Download Introduction - Albemarle County Public Schools

ACPS Curriculum Framework – Math 8
f
2014-15
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)
Page 1 of 55
ACPS Curriculum Framework – Math 8
2014-15
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.
Page 2 of 55
ACPS Curriculum Framework – Math 8
2014-15
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.
Page 3 of 55
ACPS Curriculum Framework – Math 8
2014-15
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.
Page 4 of 55
ACPS Curriculum Framework – Math 8
2014-15
Geometry
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 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.
Page 5 of 55
ACPS Curriculum Framework – Math 8
2014-15
• 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
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 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.
Page 6 of 55
ACPS Curriculum Framework – Math 8
2014-15
Page 7 of 55
ACPS Curriculum Framework – Math 8
2014-15
Patterns Functions and Algebra
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 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.
Page 8 of 55
ACPS Curriculum Framework – Math 8
2014-15
Math 8 Sample Pace
First Marking Period
Second Marking Period
Third Marking Period
Fourth Marking Period
8.1 - Expressions
8.15b - Inequalities
8.17 - Function Vocabulary
8.7 - 3-D Figures (start)
8.2 - Real Numbers
8.16 - Graphing Equations
8.6 - Angles
8.7 - 3-D Figures (finish)
8.5 - Squares and Square Roots
8.3 ab - Problem Solving
8.10 - Pythagorean Theorem
8.9 - 3-D Models
8.14 - Functions
8.11 - Composite Plane Figures
8.8 - Transformations
8.15c - Equations
8.12 - Probability
8.4 – Order of Operations
8.13 – Graphical Methods
8.15a - Equations
SOL Review
Page 9 of 55
ACPS Curriculum Framework – Math 8
2014-15
Vertical Articulation
Grade 6 Grade 7 6.2 a) frac/dec/% -­‐ a) describe as ratios; b) ID 7.1 b) determine scientific notation for from representation; c) equiv relationships; numbers > zero; c) compare/order d) compare/order fract/dec/%, and scientific notation e) ID/describe absolute value for rational numbers 6.3 a) ID/represent integers; b) order/compare integers; c) ID/describe absolute value of integers Grade 8 8.1 b) compare/order fract/dec/%, and scientific notation Algebra 1 A.1 represent verbal quantitative situations algebraically/evaluate expressions for given replacement values of variables 7.3 a) model operations (add/sub/mult/div) w/ integers Exponents/ Squares/ Square Roots Expressions/ Opations Ratios/ Proportions Solve Practical Problems 6.4 Represent mult and div of fract 6.7 solve practical problems involving add/sub/mult/div decimals 8.2 describe orally/in writing relationships between subsets of the real number system -­‐ 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 7.4 single and multistep practical problems 8.3 a) solve practical problems involving with proportional reasoning rational numbers, percent, ratios, and prop 6.2 frac/dec/% -­‐ a) describe as ratios; b) ID 7.6 determine similarity of plane figures and from representation; c) equiv relationships; write proportions to express relationships between similar quads and triangles 6.8 evaluate whole number expressions 7.13 a) write verbal expressions as algebraic 8.1 a) simplify numerical expressions A.1 represent verbal quantitative situations using order of operations expressions and sentences as equations and involving positive exponents, using rational algebraically/evaluate expressions for given vice versa; b) evaluate algebraic expressions numbers, order of operations, properties replacement values of variables 6.1 describe/compare data using ratios 6.5 investigate/describe positive exponents, 7.1 a) investigate/describe negative perfect squares exponents; d) determine square roots 8.4 evaluate algebraic expressions using order of operations 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) Page 10 of 55
ACPS Curriculum Framework – Math 8
2014-15
Grade 6 Grade 7 Plane and Solid Figures 6.13 ID/describe properties of quadrilaterals 7.7 compare/contrast quadrilaterals based on properties Grade 8 8.6 a) verify/describe relationships among vertical/adjacent/supplementary/compleme
ntary angles; b) measure angles < 360° 6.11 a) ID coordinates of a point in a 7.8 represent transformations of polygons in 8.8 a) apply transformations to plane figures; coordinate plane; b) graph ordered pairs in the coordinate plane by graphing b) ID applications of transformations coordinate plane 6.12 determine congruence of segments/angles/polygons 7.6 determine similarity of plane figures and 8.10 a) verify the Pythagorean Theorem; b) write proportions to express relationships apply the Pythagorean Theorem between similar quads and triangles Algebra 1 8.9 construct a 3-­‐D model given top or bottom/side/front views Collect/Repr
esent Data Probability Measurement Apps -­‐ Geom Figures 6.9 make ballpark comparisons between U.S. 7.5 a) describe volume/surface area of Cust/metric system 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 6.16 a) compare/contrast dep/indep events; 7.9 investigate/describe the difference b) determine probabilities for dep/indep between the experimental/theoretical events probability 8.12 determine probability of indep/dep events with and without replacement 6.14 a) construct circle graphs; b) draw conclusions/make predictions, using circle graphs; c) compare/contrast graphs 7.10 determine the probability of compound events, Basic Counting Principle 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 Page 11 of 55
ACPS Curriculum Framework – Math 8
Grade 6 Equations and Inequalities Properties 2014-15
Grade 7 7.12 represent relationships with tables, graphs, rules, and words Grade 8 Algebra 1 8.14 make connections between any two A.7 investigate/analyze functions representations (tables, graphs, words, rules) (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 7.14 a) solve one-­‐ and two-­‐step linear 8.15 a) solve multistep linear equations in A.4 solve multistep linear/quad equation (in variable equations; b) solve practical problems in one one variable (variable on one and two sides 2 variables) -­‐ a) solve leteral equation; b) variable of equations); b) solve two-­‐step linear justify steps used in simplifying expressions inequalities and graph results on number and solving equations; c) solve quad line; c) ID properties of operations used to equations (algebraically/graphically); d) solve 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 8.16 graph linear equation in two variables A.5 solve multistep linear inequalities (2 solutions on number line 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 6.19 a) investigate/recognize identity 7.16 a) apply properties w/ real numbers: 8.15 c) ID properties of operations used to properties for add/mult; b) multiplicative commutative and associative properties for solve equations property of zero; c) inverse preperty for mult add/mult; b) distributive property; c) additive/ multiplicative identity properties; d) additive/ multiplicative inverse properties; e) multiplicative property of zero 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) Virginia Department of Education -­‐ Fall 2010 DRAFT -­‐Vertical Articulation of the 2009 Mathematics Standards of Learning Page 12 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
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
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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
a
where a and b are integers and b ≠ 0.
b
1
25, , − 2.3, 75%, 4.59, − 45 are rational numbers.
4
o
Example:
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.
Page 13 of 55
ACPS Curriculum Framework – Math 8
Key Vocabulary
absolute value
base
exponent
grouping symbols
identity elements
inverses
numerical
expression
order of operations
perfect square
radical
rational number
simplify
square root
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.
2014-15
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.
Return to Course Outline
¨ The additive inverse property states that the sum of a number and its additive inverse always equals zero.
o Example: 5 + −5 = 0
Page 14 of 55
ACPS Curriculum Framework – Math 8
2014-15
¨ 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)
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 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).
Page 15 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
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
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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
a
expressed as fractions in the form
where a and b are integers and b ≠ 0.
b
1
§ Examples of rational numbers include: 25 , , −2.3 , 75%, 4.59
4
1
§ Fractions such as , can be represented as terminating decimals
8
1
(e.g.,
= 0.125, which has a finite number of decimal places) and
8
Page 16 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
(e.g.,
2
, can be represented as repeating decimals
9
2
= 0.222... , whose decimal representation does not end but
9
continues to repeat). The repeating decimal can be written with
ellipses (three dots) as in 0.222… or denoted with a bar above the
o
digit(s) that repeat as in 0.2 .
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)
Return to Course Outline
Page 17 of 55
ACPS Curriculum Framework – Math 8
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.
2014-15
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.
Return to Course Outline
Page 18 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
Curriculum Information
SOL 8.5
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.
Essential Knowledge and Skills
Key Vocabulary
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
57 lies between 7 and 8
2
7
since
= 49 and 8 = 64).
Teacher Notes and Elaborations
¨ Define a perfect square.
Estimate the square root of a nonperfect 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
¨ 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).
lies (e.g.,
2
•
•
•
Essential Questions and Understandings
Teacher Notes and Elaborations
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).
¨ 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.
to ask for the positive root and
–
Return to Course Outline
when asking for the negative root.)
Key Vocabulary
consecutive
irrational number
negative root
perfect square
¨ 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
Page 19 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
¨ 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).
Focus
Practical Applications of Operations
with Real Numbers
¨ Find the positive or positive and negative square roots of a given whole number from 0 to 400. (Use the symbol
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.
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
o
(e.g., 2 is an irrational number).
Estimation can be used to express a non-perfect square root to the nearest whole number
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.
(e.g.,
Return to Course Outline
Page 20 of 55
ACPS Curriculum Framework – Math 8
2014-15
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.
Return to Course Outline
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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 x + 4 = x − 17
¨ In an equation, the equal sign indicates that the value on the left is the same as the
value on the right.
Page 21 of 55
ACPS Curriculum Framework – Math 8
− the addition, subtraction,
multiplication and division
properties of inequality
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.
Return to Course Outline
2014-15
¨ 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)
Page 22 of 55
ACPS Curriculum Framework – Math 8
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.
2014-15
¨ 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.
1
o Example: 4 ⋅ = 1
4
o Zero has no multiplicative inverse.
¨ The zero property of multiplication states that the product of any real number and zero is zero.
Return to Course Outline
Page 23 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
Curriculum Information
SOL 8.4
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
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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.
Page 24 of 55
ACPS Curriculum Framework – Math 8
perfect square
radical
simplify
square root
substitution
variable
2014-15
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.
Return to Course Outline
Page 25 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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.
Page 26 of 55
ACPS Curriculum Framework – Math 8
2014-15
like terms
linear equation
linear inequality
multiplicative identity property
multiplicative inverse property
multiplication property of inequality
subtraction property of inequality
zero property of multiplication
Return to Course Outline
Page 27 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
Curriculum Information
SOL 8.16
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.
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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.
Return to Course Outline
Page 28 of 55
ACPS Curriculum Framework – Math 8
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.
2014-15
(continued)
¨ 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.
Return to Course Outline
Page 29 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
Curriculum Information
SOL 8.3
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.
Return to Course Outline
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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.
Page 30 of 55
ACPS Curriculum Framework – Math 8
•
2014-15
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
rational numbers, percents, ratios,
and proportions; and
b. determine the percent increase or
decrease for a given situation.
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
o
Percent of change (rate of change) is the percent that a quantity increases or
decreases.
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.
Return to Course Outline
¨ Percent decrease determines the rate of decline and may be calculated using the same
ratio as percent increase.
Page 31 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
o
a c
= .
b d
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
¨ 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 ).
Focus
Practical Applications of Operations
with Real Numbers
¨ 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.
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.
Return to Course Outline
Page 32 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
Curriculum Information
SOL 8.14
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.
Essential Knowledge and Skills
Key Vocabulary
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
Return to Course Outline
Essential Questions and Understandings
Teacher Notes and Elaborations
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
Page 33 of 55
ACPS Curriculum Framework – Math 8
2014-15
o
SOL Reporting Category
Probability, Statistics, Patterns,
Functions, and Algebra
interpreted.
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.
¨ The following function is represented below as a table, graph, rule, and in words.
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
Page 34 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
Curriculum Information
SOL 8.17
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.
Essential Knowledge and Skills
Key Vocabulary
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.
Essential Questions and Understandings
Teacher Notes and Elaborations
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.
Key Vocabulary
dependent variable
domain
independent variable
range
output
input
¨ Below is a table of values for finding circumference of circles, C = πd, where the value
of π is approximated as 3.14.
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.
Return to Course Outline
Page 35 of 55
ACPS Curriculum Framework – Math 8
2014-15
o
o
o
o
The independent variable, or input, is the diameter of the circle (d).
The set of values for the diameter make up the domain.
The dependent variable, or output, is the circumference (C) of the circle.
The set of values for the circumference makes up the range.
Return to Course Outline
Page 36 of 55
ACPS Curriculum Framework – Math 8
Interdisciplinary Concept: Systems; Communication
2014-15
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
Curriculum Information
SOL 8.6
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
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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.
Page 37 of 55
ACPS Curriculum Framework – Math 8
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
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
2014-15
¨ 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.
Page 38 of 55
ACPS Curriculum Framework – Math 8
SOL Reporting Category
Measurement and Geometry
Focus
Problem Solving
2014-15
¨ 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.
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°.
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 = x + 30 , what is the
value of x? What is the measure of ∠3 ?)
Return to Course Outline
Page 39 of 55
ACPS Curriculum Framework – Math 8
Interdisciplinary Concept: Systems
2014-15
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
Curriculum Information
SOL 8.10
SOL Reporting Category
Measurement and Geometry
Focus
Problem Solving with 2- and 3Dimensional Figures
Virginia SOL 8.10
The student will
a. verify the Pythagorean Theorem;
and
b. apply the Pythagorean Theorem.
Return to Course Outline
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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.
Page 40 of 55
ACPS Curriculum Framework – Math 8
SOL Reporting Category
Measurement and Geometry
Focus
Problem Solving with 2- and 3Dimensional Figures
2014-15
¨ 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.
Virginia SOL 8.10
The student will
a. verify the Pythagorean Theorem;
and
b. apply the Pythagorean Theorem.
¨ 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.
Return to Course Outline
Page 41 of 55
ACPS Curriculum Framework – Math 8
Curriculum Information
SOL 8.11
SOL Reporting Category
Probability, Statistics, Patterns,
Functions, and Algebra
Focus
Problem Solving with 2- and 3Dimensional Figures
Virginia SOL 8.11
The student will solve practical area
and perimeter problems involving
composite plane figures.
Essential Knowledge and Skills
Key Vocabulary
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
2014-15
Essential Questions and Understandings
Teacher Notes and Elaborations
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.
1
( A = bh )
2
Return to Course Outline
Page 42 of 55
ACPS Curriculum Framework – Math 8
2014-15
SOL Reporting Category
Probability, Statistics, Patterns,
Functions, and Algebra
¨ The area of a parallelogram is computed by multiplying the measure of its base by the measure of its height.
( A = bh )
Focus
Problem Solving with 2- and 3Dimensional Figures
¨ 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
[ A = h (b1 + b2 ) ]
2
Virginia SOL 8.11
The student will solve practical area
and perimeter problems involving
composite plane figures.
¨ The area of a circle is computed by multiplying Pi times the radius squared.
( A = π r2 )
¨ The circumference of a circle is found by multiplying Pi by the diameter or multiplying Pi by 2 times the radius,
( C = π d or C = 2π r )
¨ 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.
Return to Course Outline
Page 43 of 55
ACPS Curriculum Framework – Math 8
Interdisciplinary Concept: Systems
2014-15
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
Curriculum Information
SOL 8.7
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.
Return to Course Outline
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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?
1
The volume of a cone is
the volume of a cylinder with the same size base and
3
height.
1
The volume of a pyramid is
the volume of a prism with the same size base and
3
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.
Page 44 of 55
ACPS Curriculum Framework – Math 8
•
•
2014-15
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
¨ 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.
by a factor of 1 , 2, 3, 5, or 10.
2
Return to Course Outline
(continued)
Page 45 of 55
ACPS Curriculum Framework – Math 8
2014-15
(continued from previous page)
(continued from previous page)
(continued from previous page)
SOL Reporting Category
Measurement and Geometry
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to:
• Describe the two-dimensional figures
that result from slicing threedimensional figures parallel to the base.
(e.g., as in plane sections of right
rectangular prisms and right rectangular
pyramids).
Teacher Notes and Elaborations
¨ Investigate and compute the surface area of a right cylinder using concrete objects,
nets, diagrams and formulas.
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.
Key Vocabulary
base
cone
cylinder
face
height
net
polyhedron
prism
pyramid
rectangular prism
scale factor
slant height
surface area
volume
width
length
¨ 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
1
Bh, where B is the area of the base and h is the height.
3
¨ The area of the base of a circular cone is πr2.
Return to Course Outline
(continued)
Page 46 of 55
ACPS Curriculum Framework – Math 8
2014-15
Curriculum Information
SOL 8.7
SOL Reporting Category
Measurement and Geometry
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.
Focus
Problem Solving
¨ The volume of a cone is
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.
1 2
πr h, where h is the height and πr2 is the area of the base.
3
¨ The surface area of a right circular cylinder is 2π r 2 + 2π rh .
¨ 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.
1
¨ How are the formulas for the volume of cones and pyramids similar? For cones you are finding 3 of the volume of the cylinder
1
with the same size base and height. For pyramids you are finding 3 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.
Return to Course Outline
(continued)
Page 47 of 55
ACPS Curriculum Framework – Math 8
Curriculum Information
SOL 8.7
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.
2014-15
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.
Return to Course Outline
Page 48 of 55
ACPS Curriculum Framework – Math 8
Interdisciplinary Concept: Communication
2014-15
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
Curriculum Information
SOL 8.9
SOL Reporting Category
Measurement and Geometry
Focus
Problem Solving with 2- and 3Dimensional Figures
Virginia SOL 8.9
The student will construct a threedimensional model given the top or
bottom, side and front views.
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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 twodimensional 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.
Return to Course Outline
¨ Use snap cubes or pop cubes to construct figures given the top or bottom, side and front
view of figures and vice versa.
Page 49 of 55
ACPS Curriculum Framework – Math 8
2014-15
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
Curriculum Information
SOL 8.8
SOL Reporting Category
Measurement and Geometry
Focus
Problem Solving with 2- and 3Dimensional Figures
Virginia SOL 8.8
The student will
a. apply transformations to plane
figures; and
b. identify applications of
transformations.
Return to Course Outline
Essential Knowledge and Skills
Key Vocabulary
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.
Essential Questions and Understandings
Teacher Notes and Elaborations
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.
Key Vocabulary
angle of rotation
center of rotation
clockwise
counterclockwise
¨ Demonstrate the translation of a polygon on a coordinate grid.
preimage
reflection
rotation
scale factor
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.
¨ A translation of a geometric figure moves all the points on the figure the same
distance in the same direction.
Page 50 of 55
ACPS Curriculum Framework – Math 8
dilation
image
line of reflection
orientation
origin
original figure
2014-15
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 3Dimensional Figures
Virginia SOL 8.8
The student will
a. apply transformations to plane
figures; and
b. identify applications of
transformations.
¨ 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”).
Return to Course Outline
Page 51 of 55
ACPS Curriculum Framework – Math 8
Interdisciplinary Concept: Properties and Models; Communication
2014-15
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
Curriculum Information
SOL 8.12
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
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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
Page 52 of 55
ACPS Curriculum Framework – Math 8
2014-15
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 B and C) = 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
P(3 and 4 and 5) = P(3) ⋅ P(4) ⋅ P(5) = ⋅ ⋅ =
6 6 6 216
(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.
¨ 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( B after 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
P(blue and red) = P(blue) ⋅ P(red after blue) = ⋅ =
3 2 6
¨ Compare the outcomes of events with and without replacement.
¨ Two events are either dependent or independent.
Return to Course Outline
Page 53 of 55
ACPS Curriculum Framework – Math 8
Interdisciplinary Concept: Properties and Models
2014-15
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
Curriculum Information
SOL 8.13
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
Essential Knowledge and Skills
Key Vocabulary
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
Teacher Notes and Elaborations
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.
Page 54 of 55
ACPS Curriculum Framework – Math 8
prediction
scatterplot
correlation
trend
horizontal axis
vertical axis
2014-15
¨ 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.
¨ 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.
Return to Course Outline
Page 55 of 55