Download connected math - Orange Public Schools

Orange School District
Mathematics
Curriculum Guide – Grade 8
2011 Edition
APPROVED: January, 2011
BOARD OF EDUCATION
Patricia A. Arthur
1
President
Arthur Griffa
Vice-President
Members
Stephanie Brown
Eunice Y. Mitchell
Rev. Reginald T. Jackson
Maxine G. Johnson
David Wright
SUPERINTENDENT OF SCHOOLS
Ronald Lee
DEPUTY
SUPERINTENDENT
Dr. Paula Howard
Curriculum and Instructional Services
ADMINISTRATIVE ASSISTANT TO THE
SUPERINTENDENT
Belinda Scott-Smiley
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
2
Table of Contents
BOARD OF EDUCATION
PHILOSOPHY, VISION & PURPOSE
PROCESS GOALS
PHASES OF INSTRUCTION
TARGET GOALS
DESCRIPTION OF STUDENT UNITS
CONNECTED MATH FRAMEWORK
MATHEMATICS LEARNING GOALS
CONTENT GOALS IN EACH UNIT
ALIGNMENT WITH STANDARDS
PROCESS STANDARDS
COMMON CORE STANDARDS AND BLUEPRINT
NEW JERSEY CORE CURRICULUM CONTENT STANDARDS
1
4
5
6
8
10
11
14
18
21
21
23
42
3
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.
4
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
5
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?
6
•
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?
7
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
8




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
9
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
10
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
11
o
o
o
Generalization of Guass’s Method
Triangular Numbers
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
12
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 •
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
A Proof of the Pythagorean Theorem
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
13
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
14

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
15

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
16

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
17
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
18

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
19
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
20
Alignment with Standards
Number and Operations
•
•
Algebra
Looking for Pythagoras (Grade 8)
Clever Counting (Grade 8)
•
•
•
•
•
•
Geometry
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)
•
•
Measurement
Looking for Pythagoras (Grade 8)
Kaleidoscopes, Hubcaps, and Mirrors (Grade 8)
•
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.
21
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.
22
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
24
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.
25
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
Numbers, Number Systems and Number
Relationships
A.
B.
Represent and use numbers in
equivalent forms (e.g., integers,
fractions, decimals, percents,
exponents, scientific notation, square
roots).
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 realworld situations.
26
C.
Distinguish between and order
rational and irrational numbers.
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.
D.
E.
F.
G.
Apply ratio and proportion to
mathematical problem situations
involving distance, rate, time and
similar triangles.
Simplify and expand algebraic
expressions using exponential forms.
Use the number line model to
demonstrate integers and their
applications.
Use the inverse relationships between
addition, subtraction, multiplication,
division, exponentiation and root
extraction to determine unknown
quantities in equations.
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.
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.
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
1. Know the effects of the four basic operations on whole
numbers, fractions, mixed numbers, decimals, and integers.
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
27
C. Estimate the value of irrational numbers.
D. Estimate amount of tips and discounts using
ratios, proportions and percents.
inverse, and closure).
decimal number system.
3. Solve real-world problems involving percents (for
example, discounts, simple interest, taxes, tips).
Students understand and apply properties of numbers and
operations.
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
1. Know the inverse relationship of positive and negative
numbers.
Understand and apply the standard algebraic order of
operations, including appropriate use of parentheses.
2. Know the appropriate operation to solve real-world
problems involving integers, ratios, rates, proportions,
numbers expressed as percents, decimals, fractions, and
square roots.
Estimate square roots and cube roots of numbers
3. Solve multi-step, real-world problems involving integers,
ratios, proportions, numbers expressed as percents,
decimals, and fractions.
Recognize the limitations of estimation and assess the
amount of error resulting from estimation
1.
E.
F.
Determine the appropriateness of
overestimating or underestimating in
computation.
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
Know proportional relationships.
4. Solve real-world problems involving percents including
percents greater than 100% (for example percent of change,
commission).
Use equivalent representation of numbers such as
fractions, decimals, and percents to facilitate 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.
B.
Develop formulas and procedures for
determining measurements (e.g.,
area, volume, distance).
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.
28
C.
D.
E.
F.
G.
Measure angles in degrees and
determine relations of angles.
Estimate, use and describe measures
of distance, rate, perimeter, area,
volume, weight, mass and angles.
Describe how a change in linear
dimension of an object affects its
perimeter, area and volume.
Use scale measurements to interpret
maps or drawings.
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 or
to 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 weighs
one 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.
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.
3. Determine the appropriate precision unit for a given
situation.
29
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.
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.
B. Combine numeric relationships to
arrive at a conclusion.
2. Explain observed difference between mathematical and
experimental results.
C. Use if...then statements to construct
simple, valid arguments.
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.
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).
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:
30
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
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
Analyze the relationship of school
31
Calculate simple mathematical probabilities for independent
and dependent events.
subjects, extracurricular activities, and
community experiences to career
preparation
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-andleaf and box-and-whisker plots.
F. Use scientific and graphing calculators
and computer spreadsheets to
organize and analyze data.
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 boxand- whisker graphs, to display a data set.
DA3: Students use statistical methods to make inferences
and valid arguments about real-world situations.
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.
1. Understand the application of statistics in the formation,
testing and evaluation of a hypothesis.
G. Determine the validity of the
sampling method described in studies
32
published in local or national
newspapers.
Probability and Predictions
A.
B.
Determine the number of
combinations and permutations for
an event.
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.
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
Make valid inferences, predictions
and arguments based on probability.
Play and analyze probability-based games, and discuss the
concepts of fairness and expected value.
Algebra and Functions
A.
B.
Apply simple algebraic patterns to
basic number theory and to spatial
relations
Discover, describe and generalize
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.
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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patterns, including linear, exponential
and simple quadratic relationships.
C.
D.
E.
F.
G.
Create and interpret expressions,
equations or inequalities that model
problem situations.
Use concrete objects to model
algebraic concepts.
Select and use a strategy to solve an
equation or inequality, explain the
solution and check the solution for
accuracy.
Solve and graph equations and
inequalities using scientific and
graphing calculators and computer
spreadsheets.
Represent relationships with tables or
graphs in the coordinate plane and
verbal or symbolic rules.
3. Recognize the independent variable and the dependent
variable in a real world problem.
4. Know function rules to describe tables of related inputoutput.
Students use expressions, equations, inequalities, graphs,
and formulas to represent and interpret situations.
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.
Perform experiments in order to generate data tables that
graph functions.
Graph equations and inequalities in order to explain causeand-effect relationships.
1. Interpret and create tables, function tables, and graphs
(function tables).
2. Graph solutions to linear equations on the coordinate
plane.
Interpret the meaning of the slope of a line from a graph
depicting a real-world situation.
3. Write equations and inequalities to express relationships.
Translate algebraic expressions, equations, or inequalities
representing real-world relationships into verbal expressions
or sentences.
4. Interpret the meaning of the slope of a line from a graph
depicting a real-world situation.
Graph linear equations on the coordinate plane using tables
of values.
5. Translate verbal expressions and sentences into algebraic
expressions, equations, and inequalities.
Graphically display real-world situations represented by
algebraic equations or inequalities.
6. Solve single- and multiple-step linear equations and
inequalities in concrete or abstract form.
Evaluate algebraic expressions, equations, and inequalities
by substituting integral values for variables and simplifying
the results.
7. Know the relationships represented by algebraic
equations or inequalities and their graphic representations.
8. Know how to evaluate algebraic expressions, equations,
and inequalities.
H.
Predict outcomes from given tables of related input-output,
based upon function rules.
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.
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
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properties of real numbers.
J.
K.
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.
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)
Collecting, organizing, and representing data about
the relationship between two variables
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.
B.
C.
Construct figures incorporating
perpendicular and parallel lines, the
perpendicular bisector of a line
segment and an angle bisector using
computer software.
Draw, label, measure and list the
properties of complementary,
supplementary and vertical angles.
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 a
straight edge and a compass.
4. Recognize and draw two-dimensional representations of
Classify familiar polygons as regular or three-dimensional objects (perspective drawings
irregular up to a decagon.
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).
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D.
E.
F.
G.
H.
I.
J.
K.
Identify, name, draw and list all
properties of squares, cubes,
pyramids, parallelograms,
quadrilaterals, trapezoids, polygons,
rectangles, rhombi, circles, spheres,
triangles, prisms and cylinders.
Students use coordinate geometry to locate objects in
both two- and three-dimensions and to describe objects
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 and
horizontal lines.
Construct parallel lines, draw a
transversal and measure and compare
angles formed (e.g., alternate interior 3. Know the relationships of linear equations as they apply
to: properties of lines on a graph, including parallelism,
and exterior angles).
Distinguish between similar and
congruent polygons.
Approximate the value of  (pi)
through experimentation.
Use simple geometric figures (e.g.,
triangles, squares) to create, through
rotation, transformational figures in
three dimensions.
Generate transformations using
computer software.
Analyze geometric patterns (e.g.,
tessellations, sequences of shapes)
and develop descriptions of the
patterns.
Analyze objects to determine whether
they illustrate tessellations,
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.
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).
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.
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symmetry, congruence, similarity and
scale.
Trigonometry
A.
B.
Compute measures of sides and
angles using proportions, the
Pythagorean Theorem and right
triangle relationships.
Solve problems requiring indirect
measurement for lengths of sides of
triangles.
Create an original tessellating tile and tessellation pattern
using a combination of transformations.
Students use coordinate geometry to locate objects in
both two- and three-dimensions and to describe objects
algebraically.
By the end of 8th grade students will be able to:
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.
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.
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.
Given an equation or its graph, finds ordered-pair solutions
(for example, y = 2x).
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.
B.
C.
Analyze graphs of related quantities
for minimum and maximum values
and justify the findings.
Describe the concept of unit rate,
ratio and slope in the context of rate
of change.
Students understand and apply theories related to
numbers.
1. Determine the appropriate use of number theory
concepts, including divisibility rules, to solve real- world or
mathematical 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.
Continue a pattern of numbers or
objects that could be extended
infinitely.
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