Orange School District
Orange School District [pic]
[pic]
[pic]
Mathematics
Curriculum Guide – Grade 8
2011 Edition
APPROVED: January, 2011
| |
| |
|BOARD OF EDUCATION |
| |
|Patricia A. Arthur |
|President |
| |
|Arthur Griffa |
|Vice-President |
| |
|Members |
|Stephanie Brown |Rev. Reginald T. Jackson |Maxine G. Johnson |
|Eunice Y. Mitchell | |David Wright |
| |
|SUPERINTENDENT OF SCHOOLS |
|Ronald Lee |
| |
|DEPUTY |ADMINISTRATIVE ASSISTANT TO THE SUPERINTENDENT |
|SUPERINTENDENT | |
|Dr. Paula Howard |Belinda Scott-Smiley |
|Curriculum and Instructional Services |Operations/Human Resources |
| |
|BUSINESS ADMINISTRATOR |
|Adekunle O. James |
| |
|DIRECTORS |
|Barbara L. Clark, Special Services |
|Candace Goldstein, Special Programs |
|Candace Wallace, Curriculum & Testing |
| |
| |
|CURRICULUM TEAM |
|Candace Wallace |
|Ron Nelkin |
|Mengli Chiliu |
|Ann Burgunder |
| |
| |
| |
Table of Contents
BOARD OF EDUCATION 2
Philosophy, Vision & Purpose 4
Process Goals 5
Phases of Instruction 6
Target Goals 8
Description of Student Units 10
Connected Math Framework 11
Mathematics Learning Goals 14
Content Goals in Each Unit 18
Alignment with Standards 21
Process Standards 21
Common Core Standards and Blueprint 23
New Jersey Core Curriculum Content Standards 42
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.
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.
Counting
Determining 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
Visualizing
Recognizing 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
Comparing
Describing 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
Estimating
Determining reasonableness of answers; using "benchmarks" to estimate measures; using various strategies to approximate a calculation and to compare estimates
Measuring
Assigning 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
Modeling
Constructing, 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
Reasoning
Bringing 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
Connecting
Identifying ways in which problems, situations, and mathematical ideas are interrelated and applying knowledge gained in solving one problem to other problems
Representing
Moving flexibly among graphic, numeric, symbolic, and verbal representations and recognizing the importance of having various representations of information in a situation
Using Tools
Selecting and intelligently using calculators, computers, drawing tools, and physical models to represent, simulate, and manipulate patterns and relationships in problem settings
Becoming Mathematicians
Having the disposition and imagination to inquire, investigate, tinker, dream, conjecture, invent, and communicate with others about mathematical ideas
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?
• 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?
• What definitions or strategies do we need to generalize?
Target Goals
Number and Operation Goals
Number 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 Goals
Formulating 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 Goals
Shapes 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
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 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
• 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
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
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
o Triangular Numbers
o Equations Modeling Projectile Motion
Say It With Symbols
• Equivalent Expressions
o Verifying Equivalence
o Notes on the Distributive Property
o Interpreting Expressions
o A Note on the Use of Expression and Equations
• Combining Expressions
o Adding Expressions
o Creating New Expressions by Substitution
• Solving Equations
o Solving Linear Equations
o Solving Quadratic Equations
o A Note on Factoring
• Predicting the Underlying Patterns of Change
o Patterns of Change
o Predicting Linear Patterns of Change
o Writing Equations for Linear, Exponential, and Quadratic Functions Given Two Points
• Reasoning With Symbols
o 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 Substitution
o Solving Systems by Linear Combination
• · Types of Solutions
o Solving Linear Inequalities in Two Variables
o Solving Systems of Linear Inequalities
DATA ANALYSIS AND PROBABILITY
Samples 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 sets
o Exploring the concept of sampling
• Exploring the concept of covariation or association
GEOMETRY
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
• 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
Mathematics Learning Goals
Connected Mathematics develops four mathematical strands:
• Number and Operation
• Geometry and Measurement
• Data Analysis and Probability
• Algebra.
Goals by Mathematical Strand
Number and Operation Goals
Number 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
• 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
• 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
• 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
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
• 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)
• 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
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 Solving
All units
Because Connected Mathematics is a problem- centered curriculum, problem solving is an important part of every unit.
Reasoning and Proof
All units
Throughout 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.
Communication
All units
As 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.
Connections
All units
In 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.
Representation
All units
Throughout 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.
Common Core Standards » Mathematics » Grade 8
|THE NUMBER SYSTEM |
|Know 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 EQUATIONS |
|Expressions 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. |
| |
|FUNCTIONS |
|Define, 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 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. |
| |
|GEOMETRY |
|Understand 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. |
| |
|STATISTICS AND PROBABILITY |
|Investigate 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 | Students understand the different ways numbers and number systems are used|By the end of 8th grade students will: |
| |in the real world. | |
|Numbers, Number Systems and Number Relationships | |Associate verbal names, written word names and standard numerals with |
| | |integers, fractions, decimals; numbers expressed as percents; numbers with |
|Represent and use numbers in equivalent forms (e.g., integers, |1. Know word names and standard numerals for integers, fractions, decimals,|exponents; numbers in scientific notation; radicals; absolute value; and |
|fractions, decimals, percents, exponents, scientific notation, |ratios, numbers expressed as percents, numbers with exponents, numbers |ratios. (make use of the game concentration with index cards) |
|square roots). |expressed in scientific notation, absolute value, radicals, and ratios. | |
| | |Order (on a number line or using graphic models, number lines, and symbols) |
|Simplify numerical expressions involving exponents, scientific |2. Read and write whole numbers and decimals in expanded form, including |and diagram the relative size of integers, fractions, and decimals; numbers |
|notation and using order of operations. |exponential notation. |expressed as percents; numbers with exponents; numbers in scientific |
| | |notation; radicals; absolute value; and ratios. |
|Distinguish between and order rational and irrational numbers. |3. Compare and order fractions, decimals, integers, numbers expressed in | |
| |absolute value, scientific notation, percents, numbers with exponents, |Describe the meanings of rational and irrational numbers using physical or |
| |ratios and radicals. |graphical displays. |
| | | |
|Apply ratio and proportion to mathematical problem situations |4. Understand the meanings of rational and irrational numbers. |Give examples of rational and irrational numbers in real-world situations. |
|involving distance, rate, time and similar triangles. | | |
| |5. Know the relationships among fractions, decimals, and percents given a |Construct models to represent rational and irrational numbers. |
|Simplify and expand algebraic expressions using exponential |real-world context. | |
|forms. | |Express a given quantity in a variety of ways (for example, integers, |
| |6. Know how to simplify expressions using integers, exponents, and |fractions, decimals, numbers expressed as a percent, numbers expressed in |
| |radicals. |scientific notation, ratios). |
|Use the number line model to demonstrate integers and their | | |
|applications. |7. Understand the concept of the absolute value of a number. |Evaluate numerical or algebraic expressions that contain exponential |
| | |notation. |
| |8. Know the relationship between rational numbers and negative exponents by| |
|Use the inverse relationships between addition, subtraction, |investigating the powers of 10 from 104 through 10 -4. |Express the real-world applications to Absolute Value. Give concrete |
|multiplication, division, exponentiation and root extraction to| |examples in science, society and technology. |
|determine unknown quantities in equations. |9. Know the different properties of the numbers between 0 and 1, including | |
| |the differences in these numbers when squared, inverted, and multiplied by |Express rational numbers in exponential notation including negative |
| |whole numbers and negative fractions. |exponents (for example, 2 -3 = 1/23= 1/8). |
| | | |
| |10. Understand the structure of number systems other than the decimal |Express numbers in scientific or standard notation including decimals |
| |number system, for example, bases other than 10 and the binary number |between 0 and 1. |
| |system. | |
| | |Understand and use ratios, proportions, and percents in a variety of |
| |Students understand the effects of operations on numbers and the |situations |
| |relationships among these operations; select appropriate operations, and | |
| |are able to compute for various problem-solving situations. |Use whole numbers, fractions, decimals, and percents to represent equivalent|
| | |forms of the same number |
| | | |
| |1. Know the effects of the four basic operations on whole numbers, | |
| |fractions, mixed numbers, decimals, and integers. | |
|Computation and Estimation |Students understand the effects of operations on numbers and the |By the end of 8th grade students will be able to: |
| |relationships among these operations; select appropriate operations, and | |
|Complete calculations by applying the order of operations. |are able to compute for various problem-solving situations. |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 |
|Add, subtract, multiply and divide different kinds and forms of|1. Know the effects of the four basic operations on whole numbers, |equivalent numbers in base ten. Discuss the application of the binary (base |
|rational numbers including integers, decimal fractions, |fractions, mixed numbers, decimals, and integers. |two) number system in computer technology. |
|percents and proper and improper fractions. | | |
| |2. Apply the properties of real numbers to solve problems (commutative, |Investigate the structure of number systems other than the decimal number |
|Estimate the value of irrational numbers. |associative, distributive, identity, equality, inverse, and closure). |system. |
|Estimate amount of tips and discounts using ratios, proportions| | |
|and percents. |3. Solve real-world problems involving percents (for example, discounts, |Use and explain procedures for performing calculations involving addition, |
| |simple interest, taxes, tips). |subtraction, multiplication, division, and exponentiation with integers and |
| | |all number types named above with: |
|Determine the appropriateness of overestimating or |Know proportional relationships. |Pencil-and-paper |
|underestimating in computation. | |Mental math |
| |Students understand and apply properties of numbers and operations. |Calculator |
|Identify the difference between exact value and approximation | | |
|and determine which is appropriate for a given situation. |1. Know the inverse relationship of positive and negative numbers. |Understand and apply the standard algebraic order of operations, including |
| | |appropriate use of parentheses. |
|Career |2. Know the appropriate operation to solve real-world problems involving | |
| |integers, ratios, rates, proportions, numbers expressed as percents, |Estimate square roots and cube roots of numbers |
|• Analyze budgets and pay statements, such as, but not limited |decimals, fractions, and square roots. | |
|to: | |Use equivalent representation of numbers such as fractions, decimals, and |
|Charitable contributions |3. Solve multi-step, real-world problems involving integers, ratios, |percents to facilitate estimation |
|Expenses |proportions, numbers expressed as percents, decimals, and fractions. | |
|Gross pay | |Recognize the limitations of estimation and assess the amount of error |
|Net pay |4. Solve real-world problems involving percents including percents greater |resulting from estimation |
|Other income |than 100% (for example percent of change, | |
|Savings |commission). |Use models or pictures to show the effects of addition, subtraction, |
|Taxes | |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 |Students measure quantities in the real world and use these measures to |By the end of 8th grade students will be able to: |
| |solve problems. | |
|Develop formulas and procedures for determining measurements | |Use equivalent representation of numbers such as fractions, decimals, and |
|(e.g., area, volume, distance). | |percents to facilitate estimation |
| |1. Understand strategies used to solve real-world problems involving | |
|Solve rate problems (e.g., rate ( time = distance, principal ( |surface area and volume of three-dimensional shapes. |Recognize the limitations of estimation and assess the amount of error |
|interest rate = interest). | |resulting from estimation |
| |2. Explore and derive formulas for surface area and volume of | |
|Measure angles in degrees and determine relations of angles. |three-dimensional regular shapes, including pyramids, prisms, and cones. |Build three-dimensional solids using the two-dimensional models as faces. |
| | |Predict the surface areas, and then test predictions. |
|Estimate, use and describe measures of distance, rate, |3. Know and apply formulas for finding rates, distance, time and angle | |
|perimeter, area, volume, weight, mass and angles. |measures. |Using centimeter cubes as a guide, estimate the volume of each model. |
| | |Working with a group, tests estimations and contribute to group consensus on|
|Describe how a change in linear dimension of an object affects |4. Develop an understanding of rate of change as it applies to real-world |a working formula for finding the volume of various three-dimensional |
|its perimeter, area and volume. |problems. |models. |
| | | |
|Use scale measurements to interpret maps or drawings. |5. Know that new figures can be created by increasing or decreasing the |Describe and use rates of change (for example, temperature as it changes |
| |original dimensions. |throughout the day, or speed as the rate of change in distance over time) |
|Create and use scale models. | |and other derived measures. |
| |6. Know how changes in the volume, surface area, area, or perimeter of a | |
| |figure affect the dimensions of the figure. |Describe how a change in a figure’s dimensions affects its perimeter, area, |
| | |circumference, surface area, or volume. |
| |7. Solve real world or mathematical problems involving the effects of | |
| |changes either to the dimensions of a figure or |Investigate congruent figures with respect to volume and surface area, and |
| |to the volume, surface area, area, perimeter, or circumference of figures. |describe the differences in their dimensions. |
| | | |
| |Students compare, contrast, and convert within systems of measurement; both|Describe how the change of a figure in dimensions such as length, width, |
| |standard with nonstandard and metric with customary. |height, or radius affects its other measurements such as perimeter, area, |
| | |surface area, and volume. |
| |1. Know relationships between metric units of mass and capacity (for | |
| |example, one cubic centimeter of water weighs |Measure length, weight or mass, and capacity or volume using customary or |
| |one gram). |metric units. |
| | | |
| |2. Find measures of length, weight or mass, and capacity or volume using |Apply properties of similarity with shadow measurement and properties of |
| |proportional relationships and properties of similar geometric figures. |similar triangles to find the height of a flagpole. |
| | | |
| |Students select and use appropriate units and instruments for measurement |Select appropriate units of measurement in a real-world context. |
| |to achieve the degree of precision and accuracy required in real-world | |
| |situations. |Apply the conversion of measurements within the customary system to real |
| | |world problems. |
| | | |
| |1. Solve problems using mixed units within each system, such as feet and |Select and use appropriate instruments, technology, and techniques to |
| |inches, hours and minutes. |measure quantities and dimensions to a specified degree of accuracy. |
| | | |
| |2. Solve problems using the conversion of measurements within the customary| |
| |system. | |
| | | |
| |3. Determine the appropriate precision unit for a given situation. | |
| | | |
| |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 | |By the end of 8th grade students will be able to: |
| |Students identify patterns and makes predictions from an orderly display of| |
|Make conjectures based on logical reasoning and test |data using concepts of probability and statistics. |Calculate simple mathematical probabilities for independent and dependent |
|conjectures by using counter-examples. | |events. |
| | | |
|Combine numeric relationships to arrive at a conclusion. |1. Compare and explain the results of an experiment with the mathematically|Compare the results of an experiment with the expected theoretical outcomes.|
| |expected outcomes. | |
|Use if...then statements to construct simple, valid arguments. | |Design experiments of chance and predict outcomes of odds, for and against. |
| |2. Explain observed difference between mathematical and experimental | |
|Construct, use and explain algorithmic procedures for computing|results. | |
|and estimating with whole numbers, fractions, decimals and | | |
|integers. |3. Predict the mathematical odds for and against a specified outcome in a | |
| |given real-world situation. | |
|Distinguish between inductive and deductive reasoning. | | |
| | | |
| | | |
|Use measurements and statistics to quantify issues (e.g., in | | |
|family, consumer science situations). | | |
| | | |
| | | |
| | | |
|Mathematical Problem Solving and Communication |Students understand the effects of operations on numbers and the |By the end of 8th grade students will be able to: |
| |relationships among these operations; select appropriate operations, and | |
|Invent, select, use and justify the appropriate methods, |are able to compute for various problem-solving situations. |Express base ten numbers as equivalent numbers in different bases, such as |
|materials and strategies to solve problems. | |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 |
|Verify and interpret results using precise mathematical |1. Know the effects of the four basic operations on whole numbers, |two) number system in computer technology. |
|language, notation and representations, including numerical |fractions, mixed numbers, decimals, and integers. | |
|tables and equations, simple algebraic equations and formulas, | |Investigate the structure of number systems other than the decimal number |
|charts, graphs and diagrams. |2. Apply the properties of real numbers to solve problems (commutative, |system. |
| |associative, distributive, identity, equality, inverse, and closure). | |
|Justify strategies and defend approaches used and conclusions | |Use and explain procedures for performing calculations involving addition, |
|reached. |3. Solve real-world problems involving percents (for example, discounts, |subtraction, multiplication, division, and exponentiation with integers and |
| |simple interest, taxes, tips). |all number types named above with: |
|Determine pertinent information in problem situations and | |Pencil-and-paper |
|whether any further information is needed for solution. |Students make reasonable estimates. |Mental math |
| | |Calculator |
|Career Education Work |1. Know an appropriate estimation technique for a given situation using | |
|• Analyze budgets and pay statements, such as, but not limited |whole numbers, fractions and decimals. |Understand and apply the standard algebraic order of operations, including |
|to: | |appropriate use of parentheses. |
|Charitable contributions |2. Estimate to predict results and check reasonableness of results. | |
|Expenses | |Write and simplify expressions from real-world situations using the order of|
|Gross pay |3. Determine whether an exact answer is needed or whether an estimate would|operations. |
|Net pay |be sufficient. | |
|Other income | |Use appropriate methods of computation, such as mental computation, paper |
|Savings |Students identify patterns and makes predictions from an orderly display of|and pencil, and calculator. |
|Taxes |data using concepts of probability and statistics. | |
| | |Justify the choice of method for calculations, such as mental computation, |
|Analyze the relationship of school subjects, extracurricular | |concrete materials, algorithms, or calculators. |
|activities, and community experiences to career preparation |1. Compare and explain the results of an experiment with the mathematically| |
| |expected outcomes. |Explain a variety of estimation techniques including clustering, compatible |
| | |number, and front-end. |
| |2. Explain observed difference between mathematical and experimental | |
| |results. |Execute the 4 steps of problem solving. |
| | | |
| |3. Predict the mathematical odds for and against a specified outcome in a |Give examples in real world situations where estimation is sufficient for |
| |given real-world situation. |the situation. |
| | | |
| | |Recognize the limitations of estimation and assess the amount of error |
| | |resulting from estimation |
| | | |
| | |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. |
|Statistics and Data Analysis |Students understand the art of managing information for the purpose of data|By the end of 8th grade students will be able to: |
| |analysis. | |
|Compare and contrast different plots of data using values of | | |
|mean, median, mode, quartiles and range. | |Interpret and analyze data displayed in a variety of forms including |
| |1. Read and interpret data displayed in a variety of forms including |histograms. |
|Explain effects of sampling procedures and missing or incorrect|histograms. | |
|information on reliability. | |Determine appropriate measures of central tendency for a given situation or |
| |2. Interpret measures of dispersion (range) and of central tendency. |set of data. |
|Fit a line to the scatter plot of two quantities and describe | | |
|any correlation of the variables. |3. Find the mean, median, and mode of a set of data using raw data, tables,|Determine the mean, median, mode, and range of a set of real-world data |
| |charts, or graphs. |using appropriate technology. |
|Design and carry out a random sampling procedure. | | |
| |4. Describe a set of data by using the measures of central tendency. |Organize graphs and analyze a set of real-world data using appropriate |
|Analyze and display data in stem-and-leaf and box-and-whisker | |technology. |
|plots. |5. Construct various graphs, including scatterplots and box-and- whisker | |
| |graphs, to display a data set. |Design an experiment, perform the experiment and collect, organize, and |
|Use scientific and graphing calculators and computer | |display the data. Evaluate the hypothesis by making inferences and drawing |
|spreadsheets to organize and analyze data. |DA3: Students use statistical methods to make inferences and valid |conclusions based on statistical results. |
| |arguments about real-world situations. | |
|Determine the validity of the sampling method described in | |Perform an experiment and collect, organize, and display the data |
|studies published in local or national newspapers. | |conclusions based on statistical results. |
| |1. Understand the application of statistics in the formation, testing and | |
| |evaluation of a hypothesis. | |
|Probability and Predictions |Students identify the common uses and misuses of probability or statistical|By the end of 8th grade students will be able to: |
| |analysis in the everyday world. | |
|Determine the number of combinations and permutations for an | |Identify instances in which statistics and probability are used in |
|event. |1. Know appropriate uses of statistics and probability in real-world |advertising to mislead the public. |
| |situations. | |
|Present the results of an experiment using visual | |Design several different surveys and use the various sampling techniques for|
|representations (e.g., tables, charts, graphs). |2. Know when statistics and probability are used in misleading ways. |obtaining survey results. |
| | | |
|Analyze predictions (e.g., election polls). |3. Identify and use different types of sampling techniques (for example, |Interpret probabilities as ratios, percents, and decimals. |
| |random, systematic, stratified). | |
|Compare and contrast results from observations and mathematical| |Determine probabilities of compound events. |
|models. |4. Know whether a sample is biased. | |
| | |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 |
|Make valid inferences, predictions and arguments based on | |without replacement, are both red). |
|probability. | | |
| | |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 |Students recognize, describe, analyze and extend patterns, relations and |By the end of 8th grade students will be able to: |
| |functions. | |
|Apply simple algebraic patterns to basic number theory and to | |Read, analyze, and describe graphs of linear relationships. |
|spatial relations |1. Know the graphical representation of a linear relationship. | |
| | |Justify the reason for determining if a function is linear. |
|Discover, describe and generalize patterns, including linear, |2. Determine if a function is linear by making use of the information | |
|exponential and simple quadratic relationships. |provided in a table, graph, or rule. |Use variables to represent unknown quantities in real-world problems. |
| | | |
|Create and interpret expressions, equations or inequalities |3. Recognize the independent variable and the dependent variable in a real |Predict outcomes from given tables of related input-output, based upon |
|that model problem situations. |world problem. |function rules. |
| | | |
|Use concrete objects to model algebraic concepts. |4. Know function rules to describe tables of related input-output. | |
| | |Perform experiments in order to generate data tables that graph functions. |
| |Students use expressions, equations, inequalities, graphs, and formulas to | |
|Select and use a strategy to solve an equation or inequality, |represent and interpret situations. |Graph equations and inequalities in order to explain cause-and-effect |
|explain the solution and check the solution for accuracy. | |relationships. |
| | | |
|Solve and graph equations and inequalities using scientific and|1. Interpret and create tables, function tables, and graphs (function |Interpret the meaning of the slope of a line from a graph depicting a |
|graphing calculators and computer spreadsheets. |tables). |real-world situation. |
| | | |
| |2. Graph solutions to linear equations on the coordinate plane. |Translate algebraic expressions, equations, or inequalities representing |
|Represent relationships with tables or graphs in the coordinate| |real-world relationships into verbal expressions or sentences. |
|plane and verbal or symbolic rules. |3. Write equations and inequalities to express relationships. | |
| | |Graph linear equations on the coordinate plane using tables of values. |
| |4. Interpret the meaning of the slope of a line from a graph depicting a | |
|Graph a linear function from a rule or table. |real-world situation. |Graphically display real-world situations represented by algebraic equations|
| | |or inequalities. |
|Generate a table or graph from a function and use graphing |5. Translate verbal expressions and sentences into algebraic expressions, | |
|calculators and computer spreadsheets to graph and analyze |equations, and inequalities. |Evaluate algebraic expressions, equations, and inequalities by substituting |
|functions. | |integral values for variables and simplifying the results. |
| |6. Solve single- and multiple-step linear equations and inequalities in | |
|Show that an equality relationship between two quantities |concrete or abstract form. |Simplify algebraic expressions that represent real-world situations by |
|remains the same as long as the same change is made to both | |combining like terms and applying the properties of real numbers. |
|quantities; explain how a change in one quantity determines |7. Know the relationships represented by algebraic equations or | |
|another quantity in a functional relationship. |inequalities and their graphic representations. |Translate algebraic expressions, equations, or inequalities representing |
|Collecting, organizing, and representing data about the | |real-world relationships into verbal expressions or sentences. |
|relationship between two variables |8. Know how to evaluate algebraic expressions, equations, and inequalities.| |
| | |Graph linear equations on the coordinate plane using tables of values. |
| |9. Know the relationships between the properties of algebraic expressions | |
| |and real numbers. |Simplify algebraic expressions that represent real-world situations by |
| | |combining like terms and applying the properties of real numbers. |
| |10. Evaluate algebraic expressions, equations, and inequalities by | |
| |substituting integral values for variables and simplifying the results. |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 |Students describe, draw, identify, and analyze two and three-dimensional |By the end of 8th grade students will be able to: |
| |shapes. | |
|Construct figures incorporating perpendicular and parallel | |Draw and build three-dimensional figures from various perspectives (e.g., |
|lines, the perpendicular bisector of a line segment and an |1. Compare regular and irregular polygons and two- and three dimensional |flat patterns, isometric drawings, and nets). |
|angle bisector using computer software. |shapes. | |
| | |Draw angles (including acute, obtuse, right, straight, complementary, |
|Draw, label, measure and list the properties of complementary, |2. Determine the measures of various types of angles based upon geometric |supplementary, and vertical angles). |
|supplementary and vertical angles. |relationships in two- and three-dimensional shapes. | |
| | |Draw three-dimensional figures (including pyramid, cone, sphere, hemisphere,|
|Classify familiar polygons as regular or irregular up to a |3. Represent the properties of two- and three- dimensional figures by |rectangular solids and cylinders). |
|decagon. |drawing them with appropriate tools including a | |
| |straight edge and a compass. |Given an equation or its graph, finds ordered-pair solutions (for example, y|
|Identify, name, draw and list all properties of squares, cubes,| |= 2x). |
|pyramids, parallelograms, quadrilaterals, trapezoids, polygons,|4. Recognize and draw two-dimensional representations of three-dimensional | |
|rectangles, rhombi, circles, spheres, triangles, prisms and |objects (perspective drawings |Given the graph of a line, identifies the slope of the line (including the |
|cylinders. | |slope of vertical and horizontal lines). |
| |Students use coordinate geometry to locate objects in both two- and | |
|Construct parallel lines, draw a transversal and measure and |three-dimensions and to describe objects algebraically. |Apply and explain the simple properties of lines on a graph, including |
|compare angles formed (e.g., alternate interior and exterior | |parallelism, perpendicularity, and identifying the x and y intercepts, the |
|angles). |1. Know how to find a minimum of three ordered-pair solutions for a given |midpoint of a horizontal or vertical line segment, and the intersection |
| |equation. |point of two lines. |
|Distinguish between similar and congruent polygons. | | |
| |2. Know the formula for the graph of a line, including the slope of the |Observe, explain, make and test conjectures regarding geometric properties |
|Approximate the value of ( (pi) through experimentation. |line and the intercept of the line, including vertical and |and relationships (among regular and irregular shapes of two and three |
| |horizontal lines. |dimensions). |
|Use simple geometric figures (e.g., triangles, squares) to | | |
|create, through rotation, transformational figures in three |3. Know the relationships of linear equations as they apply to: properties |Apply the Pythagorean Theorem in real-world problems (for example, finds the|
|dimensions. |of lines on a graph, including parallelism, |relationship among sides in 45 – 45 – 90 and 30 – 60 – 90 right triangles). |
| |perpendicularity, the x and y intercepts, the midpoint of a horizontal or |Use models or diagrams (manipulatives, dot, graph, or isometric paper). |
|Generate transformations using computer software. |vertical line segment, and the intersection point of two lines. |Understand and apply concepts involving lines, angles, and planes |
| | |Complementary and supplementary angles |
|Analyze geometric patterns (e.g., tessellations, sequences of |4. Know the geometric properties and relationships (among regular and |Vertical angles |
|shapes) and develop descriptions of the patterns. |irregular shapes of two and three dimensions). |Parallel, perpendicular, and intersecting planes |
| | | |
|Analyze objects to determine whether they illustrate |5. Understand the Pythagorean relationship in special right triangles (45 –|Understand and apply properties of polygons |
|tessellations, symmetry, congruence, similarity and scale. |45 – 90 and 30 – 60 – 90). |Quadrilaterals, including squares, rectangles, parallelograms, trapezoids |
| | |Regular polygons |
| |Students visualize and illustrate ways in which shapes can be combined, |Sum of measures of interior angles of a polygon |
| |subdivided, and changed. | |
| | |Understand and apply transformations |
| | |Finding the image, given the pre-image and vice versa |
| |1. Know and apply the properties of parallelism, perpendicularity and |Sequence of transformations needed to map one figure unto another |
| |symmetry in real-world contexts. |Reflections, rotations, and translations result in images congruent to the |
| | |pre-image |
| |2. Identify congruent and similar figures in real-world situations. |Dilations (stretching/shrinking) result in images similar to the pre image |
| | | |
| |3. Continue a tessellation pattern using the needed transformations. |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. |
|Trigonometry |Students use coordinate geometry to locate objects in both two- and |By the end of 8th grade students will be able to: |
| |three-dimensions and to describe objects algebraically. | |
|Compute measures of sides and angles using proportions, the | |Given an equation or its graph, finds ordered-pair solutions (for example, y|
|Pythagorean Theorem and right triangle relationships. | |= 2x). |
| |1. Know the formula for the graph of a line, including the slope of the | |
|Solve problems requiring indirect measurement for lengths of |line and the intercept of the line, including vertical and horizontal |Apply and explain the simple properties of lines on a graph, including |
|sides of triangles. |lines. |parallelism, perpendicularity, and identifying the x and y intercepts, the |
| | |midpoint of a horizontal or vertical line segment, and the intersection |
| |3. Know the relationships of linear equations as they apply to: properties |point of two lines. |
| |of lines on a graph, including parallelism, | |
| |perpendicularity, the x and y intercepts, the midpoint of a horizontal or |Observe, explain, make and test conjectures regarding geometric properties |
| |vertical line segment, and the intersection point of two lines. |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 |Students understand and apply theories related to numbers. |By the end of 8th grade students will be able to: |
| | | |
|Analyze graphs of related quantities for minimum and maximum |1. Determine the appropriate use of number theory concepts, including |Find the greatest common factor and least common multiple of two or more |
|values and justify the findings. |divisibility rules, to solve real- world or |numbers. |
| |mathematical problems. Describe the concept of unit rate, ratio and slope | |
|Describe the concept of unit rate, ratio and slope in the |in the context of rate of change. |Apply number theory concepts to determine the terms in a real number |
|context of rate of change. | |sequence. |
| |2. Continue a pattern of numbers or objects that could be extended | |
|Continue a pattern of numbers or objects that could be extended|infinitely. |Apply number theory concepts, including divisibility rules, to solve |
|infinitely. | |real-world or mathematical problems. |
| | | |
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
Related searches
- wisconsin school district report cards
- university school district jobs
- texas school district report cards
- university place school district jobs
- tea school district report card
- university city school district calendar
- school district of university city
- school district report cards texas
- u city school district employment
- university city school district employment
- university city school district website
- university place school district employment