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Keystone Exams: Algebra IAssessment Anchors and Eligible Content

with Sample Questions and Glossary

Pennsylvania Department of Education

www.education.state.pa.us

January 2013

Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 2

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 Common 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 Common Core Standard: In the column to the right of each Eligible Content statement is a code representing one or more PA Common 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

“i.e.” (“that is”)—specific limit to the Eligible Content

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

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

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

Standards Aligned System—www.pdesas.org

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


Keystone Exams: Algebra I

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.

A = lw

l

w

V = lwh

lw

h

Arithmetic Properties

Additive Inverse: a + (ˉa) = 0

Multiplicative Inverse: a · = 1

Commutative Property: a + b = b + a a · b = b · a

Associative Property: (a + b) + c = a + (b + c) (a · b) · c = a · (b · c)

Identity Property: a + 0 = a a · 1 = a

Distributive Property: a · (b + c) = a · b + a · c

Multiplicative Property of Zero: a · 0 = 0

Additive Property of Equality: If a = b, then a + c = b + c

Multiplicative Property of Equality: If a = b, then a · c = b · c

1a

Linear Equations

Slope: m =

Point-Slope Formula: (y – y 1) = m(x – x 1)

Slope-Intercept Formula: y = mx + b

Standard Equation of a Line: Ax + By = C

y 2 – y 1x 2 – x 1



MODULE 1—Operations and Linear Equations & Inequalities

ASSESSMENT ANCHOR

A1.1.1 Operations with Real Numbers and Expressions

Anchor Descriptor Eligible Content

PA Common

Core

Standards

A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents).

A1.1.1.1.1 Compare and/or order any real numbers.Note: Rational and irrational may be mixed.

CC.2.1.8.E.1

CC.2.1.8.E.4

CC.2.1.HS.F.1

CC.2.1.HS.F.2

A1.1.1.1.2 Simplify square roots (e.g., Ï}

24 = 2 Ï}

6 ).

Sample Exam Questions

Standard A1.1.1.1.1

Which of the following inequalities is true for all real values of x?

A. x3 ≥ x2

B. 3x2 ≥ 2x3

C. (2x)2 ≥ 3x2

D. 3(x – 2)2 ≥ 3x2 – 2

Standard A1.1.1.1.2

An expression is shown below.

2 Ï}

51x

Which value of x makes the expression equivalent to 10 Ï

}

51 ?

A. 5

B. 25

C. 50

D. 100

An expression is shown below.

Ï}

87x

For which value of x should the expression be further simplified?

A. x = 10

B. x = 13

C. x = 21

D. x = 38




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.1.1.2 Apply number theory concepts to show relationships between real numbers in problem-solving settings.

A1.1.1.2.1 Find the Greatest Common Factor (GCF) and/or the Least Common Multiple (LCM) for sets of monomials.

CC.2.1.6.E.3

CC.2.1.HS.F.2

Sample Exam Question

Standard A1.1.1.2.1

Two monomials are shown below.

450x2y5 3,000x4y3

What is the least common multiple (LCM) of these monomials?

A. 2xy

B. 30xy

C. 150x2y3

D. 9,000x4y5




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.1.1.3 Use exponents, roots, and/or absolute values to solve problems.

A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots, and/or absolute values to solve problems. Note: Exponents should be integers from –10 to 10.

CC.2.1.HS.F.1

CC.2.1.HS.F.2

CC.2.2.8.B.1


Standard A1.1.1.3.1

Simplify:

2(2 Ï}

4 ) –2

A. 1 __ 8

B. 1 __ 4

C. 16

D. 32




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.1.1.4 Use estimation strategies in problem-solving situations.

A1.1.1.4.1 Use estimation to solve problems. CC.2.2.7.B.3

CC.2.2.HS.D.9


Standard A1.1.1.4.1

A theme park charges $52 for a day pass and $110 for a week pass. Last month, 4,432 day passes were sold and 979 week passes were sold. Which is the closest estimate of the total amount of money paid for the day and week passes for last month?

A. $300,000

B. $400,000

C. $500,000

D. $600,000




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.1.1.5 Simplify expressions involving polynomials.

A1.1.1.5.1 Add, subtract, and/or multiply polynomial expressions (express answers in simplest form). Note: Nothing larger than a binomial multiplied by a trinomial.

CC.2.2.HS.D.1 CC.2.2.HS.D.2 CC.2.2.HS.D.3

CC.2.2.HS.D.5

CC.2.2.HS.D.6

A1.1.1.5.2 Factor algebraic expressions, including difference of squares and trinomials.Note: Trinomials are limited to the form ax2 + bx + c where a is equal to 1 after factoring out all monomial factors.

A1.1.1.5.3 Simplify/reduce a rational algebraic expression.


Standard A1.1.1.5.1

A polynomial expression is shown below.

(mx3 + 3 ) (2x2 + 5x + 2) – (8x5 + 20x4 )

The expression is simplified to 8x3 + 6x2 + 15x + 6. What is the value of m?

A. –8

B. –4

C. 4

D. 8

Standard A1.1.1.5.2

When the expression x2 – 3x – 18 is factored completely, which is one of its factors?

A. (x – 2)

B. (x – 3)

C. (x – 6)

D. (x – 9)

Standard A1.1.1.5.3

Simplify:

–3x3 + 9 x2 + 30x }} –3x3 – 18 x2 – 24x ; x ≠ –4, –2, 0

A. – 1 } 2 x2 – 5 }

4 x

B. x3 – 1 } 2

x2 – 5 } 4 x

C. x + 5 } x – 4

D. x – 5 } x + 4




Standard A1.1.1

Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below.

h

h + 4

A. Write a polynomial expression, in simplifi ed form, that represents the area of the canvas.

Keng adds a 3-inch-wide frame around all sides of his canvas.

B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame.

Continued on next page.

ASSESSMENT ANCHOR






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

Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h2 + 8h + 12, where h represents the height of the canvas.

C. Determine the width of the new frame. Show all your work. Explain why you did each step.




Standard A1.1.1

The results of an experiment were listed in several numerical forms as listed below.

5–3 4 } 7 Ï

}

5 3 } 8 0.003

A. Order the numbers listed from least to greatest.

Another experiment required evaluating the expression shown below.

1 } 6 ( Ï

}

36 ÷ 3–2) + 43 ÷ z–8z

B. What is the value of the expression?

value of the expression:






The final experiment required simplifying 7 Ï}

425 . The steps taken are shown below.

7 425

step 1: 7 400 25)

step 2: 7(20 + 5)

step 3: 7(25)

step 4: 175

+

One of the steps shown is incorrect.

C. Rewrite the incorrect step so that it is correct.

correction:

D. Using the corrected step from part C, simplify 7 Ï}

425 .

7 Ï}

425 =




ASSESSMENT ANCHOR

A1.1.2 Linear Equations


PA Common

Core

Standards

A1.1.2.1 Write, solve, and/or graph linear equations using various methods.

A1.1.2.1.1 Write, solve, and/or apply a linear equation (including problem situations).

CC.2.1.HS.F.3

CC.2.1.HS.F.4

CC.2.1.HS.F.5

CC.2.2.8.B.3

CC.2.2.8.C.1

CC.2.2.8.C.2

CC.2.2.HS.C.3

CC.2.2.HS.D.7

CC.2.2.HS.D.8

CC.2.2.HS.D.9

CC.2.2.HS.D.10

A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation-solving process.Note: Linear equations only.

A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation.Note: Linear equations only.


Standard A1.1.2.1.1

Jenny has a job that pays her $8 per hour plus tips (t). Jenny worked for 4 hours on Monday and made $65 in all. Which equation could be used to find t, the amount Jenny made in tips?

A. 65 = 4t + 8

B. 65 = 8t ÷ 4

C. 65 = 8t + 4

D. 65 = 8(4) + t

Standard A1.1.2.1.2

One of the steps Jamie used to solve an equation is shown below.

–5(3x + 7) = 10–15x + –35 = 10

Which statements describe the procedure Jamie used in this step and identify the property that justifies the procedure?

A. Jamie added –5 and 3x to eliminate the parentheses. This procedure is justifi ed by the associative property.

B. Jamie added –5 and 3x to eliminate the parentheses. This procedure is justifi ed by the distributive property.

C. Jamie multiplied 3x and 7 by –5 to eliminate the parentheses. This procedure is justifi ed by the associative property.

D. Jamie multiplied 3x and 7 by –5 to eliminate the parentheses. This procedure is justifi ed by the distributive property.





Standard A1.1.2.1.3

Francisco purchased x hot dogs and y hamburgers at a baseball game. He spent a total of $10. The equation below describes the relationship between the number of hot dogs and the number of hamburgers purchased.

3x + 4y = 10

The ordered pair (2, 1) is a solution of the equation. What does the solution (2, 1) represent?

A. Hamburgers cost 2 times as much as hot dogs.

B. Francisco purchased 2 hot dogs and 1 hamburger.

C. Hot dogs cost $2 each, and hamburgers cost $1 each.

D. Francisco spent $2 on hot dogs and $1 on hamburgers.




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.1.2.2 Write, solve, and/or graph systems of linear equations using various methods.

A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution, and/or elimination.Note: Limit systems to two linear equations.

CC.2.1.HS.F.5

CC.2.2.8.B.3

CC.2.2.HS.D.7

CC.2.2.HS.D.9

CC.2.2.HS.D.10

A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation.Note: Limit systems to two linear equations.


Standard A1.1.2.2.1

Anna burned 15 calories per minute running for x minutes and 10 calories per minute hiking for y minutes. She spent a total of 60 minutes running and hiking and burned 700 calories. The system of equations shown below can be used to determine how much time Anna spent on each exercise.

15x + 10y = 700

x + y = 60

What is the value of x, the minutes Anna spent running?

A. 10

B. 20

C. 30

D. 40

Standard A1.1.2.2.2

Samantha and Maria purchased flowers. Samantha purchased 5 roses for x dollars each and 4 daisies for y dollars each and spent $32 on the flowers. Maria purchased 1 rose for x dollars and 6 daisies for y dollars each and spent $22. The system of equations shown below represents this situation.

5x + 4y = 32

x + 6y = 22

Which statement is true?

A. A rose costs $1 more than a daisy.

B. Samantha spent $4 on each daisy.

C. Samantha spent more on daisies than she did on roses.

D. Samantha spent over 4 times as much on daisies as she did on roses.




Standard A1.1.2

Nolan has $15.00. He earns $6.00 an hour babysitting. The equation below can be used to determine how much money in dollars (m) Nolan has after any number of hours of babysitting (h).

m = 6h + 15

A. After how many hours of babysitting will Nolan have $51.00?

hours:

Claire has $9.00. She makes $8.00 an hour babysitting.

B. Use the system of linear equations below to find the number of hours of babysitting after which Nolan and Claire will have the same amount of money.

m = 6h + 15

m = 8h + 9

hours:


ASSESSMENT ANCHOR







The graph below displays the amount of money Alex and Pat will each have saved after their hours of babysitting.

0Hours

Mon

ey S

aved

(in

Dol

lars

)

h

m

504540353025201510

5

1 2 3 4 5 6 7 8 9 10

Money Saved

KeyAlex’s money savedPat’s money saved

C. Based on the graph, for what domain (h) will Alex have more money saved than Pat? Explain your reasoning.




Standard A1.1.2

The diagram below shows 5 identical bowls stacked one inside the other.

2 inches

5 inches

Bowls

The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches.

A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls.

equation:






B. Describe what the x and y variables represent.

x-variable:

y-variable:

C. What is the height, in inches, of a stack of 10 bowls?

height: inches




ASSESSMENT ANCHOR

A1.1.3 Linear Inequalities


PA Common

Core

Standards

A1.1.3.1 Write, solve, and/or graph linear inequalities using various methods.

A1.1.3.1.1 Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities).

CC.2.1.HS.F.5

CC.2.2.HS.D.7

CC.2.2.HS.D.9

CC.2.2.HS.D.10

A1.1.3.1.2 Identify or graph the solution set to a linear inequality on a number line.

A1.1.3.1.3 Interpret solutions to problems in the context of the problem situation.Note: Linear inequalities only.


Standard A1.1.3.1.1

A compound inequality is shown below.

5 < 2 – 3y < 14

What is the solution of the compound inequality?

A. –4 > y > –1

B. –4 < y < –1

C. 1 > y > 4

D. 1 < y < 4

Standard A1.1.3.1.3

A baseball team had $1,000 to spend on supplies. The team spent $185 on a new bat. New baseballs cost $4 each. The inequality 185 + 4b ≤ 1,000 can be used to determine the number of new baseballs (b) that the team can purchase. Which statement about the number of new baseballs that can be purchased is true?

A. The team can purchase 204 new baseballs.

B. The minimum number of new baseballs that can be purchased is 185.

C. The maximum number of new baseballs that can be purchased is 185.

D. The team can purchase 185 new baseballs, but this number is neither the maximum nor the minimum.

Standard A1.1.3.1.2

The solution set of an inequality is graphed on the number line below.

–4–5 –3 –2 –1 0 1 2

The graph shows the solution set of which inequality?

A. 2x + 5 < –1

B. 2x + 5 ≤ –1

C. 2x + 5 > –1

D. 2x + 5 ≥ –1




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.1.3.2 Write, solve, and/or graph systems of linear inequalities using various methods.

A1.1.3.2.1 Write and/or solve a system of linear inequalities using graphing.Note: Limit systems to two linear inequalities.

CC.2.1.HS.F.5

CC.2.2.HS.D.7

CC.2.2.HS.D.10

A1.1.3.2.2 Interpret solutions to problems in the context of the problem situation.Note: Limit systems to two linear inequalities.





Standard A1.1.3.2.1

A system of inequalities is shown below.

y < x – 6

y > –2x

Which graph shows the solution set of the system of inequalities?

987654321

–1–2–3–4–5–6–7–8–9

1 2 3 4 5 6 7 8 9–9–8–7–6–5–4–3–2–1

y

x

987654321

–1–2–3–4–5–6–7–8–9

1 2 3 4 5 6 7 8 9–9–8–7–6–5–4–3–2–1

y

x

987654321

–1–2–3–4–5–6–7–8–9

1 2 3 4 5 6 7 8 9–9–8–7–6–5–4–3–2–1

y

x

987654321

–1–2–3–4–5–6–7–8–9

1 2 3 4 5 6 7 8 9–9–8–7–6–5–4–3–2–1

y

x

A. B.

C. D.





Standard A1.1.3.2.2

Tyreke always leaves a tip of between 8% and 20% for the server when he pays for his dinner. This can be represented by the system of inequalities shown below, where y is the amount of tip and x is the cost of dinner.

y > 0.08x

y < 0.2x

Which of the following is a true statement?

A. When the cost of dinner ( x) is $10, the amount of tip ( y) must be between $2 and $8.

B. When the cost of dinner ( x) is $15, the amount of tip ( y) must be between $1.20 and $3.00.

C. When the amount of tip ( y) is $3, the cost of dinner ( x) must be between $11 and $23.

D. When the amount of tip ( y) is $2.40, the cost of dinner ( x) must be between $3 and $6.




Standard A1.1.3

An apple farm owner is deciding how to use each day’s harvest. She can use the harvest to produce apple juice or apple butter. The information she uses to make the decision is listed below.

• A bushel of apples will make 16 quarts of apple juice. • A bushel of apples will make 20 pints of apple butter. • The apple farm can produce no more than 180 pints of apple butter each day. • The apple farm harvests no more than 15 bushels of apples each day.

The information given can be modeled with a system of inequalities. When x is the number of quarts of apple juice and y is the number of pints of apple butter, two of the inequalities that model the situation are x ≥ 0 and y ≥ 0.

A. Write two more inequalities to complete the system of inequalities modeling the information.

inequalities:

B. Graph the solution set of the inequalities from part A below. Shade the area that represents the solution set.

0Quarts of Apple Juice

Pint

s of

App

le B

utte

r

300

240

180

120

60

60 120 180 240 300

Apple Farm Production

x

y


ASSESSMENT ANCHOR







The apple farm makes a profit of $2.25 on each pint of apple butter and $2.50 on each quart of apple juice.

C. Explain how you can be certain the maximum profit will be realized when the apple farm produces 96 quarts of apple juice and 180 pints of apple butter.




Standard A1.1.3

David is solving problems with inequalities.

One of David’s problems is to graph the solution set of an inequality.

A. Graph the solution set to the inequality 4x + 3 < 7x – 9 on the number line below.

–8–10 –9 –7 –6 104–5 7–2 –1 0 1 2 3 5 6 8 9–4 –3

David correctly graphed an inequality as shown below.

104 71 2 3 5 6 8 9–8–10 –9 –7 –6 –5 –2 –1 0–4 –3

The inequality David graphed was written in the form 7 ≤ ? ≤ 9.

B. What is an expression that could be put in place of the question mark so that the inequality would have the same solution set as shown in the graph?

7 ≤ ≤ 9






The solution set to a system of linear inequalities is graphed below.

x

y

1 2 3 4 5–5 –4 –3 –2 –1

54321

–1–2–3–4–5

C. Write a system of two linear inequalities that would have the solution set shown in the graph.

linear inequality 1:

linear inequality 2:



MODULE 2—Linear Functions and Data Organizations

ASSESSMENT ANCHOR

A1.2.1 Functions


PA Common

Core

Standards

A1.2.1.1 Analyze and/or use patterns or relations.

A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically.

CC.2.2.8.C.1

CC.2.2.8.C.2

CC.2.2.HS.C.1

CC.2.2.HS.C.2

CC.2.2.HS.C.3

CC.2.4.HS.B.2

A1.2.1.1.2 Determine whether a relation is a function, given a set of points or a graph.

A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table).


Standard A1.2.1.1.1

Tim’s scores the first 5 times he played a video game are listed below.

4,526 4,599 4,672 4,745 4,818

Tim’s scores follow a pattern. Which expression can be used to determine his score after he played the video game n times?

A. 73n + 4,453

B. 73(n + 4,453)

C. 4,453n + 73

D. 4,526n





Standard A1.2.1.1.2

Which graph shows y as a function of x?

x

y

21 3 4–2–3–4

–4–3–2

1234

–1–1

x

y

21 3 4–2–3–4

–4–3–2

1234

–1–1

x

y

21 3 4–2–3–4

–4–3–2

1234

–1–1 x

y

21 3 4–2–3–4

–4–3–2

1234

–1–1

A. B.

C. D.





Standard A1.2.1.1.3

The graph of a function is shown below.

x2 4 6–6 –4 –2

8

6

4

2

–2

y

Which value is not in the range of the function?

A. 0

B. 3

C. 4

D. 5




ASSESSMENT ANCHOR

A1.2.1 Functions


PA Common

Core

Standards

A1.2.1.2 Interpret and/or use linear functions and their equations, graphs, or tables.

A1.2.1.2.1 Create, interpret, and/or use the equation, graph, or table of a linear function.

CC.2.1.HS.F.3

CC.2.1.HS.F.4

CC.2.2.8.B.2

CC.2.2.8.C.1

CC.2.2.8.C.2

CC.2.2.HS.C.2

CC.2.2.HS.C.3

CC.2.2.HS.C.4

CC.2.2.HS.C.6

CC.2.4.HS.B.2

A1.2.1.2.2 Translate from one representation of a linear function to another (i.e., graph, table, and equation).


Standard A1.2.1.2.1

A pizza restaurant charges for each pizza and adds a delivery fee. The cost (c), in dollars, to have any number of pizzas (p) delivered to a home is described by the function c = 8p + 3. Which statement is true?

A. The cost of 8 pizzas is $11.

B. The cost of 3 pizzas is $14.

C. Each pizza costs $8, and the delivery fee is $3.

D. Each pizza costs $3, and the delivery fee is $8.

Standard A1.2.1.2.2

The table below shows values of y as a function of x.

x y

2 10

6 25

14 55

26 100

34 130

Which linear equation describes the relationship between x and y?

A. y = 2.5x + 5

B. y = 3.75x + 2.5

C. y = 4x + 1

D. y = 5x




Standard A1.2.1

Hector’s family is on a car trip.

When they are 84 miles from home, Hector begins recording the distance they have driven (d ), in miles, after h hours as shown in the table below.

Distance from Home

Time

in Hours

(h)

Distance

in Miles

(d )

0 84

1 146

2 208

3 270

The pattern continues.

A. Write an equation to find the distance driven (d ), in miles, after a given number of hours ( h).


ASSESSMENT ANCHOR

A1.2.1 Functions






B. Hector also kept track of the remaining gasoline. The equation shown below can be used to find the gallons of gasoline remaining ( g) after driving a distance of d miles.

g = 16 – 1 } 20

d

Use the equation to find the missing values for gallons of gasoline remaining.

Gasoline Remaining by Mile

Distance

in Miles

(d )

Gallons of Gasoline

Remaining

( g)

100

200

300

C. Draw the graph of the line formed by the points in the table from part B.

g

d25 50 75 100 125 150 175 200 225 250 275 300

20

18

16

14

12

10

8

6

4

2

0

Gasoline Remaining by Mile

Ga

llo

ns o

f G

aso

lin

e R

em

ain

ing

Distance in Miles






D. Explain why the slope of the line drawn in part C must be negative.




Standard A1.2.1

Last summer Ben purchased materials to build model airplanes and then sold the finished models. He sold each model for the same amount of money. The table below shows the relationship between the number of model airplanes sold and the running total of Ben’s profit.

Ben’s Model-Airplane Sales

Model

Airplanes

Sold

Total Profi t

12 $68

15 $140

20 $260

22 $308

A. Write a linear equation, in slope-intercept form, to represent the amount of Ben’s total profit (y) based on the number of model airplanes (x) he sold.

y =

B. Determine a value of y that represents a situation where Ben did not make a profit from building model airplanes.

y-value:






C. How much did Ben spend on the materials he needed to build his models?

$

D. What is the least number of model airplanes Ben needed to sell in order to make a profit?

least number:




ASSESSMENT ANCHOR

A1.2.2 Coordinate Geometry


PA Common

Core

Standards

A1.2.2.1 Describe, compute, and/or use the rate of change (slope) of a line.

A1.2.2.1.1 Identify, describe, and/or use constant rates of change.

CC.2.2.8.C.2

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.2.4.HS.B.1

A1.2.2.1.2 Apply the concept of linear rate of change (slope) to solve problems.

A1.2.2.1.3 Write or identify a linear equation when given • the graph of the line, • two points on the line, or • the slope and a point on the line.

Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.

A1.2.2.1.4 Determine the slope and/or y-intercept represented by a linear equation or graph.


Standard A1.2.2.1.1

Jeff’s restaurant sells hamburgers. The amount charged for a hamburger ( h) is based on the cost for a plain hamburger plus an additional charge for each topping (t ) as shown in the equation below.

h = 0.60t + 5

What does the number 0.60 represent in the equation?

A. the number of toppings

B. the cost of a plain hamburger

C. the additional cost for each topping

D. the cost of a hamburger with 1 topping





Standard A1.2.2.1.2

A ball rolls down a ramp with a slope of 2 } 3 . At one

point the ball is 10 feet high, and at another point

the ball is 4 feet high, as shown in the diagram

below.

x ft

4 ft

10 ft

What is the horizontal distance (x), in feet, the ball travels as it rolls down the ramp from 10 feet high to 4 feet high?

A. 6

B. 9

C. 14

D. 15

Standard A1.2.2.1.3

A graph of a linear equation is shown below.

x

y

8642

–2–4–6–8

2 4 6 8–8 –6 –4 –2

Which equation describes the graph?

A. y = 0.5x – 1.5

B. y = 0.5x + 3

C. y = 2x – 1.5

D. y = 2x + 3

Standard A1.2.2.1.4

A juice machine dispenses the same amount of juice into a cup each time the machine is used. The equation below describes the relationship between the number of cups (x) into which juice is dispensed and the gallons of juice (y) remaining in the machine.

x + 12y = 180

How many gallons of juice are in the machine when it is full?

A. 12

B. 15

C. 168

D. 180




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.2.2.2 Analyze and/or interpret data on a scatter plot.

A1.2.2.2.1 Draw, identify, fi nd, and/or write an equation for a line of best fi t for a scatter plot.

CC.2.2.HS.C.6

CC.2.4.8.B.1

CC.2.4.HS.B.2

CC.2.4.HS.B.3


Standard A1.2.2.2.1

The scatter plot below shows the cost ( y) of ground shipping packages from Harrisburg, Pennsylvania, to Minneapolis, Minnesota, based on the package weight ( x).

x

y

Weight (in pounds)

Cos

t (in

dol

lars

)

10 20 30 40

Ground Shipping

0

30

20

10

Which equation best describes the line of best fit?

A. y = 0.37x + 1.57

B. y = 0.37x + 10.11

C. y = 0.68 x + 2.32

D. y = 0.68 x + 6.61




Standard A1.2.2

Georgia is purchasing treats for her classmates. Georgia can spend exactly $10.00 to purchase 25 fruit bars, each equal in price. Georgia can also spend exactly $10.00 to purchase 40 granola bars, each equal in price.

A. Write an equation that can be used to find all combinations of fruit bars ( x) and granola bars ( y) that will cost exactly $10.00.

equation:

B. Graph the equation from part A below.

Fruit Bars0

Gra

nola

Bar

s

50

40

30

20

10

10 20 30 40 50

Purchasing Treats

x

y


ASSESSMENT ANCHOR







C. What is the slope of the line graphed in part B?

slope:

D. Explain what the slope from part C means in the context of Georgia purchasing treats.




Standard A1.2.2

Ahava is traveling on a train.

The train is going at a constant speed of 80 miles per hour.

A. How many hours will it take for the train to travel 1,120 miles?

hours:

Ahava also considered taking an airplane. The airplane can travel the same 1,120 miles in 12 hours less time than it takes the train.

B. What is the speed of the airplane in miles per hour (mph)?

speed of the airplane: mph






Ahava is very concerned about the environment. The graph below displays the carbon dioxide (CO2), in metric tons, for each traveler on an airplane and each traveler on a train.

0Miles Traveled

Met

ric T

ons

of C

O2

x

y

0.40

0.32

0.24

0.16

0.08

200 400 600 800 1000

Carbon Footprint

KeyTraveler on anairplaneTraveler on atrain

C. What equation could be used to find the metric tons of CO2 produced (y) by a traveler on an airplane for x miles traveled?

equation:






On another trip, Ahava traveled to her destination on a train and returned home on an airplane. Her total carbon footprint for the trip was 0.42 metric tons of CO2 produced.

D. How far, in miles, is Ahava’s destination from her home?

miles:




ASSESSMENT ANCHOR

A1.2.3 Data Analysis


PA Common

Core

Standards

A1.2.3.1 Use measures of dispersion to describe a set of data.

A1.2.3.1.1 Calculate and/or interpret the range, quartiles, and interquartile range of data.

CC.2.4.HS.B.1

CC.2.4.HS.B.3


Standard A1.2.3.1.1

The daily high temperatures, in degrees Fahrenheit (°F), of a town are recorded for one year. The median high temperature is 62°F. The interquartile range of high temperatures is 32. Which statement is most likely true?

A. Approximately 25% of the days had a high temperature less than 30°F.

B. Approximately 25% of the days had a high temperature greater than 62°F.

C. Approximately 50% of the days had a high temperature greater than 62°F.

D. Approximately 75% of the days had a high temperature less than 94°F.




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.2.3.2 Use data displays in problem-solving settings and/or to make predictions.

A1.2.3.2.1 Estimate or calculate to make predictions based on a circle, line, bar graph, measure of central tendency, or other representation.

CC.2.4.HS.B.1

CC.2.4.HS.B.3

CC.2.4.HS.B.5

A1.2.3.2.2 Analyze data, make predictions, and/or answer questions based on displayed data (box-and-whisker plots, stem-and-leaf plots, scatter plots, measures of central tendency, or other representations).

A1.2.3.2.3 Make predictions using the equations or graphs of best-fi t lines of scatter plots.


Standard A1.2.3.2.1

Vy asked 200 students to select their favorite sport and then recorded the results in the bar graph below.

Favorite Sport

Sports

Vote

s

8070605040302010

0football basketball baseball hockey volleyball track

and field

Vy will ask another 80 students to select their favorite sport. Based on the information in the bar graph, how many more students of the next 80 asked are likely to select basketball rather than football as their favorite sport?

A. 10

B. 20

C. 25

D. 30





Standard A1.2.3.2.2

The points scored by a football team are shown in the stem-and-leaf plot below.

Football-Team Points

0123

6 2 3 4 7 0 3 4 4 7 8 8 80 7 8

Key1 | 3 = 13 points

What was the median number of points scored by the football team?

A. 24

B. 27

C. 28

D. 32





Standard A1.2.3.2.3

John recorded the weight of his dog Spot at different ages as shown in the scatter plot below.

Spot’s Weight

Age (in months)

Wei

ght (

in p

ound

s)

504540353025201510

5

20 4 6 8 1210 14 16

Based on the line of best fit, what will be Spot’s weight after 18 months?

A. 27 pounds

B. 32 pounds

C. 36 pounds

D. 50 pounds




ASSESSMENT ANCHOR



PA Common

Core

Standards

A1.2.3.3 Apply probability to practical situations.

A1.2.3.3.1 Find probabilities for compound events (e.g., fi nd probability of red and blue, fi nd probability of red or blue) and represent as a fraction, decimal, or percent.

CC.2.4.7.B.3

CC.2.4.HS.B.4

CC.2.4.HS.B.7


Standard A1.2.3.3.1

A number cube with sides labeled 1 through 6 is rolled two times, and the sum of the numbers that end face up is calculated. What is the probability that the sum of the numbers is 3?

A. 1 } 18

B. 1 } 12

C. 1 } 9

D. 1 } 2




Standard A1.2.3

The box-and-whisker plot shown below represents students’ test scores on Mr. Ali’s history test.

52 60 68 76 84 92 100

History-Test Scores

A. What is the range of scores for the history test?

range:

B. What is the best estimate for the percent of students scoring greater than 92 on the test?

percent: %


ASSESSMENT ANCHOR







Mr. Ali wanted more than half of the students to score 75 or greater on the test.

C. Explain how you know that more than half of the students did not score greater than 75.

Michael is a student in Mr. Ali’s class. The scatter plot below shows Michael’s test scores for each test given by Mr. Ali.

0Test Number

Te

st

Sc

ore

100908070605040302010

1 2 3 4 5 6 7 8

Michael’s Test Scores

D. Draw a line of best fit on the scatter plot above.




Standard A1.2.3

The weight, in pounds, of each wrestler on the high school wrestling team at the beginning of the season is listed below.

178 142 112 150 206 130

A. What is the median weight of the wrestlers?

median: pounds

B. What is the mean weight of the wrestlers?

mean: pounds






Two more wrestlers join the team during the season. The addition of these wrestlers has no effect on the mean weight of the wrestlers, but the median weight of the wrestlers increases 3 pounds.

C. Determine the weights of the two new wrestlers.

new wrestlers: pounds and pounds


KEYSTONE ALGEBRA I ASSESSMENT ANCHORS

KEY TO SAMPLE MULTIPLE-CHOICE ITEMS

Algebra I

Eligible Content Key

A1.1.1.1.1 C

A1.1.1.1.2 (top) B

A1.1.1.1.2 (bottom) C

A1.1.1.2.1 D

A1.1.1.3.1 A

A1.1.1.4.1 A

A1.1.1.5.1 C

A1.1.1.5.2 C

A1.1.1.5.3 D


A1.1.2.1.1 D

A1.1.2.1.2 D

A1.1.2.1.3 B

A1.1.2.2.1 B

A1.1.2.2.2 A


A1.1.3.1.1 B

A1.1.3.1.2 D

A1.1.3.1.3 D

A1.1.3.2.1 A

A1.1.3.2.2 B


A1.2.1.1.1 A

A1.2.1.1.2 B

A1.2.1.1.3 B

A1.2.1.2.1 C

A1.2.1.2.2 B


A1.2.2.1.1 C

A1.2.2.1.2 B

A1.2.2.1.3 D

A1.2.2.1.4 B

A1.2.2.2.1 D


A1.2.3.1.1 C

A1.2.3.2.1 A

A1.2.3.2.2 A

A1.2.3.2.3 B

A1.2.3.3.1 A

Key

ston

e Ex

ams:

Alg

ebra

Gl

ossa

ry to

the

As

sess

men

t Anc

hor &

Elig

ible

Con

tent

Th

e Ke

ysto

ne G

loss

ary

incl

udes

ter

ms

and

defin

itio

ns a

ssoc

iate

d w

ith

the

Keys

tone

Ass

essm

ent

Anc

hors

and

El

igib

le C

onte

nt.

The

term

s an

d de

finit

ions

inc

lude

d in

the

glo

ssar

y ar

e in

tend

ed t

o as

sist

Pen

nsyl

vani

a ed

ucat

ors

in b

ette

r un

ders

tand

ing

the

Keys

tone

Ass

essm

ent

Anc

hors

and

Elig

ible

Con

tent

. Th

e gl

ossa

ry d

oes

not

defin

e al

l pos

sibl

e te

rms

incl

uded

on

an a

ctua

l Key

ston

e Ex

am, a

nd it

is n

ot in

tend

ed t

o de

fine

term

s fo

r us

e in

cla

ssro

om in

stru

ctio

n fo

r a

part

icul

ar g

rade

leve

l or

cour

se.

Penn

sylv

ania

Dep

artm

ent o

f Edu

cati

on

ww

w.e

du

cati

on.s

tate

.pa.

us

Janu

ary

2013

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

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

2

Janu

ary

2013

Abs

olut

e Va

lue

A n

umbe

r’s d

ista

nce

from

zer

o on

the

num

ber l

ine.

It is

writ

ten

|a| a

nd is

read

“the

abs

olut

e va

lue

of

a.” I

t res

ults

in a

num

ber g

reat

er th

an o

r equ

al to

zer

o (e

.g.,

|4| =

4 a

nd |– 4|

= 4

). E

xam

ple

of a

bsol

ute

valu

es o

f – 4 an

d 4

on a

num

ber l

ine:

Add

itive

Inve

rse

The

oppo

site

of a

num

ber (

i.e.,

for a

ny n

umbe

r a, t

he a

dditi

ve in

vers

e is

– a). A

ny n

umbe

r and

its

addi

tive

inve

rse

will

hav

e a

sum

of z

ero

(e.g

., – 4

is th

e ad

ditiv

e in

vers

e of

4 s

ince

4 +

– 4 =

0; li

kew

ise,

th

e ad

ditiv

e in

vers

e of

– 4 is

4 s

ince

– 4 +

4 =

0).

Arit

hmet

ic S

eque

nce

An

orde

red

list o

f num

bers

that

incr

ease

s or

dec

reas

es a

t a c

onst

ant r

ate

(i.e.

, the

diff

eren

ce b

etw

een

num

bers

rem

ains

the

sam

e). E

xam

ple:

1, 7

, 13,

19,

… is

an

arith

met

ic s

eque

nce

as it

has

a c

onst

ant

diffe

renc

e of

+6

(i.e.

, 6 is

add

ed o

ver a

nd o

ver).

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

3

Janu

ary

2013

Asy

mpt

ote

A s

traig

ht li

ne to

whi

ch th

e cu

rve

of a

gra

ph c

omes

clo

ser a

nd c

lose

r. Th

e di

stan

ce b

etw

een

the

curv

e an

d th

e as

ympt

ote

appr

oach

es z

ero

as th

ey te

nd to

infin

ity. T

he a

sym

ptot

e is

den

oted

by

a da

shed

lin

e on

a g

raph

. The

mos

t com

mon

asy

mpt

otes

are

hor

izon

tal a

nd v

ertic

al. E

xam

ple

of a

hor

izon

tal

asym

ptot

e:

Bar

Gra

ph

A g

raph

that

sho

ws

a se

t of f

requ

enci

es u

sing

bar

s of

equ

al w

idth

, but

hei

ghts

that

are

pro

porti

onal

to

the

frequ

enci

es. I

t is

used

to s

umm

ariz

e di

scre

te d

ata.

Exa

mpl

e of

a b

ar g

raph

:

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

4

Janu

ary

2013

Bin

omia

l A

pol

ynom

ial w

ith tw

o un

like

term

s (e

.g.,

3x +

4y

or a

3 – 4

b2 ). Ea

ch te

rm is

a m

onom

ial,

and

the

mon

omia

ls a

re jo

ined

by

an a

dditi

on s

ymbo

l (+)

or a

sub

tract

ion

sym

bol (

–). I

t is

cons

ider

ed a

n al

gebr

aic

expr

essi

on.

Box

-and

-Whi

sker

Plo

t A

gra

phic

met

hod

for s

how

ing

a su

mm

ary

and

dist

ribut

ion

of d

ata

usin

g m

edia

n, q

uarti

les,

and

ex

trem

es (i

.e.,

min

imum

and

max

imum

) of d

ata.

Thi

s sh

ows

how

far a

part

and

how

eve

nly

data

is

dist

ribut

ed. I

t is

help

ful w

hen

a vi

sual

is n

eede

d to

see

if a

dis

tribu

tion

is s

kew

ed o

r if t

here

are

any

ou

tlier

s . E

xam

ple

of a

box

-and

-whi

sker

plo

t:

Circ

le G

raph

(or P

ie C

hart

) A

circ

ular

dia

gram

usi

ng d

iffer

ent-s

ized

sec

tors

of a

circ

le w

hose

ang

les

at th

e ce

nter

are

pro

porti

onal

to

the

frequ

ency

. Sec

tors

can

be

visu

ally

com

pare

d to

sho

w in

form

atio

n (e

.g.,

stat

istic

al d

ata)

. Sec

tors

re

sem

ble

slic

es o

f a p

ie. E

xam

ple

of a

circ

le g

raph

:

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

5

Janu

ary

2013

Coe

ffici

ent

The

num

ber,

usua

lly a

con

stan

t, th

at is

mul

tiplie

d by

a v

aria

ble

in a

term

(e.g

., 35

is th

e co

effic

ient

of

35x2 y)

; the

abs

ence

of a

coe

ffici

ent i

s th

e sa

me

as a

1 b

eing

pre

sent

(e.g

., x

is th

e sa

me

as 1

x).

Com

bina

tion

An

unor

dere

d ar

rang

emen

t, lis

ting

or s

elec

tion

of o

bjec

ts (e

.g.,

two-

lette

r com

bina

tions

of t

he th

ree

lette

rs X

, Y, a

nd Z

wou

ld b

e XY

, XZ,

and

YZ;

XY

is th

e sa

me

as Y

X an

d is

not

cou

nted

as

a di

ffere

nt

com

bina

tion)

. A c

ombi

natio

n is

sim

ilar t

o, b

ut n

ot th

e sa

me

as, a

per

mut

atio

n.

Com

mon

Log

arith

m

A lo

garit

hm w

ith b

ase

10. I

t is

writ

ten

log

x. T

he c

omm

on lo

garit

hm is

the

pow

er o

f 10

nece

ssar

y to

eq

ual a

giv

en n

umbe

r (i.e

., lo

g x

= y

is e

quiv

alen

t to

10y =

x).

Com

plex

Num

ber

The

sum

or d

iffer

ence

of a

real

num

ber a

nd a

n im

agin

ary

num

ber.

It is

writ

ten

in th

e fo

rm a

+ b

i,

whe

re a

and

b a

re re

al n

umbe

rs a

nd i

is th

e im

agin

ary

unit

(i.e.

, i =

1

−).

The

a is

cal

led

the

real

par

t,

and

the

bi is

cal

led

the

imag

inar

y pa

rt.

Com

posi

te N

umbe

r A

ny n

atur

al n

umbe

r with

mor

e th

an tw

o fa

ctor

s (e

.g.,

6 is

a c

ompo

site

num

ber s

ince

it h

as fo

ur

fact

ors:

1, 2

, 3, a

nd 6

). A

com

posi

te n

umbe

r is

not a

prim

e nu

mbe

r.

Com

poun

d (o

r Com

bine

d)

Even

t A

n ev

ent t

hat i

s m

ade

up o

f tw

o or

mor

e si

mpl

e ev

ents

, suc

h as

the

flipp

ing

of tw

o or

mor

e co

ins.

Com

poun

d In

equa

lity

Whe

n tw

o or

mor

e in

equa

litie

s ar

e ta

ken

toge

ther

and

writ

ten

with

the

ineq

ualit

ies

conn

ecte

d by

the

wor

ds a

nd o

r or (

e.g.

, x >

6 a

nd x

< 1

2, w

hich

can

als

o be

writ

ten

as 6

< x

< 1

2).

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

6

Janu

ary

2013

Con

stan

t A

term

or e

xpre

ssio

n w

ith n

o va

riabl

e in

it. I

t has

the

sam

e va

lue

all t

he ti

me.

Coo

rdin

ate

Plan

e A

pla

ne fo

rmed

by

perp

endi

cula

r num

ber l

ines

. The

hor

izon

tal n

umbe

r lin

e is

the

x-ax

is, a

nd th

e ve

rtica

l num

ber l

ine

is th

e y-

axis

. The

poi

nt w

here

the

axes

mee

t is

calle

d th

e or

igin

. Exa

mpl

e of

a

coor

dina

te p

lane

:

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

7

Janu

ary

2013

Cub

e R

oot

One

of t

hree

equ

al fa

ctor

s (ro

ots)

of a

num

ber o

r exp

ress

ion;

a ra

dica

l exp

ress

ion

with

a d

egre

e of

3

(e.g

., 3

a).

The

cube

root

of a

num

ber o

r exp

ress

ion

has

the

sam

e si

gn a

s th

e nu

mbe

r or e

xpre

ssio

n

unde

r the

radi

cal (

e.g.

, 3

_6

343x

= – (7

x2 ) and

36

343x

= 7

x2 ).

Cur

ve o

f Bes

t Fit

(for a

Sc

atte

r Plo

t) S

ee li

ne o

r cur

ve o

f bes

t fit

(for a

sca

tter p

lot).

Deg

ree

(of a

Pol

ynom

ial)

The

valu

e of

the

grea

test

exp

onen

t in

a po

lyno

mia

l.

Dep

ende

nt E

vent

s Tw

o or

mor

e ev

ents

in w

hich

the

outc

ome

of o

ne e

vent

affe

cts

or in

fluen

ces

the

outc

ome

of th

e ot

her

even

t(s).

Dep

ende

nt V

aria

ble

The

outp

ut n

umbe

r or v

aria

ble

in a

rela

tion

or fu

nctio

n th

at d

epen

ds u

pon

anot

her v

aria

ble,

cal

led

the

inde

pend

ent v

aria

ble,

or i

nput

num

ber (

e.g.

, in

the

equa

tion

y =

2x +

4, y

is th

e de

pend

ent v

aria

ble

sinc

e its

val

ue d

epen

ds o

n th

e va

lue

of x

). It

is th

e va

riabl

e fo

r whi

ch a

n eq

uatio

n is

sol

ved.

Its

valu

es

mak

e up

the

rang

e of

the

rela

tion

or fu

nctio

n.

Dom

ain

(of a

Rel

atio

n or

Fu

nctio

n)

The

set o

f all

poss

ible

val

ues

of th

e in

depe

nden

t var

iabl

e on

whi

ch a

func

tion

or re

latio

n is

allo

wed

to

oper

ate.

Als

o, th

e fir

st n

umbe

rs in

the

orde

red

pairs

of a

rela

tion;

the

valu

es o

f the

x-c

oord

inat

es in

(x

, y).

Elim

inat

ion

Met

hod

See

line

ar c

ombi

natio

n.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

8

Janu

ary

2013

Equa

tion

A m

athe

mat

ical

sta

tem

ent o

r sen

tenc

e th

at s

ays

one

mat

hem

atic

al e

xpre

ssio

n or

qua

ntity

is e

qual

to

anot

her (

e.g.

, x +

5 =

y –

7).

An

equa

tion

will

alw

ays

cont

ain

an e

qual

sig

n (=

).

Estim

atio

n St

rate

gy

An

appr

oxim

atio

n ba

sed

on a

judg

men

t; m

ay in

clud

e de

term

inin

g ap

prox

imat

e va

lues

, est

ablis

hing

th

e re

ason

able

ness

of a

nsw

ers,

ass

essi

ng th

e am

ount

of e

rror r

esul

ting

from

est

imat

ion,

and

/or

dete

rmin

ing

if an

erro

r is

with

in a

ccep

tabl

e lim

its.

Expo

nent

Th

e po

wer

to w

hich

a n

umbe

r or e

xpre

ssio

n is

rais

ed. W

hen

the

expo

nent

is a

frac

tion,

the

num

ber o

r

expr

essi

on c

an b

e re

writ

ten

with

a ra

dica

l sig

n (e

.g.,

x3/4 =

43 x

). S

ee a

lso

posi

tive

expo

nent

and

nega

tive

expo

nent

.

Expo

nent

ial E

quat

ion

An

equa

tion

with

var

iabl

es in

its

expo

nent

s (e

.g.,

4x = 5

0). I

t can

be

solv

ed b

y ta

king

loga

rithm

s of

bo

th s

ides

.

Expo

nent

ial E

xpre

ssio

n A

n ex

pres

sion

in w

hich

the

varia

ble

occu

rs in

the

expo

nent

(suc

h as

4x ra

ther

than

x4 ).

Ofte

n it

occu

rs

whe

n a

quan

tity

chan

ges

by th

e sa

me

fact

or fo

r eac

h un

it of

tim

e (e

.g.,

“dou

bles

eve

ry y

ear”

or

“dec

reas

es 2

% e

ach

mon

th”).

Expo

nent

ial F

unct

ion

(or

Mod

el)

A fu

nctio

n w

hose

gen

eral

equ

atio

n is

y =

a •

bx whe

re a

and

b a

re c

onst

ants

.

Expo

nent

ial G

row

th/D

ecay

A

situ

atio

n w

here

a q

uant

ity in

crea

ses

or d

ecre

ases

exp

onen

tially

by

the

sam

e fa

ctor

ove

r tim

e; it

is

used

for s

uch

phen

omen

a as

infla

tion,

pop

ulat

ion

grow

th, r

adio

activ

ity o

r dep

reci

atio

n.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

9

Janu

ary

2013

Expr

essi

on

A m

athe

mat

ical

phr

ase

that

incl

udes

ope

ratio

ns, n

umbe

rs, a

nd/o

r var

iabl

es (e

.g.,

2x +

3y

is a

n al

gebr

aic

expr

essi

on, 1

3.4

– 4.

7 is

a n

umer

ic e

xpre

ssio

n). A

n ex

pres

sion

doe

s no

t con

tain

an

equa

l si

gn (=

) or a

ny ty

pe o

f ine

qual

ity s

ign.

Fact

or (n

oun)

Th

e nu

mbe

r or e

xpre

ssio

n th

at is

mul

tiplie

d by

ano

ther

to g

et a

pro

duct

(e.g

., 6

is a

fact

or o

f 30,

and

6x

is a

fact

or o

f 42x

2 ).

Fact

or (v

erb)

To

exp

ress

or w

rite

a nu

mbe

r, m

onom

ial,

or p

olyn

omia

l as

a pr

oduc

t of t

wo

or m

ore

fact

ors.

Fact

or a

Mon

omia

l To

exp

ress

a m

onom

ial a

s th

e pr

oduc

t of t

wo

or m

ore

mon

omia

ls.

Fact

or a

Pol

ynom

ial

To e

xpre

ss a

pol

ynom

ial a

s th

e pr

oduc

t of m

onom

ials

and

/or p

olyn

omia

ls (e

.g.,

fact

orin

g th

e po

lyno

mia

l x2 +

x –

12

resu

lts in

the

prod

uct (

x –

3)(x

+ 4

)).

Freq

uenc

y H

ow o

ften

som

ethi

ng o

ccur

s (i.

e., t

he n

umbe

r of t

imes

an

item

, num

ber,

or e

vent

hap

pens

in a

set

of

dat

a).

Func

tion

A re

latio

n in

whi

ch e

ach

valu

e of

an

inde

pend

ent v

aria

ble

is a

ssoc

iate

d w

ith a

uni

que

valu

e of

a

depe

nden

t var

iabl

e (e

.g.,

one

elem

ent o

f the

dom

ain

is p

aire

d w

ith o

ne a

nd o

nly

one

elem

ent o

f the

ra

nge)

. It i

s a

map

ping

whi

ch in

volv

es e

ither

a o

ne-to

-one

cor

resp

onde

nce

or a

man

y-to

-one

co

rresp

onde

nce,

but

not

a o

ne-to

-man

y co

rresp

onde

nce.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

10

Ja

nuar

y 20

13

Fund

amen

tal C

ount

ing

Prin

cipl

e A

way

to c

alcu

late

all

of th

e po

ssib

le c

ombi

natio

ns o

f a g

iven

num

ber o

f eve

nts.

It s

tate

s th

at if

ther

e ar

e x

diffe

rent

way

s of

doi

ng o

ne th

ing

and

y di

ffere

nt w

ays

of d

oing

ano

ther

thin

g, th

en th

ere

are

xy d

iffer

ent w

ays

of d

oing

bot

h th

ings

. It u

ses

the

mul

tiplic

atio

n ru

le.

Geo

met

ric S

eque

nce

A

n or

dere

d lis

t of n

umbe

rs th

at h

as th

e sa

me

ratio

bet

wee

n co

nsec

utiv

e te

rms

(e.g

., 1,

7, 4

9, 3

43, …

is

a g

eom

etric

seq

uenc

e th

at h

as a

ratio

of 7

/1 b

etw

een

cons

ecut

ive

term

s; e

ach

term

afte

r the

firs

t te

rm c

an b

e fo

und

by m

ultip

lyin

g th

e pr

evio

us te

rm b

y a

cons

tant

, in

this

cas

e th

e nu

mbe

r 7 o

r 7/1

).

Gre

ates

t Com

mon

Fac

tor

(GC

F)

The

larg

est f

acto

r tha

t tw

o or

mor

e nu

mbe

rs o

r alg

ebra

ic te

rms

have

in c

omm

on. I

n so

me

case

s th

e G

CF

may

be

1 or

one

of t

he a

ctua

l num

bers

(e.g

., th

e G

CF

of 1

8x3 a

nd 2

4x5 is

6x3 ).

Imag

inar

y N

umbe

r Th

e sq

uare

root

of a

neg

ativ

e nu

mbe

r, or

the

oppo

site

of t

he s

quar

e ro

ot o

f a n

egat

ive

num

ber.

It is

writ

ten

in th

e fo

rm b

i, w

here

b is

a re

al n

umbe

r and

i is

the

imag

inar

y ro

ot (i

.e.,

i =

1−

or i

2 = – 1)

.

Inde

pend

ent E

vent

(s)

Two

or m

ore

even

ts in

whi

ch th

e ou

tcom

e of

one

eve

nt d

oes

not a

ffect

the

outc

ome

of th

e ot

her

even

t(s) (

e.g.

, tos

sing

a c

oin

and

rollin

g a

num

ber c

ube

are

inde

pend

ent e

vent

s). T

he p

roba

bilit

y of

tw

o in

depe

nden

t eve

nts

(A a

nd B

) occ

urrin

g is

writ

ten

P(A

and

B) o

r P(A

B

) and

equ

als

P(A

) • P

(B)

(i.e.

, the

pro

duct

of t

he p

roba

bilit

ies

of th

e tw

o in

divi

dual

eve

nts)

.

Inde

pend

ent V

aria

ble

The

inpu

t num

ber o

r var

iabl

e in

a re

latio

n or

func

tion

who

se v

alue

is s

ubje

ct to

cho

ice.

It is

not

de

pend

ent u

pon

any

othe

r val

ues.

It is

usu

ally

the

x-va

lue

or th

e x

in f(

x). I

t is

grap

hed

on th

e x-

axis

. Its

val

ues

mak

e up

the

dom

ain

of th

e re

latio

n or

func

tion.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

11

Ja

nuar

y 20

13

Ineq

ualit

y A

mat

hem

atic

al s

ente

nce

that

con

tain

s an

ineq

ualit

y sy

mbo

l (i.e

., >,

<, ≥

, ≤, o

r ≠).

It co

mpa

res

two

quan

titie

s. T

he s

ymbo

l > m

eans

gre

ater

than

, the

sym

bol <

mea

ns le

ss th

an, t

he s

ymbo

l ≥ m

eans

gr

eate

r tha

n or

equ

al to

, the

sym

bol ≤

mea

ns le

ss th

an o

r equ

al to

, and

the

sym

bol ≠

mea

ns n

ot

equa

l to.

Inte

ger

A n

atur

al n

umbe

r, th

e ad

ditiv

e in

vers

e of

a n

atur

al n

umbe

r, or

zer

o. A

ny n

umbe

r fro

m th

e se

t of

num

bers

repr

esen

ted

by {…

, – 3, – 2,

– 1, 0

, 1, 2

, 3, …

}.

Inte

rqua

rtile

Ran

ge (o

f Dat

a)

The

diffe

renc

e be

twee

n th

e fir

st (l

ower

) and

third

(upp

er) q

uarti

le. I

t rep

rese

nts

the

spre

ad o

f the

m

iddl

e 50

% o

f a s

et o

f dat

a.

Inve

rse

(of a

Rel

atio

n)

A re

latio

n in

whi

ch th

e co

ordi

nate

s in

eac

h or

dere

d pa

ir ar

e sw

itche

d fro

m a

giv

en re

latio

n. T

he p

oint

(x

, y) b

ecom

es (y

, x),

so (3

, 8) w

ould

bec

ome

(8, 3

).

Irrat

iona

l Num

ber

A re

al n

umbe

r tha

t can

not b

e w

ritte

n as

a s

impl

e fra

ctio

n (i.

e., t

he ra

tio o

f tw

o in

tege

rs).

It is

a n

on-

term

inat

ing

(infin

ite) a

nd n

on-re

peat

ing

deci

mal

. The

squ

are

root

of a

ny p

rime

num

ber i

s irr

atio

nal,

as

are π

and

e.

Leas

t (or

Low

est)

Com

mon

M

ultip

le (L

CM

) Th

e sm

alle

st n

umbe

r or e

xpre

ssio

n th

at is

a c

omm

on m

ultip

le o

f tw

o or

mor

e nu

mbe

rs o

r alg

ebra

ic

term

s, o

ther

than

zer

o.

Like

Ter

ms

Mon

omia

ls th

at c

onta

in th

e sa

me

varia

bles

and

cor

resp

ondi

ng p

ower

s an

d/or

root

s. O

nly

the

coef

ficie

nts

can

be d

iffer

ent (

e.g.

, 4x3 a

nd 1

2x3 ).

Like

term

s ca

n be

add

ed o

r sub

tract

ed.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

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ia D

epar

tmen

t of E

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tion

Page

12

Ja

nuar

y 20

13

Line

Gra

ph

A g

raph

that

use

s a

line

or li

ne s

egm

ents

to c

onne

ct d

ata

poin

ts, p

lotte

d on

a c

oord

inat

e pl

ane,

us

ually

to s

how

tren

ds o

r cha

nges

in d

ata

over

tim

e. M

ore

broa

dly,

a g

raph

to re

pres

ent t

he

rela

tions

hip

betw

een

two

cont

inuo

us v

aria

bles

.

Line

or C

urve

of B

est F

it (fo

r a

Scat

ter P

lot)

A li

ne o

r cur

ve d

raw

n on

a s

catte

r plo

t to

best

est

imat

e th

e re

latio

nshi

p be

twee

n tw

o se

ts o

f dat

a. It

de

scrib

es th

e tre

nd o

f the

dat

a. D

iffer

ent m

easu

res

are

poss

ible

to d

escr

ibe

the

best

fit.

The

mos

t co

mm

on is

a li

ne o

r cur

ve th

at m

inim

izes

the

sum

of t

he s

quar

es o

f the

erro

rs (v

ertic

al d

ista

nces

) fro

m

the

data

poi

nts

to th

e lin

e. T

he li

ne o

f bes

t fit

is a

sub

set o

f the

cur

ve o

f bes

t fit.

Exa

mpl

es o

f a li

ne o

f be

st fi

t and

a c

urve

of b

est f

it:

Line

ar C

ombi

natio

n A

met

hod

by w

hich

a s

yste

m o

f lin

ear e

quat

ions

can

be

solv

ed. I

t use

s ad

ditio

n or

sub

tract

ion

in

com

bina

tion

with

mul

tiplic

atio

n or

div

isio

n to

elim

inat

e on

e of

the

varia

bles

in o

rder

to s

olve

for t

he

othe

r var

iabl

e.

Line

ar E

quat

ion

An

equa

tion

for w

hich

the

grap

h is

a s

traig

ht li

ne (i

.e.,

a po

lyno

mia

l equ

atio

n of

the

first

deg

ree

of th

e fo

rm A

x +

By

= C

, whe

re A

, B, a

nd C

are

real

num

bers

and

whe

re A

and

B a

re n

ot b

oth

zero

; an

equa

tion

in w

hich

the

varia

bles

are

not

mul

tiplie

d by

one

ano

ther

or r

aise

d to

any

pow

er o

ther

than

1).

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

13

Ja

nuar

y 20

13

Line

ar F

unct

ion

A fu

nctio

n fo

r whi

ch th

e gr

aph

is a

non

-ver

tical

stra

ight

line

. It i

s a

first

deg

ree

poly

nom

ial o

f the

co

mm

on fo

rm f(

x) =

mx

+ b,

whe

re m

and

b a

re c

onst

ants

and

x is

a re

al v

aria

ble.

The

con

stan

t m is

ca

lled

the

slop

e an

d b

is c

alle

d th

e y-

inte

rcep

t. It

has

a co

nsta

nt ra

te o

f cha

nge.

Line

ar In

equa

lity

Th

e re

latio

n of

two

expr

essi

ons

usin

g th

e sy

mbo

ls <

, >, ≤

, ≥, o

r ≠ a

nd w

hose

bou

ndar

y is

a s

traig

ht

line.

The

line

div

ides

the

coor

dina

te p

lane

into

two

parts

. If t

he in

equa

lity

is e

ither

≤ o

r ≥, t

hen

the

boun

dary

is s

olid

. If t

he in

equa

lity

is e

ither

< o

r >, t

hen

the

boun

dary

is d

ashe

d. If

the

ineq

ualit

y is

≠,

then

the

solu

tion

cont

ains

eve

ryth

ing

exce

pt fo

r the

bou

ndar

y.

Loga

rithm

Th

e ex

pone

nt re

quire

d to

pro

duce

a g

iven

num

ber (

e.g.

, sin

ce 2

rais

ed to

a p

ower

of 5

is 3

2, th

e lo

garit

hm b

ase

2 of

32

is 5

; thi

s is

writ

ten

as lo

g 2 3

2 =

5). T

wo

frequ

ently

use

d ba

ses

are

10 (c

omm

on

loga

rithm

) and

e (n

atur

al lo

garit

hm).

Whe

n a

loga

rithm

is w

ritte

n w

ithou

t a b

ase,

it is

und

erst

ood

to b

e ba

se 1

0.

Loga

rithm

ic E

quat

ion

An

equa

tion

whi

ch c

onta

ins

a lo

garit

hm o

f a v

aria

ble

or n

umbe

r. S

omet

imes

it is

sol

ved

by re

writ

ing

the

equa

tion

in e

xpon

entia

l for

m a

nd s

olvi

ng fo

r the

var

iabl

e (e

.g.,

log 2

32

= 5

is th

e sa

me

as 2

5 = 3

2).

It is

an

inve

rse

func

tion

of th

e ex

pone

ntia

l fun

ctio

n.

Map

ping

Th

e m

atch

ing

or p

airin

g of

one

set

of n

umbe

rs to

ano

ther

by

use

of a

rule

. A n

umbe

r in

the

dom

ain

is

mat

ched

or p

aire

d w

ith a

num

ber i

n th

e ra

nge

(or a

rela

tion

or fu

nctio

n). I

t may

be

a on

e-to

-one

co

rresp

onde

nce,

a o

ne-to

-man

y co

rresp

onde

nce,

or a

man

y-to

-one

cor

resp

onde

nce.

Max

imum

Val

ue (o

f a G

raph

) Th

e va

lue

of th

e de

pend

ent v

aria

ble

for t

he h

ighe

st p

oint

on

the

grap

h of

a c

urve

.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

14

Ja

nuar

y 20

13

Mea

n A

mea

sure

of c

entra

l ten

denc

y th

at is

cal

cula

ted

by a

ddin

g al

l the

val

ues

of a

set

of d

ata

and

divi

ding

th

at s

um b

y th

e to

tal n

umbe

r of v

alue

s. U

nlik

e m

edia

n, th

e m

ean

is s

ensi

tive

to o

utlie

r val

ues.

It is

al

so c

alle

d “a

rithm

etic

mea

n” o

r “av

erag

e”.

Mea

sure

of C

entr

al

Tend

ency

A

mea

sure

of l

ocat

ion

of th

e m

iddl

e (c

ente

r) of

a d

istri

butio

n of

a s

et o

f dat

a (i.

e., h

ow d

ata

clus

ters

). Th

e th

ree

mos

t com

mon

mea

sure

s of

cen

tral t

ende

ncy

are

mea

n, m

edia

n, a

nd m

ode.

Mea

sure

of D

ispe

rsio

n A

mea

sure

of t

he w

ay in

whi

ch th

e di

strib

utio

n of

a s

et o

f dat

a is

spr

ead

out.

In g

ener

al th

e m

ore

spre

ad o

ut a

dis

tribu

tion

is, t

he la

rger

the

mea

sure

of d

ispe

rsio

n. R

ange

and

inte

rqua

rtile

rang

e ar

e tw

o m

easu

res

of d

ispe

rsio

n.

Med

ian

A m

easu

re o

f cen

tral t

ende

ncy

that

is th

e m

iddl

e va

lue

in a

n or

dere

d se

t of d

ata

or th

e av

erag

e of

the

two

mid

dle

valu

es w

hen

the

set h

as tw

o m

iddl

e va

lues

(occ

urs

whe

n th

e se

t of d

ata

has

an e

ven

num

ber o

f dat

a po

ints

). It

is th

e va

lue

halfw

ay th

roug

h th

e or

dere

d se

t of d

ata,

bel

ow a

nd a

bove

whi

ch

ther

e is

an

equa

l num

ber o

f dat

a va

lues

. It i

s ge

nera

lly a

goo

d de

scrip

tive

mea

sure

for s

kew

ed d

ata

or

data

with

out

liers

.

Min

imum

Val

ue (o

f a G

raph

) Th

e va

lue

of th

e de

pend

ent v

aria

ble

for t

he lo

wes

t poi

nt o

n th

e gr

aph

of a

cur

ve.

Mod

e A

mea

sure

of c

entra

l ten

denc

y th

at is

the

valu

e or

val

ues

that

occ

ur(s

) mos

t ofte

n in

a s

et o

f dat

a. A

se

t of d

ata

can

have

one

mod

e, m

ore

than

one

mod

e, o

r no

mod

e.

Mon

omia

l A

pol

ynom

ial w

ith o

nly

one

term

; it c

onta

ins

no a

dditi

on o

r sub

tract

ion.

It c

an b

e a

num

ber,

a va

riabl

e,

or a

pro

duct

of n

umbe

rs a

nd/o

r mor

e va

riabl

es (e

.g.,

2 • 5

or x

3 y4 or

24 3

rπ).

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

15

Ja

nuar

y 20

13

Mul

tiplic

ativ

e In

vers

e Th

e re

cipr

ocal

of a

num

ber (

i.e.,

for a

ny n

on-z

ero

num

ber a

, the

mul

tiplic

ativ

e in

vers

e is

1 a; f

or a

ny

ratio

nal n

umbe

r b c

, whe

re b

≠ 0

and

c ≠

0, t

he m

ultip

licat

ive

inve

rse

is c b

). A

ny n

umbe

r and

its

mul

tiplic

ativ

e in

vers

e ha

ve a

pro

duct

of 1

(e.g

., 1 4

is th

e m

ultip

licat

ive

inve

rse

of 4

sin

ce 4

• 1 4

= 1

;

likew

ise,

the

mul

tiplic

ativ

e in

vers

e of

1 4 is

4 s

ince

1 4 •

4 =

1).

Mut

ually

Exc

lusi

ve E

vent

s Tw

o ev

ents

that

can

not o

ccur

at t

he s

ame

time

(i.e.

, eve

nts

that

hav

e no

out

com

es in

com

mon

). If

two

even

ts A

and

B a

re m

utua

lly e

xclu

sive

, the

n th

e pr

obab

ility

of A

or B

occ

urrin

g is

the

sum

of t

heir

indi

vidu

al p

roba

bilit

ies:

P(A

U B

) = P

(A) +

P(B

). A

lso

defin

ed a

s w

hen

the

inte

rsec

tion

of tw

o se

ts is

em

pty,

writ

ten

as A

∩ B

= Ø

.

Nat

ural

Log

arith

m

A lo

garit

hm w

ith b

ase

e. It

is w

ritte

n ln

x. T

he n

atur

al lo

garit

hm is

the

pow

er o

f e n

eces

sary

to e

qual

a

give

n nu

mbe

r (i.e

., ln

x =

y is

equ

ival

ent t

o e y

= x

). Th

e co

nsta

nt e

is a

n irr

atio

nal n

umbe

r who

se v

alue

is

app

roxi

mat

ely

2.71

828…

.

Nat

ural

Num

ber

A c

ount

ing

num

ber.

A n

umbe

r rep

rese

ntin

g a

posi

tive,

who

le a

mou

nt. A

ny n

umbe

r fro

m th

e se

t of

num

bers

repr

esen

ted

by {1

, 2, 3

, …}.

Som

etim

es, i

t is

refe

rred

to a

s a

“pos

itive

inte

ger”.

Neg

ativ

e Ex

pone

nt

An

expo

nent

that

indi

cate

s a

reci

proc

al th

at h

as to

be

take

n be

fore

the

expo

nent

can

be

appl

ied

(e.g

., 2

215

5−=

or

1x

xa

a−=

). It

is u

sed

in s

cien

tific

not

atio

n fo

r num

bers

bet

wee

n – 1

and

1.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

16

Ja

nuar

y 20

13

Num

ber L

ine

A g

radu

ated

stra

ight

line

that

repr

esen

ts th

e se

t of a

ll re

al n

umbe

rs in

ord

er. T

ypic

ally

, it i

s m

arke

d sh

owin

g in

tege

r val

ues.

Odd

s A

com

paris

on, i

n ra

tio fo

rm (a

s a

fract

ion

or w

ith a

col

on),

of o

utco

mes

. “O

dds

in fa

vor”

(or s

impl

y “o

dds”

) is

the

ratio

of f

avor

able

out

com

es to

unf

avor

able

out

com

es (e

.g.,

the

odds

in fa

vor o

f pic

king

a

red

hat w

hen

ther

e ar

e 3

red

hats

and

5 n

on-re

d ha

ts is

3:5

). “O

dds

agai

nst”

is th

e ra

tio o

f unf

avor

able

ou

tcom

es to

favo

rabl

e ou

tcom

es (e

.g.,

the

odds

aga

inst

pic

king

a re

d ha

t whe

n th

ere

are

3 re

d ha

ts

and

5 no

n-re

d ha

ts is

5:3

).

Ord

er o

f Ope

ratio

ns

Rul

es d

escr

ibin

g w

hat o

rder

to u

se in

eva

luat

ing

expr

essi

ons:

(1

) Per

form

ope

ratio

ns in

gro

upin

g sy

mbo

ls (p

aren

thes

es a

nd b

rack

ets)

, (2

) Eva

luat

e ex

pone

ntia

l exp

ress

ions

and

radi

cal e

xpre

ssio

ns fr

om le

ft to

righ

t, (3

) Mul

tiply

or d

ivid

e fro

m le

ft to

righ

t, (4

) Add

or s

ubtra

ct fr

om le

ft to

righ

t.

Ord

ered

Pai

r A

pai

r of n

umbe

rs u

sed

to lo

cate

a p

oint

on

a co

ordi

nate

pla

ne, o

r the

sol

utio

n of

an

equa

tion

in tw

o va

riabl

es. T

he fi

rst n

umbe

r tel

ls h

ow fa

r to

mov

e ho

rizon

tally

, and

the

seco

nd n

umbe

r tel

ls h

ow fa

r to

mov

e ve

rtica

lly; w

ritte

n in

the

form

(x-c

oord

inat

e, y

-coo

rdin

ate)

. Ord

er m

atte

rs: t

he p

oint

(x, y

) is

not

the

sam

e as

(y, x

).

Orig

in

The

poin

t (0,

0) o

n a

coor

dina

te p

lane

. It i

s th

e po

int o

f int

erse

ctio

n fo

r the

x-a

xis

and

the

y-ax

is.

Out

lier

A v

alue

that

is m

uch

grea

ter o

r muc

h le

ss th

an th

e re

st o

f the

dat

a. It

is d

iffer

ent i

n so

me

way

from

the

gene

ral p

atte

rn o

f dat

a. It

dire

ctly

sta

nds

out f

rom

the

rest

of t

he d

ata.

Som

etim

es it

is re

ferre

d to

as

any

data

poi

nt m

ore

than

1.5

inte

rqua

rtile

rang

es g

reat

er th

an th

e up

per (

third

) qua

rtile

or l

ess

than

th

e lo

wer

(firs

t) qu

artil

e.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

17

Ja

nuar

y 20

13

Patte

rn (o

r Seq

uenc

e)

A s

et o

f num

bers

arra

nged

in o

rder

(or i

n a

sequ

ence

). Th

e nu

mbe

rs a

nd th

eir a

rrang

emen

t are

de

term

ined

by

a ru

le, i

nclu

ding

repe

titio

n an

d gr

owth

/dec

ay ru

les.

See

arit

hmet

ic s

eque

nce

and

geom

etric

seq

uenc

e.

Perf

ect S

quar

e A

num

ber w

hose

squ

are

root

is a

who

le n

umbe

r (e.

g., 2

5 is

a p

erfe

ct s

quar

e si

nce

25 =

5).

A

perfe

ct s

quar

e ca

n be

foun

d by

rais

ing

a w

hole

num

ber t

o th

e se

cond

pow

er (e

.g.,

52 = 2

5).

Perm

utat

ion

An

orde

red

arra

ngem

ent o

f obj

ects

from

a g

iven

set

in w

hich

the

orde

r of t

he o

bjec

ts is

sig

nific

ant

(e.g

., tw

o-le

tter p

erm

utat

ions

of t

he th

ree

lette

rs X

, Y, a

nd Z

wou

ld b

e XY

, YX,

XZ,

ZX,

YZ,

and

ZY)

. A

perm

utat

ion

is s

imila

r to,

but

not

the

sam

e as

, a c

ombi

natio

n.

Poin

t-Slo

pe F

orm

(of a

Li

near

Equ

atio

n)

An

equa

tion

of a

stra

ight

, non

-ver

tical

line

writ

ten

in th

e fo

rm y

– y

1 = m

(x –

x1)

, whe

re m

is th

e sl

ope

of th

e lin

e an

d (x

1, y 1

) is

a gi

ven

poin

t on

the

line.

Poly

nom

ial

An

alge

brai

c ex

pres

sion

that

is a

mon

omia

l or t

he s

um o

r diff

eren

ce o

f tw

o or

mor

e m

onom

ials

(e.g

., 6a

or 5

a2 + 3

a –

13 w

here

the

expo

nent

s ar

e na

tura

l num

bers

).

Poly

nom

ial F

unct

ion

A fu

nctio

n of

the

form

f(x)

= a

nxn +

an–

1xn–

1 + …

+ a

1x +

a0,

whe

re a

n ≠ 0

and

nat

ural

num

ber n

is th

e de

gree

of t

he p

olyn

omia

l.

Posi

tive

Expo

nent

In

dica

tes

how

man

y tim

es a

bas

e nu

mbe

r is

mul

tiplie

d by

itse

lf. In

the

expr

essi

on x

n , n is

the

posi

tive

expo

nent

, and

x is

the

base

num

ber (

e.g.

, 23 =

2 •

2 • 2

).

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

18

Ja

nuar

y 20

13

Pow

er

The

valu

e of

the

expo

nent

in a

term

. The

exp

ress

ion

an is re

ad “a

to th

e po

wer

of n

.” To

rais

e a

num

ber,

a, to

the

pow

er o

f ano

ther

who

le n

umbe

r, n,

is to

mul

tiply

a b

y its

elf n

tim

es (e

.g.,

the

num

ber

43 is re

ad “f

our t

o th

e th

ird p

ower

” and

repr

esen

ts 4

• 4

• 4).

Pow

er o

f a P

ower

A

n ex

pres

sion

of t

he fo

rm (a

m)n . I

t can

be

foun

d by

mul

tiply

ing

the

expo

nent

s (e

.g.,

(23 )4 =

23•

4 = 2

12 =

4,0

96).

Pow

ers

of P

rodu

cts

An

expr

essi

on o

f the

form

am •

an . It c

an b

e fo

und

by a

ddin

g th

e ex

pone

nts

whe

n m

ultip

lyin

g po

wer

s th

at h

ave

the

sam

e ba

se (e

.g.,

23 • 24

= 23+

4 = 2

7 = 1

28).

Prim

e N

umbe

r A

ny n

atur

al n

umbe

r with

exa

ctly

two

fact

ors,

1 a

nd it

self

(e.g

., 3

is a

prim

e nu

mbe

r sin

ce it

has

onl

y tw

o fa

ctor

s: 1

and

3).

[Not

e: S

ince

1 h

as o

nly

one

fact

or, i

tsel

f, it

is n

ot a

prim

e nu

mbe

r.] A

prim

e nu

mbe

r is

not a

com

posi

te n

umbe

r.

Prob

abili

ty

A n

umbe

r fro

m 0

to 1

(or 0

% to

100

%) t

hat i

ndic

ates

how

like

ly a

n ev

ent i

s to

hap

pen.

A v

ery

unlik

ely

even

t has

a p

roba

bilit

y ne

ar 0

(or 0

%) w

hile

a v

ery

likel

y ev

ent h

as a

pro

babi

lity

near

1 (o

r 100

%).

It is

w

ritte

n as

a ra

tio (f

ract

ion,

dec

imal

, or e

quiv

alen

t per

cent

). Th

e nu

mbe

r of w

ays

an e

vent

cou

ld

happ

en (f

avor

able

out

com

es) i

s pl

aced

ove

r the

tota

l num

ber o

f eve

nts

(tota

l pos

sibl

e ou

tcom

es) t

hat

coul

d ha

ppen

. A p

roba

bilit

y of

0 m

eans

it is

impo

ssib

le, a

nd a

pro

babi

lity

of 1

mea

ns it

is c

erta

in.

Prob

abili

ty o

f a C

ompo

und

(or C

ombi

ned)

Eve

nt

Ther

e ar

e tw

o ty

pes:

1.

Th

e un

ion

of tw

o ev

ents

A a

nd B

, whi

ch is

the

prob

abilit

y of

A o

r B o

ccur

ring.

Thi

s is

re

pres

ente

d as

P(A

U B

) = P

(A) +

P(B

) – P

(A) •

P(B

). 2.

Th

e in

ters

ectio

n of

two

even

ts A

and

B, w

hich

is th

e pr

obab

ility

of A

and

B o

ccur

ring.

Thi

s is

re

pres

ente

d as

P(A

∩ B

) = P

(A) •

P(B

).

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

19

Ja

nuar

y 20

13

Qua

dran

ts

The

four

regi

ons

of a

coo

rdin

ate

plan

e th

at a

re s

epar

ated

by

the

x-ax

is a

nd th

e y-

axis

, as

show

n be

low

.

(1

) The

firs

t qua

dran

t (Q

uadr

ant I

) con

tain

s al

l the

poi

nts

with

pos

itive

x a

nd p

ositi

ve y

coo

rdin

ates

(e

.g.,

(3, 4

)).

(2) T

he s

econ

d qu

adra

nt (Q

uadr

ant I

I) co

ntai

ns a

ll th

e po

ints

with

neg

ativ

e x

and

posi

tive

y co

ordi

nate

s (e

.g.,

(– 3, 4

)).

(3) T

he th

ird q

uadr

ant (

Qua

dran

t III)

con

tain

s al

l the

poi

nts

with

neg

ativ

e x

and

nega

tive

y co

ordi

nate

s (e

.g.,

(– 3, – 4)

). (4

) The

four

th q

uadr

ant (

Qua

dran

t IV

) con

tain

s al

l the

poi

nts

with

pos

itive

x a

nd n

egat

ive

y co

ordi

nate

s (e

.g.,

(3, – 4)

).

Qua

drat

ic E

quat

ion

An

equa

tion

that

can

be

writ

ten

in th

e st

anda

rd fo

rm a

x2 + b

x +

c =

0, w

here

a, b

, and

c a

re re

al

num

bers

and

a d

oes

not e

qual

zer

o. T

he h

ighe

st p

ower

of t

he v

aria

ble

is 2

. It h

as, a

t mos

t, tw

o so

lutio

ns. T

he g

raph

is a

par

abol

a.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

20

Ja

nuar

y 20

13

Qua

drat

ic F

orm

ula

The

solu

tions

or r

oots

of a

qua

drat

ic e

quat

ion

in th

e fo

rm a

x2 + b

x +

c =

0, w

here

a ≠

0, a

re g

iven

by

the

form

ula

24

2b

bac

xa

−±

−=

.

Qua

drat

ic F

unct

ion

A fu

nctio

n th

at c

an b

e ex

pres

sed

in th

e fo

rm f(

x) =

ax2 +

bx

+ c,

whe

re a

≠ 0

and

the

high

est p

ower

of

the

varia

ble

is 2

. The

gra

ph is

a p

arab

ola.

Qua

rtile

O

ne o

f thr

ee v

alue

s th

at d

ivid

es a

set

of d

ata

into

four

equ

al p

arts

: 1.

M

edia

n di

vide

s a

set o

f dat

a in

to tw

o eq

ual p

arts

. 2.

Lo

wer

qua

rtile

(25th

per

cent

ile) i

s th

e m

edia

n of

the

low

er h

alf o

f the

dat

a.

3.

Upp

er q

uarti

le (7

5th p

erce

ntile

) is

the

med

ian

of th

e up

per h

alf o

f the

dat

a.

Rad

ical

Exp

ress

ion

An

expr

essi

on c

onta

inin

g a

radi

cal s

ymbo

l (n

a).

The

expr

essi

on o

r num

ber i

nsid

e th

e ra

dica

l (a)

is

calle

d th

e ra

dica

nd, a

nd th

e nu

mbe

r app

earin

g ab

ove

the

radi

cal (

n) is

the

degr

ee. T

he d

egre

e is

al

way

s a

posi

tive

inte

ger.

Whe

n a

radi

cal i

s w

ritte

n w

ithou

t a d

egre

e, it

is u

nder

stoo

d to

be

a de

gree

of

2 a

nd is

read

as

“the

squa

re ro

ot o

f a.”

Whe

n th

e de

gree

is 3

, it i

s re

ad a

s “th

e cu

be ro

ot o

f a.”

For

any

othe

r deg

ree,

the

expr

essi

on n

a is

read

as

“the

nth

root

of a

.” W

hen

the

degr

ee is

an

even

nu

mbe

r, th

e ra

dica

l exp

ress

ion

is a

ssum

ed to

be

the

prin

cipa

l (po

sitiv

e) ro

ot (e

.g.,

alth

ough

(– 7)2 =

49,

49

= 7

).

Ran

ge (o

f a R

elat

ion

or

Func

tion)

Th

e se

t of a

ll po

ssib

le v

alue

s fo

r the

out

put (

depe

nden

t var

iabl

e) o

f a fu

nctio

n or

rela

tion;

the

set o

f se

cond

num

bers

in th

e or

dere

d pa

irs o

f a fu

nctio

n or

rela

tion;

the

valu

es o

f the

y-c

oord

inat

es in

(x, y

).

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

21

Ja

nuar

y 20

13

Ran

ge (o

f Dat

a)

In s

tatis

tics,

a m

easu

re o

f dis

pers

ion

that

is th

e di

ffere

nce

betw

een

the

grea

test

val

ue (m

axim

um

valu

e) a

nd th

e le

ast v

alue

(min

imum

val

ue) i

n a

set o

f dat

a.

Rat

e A

ratio

that

com

pare

s tw

o qu

antit

ies

havi

ng d

iffer

ent u

nits

(e.g

., 16

8 m

iles

3.5

hour

s o

r 12

2.5

calo

ries

5 cu

ps).

Whe

n

the

rate

is s

impl

ified

so

that

the

seco

nd (i

ndep

ende

nt) q

uant

ity is

1, i

t is

calle

d a

unit

rate

(e.g

.,

48 m

iles

per h

our o

r 24.

5 ca

lorie

s pe

r cup

).

Rat

e (o

f Cha

nge)

Th

e am

ount

a q

uant

ity c

hang

es o

ver t

ime

(e.g

., 3.

2 cm

per

yea

r). A

lso

the

amou

nt a

func

tion’

s ou

tput

ch

ange

s (in

crea

ses

or d

ecre

ases

) for

eac

h un

it of

cha

nge

in th

e in

put.

See

slo

pe.

Rat

e (o

f Int

eres

t) Th

e pe

rcen

t by

whi

ch a

mon

etar

y ac

coun

t acc

rues

inte

rest

. It i

s m

ost c

omm

on fo

r the

rate

of i

nter

est

to b

e m

easu

red

on a

n an

nual

bas

is (e

.g.,

4.5%

per

yea

r), e

ven

if th

e in

tere

st is

com

poun

ded

perio

dica

lly (i

.e.,

mor

e fre

quen

tly th

an o

nce

per y

ear).

Rat

io

A c

ompa

rison

of t

wo

num

bers

, qua

ntiti

es o

r exp

ress

ions

by

divi

sion

. It i

s of

ten

writ

ten

as a

frac

tion,

but n

ot a

lway

s (e

.g.,

2 3, 2

:3, 2

to 3

, 2 ÷

3 a

re a

ll th

e sa

me

ratio

s).

Rat

iona

l Exp

ress

ion

An

expr

essi

on th

at c

an b

e w

ritte

n as

a p

olyn

omia

l div

ided

by

a po

lyno

mia

l, de

fined

onl

y w

hen

the

latte

r is

not e

qual

to z

ero.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

22

Ja

nuar

y 20

13

Rat

iona

l Num

ber

Any

num

ber t

hat c

an b

e w

ritte

n in

the

form

a b w

here

a is

any

inte

ger a

nd b

is a

ny in

tege

r exc

ept z

ero.

All

repe

atin

g de

cim

al a

nd te

rmin

atin

g de

cim

al n

umbe

rs a

re ra

tiona

l num

bers

.

Rea

l Num

ber

The

com

bine

d se

t of r

atio

nal a

nd ir

ratio

nal n

umbe

rs. A

ll nu

mbe

rs o

n th

e nu

mbe

r lin

e. N

ot a

n im

agin

ary

num

ber.

Reg

ress

ion

Cur

ve

The

line

or c

urve

of b

est f

it th

at re

pres

ents

the

leas

t dev

iatio

n fro

m th

e po

ints

in a

sca

tter p

lot o

f dat

a.

Mos

t com

mon

ly it

is li

near

and

use

s a

“leas

t squ

ares

” met

hod.

Exa

mpl

es o

f reg

ress

ion

curv

es:

Rel

atio

n A

set

of p

airs

of v

alue

s (e

.g.,

{(1, 2

), (2

, 3) (

3, 2

)}). T

he fi

rst v

alue

in e

ach

pair

is th

e in

put

(inde

pend

ent v

alue

), an

d th

e se

cond

val

ue in

the

pair

is th

e ou

tput

(dep

ende

nt v

alue

). In

a re

latio

n,

neith

er th

e in

put v

alue

s no

r the

out

put v

alue

s ne

ed to

be

uniq

ue.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

23

Ja

nuar

y 20

13

Rep

eatin

g D

ecim

al

A d

ecim

al w

ith o

ne o

r mor

e di

gits

that

repe

ats

endl

essl

y (e

.g.,

0.66

6…, 0

.727

272…

, 0.0

8333

…).

To

indi

cate

the

repe

titio

n, a

bar

may

be

writ

ten

abov

e th

e re

peat

ed d

igits

(e.g

., 0.

666…

= 0

.6,

0.72

7272

… =

0.7

2, 0

.083

33…

= 0

.083

). A

dec

imal

that

has

eith

er a

0 o

r a 9

repe

atin

g en

dles

sly

is

equi

vale

nt to

a te

rmin

atin

g de

cim

al (e

.g.,

0.37

5000

… =

0.3

75, 0

.199

9… =

0.2

). A

ll re

peat

ing

deci

mal

s ar

e ra

tiona

l num

bers

.

Ris

e Th

e ve

rtica

l (up

and

dow

n) c

hang

e or

diff

eren

ce b

etw

een

any

two

poin

ts o

n a

line

on a

coo

rdin

ate

plan

e (i.

e., f

or p

oint

s (x

1, y 1

) and

(x2,

y 2),

the

rise

is y

2 – y

1). S

ee s

lope

.

Run

Th

e ho

rizon

tal (

left

and

right

) cha

nge

or d

iffer

ence

bet

wee

n an

y tw

o po

ints

on

a lin

e on

a c

oord

inat

e pl

ane

(i.e.

, for

poi

nts

(x1,

y 1) a

nd (x

2, y 2

), th

e ru

n is

x2 –

x1)

. See

slo

pe.

Scat

ter P

lot

A g

raph

that

sho

ws

the

“gen

eral

” rel

atio

nshi

p be

twee

n tw

o se

ts o

f dat

a. F

or e

ach

poin

t tha

t is

bein

g pl

otte

d th

ere

are

two

sepa

rate

pie

ces

of d

ata.

It s

how

s ho

w o

ne v

aria

ble

is a

ffect

ed b

y an

othe

r. E

xam

ple

of a

sca

tter p

lot:

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

24

Ja

nuar

y 20

13

Sim

ple

Even

t W

hen

an e

vent

con

sist

s of

a s

ingl

e ou

tcom

e (e

.g.,

rollin

g a

num

ber c

ube)

.

Sim

ples

t For

m (o

f an

Expr

essi

on)

Whe

n al

l lik

e te

rms

are

com

bine

d (e

.g.,

8x +

2(6

x –

22) b

ecom

es 2

0x –

44

whe

n in

sim

ples

t for

m).

The

form

whi

ch n

o lo

nger

con

tain

s an

y lik

e te

rms,

par

enth

eses

, or r

educ

ible

frac

tions

.

Sim

plify

To

writ

e an

exp

ress

ion

in it

s si

mpl

est f

orm

(i.e

., re

mov

e an

y un

nece

ssar

y te

rms,

usu

ally

by

com

bini

ng

seve

ral o

r man

y te

rms

into

few

er te

rms

or b

y ca

ncel

ling

term

s).

Slop

e (o

f a L

ine)

A

rate

of c

hang

e. T

he m

easu

rem

ent o

f the

ste

epne

ss, i

nclin

e, o

r gra

de o

f a li

ne fr

om le

ft to

righ

t. It

is

the

ratio

of v

ertic

al c

hang

e to

hor

izon

tal c

hang

e. M

ore

spec

ifica

lly, i

t is

the

ratio

of t

he c

hang

e in

the

y-co

ordi

nate

s (ri

se) t

o th

e co

rresp

ondi

ng c

hang

e in

the

x- c

oord

inat

es (r

un) w

hen

mov

ing

from

one

poin

t to

anot

her a

long

a li

ne. I

t als

o in

dica

tes

whe

ther

a li

ne is

tilte

d up

war

d (p

ositi

ve s

lope

) or

dow

nwar

d (n

egat

ive

slop

e) a

nd is

writ

ten

as th

e le

tter m

whe

re m

= ris

eru

n =

2

1

21

yy

xx

− −. E

xam

ple

of s

lope

:

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

25

Ja

nuar

y 20

13

Slop

e-In

terc

ept F

orm

A

n eq

uatio

n of

a s

traig

ht, n

on-v

ertic

al li

ne w

ritte

n in

the

form

y =

mx

+ b,

whe

re m

is th

e sl

ope

and

b is

th

e y-

inte

rcep

t.

Squa

re R

oot

One

of t

wo

equa

l fac

tors

(roo

ts) o

f a n

umbe

r or e

xpre

ssio

n; a

radi

cal e

xpre

ssio

n (

a) w

ith a

n un

ders

tood

deg

ree

of 2

. The

squ

are

root

of a

num

ber o

r exp

ress

ion

is a

ssum

ed to

be

the

prin

cipa

l

(pos

itive

) roo

t (e.

g.,

449

x =

7x2 ).

The

squa

re ro

ot o

f a n

egat

ive

num

ber r

esul

ts in

an

imag

inar

y

num

ber (

e.g.

, _49

= 7

i).

Stan

dard

For

m (o

f a L

inea

r Eq

uatio

n)

An

equa

tion

of a

stra

ight

line

writ

ten

in th

e fo

rm A

x +

By

= C

, whe

re A

, B, a

nd C

are

real

num

bers

and

w

here

A a

nd B

are

not

bot

h ze

ro. I

t inc

lude

s va

riabl

es o

n on

e si

de o

f the

equ

atio

n an

d a

cons

tant

on

the

othe

r sid

e.

Stem

-and

-Lea

f Plo

t A

vis

ual w

ay to

dis

play

the

shap

e of

a d

istri

butio

n th

at s

how

s gr

oups

of d

ata

arra

nged

by

plac

e va

lue;

a

way

to s

how

the

frequ

ency

with

whi

ch c

erta

in c

lass

es o

f dat

a oc

cur.

The

stem

con

sist

s of

a c

olum

n of

the

larg

er p

lace

val

ue(s

); th

ese

num

bers

are

not

repe

ated

. The

leav

es c

onsi

st o

f the

sm

alle

st p

lace

va

lue

(usu

ally

the

ones

pla

ce) o

f eve

ry p

iece

of d

ata;

thes

e nu

mbe

rs a

re a

rrang

ed in

num

eric

al o

rder

in

the

row

of t

he a

ppro

pria

te s

tem

(e.g

., th

e nu

mbe

r 36

wou

ld b

e in

dica

ted

by a

leaf

of 6

app

earin

g in

th

e sa

me

row

as

the

stem

of 3

). E

xam

ple

of a

ste

m-a

nd-le

af p

lot:

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

26

Ja

nuar

y 20

13

Subs

titut

ion

The

repl

acem

ent o

f a te

rm o

r var

iabl

e in

an

expr

essi

on o

r equ

atio

n by

ano

ther

that

has

the

sam

e va

lue

in o

rder

to s

impl

ify o

r eva

luat

e th

e ex

pres

sion

or e

quat

ion.

Syst

em o

f Lin

ear E

quat

ions

A

set

of t

wo

or m

ore

linea

r equ

atio

ns w

ith th

e sa

me

varia

bles

. The

sol

utio

n to

a s

yste

m o

f lin

ear

equa

tions

may

be

foun

d by

line

ar c

ombi

natio

n, s

ubst

itutio

n, o

r gra

phin

g. A

sys

tem

of t

wo

linea

r eq

uatio

ns w

ill ei

ther

hav

e on

e so

lutio

n, in

finite

ly m

any

solu

tions

, or n

o so

lutio

ns.

Syst

em o

f Lin

ear

Ineq

ualit

ies

Two

or m

ore

linea

r ine

qual

ities

with

the

sam

e va

riabl

es. S

ome

syst

ems

of in

equa

litie

s m

ay in

clud

e eq

uatio

ns a

s w

ell a

s in

equa

litie

s. T

he s

olut

ion

regi

on m

ay b

e cl

osed

or b

ound

ed b

ecau

se th

ere

are

lines

on

all s

ides

, whi

le o

ther

sol

utio

ns m

ay b

e op

en o

r unb

ound

ed.

Syst

ems

of E

quat

ions

A

set

of t

wo

or m

ore

equa

tions

con

tain

ing

a se

t of c

omm

on v

aria

bles

.

Term

A

par

t of a

n al

gebr

aic

expr

essi

on. T

erm

s ar

e se

para

ted

by e

ither

an

addi

tion

sym

bol (

+) o

r a

subt

ract

ion

sym

bol (

–). I

t can

be

a nu

mbe

r, a

varia

ble,

or a

pro

duct

of a

num

ber a

nd o

ne o

r mor

e va

riabl

es (e

.g.,

in th

e ex

pres

sion

4x2 +

6y, 4

x2 and

6y

are

both

term

s).

Term

inat

ing

Dec

imal

A

dec

imal

with

a fi

nite

num

ber o

f dig

its. A

dec

imal

for w

hich

the

divi

sion

ope

ratio

n re

sults

in e

ither

re

peat

ing

zero

es o

r rep

eatin

g ni

nes

(e.g

., 0.

3750

00…

= 0

.375

, 0.1

999…

= 0

.2).

It is

gen

eral

ly w

ritte

n to

the

last

non

-zer

o pl

ace

valu

e, b

ut c

an a

lso

be w

ritte

n w

ith a

dditi

onal

zer

oes

in s

mal

ler p

lace

val

ues

as n

eede

d (e

.g.,

0.25

can

als

o be

writ

ten

as 0

.250

0). A

ll te

rmin

atin

g de

cim

als

are

ratio

nal n

umbe

rs.

Trin

omia

l A

pol

ynom

ial w

ith th

ree

unlik

e te

rms

(e.g

., 7a

+ 4

b +

9c).

Eac

h te

rm is

a m

onom

ial,

and

the

mon

omia

ls a

re jo

ined

by

an a

dditi

on s

ymbo

l (+)

or a

sub

tract

ion

sym

bol (

–). I

t is

cons

ider

ed a

n al

gebr

aic

expr

essi

on.

K

eyst

one

Exam

s: A

lgeb

ra

Ass

essm

ent

An

chor

& E

ligi

ble

Con

ten

t G

loss

ary

Jan

uar

y 2

01

3

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

27

Ja

nuar

y 20

13

Uni

t Rat

e A

rate

in w

hich

the

seco

nd (i

ndep

ende

nt) q

uant

ity o

f the

ratio

is 1

(e.g

., 60

wor

ds p

er m

inut

e, $

4.50

pe

r pou

nd, 2

1 st

uden

ts p

er c

lass

).

Varia

ble

A le

tter o

r sym

bol u

sed

to re

pres

ent a

ny o

ne o

f a g

iven

set

of n

umbe

rs o

r oth

er o

bjec

ts (e

.g.,

in th

e eq

uatio

n y

= x

+ 5,

the

y an

d x

are

varia

bles

). S

ince

it c

an ta

ke o

n di

ffere

nt v

alue

s, it

is th

e op

posi

te o

f a

cons

tant

.

Who

le N

umbe

r A

nat

ural

num

ber o

r zer

o. A

ny n

umbe

r fro

m th

e se

t of n

umbe

rs re

pres

ente

d by

{0, 1

, 2, 3

, …}.

Som

etim

es it

is re

ferre

d to

as

a “n

on-n

egat

ive

inte

ger”.

x-A

xis

The

horiz

onta

l num

ber l

ine

on a

coo

rdin

ate

plan

e th

at in

ters

ects

with

a v

ertic

al n

umbe

r lin

e, th

e y-

axis

; the

line

who

se e

quat

ion

is y

= 0

. The

x-a

xis

cont

ains

all

the

poin

ts w

ith a

zer

o y-

coor

dina

te

(e.g

., (5

, 0)).

x-In

terc

ept(s

) Th

e x-

coor

dina

te(s

) of t

he p

oint

(s) a

t whi

ch th

e gr

aph

of a

n eq

uatio

n cr

osse

s th

e x-

axis

(i.e

., th

e va

lue(

s) o

f the

x-c

oord

inat

e w

hen

y =

0). T

he s

olut

ion(

s) o

r roo

t(s) o

f an

equa

tion

that

is s

et e

qual

to

0.

y-A

xis

The

verti

cal n

umbe

r lin

e on

a c

oord

inat

e pl

ane

that

inte

rsec

ts w

ith a

hor

izon

tal n

umbe

r lin

e, th

e x-

axis

; the

line

who

se e

quat

ion

is x

= 0

. The

y-a

xis

cont

ains

all

the

poin

ts w

ith a

zer

o x-

coor

dina

te

(e.g

., (0

, 7)).

y-In

terc

ept(s

) Th

e y-

coor

dina

te(s

) of t

he p

oint

(s) a

t whi

ch th

e gr

aph

of a

n eq

uatio

n cr

osse

s th

e y-

axi

s (i.

e., t

he

valu

e(s)

of t

he y

-coo

rdin

ate

whe

n x

= 0)

. For

a li

near

equ

atio

n in

slo

pe-in

terc

ept f

orm

(y =

mx

+ b)

, it i

s in

dica

ted

by b

.

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

Copyright © 2013 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

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January 2013

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