acps curriculum framework - 8th grade math curriculum framework – math 8 2014-15 page 2 of 55...

ACPS Curriculum Framework – Math 8 2014-15

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Introduction

The Mathematics Curriculum Framework serves as a guide for teachers when planning instruction and assessments. It defines the content knowledge, skills, and understandings that are measured by the Virginia Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Framework delineates in greater specificity the content that all teachers should teach and all students should learn. The format of the Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Framework is divided by unit and ordered to match a sample pacing. Each unit is divided into three parts: ACPS Standards, Curriculum Overview and a Teacher Notes. The ACPS Standards section is divided into concepts, enduring understandings, essential standards, and lifelong learner standards. The Curriculum Overview contains the DOE curriculum framework information including the related SOL(s), strands, Essential Knowledge and Skills, and Essential Understandings. The Teacher Notes section is divided by Key Vocabulary, Essential Questions, Teacher Notes and Elaborations, Extensions, and Sample Instructional Strategies and Activities. The purpose of each section is explained below. Vertical Articulation (VDOE): This section includes the foundational objectives and the future objectives correlated to each SOL. ACPS Standards:

• Concepts: Interdisciplinary concepts and mathematics concepts specific to the associated SOL’s are listed in support of a concept centered approach to learning encouraged by ACPS. Mental construct or organizing idea that categorizes a variety of examples. Concepts are timeless, universal, abstract, and broad.

• Enduring Understandings: Broad generalizations and principles that connect two or more concepts in a statement of relationship. These understandings are build upon K-12.

• Essential Standards: Essential standards are derived through making connections between topics and enduring understandings. These standards are associated with particular gradebands.

• Lifelong Learner Standards: A standard designed to provide students with a foundation for lifelong inquiry and learning. Unit Overview:

• Curriculum Information: This section includes the SOL and SOL Reporting Category and focus or topic. • Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined.

This is not meant to be an exhaustive list nor is a list that limits what taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective. (Taken from the VDOE Curriculum Framework)

• Essential Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives. (Taken from the VDOE Curriculum Framework)


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Teacher Notes:

• Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills.

• Essential Questions: This section explains what is meant to be the key knowledge and skills that define the standard. • Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this

objective and may extend the teachers’ knowledge of the objective beyond the current grade level. • Extensions: This section provides content and suggestions to differentiate for honors level classes. • Sample Instructional Strategies and Activities: This section provides suggestions for varying instructional techniques within the classroom.

Strands Number and Number Sense In the middle grades, the focus of mathematics learning is to • build on students’ concrete reasoning experiences developed in the elementary grades; • construct a more advanced understanding of mathematics through active learning experiences; • develop deep mathematical understandings required for success in abstract learning experiences; and • apply mathematics as a tool in solving practical problems. Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.

• Students in the middle grades focus on mastering rational numbers. Rational numbers play a critical role in the development of proportional reasoning and advanced mathematical thinking. The study of rational numbers builds on the understanding of whole numbers, fractions, and decimals developed by students in the elementary grades. Proportional reasoning is the key to making connections to most middle school mathematics topics.

• Students develop an understanding of integers and rational numbers by using concrete, pictorial, and abstract representations. They learn how to use equivalent representations of fractions, decimals, and percents and recognize the advantages and disadvantages of each type of representation. Flexible thinking about rational-number representations is encouraged when students solve problems.

• Students develop an understanding of the properties of operations on real numbers through experiences with rational numbers and by applying the order of operations.

• Students use a variety of concrete, pictorial, and abstract representations to develop proportional reasoning skills. Ratios and proportions are a major focus of mathematics learning in the middle grades.


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Computation and Estimation

In the middle grades, the focus of mathematics learning is to • build on students’ concrete reasoning experiences developed in the elementary grades; • construct through active learning experiences a more advanced understanding of mathematics; • develop deep mathematical understandings required for success in abstract learning experiences; and • apply mathematics as a tool in solving practical problems. Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.

• Students develop conceptual and algorithmic understanding of operations with integers and rational numbers through concrete activities and discussions that bring meaning to why procedures work and make sense.

• Students develop and refine estimation strategies and develop an understanding of when to use algorithms and when to use calculators. Students learn when exact answers are appropriate and when, as in many life experiences, estimates are equally appropriate.

• Students learn to make sense of the mathematical tools they use by making valid judgments of the reasonableness of answers.

• Students reinforce skills with operations with whole numbers, fractions, and decimals through problem solving and application activities.


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Measurement

In the middle grades, the focus of mathematics learning is to • build on students’ concrete reasoning experiences developed in the elementary grades; • construct a more advanced understanding of mathematics through active learning experiences; • develop deep mathematical understandings required for success in abstract learning experiences; and • apply mathematics as a tool in solving practical problems. Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.

• Students develop the measurement skills that provide a natural context and connection among many mathematics concepts. Estimation skills are developed in determining length, weight/mass, liquid volume/capacity, and angle measure. Measurement is an essential part of mathematical explorations throughout the school year.

• Students continue to focus on experiences in which they measure objects physically and develop a deep understanding of the concepts and processes of measurement. Physical experiences in measuring various objects and quantities promote the long-term retention and understanding of measurement. Actual measurement activities are used to determine length, weight/mass, and liquid volume/capacity.

• Students examine perimeter, area, and volume, using concrete materials and practical situations. Students focus their study of surface area and volume on rectangular prisms, cylinders, pyramids, and cones.


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Geometry


• Students expand the informal experiences they have had with geometry in the elementary grades and develop a solid foundation for the exploration of geometry in high school. Spatial reasoning skills are essential to the formal inductive and deductive reasoning skills required in subsequent mathematics learning.

• Students learn geometric relationships by visualizing, comparing, constructing, sketching, measuring, transforming, and classifying geometric figures. A variety of tools such as geoboards, pattern blocks, dot paper, patty paper, miras, and geometry software provides experiences that help students discover geometric concepts. Students describe, classify, and compare plane and solid figures according to their attributes. They develop and extend understanding of geometric transformations in the coordinate plane.

• Students apply their understanding of perimeter and area from the elementary grades in order to build conceptual understanding of the surface area and volume of prisms, cylinders, pyramids, and cones. They use visualization, measurement, and proportional reasoning skills to develop an understanding of the effect of scale change on distance, area, and volume. They develop and reinforce proportional reasoning skills through the study of similar figures.

• Students explore and develop an understanding of the Pythagorean Theorem. Mastery of the use of the Pythagorean Theorem has far-reaching impact on subsequent mathematics learning and life experiences.

The van Hiele theory of geometric understanding describes how students learn geometry and provides a framework for structuring student experiences that should lead to conceptual growth and understanding.

• Level 0: Pre-recognition. Geometric figures are not recognized. For example, students cannot differentiate between three-sided and four-sided polygons.


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• Level 1: Visualization. Geometric figures are recognized as entities, without any awareness of parts of figures or relationships between components of a figure. Students should recognize and name figures and distinguish a given figure from others that look somewhat the same. (This is the expected level of student performance during grades K and 1.)

• Level 2: Analysis. Properties are perceived but are isolated and unrelated. Students should recognize and name properties of geometric figures. (Students are expected to transition to this level during grades 2 and 3.)

• Level 3: Abstraction. Definitions are meaningful, with relationships being perceived between properties and between figures. Logical implications and class inclusions are understood, but the role and significance of deduction is not understood. (Students should transition to this level during grades 5 and 6 and fully attain it before taking algebra.)

• Level 4: Deduction. Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. Students should be able to supply reasons for steps in a proof. (Students should transition to this level before taking geometry.)

Probability and Statistics


• Students develop an awareness of the power of data analysis and probability by building on their natural curiosity about data and making predictions.

• Students explore methods of data collection and use technology to represent data with various types of graphs. They learn that different types of graphs represent different types of data effectively. They use measures of center and dispersion to analyze and interpret data.

• Students integrate their understanding of rational numbers and proportional reasoning into the study of statistics and probability.

• Students explore experimental and theoretical probability through experiments and simulations by using concrete, active learning activities.


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Patterns Functions and Algebra


• Students extend their knowledge of patterns developed in the elementary grades and through life experiences by investigating and describing functional relationships.

• Students learn to use algebraic concepts and terms appropriately. These concepts and terms include variable, term, coefficient, exponent, expression, equation, inequality, domain, and range. Developing a beginning knowledge of algebra is a major focus of mathematics learning in the middle grades.

• Students learn to solve equations by using concrete materials. They expand their skills from one-step to two-step equations and inequalities.

• Students learn to represent relations by using ordered pairs, tables, rules, and graphs. Graphing in the coordinate plane linear equations in two variables is a focus of the study of functions.

Special thanks to Henrico County Public Schools for allowing information from their curriculum documents to be included in this document.


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Math 8 Sample Pace

First Marking Period Second Marking Period Third Marking Period Fourth Marking Period

8.1 - Expressions

8.2 - Real Numbers

8.5 - Squares and Square Roots

8.15c - Equations

8.4 – Order of Operations

8.15a - Equations

8.15b - Inequalities

8.16 - Graphing Equations

8.3 ab - Problem Solving

8.14 - Functions

8.17 - Function Vocabulary

8.6 - Angles

8.10 - Pythagorean Theorem

8.11 - Composite Plane Figures

8.7 - 3-D Figures (start)

8.7 - 3-D Figures (finish)

8.9 - 3-D Models

8.8 - Transformations

8.12 - Probability

8.13 – Graphical Methods

SOL Review


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Vertical Articulation Grade 6 Grade 7 Grade 8 Algebra 1

6.2 a) frac/dec/% -‐ a) describe as ratios; b) ID from representation; c) equiv relationships; d) compare/order

7.1 b) determine scientific notation for numbers > zero; c) compare/order fract/dec/%, and scientific notation e) ID/describe absolute value for rational numbers

8.1 b) compare/order fract/dec/%, and scientific notation

A.1 represent verbal quantitative situations algebraically/evaluate expressions for given replacement values of variables

6.3 a) ID/represent integers; b) order/compare integers; c) ID/describe absolute value of integers

7.3 a) model operations (add/sub/mult/div) w/ integers

6.4 Represent mult and div of fract 8.2 describe orally/in writing relationships

between subsets of the real number system

-‐

Solve Practical

Prob

lems

6.7 solve practical problems involving add/sub/mult/div decimals

8.3 a) solve practical problems involving rational numbers, percent, ratios, and prop; b) determine percent inc/dec

6.6 b) solve practical problems involving add/sub/mult/div fractions

Ratio

s/

Prop

ortio

ns 6.1 describe/compare data using ratios 7.4 single and multistep practical problems

with proportional reasoning 8.3 a) solve practical problems involving rational numbers, percent, ratios, and prop

6.2 frac/dec/% -‐ a) describe as ratios; b) ID from representation; c) equiv relationships;

7.6 determine similarity of plane figures and write proportions to express relationships between similar quads and triangles

Expressio

ns/

Opa

tions

6.8 evaluate whole number expressions using order of operations

7.13 a) write verbal expressions as algebraic expressions and sentences as equations and vice versa; b) evaluate algebraic expressions

8.1 a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, properties

A.1 represent verbal quantitative situations algebraically/evaluate expressions for given replacement values of variables

8.4 evaluate algebraic expressions using order of operations

Expo

nents/

Squa

res/

Squa

re

Roots

6.5 investigate/describe positive exponents, perfect squares

7.1 a) investigate/describe negative exponents; d) determine square roots

8.5 a) determine if a number is a perfect square; b) find two consecutive whole numbers between which a square root lies

A.3 express square roots/cube roots of whole numbers/the square root of monomial algebraic expression (simplest radical form)


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Grade 6 Grade 7 Grade 8 Algebra 1 Plan

e an

d Solid Figures

6.13 ID/describe properties of quadrilaterals 7.7 compare/contrast quadrilaterals based on properties

8.6 a) verify/describe relationships among vertical/adjacent/supplementary/complementary angles; b) measure angles < 360°

6.11 a) ID coordinates of a point in a coordinate plane; b) graph ordered pairs in coordinate plane

7.8 represent transformations of polygons in the coordinate plane by graphing

8.8 a) apply transformations to plane figures; b) ID applications of transformations

8.9 construct a 3-‐D model given top or

bottom/side/front views

6.12 determine congruence of segments/angles/polygons

7.6 determine similarity of plane figures and write proportions to express relationships between similar quads and triangles

8.10 a) verify the Pythagorean Theorem; b) apply the Pythagorean Theorem

Measuremen

t App

s -‐ Geo

m Figures 6.9 make ballpark comparisons between U.S.

Cust/metric system 7.5 a) describe volume/surface area of cylinders; b) solve practical problems involving volume/surface area of rect. prims and cylinders; c) describe how changes in measured attribute affects volume/surface area

8.7 a) investigate/solve practical problems involving volume/surface area of prisms, cylinders, cones, pyramids; b) describe how changes in measured attribute affects volume/surface area

6.10 a) define π; b) solve practical problems w/ circumference/area of circle; c) solve practical problems involving area and perimeter given radius/diameter; d) describe/determine volume/surface area of rect. prism

8.11 solve practical area/perimeter problems involving composite plane figures

Prob

ability 6.16 a) compare/contrast dep/indep events;

b) determine probabilities for dep/indep events

7.9 investigate/describe the difference between the experimental/theoretical probability

8.12 determine probability of indep/dep events with and without replacement

7.10 determine the probability of compound

events, Basic Counting Principle

Collect/Rep

resen

t Data 6.14 a) construct circle graphs; b) draw

conclusions/make predictions, using circle graphs; c) compare/contrast graphs

7.11 a) construct/analyze histograms; b) compare/contrast histograms

8.13 a) make comparisons/predictions/inferences, using information displayed in graphs; b) construct/analyze scatterplots

A.10 compare/contrast multiple univariate data sets with box-‐and-‐whisker plots


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Virginia Department of Education -‐ Fall 2010 DRAFT -‐Vertical Articulation of the 2009 Mathematics Standards of Learning

Grade 6 Grade 7 Grade 8 Algebra 1 Eq

uatio

ns and

Ineq

ualities

7.12 represent relationships with tables, graphs, rules, and words

8.14 make connections between any two representations (tables, graphs, words, rules)

A.7 investigate/analyze functions (linear/quadratic) families and characteristics (algebraically/graphically) -‐ a) determine relation is function; b) domain/range; c) zeros; d) x-‐ and y-‐intercepts; e) find values of function for elements in domain; f) make connect between/among multiple representation of functions (concrete/verbal/numeric/graphic/algebraic)

6.18 solve one-‐step linear equations in one variable

7.14 a) solve one-‐ and two-‐step linear equations; b) solve practical problems in one variable

8.15 a) solve multistep linear equations in one variable (variable on one and two sides of equations); b) solve two-‐step linear inequalities and graph results on number line; c) ID properties of operations used to solve

A.4 solve multistep linear/quad equation (in 2 variables) -‐ a) solve leteral equation; b) justify steps used in simplifying expressions and solving equations; c) solve quad equations (algebraically/graphically); d) solve multistep linear equations (algebraically/graphically); e) solve systems of two linear equation (2 variable-‐algebraically/graphically); f) solve real-‐world problems involving equations and systems of equations

6.20 graph inequalities on number line 7.15 a) solve one-‐step inequalities; b) graph solutions on number line

8.16 graph linear equation in two variables A.5 solve multistep linear inequalities (2 variables) -‐ a) solve multistep linear inequalities (algebraically/graphically); b) justify steps used in solving inequalities; c) solve real-‐world problems involving inequalities; d) solve systems of inequalities

8.17 ID domain, range, indep/dep variable A.6 graph linear equations/linear inequalities (in 2 variables) -‐ a) determine slope of line given equation of line/graph of line or two points on line -‐ slope as rate of change; b) write equation of line given graph of line, two points on line or slope & point on line

Prop

ertie

s

6.19 a) investigate/recognize identity properties for add/mult; b) multiplicative property of zero; c) inverse preperty for mult

7.16 a) apply properties w/ real numbers: commutative and associative properties for add/mult; b) distributive property; c) additive/ multiplicative identity properties; d) additive/ multiplicative inverse properties; e) multiplicative property of zero

8.15 c) ID properties of operations used to solve equations

A.2 perform operations on polynomials -‐ a) apply laws of exponents to perform ops on expressions; b) add/subtract/multiply/divide polynomials; c) factor first and second degree binomials/trinomials (1 or 2 variables)


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Interdisciplinary Concept: Systems Math Concept: Relationships ACPS Mathematics Enduring Understandings: 1 - Relationships among numbers and number systems form the foundations of number sense and mathematics communication. ACPS Essential Standard in grade band 6-8: Use strategies to build fluency and extend knowledge of the number system. Life Long Learner Standards

Curriculum Information SOL 8.1

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Relationships within the Real Number System Virginia SOL 8.1 The student will a. simplify numerical expressions

involving positive exponents, using rational numbers, order of operations and properties of operations with real numbers; and

b. compare and order decimals, fractions, percents, and numbers written in scientific notation.

Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Simplify numerical expressions

containing: 1. exponents (where the base is a

rational number and the exponent is a positive whole number);

2. fractions, decimals, integers and square roots of perfect squares; and

3. grouping symbols (no more than 2 embedded grouping symbols). Order of operations and properties of operations with real numbers should be used.

• Compare and order no more than five fractions, decimals, percents, and numbers written in scientific notation using positive and negative exponents. Ordering may be in ascending or descending order.

Essential Questions and Understandings • What is the role of the order of operations when simplifying numerical expressions?

The order of operations prescribes the order to use to simplify a numerical expression.

• How does the different ways rational numbers can be represented help us compare and order rational numbers?

Numbers can be represented as decimals, fractions, percents, and in scientific notation. It is often useful to convert numbers to be compared and/or ordered to one representation (e.g., fractions, decimals or percents).

• What is a rational number? A rational number is any number that can be written in fraction form.

• When are numbers written in scientific notation? Scientific notation is used to represent very large and very small numbers.

Teacher Notes and Elaborations ¨ The set of rational numbers includes the set of all numbers that can be expressed as

fractions in the form ab

where a and b are integers and b ≠ 0.

o Example: 125, , 2.3, 75%, 4.59, 454

− − are rational numbers.

o A rational number is any number that can be written in fraction form.

¨ Expression is a word used to designate any symbolic mathematical phrase that may contain numbers and/or variables.

o Expressions do not contain equal or inequality signs.


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Key Vocabulary

absolute value base exponent grouping symbols identity elements inverses numerical expression order of operations perfect square radical rational number simplify square root

Properties additive identity additive inverse associative of add./mult. commutative of add./mult. distributive multiplicative identity multiplicative inverse multiplicative property of zero

¨ Expressions are simplified using the order of operations and the properties for

operations with real numbers o Properties include the associative, commutative, distributive, and inverse

properties. ¨ The commutative property of addition states that changing the order of the addends

does not change the sum. o Example: 5 + 4 = 4 + 5

¨ The commutative property of multiplication states that changing the order of the factors

does not change the product. o Example: 5 · 4 = 4 · 5





¨ The associative property of addition states that regrouping the addends does not change the sum. o Example: 5 + (4 + 3) = (5 + 4) + 3

¨ The associative property of multiplication states that regrouping the factors does not change the product.

o Example: 5 · (4 · 3) = (5 · 4) · 3 ¨ The distributive property states that the product of a number and the sum (or difference) of two other numbers equals the sum (or

difference) of the products of the number and each other number. o Example: 5 · (3 + 7) = (5 · 3) + (5 · 7), or 5 · (3 – 7) = (5 · 3) – (5 · 7)

¨ Identity elements are numbers that combine with other numbers without changing the other numbers.

o Zero (0) is the identity element for addition. o One (1) is the identity element for multiplication..

¨ The additive identity property states that the sum of any real number and zero is equal to the given real number.

o Example: 5 + 0 = 5

¨ The multiplicative identity property states that the product of any real number and one is equal to the given real number. o Example: 8 · 1 = 8.

¨ Inverses are numbers that combine with other numbers and result in identity elements.

¨ The additive inverse property states that the sum of a number and its additive inverse always equals zero.

o Example: 5 + −5 = 0


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¨ The multiplicative inverse property states that the product of a number and its multiplicative inverse (reciprocal) always equals one. o Example: !

!∙ 5 = 1

o Zero has no multiplicative inverse. ¨ The multiplicative property of zero states that the product of zero and any real number is zero.

(continued)





¨ The order of operations, a mathematical convention, defines the order in which operations are performed to simplify an expression. o To simplify an expression, regroup and combine like terms (integers and/or terms with the same variable) The order of

operations is as follows: 1. Complete all operations within grouping symbols.* If there are grouping symbols within other grouping symbols,

(embedded), do the innermost operation first. 2. Evaluate all exponential expressions. 3. Multiply and/or divide in order from left to right. 4. Add and/or subtract in order from left to right.

o Parentheses ( ), brackets [ ], braces { }, absolute value , division/fraction bar – , and the square root symbol , should be

treated as grouping symbols. ¨ Any real number raised to the zero power is 1.

o The only exception to this rule is zero itself ( 00 1≠ ). o Zero raised to the zero power is undefined.

¨ A power of a number represents repeated multiplication of the number.

o For example, −5 4 means (−5) ∙ (−5) ∙ (−5) ∙ (−5). § The product is 625.

o The base is the number that is multiplied, and the exponent represents the number of times the base is used as a factor. o Notice that the base appears inside the grouping symbols.

§ The meaning changes with the removal of the grouping symbols. § For example, −54 means 5·5·5·5 negated which results in a product of −625. § The expression −(5)4 means to take the opposite of 5·5·5·5 which is −625. § Students should be exposed to all three representations.

¨ Scientific notation is used to represent very large or very small numbers.

o A number written in scientific notation is the product of two factors: a decimal greater than or equal to 1 but less than 10, multiplied by a power of 10.

§ Example: 3.1 × 105 = 310,000 and 3.1 × 10-5 = 0.000031 ¨ All state approved scientific calulators use algebraic logic (follow the order of operations).


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Interdisciplinary Concept: Systems Math Concept: Relationships ACPS Mathematics Enduring Understandings: 1 - Relationships among numbers and number systems form the foundations of number sense and mathematics communication. ACPS Essential Standard in grade band 6-8: Use strategies to build fluency and extend knowledge of the number system. Life Long Learner Standards

Curriculum Information

SOL 8.2 Essential Knowledge and Skills

Key Vocabulary Essential Questions and Understandings

Teacher Notes and Elaborations SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Relationships within the Real Number System Virginia SOL 8.2 The student will describe orally and in writing the relationships between the subsets of the real number system. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Describe orally and in writing the

relationships among the sets of natural or counting numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

• Illustrate the relationships among the subsets of the real number system by using graphic organizers such as Venn diagrams. Subsets include rational numbers, irrational numbers, integers, whole numbers, and natural or counting numbers.

• Identify the subsets of the real number system to which a given number belongs.

• Determine whether a given number is a member of a particular subset of the real number system and explain why.

• Describe each subset of the set of real numbers and include examples and non-examples.

• Recognize that the sum or product of two rational numbers is rational; that the sum of a rational number and an

Essential Questions and Understandings • How are the real numbers related?

Some numbers can appear in more than one subset (e.g., 4 is an integer, a whole number, a counting or natural number, and a rational number.). The attributes (characteristics) of one subset can be contained in whole or in part in another subset.

Teacher Notes and Elaborations ¨ The set of real numbers includes the subsets (parts of sets) natural or counting

numbers, whole numbers, integers, rational, and irrational numbers. o The set of natural numbers is the set of counting numbers

§ Example: {1, 2, 3, 4, …}. o The set of whole numbers includes the set of all the natural numbers or

counting numbers and zero § Example: {0, 1, 2, 3, …}.

o The set of integers includes the set of whole numbers and their opposites § Example: {…, 3− , 2− , 1− , 0, 1, 2, 3, …}.

o The set of rational numbers includes the set of all numbers that can be

expressed as fractions in the form ab

where a and b are integers and b ≠ 0.

§ Examples of rational numbers include: 25 , 14

, 2.3− , 75%, 4.59

§ Fractions such as 18

, can be represented as terminating decimals

(e.g., 18

= 0.125, which has a finite number of decimal places) and


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irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Key Vocabulary attributes integers irrational numbers natural numbers non-repeating dec. non-terminating decimals

rational numbers real numbers repeating decimals terminating decimals Venn diagram whole numbers

fractions such as 29

, can be represented as repeating decimals

(e.g., 2 0.222...9= , whose decimal representation does not end but

continues to repeat). The repeating decimal can be written with ellipses (three dots) as in 0.222… or denoted with a bar above the digit(s) that repeat as in 0.2 .

o The set of irrational numbers is the set of all non-repeating, non-terminating decimals. An irrational number cannot be written in fraction form

§ Examples of irrational numbers include:π , 2 , 1.232332333…

(continued)


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SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Relationships within the Real Number System Virginia SOL 8.2 The student will describe orally and in writing the relationships between the subsets of the real number system. Return to Course Outline

Knowledge/Comprehension Level: � The diagram shows how some of the subsets of the set of real numbers are related.

The letters represent members of the sets. Terrie wants to replace the letters with actual numbers. Which letter could be replaced with -3.

Application/Analysis Level: � Bob is a mechanic for 16 wheel trucks. One of Bob‚s tools is a machinist caliper,

which measures distances to an accuracy of 10-4 (one ten-thousandth) in. Bob was working on a broken piston for a Freightliner. The cylinder of the piston was OK, but the piston head was broken. He had three piston heads to choose from. To rebuild the piston and make it as efficient as possible he had to choose the one that fit the best. He measured each head. One is 1.32 x 10-3 inches smaller, the next is 8.68 x 10-4 inches smaller, and the third is 4.764 x10-4 inches smaller. If he needs the smallest one, then which one does he choose?

Synthesis/Evaluation Level: � Peter says that all fractions are rational numbers, but Taryn argued that all

rational numbers are fractions. Who is correct? Justify your response. � Create an analogy that represents the relationship among the Real Number System and its subsets.


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Interdisciplinary Concept: Change and Interactions Math Concept: Patterns ACPS Mathematics Enduring Understandings: 8 - Patterns and relationships among operations are essential to making estimates and computing fluently. ACPS Essential Standard in grade band 6-8: Investigate the properties and obtain computational fluency within the real number system Life Long Learner Standards




SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Practical Applications of Operations with Real Numbers Virginia SOL 8.5 The student will a. determine whether a given number

is a perfect square; and b. find the two consecutive whole

numbers between which a square root lies.


The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Identify the perfect squares from 0 to

400. • Identify the two consecutive whole

numbers between which the square root of a given whole number from 0 to 400 lies (e.g., 57 lies between 7 and 8 since 27 = 49 and 28 = 64).

• Estimate the square root of a non-perfect square to the nearest whole number.

• Define a perfect square. • Find the positive or positive and

negative square roots of a given whole number from 0 to 400. (Use the symbol

to ask for the positive root and – when asking for the negative root.)

Key Vocabulary consecutive irrational number negative root perfect square

Essential Questions and Understandings • How does the area of a square relate to the square of a number?

The area determines the perfect square number. If it is not a perfect square, the area provides a means for estimation.

• Why do numbers have both positive and negative roots? The square root of a number is any number which when multiplied by itself equals the number. A product, when multiplying two positive factors, is always the same as the product when multiplying their opposites (e.g., 7 · 7 = 49 and 7 7− ⋅− = 49).

Teacher Notes and Elaborations ¨ Define a perfect square.

¨ A perfect square is a whole number whose square root is an integer (e.g., The square root of 25 is 5 and -5; thus, 25 is a perfect square).

¨ The square root of a number is any number which when multiplied by itself equals the number.

¨ Identify the perfect squares from 0 to 400.

¨ Identify the two consecutive whole numbers between which the square root of a given whole number from 0 to 400 lies (e.g., 57 lies between 7 and 8 since 72 = 49 and 82 = 64).

¨ Whole numbers have both positive and negative roots.

¨ Any whole number other than a perfect square has a square root that lies between two


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positive root rational number square root whole number integer

consecutive whole numbers.

¨ The square root of a whole number that is not a perfect square is an irrational number

(e.g., 2 is an irrational number). An irrational number cannot be expressed exactly as a ratio.

¨ Students can use grid paper and estimation to determine what is needed to build a perfect square.

(continued)

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Practical Applications of Operations with Real Numbers Virginia SOL 8.5 The student will a. determine whether a given number

is a perfect square; and b. find the two consecutive whole

numbers between which a square root lies.


¨ Why do numbers have both positive and negative roots? The square root of a number is any number which when multiplied by itself equals the number. A product, when multiplying two positive factors, is always the same as the product when multiplying their opposites (e.g., 7 � 7 = 49 and -7 � -7 = 49).

¨ Find the positive or positive and negative square roots of a given whole number from 0 to 400. (Use the symbol to ask for the positive root and − when asking for the negative root.)

o The square root of a whole number that is not a perfect square is an irrational number (e.g., 2 is an irrational number).

o Estimation can be used to express a non-perfect square root to the nearest whole number (e.g., 11 is between 9 and 16 . The 11 is a little more than 3 because 11 is closer to 9 than to 16. Therefore 11 estimated to the nearest whole number is 3. A number line can be used to illustrate this example.


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Interdisciplinary Concept: Properties and Models Math Concept: Models ACPS Mathematics Enduring Understandings: 4 - Situations and structures can be represented, modeled and analyzed using algebraic symbols. ACPS Essential Standard in grade band 6-8: Solve problems and understand that relationships among quantities can often be expressed symbolically and represented in more than one way. Life Long Learner Standards

Curriculum Information SOL 8.15a,c



SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Linear Relationships Virginia SOL 8.15a, c The student will a. solve multi-step linear equations in

one variable on one and two sides of the equation;

c. identify properties of operations used to solve an equation.


The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Combine like terms to simplify

expressions. • Solve two- to four-step linear equations

in one variable using concrete materials, pictorial representations and paper and pencil illustrating the steps performed.

• Identify properties of operations used to solve an equation/inequality from among: − the commutative properties of

addition and multiplication; − the associative properties of

addition and multiplication; − the distributive property; − the identity properties of addition

and multiplication; − the zero property of multiplication; − the additive inverse property; and − the multiplicative inverse property. − the addition, subtraction,

multiplication and division properties of equality

Essential Questions and Understandings • How does the solution to an equation differ from the solution to an inequality?

While a linear equation has only one replacement value for the variable that makes the equation true, an inequality can have more than one.

Teacher Notes and Elaborations ¨ A linear equation is an equation in which the variables are raised to the first power.

o A linear equation may have one variable or several variables. o An equation in one variable can be of the form ax + b = 0 where x is the

variable, a is the numerical coefficient, and b is the constant. ¨ A multi-step equation is an equation that requires more than one different mathematical

operation to solve. o Sometimes terms contain the same variable and must be combined.

¨ Combining like terms means to combine terms that have the same variable and the

same exponent. o Example: 8x + 11 – 3x can be 5x +11 by combining the like terms of 8x and

3x− . § Note: In this example 8 and 3− are coefficients (numerical factors)

of the terms.

¨ Variables can also be on both sides of the equation. o Example: 3 4 17x x+ = −

¨ In an equation, the equal sign indicates that the value on the left is the same as the value on the right.


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− the addition, subtraction, multiplication and division properties of inequality

¨ To maintain equality, an operation that is performed on one side of an equation must be performed on the other side.





Key Vocabulary addition property of equality additive identity property additive inverse property associative property of addition associative property of multiplication coefficient commutative property of addition commutative property of multiplication distributive property division property of equality identity elements inverses like terms linear equation multiplicative identity property multiplicative inverse property multiplication property of equality subtraction property of equality zero property of multiplication

(continued)


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¨ The commutative property of addition states that changing the order of the addends does not change the sum. o Example: 5 + 4 = 4 + 5

¨ The commutative property of multiplication states that changing the order of the factors does not change the product.

o Example: 5 • 4 = 4 • 5 ¨ The associative property of addition states that regrouping the addends does not change the sum.

o Example: 5 + (4 + 3) = (5 + 4) + 3

¨ The associative property of multiplication states that regrouping the factors does not change the product o Example: 5 • (4 • 3) = (5 • 4) • 3

¨ The distributive property states that the product of a number and the sum (or difference) of two other numbers equals the sum (or

difference) of the products of the number and each other number. o Examples: 5(3 + 7) = (5 · 3) + (5 · 7), or 5(3 – 7) = (5 · 3) – (5 · 7)

¨ Identity elements are numbers that combine with other numbers without changing the other numbers.

o The additive identity is zero (0). o The multiplicative identity is one (1). o There are no identity elements for subtraction and division.

¨ The additive identity property states that the sum of any real number and zero is equal to the given real number.

o Example: 5 + 0 = 5 ¨ The multiplicative identity property states that the product of any real number and one is equal to the given real number.

o Example: 8 · 1 = 8 ¨ The additive inverse property states that the sum of a number and its additive inverse always equals zero.

o Example: 5 + (–5) = 0 ¨ The multiplicative inverse property states that the product of a number and its multiplicative inverse (or reciprocal) always equals one.

o Example: 14 14⋅ =

o Zero has no multiplicative inverse. ¨ The zero property of multiplication states that the product of any real number and zero is zero.


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Interdisciplinary Concept: Change and Interactions Math Concept: Patterns ACPS Mathematics Enduring Understandings: 8 - Patterns and relationships among operations are essential to making estimates and computing fluently. ACPS Essential Standard in grade band 6-8: Investigate the properties and obtain computational fluency within the real number system Life Long Learner Standards




Teacher Notes and Elaborations SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Practical Applications of Operations with Real Numbers Virginia SOL 8.4 The student will apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Substitute rational numbers for

variables in algebraic expressions and simplify the expressions by using the order of operations. Exponents are positive and limited to whole numbers less than 4. Square roots are limited to perfect squares.

• Apply the order of operations to evaluate formulas. Problems will be limited to positive exponents. Square roots may be included in the expressions but limited to perfect squares.

Key Vocabulary absolute value algebraic expression coefficient evaluate exponent grouping symbols numerical expression order of operations

Essential Questions and Understandings • What is the role of the order of operations when evaluating expressions?

Using the order of operations assures only one correct answer for an expression. Teacher Notes and Elaborations ¨ An expression is a word used to designate any symbolic mathematical phrase that may

contain numbers and/or variables. o Expressions do not contain equal or inequality signs and cannot be solved. o A numerical expression contains only numbers and the operations on those

numbers. o An algebraic expression consists of one or more terms. Algebraic expressions

use operations with algebraic symbols (variables) and numbers.

¨ A variable is a letter or other symbol that represents a number. o Substitution is replacing one symbol by another.

¨ A coefficient is the numerical factor of a term

o Examples: The numerical coefficient of 2x is 2, the numerical coefficient of 5y2 is 5, and the numerical coefficient of n is 1.

¨ The order of operations, a mathematical convention, defines the order in which

operations are performed to simplify an expression. o To simplify an expression, regroup and combine like terms (integers and/or

terms with the same variable) The order of operations is as follows: 1. Complete all operations within grouping symbols. If there are grouping

symbols within other grouping symbols, (embedded), do the innermost operation first.


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perfect square radical simplify square root substitution variable

2. Evaluate all exponential expressions. 3. Multiply and/or divide in order from left to right. 4. Add and/or subtract in order from left to right.

o Make sure students explain “left to right” when performing order of operations.

¨ Parentheses ( ), brackets [ ], braces { }, absolute value , division/fraction bar – , and

the square root symbol , should be treated as grouping symbols.


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Interdisciplinary Concept: Properties and Models Math Concept: Models ACPS Mathematics Enduring Understandings: 4 - Situations and structures can be represented, modeled and analyzed using algebraic symbols. ACPS Essential Standard in grade band 6-8: Solve problems and understand that relationships among quantities can often be expressed symbolically and represented in more than one way. Life Long Learner Standards

Curriculum Information SOL 8.15b



SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Relationships Virginia SOL 8.15b The student will b. solve two-step linear inequalities and graph the results on a number line Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Combine like terms to simplify

expressions. • Solve two-step inequalities in one

variable by showing steps and using algebraic sentences.

• Graph solutions to two-step linear inequalities on a number line.

• Identify properties of operations used to solve an equation/inequality.

Key Vocabulary addition property of inequality additive identity property additive inverse property associative property of addition associative property of multiplication coefficient commutative property of addition commutative property of multiplication distributive property division property of inequality identity elements inverses

Essential Questions and Understandings • When solving an equation, why is it important to perform identical operations on each

side of the equal sign? An operation that is performed on one side of an equation must be performed on the other side to maintain equality.

Teacher Notes and Elaborations ¨ A linear inequality is an inequality in which the variables are raised to the first power.

o A linear inequality in one variable can be of the form ax + b > 0, or 0 < ax + b, or ax + b ≥ 0, or 0 ≤ ax + b where x is the variable, a is the numerical coefficient, and b is the constant.

¨ A two-step inequality is defined as an inequality that requires the use of two different operations to solve.

o Example: 3x – 4 > 9

¨ When both expressions of an inequality are multiplied or divided by a negative number, the inequality sign reverses.

¨ Review Teacher Notes and Elaborations in SOL 8.15a,c for additional notes.


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like terms linear equation linear inequality multiplicative identity property multiplicative inverse property multiplication property of inequality subtraction property of inequality zero property of multiplication


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Interdisciplinary Concept: Systems; Change and Interactions Math Concept: Relationships; Cause and Effect ACPS Mathematics Enduring Understandings: 10 - Change, in various contexts, both quantitative and qualitative, can be identified and analyzed. ACPS Essential Standard in grade band 6-8: Use graphs to analyze the nature of changes in quantities in linear relationships. Life Long Learner Standards




SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Relationships Virginia SOL 8.16 The student will graph a linear equation in two variables. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Construct a table of ordered pairs by

substituting values for x in a linear equation to find values for y.

• Plot in the coordinate plane ordered pairs (x, y) from a table.

• Connect the ordered pairs to form a straight line (a continuous function).

• Interpret the unit rate of the proportional relationship graphed as the slope of the graph, and compare two different proportional relationships represented in different ways.

Key Vocabulary continuous function coordinate plane linear equation ordered pair

Essential Questions and Understandings • What types of real life situations can be represented with linear equations?

Any situation with a constant rate can be represented by a linear equation.

Teacher Notes and Elaborations ¨ A linear equation is an equation in two variables whose graph is a straight line, a type

of continuous function.

¨ A linear equation represents a situation with a constant rate. o For example, when driving at a rate of 35 mph, the distance increases as the

time increases, but the rate of speed remains the same.

¨ The slope of a nonvertical line is the ratio of the rise (vertical change) to the run (horizontal change) between any two points on the line.

¨ Compare two different proportional relationships represented in different ways. o For example compare a graph of one relationship to an equation of another

relationship. § Compare a distance-time graph to a distance-time equation to

determine which of two moving objects has greater speed.

¨ In the following example y represents the distance traveled and x represents the time in hours.


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(continued) SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Relationships Virginia SOL 8.16 The student will graph a linear equation in two variables. Return to Course Outline

¨ Based on the comparison of these relationships, Train B at a unit rate of 52 mph has a greater speed than Train A at a unit rate of 50 mph.

¨ The axes of a coordinate plane are generally labeled x and y; however, any letters may be used that are appropriate for the function. ¨ Example: If the constant rate is represented in mph, then t for time and d for distance might be used.

¨ Graphing a linear equation requires determining a table of ordered pairs by substituting into the equation values for one variable and

solving for the other variable, plotting the ordered pairs in the coordinate plane, and connecting the points to form a straight line using a straightedge.


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Interdisciplinary Concept: Change and Interactions Math Concept: Patterns; Cause and Effect ACPS Mathematics Enduring Understandings: 8 - Patterns and relationships among operations are essential to making estimates and computing fluently. 10 - Change, in various contexts, both quantitative and qualitative, can be identified and analyzed. ACPS Essential Standard in grade band 6-8: Investigate the properties and obtain computational fluency within the real number system Use graphs to analyze the nature of changes in quantities in linear relationships. Life Long Learner Standards




Teacher Notes and Elaborations SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Practical Applications of Operations with Real Numbers Virginia SOL 8.3 The student will a. solve practical problems involving

rational numbers, percents, ratios, and proportions; and

b. determine the percent increase or decrease for a given situation.


The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Solve practical problems by using

computation procedures for whole numbers, integers, fractions, percents, ratios, and proportions. Some problems may require the application of a formula.

• Maintain a checkbook and check registry for five or fewer transactions.

• Compute a discount or markup and the resulting sale price for one discount or markup.

• Write a proportion given the relationship of equality between two ratios.

• Compute the percent increase or decrease for a one-step equation found in a real life situation.

• Compute the sales tax and/or tip and resulting total.

• Substitute values for variables in given formulas. For example, use the simple

Essential Questions and Understandings • What is a percent?

A percent is a special ratio with a denominator of 100. • What is the difference between percent increase and percent decrease?

Percent increase and percent decrease are both percents of change measuring the percent a quantity increases or decreases. Percent increase shows a growing change in the quantity while percent decrease shows a lessening change.

Teacher Notes and Elaborations ¨ A rate is a ratio that compares two quantities measured in different units. ¨ A unit rate is a rate with a denominator of 1.

o Examples of unit rates include miles/hour and revolutions/minute. ¨ A discount rate is the percent off an item.

o Example: If an item is reduced in price by 20%, 20% is the discount rate. o The amount of discount (discount) is how much is subtracted from the

original amount. o The sale price (discount price) is the result of subtracting the discount from

the original price. ¨ A sales tax rate is the percent of tax.

o Example: Virginia has a 5% tax rate on most items purchased. o Sales tax is the amount added to the price of an item based on the tax rate.


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interest formula I=prt to determine the value of any missing variable when given specific information.

• Compute the simple interest and new balance earned in an investment or on a loan for a given number of years.

¨ A tip is a small sum of money given as acknowledgment of services rendered, (a gratuity).

o It is often times computed as a percent of the bill or service. ¨ A percent is a special ratio with a denominator of 100. ¨ A markup is a price increase.

o It is the difference between a cost of an item and its selling price. ¨ Interest is an amount of money paid for the use of money.

o The percent of the invested or borrowed amount on which the interest is based is called the interest rate.

o Simple interest for a number of years is determined by multiplying the principal (loan amount) by the rate of interest by the number of years of the loan or investment (I=prt).

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Practical Applications of Operations with Real Numbers Virginia SOL 8.3 The student will a. solve practical problems involving




Key Vocabulary

amount of discount discount price (sale price) discount rate formula interest markup percent

percent of change (rate of change) principal proportion rate (interest rate, tax rate, unit rate) sales tax simple interest tip

¨ Practical problems may include, but not be limited to, those related to economics,

sports, science, social sciences, transportation, and health. o Some examples include problems involving the amount of a paycheck per

month, balancing a checkbook, the discount price on a product, temperature, simple interest, sales tax, and installment buying.

¨ The total value of an investment is equal to the sum of the original investment and the

interest earned. ¨ The total cost of a loan is equal to the sum of the original cost and the interest paid. ¨ Percent increase and percent decrease are both percents of change.

o Percent of change (rate of change) is the percent that a quantity increases or

decreases. o Percent increase determines the rate of growth and may be calculated using a

ratio: amount of change (new original)

original−

¨ For percent increase the change will result in a positive number.

¨ Percent decrease determines the rate of decline and may be calculated using the same

ratio as percent increase.


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o However, the change will result in a negative number (e.g., a 12% decrease is equal to -12%).

¨ A proportion is an equation stating that two ratios are equal.

¨ Proportions are widely used as a problem-solving method. A proportion may be

denoted by a:b = c:d or by a cb d= .

o A proportional situation is based on a multiplicative relationship. ¨ Equal ratios result from multiplication or division, not from addition or subtraction.

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Practical Applications of Operations with Real Numbers Virginia SOL 8.3 The student will a. solve practical problems involving




¨ The first and fourth terms, a and d, are called the extremes of the proportion, and the second and third, b and c, the means of the proportion.

o In a proportion, the product of the means equals the product of the extremes ( a d b c⋅ = ⋅ ). ¨ A formula is an equation that shows a mathematical relationship.

o Example: The volume of a rectangular prism, V = lwh o When given formulas, students must determine the value of any missing variable when given specific information.


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Interdisciplinary Concept: Change and Interactions Math Concept: Patterns ACPS Mathematics Enduring Understandings: 9 - Patterns, relations, and functions can be recognized and understood mathematically. ACPS Essential Standard in grade band 6-8: The study of patterns and relationships should focus on patterns that arise when there is a rate of change. Life Long Learner Standards




SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Relationships Virginia SOL 8.14 The student will make connections between any two representations (tables, graphs, words, and rules) of a given relationship. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Graph in a coordinate plane ordered

pairs that represent a relation. • Describe and represent relations and

functions using tables, graphs, words, and rules. Given one representation, students will be able to represent the relation in another form.

• Relate and compare different representations for the same relation.

Key Vocabulary continuous function discrete function function relation ordered pair origin quadrant horizontal vertical

Essential Questions and Understandings • What is the relationship among tables, graphs, words, and rules in modeling a given

situation? Any given relationship can be represented by all four.

Teacher Notes and Elaborations ¨ A relation is any set of ordered pairs.

o For each first member (domain), there may be many second members (range). ¨ A function is a relation in which there is one and only one second member (range) for

each first member (domain).

¨ As a table of values, a function has a unique value assigned to the second variable for each value of the first variable.

¨ As a graph, a function is any curve (including straight lines) such that any vertical line would pass through the curve only once. Some relations are functions; all functions are relations.

¨ Functions can be represented as tables, graphs, equations, physical models, or in words. o Information given in any one of these ways can be represented in the other

ways. ¨ Graphs of functions can be discrete or continuous.

o In a discrete function graph, there are separate, distinct points. § A line is not used to connect these points on a graph. § The points between the plotted points have no meaning and cannot be


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interpreted. o In a graph of a continuous function every point in the domain can be

interpreted, therefore it is possible to connect the points on the graph with a continuous line as every point on the line answers the original question being asked.

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Relationships Virginia SOL 8.14 The student will make connections between any two representations (tables, graphs, words, and rules) of a given relationship. Return to Course Outline

¨ The following function is represented below as a table, graph, rule, and in words.


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Interdisciplinary Concept: Change and Interactions Math Concept: Patterns ACPS Mathematics Enduring Understandings: 9 - Patterns, relations, and functions can be recognized and understood mathematically. ACPS Essential Standard in grade band 6-8: The study of patterns and relationships should focus on patterns that arise when there is a rate of change. Life Long Learner Standards




SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Relationships Virginia SOL 8.17 The student will identify the domain, range, independent variable, or dependent variable in a given situation. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Apply the following algebraic terms

appropriately: domain, range, independent variable, and dependent variable.

• Identify examples of domain, range, independent variable, and dependent variable.

• Determine the domain of a function. • Determine the range of a function. • Determine the independent variable of a

relationship. Determine the dependent variable of a relationship. Key Vocabulary dependent variable domain independent variable range output input

Essential Questions and Understandings • What are the similarities and differences among the terms domain, range, independent

variable, and dependent variable? The value of the dependent variable changes as the independent variable changes. The domain is the set of all input values for the independent variable. The range is the set of all possible values for the dependent variable.

Teacher Notes and Elaborations ¨ The domain is the set of all the input values for the independent variable in a given

situation. ¨ The range is the set of all the output values for the dependent variable in a given

situation. ¨ The independent variable is the input value. ¨ The dependent variable depends on the independent variable and is the output value. ¨ Below is a table of values for finding circumference of circles, C = πd, where the value

of π is approximated as 3.14.


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o The independent variable, or input, is the diameter of the circle (d). o The set of values for the diameter make up the domain. o The dependent variable, or output, is the circumference (C) of the circle. o The set of values for the circumference makes up the range.


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Interdisciplinary Concept: Systems; Communication Math Concept: Quantifying Representation; Reasoning and Justification ACPS Mathematics Enduring Understandings: 3 - Attributes of objects can be measured using processes and quantified units, and using appropriate techniques, tools, and formulas. 11 - Analyze characteristics and properties of 2- and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships ACPS Essential Standard in grade band 6-8: Become proficient in selecting the appropriate size and type of unit for a given measurement situation, including length, area, and volume Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects Life Long Learner Standards




SOL Reporting Category Measurement and Geometry Focus Problem Solving Virginia SOL 8.6 The student will a. verify by measuring and describe

the relationships among vertical angles, adjacent angles, supplementary angles, and complementary angles; and

b. measure angles of less than 360°. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Measure and draw angles of less than

360° to the nearest degree, using appropriate tools.

• Identify and describe the relationships between angles formed by two intersecting lines.

• Classify the types of angles formed by two lines and a transversal.

• Identify and describe the relationship between pairs of angles that are vertical.

• Identify and describe the relationship between pairs of angles that are alternate interior angles and same side interior angles.

• Identify and describe the relationship between pairs of angles that are supplementary.

Essential Questions and Understandings • How are vertical, adjacent, complementary and supplementary angles related?

Adjacent angles are any two non-overlapping angles that share a common side and a common vertex. Vertical angles will always be nonadjacent angles. Supplementary and complementary angles may or may not be adjacent.

• What are the relationships between the angles formed when two parallel lines are cut by a transversal?

When two parallel lines are cut by a transversal, several pairs of angles are formed. Pairs of alternate interior angles, alternate exterior angles, and vertical angles are congruent. Adjacent angles, and same side (consecutive) interior angles are supplementary.

Teacher Notes and Elaborations ¨ Identify and describe the relationships between angles formed by two intersecting

lines.

¨ Vertical angles are (all nonadjacent angles) formed by two intersecting lines. Vertical angles are congruent and share a common vertex.

¨ Identify and describe the relationship between pairs of angles that are vertical.


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• Identify and describe the relationship between pairs of angles that are complementary.

• Identify and describe the relationship between pairs of angles that are adjacent.

• Use the relationships among supplementary, complementary, vertical, and adjacent angles to solve practical problems.

• Solve practical problems by using the relationship between pairs of angles such as vertical angles, alternate interior angles, same side interior angles, complementary and supplementary angles.

• Identify lines as parallel, intersecting, or perpendicular.

Key Vocabulary adjacent angles alternate interior angles complementary angles congruent intersecting lines nonadjacent angles parallel lines perpendicular lines protractor reflex angles same side interior angles straight angle supplementary angles transversal vertex vertical angles ray acute obtuse straight angle

¨ Identify and describe the relationship between pairs of angles that are complementary.

¨ Complementary angles are any two angles such that the sum of their measures is 90°.

¨ Identify and describe the relationship between pairs of angles that are supplementary.

¨ Supplementary angles are any two angles such that the sum of their measures is 180°.

¨ Reflex angles measure more than 180°.

¨ Identify and describe the relationship between pairs of angles that are adjacent. Teacher Notes and Elaborations ¨ Adjacent angles are any two non-overlapping angles that share a common side and a

common vertex.

¨ Measure angles of less than 360° to the nearest degree, using appropriate tools.

¨ Use the relationships among supplementary, complementary, vertical, and adjacent angles to solve practical problems

¨ A transversal is a line that intersects two or more coplanar lines in different points forming eight angles.

¨ Interior angles lie between the two lines.

¨ Alternate interior angles are on opposite sides of the transversal.

¨ Consecutive interior angles are on the same side of the transversal.

¨ Exterior angles lie outside the two lines.

¨ Alternate exterior angles are on opposite sides of the transversal.


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¨ If two parallel lines are cut by a transversal, then alternate interior angles are congruent. If two parallel lines are cut by a transversal, then same side (consecutive) interior angles are supplementary.

o Given parallel lines and the transversal (t), students should identify, classify, and describe angle relationships.

Examples: 1 and 2∠ ∠ are adjacent supplementary angles.

4 and 6∠ ∠ and 3 and 5∠ ∠ are pairs of same side interior supplementary angles.

1 and 4∠ ∠ , 2 and 3∠ ∠ , 5 and 8∠ ∠ , 6 and 7∠ ∠ are pairs of vertical angles.

3 and 6∠ ∠ and 4 and 5∠ ∠ are pairs of alternate interior angles.

o Using angle relationships when two parallel lines are cut by a transversal, students are expected to determine angle measures given the measure of one angle.

o Problems should include algebraic expressions and equations (e.g., Given 3 3x∠ = and 6 30x∠ = + , what is the value of x? What is the measure of 3∠ ?)


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Interdisciplinary Concept: Systems Math Concept: Quantifying Representation ACPS Mathematics Enduring Understandings: 3 - Attributes of objects can be measured using processes and quantified units, and using appropriate techniques, tools, and formulas. ACPS Essential Standard in grade band 6-8: Become proficient in selecting the appropriate size and type of unit for a given measurement situation, including length, area, and volume Life Long Learner Standards




SOL Reporting Category Measurement and Geometry Focus Problem Solving with 2- and 3-Dimensional Figures Virginia SOL 8.10 The student will a. verify the Pythagorean Theorem;

and b. apply the Pythagorean Theorem. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Identify the parts of a right triangle (the

hypotenuse and the legs). • Verify a triangle is a right triangle

given the measures of its three sides. • Verify the Pythagorean Theorem, using

diagrams, concrete materials, and measurement.

• Find the measure of a side of a right triangle given the measures of the other two sides.

• Solve practical problems involving right triangles by using the Pythagorean Theorem.

Key Vocabulary hypotenuse leg Pythagorean Theorem Pythagorean triples right triangle square root

Essential Questions and Understandings • How can the area of squares generated by the legs and the hypotenuse of a right triangle

be used to verify the Pythagorean Theorem? For a right triangle, the area of a square with one side equal to the measure of the hypotenuse equals the sum of the areas of the squares with one side each equal to the measures of the legs of the triangle.

Teacher Notes and Elaborations ¨ Verify the Pythagorean Theorem, using diagrams, concrete materials, and

measurement.

¨ In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the legs (altitude and base). This relationship is known as the Pythagorean Theorem: a2 + b2 = c2.

¨ Find the measure of a side of a right triangle, given the measures of the other two sides.


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SOL Reporting Category Measurement and Geometry Focus Problem Solving with 2- and 3-Dimensional Figures Virginia SOL 8.10 The student will a. verify the Pythagorean Theorem;

and b. apply the Pythagorean Theorem. Return to Course Outline

¨ The Pythagorean Theorem is used to find the measure of any one of the three sides of a right triangle if the measures of the other two sides are known.

o For example: Given the following right triangle, find the missing leg.

¨ Verify a triangle is a right triangle given the measures of its three sides.

¨ Whole number triples that are the measures of the sides of right triangles, such as (3,4,5), (6,8,10), (9,12,15), and (5,12,13), are commonly known as Pythagorean triples.

¨ Identify the parts of a right triangle (the hypotenuse and the legs).

¨ The hypotenuse of a right triangle is the side opposite the right angle.

¨ The hypotenuse of a right triangle is always the longest side of the right triangle.

¨ The legs of a right triangle form the right angle.

¨ Solve practical problems involving right triangles by using the Pythagorean Theorem.


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Teacher Notes and Elaborations SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Problem Solving with 2- and 3-Dimensional Figures Virginia SOL 8.11 The student will solve practical area and perimeter problems involving composite plane figures. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Subdivide a figure into triangles,

rectangles, squares, trapezoids and semicircles. Find the area of subdivisions and combine to determine the area of the composite figure.

• Use the attributes of the subdivisions to determine the perimeter and circumference of a figure.

• Apply perimeter, circumference, and area formulas to solve practical problems.

Key Vocabulary area circumference composite figure perimeter plane figure semicircle adjacent trapezoid pi radius diameter polygon parallelogram base height length

Essential Questions and Understandings • How does knowing the perimeter and/or circumference and areas of polygons and

circles assist in calculating the perimeters and areas of composite figures? The perimeter of a composite figure can be found by subdividing the figure into triangles, rectangles, squares, trapezoids and/or semi-circles, and calculating the perimeter using the appropriate measurements. The area of a composite figure can be found by subdividing the figure into triangles, rectangles, squares, trapezoids and/or semi-circles, calculating their areas, and adding the areas together.

Teacher Notes and Elaborations ¨ Subdivide a figure into triangles, rectangles, squares, trapezoids and semicircles.

o Estimate the area of subdivisions and combine to determine the area of the composite figure.

¨ Use the attributes of the subdivisions to determine the perimeter and circumference of a figure.

¨ Apply perimeter, circumference and area formulas to solve practical problems.

¨ A polygon is a simple, closed plane figure with sides that are line segments.

¨ The perimeter of a polygon is the distance around the figure.

¨ The area of any composite figure is based upon knowing how to find the area of the composite parts such as triangles and rectangles.

¨ The area of a rectangle is computed by multiplying the lengths of two adjacent sides. ( A lw= )

¨ The area of a triangle is computed by multiplying the measure of its base by the measure of its height and dividing the product by 2.

( 12

A bh= )


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SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Problem Solving with 2- and 3-Dimensional Figures Virginia SOL 8.11 The student will solve practical area and perimeter problems involving composite plane figures. Return to Course Outline

¨ The area of a parallelogram is computed by multiplying the measure of its base by the measure of its height. ( A bh= )

¨ The area of a trapezoid is computed by taking the average of the measures of the two bases and multiplying this average by the height.

[ ( )1 212

A h b b= + ]

¨ The area of a circle is computed by multiplying Pi times the radius squared. ( 2A rπ= )

¨ The circumference of a circle is found by multiplying Pi by the diameter or multiplying Pi by 2 times the radius, (C dπ= or 2C rπ= )

¨ An estimate of the area of a composite figure can be made by subdividing the polygon into triangles, rectangles, squares, trapezoids and semicircles, estimating their areas, and adding the areas together.


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Interdisciplinary Concept: Systems Math Concept: Quantifying Representation ACPS Mathematics Enduring Understandings: 3 - Attributes of objects can be measured using processes and quantified units, and using appropriate techniques, tools, and formulas. ACPS Essential Standard in grade band 6-8: Become proficient in selecting the appropriate size and type of unit for a given measurement situation, including length, area, and volume Life Long Learner Standards




SOL Reporting Category Measurement and Geometry Focus Problem Solving Virginia SOL 8.7 The student will a. investigate and solve practical

problems involving volume and surface area of prisms, cylinders, cones, and pyramids; and

b. describe how changing one measured attribute of the figure affects the volume and surface area.


The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Distinguish between situations that are

applications of surface area and those that are applications of volume.

• Investigate and compute the surface area of a square or triangular pyramid by finding the sum of the areas of the triangular faces and the base using concrete objects, nets, diagrams, and formulas.

• Investigate and compute the surface area of a cone by calculating the sum of the areas of the sides and base, using concrete objects, nets, diagrams, and formulas.

• Investigate and compute the surface area of a right cylinder using concrete objects, nets, diagrams, and formulas.

• Investigate and compute the surface area of a rectangular prism using concrete objects, nets, diagrams, and formulas.

• Investigate and compute the volume of prisms, cylinders, cones, and pyramids using concrete objects, nets, diagrams,

Essential Questions and Understandings • How does the volume of a three-dimensional figure differ from its surface area?

Volume is the amount a container holds. Surface area of a figure is the sum of the area on surfaces of the figure.

• How are the formulas for the volume of prisms and cylinders similar? In both formulas, the area of the base is multiplied by the height to find the volume.

• How are the formulas for the volume of cones and pyramids similar?

The volume of a cone is 13

the volume of a cylinder with the same size base and

height.

The volume of a pyramid is 13

the volume of a prism with the same size base and

height. • What effect does changing one attribute of a rectangular prism by a scale factor have on

the surface area of the prism? There is no direct relationship for surface area as there is for volume (e.g., If width triples, surface area will increase but it will not triple.).

• What effect does changing one attribute of a rectangular prism by a scale factor have on the volume of the prism?

When the length, width or height of a rectangular prism is increased or decreased by a factor, the volume of the prism is also increased or decreased by that factor (e.g., If width triples, volume triples.).

Teacher Notes and Elaborations ¨ Distinguish between situations that are applications of surface area and those that are

applications of volume.


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and formulas. • Solve practical problems involving

volume and surface area of prisms, cylinders, cones, and pyramids.

• Compare and contrast the volume and surface area of a prism with a given set of attributes with the volume and surface area of a prism where one of the attributes has been increased/decreased

by a factor of 12

, 2, 3, 5, or 10.

¨ Investigate and compute the surface area of a square or triangular pyramid by finding the sum of the areas of the triangular faces and the base using concrete objects, nets, diagrams and formulas.

¨ Investigate and compute the surface area of a cone by calculating the sum of the areas of the side and the base, using concrete objects, nets, diagrams and formulas.

(continued)


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(continued from previous page) SOL Reporting Category Measurement and Geometry Focus Problem Solving Virginia SOL 8.7 The student will a. investigate and solve practical




(continued from previous page) The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Describe the two-dimensional figures

that result from slicing three-dimensional figures parallel to the base. (e.g., as in plane sections of right rectangular prisms and right rectangular pyramids).

Key Vocabulary base cone cylinder face height net polyhedron prism pyramid rectangular prism scale factor slant height surface area volume width length

(continued from previous page) Teacher Notes and Elaborations ¨ Investigate and compute the surface area of a right cylinder using concrete objects,

nets, diagrams and formulas.

¨ Investigate and compute the surface area of a rectangular prism using concrete objects, nets, diagrams and formulas.

¨ Investigate and compute the volume of prisms, cylinders, cones, and pyramids, using concrete objects, nets, diagrams, and formulas.

¨ Solve practical problems involving volume and surface area of prisms, cylinders, cones, and pyramids.

¨ Nets are two-dimensional representations that can be folded into three-dimensional figures.

¨ Describe the two-dimensional figures that result from slicing three-dimensional figures parallel to the base (e.g., as in plane sections of right rectangular prisms and right rectangular pyramids).

¨ A polyhedron is a solid figure whose faces are all polygons.

¨ A pyramid is a polyhedron with a base that is a polygon and other faces that are triangles with a common vertex.

¨ The area of the base of a pyramid is the area of the polygon which is the base.

¨ The total surface area of a pyramid is the sum of the areas of the triangular faces and the area of the base.

¨ The volume of a pyramid is 13 Bh, where B is the area of the base and h is the height.

¨ The area of the base of a circular cone is πr2.

(continued)


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Essential Questions and Understandings Teacher Notes and Elaborations (continued)





¨ The surface area of a right circular cone is πr2 + πrl, where l represents the slant height of the cone.

¨ The volume of a cone is 13 πr2h, where h is the height and πr2 is the area of the base.

¨ The surface area of a right circular cylinder is 22 2r rhπ π+ .

¨ The volume of a cylinder is the area of the base of the cylinder multiplied by the height.

¨ A prism is a solid figure that has a congruent pair of parallel bases and faces that are parallelograms. The surface area of a prism is the sum of the areas of the faces and bases.

¨ The surface area of a rectangular prism is the sum of the areas of the six faces.

¨ The volume of a prism is Bh, where B is the area of the base and h is the height of the prism.

¨ The volume of a rectangular prism is calculated by multiplying the length, width and height of the prism.

¨ How are the formulas for the volume of prisms and cylinders similar? For both formulas you are finding the area of the base and multiplying that by the height.

¨ How are the formulas for the volume of cones and pyramids similar? For cones you are finding 13 of the volume of the cylinder

with the same size base and height. For pyramids you are finding 13 of the volume of the prism with the same size base and height.

¨ In general what effect does changing one attribute of a prism by a scale factor have on the volume of the prism? When you increase or decrease the length, width or height of a prism by a factor greater than 1, the volume of the prism is also increased by that factor.

¨ Compare and contrast the volume and surface area of a prism with a given set of attributes with the volume of a prism where one of the attributes has been increased by a factor of 2, 3, 5 or 10.

(continued)


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Essential Questions and Understandings Teacher Notes and Elaborations (continued)





¨ When one attribute of a prism is changed through multiplication or division the volume increases by the same factor that the attribute increased by. For example, if a prism has a volume of 2 x 3 x 4, the volume is 24. However, if one of the attributes are doubled, the volume doubles.

o Example: Given a rectangular prism with the following dimensions: l = 5 meters, w = 4 meters and h = 3 meters. Students should describe how the volume and surface area of a rectangular prism is affected when one attribute is multiplied by a scale factor.


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Interdisciplinary Concept: Communication Math Concept: Reasoning and Justification ACPS Mathematics Enduring Understandings: 11 - Analyze characteristics and properties of 2- and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships ACPS Essential Standard in grade band 6-8: Investigate relationships of polygons by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects Life Long Learner Standards




SOL Reporting Category Measurement and Geometry Focus Problem Solving with 2- and 3-Dimensional Figures Virginia SOL 8.9 The student will construct a three-dimensional model given the top or bottom, side and front views. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Construct three-dimensional models

given the top or bottom, side, and front views.

• Identify three-dimensional models given a two-dimensional perspective.

Key Vocabulary perspective three-dimensional model isometric

Essential Questions and Understandings • How does knowledge of two-dimensional figures inform work with three-dimensional

objects? It is important to know that a three-dimensional object can be represented as a two-dimensional model with views of the object from different perspectives.

Teacher Notes and Elaborations ¨ Construct three-dimensional models, given the top or bottom, side, and front views.

¨ Identify three-dimensional models given a two-dimensional perspective.

¨ Three-dimensional models of geometric solids can be used to understand perspective

and provide tactile experiences in determining two-dimensional perspectives.

¨ Three-dimensional models of geometric solids can be represented on isometric paper.

¨ The top view is a mirror image of the bottom view.

¨ Use snap cubes or pop cubes to construct figures given the top or bottom, side and front view of figures and vice versa.


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Interdisciplinary Concept: Communication Math Concept: Reasoning and Justification ACPS Mathematics Enduring Understandings: 12 - Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations. ACPS Essential Standard in grade band 6-8: Create and quantify the results of various transformations, including dilation Life Long Learner Standards




SOL Reporting Category Measurement and Geometry Focus Problem Solving with 2- and 3-Dimensional Figures Virginia SOL 8.8 The student will a. apply transformations to plane

figures; and b. identify applications of

transformations. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Demonstrate the reflection of a polygon

over the vertical or horizontal axis on a coordinate grid.

• Demonstrate 90°, 180°, 270°, and 360° clockwise and counterclockwise rotations of a figure on a coordinate grid. The center of rotation will be limited to the origin.

• Demonstrate the translation of a polygon on a coordinate grid.

• Demonstrate the dilation of a polygon from a fixed point on a coordinate grid.

• Identify practical applications of transformations including, but not limited, to, tiling, fabric, and wallpaper designs, art and scale drawings.

• Identify the type of transformation in a given example.

Key Vocabulary

angle of rotation center of rotation clockwise counterclockwise

preimage reflection rotation scale factor

Essential Questions and Understandings • How does the transformation of a figure on the coordinate grid affect the congruency,

orientation, and location of an image? Translations, rotations, and reflections maintain congruence between the preimage and image but change location. Dilations by a scale factor other than 1 produce an image that is not congruent to the preimage but is similar. Rotations and reflections change the orientation of the image.

Teacher Notes and Elaborations ¨ Demonstrate the reflection of a polygon over the vertical or horizontal axis on a

coordinate grid.

¨ A reflection of a geometric figure moves all of the points of the figure across an axis. Each point on the reflected figure is the same distance from the axis as the corresponding point in the original figure.

¨ Demonstrate 90°, 180°, 270°, and 360°clockwise and counterclockwise rotations of a figure on a coordinate grid. The center of rotation will be limited to the origin.

¨ A rotation of a geometric figure is a clockwise or counterclockwise turn of the figure around a fixed point. The point may or may not be on the figure. The fixed point is called the center of rotation.

¨ Demonstrate the translation of a polygon on a coordinate grid.

¨ A translation of a geometric figure moves all the points on the figure the same distance in the same direction.


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dilation image line of reflection orientation origin original figure

similar figure transformation translation ordered pair polygon quadrant

¨ Demonstrate the dilation of a polygon from a fixed point on a coordinate grid.

¨ A dilation of a geometric figure is a transformation that changes the size of a figure by a scale factor to create a similar figure.

¨ Identify practical applications of transformations including, but not limited to, tiling, fabric, and wallpaper designs, art and scale drawings.

¨ Identify the type of transformation in a given example.

(continued)

SOL Reporting Category Measurement and Geometry Focus Problem Solving with 2- and 3-Dimensional Figures Virginia SOL 8.8 The student will a. apply transformations to plane

figures; and b. identify applications of

transformations. Return to Course Outline

¨ Practical applications may include, but are not limited to, the following: o A rotation of the hour hand of a clock from 2:00 to 3:00 shows a turn of 30° clockwise; o A reflection of a boat in water shows an image of the boat flipped upside down with the water line being the line of

reflection; o A translation of a figure on a wallpaper pattern shows the same figure slid the same distance in the same direction; and o A dilation of a model airplane is the production model of the airplane.

¨ The image of a polygon is the resulting polygon after a transformation. The preimage is the original polygon before the transformation.

¨ A transformation of preimage point A can be denoted as the image Aʹ′ (read as “A prime”).


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Interdisciplinary Concept: Properties and Models; Communication Math Concept: Analysis and Evaluation; Theory ACPS Mathematics Enduring Understandings: 6 - Data can be collected, organized, and displayed in purposeful ways 13 - Probability and data analysis can be used to make predictions ACPS Essential Standard in grade band 6-8: Formulate questions, design studies, collect relevant data, and create and use appropriate graphical representations of data Use a basic understanding of probability to make and test conjectures about the results of experiments and simulations. Life Long Learner Standards




SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Statistical Analysis of Graphs and Problem Situations Virginia SOL 8.12 The student will determine the probability of independent and dependent events with and without replacement. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Determine the probability of no more

than three independent events. • Determine the probability of no more

than two dependent events without replacement.

• Compare the outcomes of events with and without replacement.

Key Vocabulary compound events dependent event independent event outcome probability replacement occurrence

Essential Questions and Understandings • How are the probabilities of dependent and independent events similar? Different?

If events are dependent then the second event is considered only if the first event has already occurred. If events are independent, then the second event occurs regardless of whether or not the first occurs.

Teacher Notes and Elaborations ¨ In probability, the outcome is the result of performing an experiment. The probability

of an event occurring is the ratio of the desired outcomes to the total number of possible outcomes.

o The probability that an event is likely to occur is close to one. o The probability that an event is not likely to occur is close to zero. o The probability that an event is as likely to occur as it is not to occur is close

to one half.

¨ Events that contain more than one outcome are called compound events

¨ Determine the probability of no more than three independent events.

¨ If the outcome of one event does not influence the occurrence of the other event, they are called independent. If events are independent, then the second event occurs regardless of whether or not the first occurs. For example, the first roll of a number cube does not influence the second roll of the number cube. Other examples of independent events are, but not limited to: flipping two coins; spinning a spinner and


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rolling a number cube; flipping a coin and selecting a card; and choosing a card from a deck, replacing the card and selecting again.

¨ The probability of three independent events is found by using the following formula: ( ) ( ) ( ) ( )P Aand BandC P A P B P C= ⋅ ⋅ o Ex: When rolling three number cubes simultaneously, what is the probability

of rolling a 3 on one cube, a 4 on one cube, and a 5 on the third?

1 1 1 1(3 4 5) (3) (4) (5)6 6 6 216

P and and P P P= ⋅ ⋅ = ⋅ ⋅ =

(continued) SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Statistical Analysis of Graphs and Problem Situations Virginia SOL 8.12 The student will determine the probability of independent and dependent events with and without replacement. Return to Course Outline

¨ Determine the probability of no more than two dependent events without replacement.

¨ If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. For example, if you are dealt a King from a deck of cards and you do not place the King back into the deck before selecting a second card, the chance of selecting a King the second time is diminished because there are now only three Kings remaining in the deck. Other examples of dependent events are, but not limited to: choosing two marbles from a bag but not replacing the first after selecting it; and picking a sock out of a drawer and then picking a second sock without replacing the first.

¨ The probability of two dependent events is found by using the following formula ( ) ( ) ( )P Aand B P A P Bafter A= ⋅

o Ex: You have a bag holding a blue ball, a red ball, and a yellow ball. What is the probability of picking a blue ball out of the

bag on the first pick then without replacing the blue ball in the bag, picking a red ball on the second pick?

1 1 1(blue red) (blue) (red blue)3 2 6

P and P P after= ⋅ = ⋅ =

¨ Compare the outcomes of events with and without replacement.

¨ Two events are either dependent or independent.


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Interdisciplinary Concept: Properties and Models Math Concept: Analysis and Evaluation ACPS Mathematics Enduring Understandings: 7 - Various statistical methods can be used to observe, analyze, predict, and make inferences about data. ACPS Essential Standard in grade band 6-8: Discuss and understand the correspondence between data sets and their graphic representations, and find, use, and interpret their measures of central tendency Life Long Learner Standards




SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Statistical Analysis of Graphs and Problem Situations Virginia SOL 8.13 The student will a. make comparisons, predictions, and

inferences, using information displayed in graphs; and

b. construct and analyze scatterplots. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Collect, organize and interpret a data

set of no more than 20 items using scatterplots. Predict from the trend an estimate of the line of best fit with a drawing.

• Interpret a set of data points in a scatterplot as having a positive relationship, a negative relationship, or no relationship.

• Make comparisons, predictions, and inferences, given data sets that are displayed in frequency distributions, scatterplots, line, bar, circle, picture graphs and histograms.

Key Vocabulary comparison inference line of best fit negative relationship no relationship positive relationship

Essential Questions and Understandings • What is a line of best fit?

A line of best fit (curve of best fit) is a trend line that shows the relationship between two sets of data most accurately.

• Why do we estimate a line of best fit for a scatterplot? A line of best fit helps in making interpretations and predictions about the situation modeled in the data set.

• What are the inferences that can be drawn from sets of data points having a positive relationship, a negative relationship, and no relationship?

Sets of data points with positive relationships demonstrate that the values of the two variables are increasing. A negative relationship indicates that as the value of the independent variable increases, the value of the dependent variable decreases.

Teacher Notes and Elaborations ¨ A scatterplot illustrates the relationship between two sets of data. A scatterplot

consists of points. The coordinates of the point represent the measures of the two attributes of the point. No lines are drawn to connect the points. The coordinates of the point represent the measures of the two attributes of the point. Scatterplots can be used to predict trends and to estimate a line of best fit.

¨ Scatterplots can be used to predict trends and estimate a line of best fit.

¨ In a scatterplot, each point is represented by an independent and dependent variable. The independent variable is graphed on the horizontal axis and the dependent is graphed on the vertical axis.


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prediction scatterplot correlation trend horizontal axis vertical axis

¨ Collect, organize, and interpret a data set of no more than 20 items using scatterplots. Predict from the trend an estimate of the line of best fit with a drawing.

(continued)

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Statistical Analysis of Graphs and Problem Situations Virginia SOL 8.13 The student will a. make comparisons, predictions, and

inferences, using information displayed in graphs; and

b. construct and analyze scatterplots. Return to Course Outline

¨ Interpret a set of data points in a scatterplot as having a positive relationship, a negative relationship, or no relationship.

¨ Comparisons, predictions, and inferences are made by examining characteristics of a data set displayed in a variety of graphical representations to draw conclusions.

¨ The information displayed in different graphs may be examined to determine how data are or are not related, ascertaining differences between characteristics (comparisons), trends that suggest what new data might be like (predictions), and/or “what could happen if” (inferences).

¨ Make comparisons, predictions, and inferences, given data sets that are displayed in frequency distributions, scatterplots, line, bar, circle, picture graphs and histograms.

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