th

e y

-axis

. T

he

po

int

wh

ere

th

e a

xe

s m

ee

t is

ca

lled

th

e o

rigin

. E

xa

mp

le o

f a

co

ord

inate

pla

ne

:

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 7

A

pri

l 20

14

Cu

be

Ro

ot

On

e o

f th

ree

equ

al fa

cto

rs (

roo

ts)

of

a n

um

be

r or

exp

ressio

n; a

rad

ica

l e

xp

ressio

n w

ith

a d

egre

e o

f 3

(e.g

.,

3a

). T

he

cu

be r

oo

t of

a n

um

be

r or

exp

ressio

n h

as t

he

sa

me

sig

n a

s t

he

nu

mb

er

or

exp

ressio

n

un

de

r th

e r

ad

ical (e

.g.,

3

_6

343

x =

–(7

x2)

an

d

36

343

x =

7x

2).

Cu

rve

of

Bes

t F

it (

for

a

Sc

att

er

Plo

t)

Se

e lin

e o

r curv

e o

f b

est fit

(fo

r a s

ca

tte

r plo

t).

De

gre

e (

of

a P

oly

no

mia

l)

The

va

lue

of

the

gre

ate

st

exp

one

nt in

a p

oly

no

mia

l.

De

pe

nd

en

t E

ve

nts

T

wo

or

mo

re e

ve

nts

in

wh

ich t

he

outc

om

e o

f o

ne

eve

nt

aff

ects

or

influ

en

ce

s t

he o

utc

om

e o

f th

e o

the

r e

ve

nt(

s).

De

pe

nd

en

t V

ari

ab

le

The

ou

tpu

t nu

mb

er

or

va

ria

ble

in a

rela

tio

n o

r fu

nctio

n th

at d

ep

end

s u

pon

ano

the

r variab

le, ca

lled

th

e

ind

epe

nde

nt va

ria

ble

, o

r in

pu

t n

um

be

r (e

.g.,

in

the

equ

atio

n y

= 2

x +

4,

y is th

e d

ep

end

ent va

ria

ble

sin

ce

its

va

lue

de

pe

nds o

n th

e v

alu

e o

f x).

It

is t

he

va

ria

ble

fo

r w

hic

h a

n e

qu

atio

n is s

olv

ed

. It

s v

alu

es

ma

ke

up

the

ran

ge

of

the

rela

tio

n o

r fu

nctio

n.

Do

main

(o

f a

Rela

tio

n o

r F

un

cti

on

) T

he

se

t of

all

po

ssib

le v

alu

es o

f th

e ind

epe

nde

nt

va

ria

ble

on

wh

ich a

fu

nctio

n o

r re

lation is a

llow

ed

to

o

pe

rate

. A

lso,

the

first n

um

be

rs in

th

e o

rde

red

pa

irs o

f a

re

lation

; th

e v

alu

es o

f th

e x

-co

ord

ina

tes in

(x

, y).

Eli

min

ati

on

Me

tho

d

Se

e lin

ea

r com

bin

ation

.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 8

A

pri

l 20

14

Eq

ua

tio

n

A m

ath

em

atica

l sta

tem

en

t o

r sen

ten

ce

tha

t says o

ne

ma

the

ma

tica

l e

xp

ressio

n o

r qu

an

tity

is e

qu

al to

a

no

the

r (e

.g.,

x +

5 =

y –

7).

An

equ

atio

n w

ill a

lwa

ys c

on

tain

an

equ

al sig

n (

=).

Es

tim

ati

on

Str

ate

gy

An

ap

pro

xim

atio

n b

ased

on a

ju

dgm

en

t; m

ay in

clu

de d

ete

rmin

ing a

pp

roxim

ate

va

lue

s, e

sta

blis

hin

g

the

rea

so

na

ble

ne

ss o

f a

nsw

ers

, a

sse

ssin

g t

he

am

ou

nt of

err

or

resu

ltin

g f

rom

estim

atio

n,

an

d/o

r d

ete

rmin

ing if

an

err

or

is w

ith

in a

cce

pta

ble

lim

its.

Ex

po

nen

t T

he

po

we

r to

wh

ich a

nu

mb

er

or

exp

ressio

n is r

ais

ed

. W

hen

th

e e

xpo

nen

t is

a f

raction

, th

e n

um

be

r or

exp

ressio

n c

an b

e r

ew

ritt

en

with

a r

ad

ical sig

n (

e.g

., x

3/4

=

43

x).

Se

e a

lso

po

sitiv

e e

xpo

ne

nt a

nd

ne

ga

tive

exp

on

ent.

Ex

po

nen

tia

l E

qu

ati

on

A

n e

qu

atio

n w

ith

va

ria

ble

s in its

exp

on

en

ts (

e.g

., 4

x =

50

). It

can

be

so

lved

by t

akin

g loga

rith

ms o

f b

oth

sid

es.

Ex

po

nen

tia

l E

xp

res

sio

n

An

exp

ressio

n in

wh

ich

th

e v

aria

ble

occu

rs in

th

e e

xp

on

ent

(such a

s 4

x r

ath

er

than

x4).

Oft

en

it

occu

rs

wh

en

a q

ua

ntity

ch

an

ge

s b

y t

he

sa

me

fa

cto

r fo

r ea

ch u

nit o

f tim

e (

e.g

., “

do

ub

les e

ve

ry y

ea

r” o

r “d

ecre

ase

s 2

% e

ach m

on

th”)

.

Ex

po

nen

tia

l F

un

cti

on

(o

r M

od

el)

A

fu

nctio

n w

ho

se

ge

nera

l e

qu

atio

n is y

= a

• b

x w

he

re a

an

d b

are

con

sta

nts

.

Ex

po

nen

tia

l G

row

th/D

ec

ay

A s

itu

atio

n w

he

re a

qu

an

tity

in

cre

ase

s o

r d

ecre

ase

s e

xp

on

entia

lly b

y t

he

sa

me

fa

cto

r o

ve

r tim

e;

it is

use

d fo

r such

ph

eno

me

na

as inflatio

n, p

op

ula

tio

n g

row

th,

rad

ioa

ctivity o

r de

pre

cia

tion

.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 9

A

pri

l 20

14

Ex

pre

ss

ion

A

ma

the

ma

tica

l p

hra

se t

ha

t in

clu

de

s o

pe

ration

s,

nu

mb

ers

, a

nd

/or

varia

ble

s (

e.g

., 2

x +

3y is a

n

alg

eb

raic

exp

ressio

n,

13

.4 –

4.7

is a

nu

me

ric e

xp

ressio

n).

An e

xp

ressio

n d

oe

s n

ot

co

nta

in a

n e

qu

al

sig

n (

=)

or

an

y t

yp

e o

f in

equ

alit

y s

ign

.

Fa

cto

r (n

ou

n)

The

nu

mb

er

or

exp

ressio

n t

ha

t is

mu

ltip

lied

by a

no

the

r to

ge

t a

pro

du

ct

(e.g

., 6

is a

fa

cto

r of

30

, a

nd

6x is a

fa

cto

r of

42

x2).

Fa

cto

r (v

erb

) T

o e

xp

ress o

r w

rite

a n

um

be

r, m

on

om

ial, o

r po

lyn

om

ial a

s a

pro

du

ct of

two

or

mo

re f

acto

rs.

Fa

cto

r a

Mo

no

mia

l T

o e

xp

ress a

mo

no

mia

l a

s t

he

pro

du

ct of

two

or

mo

re m

on

om

ials

.

Fa

cto

r a

Po

lyn

om

ial

To e

xp

ress a

po

lyn

om

ial a

s th

e p

rod

uct

of

mo

no

mia

ls a

nd

/or

po

lyn

om

ials

(e.g

., f

acto

rin

g t

he

po

lyn

om

ial x

2 +

x –

12

resu

lts in

th

e p

rod

uct

(x –

3)(

x +

4))

.

Fre

qu

en

cy

Ho

w o

ften

so

me

thin

g o

ccu

rs (

i.e

., th

e n

um

be

r of

tim

es a

n ite

m,

nu

mb

er,

or

eve

nt h

ap

pen

s in a

se

t of

da

ta).

Fu

nc

tio

n

A r

ela

tion

in w

hic

h e

ach

va

lue

of

an ind

ep

en

de

nt

va

ria

ble

is a

sso

cia

ted

with

a u

niq

ue

va

lue

of

a

de

pe

nde

nt

va

ria

ble

(e

.g.,

on

e e

lem

en

t of

the

do

ma

in is p

aire

d w

ith

on

e a

nd

on

ly o

ne

ele

me

nt

of

the

ra

nge

). I

t is

a m

ap

pin

g w

hic

h invo

lves e

ithe

r a o

ne

-to

-on

e c

orr

espo

nd

en

ce

or

a m

an

y-t

o-o

ne

co

rre

sp

ond

en

ce

, b

ut no

t a

one

-to

-ma

ny c

orr

esp

ond

en

ce

.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

0

Ap

ril 2

01

4

Fu

nd

am

en

tal

Co

un

tin

g

Pri

nc

iple

A

wa

y t

o c

alc

ula

te a

ll of

the p

ossib

le c

om

bin

ation

s o

f a

giv

en

nu

mb

er

of

eve

nts

. It

sta

tes t

hat

if t

he

re

are

x d

iffe

rent

wa

ys o

f d

oin

g o

ne

th

ing a

nd y

diffe

rent

wa

ys o

f do

ing a

noth

er

thin

g,

the

n the

re a

re

xy d

iffe

ren

t w

ays o

f d

oin

g b

oth

th

ings.

It u

se

s th

e m

ultip

lication

rule

.

Ge

om

etr

ic S

eq

uen

ce

A

n o

rde

red

lis

t of

nu

mb

ers

tha

t ha

s th

e s

am

e r

atio

be

twe

en

co

nse

cu

tive

te

rms (

e.g

., 1

, 7

, 4

9, 3

43, …

is

a g

eo

me

tric

se

qu

ence

tha

t ha

s a

ratio o

f 7/1

betw

ee

n c

on

se

cu

tive t

erm

s;

ea

ch

te

rm a

fte

r th

e f

irst

term

ca

n b

e fo

un

d b

y m

ultip

lyin

g t

he

pre

vio

us t

erm

by a

co

nsta

nt,

in

th

is c

ase

th

e n

um

be

r 7 o

r 7/1

).

Gre

ate

st

Co

mm

on

Fac

tor

(GC

F)

The

la

rge

st fa

cto

r th

at tw

o o

r m

ore

nu

mb

ers

or

alg

eb

raic

te

rms h

ave

in

co

mm

on

. In

so

me

ca

se

s th

e

GC

F m

ay b

e 1

or

on

e o

f th

e a

ctu

al n

um

be

rs (

e.g

., t

he

GC

F o

f 1

8x

3 a

nd

24

x5 is 6

x3).

Ima

gin

ary

Nu

mb

er

The

squ

are

roo

t of

a n

ega

tive

nu

mb

er,

or

the o

ppo

site

of

the

squ

are

roo

t of

a n

ega

tive n

um

be

r. It is

wri

tte

n in

th

e fo

rm b

i, w

he

re b

is a

rea

l n

um

be

r a

nd i

is th

e im

agin

ary

roo

t (i.e

., i

=

1

or

i2 =

–1

).

Ind

ep

en

de

nt

Eve

nt(

s)

Tw

o o

r m

ore

eve

nts

in

wh

ich t

he

outc

om

e o

f o

ne

eve

nt

do

es n

ot aff

ect

the

ou

tco

me

of

the o

the

r e

ve

nt(

s)

(e.g

., to

ssin

g a

co

in a

nd r

olli

ng a

num

be

r cub

e a

re in

dep

en

den

t e

ve

nts

). T

he

pro

ba

bili

ty o

f tw

o in

de

pen

den

t e

ve

nts

(A

and

B)

occu

rrin

g is w

ritte

n P

(A a

nd

B)

or

P(A

B

) an

d e

qu

als

P(A

) •

P(B

)

(i.e

., t

he

pro

du

ct of

the p

rob

ab

ilities o

f th

e t

wo

in

div

idu

al e

ven

ts).

Ind

ep

en

de

nt

Va

ria

ble

T

he

in

pu

t n

um

be

r or

va

ria

ble

in a

rela

tio

n o

r fu

nction

wh

ose

va

lue

is s

ub

ject

to c

ho

ice.

It is n

ot

de

pe

nde

nt u

po

n a

ny o

the

r valu

es.

It is u

sua

lly t

he

x-v

alu

e o

r th

e x

in f(x

). I

t is

gra

ph

ed

on

the

x-a

xis

. It

s v

alu

es m

ake

up

th

e d

om

ain

of

the

re

lation o

r fu

nctio

n.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

1

Ap

ril 2

01

4

Ine

qu

ali

ty

A m

ath

em

atica

l se

nte

nce

tha

t con

tain

s a

n in

equ

alit

y s

ym

bo

l (i.e

., >

, <

, ≥, ≤,

or

≠).

It

co

mp

are

s t

wo

qu

an

titie

s.

Th

e s

ym

bo

l >

me

an

s g

rea

ter

than

, th

e s

ym

bo

l <

me

an

s le

ss t

ha

n, th

e s

ym

bo

l

me

an

s

gre

ate

r th

an

or

equ

al to

, th

e s

ym

bo

l m

ea

ns le

ss t

ha

n o

r equ

al to

, a

nd

th

e s

ym

bo

l m

ea

ns n

ot

equ

al to

.

Inte

ge

r A

na

tura

l nu

mb

er,

th

e a

dd

itiv

e in

ve

rse

of

a n

atu

ral nu

mb

er,

or

ze

ro.

An

y n

um

be

r fr

om

th

e s

et

of

nu

mb

ers

re

pre

se

nte

d b

y {

…,

–3

, –2

, –1,

0, 1

, 2

, 3

, …

}.

Inte

rqu

art

ile

Ra

ng

e (

of

Da

ta)

The

diffe

ren

ce

be

twe

en

the

first

(lo

we

r) a

nd

th

ird

(up

pe

r) q

ua

rtile

. It

rep

rese

nts

the

sp

rea

d o

f th

e

mid

dle

50%

of

a s

et

of d

ata

.

Inve

rse

(o

f a

Re

lati

on

) A

rela

tion

in w

hic

h t

he

co

ord

inate

s in e

ach

ord

ere

d p

air

are

sw

itch

ed

fro

m a

giv

en

rela

tion

. T

he

po

int

(x,

y)

be

co

me

s (

y,

x),

so

(3

, 8

) w

ou

ld b

eco

me

(8,

3).

Irra

tio

nal

Nu

mb

er

A r

ea

l nu

mb

er

tha

t ca

nn

ot b

e w

ritte

n a

s a

sim

ple

fra

ction

(i.e

., t

he

ratio

of

two

in

tege

rs).

It

is a

no

n-

term

ina

tin

g (

infinite

) and

non

-rep

ea

tin

g d

ecim

al. T

he

squa

re r

oot

of a

ny p

rim

e n

um

be

r is

irr

atio

na

l, a

s

are

π a

nd

e.

Le

as

t (o

r L

ow

es

t) C

om

mo

n

Mu

ltip

le (

LC

M)

The

sm

alle

st

nu

mb

er

or

exp

ressio

n th

at

is a

co

mm

on

mu

ltip

le o

f tw

o o

r m

ore

nu

mb

ers

or

alg

eb

raic

te

rms,

oth

er

tha

n z

ero

.

Lik

e T

erm

s

Mo

no

mia

ls t

ha

t con

tain

the

sa

me

va

ria

ble

s a

nd

co

rre

sp

ond

ing p

ow

ers

an

d/o

r ro

ots

. O

nly

th

e

co

eff

icie

nts

can

be

diffe

ren

t (e

.g.,

4x

3 a

nd

12x

3).

Lik

e t

erm

s c

an

be

ad

de

d o

r su

btr

acte

d.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

2

Ap

ril 2

01

4

Lin

e G

rap

h

A g

rap

h th

at u

se

s a

lin

e o

r lin

e s

egm

en

ts t

o c

on

ne

ct

da

ta p

oin

ts,

plo

tte

d o

n a

coo

rdin

ate

pla

ne

, u

su

ally

to

sh

ow

tre

nd

s o

r cha

nge

s in

da

ta o

ver

tim

e.

Mo

re b

roa

dly

, a

gra

ph

to

re

pre

sen

t th

e

rela

tion

sh

ip b

etw

ee

n t

wo

co

ntin

uou

s v

aria

ble

s.

Lin

e o

r C

urv

e o

f B

es

t F

it (

for

a S

ca

tte

r P

lot)

A

lin

e o

r cu

rve d

raw

n o

n a

sca

tte

r p

lot

to b

est e

stim

ate

th

e r

ela

tio

nship

be

twe

en

tw

o s

ets

of d

ata

. It

d

escrib

es th

e tre

nd o

f th

e d

ata

. D

iffe

ren

t m

ea

su

res a

re p

ossib

le t

o d

escrib

e t

he

be

st fit. T

he

mo

st

co

mm

on

is a

lin

e o

r curv

e t

ha

t m

inim

ize

s t

he

su

m o

f th

e s

qu

are

s o

f th

e e

rro

rs (

ve

rtic

al d

ista

nce

s)

from

th

e d

ata

po

ints

to

th

e lin

e.

The

lin

e o

f b

est fit is

a s

ub

se

t of

the

cu

rve o

f be

st fit. E

xa

mp

les o

f a

lin

e o

f b

est fit a

nd

a c

urv

e o

f b

est fit:

Lin

ea

r C

om

bin

ati

on

A

me

tho

d b

y w

hic

h a

syste

m o

f lin

ea

r e

qu

ation

s c

an b

e s

olv

ed.

It u

se

s a

dd

itio

n o

r sub

tractio

n in

co

mb

ina

tio

n w

ith

mu

ltip

lica

tio

n o

r div

isio

n to

elim

ina

te o

ne

of

the

varia

ble

s in o

rde

r to

so

lve

fo

r th

e

oth

er

va

ria

ble

.

Lin

ea

r E

qu

ati

on

A

n e

qu

atio

n fo

r w

hic

h t

he

gra

ph

is a

str

aig

ht

line

(i.e., a

po

lyn

om

ial e

qu

atio

n o

f th

e f

irst d

egre

e o

f th

e

form

Ax +

By =

C,

wh

ere

A,

B,

an

d C

are

re

al n

um

be

rs a

nd

wh

ere

A a

nd

B a

re n

ot

both

ze

ro;

an

e

qu

atio

n in

wh

ich t

he

va

ria

ble

s a

re n

ot

mu

ltip

lied

by o

ne

ano

the

r or

rais

ed

to

an

y p

ow

er

oth

er

than

1).

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

3

Ap

ril 2

01

4

Lin

ea

r F

un

cti

on

A

fu

nctio

n fo

r w

hic

h t

he

gra

ph

is a

non

-ve

rtic

al str

aig

ht

line

. It

is a

first

de

gre

e p

oly

no

mia

l o

f th

e

co

mm

on

fo

rm f

(x)

= m

x +

b,

wh

ere

m a

nd

b a

re c

on

sta

nts

and

x is a

rea

l va

ria

ble

. T

he

consta

nt m

is

ca

lled

th

e s

lop

e a

nd b

is c

alle

d th

e y

-inte

rcep

t. I

t ha

s a

co

nsta

nt

rate

of

cha

nge

.

Lin

ea

r In

eq

uali

ty

The

rela

tio

n o

f tw

o e

xp

ressio

ns u

sin

g th

e s

ym

bo

ls <

, >

, ≤,

≥,

or

≠ a

nd

wh

ose

bou

nd

ary

is a

str

aig

ht

line

. T

he lin

e d

ivid

es the

co

ord

ina

te p

lan

e into

tw

o p

art

s.

If t

he

ine

qua

lity is e

ith

er

≤ o

r ≥, th

en

th

e

bo

un

da

ry is s

olid

. If

the

in

equ

alit

y is e

ith

er

< o

r >

, th

en

the

bo

un

da

ry is d

ash

ed

. If

th

e in

equ

alit

y is ≠

, th

en t

he

so

lutio

n c

on

tain

s e

ve

ryth

ing e

xce

pt fo

r th

e b

ou

nda

ry.

Lo

ga

rith

m

Th

e e

xp

one

nt

requ

ire

d t

o p

rod

uce

a g

iven n

um

be

r (e

.g.,

sin

ce

2 r

ais

ed

to

a p

ow

er

of

5 is 3

2,

the

loga

rith

m b

ase

2 o

f 3

2 is 5

; th

is is w

ritt

en

as log

2 3

2 =

5).

Tw

o f

reque

ntly u

se

d b

ase

s a

re 1

0 (

co

mm

on

loga

rith

m)

an

d e

(n

atu

ral lo

ga

rith

m).

Wh

en

a lo

ga

rith

m is w

ritte

n w

ith

ou

t a

ba

se

, it is u

nd

ers

too

d t

o b

e

ba

se

10

.

Lo

ga

rith

mic

Eq

uati

on

A

n e

qu

atio

n w

hic

h c

onta

ins a

loga

rith

m o

f a

va

ria

ble

or

nu

mb

er.

Som

etim

es it

is s

olv

ed

by r

ew

ritin

g

the

equ

atio

n in

exp

on

en

tia

l fo

rm a

nd

so

lvin

g f

or

the

va

ria

ble

(e

.g.,

log

2 3

2 =

5 is the

sa

me

as 2

5 =

32

).

It is a

n in

ve

rse fu

nctio

n o

f th

e e

xp

on

entia

l fu

nctio

n.

Ma

pp

ing

T

he

ma

tch

ing o

r pa

irin

g o

f o

ne

se

t of

nu

mb

ers

to

an

oth

er

by u

se

of

a r

ule

. A

nu

mb

er

in t

he d

om

ain

is

ma

tch

ed

or

pa

ire

d w

ith

a n

um

be

r in

th

e r

an

ge (

or

a r

ela

tion

or

fun

ctio

n).

It m

ay b

e a

on

e-t

o-o

ne

co

rre

sp

ond

en

ce

, a

on

e-t

o-m

an

y c

orr

esp

on

den

ce

, o

r a m

an

y-t

o-o

ne

co

rre

sp

ond

en

ce

.

Ma

xim

um

Va

lue

(o

f a

Gra

ph

) T

he

va

lue

of

the

de

pe

nd

en

t va

ria

ble

fo

r th

e h

igh

est

po

int

on

the

gra

ph

of

a c

urv

e.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

4

Ap

ril 2

01

4

Me

an

A

me

asu

re o

f ce

ntr

al te

nde

ncy t

ha

t is

ca

lcula

ted

by a

dd

ing a

ll th

e v

alu

es o

f a s

et of

da

ta a

nd

div

idin

g

tha

t su

m b

y t

he

to

tal nu

mb

er

of

va

lue

s.

Unlik

e m

ed

ian

, th

e m

ea

n is s

en

sitiv

e to

ou

tlie

r va

lue

s.

It is

als

o c

alle

d “

arith

me

tic m

ea

n”

or

“ave

rage

”.

Me

as

ure

of

Cen

tral

Te

nd

en

cy

A m

ea

su

re o

f lo

ca

tion

of

the

mid

dle

(cen

ter)

of

a d

istr

ibu

tion

of

a s

et

of

da

ta (

i.e., h

ow

da

ta c

luste

rs).

T

he

th

ree

mo

st

co

mm

on

me

asu

res o

f cen

tral te

nd

en

cy a

re m

ea

n,

me

dia

n,

and

mo

de

.

Me

as

ure

of

Dis

pers

ion

A

me

asu

re o

f th

e w

ay in

wh

ich t

he

dis

trib

ution o

f a s

et of

da

ta is s

pre

ad

ou

t. In

ge

ne

ral th

e m

ore

sp

rea

d o

ut a

dis

trib

ution

is,

the

la

rge

r th

e m

ea

su

re o

f d

ispe

rsio

n.

Ran

ge

an

d inte

rqu

art

ile r

an

ge

are

tw

o m

ea

su

res o

f d

ispers

ion

.

Me

dia

n

A m

ea

su

re o

f ce

ntr

al te

nde

ncy t

ha

t is

th

e m

idd

le v

alu

e in a

n o

rde

red

se

t of

da

ta o

r th

e a

vera

ge

of

the

tw

o m

idd

le v

alu

es w

he

n t

he

se

t h

as t

wo

mid

dle

va

lue

s (

occu

rs w

he

n t

he

se

t of

da

ta h

as a

n e

ve

n

nu

mb

er

of

da

ta p

oin

ts).

It

is th

e v

alu

e h

alfw

ay t

hro

ugh

th

e o

rde

red

se

t of

data

, b

elo

w a

nd

ab

ove

wh

ich

the

re is a

n e

qu

al nu

mb

er

of

da

ta v

alu

es.

It is g

en

era

lly a

go

od

de

scriptive m

ea

su

re f

or

skew

ed

da

ta o

r d

ata

with

ou

tlie

rs.

Min

imu

m V

alu

e (

of

a G

rap

h)

The

va

lue

of

the

de

pe

nd

en

t va

ria

ble

fo

r th

e low

est

po

int

on

th

e g

raph

of

a c

urv

e.

Mo

de

A

me

asu

re o

f ce

ntr

al te

nde

ncy t

ha

t is

th

e v

alu

e o

r va

lue

s th

at

occu

r(s)

mo

st

oft

en

in

a s

et of

da

ta.

A

se

t of

data

ca

n h

ave o

ne

mo

de

, m

ore

tha

n o

ne

mo

de

, o

r no m

od

e.

Mo

no

mia

l A

po

lyn

om

ial w

ith

on

ly o

ne

te

rm;

it c

on

tain

s n

o a

dd

itio

n o

r sub

traction

. It

ca

n b

e a

nu

mb

er,

a v

aria

ble

,

or

a p

rod

uct

of

nu

mb

ers

an

d/o

r m

ore

va

ria

ble

s (

e.g

., 2

• 5

or

x3y

4 o

r 2

4 3r

).

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

5

Ap

ril 2

01

4

Mu

ltip

lic

ati

ve

In

ve

rse

T

he

recip

roca

l of

a n

um

be

r (i.e

., f

or

an

y n

on

-ze

ro n

um

be

r a

, th

e m

ultip

licative inve

rse is 1 a

; fo

r an

y

ratio

na

l n

um

be

r b c

, w

he

re b

≠ 0

an

d c

≠ 0

, th

e m

ultip

licative inve

rse is c b

). A

ny n

um

be

r an

d its

mu

ltip

licative inve

rse h

ave

a p

rodu

ct of

1 (

e.g

.,

1 4 is th

e m

ultip

licative in

ve

rse o

f 4

sin

ce

4 •

1 4

= 1

;

like

wis

e,

the

mu

ltip

licative inve

rse o

f 1 4

is 4

sin

ce

1 4

• 4

= 1

).

Mu

tuall

y E

xc

lus

ive

Eve

nts

T

wo

eve

nts

tha

t ca

nnot

occu

r at

the

sa

me

tim

e (

i.e.,

eve

nts

tha

t h

ave

no o

utc

om

es in

co

mm

on

). If

two

e

ve

nts

A a

nd

B a

re m

utu

ally

exclu

siv

e,

the

n th

e p

rob

ab

ility

of

A o

r B

occu

rrin

g is t

he

su

m o

f th

eir

ind

ivid

ua

l p

rob

ab

ilities:

P(A

B

) =

P(A

) +

P(B

). A

lso d

efined

as w

he

n th

e inte

rsection

of

two

se

ts is

em

pty

, w

ritte

n a

s A

B

= Ø

.

Na

tura

l L

og

ari

thm

A

loga

rith

m w

ith

ba

se

e.

It is w

ritt

en

ln

x.

The

na

tura

l lo

ga

rith

m is th

e p

ow

er

of

e n

ece

ssa

ry t

o e

qu

al a

giv

en

nu

mb

er

(i.e

., ln

x =

y is e

qu

ivale

nt to

e y =

x).

The

con

sta

nt

e is

an

irr

atio

na

l n

um

be

r w

ho

se

va

lue

is

ap

pro

xim

ate

ly 2

.71

82

8…

.

Na

tura

l N

um

be

r A

co

un

tin

g n

um

be

r. A

nu

mb

er

rep

rese

ntin

g a

po

sitiv

e,

wh

ole

am

ou

nt.

An

y n

um

be

r fr

om

th

e s

et

of

nu

mb

ers

re

pre

se

nte

d b

y {

1,

2,

3, …

}. S

om

etim

es,

it is r

efe

rred

to a

s a

“p

ositiv

e inte

ge

r”.

Ne

ga

tive

Ex

po

nen

t A

n e

xp

one

nt th

at

ind

icate

s a

recip

roca

l th

at

ha

s t

o b

e ta

ken

befo

re th

e e

xp

one

nt ca

n b

e a

pp

lied

(e.g

.,

2

215

5

or

1x

xa

a

). It

is u

se

d in

scie

ntific n

ota

tio

n f

or

nu

mb

ers

be

twe

en

–1

and

1.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

6

Ap

ril 2

01

4

Nu

mb

er

Lin

e

A g

rad

ua

ted

str

aig

ht

line

th

at

rep

resen

ts th

e s

et

of

all

rea

l n

um

be

rs in o

rde

r. T

yp

ically

, it is m

ark

ed

sh

ow

ing inte

ge

r va

lue

s.

Od

ds

A

co

mp

ariso

n,

in r

atio

fo

rm (

as a

fra

ctio

n o

r w

ith

a c

olo

n),

of

ou

tco

me

s.

“Odd

s in f

avo

r” (

or

sim

ply

“o

dd

s”)

is th

e r

atio o

f fa

vo

rab

le o

utc

om

es t

o u

nfa

vo

rab

le o

utc

om

es (

e.g

., t

he o

dd

s in

fa

vo

r of

pic

kin

g a

re

d h

at

wh

en

th

ere

are

3 r

ed

ha

ts a

nd

5 n

on

-red

ha

ts is 3

:5).

“O

dd

s a

ga

inst”

is th

e r

atio o

f u

nfa

vo

rab

le

ou

tco

me

s t

o fa

vo

rab

le o

utc

om

es (

e.g

., th

e o

dd

s a

ga

inst p

ickin

g a

red

hat

wh

en

th

ere

are

3 r

ed

ha

ts

an

d 5

no

n-r

ed

ha

ts is 5

:3).

Ord

er

of

Op

era

tio

ns

Rule

s d

escrib

ing w

ha

t o

rde

r to

use in

eva

lua

tin

g e

xp

ressio

ns:

(1)

Pe

rfo

rm o

pe

ration

s in

gro

up

ing s

ym

bo

ls (

pa

ren

the

se

s a

nd b

racke

ts),

(2

) E

va

lua

te e

xp

on

ential exp

ressio

ns a

nd r

ad

ical e

xp

ressio

ns f

rom

le

ft to

rig

ht,

(3

) M

ultip

ly o

r d

ivid

e f

rom

left

to

rig

ht,

(4

) A

dd

or

su

btr

act fr

om

le

ft t

o r

igh

t.

Ord

ere

d P

air

A

pa

ir o

f nu

mb

ers

used

to

lo

ca

te a

po

int o

n a

co

ord

ina

te p

lane

, o

r th

e s

olu

tio

n o

f an

equ

atio

n in t

wo

va

ria

ble

s.

The

first n

um

be

r te

lls h

ow

fa

r to

move

ho

rizo

nta

lly,

an

d th

e s

eco

nd

nu

mb

er

tells

ho

w f

ar

to

mo

ve

ve

rtic

ally

; w

ritte

n in

the

fo

rm (

x-c

oo

rdin

ate

, y-c

oo

rdin

ate

). O

rde

r m

att

ers

: th

e p

oin

t (x

, y)

is n

ot

the

sa

me

as (

y,

x).

Ori

gin

T

he

po

int

(0,

0)

on

a c

oo

rdin

ate

pla

ne

. It is t

he

po

int

of

inte

rsection fo

r th

e x

-axis

and

the

y-a

xis

.

Ou

tlie

r A

va

lue t

ha

t is

mu

ch

gre

ate

r o

r m

uch

le

ss t

han

the

rest of

the

da

ta.

It is d

iffe

ren

t in

so

me

wa

y f

rom

th

e

ge

ne

ral p

att

ern

of

da

ta.

It d

ire

ctly s

tan

ds o

ut fr

om

th

e r

est

of

the d

ata

. S

om

etim

es it

is r

efe

rre

d t

o a

s

an

y d

ata

po

int

mo

re t

ha

n 1

.5 inte

rqu

art

ile r

ange

s g

rea

ter

than

the

up

pe

r (t

hird

) qu

art

ile o

r le

ss t

ha

n

the

lo

we

r (f

irst)

qu

art

ile.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

7

Ap

ril 2

01

4

Pa

tte

rn (

or

Se

qu

en

ce)

A s

et of

nu

mb

ers

arr

ange

d in

ord

er

(or

in a

sequ

en

ce

). T

he

nu

mb

ers

an

d t

he

ir a

rra

nge

me

nt

are

d

ete

rmin

ed

by a

rule

, in

clu

din

g r

epe

titio

n a

nd g

row

th/d

eca

y r

ule

s.

Se

e a

rith

me

tic s

equ

en

ce

and

ge

om

etr

ic s

equ

en

ce

.

Pe

rfec

t S

qu

are

A

nu

mb

er

wh

ose

squ

are

ro

ot

is a

wh

ole

nu

mb

er

(e.g

., 2

5 is a

pe

rfect

squ

are

sin

ce

25

= 5

). A

pe

rfe

ct

squa

re c

an

be

fo

und

by r

ais

ing a

wh

ole

nu

mb

er

to t

he

se

cond

po

we

r (e

.g.,

52 =

25

).

Pe

rmu

tati

on

A

n o

rde

red

arr

an

ge

me

nt

of

ob

jects

fro

m a

giv

en

set

in w

hic

h t

he

ord

er

of

the

ob

jects

is s

ign

ific

an

t (e

.g.,

tw

o-lett

er

pe

rmu

tatio

ns o

f th

e th

ree le

tters

X,

Y,

an

d Z

wo

uld

be

XY

, Y

X,

XZ

, Z

X,

YZ

, a

nd

ZY

). A

p

erm

uta

tio

n is s

imila

r to

, bu

t no

t th

e s

am

e a

s, a

co

mb

ina

tion

.

Po

int-

Slo

pe

Fo

rm (

of

a

Lin

ea

r E

qu

ati

on

) A

n e

qu

atio

n o

f a

str

aig

ht,

no

n-v

ert

ical lin

e w

ritt

en

in t

he

fo

rm y

– y

1 =

m(x

– x

1),

wh

ere

m is th

e s

lop

e

of

the

lin

e a

nd

(x

1,

y1)

is a

giv

en

po

int o

n t

he

lin

e.

Po

lyn

om

ial

A

n a

lge

bra

ic e

xp

ressio

n t

ha

t is

a m

on

om

ial o

r th

e s

um

or

diffe

ren

ce

of

two

or

mo

re m

on

om

ials

(e.g

.,

6a

or

5a

2 +

3a

– 1

3 w

he

re t

he

exp

one

nts

are

na

tura

l n

um

be

rs).

Po

lyn

om

ial

Fu

nc

tio

n

A f

un

ctio

n o

f th

e fo

rm f

(x)

= a

nx

n +

an–1x

n–1 +

… +

a1x +

a0,

wh

ere

an ≠

0 a

nd

na

tura

l nu

mb

er

n is the

d

egre

e o

f th

e p

oly

no

mia

l.

Po

sit

ive

Ex

po

nen

t In

dic

ate

s h

ow

ma

ny t

ime

s a

ba

se

nu

mb

er

is m

ultip

lied

by its

elf.

In th

e e

xp

ressio

n x

n,

n is th

e p

ositiv

e

exp

on

en

t, a

nd

x is th

e b

ase

nu

mb

er

(e.g

., 2

3 =

2 •

2 •

2).

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

8

Ap

ril 2

01

4

Po

we

r T

he

va

lue

of

the

exp

one

nt

in a

te

rm.

The

exp

ressio

n a

n is r

ea

d “

a to

th

e p

ow

er

of

n.”

To r

ais

e a

nu

mb

er,

a,

to th

e p

ow

er

of

ano

the

r w

ho

le n

um

be

r, n

, is

to

mu

ltip

ly a

by its

elf n

tim

es (

e.g

., th

e n

um

be

r

43 is r

ea

d “

fou

r to

th

e th

ird

po

we

r” a

nd

rep

rese

nts

4 •

4 •

4).

Po

we

r o

f a P

ow

er

An

exp

ressio

n o

f th

e form

(a

m)n

. It

ca

n b

e f

ou

nd

by m

ultip

lyin

g t

he

exp

one

nts

(e.g

.,

(23)4

= 2

3•4

= 2

12 =

4,0

96

).

Po

we

rs o

f P

rod

ucts

A

n e

xp

ressio

n o

f th

e form

am •

an. It

ca

n b

e fou

nd

by a

dd

ing t

he

expo

nen

ts w

he

n m

ultip

lyin

g p

ow

ers

tha

t h

ave

the

sa

me

base

(e.g

., 2

3 •

24 =

23+

4 =

27 =

12

8).

Pri

me

Nu

mb

er

An

y n

atu

ral n

um

be

r w

ith

exa

ctly t

wo

fa

cto

rs, 1 a

nd its

elf (

e.g

., 3

is a

prim

e n

um

be

r sin

ce

it

ha

s o

nly

tw

o f

acto

rs: 1

and

3).

[N

ote

: S

ince

1 h

as o

nly

on

e f

acto

r, its

elf, it is n

ot

a p

rim

e n

um

be

r.]

A p

rim

e

nu

mb

er

is n

ot a

co

mp

osite

nu

mb

er.

Pro

ba

bilit

y

A n

um

be

r fr

om

0 t

o 1

(o

r 0%

to

10

0%

) th

at

ind

icate

s h

ow

lik

ely

an

eve

nt

is to

ha

pp

en

. A

very

un

like

ly

eve

nt h

as a

pro

bab

ility

ne

ar

0 (

or

0%

) w

hile

a v

ery

lik

ely

eve

nt

ha

s a

pro

bab

ility

ne

ar

1 (

or

10

0%

). It

is

wri

tte

n a

s a

ratio (

fra

ctio

n, d

ecim

al, o

r e

qu

ivale

nt p

erc

ent)

. T

he n

um

be

r of

wa

ys a

n e

ve

nt

co

uld

h

app

en

(fa

vo

rab

le o

utc

om

es)

is p

lace

d o

ve

r th

e to

tal n

um

be

r of

even

ts (

tota

l p

ossib

le o

utc

om

es)

tha

t co

uld

ha

pp

en.

A p

robab

ility

of

0 m

ea

ns it

is im

po

ssib

le,

an

d a

pro

ba

bili

ty o

f 1

me

an

s it

is c

ert

ain

.

Pro

ba

bilit

y o

f a

Co

mp

ou

nd

(o

r C

om

bin

ed

) E

ve

nt

The

re a

re t

wo

typ

es:

1.

Th

e u

nio

n o

f tw

o e

ve

nts

A a

nd

B,

wh

ich is th

e p

rob

ab

ility

of

A o

r B

occu

rrin

g.

This

is

rep

rese

nte

d a

s P

(A

B)

= P

(A)

+ P

(B)

– P

(A)

• P

(B).

2.

Th

e in

ters

ection o

f tw

o e

ve

nts

A a

nd B

, w

hic

h is th

e p

roba

bili

ty o

f A

an

d B

occu

rrin

g.

This

is

rep

rese

nte

d a

s P

(A

B)

= P

(A)

• P

(B).

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 1

9

Ap

ril 2

01

4

Qu

ad

ran

ts

The

fou

r re

gio

ns o

f a

co

ord

inate

pla

ne

tha

t a

re s

epa

rate

d b

y t

he

x-a

xis

an

d th

e y

-axis

, a

s s

ho

wn

b

elo

w.

(1

) T

he

first

qu

ad

ran

t (Q

ua

dra

nt I)

con

tain

s a

ll th

e p

oin

ts w

ith

po

sitiv

e x

an

d p

ositiv

e y

coo

rdin

ate

s

(e.g

., (

3,

4))

. (2

) T

he

se

co

nd q

ua

dra

nt

(Qu

ad

ran

t II)

co

nta

ins a

ll th

e p

oin

ts w

ith

ne

ga

tive

x a

nd

po

sitiv

e y

co

ord

inate

s (

e.g

., (

–3, 4

)).

(3)

Th

e t

hird q

ua

dra

nt

(Qua

dra

nt II

I) c

on

tain

s a

ll th

e p

oin

ts w

ith

ne

ga

tive x

and

ne

ga

tive

y

co

ord

inate

s (

e.g

., (

–3,

–4

)).

(4)

Th

e f

ou

rth q

ua

dra

nt

(Qu

ad

ran

t IV

) con

tain

s a

ll th

e p

oin

ts w

ith

po

sitiv

e x

an

d n

ega

tive

y

co

ord

inate

s (

e.g

., (

3,

–4

)).

Qu

ad

rati

c E

qu

ati

on

A

n e

qu

atio

n th

at

can

be

writt

en

in

the

sta

nda

rd f

orm

ax

2 +

bx +

c =

0,

wh

ere

a,

b,

an

d c

are

rea

l

nu

mb

ers

an

d a

do

es n

ot

equ

al ze

ro.

Th

e h

ighe

st

po

we

r o

f th

e v

aria

ble

is 2

. It h

as,

at

mo

st,

tw

o

so

lution

s.

The g

rap

h is a

pa

rab

ola

.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

0

Ap

ril 2

01

4

Qu

ad

rati

c F

orm

ula

T

he

so

lutio

ns o

r ro

ots

of

a q

ua

dra

tic e

qu

atio

n in t

he

fo

rm a

x2 +

bx +

c =

0,

wh

ere

a ≠

0, a

re g

iven

by

the

fo

rmu

la

24

2

bb

ac

xa

.

Qu

ad

rati

c F

un

cti

on

A

fu

nctio

n th

at

can

be

exp

ressed

in

th

e fo

rm f

(x)

= a

x2 +

bx +

c,

wh

ere

a ≠

0 a

nd

th

e h

igh

est

po

we

r o

f

the

va

ria

ble

is 2

. T

he

gra

ph

is a

pa

rab

ola

.

Qu

art

ile

On

e o

f th

ree

va

lue

s tha

t d

ivid

es a

se

t of

da

ta in

to fo

ur

equa

l p

art

s:

1.

Me

dia

n d

ivid

es a

se

t of

da

ta into

tw

o e

qu

al p

art

s.

2.

Lo

we

r qu

art

ile (

25

th p

erc

en

tile

) is

th

e m

ed

ian

of

the

lo

we

r ha

lf o

f th

e d

ata

. 3

. U

pp

er

qu

art

ile (

75

th p

erc

en

tile

) is

th

e m

ed

ian

of

the

up

pe

r ha

lf o

f th

e d

ata

.

Ra

dic

al E

xp

res

sio

n

An

exp

ressio

n c

on

tain

ing a

rad

ical sym

bo

l (

na

). T

he

exp

ressio

n o

r nu

mb

er

insid

e t

he

rad

ical (a

) is

ca

lled

th

e r

ad

ican

d,

and

the

nu

mb

er

ap

pea

ring a

bo

ve

th

e r

ad

ical (n

) is

th

e d

egre

e.

Th

e d

egre

e is

alw

ays a

po

sitiv

e inte

ge

r. W

hen

a r

ad

ical is

writt

en

with

ou

t a

de

gre

e, it is u

nd

ers

too

d to

be

a d

egre

e

of

2 a

nd

is r

ead

as “

the s

qu

are

ro

ot of

a.”

When

the

de

gre

e is 3

, it is r

ea

d a

s “

the

cub

e r

oo

t of

a.”

Fo

r

an

y o

the

r de

gre

e,

the

exp

ressio

n

na

is r

ea

d a

s “

the

nth

roo

t of

a.”

Whe

n th

e d

egre

e is a

n e

ve

n

nu

mb

er,

th

e r

ad

ical e

xp

ressio

n is a

ssu

me

d t

o b

e th

e p

rin

cip

al (p

ositiv

e)

roo

t (e

.g.,

altho

ugh (

–7

)2 =

49

,

49

= 7

).

Ra

ng

e (

of

a R

ela

tio

n o

r F

un

cti

on

) T

he

se

t of

all

po

ssib

le v

alu

es fo

r th

e o

utp

ut

(de

pen

den

t va

riab

le)

of

a f

un

ctio

n o

r re

lation;

the

se

t of

se

co

nd

nu

mb

ers

in

th

e o

rde

red

pa

irs o

f a f

unctio

n o

r re

latio

n; th

e v

alu

es o

f th

e y

-co

ord

ina

tes in (

x,

y).

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

1

Ap

ril 2

01

4

Ra

ng

e (

of

Da

ta)

In s

tatistics,

a m

ea

su

re o

f d

isp

ers

ion

th

at

is the

diffe

ren

ce b

etw

ee

n th

e g

rea

test

va

lue

(m

axim

um

va

lue

) an

d th

e le

ast

valu

e (

min

imu

m v

alu

e)

in a

se

t of

da

ta.

Ra

te

A r

atio

th

at co

mp

are

s tw

o q

ua

ntitie

s h

avin

g d

iffe

ren

t u

nits (

e.g

., 1

68 m

iles

3.5

hours

or

122

.5 c

alo

rie

s

5 c

ups

). W

hen

the

rate

is s

imp

lifie

d s

o t

ha

t th

e s

eco

nd (

ind

epe

nde

nt)

qu

an

tity

is 1

, it is c

alle

d a

un

it r

ate

(e.g

.,

48

mile

s p

er

hou

r o

r 24

.5 c

alo

rie

s p

er

cup

).

Ra

te (

of

Ch

an

ge

) T

he

am

ou

nt

a q

uan

tity

ch

an

ge

s o

ve

r tim

e (

e.g

., 3

.2 c

m p

er

ye

ar)

. A

lso t

he

am

ou

nt a

fu

nction

’s o

utp

ut

ch

an

ge

s (

incre

ase

s o

r d

ecre

ase

s)

for

ea

ch

unit o

f cha

nge

in t

he in

put.

See

slo

pe

.

Ra

te (

of

Inte

res

t)

The

pe

rcen

t b

y w

hic

h a

mo

ne

tary

acco

un

t a

ccru

es in

tere

st.

It

is m

ost

co

mm

on

fo

r th

e r

ate

of

inte

rest

to b

e m

ea

su

red

on

an

an

nu

al ba

sis

(e.g

., 4

.5%

pe

r ye

ar)

, e

ve

n if

the

in

tere

st

is c

om

po

unde

d

pe

rio

dic

ally

(i.e

., m

ore

fre

qu

en

tly t

ha

n o

nce

pe

r ye

ar)

.

Ra

tio

A

co

mp

ariso

n o

f tw

o n

um

be

rs, qu

an

titie

s o

r exp

ressio

ns b

y d

ivis

ion

. It

is o

fte

n w

ritt

en

as a

fra

ctio

n,

bu

t n

ot a

lwa

ys (

e.g

., 2 3

, 2

:3,

2 to

3,

2 ÷

3 a

re a

ll th

e s

am

e r

atios).

Ra

tio

na

l E

xp

ress

ion

A

n e

xp

ressio

n th

at

can b

e w

ritte

n a

s a

po

lyn

om

ial d

ivid

ed

by a

po

lyn

om

ial, d

efin

ed o

nly

wh

en

th

e

latt

er

is n

ot

equa

l to

zero

.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

2

Ap

ril 2

01

4

Ra

tio

na

l N

um

be

r A

ny n

um

be

r th

at

ca

n b

e w

ritt

en

in

the

fo

rm a b

wh

ere

a is a

ny inte

ge

r a

nd

b is a

ny inte

ge

r e

xce

pt

ze

ro.

All

rep

eatin

g d

ecim

al a

nd

te

rmin

atin

g d

ecim

al n

um

be

rs a

re r

ation

al n

um

be

rs.

Re

al

Nu

mb

er

The

co

mb

ine

d s

et of

ratio

na

l a

nd

irr

ation

al n

um

be

rs. A

ll n

um

be

rs o

n t

he

nu

mb

er

line

. N

ot

an

im

agin

ary

nu

mb

er.

Re

gre

ssio

n C

urv

e

The

lin

e o

r cu

rve o

f b

est fit

tha

t re

pre

sen

ts the

le

ast d

evia

tio

n f

rom

th

e p

oin

ts in a

sca

tte

r plo

t of

da

ta.

Mo

st

co

mm

on

ly it

is lin

ea

r a

nd

use

s a

“le

ast

squ

are

s”

me

tho

d.

Exa

mp

les o

f re

gre

ssio

n c

urv

es:

Re

lati

on

A

se

t of

pa

irs o

f va

lue

s (

e.g

., {

(1,

2),

(2, 3

) (3

, 2

)}).

The

first

va

lue

in

ea

ch

pa

ir is t

he

inp

ut

(ind

ep

en

de

nt

va

lue

), a

nd

th

e s

eco

nd

va

lue

in t

he

pa

ir is t

he

ou

tpu

t (d

epe

nde

nt

va

lue

). I

n a

rela

tion

, n

eithe

r th

e in

pu

t va

lues n

or

the

ou

tpu

t va

lue

s n

eed

to

be

un

iqu

e.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

3

Ap

ril 2

01

4

Re

pe

ati

ng

De

cim

al

A d

ecim

al w

ith

on

e o

r m

ore

dig

its t

ha

t re

pea

ts e

nd

lessly

(e.g

., 0

.666

…,

0.7

27

272…

, 0.0

83

33…

). T

o

ind

icate

th

e r

epe

titio

n, a

ba

r m

ay b

e w

ritt

en

ab

ove

th

e r

ep

ea

ted

dig

its (

e.g

., 0

.66

6…

= 0

.6,

0.7

27

272…

= 0

.72

, 0

.08

333

… =

0.0

83

). A

de

cim

al th

at h

as e

ith

er

a 0

or

a 9

rep

ea

tin

g e

nd

lessly

is

equ

ivale

nt

to a

te

rmin

atin

g d

ecim

al (e

.g.,

0.3

75

000…

= 0

.37

5, 0

.19

99

… =

0.2

). A

ll re

pea

ting d

ecim

als

a

re r

ation

al nu

mb

ers

.

Ris

e

The

ve

rtic

al (u

p a

nd

dow

n)

cha

nge

or

diffe

rence

betw

ee

n a

ny t

wo

po

ints

on a

lin

e o

n a

co

ord

ina

te

pla

ne (

i.e

., fo

r po

ints

(x

1,

y1)

an

d (

x2,

y2),

the

ris

e is y

2 –

y1).

Se

e s

lope

.

Ru

n

The

ho

rizo

nta

l (left

an

d r

igh

t) c

ha

nge

or

diffe

ren

ce

be

twe

en

an

y t

wo

po

ints

on

a lin

e o

n a

co

ord

inate

p

lan

e (

i.e

., fo

r po

ints

(x

1,

y1)

an

d (

x2,

y2),

the

ru

n is x

2 –

x1).

See

slo

pe

.

Sc

att

er

Plo

t A

gra

ph

th

at

sh

ow

s t

he

“ge

ne

ral” r

ela

tio

nsh

ip b

etw

ee

n t

wo

se

ts o

f d

ata

. F

or

ea

ch p

oin

t th

at

is b

ein

g

plo

tted

th

ere

are

tw

o s

ep

ara

te p

iece

s o

f da

ta. It

sho

ws h

ow

on

e v

aria

ble

is a

ffe

cte

d b

y a

no

the

r.

Exa

mp

le o

f a s

ca

tte

r plo

t:

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

4

Ap

ril 2

01

4

Sim

ple

Eve

nt

When a

n e

ve

nt co

nsis

ts o

f a s

ingle

outc

om

e (

e.g

., r

olli

ng a

nu

mb

er

cu

be

).

Sim

ple

st

Fo

rm (

of

an

E

xp

ress

ion

) W

hen a

ll lik

e te

rms a

re c

om

bin

ed

(e.g

., 8

x +

2(6

x –

22

) b

eco

me

s 2

0x –

44

wh

en

in

sim

ple

st

form

).

The

fo

rm w

hic

h n

o lo

nge

r co

nta

ins a

ny lik

e t

erm

s,

pa

ren

the

se

s,

or

red

ucib

le f

raction

s.

Sim

plify

T

o w

rite

an

exp

ressio

n in

its

sim

ple

st

form

(i.e

., r

em

ove

an

y u

nn

ece

ssa

ry t

erm

s,

usu

ally

by c

om

bin

ing

se

ve

ral o

r m

an

y t

erm

s into

fe

we

r te

rms o

r b

y c

an

ce

lling t

erm

s).

Slo

pe

(o

f a

Lin

e)

A r

ate

of

ch

an

ge

. T

he

me

asu

rem

en

t o

f th

e s

tee

pn

ess,

inclin

e, o

r gra

de

of

a lin

e f

rom

le

ft to

rig

ht.

It

is

the

ratio

of

ve

rtic

al chan

ge

to

ho

rizo

nta

l cha

nge

. M

ore

sp

ecific

ally

, it is t

he

ratio

of

the

cha

nge

in

th

e

yc

oo

rdin

ate

s (

rise

) to

th

e c

orr

esp

ond

ing c

hange

in

th

e x

- co

ord

ina

tes (

run

) w

he

n m

ovin

g f

rom

on

e

po

int to

an

oth

er

alo

ng a

lin

e. It

als

o in

dic

ate

s w

he

the

r a lin

e is t

ilte

d u

pw

ard

(p

ositiv

e s

lop

e)

or

do

wn

wa

rd (

ne

ga

tive

slo

pe

) an

d is w

ritt

en

as th

e le

tte

r m

wh

ere

m =

rise

run

=

21

21

yy

xx

.

Exa

mp

le o

f slo

pe:

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

5

Ap

ril 2

01

4

Slo

pe

-In

terc

ep

t F

orm

A

n e

qu

atio

n o

f a

str

aig

ht,

no

n-v

ert

ical lin

e w

ritt

en

in t

he

fo

rm y

= m

x +

b,

wh

ere

m is th

e s

lop

e a

nd

b is

the

yinte

rcep

t.

Sq

ua

re R

oo

t O

ne

of

two

equ

al fa

cto

rs (

roo

ts)

of

a n

um

be

r o

r e

xp

ressio

n;

a r

ad

ical e

xp

ressio

n (

a)

with

an

un

de

rsto

od

de

gre

e o

f 2

. T

he s

qu

are

roo

t of

a n

um

be

r or

exp

ressio

n is a

ssu

me

d t

o b

e th

e p

rin

cip

al

(po

sitiv

e)

roo

t (e

.g.,

4

49

x =

7x

2).

Th

e s

qu

are

roo

t of

a n

ega

tive

nu

mb

er

resu

lts in

an

im

agin

ary

nu

mb

er

(e.g

.,

_49

= 7

i).

Sta

nd

ard

Fo

rm (

of

a L

ine

ar

Eq

ua

tio

n)

An

equ

atio

n o

f a

str

aig

ht

line

writt

en

in

th

e f

orm

Ax +

By =

C,

wh

ere

A,

B, a

nd

C a

re r

ea

l num

be

rs a

nd

w

he

re A

an

d B

are

no

t b

oth

ze

ro.

It in

clu

de

s v

aria

ble

s o

n o

ne

sid

e o

f th

e e

qu

ation

an

d a

con

sta

nt

on

th

e o

the

r sid

e.

Ste

m-a

nd

-Lea

f P

lot

A v

isua

l w

ay t

o d

ispla

y t

he

sh

ap

e o

f a d

istr

ibutio

n th

at

sh

ow

s g

rou

ps o

f da

ta a

rra

nge

d b

y p

lace

va

lue

; a

wa

y t

o s

ho

w t

he

fre

qu

en

cy w

ith

wh

ich c

ert

ain

cla

sse

s o

f da

ta o

ccur.

Th

e s

tem

co

nsis

ts o

f a

co

lum

n

of

the

la

rge

r pla

ce v

alu

e(s

); t

he

se

nu

mb

ers

are

not

repe

ate

d.

Th

e lea

ve

s c

on

sis

t of

the

sm

alle

st

pla

ce

va

lue

(usu

ally

th

e o

ne

s p

lace

) of

eve

ry p

iece

of

da

ta; th

ese

nu

mb

ers

are

arr

an

ged

in

nu

me

rica

l o

rde

r in

th

e r

ow

of

the a

pp

rop

ria

te s

tem

(e.g

., t

he

nu

mb

er

36

wo

uld

be in

dic

ate

d b

y a

lea

f of

6 a

pp

ea

rin

g in

th

e s

am

e r

ow

as t

he

ste

m o

f 3

). E

xa

mp

le o

f a s

tem

-an

d-leaf

plo

t:

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

6

Ap

ril 2

01

4

Su

bs

titu

tio

n

The

rep

lace

me

nt

of

a t

erm

or

va

ria

ble

in a

n e

xp

ressio

n o

r e

qu

ation

by a

no

the

r th

at

ha

s th

e s

am

e

va

lue

in o

rde

r to

sim

plif

y o

r eva

lua

te t

he

exp

ressio

n o

r equ

ation

.

Sys

tem

of

Lin

ea

r E

qu

ati

on

s

A s

et of

two

or

mo

re lin

ea

r equ

atio

ns w

ith

th

e s

am

e v

aria

ble

s.

The

so

lution t

o a

syste

m o

f lin

ea

r e

qu

atio

ns m

ay b

e f

ou

nd

by lin

ea

r com

bin

atio

n,

sub

stitu

tio

n, o

r gra

ph

ing

. A

syste

m o

f tw

o lin

ea

r e

qu

atio

ns w

ill e

ith

er

have

one

so

lutio

n, in

finitely

ma

ny s

olu

tion

s,

or

no

so

lutio

ns.

Sys

tem

of

Lin

ea

r In

eq

uali

ties

Tw

o o

r m

ore

lin

ea

r in

equ

alit

ies w

ith

th

e s

am

e v

aria

ble

s. S

om

e s

yste

ms o

f in

equ

alit

ies m

ay inclu

de

e

qu

atio

ns a

s w

ell

as in

equ

alit

ies.

The

so

lutio

n r

egio

n m

ay b

e c

lose

d o

r bo

un

de

d b

ecau

se

the

re a

re

line

s o

n a

ll sid

es,

wh

ile o

the

r solu

tion

s m

ay b

e o

pe

n o

r u

nb

ou

nde

d.

Sys

tem

s o

f E

qu

ati

on

s

A s

et of

two

or

mo

re e

qu

atio

ns c

on

tain

ing a

se

t of

co

mm

on

va

ria

ble

s.

Te

rm

A p

art

of

an

alg

eb

raic

exp

ressio

n.

Term

s a

re s

ep

ara

ted

by e

ith

er

an

ad

ditio

n s

ym

bo

l (+

) or

a

su

btr

action

sym

bo

l (–

). I

t can

be

a n

um

be

r, a

va

ria

ble

, o

r a p

rod

uct

of

a n

um

be

r an

d o

ne

or

mo

re

va

ria

ble

s (

e.g

., in t

he

exp

ressio

n 4

x2+

6y,

4x

2 a

nd

6y a

re b

oth

te

rms).

Te

rmin

ati

ng

De

cim

al

A d

ecim

al w

ith

a f

inite

nu

mb

er

of

dig

its.

A d

ecim

al fo

r w

hic

h t

he

div

isio

n o

pe

ration

resu

lts in e

ith

er

rep

ea

tin

g z

ero

es o

r re

pe

atin

g n

ine

s (

e.g

., 0

.37

500

0…

= 0

.375

, 0.1

99

9…

= 0

.2).

It is

ge

ne

rally

wri

tte

n

to t

he

la

st n

on

-ze

ro p

lace

va

lue

, bu

t ca

n a

lso

be

writt

en

with

ad

ditio

na

l ze

roe

s in

sm

alle

r pla

ce

va

lue

s

as n

ee

ded

(e.g

., 0

.25

ca

n a

lso

be w

ritt

en

as 0

.25

00

). A

ll te

rmin

ating d

ecim

als

are

ratio

na

l nu

mb

ers

.

Tri

no

mia

l A

po

lyn

om

ial w

ith

th

ree

un

like t

erm

s (

e.g

., 7

a +

4b

+ 9

c).

Ea

ch t

erm

is a

mo

no

mia

l, a

nd

the

mo

no

mia

ls a

re jo

ine

d b

y a

n a

dd

itio

n s

ym

bo

l (+

) or

a s

ub

tractio

n s

ym

bo

l (–

). I

t is

co

nsid

ere

d a

n

alg

eb

raic

exp

ressio

n.

K

ey

sto

ne

Ex

am

s: A

lge

bra

Ass

ess

me

nt

An

cho

r &

Eli

gib

le C

on

ten

t G

loss

ary

A

pri

l 2

01

4

Pen

nsy

lva

nia

Dep

art

men

t o

f E

du

cati

on

Pag

e 2

7

Ap

ril 2

01

4

Un

it R

ate

A

rate

in

wh

ich t

he

se

co

nd

(in

de

pe

nde

nt)

quan

tity

of

the

ratio

is 1

(e.g

., 6

0 w

ord

s p

er

min

ute

, $4

.50

p

er

pou

nd

, 21

stu

den

ts p

er

cla

ss).

Va

ria

ble

A

le

tte

r o

r sym

bo

l u

se

d t

o r

ep

rese

nt

an

y o

ne

of

a g

iven s

et of

nu

mb

ers

or

oth

er

ob

jects

(e

.g.,

in t

he

equ

atio

n y

= x

+ 5

, th

e y

an

d x

are

va

ria

ble

s).

Sin

ce

it

ca

n ta

ke

on

diffe

ren

t va

lue

s,

it is th

e o

ppo

site

of

a c

on

sta

nt.

Wh

ole

Nu

mb

er

A n

atu

ral nu

mb

er

or

ze

ro.

An

y n

um

be

r fr

om

th

e s

et of

nu

mb

ers

re

pre

se

nte

d b

y {

0,

1,

2, 3

, …

}.

So

me

tim

es it

is r

efe

rred

to

as a

“no

n-n

ega

tive inte

ge

r”.

x-A

xis

T

he

ho

rizo

nta

l n

um

be

r lin

e o

n a

co

ord

ina

te p

lan

e th

at

inte

rsects

with

a v

ert

ical n

um

be

r lin

e, th

e

ya

xis

; th

e lin

e w

ho

se

equ

atio

n is y

= 0

. T

he

x-a

xis

co

nta

ins a

ll th

e p

oin

ts w

ith

a z

ero

y-c

oo

rdin

ate

(e

.g.,

(5,

0))

.

x-I

nte

rce

pt(

s)

The

x-c

oo

rdin

ate

(s)

of

the

po

int(

s)

at

wh

ich t

he

gra

ph

of

an

equ

atio

n c

rosse

s t

he

x-a

xis

(i.e

., t

he

va

lue

(s)

of

the x

-co

ord

ina

te w

he

n y

= 0

). T

he

so

lution

(s)

or

roo

t(s)

of

an

equ

atio

n t

ha

t is

set e

qu

al

to 0

.

y-A

xis

T

he

ve

rtic

al n

um

be

r lin

e o

n a

co

ord

ina

te p

lane

tha

t in

ters

ects

with

a h

orizo

nta

l n

um

be

r lin

e, th

e

xa

xis

; th

e lin

e w

ho

se

equ

atio

n is x

= 0

. T

he

y-a

xis

co

nta

ins a

ll th

e p

oin

ts w

ith

a z

ero

x-c

oo

rdin

ate

(e

.g.,

(0,

7))

.

y-I

nte

rce

pt(

s)

The

y-c

oo

rdin

ate

(s)

of

the

po

int(

s)

at

wh

ich t

he

gra

ph

of

an

equ

atio

n c

rosse

s t

he

y-

axis

(i.e.,

the

va

lue

(s)

of

the y

-co

ord

ina

te w

he

n x

= 0

). F

or

a lin

ea

r e

qu

atio

n in s

lop

e-inte

rcep

t fo

rm (

y =

mx +

b),

it is

in

dic

ate

d b

y b

.


Assessment Anchors and Eligible Contentwith Sample Questions and Glossary

April 2014

Cover photo © Hill Street Studios/Harmik Nazarian/Blend Images/Corbis.

Copyright © 2014 by the Pennsylvania Department of Education. The materials contained in this publication may be

duplicated by Pennsylvania educators for local classroom use. This permission does not extend to the duplication

of materials for commercial use.

keystone exams: algebra ii · keystone exams: algebra ii module 1—number systems and non-linear...

Documents