Orange School District

Mathematics

Curriculum Guide – Grade 82011 Edition

APPROVED: January, 2011

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BOARD OF EDUCATION

Patricia A. ArthurPresident

Arthur GriffaVice-President

MembersStephanie Brown Rev. Reginald T. Jackson Maxine G. JohnsonEunice Y. Mitchell David Wright

SUPERINTENDENT OF SCHOOLSRonald Lee

DEPUTYSUPERINTENDENT

ADMINISTRATIVE ASSISTANT TO THE SUPERINTENDENT

Dr. Paula HowardCurriculum and Instructional Services

Belinda Scott-SmileyOperations/Human Resources

BUSINESS ADMINISTRATORAdekunle O. James

DIRECTORSBarbara L. Clark, Special Services

Candace Goldstein, Special ProgramsCandace Wallace, Curriculum & Testing

CURRICULUM TEAMCandace Wallace

Ron NelkinMengli Chiliu

Ann Burgunder

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Table of Contents

BOARD OF EDUCATION 2

PHILOSOPHY, VISION & PURPOSE 4PROCESS GOALS 5PHASES OF INSTRUCTION 6TARGET GOALS 8DESCRIPTION OF STUDENT UNITS 10CONNECTED MATH FRAMEWORK 11MATHEMATICS LEARNING GOALS 14CONTENT GOALS IN EACH UNIT 18ALIGNMENT WITH STANDARDS 21PROCESS STANDARDS 21COMMON CORE STANDARDS AND BLUEPRINT 23NEW JERSEY CORE CURRICULUM CONTENT STANDARDS 42

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PHILOSOPHY The philosophy upon which the Mathematics Curriculum Guide is to encourage and support the enjoyment of learning mathematics, as a way to make sense of the world in students’ everyday lives. Mathematics is everywhere, from the practicalities of counting, to find easier ways of organizing numbers and data to model and represent daily life experiences. Mathematics involves other disciplines, and is a way in which ideas are communicated, such as in tables and graphs.

Mathematics is developmental by nature. Therefore it is important that should any concerns arise related to mathematics understanding, that this is communicated with the student’s teacher as soon as possible. There are varied approaches used to teach and learn mathematics, which is referred to as a balanced mathematics approach. This includes traditional algorithms to approaching the study of mathematics that have been used for many years, along with newer and varied approaches, to provide multiple representations to model solving a problem.

The study of mathematics provides pathways to higher level thinking skills. As students learn mathematics, specialized terminology assist their development. This enables students to not only learn mathematics in a routine way, but to enable them to become problem solvers in novel situations, able to draw on a repertoire of skills and approaches.

We hope these beliefs will assist students to develop their understanding to use mathematics to make meaning, as well as to promote their critical thinking and development as lifelong learners. The goals are to promote problem-solving, and communication, to foster an understanding of the world, that has a conceptual foundation in the study of mathematics.

Vision

In Orange, we recognize that each student is unique and that the purpose of education is to enable every student to acquire the learning skills necessary to compete in the global community. It is essential that we provide a rigorous, high-quality Mathematics curriculum that allows each student’s talents and abilities to be developed to their full potential.

Purpose

The Curriculum Guide was prepared by teachers and administrators with input from consultants who have expertise in Mathematics. Students and parents are welcome to read, review, and ask questions about the curriculum, to understand what they and their children are learning.

The Mathematics Curriculum Guide is based on an alignment with the New Jersey Core Content Curriculum Standards, and the Common Core State Standards which are a national set of shared standards which adopted by over 30 states. It is also based on national standards shared through the National Council of Teachers of Mathematics, which develops agreed upon content at each grade level.

Content was designed with a student development perspective across each grade, as well as a vertical articulation, with spirals learning upward, based on the foundation that is developed.

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 Mathematic Process Goals

In setting mathematical goals for a school curriculum, the choice of content topics must be accompanied by an analysis of the kinds of thinking students will be able to demonstrate upon completion of the curriculum. The text below describes the eleven key mathematical processes developed in all the main content strands used in the Mathematics program.

CountingDetermining the number of elements in finite data sets, trees, graphs, or combinations by application of mental computation, estimation, counting principles, calculators and computers, and formal algorithms

VisualizingRecognizing and describing shape, size, and position of one-, two-, and three-dimensional objects and their images under transformations; interpreting graphical representations of data, functions, relations, and symbolic expressions

ComparingDescribing relationships among quantities and shapes using concepts such as equality and inequality, order of magnitude, proportion, congruence, similarity, parallelism, perpendicularity, symmetry, and rates of growth or change

EstimatingDetermining reasonableness of answers; using "benchmarks" to estimate measures; using various strategies to approximate a calculation and to compare estimates

MeasuringAssigning numbers as measures of geometric objects and probabilities of events; choosing appropriate measures in a decision-making problem, choosing appropriate units or scales and making approximate measurements or applying formal rules to find measures

ModelingConstructing, making inferences from, and interpreting concrete, symbolic, graphic, verbal, and algorithmic models of quantitative, visual, statistical, probabilistic, and algebraic relationships in problem situations; translating information from one model to another

ReasoningBringing to any problem situation the disposition and ability to observe, experiment, analyze, abstract, induce, deduce, extend, generalize, relate, and manipulate in order to find solutions or prove conjectures involving interesting and important patterns

ConnectingIdentifying ways in which problems, situations, and mathematical ideas are interrelated and applying knowledge gained in solving one problem to other problems

RepresentingMoving flexibly among graphic, numeric, symbolic, and verbal representations and recognizing the importance of having various representations of information in a situation

Using ToolsSelecting and intelligently using calculators, computers, drawing tools, and physical models to represent, simulate, and manipulate patterns and relationships in problem settings

Becoming MathematiciansHaving the disposition and imagination to inquire, investigate, tinker, dream, conjecture, invent, and communicate with others about mathematical ideas

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Phases of Instruction

Problem-centered teaching opens the mathematics classroom to exploring, conjecturing, reasoning, and communicating. For this model of instruction, there are three phases: Launch, Explore, and Summarize.

Launch

In the first phase, the teacher launches the problem with the whole class. This involves helping students understand the problem setting, the mathematical context, and the challenge. The following questions can help the teacher prepare for the launch:

• What are students expected to do?

• What do the students need to know to understand the context of the story and the challenge of the problem?

• What difficulties can I foresee for students?

• How can I keep from giving away too much of the problem solution?

The launch phase is also the time when the teacher introduces new ideas, clarifies definitions, reviews old concepts, and connects the problem to past experiences of the students. It is critical that, while giving students a clear picture of what is expected, the teacher leaves the potential of the task intact. He or she must be careful to not tell too much and consequently lower the challenge of the task to something routine, or to cut off the rich array of strategies that may evolve from a more open launch of the problem.

Explore

The nature of the problem suggests whether students work individually, in pairs, in small groups, or occasionally as a whole class to solve the problem during the explore phase. The Teacher's Guide suggests an appropriate grouping. As students work, they gather data, share ideas, look for patterns, make conjectures, and develop problem-solving strategies.

It is inevitable that students will exhibit variation in their progress. The teacher's role during this phase is to move about the classroom, observing individual performance and encouraging on-task behavior. The teacher helps students persevere in their work by asking appropriate questions and providing confirmation and redirection where needed. For students who are interested in and capable of deeper investigation, the teacher may provide extra questions related to the problem. These questions are called Going Further and are provided in the explore discussion in the Teacher's Guide. Suggestions for helping students who may be struggling are also provided in the Teacher's Guide. The explore part of the instruction is an appropriate place to attend to differentiated learning.

The following questions can help the teacher prepare for the explore phase:

• How will I organize the students to explore this problem? (Individuals? Pairs? Groups? Whole class?)

• What materials will students need?

• How should students record and report their work?

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• What different strategies can I anticipate they might use?

• What questions can I ask to encourage student conversation, thinking, and learning?

• What questions can I ask to focus their thinking if they become frustrated or off-task?

• What questions can I ask to challenge students if the initial question is "answered"?

As the teacher moves about the classroom during the explore, she or he should attend to the following questions:

• What difficulties are students having?

• How can I help without giving away the solution?

• What strategies are students using? Are they correct?

• How will I use these strategies during the summary?

Summarize

It is during the summary that the teacher guides the students to reach the mathematical goals of the problem and to connect their new understanding to prior mathematical goals and problems in the unit. The summarize phase of instruction begins when most students have gathered sufficient data or made sufficient progress toward solving the problem. In this phase, students present and discuss their solutions as well as the strategies they used to approach the problem, organize the data, and find the solution. During the discussion, the teacher helps students enhance their conceptual understanding of the mathematics in the problem and guides them in refining their strategies into efficient, effective, generalizable problem-solving techniques or algorithms.

Although the summary discussion is led by the teacher, students play a significant role. Ideally, they should pose conjectures, question each other, offer alternatives, provide reasons, refine their strategies and conjectures, and make connections. As a result of the discussion, students should become more skillful at using the ideas and techniques that come out of the experience with the problem.

If it is appropriate, the summary can end by posing a problem or two that checks students' understanding of the mathematical goal(s) that have been developed at this point in time. Check for Understanding questions occur occasionally in the summary in the Teacher's Guide. These questions help the teacher to assess the degree to which students are developing their mathematical knowledge. The following questions can help the teacher prepare for the summary:

• How can I help the students make sense of and appreciate the variety of methods that may be used?

• How can I orchestrate the discussion so that students summarize their thinking about the problem?

• What questions can guide the discussion?

• What concepts or strategies need to be emphasized?

• What ideas do not need closure at this time?

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• What definitions or strategies do we need to generalize?

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Target Goals

Number and Operation GoalsNumber Sense Use numbers in various forms to solve problems Understand and use large numbers, including in exponential and scientific notation Reason proportionally in a variety of contexts using geometric and numerical reasoning, including scaling and solving proportions Compare numbers in a variety of ways, including differences, rates, ratios, and percents and choose when each comparison is appropriate Order positive and/or negative rational numbers Make estimates and use benchmarks

Operations and Algorithms Use the order of operations to write, evaluate, and simplify numerical expressions

Properties Use the commutative and distributive properties to write equivalent numerical expressions

Data and Probability GoalsFormulating Questions Formulate questions that can be answered through data collection and analysis Design data collection strategies to gather data to answer these questions Design experiments and simulations to test hypotheses about probability situations

Data Collection Carry out data collection strategies to answer questions Distinguish between samples and populations Characterize samples as representative or non- representative, as random Use these characterizations to evaluate the quality of the collected data

Data Analysis Organize, analyze, and interpret data to make predictions, construct arguments, and make decisions Informally evaluate the significance of differences between sets of data Use information from samples to draw conclusions about populations

Probability Compute and compare the chances of various outcomes, including two-stage outcomes

Geometry and Measurement GoalsShapes and Their Properties Build and visualize three-dimensional figures from various two-dimensional representations and vice versa Recognize and use shapes and their properties to make mathematical arguments and to solve problems Use the Pythagorean Theorem and properties of special triangles (e.g. isosceles right triangles) to solve problems Use a coordinate grid to describe and investigate relationships among shapes Recognize and use standard, essential geometric vocabulary

Transformations-Symmetry, Similarity, and Congruence Recognize line, rotational, and translational symmetries and use them to solve problems Predict ways that similarity and congruence transformations affect lengths, angle measures, perimeters, areas, volume, and orientation Investigate the effects of combining one or more transformations of a shape Identify and use congruent triangles and/or quadrilaterals to solve problems about shapes and measurement Use a coordinate grid to explore and verify similarity and congruence relationships

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Measurement Use measurement concepts to solve problems

Geometric Connections Use geometric concepts to build understanding of concepts in other areas of mathematics Connect geometric concepts to concepts in other areas of mathematics

Algebra GoalsPatterns of Change-Functions Identify and use variables to describe relationships between quantitative variables in order to solve problems or make decisions Recognize and distinguish among patterns of change associated with linear, inverse, exponential and quadratic functions

Representation Construct tables, graphs, symbolic expressions and verbal descriptions and use them to describe and predict patterns of change in variables Move easily among tables, graphs, symbolic expressions, and verbal descriptions Describe the advantages and disadvantages of each representation and use these descriptions to make choices when solving problems Use linear, inverse, exponential and quadratic equations and inequalities as mathematical models of situations involving variables

Symbolic Reasoning Connect equations to problem situations Connect solving equations in one variable to finding specific values of functions Solve linear equations and inequalities and simple quadratic equations using symbolic methods Find equivalent forms of many kinds of equations, including factoring simple quadratic equations Use the distributive and commutative properties to write equivalent expressions and equations Solve systems of linear equations Solve systems of linear inequalities by graphing

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CONNECTED MATH

Description of Units

Thinking With Mathematical Models

Linear and Inverse Variation - introduction to functions and modeling; finding the equation of a line; inverse functions; inequalities

Looking for Pythagoras

The Pythagorean Theorem - square roots; the Pythagorean Theorem; connections amongcoordinates, slope, distance, and area; distances in the plane

Growing, Growing, Growing

Exponential Relationships - recognize and represent exponential growth and decay in tables, graphs, words, and symbols; rules of exponents; scientific notation

Frogs, Fleas and Painted Cubes

Quadratic Relationships - recognize and represent quadratic functions in tables, graphs, words and symbols; factor simple quadratic expressions

Kaleidoscopes, Hubcaps and Mirrors

Symmetry and Transformations - symmetries of designs, symmetry transformations, congruence, congruence rules for triangles

Say It With Symbols

Making Sense of Symbols - equivalent expressions, substitute and combine expressions, solve quadratic equations, the quadratic formula

Shapes of Algebra

Linear Systems and Inequalities - coordinate geometry, solve inequalities, standard form of linear equations, solve systems of linear equations and linear equalities.

Samples and Populations

Data and Statistics - use samples to reason about populations and make predictions, compare samples and sample distributions, relationships among attributes in data sets

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CONNECTED MATH FRAMEWORK

ALGEBRA

Thinking With Mathematical Models – Grade 8 Linear Functions, Equations, and inequalities Inverse Variation Mathematical Modeling

Looking for Pythagoras Finding Area and Distance Square Roots Using Squares to Find Lengths of Segments Developing and Using the Pythagorean Theorem A Proof of the Pythagorean Theorem Using the Pythagorean Theorem to Find Lengths The Converse of the Pythagorean Theorem Special Right Triangles Rational and Irrational Numbers Converting Repeating Decimals to Fractions Proof that √2 is irrational Square Root Versus Decimal Approximation Number Systems

Growing, Growing, Growing Exponential Growth Growth Factor Exponential Equations y‐Intercept or Initial Value Growth Rates Exponential Decay Graphs of Exponential Relationships Tables of Exponential Relationships: Recursive or Iterative Processes Equivalence of Two Forms of Exponential Functions Logarithms Rules of Exponents

Frogs, Fleas & Painted Cubes Representing Quadratic Functions with Equations Representing Quadratic Patterns of Change with Tables Connecting Patterns of Change to Calculus Extending Patterns of Change to Cubic Functions or Polynomial Functions Representing Quadratic Patterns of Change with Graphs Maximum/Minimum Points The Line of Symmetry · x Intercepts ‐ The Distributive Property and Equivalent Quadratic Expressions A Note on Terminology Other Contexts for Quadratic Functions

o Counting Handshakes o Sum of the First n Counting Numbers o Generalization of Guass’s Method

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o Triangular Numbers o Equations Modeling Projectile Motion

Say It With Symbols Equivalent Expressions

o Verifying Equivalenceo Notes on the Distributive Property o Interpreting Expressions o A Note on the Use of Expression and Equations

Combining Expressions o Adding Expressionso Creating New Expressions by Substitution

Solving Equationso Solving Linear Equations o Solving Quadratic Equations o A Note on Factoring

Predicting the Underlying Patterns of Change o Patterns of Changeo Predicting Linear Patterns of Changeo Writing Equations for Linear, Exponential, and Quadratic Functions Given Two Points

Reasoning With Symbolso Using Symbolic Statements to Confirm a Conjecture

The Shapes of Algebra Equations of Circles Linear Inequalities Systems of Linear Equations

o Graphic Solution of Systems o Equivalent Form o Solving Systems by Substitutiono Solving Systems by Linear Combination

· Types of Solutions o Solving Linear Inequalities in Two Variables o Solving Systems of Linear Inequalities

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DATA ANALYSIS AND PROBABILITYSamples and Populations The process of statistical investigation (doing meaningful statistics) Distinguishing different types of data

o Attributes and values o Categorical or numerical values o Understanding the concept of distribution o Exploring the concept of variability o Making sense of a data set

§ Using standard graphical representations Line plot Histogram Box and whisker plot ‐ ‐ Scatter plot

§ Reading Standard Graphs§ Using Summary Statistics

o Comparing data setso Exploring the concept of sampling

Exploring the concept of covariation or association

GEOMETRYLooking for Pythagoras Finding Area and Distance Square Roots Using Squares to Find Lengths of Segments Developing and Using the Pythagorean Theorem • A Proof of the Pythagorean Theorem Using the Pythagorean Theorem The Converse of the Pythagorean Theorem Special Right Triangles Rational and Irrational Numbers Converting and Repeating Decimals to Fractions Proof that √2 Is Irrational Square Root Versus Decimal Approximation Number Systems

Kaleidoscopes, Hubcaps, and Mirrors Types of Symmetry Making Symmetric Designs Using Tools to Investigate Symmetries

o Transparent Reflection o Hinged Mirrors o Finding perpendicular bisectors

Symmetric Transformations Congruent Figures Reasoning From Symmetry and Congruence Coordinate Rules for Symmetry Transformations Combining Transformations

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Mathematics Learning GoalsConnected Mathematics develops four mathematical strands:

Number and Operation

Geometry and Measurement

Data Analysis and Probability

Algebra.

Goals by Mathematical Strand

Number and Operation GoalsNumber Sense

Use numbers in various forms to solve problems

Understand and use large numbers, including in exponential and scientific notation

Reason proportionally in a variety of contexts using geometric and numerical reasoning, including scaling and solving proportions

Compare numbers in a variety of ways, including differences, rates, ratios, and percents and choose when each comparison is appropriate

Order positive and/or negative rational numbers

Express rational numbers in equivalent forms

Make estimates and use benchmarks

Operations and Algorithms

Develop understanding and skill with all four arithmetic operations on fractions and decimals (6)

Develop understanding and skill in solving a variety of percent problems

Use the order of operations to write, evaluate, and simplify numerical expressions

Develop fluency with paper and pencil computation, calculator use, mental calculation, estimation; and choose among these when solving problems

Properties

Understand the multiplicative structure of numbers, including the concepts of prime and composite numbers, evens, odds, and prime factorizations

Use the commutative and distributive properties to write equivalent numerical expressions

Data and Probability Goals

Formulating Questions

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Formulate questions that can be answered through data collection and analysis

Design data collection strategies to gather data to answer these questions

Design experiments and simulations to test hypotheses about probability situations

Data Collection

Carry out data collection strategies to answer questions

Distinguish between samples and populations

Characterize samples as representative or non- representative, as random

Use these characterizations to evaluate the quality of the collected data

Data Analysis

Organize, analyze, and interpret data to make predictions, construct arguments, and make decisions

Use measures of center and spread to describe and to compare data sets

Be able to read, create, and choose data representations, including bar graphs, line plots, coordinate graphs, box and whisker plots, histograms, and stem and leaf plots

Informally evaluate the significance of differences between sets of data

Use information from samples to draw conclusions about populations

Probability

Distinguish between theoretical and experimental probabilities and understand the relationship between them

Use probability concepts to make decisions

Find and interpret expected value

Compute and compare the chances of various outcomes, including two-stage outcomes

Geometry and Measurement Goals

Shapes and Their Properties

Generate important examples of angles, lines, and two- and three-dimensional shapes (6)

Categorize, define, and relate figures in a variety of representations

Understand principles governing the construction of shapes with reasons why certain shapes serve special purposes(e.g. t riangles for trusses)

Build and visualize three-dimensional figures from various two-dimensional representations and vice versa

Recognize and use shapes and their properties to make mathematical arguments and to solve problems

Use the Pythagorean Theorem and properties of special triangles (e.g. isosceles right triangles) to solve problems

Use a coordinate grid to describe and investigate relationships among shapes

Recognize and use standard, essential geometric vocabulary

• Transformations-Symmetry, Similarity, and Congruence

Recognize line, rotational, and translational symmetries and use them to solve problems

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Use scale factor and ratios to create similar figures or determine whether two or more shapes are similar or congruent

Predict ways that similarity and congruence transformations affect lengths, angle measures, perimeters, areas, volume, and orientation

Investigate the effects of combining one or more transformations of a shape

Identify and use congruent triangles and/or quadrilaterals to solve problems about shapes and measurement

Use properties of similar figures to solve problems about shapes and measurement

Use a coordinate grid to explore and verify similarity and congruence relationships

Measurement

Understand what it means to measure an attribute of a figure or a phenomenon

Estimate and measure angles, line segments, areas, and volumes using tools and formulas

Relate angle measure and side lengths to the shape of a polygon

Find area and perimeter of rectangles, parallelograms, triangles, circles, and irregular figures

Find surface area and volume of rectangular solids, cylinders, prisms, cones, and pyramids and find the volume of spheres

Relate units within and between the customary and metric systems

Use ratios and proportions to derive indirect measurements

Use measurement concepts to solve problems

Geometric Connections

Use geometric concepts to build understanding of concepts in other areas of mathematics

Connect geometric concepts to concepts in other areas of mathematics

Algebra Goals

Patterns of Change-Functions

Identify and use variables to describe relationships between quantitative variables in order to solve problems or make decisions

Recognize and distinguish among patterns of change associated with linear, inverse, exponential and quadratic functions (

Representation

Construct tables, graphs, symbolic expressions and verbal descriptions and use them to describe and predict patterns of change in variables

Move easily among tables, graphs, symbolic expressions, and verbal descriptions

Describe the advantages and disadvantages of each representation and use these descriptions to make choices when solving problems

Use linear, inverse, exponential and quadratic equations and inequalities as mathematical models of situations involving variables

Symbolic Reasoning

Connect equations to problem situations

Connect solving equations in one variable to finding specific values of functions

Solve linear equations and inequalities and simple quadratic equations using symbolic methods

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Find equivalent forms of many kinds of equations, including factoring simple quadratic equations

Use the distributive and commutative properties to write equivalent expressions and equations

Solve systems of linear equations

Solve systems of linear inequalities by graphing

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CONTENT GOALS IN EACH UNIT

Looking For Pythagoras (Algebra)

Relate the area of a square to the length of a side of the square

Estimate square roots

Develop strategies for finding the distance between two points on a coordinate grid

Understand and apply the Pythagorean Theorem

Use the Pythagorean Theorem to solve a variety of problems

Kaleidoscopes, Hubcaps, and Mirrors (Geometry)

Understand important properties of symmetry

Recognize and describe symmetries of figures

Use tools to examine symmetries and transformations

Make figures with specified symmetries

Identify basic design elements that can be used to replicate a given design

Perform symmetry transformations of figures, including reflections, translations, and rotations

Examine and describe the symmetries of a design made from a figure and its image(s) under a symmetry transformation

Give precise mathematical directions for performing reflections, rotations, and translations

Draw conclusions about a figure, such as measures of sides and angles, lengths of diagonals, or intersection points of diagonals, based on symmetries of the figure

Understand that figures with the same shape and size are congruent

Use symmetry transformations to explore whether two figures are congruent

Give examples of minimum sets of measures of angles and sides that will guarantee that two triangles are congruent

Use congruence of triangles to explore congruence of two quadrilaterals

Use symmetry and congruence to deduce properties of figures

Write coordinate rules for specifying the image of a general point (x, y) under particular transformations

Use transformational geometry to describe motions, patterns, designs, and properties of shapes in the real world

Thinking With Mathematical Models (Algebra)

Recognize linear and non-linear patterns in contexts, tables and graphs and describe those patterns using words and symbolic expressions

Write equations to express linear patterns appearing in tables, graphs, and verbal contexts

Write linear equations when specific information, such as two points or a point and a slope, is given for a line

Approximate linear data patterns with graph and equation models

Solve linear equations

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Interpret inequalities

Write equations describing inverse variation

Use linear and inverse variation equations to solve problems and to make predictions and decisions

Frogs Fleas and Painted Cubes (Algebra)

Recognize the patterns of change for quadratic relationships in a table, graph, equation, and problem situation

Construct equations to express quadratic relationships that appear in tables, graphs and problem situations

Recognize the connections between quadratic equations and patterns in tables and graphs of those relationships

Use tables, graphs, and equations of quadratic relationships to locate maximum and minimum values of a dependent variable and the x- and y-intercepts and other important features of parabolas.

Recognize equivalent symbolic expressions for the dependent variable in quadratic relationships

Use the distributive property to write equivalent quadratic expressions in factored form or expanded form

Use tables, graphs, and equations of quadratic relations to solve problems in a variety of situations from geometry, science, and business

Compare properties of quadratic, linear, and exponential relationships

Say It With Symbols (Algebra)

Model situations with symbolic statements Write equivalent expressions Determine if different symbolic expressions are mathematically equivalent Interpret the information equivalent expressions represent in a given context Determine which equivalent expression to use to answer particular questions; Solve linear equations involving parentheses Solve quadratic equations by factoring Use equations to make predictions and decisions Analyze equations to determine the patterns of change in the tables and graphs that the equation represents Understand how and when symbols should be used to display relationships, generalizations, and proofs

The Shapes of Algebra (Algebra)

Write and use equations of circles Determine lines are parallel or perpendicular by looking at patterns in their graphs, coordinates, and equations Find coordinates of points that divide line segments in various ratios Write inequalities that satisfy given situations Find solutions to inequalities represented by a graph or an equation Solve systems of linear equations by graphing, combining equations, and by substitution Write linear inequalities in two variables to match constraints in problem conditions Graph linear inequalities and systems of inequalities and use the results to solve problems

Samples and Populations (Data Analysis)

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Revisit and use the process of statistical investigation to explore problems Distinguish between samples and populations and use information drawn from samples to draw conclusions about populations Explore the influence of sample size and of random or nonrandom sample selection Apply concepts from probability to select random samples from populations Compare sample distributions using measures of center (mean or median), measures of dispersion (range or percentiles), and

data displays that group data (histograms and box-and-whisker plots) Explore relationships between paired values of numerical attributes

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Alignment with Standards

Number and Operations

• Looking for Pythagoras (Grade 8)• Clever Counting (Grade 8) Algebra

• Thinking With Mathematical Models (Grade 8)• Looking for Pythagoras (Grade 8)• Growing, Growing, Growing (Grade 8)• Frogs, Fleas, and Painted Cubes (Grade 8)• Say It With Symbols (Grade 8)• Shapes of Algebra (Grade 8)Geometry

• Looking for Pythagoras (Grade 8)• Kaleidoscopes, Hubcaps, and Mirrors (Grade 8)Measurement

• Looking for Pythagoras (Grade 8)

Data Analysis and Probability

• Distributions (Grade 8)• Samples and Populations (Grade 8)• Clever Counting (Grade 8)

Process Standards

Problem SolvingAll unitsBecause Connected Mathematics is a problem- centered curriculum, problem solving is an important part of every unit.

Reasoning and ProofAll unitsThroughout the curriculum, students are encouraged to look for patterns, make conjectures, provide evidence for their conjectures, refine their conjectures and strategies, connect their knowledge, and extend their findings. Informal reasoning evolves into more deductive arguments as students proceed from Grade 6 through Grade 8.

CommunicationAll unitsAs students work on the problems, they must communicate ideas with others. Emphasis is placed on students' discussing problems in class, talking through their solutions, formalizing their conjectures and strategies, and learning to communicate their ideas to a more general audience. Students learn to express their ideas, solutions, and strategies using written explanations, graphs, tables, and equations.

ConnectionsAll unitsIn all units, the mathematical content is connected to other units, to other areas of mathematics, to other school subjects, and to applications in the real world. Connecting and building on prior knowledge is important for building and retaining new knowledge.

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RepresentationAll unitsThroughout the units, students organize, record, and communicate information and ideas using words, pictures, graphs, tables, and symbols. They learn to choose appropriate representations for given situations and to translate among representations. Students also learn to interpret information presented in various forms.

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Common Core Standards » Mathematics » Grade 8THE NUMBER SYSTEMKnow that there are numbers that are not rational, and approximate them by rational numbers.

• 1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

• 2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

EXPRESSIONS AND EQUATIONSExpressions and Equations work with radicals and integer exponents.

• 1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.• 2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number.

Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.• 3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to

express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.

• 4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Understand the connections between proportional relationships, lines, and linear equations.

• 5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

• 6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Analyze and solve linear equations and pairs of simultaneous linear equations.

• 7. Solve linear equations in one variable.

• Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

• Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

• 8. Analyze and solve pairs of simultaneous linear equations.

• Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

• Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

FUNCTIONSDefine, evaluate, and compare functions.

• 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1

• 2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

• 3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Use functions to model relationships between quantities.

• 4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from

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a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

GEOMETRYUnderstand congruence and similarity using physical models, transparencies, or geometry software.

• 1. Verify experimentally the properties of rotations, reflections, and translations:• a. Lines are taken to lines, and line segments to line segments of the same length.• b. Angles are taken to angles of the same measure.• c. Parallel lines are taken to parallel lines.• 2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations,

reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.• 3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.• 4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations,

reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

• 5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Understand and apply the Pythagorean Theorem.

• 6. Explain a proof of the Pythagorean Theorem and its converse.• 7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and

three dimensions.• 8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

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STATISTICS AND PROBABILITYInvestigate patterns of association in bivariate data.

• 1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

• 2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

• 3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

• 4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

GRADE 8 Math

Numbers, Number Systems and Number Relationships

A. Represent and use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, exponents, scientific notation, square roots).

B. Simplify numerical expressions involving exponents, scientific notation and using order of operations.

Students understand the different ways numbers and number systems are used in the real world.

1. Know word names and standard numerals for integers, fractions, decimals, ratios, numbers expressed as percents, numbers with exponents, numbers expressed in scientific notation, absolute value, radicals, and ratios.

2. Read and write whole numbers and decimals in expanded form, including exponential notation.

3. Compare and order fractions, decimals, integers, numbers expressed in absolute value, scientific notation, percents, numbers with exponents, ratios and radicals.

4. Understand the meanings of rational and irrational numbers.

By the end of 8th grade students will:

Associate verbal names, written word names and standard numerals with integers, fractions, decimals; numbers expressed as percents; numbers with exponents; numbers in scientific notation; radicals; absolute value; and ratios. (make use of the game concentration with index cards)

Order (on a number line or using graphic models, number lines, and symbols) and diagram the relative size of integers, fractions, and decimals; numbers expressed as percents; numbers with exponents; numbers in scientific notation; radicals; absolute value; and ratios.

Describe the meanings of rational and irrational numbers using physical or graphical displays.

Give examples of rational and irrational numbers in real-world situations.

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C. Distinguish between and order rational and irrational numbers.

D. Apply ratio and proportion to mathematical problem situations involving distance, rate, time and similar triangles.

E. Simplify and expand algebraic expressions using exponential forms.

F. Use the number line model to demonstrate integers and their applications.

G. Use the inverse relationships between addition, subtraction, multiplication, division, exponentiation and root extraction to determine unknown quantities in equations.

5. Know the relationships among fractions, decimals, and percents given a real-world context.

6. Know how to simplify expressions using integers, exponents, and radicals.

7. Understand the concept of the absolute value of a number.

8. Know the relationship between rational numbers and negative exponents by investigating the powers of 10 from 104 through 10 -4.

9. Know the different properties of the numbers between 0 and 1, including the differences in these numbers when squared, inverted, and multiplied by whole numbers and negative fractions.

10. Understand the structure of number systems other than the decimal number system, for example, bases other than 10 and the binary number system.

Students understand the effects of operations on numbers and the relationships among these operations; select appropriate operations, and are able to compute for various problem-solving situations.

1. Know the effects of the four basic operations on whole numbers, fractions, mixed numbers, decimals, and integers.

Construct models to represent rational and irrational numbers.

Express a given quantity in a variety of ways (for example, integers, fractions, decimals, numbers expressed as a percent, numbers expressed in scientific notation, ratios).

Evaluate numerical or algebraic expressions that contain exponential notation.

Express the real-world applications to Absolute Value. Give concrete examples in science, society and technology.

Express rational numbers in exponential notation including negative exponents (for example, 2 -3 = 1/23= 1/8).

Express numbers in scientific or standard notation including decimals between 0 and 1.

Understand and use ratios, proportions, and percents in a variety of situations

Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number

Computation and Estimation

A. Complete calculations by applying the order of operations.

B. Add, subtract, multiply and divide different kinds and forms of rational numbers including integers, decimal fractions, percents and proper and improper fractions.

Students understand the effects of operations on numbers and the relationships among these operations; select appropriate operations, and are able to compute for various problem-solving situations.

1. Know the effects of the four basic operations on whole numbers, fractions, mixed numbers, decimals, and integers.

2. Apply the properties of real numbers to solve problems (commutative, associative, distributive, identity, equality,

By the end of 8th grade students will be able to:

Express base ten numbers as equivalent numbers in different bases, such as base two, base five, and base eight. Express non-base ten numbers as equivalent numbers in base ten. Discuss the application of the binary (base two) number system in computer technology.

Investigate the structure of number systems other than the

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C. Estimate the value of irrational numbers.D. Estimate amount of tips and discounts using

ratios, proportions and percents.

E. Determine the appropriateness of overestimating or underestimating in computation.

F. Identify the difference between exact value and approximation and determine which is appropriate for a given situation.

Career

• Analyze budgets and pay statements, such as, but not limited to: Charitable contributions Expenses Gross pay Net pay Other income Savings Taxes

inverse, and closure).

3. Solve real-world problems involving percents (for example, discounts, simple interest, taxes, tips).

1. Know proportional relationships.

Students understand and apply properties of numbers and operations.

1. Know the inverse relationship of positive and negative numbers.

2. Know the appropriate operation to solve real-world problems involving integers, ratios, rates, proportions, numbers expressed as percents, decimals, fractions, and square roots.

3. Solve multi-step, real-world problems involving integers, ratios, proportions, numbers expressed as percents, decimals, and fractions.

4. Solve real-world problems involving percents including percents greater than 100% (for example percent of change,commission).

decimal number system.

Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

Pencil-and-paper Mental math Calculator

Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.

Estimate square roots and cube roots of numbers

Use equivalent representation of numbers such as fractions, decimals, and percents to facilitate estimation

Recognize the limitations of estimation and assess the amount of error resulting from estimation

Use models or pictures to show the effects of addition, subtraction, multiplication, and division on whole numbers, decimals, fractions, mixed numbers, and integers.

Solve real-world problems involving decimals and fractions using two- or three-step problems.

Use tables & graphs relationships to explain problems.Measurement and Estimation

A. Develop formulas and procedures for determining measurements (e.g., area, volume, distance).

B. Solve rate problems (e.g., rate time = distance, principal interest rate = interest).

Students measure quantities in the real world and use these measures to solve problems.

1. Understand strategies used to solve real-world problems involving surface area and volume of three-dimensional shapes.

2. Explore and derive formulas for surface area and volume of three-dimensional regular shapes, including pyramids, prisms, and cones.

By the end of 8th grade students will be able to:

Use equivalent representation of numbers such as fractions, decimals, and percents to facilitate estimation

Recognize the limitations of estimation and assess the amount of error resulting from estimation

Build three-dimensional solids using the two-dimensional models as faces. Predict the surface areas, and then test predictions.

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C. Measure angles in degrees and determine relations of angles.

D. Estimate, use and describe measures of distance, rate, perimeter, area, volume, weight, mass and angles.

E. Describe how a change in linear dimension of an object affects its perimeter, area and volume.

F. Use scale measurements to interpret maps or drawings.

G. Create and use scale models.

3. Know and apply formulas for finding rates, distance, time and angle measures.

4. Develop an understanding of rate of change as it applies to real-world problems.

5. Know that new figures can be created by increasing or decreasing the original dimensions.

6. Know how changes in the volume, surface area, area, or perimeter of a figure affect the dimensions of the figure.

7. Solve real world or mathematical problems involving the effects of changes either to the dimensions of a figure orto the volume, surface area, area, perimeter, or circumference of figures.

Students compare, contrast, and convert within systems of measurement; both standard with nonstandard and metric with customary.

1. Know relationships between metric units of mass and capacity (for example, one cubic centimeter of water weighsone gram).

2. Find measures of length, weight or mass, and capacity or volume using proportional relationships and properties of similar geometric figures.

Students select and use appropriate units and instruments for measurement to achieve the degree of precision and accuracy required in real-world situations.

1. Solve problems using mixed units within each system, such as feet and inches, hours and minutes.

2. Solve problems using the conversion of measurements within the customary system.

3. Determine the appropriate precision unit for a given situation.

Using centimeter cubes as a guide, estimate the volume of each model. Working with a group, tests estimations and contribute to group consensus on a working formula for finding the volume of various three-dimensional models.

Describe and use rates of change (for example, temperature as it changes throughout the day, or speed as the rate of change in distance over time) and other derived measures.

Describe how a change in a figure’s dimensions affects its perimeter, area, circumference, surface area, or volume.

Investigate congruent figures with respect to volume and surface area, and describe the differences in their dimensions.

Describe how the change of a figure in dimensions such as length, width, height, or radius affects its other measurements such as perimeter, area, surface area, and volume.

Measure length, weight or mass, and capacity or volume using customary or metric units.

Apply properties of similarity with shadow measurement and properties of similar triangles to find the height of a flagpole.

Select appropriate units of measurement in a real-world context.

Apply the conversion of measurements within the customary system to real world problems.

Select and use appropriate instruments, technology, and techniques to measure quantities and dimensions to a specified degree of accuracy.

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4. Measure accurately with measurement tools to the specified degree of accuracy for the task and in keeping with the precision of the measurement tool.

Mathematical Reasoning and Connections

A. Make conjectures based on logical reasoning and test conjectures by using counter-examples.

B. Combine numeric relationships to arrive at a conclusion.

C. Use if...then statements to construct simple, valid arguments.

D. Construct, use and explain algorithmic procedures for computing and estimating with whole numbers, fractions, decimals and integers.

E. Distinguish between inductive and deductive reasoning.

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

Students identify patterns and makes predictions from an orderly display of data using concepts of probability and statistics.

1. Compare and explain the results of an experiment with the mathematically expected outcomes.

2. Explain observed difference between mathematical and experimental results.

3. Predict the mathematical odds for and against a specified outcome in a given real-world situation.

By the end of 8th grade students will be able to:

Calculate simple mathematical probabilities for independent and dependent events.

Compare the results of an experiment with the expected theoretical outcomes.

Design experiments of chance and predict outcomes of odds, for and against.

Mathematical Problem Solving and Communication

Students understand the effects of operations on numbers and the relationships among these operations; select

By the end of 8th grade students will be able to:

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A. Invent, select, use and justify the appropriate methods, materials and strategies to solve 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.

C. Justify strategies and defend approaches used and conclusions reached.

D. Determine pertinent information in problem situations and whether any further information is needed for solution.

Career Education Work • Analyze budgets and pay statements,

such as, but not limited to: Charitable contributions Expenses Gross pay Net pay Other income Savings Taxes

Analyze the relationship of school

appropriate operations, and are able to compute for various problem-solving situations.

1. Know the effects of the four basic operations on whole numbers, fractions, mixed numbers, decimals, and integers.

2. Apply the properties of real numbers to solve problems (commutative, associative, distributive, identity, equality, inverse, and closure).

3. Solve real-world problems involving percents (for example, discounts, simple interest, taxes, tips).

Students make reasonable estimates.

1. Know an appropriate estimation technique for a given situation using whole numbers, fractions and decimals.

2. Estimate to predict results and check reasonableness of results.

3. Determine whether an exact answer is needed or whether an estimate would be sufficient.

Students identify patterns and makes predictions from an orderly display of data using concepts of probability and statistics.

1. Compare and explain the results of an experiment with the mathematically expected outcomes.

2. Explain observed difference between mathematical and experimental results.

3. Predict the mathematical odds for and against a specified outcome in a given real-world situation.

Express base ten numbers as equivalent numbers in different bases, such as base two, base five, and base eight. Express non-base ten numbers as equivalent numbers in base ten. Discuss the application of the binary (base two) number system in computer technology.

Investigate the structure of number systems other than the decimal number system.

Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

Pencil-and-paper Mental math Calculator

Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.

Write and simplify expressions from real-world situations using the order of operations.

Use appropriate methods of computation, such as mental computation, paper and pencil, and calculator.

Justify the choice of method for calculations, such as mental computation, concrete materials, algorithms, or calculators.

Explain a variety of estimation techniques including clustering, compatible number, and front-end.

Execute the 4 steps of problem solving.

Give examples in real world situations where estimation is sufficient for the situation.

Recognize the limitations of estimation and assess the amount of error resulting from estimation

Calculate simple mathematical probabilities for independent 32

subjects, extracurricular activities, and community experiences to career preparation

and dependent events.

Compare the results of an experiment with the expected theoretical outcomes.

Design experiments of chance and predict outcomes of odds, for and against.

Statistics and Data Analysis

A. Compare and contrast different plots of data using values of mean, median, mode, quartiles and range.

B. Explain effects of sampling procedures and missing or incorrect information on reliability.

C. Fit a line to the scatter plot of two quantities and describe any correlation of the variables.

D. Design and carry out a random sampling procedure.

E. Analyze and display data in stem-and-leaf and box-and-whisker plots.

F. Use scientific and graphing calculators and computer spreadsheets to organize and analyze data.

G. Determine the validity of the sampling method described in studies published in local or national

Students understand the art of managing information for the purpose of data analysis.

1. Read and interpret data displayed in a variety of forms including histograms.

2. Interpret measures of dispersion (range) and of central tendency.

3. Find the mean, median, and mode of a set of data using raw data, tables, charts, or graphs.

4. Describe a set of data by using the measures of central tendency.

5. Construct various graphs, including scatterplots and box-and- whisker graphs, to display a data set.

DA3: Students use statistical methods to make inferences and valid arguments about real-world situations.

1. Understand the application of statistics in the formation, testing and evaluation of a hypothesis.

By the end of 8th grade students will be able to:

Interpret and analyze data displayed in a variety of forms including histograms.

Determine appropriate measures of central tendency for a given situation or set of data.

Determine the mean, median, mode, and range of a set of real-world data using appropriate technology.

Organize graphs and analyze a set of real-world data using appropriate technology.

Design an experiment, perform the experiment and collect, organize, and display the data. Evaluate the hypothesis by making inferences and drawing conclusions based on statistical results.

Perform an experiment and collect, organize, and display the data conclusions based on statistical results.

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newspapers.

Probability and Predictions

A. Determine the number of combinations and permutations for an event.

B. Present the results of an experiment using visual representations (e.g., tables, charts, graphs).

C. Analyze predictions (e.g., election polls).

D. Compare and contrast results from observations and mathematical models.

E. Make valid inferences, predictions and arguments based on probability.

Students identify the common uses and misuses of probability or statistical analysis in the everyday world.

1. Know appropriate uses of statistics and probability in real-world situations.

2. Know when statistics and probability are used in misleading ways.

3. Identify and use different types of sampling techniques (for example, random, systematic, stratified).

4. Know whether a sample is biased.

By the end of 8th grade students will be able to:

Identify instances in which statistics and probability are used in advertising to mislead the public.

Design several different surveys and use the various sampling techniques for obtaining survey results.

Interpret probabilities as ratios, percents, and decimals.

Determine probabilities of compound events.

Explore the probabilities of conditional events (e.g., if there are seven marbles in a bag, three red and four green, what is the probability that two marbles picked from the bag without replacement, are both red).

Model situations involving probability with simulations (using spinners, dice, calculators and computers) and rhetorical models

Frequency, relative frequency

Estimate probabilities and make predictions based on experimental and rhetorical probabilities

Play and analyze probability-based games, and discuss the concepts of fairness and expected value.

Algebra and Functions

A. Apply simple algebraic patterns to basic number theory and to spatial relations

B. Discover, describe and generalize patterns, including linear, exponential

Students recognize, describe, analyze and extend patterns, relations and functions.

1. Know the graphical representation of a linear relationship.

2. Determine if a function is linear by making use of the information provided in a table, graph, or rule.

3. Recognize the independent variable and the dependent

By the end of 8th grade students will be able to:

Read, analyze, and describe graphs of linear relationships.

Justify the reason for determining if a function is linear.

Use variables to represent unknown quantities in real-world problems.

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and simple quadratic relationships.

C. Create and interpret expressions, equations or inequalities that model problem situations.

D. Use concrete objects to model algebraic concepts.

E. Select and use a strategy to solve an equation or inequality, explain the solution and check the solution for accuracy.

F. Solve and graph equations and inequalities using scientific and graphing calculators and computer spreadsheets.

G. Represent relationships with tables or graphs in the coordinate plane and verbal or symbolic rules.

H. Graph a linear function from a rule or table.

I. Generate a table or graph from a function and use graphing calculators and computer spreadsheets to graph and analyze functions.

variable in a real world problem.

4. Know function rules to describe tables of related input-output.

Students use expressions, equations, inequalities, graphs, and formulas to represent and interpret situations.

1. Interpret and create tables, function tables, and graphs (function tables).

2. Graph solutions to linear equations on the coordinate plane.

3. Write equations and inequalities to express relationships.

4. Interpret the meaning of the slope of a line from a graph depicting a real-world situation.

5. Translate verbal expressions and sentences into algebraic expressions, equations, and inequalities.

6. Solve single- and multiple-step linear equations and inequalities in concrete or abstract form.

7. Know the relationships represented by algebraic equations or inequalities and their graphic representations.

8. Know how to evaluate algebraic expressions, equations, and inequalities.

9. Know the relationships between the properties of algebraic expressions and real numbers.

10. Evaluate algebraic expressions, equations, and inequalities by substituting integral values for variables and simplifying the results.

Predict outcomes from given tables of related input-output, based upon function rules.

Perform experiments in order to generate data tables that graph functions.

Graph equations and inequalities in order to explain cause-and-effect relationships.

Interpret the meaning of the slope of a line from a graph depicting a real-world situation.

Translate algebraic expressions, equations, or inequalities representing real-world relationships into verbal expressions or sentences.

Graph linear equations on the coordinate plane using tables of values.

Graphically display real-world situations represented by algebraic equations or inequalities.

Evaluate algebraic expressions, equations, and inequalities by substituting integral values for variables and simplifying the results.

Simplify algebraic expressions that represent real-world situations by combining like terms and applying the properties of real numbers.

Translate algebraic expressions, equations, or inequalities representing real-world relationships into verbal expressions or sentences.

Graph linear equations on the coordinate plane using tables of values.

Simplify algebraic expressions that represent real-world situations by combining like terms and applying the properties of real numbers.

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J. Show that an equality relationship between two quantities remains the same as long as the same change is made to both quantities; explain how a change in one quantity determines another quantity in a functional relationship.

K. Collecting, organizing, and representing data about the relationship between two variables

Graph functions, and understand and describe their general behavior

Equations involving two variables

Use graphing techniques on a number line Absolute value Arithmetic operations represented by vectors

(arrows)

Solve simple linear inequalities

Create, evaluate, and simplify algebraic expressions involving variables

Order of operations, including appropriate use of parentheses

Distributive property Substitution of a number for a variable Translation of a verbal phrase or sentence into

algebraic expression, equation, or inequality, and vice versa

Geometry

A. Construct figures incorporating perpendicular and parallel lines, the perpendicular bisector of a line segment and an angle bisector using computer software.

B. Draw, label, measure and list the properties of complementary, supplementary and vertical angles.

C. Classify familiar polygons as regular or irregular up to a decagon.

Students describe, draw, identify, and analyze two and three-dimensional shapes.

1. Compare regular and irregular polygons and two- and three dimensional shapes.

2. Determine the measures of various types of angles based upon geometric relationships in two- and three-dimensional shapes.

3. Represent the properties of two- and three- dimensional figures by drawing them with appropriate tools including astraight edge and a compass.

4. Recognize and draw two-dimensional representations of three-dimensional objects (perspective drawings

Students use coordinate geometry to locate objects in both two- and three-dimensions and to describe objects

By the end of 8th grade students will be able to:

Draw and build three-dimensional figures from various perspectives (e.g., flat patterns, isometric drawings, and nets).

Draw angles (including acute, obtuse, right, straight, complementary, supplementary, and vertical angles).

Draw three-dimensional figures (including pyramid, cone, sphere, hemisphere, rectangular solids and cylinders).

Given an equation or its graph, finds ordered-pair solutions (for example, y = 2x).

Given the graph of a line, identifies the slope of the line (including the slope of vertical and horizontal lines).

Apply and explain the simple properties of lines on a graph,

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D. Identify, name, draw and list all properties of squares, cubes, pyramids, parallelograms, quadrilaterals, trapezoids, polygons, rectangles, rhombi, circles, spheres, triangles, prisms and cylinders.

E. Construct parallel lines, draw a transversal and measure and compare angles formed (e.g., alternate interior and exterior angles).

F. Distinguish between similar and congruent polygons.

G. Approximate the value of (pi) through experimentation.

H. Use simple geometric figures (e.g., triangles, squares) to create, through rotation, transformational figures in three dimensions.

I. Generate transformations using computer software.

J. Analyze geometric patterns (e.g., tessellations, sequences of shapes) and develop descriptions of the patterns.

K. Analyze objects to determine whether they illustrate tessellations, symmetry, congruence, similarity and

algebraically.

1. Know how to find a minimum of three ordered-pair solutions for a given equation.

2. Know the formula for the graph of a line, including the slope of the line and the intercept of the line, including vertical andhorizontal lines.

3. Know the relationships of linear equations as they apply to: properties of lines on a graph, including parallelism,perpendicularity, the x and y intercepts, the midpoint of a horizontal or vertical line segment, and the intersection point of two lines.

4. Know the geometric properties and relationships (among regular and irregular shapes of two and three dimensions).

5. Understand the Pythagorean relationship in special right triangles (45 – 45 – 90 and 30 – 60 – 90).

Students visualize and illustrate ways in which shapes can be combined, subdivided, and changed.

1. Know and apply the properties of parallelism, perpendicularity and symmetry in real-world contexts.

2. Identify congruent and similar figures in real-world situations.

3. Continue a tessellation pattern using the needed transformations.

including parallelism, perpendicularity, and identifying the x and y intercepts, the midpoint of a horizontal or vertical line segment, and the intersection point of two lines.

Observe, explain, make and test conjectures regarding geometric properties and relationships (among regular and irregular shapes of two and three dimensions).

Apply the Pythagorean Theorem in real-world problems (for example, finds the relationship among sides in 45 – 45 – 90 and 30 – 60 – 90 right triangles). Use models or diagrams (manipulatives, dot, graph, or isometric paper).Understand and apply concepts involving lines, angles, and planes

Complementary and supplementary angles Vertical angles Parallel, perpendicular, and intersecting planes

Understand and apply properties of polygons Quadrilaterals, including squares, rectangles,

parallelograms, trapezoids Regular polygons Sum of measures of interior angles of a polygon

Understand and apply transformations Finding the image, given the pre-image and vice

versa Sequence of transformations needed to map one

figure unto another Reflections, rotations, and translations result in

images congruent to the pre-image Dilations (stretching/shrinking) result in images

similar to the pre image

Use the properties of parallelism, perpendicularity, and symmetry in solving real-world problems.

Justify the identification of congruent and similar figures.

Create an original tessellating tile and tessellation pattern using a combination of transformations.

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scale.

Trigonometry

A. Compute measures of sides and angles using proportions, the Pythagorean Theorem and right triangle relationships.

B. Solve problems requiring indirect measurement for lengths of sides of triangles.

Students use coordinate geometry to locate objects in both two- and three-dimensions and to describe objects algebraically.

1. Know the formula for the graph of a line, including the slope of the line and the intercept of the line, including vertical and horizontal lines.

3. Know the relationships of linear equations as they apply to: properties of lines on a graph, including parallelism,perpendicularity, the x and y intercepts, the midpoint of a horizontal or vertical line segment, and the intersection point of two lines.

By the end of 8th grade students will be able to:

Given an equation or its graph, finds ordered-pair solutions (for example, y = 2x).

Apply and explain the simple properties of lines on a graph, including parallelism, perpendicularity, and identifying the x and y intercepts, the midpoint of a horizontal or vertical line segment, and the intersection point of two lines.

Observe, explain, make and test conjectures regarding geometric properties and relationships (among regular and irregular shapes of two and three dimensions).

Develop and apply strategies for finding perimeter and area

Geometric figures made by combining triangles, rectangles and circles or parts of a circle

Concepts of Calculus

A. Analyze graphs of related quantities for minimum and maximum values and justify the findings.

B. Describe the concept of unit rate, ratio and slope in the context of rate of change.

C. Continue a pattern of numbers or objects that could be extended infinitely.

Students understand and apply theories related to numbers.

1. Determine the appropriate use of number theory concepts, including divisibility rules, to solve real- world ormathematical problems. Describe the concept of unit rate, ratio and slope in the context of rate of change.

2. Continue a pattern of numbers or objects that could be extended infinitely.

By the end of 8th grade students will be able to:

Find the greatest common factor and least common multiple of two or more numbers.

Apply number theory concepts to determine the terms in a real number sequence.

Apply number theory concepts, including divisibility rules, to solve real-world or mathematical problems.

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