Pennsylvania Bulletin

PA Bulletin, Doc. No. 99-110c

[29 Pa.B. 399]

[Continued from previous Web Page]

III.  GLOSSARY

Alliteration: The repetition of initial consonant sounds in neighboring words.

Allusion: An implied or indirect reference in literature to a familiar person, place or event.

Analysis: The process or result of identifying the parts of a whole and their relationships to one another.

Antonym: A word that is the opposite of another word.

Characterization: The method an author uses to reveal characters and their various personalities.

Compare: Place together characters, situations or ideas to show common or differing features in literary selections.

Context clues: Information from the reading that identifies a word or group of words.

Conventions of    Language: Mechanics, usage and sentence completeness.

Evaluate: Examine and judge carefully.

Figurative    language: Language that cannot be taken literally since it was written to create a special effect or feeling.

Fluency: The clear, easy, written or spoken expression of ideas. Freedom from word-identification problems which might hinder comprehension in silent reading or the expression of ideas in oral reading.

Focus: The center of interest or attention.

Genre: A category used to classify literary works, usually by form, technique or content (e.g., prose, poetry).

Graphic organizer: A diagram or pictorial device that shows relationships.

Homophone: One of two or more words pronounced alike, but different in spelling or meaning (e.g., hair/hare, scale (fish)/scale (musical)).

Hyperbole: An exaggeration or overstatement (e.g., I was so embarrassed I could have died.).

Idiomatic    language: An expression peculiar to itself grammatically or that cannot be understood if taken literally (e.g., Let's get on the ball.).

Irony: The use of a word or phrase to mean the exact opposite of its literal or usual meaning; incongruity between the actual result of a sequence of events and the expected result.

Literary conflict: The struggle that grows out of the interplay of the two opposing forces in a plot.

Literary elements: The essential techniques used in literature (e.g., characterization, setting, plot, theme).

Literary devices: Tools used by the author to enliven and provide voice to the writing (e.g., dialogue, alliteration).

Literary    structures: The author's method of organizing text, (e.g., foreshadowing and flashbacks).

Metaphor: The comparison of two unlike things in which no words of comparison (like or as) are used (e.g., That new kid in class is really a squirrel.).

Meter: The repetition of stressed and unstressed syllables in a line of poetry.

Narrative: A story, actual or fictional, expressed orally or in writing.

Paraphrase: Restate text or passage in other words, often to clarify meaning or show understanding.

Pattern book: A book with a predictable plot structure and often written with predictable text; also known as predictable book.

Personification: An object or abstract idea given human qualities or human form (e.g., Flowers danced about the lawn.).

Phonics: The relationship between letters and sounds fundamental in beginning reading.

Point of view: The way in which an author reveals characters, events and ideas in telling a story; the vantage point from which the story is told.

Public document: A document that focuses on civic issues or matters of public policy at the community level and beyond.

Reading critically: Reading in which a questioning attitude, logical analysis and inference are used to judge the worth of text; evaluating relevancy and adequacy of what is read; the judgment of validity or worth of what is read, based on sound criteria.

Reading rate: The speed at which a person reads, usually silently.

Research: A systematic inquiry into a subject or problem in order to discover, verify or revise relevant facts or principles having to do with that subject or problem.

Satire: A literary tone used to ridicule or make fun of human vice or weakness.

Self-monitor: Know when what one is reading or writing is not making sense; adjust strategies for comprehension.

Semantics: The study of meaning in language.

Simile: A comparison of two unlike things in which a word of comparison (like or as) is used (e.g., She eats like a bird.).

Sources:

      Primary: Text and/or artifacts that tell or show a first-hand account of an event; original works used when researching.

      Secondary: Text and/or artifacts used when researching that are derived from something original.

Subject area: An organized body of knowledge; a discipline; a content area.

Style: How an author writes; an author's use of language, its effects and appropriateness to the author's intent and theme.

Synonym: One of two or more words in a language that have highly similar meanings (e.g., sorrow, grief, sadness).

Syntax: The pattern or structure of word order in sentences, clauses and phrases.

Theme: A topic of discussion or writing; a major idea broad enough to cover the entire scope of a literary work.

Thesis: The basic argument advanced by a speaker or writer who then attempts to prove it; the subject or major argument of a speech or composition.

Tone: The attitude of the author toward the audience and characters (e.g., serious or humorous).

Voice: The fluency, rhythm and liveliness in writing that makes it unique to the writer.

Academic Standards for Mathematics

IV.  TABLE OF CONTENTS

Introduction
V.

THE ACADEMIC STANDARDS

Numbers, Number Systems and Number
Relationships 2.1.

      Types of numbers (e.g., whole, prime, irrational,
complex)
Equivalent forms (e.g. fractions, decimals, percents)

Computation and Estimation 2.2.

      Basic functions (+, -, ×, ÷)
Reasonableness of answers
Calculators

Measurement and Estimation 2.3.

      Types of measurement (e.g., length, time)
Units and tools of measurement
Computing and comparing measurements

Mathematical Reasoning and Connections 2.4.

      Using inductive and deductive reasoning
Validating arguments (e.g., if . . . then statements, proofs)

Mathematical Problem Solving and
Communication 2.5.

      Problem solving strategies
Representing problems in various ways
Interpreting results

Statistics and Data Analysis 2.6.

      Collecting and reporting data (e.g., charts, graphs)
Analyzing data

Probability and Predictions 2.7.

      Validity of data
Calculating probability to make predictions

Algebra and Functions 2.8.

      Equations
Patterns and functions

Geometry 2.9.

      Shapes and their properties
Using geometric principles to solve problems

Trigonometry 2.10.

      Right angles
Measuring and computing with triangles
Using graphing calculators

Concepts of Calculus 2.11.

      Comparing quantities and values
Graphing rates of change
Continuing patterns infinitely

Glossary VI.

V.  INTRODUCTION

This document includes Mathematics Standards:

*  2.1.     Numbers, Number Systems and Number Relationships

*  2.2.     Computation and Estimation

*  2.3.     Measurement and Estimation

*  2.4.     Mathematical Reasoning and Connections

*  2.5.     Mathematical Problem Solving and Communication

*  2.6.     Statistics and Data Analysis

*  2.7.     Probability and Predictions

*  2.8.     Algebra and Functions

*  2.9.     Geometry

*  2.10.    Trigonometry

*  2.11.    Concepts of Calculus

   The Mathematics Standards describe what students should know and be able to do at four grade levels (third, fifth, eighth and eleventh). They reflect the increasing complexity and sophistication that students are expected to achieve as they progress through school.

   This document avoids repetition of learned skills, making an obvious progression across grade levels less explicit. Teachers shall expect that students know and can apply the concepts and skills expressed at the preceding level. Consequently, previous learning is reinforced but not retaught.

   Students who achieve these mathematical standards will be able to communicate mathematically. Although it is an interesting and enjoyable study for its own sake, mathematics is most appropriately used as a tool to help organize and understand information from other academic disciplines. Because our capacity to deal with all things mathematical is changing rapidly, students must be able to bring the most modern and effective technology to bear on their learning of mathematical concepts and skills.

   A glossary is included to assist the reader in understanding terminology contained in the standards.

2.1.  Numbers, Number Systems and Number Relationships

2.1.3.  GRADE 3 2.1.5.  GRADE 5 2.1.8.  GRADE 8 2.1.11.  GRADE 11

Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills needed to:

A. Count using whole numbers (to 10,000) and by 2's, 3's, 5's, 10's, 25's and 100's. A. Use expanded notation to represent whole numbers or decimals. A. Represent and use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, exponents, scientific notation, square roots). A. Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots, finding logarithms).

B. Use whole numbers and fractions to represent quantities. B. Apply number theory concepts to rename a number quantity (e.g., six, 6, 12/2, 3 x 2, 10 - 4). B. Simplify numerical expressions involving exponents, scientific notation and using order of operations.

C. Represent equivalent forms of the same number through the use of concrete objects, drawings, word names and symbols. C. Demonstrate that mathematical operations can represent a variety of problem situations. C. Distinguish between and order rational and irrational numbers.

D. Use drawings, diagrams or models to show the concept of fraction as part of a whole. D. Use models to represent fractions and decimals. D. Apply ratio and proportion to mathematical problem situations involving distance, rate, time and similar triangles.

E. Count, compare and make change using a collection of coins and one-dollar bills. E. Explain the concepts of prime and composite numbers. E. Simplify and expand algebraic expressions using exponential forms.

F. Apply number patterns (even and odd) and compare values of numbers on the hundred board. F. Use simple concepts of negative numbers (e.g., on a number line, in counting and in temperature). F. Use the number line model to demonstrate integers and their applications.

G. Use concrete objects to count, order and group. G. Develop and apply number theory concepts (e.g., primes, factors, multiples, composites) to represent numbers in various ways. G. Use the inverse relationships between addition, subtraction, multiplication, division, exponentiation and root extraction to determine unknown quantities in equations.

H. Demonstrate understanding of one-to-one correspondence.

I. Apply place-value concepts and numeration to counting, ordering and grouping.

J. Estimate, approximate, round or use exact numbers as appropriate.

K. Describe the inverse relationship between addition and subtraction.

L. Demonstrate knowledge of basic facts in four basic operations.

2.2.  Computation and Estimation

2.2.3.  GRADE 3 2.2.5.  GRADE 5 2.2.8.  GRADE 8 2.2.11.  GRADE 11

Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills needed to:

A. Apply addition and subtraction in everyday situations using concrete objects. A. Create and solve word problems involving addition, subtraction, multiplication and division of whole numbers. A. Complete calculations by applying the order of operations. A. Develop and use computation concepts, operations and procedures on real numbers in problem solving situations.

B. Solve single- and double-digit addition and subtraction problems with regrouping in vertical form. B. Develop and apply algorithms to solve word problems that involve addition, subtraction, and/or multiplication with decimals with and without regrouping. B. Add, subtract, multiply and divide different kinds and forms of rational numbers including integers, decimal fractions, percents and proper and improper fractions. B. Use estimation to solve problems for which exact answer is not needed.

C. Demonstrate the concept of multiplication as repeated addition and arrays. C. Develop and apply algorithms to solve word problems that involve addition, subtraction, and/or multiplication with fractions and mixed numbers that include like and unlike denominators. C. Estimate the value of irrational numbers. C. Construct and apply mathematical models, including lines and curves of best fit, to estimate values of related quantities.

D. Demonstrate the concept of division as repeated subtraction and as sharing. D. Demonstrate the ability to round numbers. D. Estimate amount of tips and discounts using ratios, proportions and percents. D. Describe and explain the amount of error that may exist in a computation using estimates.

E. Use estimation skills to arrive at conclusions. E. Determine through estimations the reasonableness of answers to problems involving addition, subtraction, multiplication and division of whole numbers. E. Determine the appropriateness of overestimating or underestimating in computation. E. Recognize that the degree of precision needed in calculating a number depends on how the results will be used and the instruments used to generate the measure.

F. Determine the reasonableness of calculated answers. F. Demonstrate skills for using fraction calculators to verify conjectures, confirm computations and explore complex problem-solving situations. F. Identify the difference between exact value and approximation and determine which is appropriate for a given situation. F. Demonstrate skills for using computer spreadsheets and scientific and graphing calculators.

G. Explain addition and subtraction algorithms with regrouping. G. Apply estimation strategies to a variety of problems including time and money.

H. Explain multiplication and division algorithms.

I. Select a method for computation and explain why it is appropriate.

2.3.  Measurement and Estimation

2.3.3.  GRADE 3 2.3.5.  GRADE 5 2.3.8.  GRADE 8 2.3.11.  GRADE 11

Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to:

A. Compare measurable characteristics of different objects on the same dimensions (e.g., time, temperature, area, length, weight, capacity, perimeter). A. Select and use appropriate instruments and units for measuring quantities (e.g., perimeter, volume, area, weight, time, temperature). A. Develop formulas and procedures for determining measurements (e.g., area, volume, distance). A. Select and use appropriate units and tools to measure to the degree of accuracy required in particular measurement situations.

B. Determine the measurement of objects with nonstandard and standard units (e.g., US customary and metric). B. Select and use standard tools to measure the size of figures with specified accuracy, including length, width, perimeter and area. B. Solve rate problems (e.g., rate x time = distance, principal x interest rate = interest). B. Measure and compare angles in degrees and radians.

C. Determine and compare elapsed times. C. Estimate, refine and verify specified measurements of objects. C. Measure angles in degrees and determine relations of angles. C. Demonstrate the ability to produce measures with specified levels of precision.

D. Tell time (analog and digital) to the minute. D. Convert linear measurements within the same system. D. Estimate, use and describe measures of distance, rate, perimeter, area, volume, weight, mass and angles.

E. Determine appropriate unit of measure. E. Add and subtract measurements. E. Describe how a change in linear dimension of an object affects its perimeter, area and volume.

F. Use concrete objects to determine area and perimeter. F. Use scale measurements to interpret maps or drawings.

G. Estimate and verify measurements. G. Create and use scale models.

Demonstrate that a single object has different attributes that can be measured in different ways (e.g. length, mass, weight, time, area, temperature, capacity, perimeter).

2.4.  Mathematical Reasoning and Connections

2.4.3.  GRADE 3 2.4.5.  GRADE 5 2.4.8.  GRADE 8 2.4.11.  GRADE 11

Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to:

A. Make, check and verify predictions about the quantity, size and shape of objects and groups of objects. A. Compare quantities and magnitudes of numbers. A. Make conjectures based on logical reasoning and test conjectures by using counter-examples. A. Use direct proofs, indirect proofs or proof by contradiction to validate conjectures.

B. Use measurements in everyday situations (e.g., determine the geography of the school building). B. Use models, number facts, properties and relationships to check and verify predictions and explain reasoning. B. Combine numeric relationships to arrive at a conclusion. B. Construct valid arguments from stated facts.

C. Draw inductive and deductive conclusions within mathematical contexts. C. Use if . . . then statements to construct simple valid arguments. C. Determine the validity of an argument.

D. Distinguish between relevant and irrelevant information in a mathematical problem. D. Construct, use and explain algorithmic procedures for computing and estimating with whole numbers, fractions, decimals and integers. D. Use truth tables to reveal the logic of mathematical statements.

E. Interpret statements made with precise language of logic (e.g., ''all'', ''or'', ''every'', ''none'', ''some'', ''or'', ''many''). E. Distinguish between inductive and deductive reasoning. E. Demonstrate mathematical solutions to problems (e.g., in the physical sciences).

F. Use statistics to quantify issues (e.g., in social studies, in science). F. Use measurements and statistics to quantify issues (e.g., in family and consumer science situations).

2.5.  Mathematical Problem Solving and Communication

2.5.3.  GRADE 3 2.5.5.  GRADE 5 2.5.8.  GRADE 8 2.5.11.  GRADE 11

Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to:

A. Use appropriate problem-solving strategies (e.g., guess and check, working backwards). A. Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense and explain how the problem was solved. A. Invent, select, use and justify the appropriate methods, materials and strategies used to solve problems. A. Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.

B. Determine when sufficient information is present to solve a problem and explain how to solve a problem. B. Use appropriate mathematical terms, vocabulary, language symbols and graphs to clearly and logically explain solutions to problems. B. Verify and interpret results using precise mathematical language, notation and representations, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs and diagrams. B. Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results.

C. Select and use an appropriate method, materials and strategy to solve problems, including mental mathematics, paper and pencil and concrete objects. C. Show ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models. C. Justify strategies and defend approaches used and conclusions reached. C. Present mathematical procedures and results clearly, systematically, succinctly and correctly.

D. Connect, extend and generalize problem solutions to other concepts, problems and circumstances in mathematics. D. Determine pertinent information in problem situations and whether any further information is needed for solution. D. Conclude a solution process with a summary of results and evaluate the degree to which the results obtained represent an acceptable response to the initial problem and why the reasoning is valid.

E. Select, use and justify the methods, materials and strategies used to solve problems.

F. Use appropriate problem-solving strategies (e.g., solving a simpler problem, drawing a picture or diagram.

2.6.  Statistics and Data Analysis

2.6.3.  GRADE 3 2.6.5.  GRADE 5 2.6.8.  GRADE 8 2.6.11.  GRADE 11

Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to:

A. Gather, organize and display data using pictures, tallies, charts, bar graphs and pictographs. A. Organize and display data using pictures, tallies, tables, charts, bar graphs and circle graphs. A. Compare and contrast different plots of data using values of mean, median, mode, quartiles and range. A. Design and conduct an experiment using random sampling. Describe the data as an example of a distribution using statistical measures of center and spread. Organize and represent the results with graphs. (Use standard deviation, variance and t-tests).

B. Formulate and answer questions based on data shown on graphs. B. Describe data sets using mean, median, mode and range. B. Explain effects of sampling procedures and missing or incorrect information on reliability. B. Use appropriate technology to organize and analyze data taken from the local community.

C. Predict the likely number of times a condition will occur based on the analyzed data. C. Sort data using Venn diagrams. C. Fit a line to the scatter plot of two quantities and describe any correlation of the variables. C. Determine the regression equation of best fit (e.g., linear, quadratic and exponential).

D. Form and justify an opinion on whether a given statement is reasonable based on a comparison to data. D. Predict the likely number of times a condition will occur based on analyzed data. D. Design and carry out a random sampling procedure. D. Make predictions using interpolation, extrapolation, regression and estimation using technology to verify them.

E. Construct and defend simple conclusions based on data. E. Analyze and display data in stem-and-leaf and box-and-whisker plots. E. Determine the validity of the sampling method described in a given study.

F. Use scientific and graphing calculators and computer spreadsheets to organize and analyze data. F. Determine the degree of dependence of two quantities specified by a two-way table.

G. Determine the validity of the sampling method described in studies published in local or National newspapers. G. Describe questions of experimental design, control groups, treatment groups, cluster sampling and reliability.

H. Use sampling techniques to draw inferences about large populations.

I. Describe the normal curve and use its properties to answer questions about sets of data that are assumed to be normally distributed.

2.7.  Probability and Predictions

2.7.3.  GRADE 3 2.7.5.  GRADE 5 2.7.8.  GRADE 8 2.7.11.  GRADE 11

Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills needed to:

A. Predict and measure the likelihood of events and recognize that the results of an experiment may not match predicted outcomes. A. Perform simulations with concrete devices (e.g., dice, spinner) to predict the chance of an event occurring. A. Determine the number of combinations and permutations for an event. A. Compare odds and probability.

B. Design a fair and an unfair spinner. B. Determine the fairness of the design of a spinner. B. Present the results of an experiment using visual representations (e.g., tables, charts, graphs). B. Apply probability and statistics to perform an experiment involving a sample and generalize its results to the entire population.

C. List or graph the possible results of an experiment. C. Express probabilities as fractions and decimals. C. Analyze predictions (e.g., election polls). C. Draw and justify a conclusion regarding the validity of a probability or statistical argument.

D. Analyze data using the concepts of largest, smallest, most often, least often and middle. D. Compare predictions based on theoretical probability and experimental results. D. Compare and contrast results from observations and mathematical models. D. Use experimental and theoretical probability distributions to make judgments about the likelihood of various outcomes in uncertain situations.

E. Calculate the probability of a simple event. E. Make valid inferences, predictions and arguments based on probability. E. Solve problems involving independent simple and compound events.

F. Determine patterns generated as a result of an experiment.

G. Determine the probability of an event involving ''and'', ''or'' or ''not''.

H. Predict and determine why some outcomes are certain, more likely, less likely, equally likely or impossible.

I. Find all possible combinations and arrangements involving a limited number of variables.

J. Develop a tree diagram and list the elements.

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Alliteration:	The repetition of initial consonant sounds in neighboring words.
Allusion:	An implied or indirect reference in literature to a familiar person, place or event.
Analysis:	The process or result of identifying the parts of a whole and their relationships to one another.
Antonym:	A word that is the opposite of another word.
Characterization:	The method an author uses to reveal characters and their various personalities.
Compare:	Place together characters, situations or ideas to show common or differing features in literary selections.
Context clues:	Information from the reading that identifies a word or group of words.
Conventions of Language:	Mechanics, usage and sentence completeness.
Evaluate:	Examine and judge carefully.
Figurative language:	Language that cannot be taken literally since it was written to create a special effect or feeling.
Fluency:	The clear, easy, written or spoken expression of ideas. Freedom from word-identification problems which might hinder comprehension in silent reading or the expression of ideas in oral reading.
Focus:	The center of interest or attention.
Genre:	A category used to classify literary works, usually by form, technique or content (e.g., prose, poetry).
Graphic organizer:	A diagram or pictorial device that shows relationships.
Homophone:	One of two or more words pronounced alike, but different in spelling or meaning (e.g., hair/hare, scale (fish)/scale (musical)).
Hyperbole:	An exaggeration or overstatement (e.g., I was so embarrassed I could have died.).
Idiomatic language:	An expression peculiar to itself grammatically or that cannot be understood if taken literally (e.g., Let's get on the ball.).
Irony:	The use of a word or phrase to mean the exact opposite of its literal or usual meaning; incongruity between the actual result of a sequence of events and the expected result.
Literary conflict:	The struggle that grows out of the interplay of the two opposing forces in a plot.
Literary elements:	The essential techniques used in literature (e.g., characterization, setting, plot, theme).
Literary devices:	Tools used by the author to enliven and provide voice to the writing (e.g., dialogue, alliteration).
Literary structures:	The author's method of organizing text, (e.g., foreshadowing and flashbacks).
Metaphor:	The comparison of two unlike things in which no words of comparison (like or as) are used (e.g., That new kid in class is really a squirrel.).
Meter:	The repetition of stressed and unstressed syllables in a line of poetry.
Narrative:	A story, actual or fictional, expressed orally or in writing.
Paraphrase:	Restate text or passage in other words, often to clarify meaning or show understanding.
Pattern book:	A book with a predictable plot structure and often written with predictable text; also known as predictable book.
Personification:	An object or abstract idea given human qualities or human form (e.g., Flowers danced about the lawn.).
Phonics:	The relationship between letters and sounds fundamental in beginning reading.
Point of view:	The way in which an author reveals characters, events and ideas in telling a story; the vantage point from which the story is told.
Public document:	A document that focuses on civic issues or matters of public policy at the community level and beyond.
Reading critically:	Reading in which a questioning attitude, logical analysis and inference are used to judge the worth of text; evaluating relevancy and adequacy of what is read; the judgment of validity or worth of what is read, based on sound criteria.
Reading rate:	The speed at which a person reads, usually silently.
Research:	A systematic inquiry into a subject or problem in order to discover, verify or revise relevant facts or principles having to do with that subject or problem.
Satire:	A literary tone used to ridicule or make fun of human vice or weakness.
Self-monitor:	Know when what one is reading or writing is not making sense; adjust strategies for comprehension.
Semantics:	The study of meaning in language.
Simile:	A comparison of two unlike things in which a word of comparison (like or as) is used (e.g., She eats like a bird.).
Sources:
*Primary:*	Text and/or artifacts that tell or show a first-hand account of an event; original works used when researching.
*Secondary:*	Text and/or artifacts used when researching that are derived from something original.
Subject area:	An organized body of knowledge; a discipline; a content area.
Style:	How an author writes; an author's use of language, its effects and appropriateness to the author's intent and theme.
Synonym:	One of two or more words in a language that have highly similar meanings (e.g., sorrow, grief, sadness).
Syntax:	The pattern or structure of word order in sentences, clauses and phrases.
Theme:	A topic of discussion or writing; a major idea broad enough to cover the entire scope of a literary work.
Thesis:	The basic argument advanced by a speaker or writer who then attempts to prove it; the subject or major argument of a speech or composition.
Tone:	The attitude of the author toward the audience and characters (e.g., serious or humorous).
Voice:	The fluency, rhythm and liveliness in writing that makes it unique to the writer.

Numbers, Number Systems and Number Relationships	2.1.
Types of numbers (e.g., whole, prime, irrational, complex) Equivalent forms (e.g. fractions, decimals, percents)
Computation and Estimation	2.2.
Basic functions (+, -, ×, ÷) Reasonableness of answers Calculators
Measurement and Estimation	2.3.
Types of measurement (e.g., length, time) Units and tools of measurement Computing and comparing measurements
Mathematical Reasoning and Connections	2.4.
Using inductive and deductive reasoning Validating arguments (e.g., if . . . then statements, proofs)
Mathematical Problem Solving and Communication	2.5.
Problem solving strategies Representing problems in various ways Interpreting results
Statistics and Data Analysis	2.6.
Collecting and reporting data (e.g., charts, graphs) Analyzing data
Probability and Predictions	2.7.
Validity of data Calculating probability to make predictions
Algebra and Functions	2.8.
Equations Patterns and functions
Geometry	2.9.
Shapes and their properties Using geometric principles to solve problems
Trigonometry	2.10.
Right angles Measuring and computing with triangles Using graphing calculators
Concepts of Calculus	2.11.
Comparing quantities and values Graphing rates of change Continuing patterns infinitely
Glossary	VI.

2.1. Numbers, Number Systems and Number Relationships
2.1.3. GRADE 3		2.1.5. GRADE 5		2.1.8. GRADE 8		2.1.11. GRADE 11
*Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills needed to:*
A.	Count using whole numbers (to 10,000) and by 2's, 3's, 5's, 10's, 25's and 100's.	A.	Use expanded notation to represent whole numbers or decimals.	A.	Represent and use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, exponents, scientific notation, square roots).	A.	Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots, finding logarithms).
B.	Use whole numbers and fractions to represent quantities.	B.	Apply number theory concepts to rename a number quantity (e.g., six, 6, 12/2, 3 x 2, 10 - 4).	B.	Simplify numerical expressions involving exponents, scientific notation and using order of operations.
C.	Represent equivalent forms of the same number through the use of concrete objects, drawings, word names and symbols.	C.	Demonstrate that mathematical operations can represent a variety of problem situations.	C.	Distinguish between and order rational and irrational numbers.
D.	Use drawings, diagrams or models to show the concept of fraction as part of a whole.	D.	Use models to represent fractions and decimals.	D.	Apply ratio and proportion to mathematical problem situations involving distance, rate, time and similar triangles.
E.	Count, compare and make change using a collection of coins and one-dollar bills.	E.	Explain the concepts of prime and composite numbers.	E.	Simplify and expand algebraic expressions using exponential forms.
F.	Apply number patterns (even and odd) and compare values of numbers on the hundred board.	F.	Use simple concepts of negative numbers (e.g., on a number line, in counting and in temperature).	F.	Use the number line model to demonstrate integers and their applications.
G.	Use concrete objects to count, order and group.	G.	Develop and apply number theory concepts (e.g., primes, factors, multiples, composites) to represent numbers in various ways.	G.	Use the inverse relationships between addition, subtraction, multiplication, division, exponentiation and root extraction to determine unknown quantities in equations.
H.	Demonstrate understanding of one-to-one correspondence.
I.	Apply place-value concepts and numeration to counting, ordering and grouping.
J.	Estimate, approximate, round or use exact numbers as appropriate.
K.	Describe the inverse relationship between addition and subtraction.
L.	Demonstrate knowledge of basic facts in four basic operations.

2.2. Computation and Estimation
2.2.3. GRADE 3		2.2.5. GRADE 5		2.2.8. GRADE 8		2.2.11. GRADE 11
*Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills needed to:*
A.	Apply addition and subtraction in everyday situations using concrete objects.	A.	Create and solve word problems involving addition, subtraction, multiplication and division of whole numbers.	A.	Complete calculations by applying the order of operations.	A.	Develop and use computation concepts, operations and procedures on real numbers in problem solving situations.
B.	Solve single- and double-digit addition and subtraction problems with regrouping in vertical form.	B.	Develop and apply algorithms to solve word problems that involve addition, subtraction, and/or multiplication with decimals with and without regrouping.	B.	Add, subtract, multiply and divide different kinds and forms of rational numbers including integers, decimal fractions, percents and proper and improper fractions.	B.	Use estimation to solve problems for which exact answer is not needed.
C.	Demonstrate the concept of multiplication as repeated addition and arrays.	C.	Develop and apply algorithms to solve word problems that involve addition, subtraction, and/or multiplication with fractions and mixed numbers that include like and unlike denominators.	C.	Estimate the value of irrational numbers.	C.	Construct and apply mathematical models, including lines and curves of best fit, to estimate values of related quantities.
D.	Demonstrate the concept of division as repeated subtraction and as sharing.	D.	Demonstrate the ability to round numbers.	D.	Estimate amount of tips and discounts using ratios, proportions and percents.	D.	Describe and explain the amount of error that may exist in a computation using estimates.
E.	Use estimation skills to arrive at conclusions.	E.	Determine through estimations the reasonableness of answers to problems involving addition, subtraction, multiplication and division of whole numbers.	E.	Determine the appropriateness of overestimating or underestimating in computation.	E.	Recognize that the degree of precision needed in calculating a number depends on how the results will be used and the instruments used to generate the measure.
F.	Determine the reasonableness of calculated answers.	F.	Demonstrate skills for using fraction calculators to verify conjectures, confirm computations and explore complex problem-solving situations.	F.	Identify the difference between exact value and approximation and determine which is appropriate for a given situation.	F.	Demonstrate skills for using computer spreadsheets and scientific and graphing calculators.
G.	Explain addition and subtraction algorithms with regrouping.	G.	Apply estimation strategies to a variety of problems including time and money.
		H.	Explain multiplication and division algorithms.
		I.	Select a method for computation and explain why it is appropriate.

2.3. Measurement and Estimation
2.3.3. GRADE 3		2.3.5. GRADE 5		2.3.8. GRADE 8		2.3.11. GRADE 11
*Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to:*
A.	Compare measurable characteristics of different objects on the same dimensions (e.g., time, temperature, area, length, weight, capacity, perimeter).	A.	Select and use appropriate instruments and units for measuring quantities (e.g., perimeter, volume, area, weight, time, temperature).	A.	Develop formulas and procedures for determining measurements (e.g., area, volume, distance).	A.	Select and use appropriate units and tools to measure to the degree of accuracy required in particular measurement situations.
B.	Determine the measurement of objects with nonstandard and standard units (e.g., US customary and metric).	B.	Select and use standard tools to measure the size of figures with specified accuracy, including length, width, perimeter and area.	B.	Solve rate problems (e.g., rate x time = distance, principal x interest rate = interest).	B.	Measure and compare angles in degrees and radians.
C.	Determine and compare elapsed times.	C.	Estimate, refine and verify specified measurements of objects.	C.	Measure angles in degrees and determine relations of angles.	C.	Demonstrate the ability to produce measures with specified levels of precision.
D.	Tell time (analog and digital) to the minute.	D.	Convert linear measurements within the same system.	D.	Estimate, use and describe measures of distance, rate, perimeter, area, volume, weight, mass and angles.
E.	Determine appropriate unit of measure.	E.	Add and subtract measurements.	E.	Describe how a change in linear dimension of an object affects its perimeter, area and volume.
F.	Use concrete objects to determine area and perimeter.			F.	Use scale measurements to interpret maps or drawings.
G.	Estimate and verify measurements.			G.	Create and use scale models.
Demonstrate that a single object has different attributes that can be measured in different ways (e.g. length, mass, weight, time, area, temperature, capacity, perimeter).

2.4. Mathematical Reasoning and Connections
2.4.3. GRADE 3		2.4.5. GRADE 5		2.4.8. GRADE 8		2.4.11. GRADE 11
*Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to:*
A.	Make, check and verify predictions about the quantity, size and shape of objects and groups of objects.	A.	Compare quantities and magnitudes of numbers.	A.	Make conjectures based on logical reasoning and test conjectures by using counter-examples.	A.	Use direct proofs, indirect proofs or proof by contradiction to validate conjectures.
B.	Use measurements in everyday situations (e.g., determine the geography of the school building).	B.	Use models, number facts, properties and relationships to check and verify predictions and explain reasoning.	B.	Combine numeric relationships to arrive at a conclusion.	B.	Construct valid arguments from stated facts.
		C.	Draw inductive and deductive conclusions within mathematical contexts.	C.	Use if . . . then statements to construct simple valid arguments.	C.	Determine the validity of an argument.
		D.	Distinguish between relevant and irrelevant information in a mathematical problem.	D.	Construct, use and explain algorithmic procedures for computing and estimating with whole numbers, fractions, decimals and integers.	D.	Use truth tables to reveal the logic of mathematical statements.
		E.	Interpret statements made with precise language of logic (e.g., ''all'', ''or'', ''every'', ''none'', ''some'', ''or'', ''many'').	E.	Distinguish between inductive and deductive reasoning.	E.	Demonstrate mathematical solutions to problems (e.g., in the physical sciences).
		F.	Use statistics to quantify issues (e.g., in social studies, in science).	F.	Use measurements and statistics to quantify issues (e.g., in family and consumer science situations).

2.5. Mathematical Problem Solving and Communication
2.5.3. GRADE 3		2.5.5. GRADE 5		2.5.8. GRADE 8		2.5.11. GRADE 11
*Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to:*
A.	Use appropriate problem-solving strategies (e.g., guess and check, working backwards).	A.	Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense and explain how the problem was solved.	A.	Invent, select, use and justify the appropriate methods, materials and strategies used to solve problems.	A.	Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.
B.	Determine when sufficient information is present to solve a problem and explain how to solve a problem.	B.	Use appropriate mathematical terms, vocabulary, language symbols and graphs to clearly and logically explain solutions to problems.	B.	Verify and interpret results using precise mathematical language, notation and representations, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs and diagrams.	B.	Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results.
C.	Select and use an appropriate method, materials and strategy to solve problems, including mental mathematics, paper and pencil and concrete objects.	C.	Show ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models.	C.	Justify strategies and defend approaches used and conclusions reached.	C.	Present mathematical procedures and results clearly, systematically, succinctly and correctly.
		D.	Connect, extend and generalize problem solutions to other concepts, problems and circumstances in mathematics.	D.	Determine pertinent information in problem situations and whether any further information is needed for solution.	D.	Conclude a solution process with a summary of results and evaluate the degree to which the results obtained represent an acceptable response to the initial problem and why the reasoning is valid.
		E.	Select, use and justify the methods, materials and strategies used to solve problems.
		F.	Use appropriate problem-solving strategies (e.g., solving a simpler problem, drawing a picture or diagram.

2.6. Statistics and Data Analysis
2.6.3. GRADE 3		2.6.5. GRADE 5		2.6.8. GRADE 8		2.6.11. GRADE 11
*Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to:*
A.	Gather, organize and display data using pictures, tallies, charts, bar graphs and pictographs.	A.	Organize and display data using pictures, tallies, tables, charts, bar graphs and circle graphs.	A.	Compare and contrast different plots of data using values of mean, median, mode, quartiles and range.	A.	Design and conduct an experiment using random sampling. Describe the data as an example of a distribution using statistical measures of center and spread. Organize and represent the results with graphs. (Use standard deviation, variance and t-tests).
B.	Formulate and answer questions based on data shown on graphs.	B.	Describe data sets using mean, median, mode and range.	B.	Explain effects of sampling procedures and missing or incorrect information on reliability.	B.	Use appropriate technology to organize and analyze data taken from the local community.
C.	Predict the likely number of times a condition will occur based on the analyzed data.	C.	Sort data using Venn diagrams.	C.	Fit a line to the scatter plot of two quantities and describe any correlation of the variables.	C.	Determine the regression equation of best fit (e.g., linear, quadratic and exponential).
D.	Form and justify an opinion on whether a given statement is reasonable based on a comparison to data.	D.	Predict the likely number of times a condition will occur based on analyzed data.	D.	Design and carry out a random sampling procedure.	D.	Make predictions using interpolation, extrapolation, regression and estimation using technology to verify them.
		E.	Construct and defend simple conclusions based on data.	E.	Analyze and display data in stem-and-leaf and box-and-whisker plots.	E.	Determine the validity of the sampling method described in a given study.
				F.	Use scientific and graphing calculators and computer spreadsheets to organize and analyze data.	F.	Determine the degree of dependence of two quantities specified by a two-way table.
				G.	Determine the validity of the sampling method described in studies published in local or National newspapers.	G.	Describe questions of experimental design, control groups, treatment groups, cluster sampling and reliability.
						H.	Use sampling techniques to draw inferences about large populations.
						I.	Describe the normal curve and use its properties to answer questions about sets of data that are assumed to be normally distributed.

2.7. Probability and Predictions
2.7.3. GRADE 3		2.7.5. GRADE 5		2.7.8. GRADE 8		2.7.11. GRADE 11
*Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills needed to:*
A.	Predict and measure the likelihood of events and recognize that the results of an experiment may not match predicted outcomes.	A.	Perform simulations with concrete devices (e.g., dice, spinner) to predict the chance of an event occurring.	A.	Determine the number of combinations and permutations for an event.	A.	Compare odds and probability.
B.	Design a fair and an unfair spinner.	B.	Determine the fairness of the design of a spinner.	B.	Present the results of an experiment using visual representations (e.g., tables, charts, graphs).	B.	Apply probability and statistics to perform an experiment involving a sample and generalize its results to the entire population.
C.	List or graph the possible results of an experiment.	C.	Express probabilities as fractions and decimals.	C.	Analyze predictions (e.g., election polls).	C.	Draw and justify a conclusion regarding the validity of a probability or statistical argument.
D.	Analyze data using the concepts of largest, smallest, most often, least often and middle.	D.	Compare predictions based on theoretical probability and experimental results.	D.	Compare and contrast results from observations and mathematical models.	D.	Use experimental and theoretical probability distributions to make judgments about the likelihood of various outcomes in uncertain situations.
		E.	Calculate the probability of a simple event.	E.	Make valid inferences, predictions and arguments based on probability.	E.	Solve problems involving independent simple and compound events.
		F.	Determine patterns generated as a result of an experiment.
		G.	Determine the probability of an event involving ''and'', ''or'' or ''not''.
		H.	Predict and determine why some outcomes are certain, more likely, less likely, equally likely or impossible.
		I.	Find all possible combinations and arrangements involving a limited number of variables.
		J.	Develop a tree diagram and list the elements.