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ACPS Curriculum Framework – Math 7 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) 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.

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ACPS Curriculum Framework – Math 7 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)

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.

ACPS Curriculum Framework – Math 7 2014-15

• 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 Overview of Each Strand: Strand: 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.

Strand 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 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 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. Strand: 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.

Strand: 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.

• 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.)

Strand: 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.

Adapted from the VDOE Standards of Learning 2009 Thanks to Henrico County Public Schools for allowing information from their curriculum documents to be included in this document.

ACPS Curriculum Framework Math 7

Page 7 of 51

Math 7 Sample Pacing Guide

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

7.1e - Absolute Value

7.3* - Integer Operations

7.1* - Perfect Squares

7.16 - Properties

7.14 - Equations

7.4 - Proportional Reasoning

7.6 - Similar Figures

7.7 - Quadrilaterals

7.9 - Probability

7.10 - Probability of Compound Events

SOL Review

* SOL test items measuring Objectives 7.1b-d and 7.3b

will be completed without the use of a calculator.

7.1a - Negative Exponents

7.1bc - Scientific Notation, Fractions, Decimals, Percents

7.13 - Expressions

7.15 - Inequalities

7.2 - Sequences

7.12 - Representations

7.8 - Transformations

7.5 - Volume and Surface Area

7.11 - Statistics

ACPS Curriculum Framework Math 7

Page 8 of 51

Vertical Articulation

    Grade  5   Grade  6   Grade  7   Grade  8   Algebra  1  

Expo

nents/  

Squa

res/  

Squa

re  Roo

ts  

   

6.5  investigate/describe  positive  exponents,  perfect  squares  

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

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

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

Mod

eling/  Com

parin

g/Ordering   5.2  a)  recognize/name  fractions  in  

their  equivalent  decimal  form  and  vice  versa;  b)  compare/order  fracts  and  decimals  

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

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

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

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

5.18  c)  model  one-­‐step  linear  equations  using  add/sub  

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

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

       

Alg  Patt/  

Seq  

5.17  describe/express  the  relationship  in  a  number  pattern  

6.17  ID/extend  geometric/arithmetic  sequences  

7.2  describe/represent  arithmetic/geometric  sequences  using  variable  expressions  

       

Ope

ratio

ns/

Recall  

5.5  a)  find  sum/diff/product/quotient  of  two  decimals  through  thousandths  

6.6  a)  mult/div  fractions   7.3  b)  add/sub/mult/div  integers  

   

   

Ratio

s/  Propo

rtions  

   

6.1  describe/compare  data  using  ratios  

7.4  single  and  multistep  practical  problems  with  proportional  reasoning  

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

   

   

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

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

       

ACPS Curriculum Framework Math 7

Page 9 of 51

    Grade  5   Grade  6   Grade  7   Grade  8   Algebra  1  Measuremen

t  App

s  -­‐  Geo

m  

Figures  

5.8  a)  find  perimeter/area/volume;  b)  differentiate  among  perimeter/area/  volume,  ID  which  measure  is  appropriate;  c)  ID  equiv  measurements  within  metric  system;  d)  estimate/measure  U.S.  Cust/metric;  e)  choose  appropriate  unit  of  measure  w/  U.S.  Cust/  metric  

6.9  make  ballpark  comparisons  between  U.S.  Cust/metric  system  

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

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

   

Plan

e  an

d  Solid  Figures  

5.13  a)  using  plane  figures  will  develop  definitions  of  plane  figures;  b)  investigate/describe  results  of  combining/subdividing  plane  figures  

6.12  determine  congruence  of  segments/angles/polygons  

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

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

   

5.12  a)  classify  angles  as  right/  acute/  obtuse/straight;  b)  triangles  as  right/  acute/obtuse/equilateral/scalene/isosceles.  

6.13  ID/describe  properties  of  quadrilaterals  

7.7  compare/contrast  quadrilaterals  based  on  properties  

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

   

5.11  measure  right/acute/obtuse/straight  angles  

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

7.8  represent  transformations  of  polygons  in  the  coordinate  plane  by  graphing  

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

   

Prob

ability  

5.14  make  predictions/determine  probability  by  constructing  a  sample  space  

6.16  a)  compare/contrast  dep/indep  events;  b)  determine  probabilities  for  dep/indep  events  

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

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

   

        7.10  determine  the  probability  of  compound  events,  Basic  Counting  Principle  

       

Collect/Rep

resent  

Data  

5.15  collect/organize/interpret  data,  using  stem-­‐and-­‐leaf  plots/line  graphs  

6.14  a)  construct  circle  graphs;  b)  draw  conclusions/make  predictions,  using  circle  graphs;  c)  compare/contrast  graphs    

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

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

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

ACPS Curriculum Framework Math 7

Page 10 of 51

Equa

tions  and

 Ineq

ualities  C

ollect/Rep

resent  Data  

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

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

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

5.18  a)  investigate/describe  concept  of  variable;  b)  write  open  sentence  using  variable;  c)  model  one-­‐step  linear  equations  using  add/sub;  d)  create  problems  based  on  open  sentence  

6.18  solve  one-­‐step  linear  equations  in  one  variable  

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

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

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

    6.20  graph  inequalities  on  number  line  

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

8.16  graph  linear  equation  in  two  variables  

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

    Grade  5   Grade  6   Grade  7   Grade  8   Algebra  1  

ACPS Curriculum Framework Math 7

Page 11 of 51

Equa

tions  and

 Ineq

ualities  

Collect/Rep

resent  Data                   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  

Expressio

ns/  

Ope

ratio

ns   5.7  evaluate  whole  number  

numerical  expressions  using  order  of  operations  

6.8  evaluate  whole  number  expressions  using  order  of  operations  

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

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

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

Prop

ertie

s  

5.19  distributive  property  of  mult  over  addition  

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

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

8.15  c)  ID  properties  of  operations  used  to  solve  equations  

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

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

    Grade  5   Grade  6   Grade  7   Grade  8   Algebra  1  

ACPS Curriculum Framework Math 7

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

Curriculum Information SOL 7.1

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.1 The student will a. investigate and describe the concept

of negative exponents for powers of ten;

b. determine scientific notation for numbers greater than zero;*

c. compare and order fractions, decimals, percents and numbers written in scientific notation;*

d. determine square roots;* and e. identify and describe absolute value

for rational numbers. *SOL test items measuring Objective 7.1b-d will be completed without the use of a calculator. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Recognize powers of 10 with negative

exponents by examining patterns. • Write a power of 10 with a negative

exponent in fraction and decimal form. • Recognize a number greater than zero

in scientific notation. • Write a number greater than zero in

scientific notation. • Compare and determine equivalent

relationships between numbers larger than zero, written in scientific notation.

• Order no more than three numbers greater than zero written in scientific notation.

• Represent a number in fraction, decimal, and percent forms.

• Compare, order, and determine equivalent relationships among fractions, decimals, and percents. Decimals are limited to the thousandths place, and percents are limited to the tenths place. Ordering is limited to no more than four numbers.

Essential Questions and Understandings • When should scientific notation be used?

Scientific notation should be used whenever the situation calls for use of very large or very small numbers.

• How are fractions, decimals and percents related? Any rational number can be represented in fraction, decimal and percent form.

• What does a negative exponent mean when the base is 10? A base of 10 raised to a negative exponent represents a number between 0 and 1.

• How is taking a square root different from squaring a number? Squaring a number and taking a square root are inverse operations.

• Why is the absolute value of a number positive? The absolute value of a number represents distance from zero on a number line regardless of direction. Distance is positive.

Teacher Notes and Elaborations ¨ Recognize powers of 10 with negative exponents by examining patterns.

o An exponent tells how many times the base is used as a factor. In the expression 32, 3 is the base and 2 is the exponent.

o Negative exponents for powers of 10 can be investigated through patterns such as:

2

1

0

10 10010 1010 1

=

=

=

11

1 110 0.11010

− = = =

ACPS Curriculum Framework Math 7

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SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.1 The student will a. investigate and describe the concept

of negative exponents for powers of ten;

b. determine scientific notation for numbers greater than zero;*

c. compare and order fractions, decimals, percents and numbers written in scientific notation;*

d. determine square roots;* and e. identify and describe absolute value

for rational numbers. *SOL test items measuring Objective 7.1b-d will be completed without the use of a calculator. Return to Course Outline

• Compare and order fractions, decimals, percents, and numbers written in scientific notation.

• Determine the square root of a perfect square less than or equal to 400 without the use of a calculator.

• Demonstrate absolute value using a number line.

• Determine the absolute value of a rational number

22

1 110 0.0110010

− = = =

¨ Write a power of 10 with a negative exponent in fraction and decimal form. o Negative exponents for powers of 10 are used to represent numbers between 0

and 1 (e.g., 33

11010

− = and 3

1 0.00110

= ).

Essential Questions and Understandings ¨ Scientific notation is used to represent very large and very small numbers.

o A number is in scientific notation when it is written in the form: a · 10n where 1 ≤ a < 10 and n is an integer.

o A number written in scientific notation is the product of two factors, a decimal greater than or equal to one but less than 10, and a power of 10 (e.g., 3.1 · 10 5 = 310,000 and 2.85 · 10-4 = 0.000285).

¨ When should scientific notation be used? Scientific notation should be used whenever the situation calls for use of very large or very small numbers.

¨ Recognize a number greater than zero in scientific notation. ¨ Write a number greater than zero in scientific notation. ¨ Compare and determine equivalent relationships between numbers larger than zero,

written in scientific notation. ¨ Order no more than three numbers greater than zero written in scientific notation. ¨ Compare and order fractions, decimals, percents, and numbers written in scientific

notation. ¨ How are fractions, decimals and percents related? Any rational number can be

represented in fraction, decimal and percent form.

¨ A rational number is any number that can be expressed in the form ab

, where a and b

are integers and b ≠ 0. ¨ Percent means “per hundred”. A number followed by a percent symbol (%) is

ACPS Curriculum Framework Math 7

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The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Show that the distance between two

rational numbers on the number line is the absolute value of their difference, and apply this principle to solve practical problems.

Key Vocabulary absolute value exponent percent perfect square rational number scientific notation square root

equivalent to that number with a denominator of 100 (e.g., 3 605 100= , 60 0.60

100= , 0.60

= 60%).

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.1 The student will a. investigate and describe the concept

of negative exponents for powers of ten;

b. determine scientific notation for numbers greater than zero;*

Return to Course Outline

¨ Equivalent relationships among fractions, decimals, and percents can be determined by using manipulatives (e.g., fraction bars, Base-10 blocks, fraction circles, graph paper, number lines and calculators).

¨ Represent a number in fraction, decimal, and percent forms.

¨ Compare, order, and determine equivalent relationships among fractions, decimals, and percents. Decimals are limited to the thousandths place, and percents are limited to the tenths place. Ordering is limited to no more than four numbers.

¨ How is taking a square root different from squaring a number? Squaring a number and taking a square root are inverse operations.

¨ A square root of a number is a number which, when multiplied by itself, produces the given number (e.g., 121 is 11 since 11 · 11 = 121). A whole number that can be named as a product of a number with itself is a perfect square (e.g., 81 = 9 · 9, where 81 is a perfect square).

¨ The square root of a number can be represented geometrically as the length of a side of the square.

¨ Determine the square root of a perfect square less than or equal to 400 without the use of a calculator.

¨ Why is the absolute value of a number positive? The absolute value of a number represents distance from zero on a number line regardless of direction. Distance is always positive.

¨ The absolute value of a number is the distance from 0 on the number line regardless of direction (e.g., 1 12 2

− = , 1 12 2−

= , 1 12 2=

−,

and 1 12 2= ).

¨ The distance between two rational numbers on the number line is the absolute value of their difference.

ACPS Curriculum Framework Math 7

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c. compare and order fractions, decimals, percents and numbers written in scientific notation;*

d. determine square roots;* and e. identify and describe absolute value

for rational numbers. *SOL test items measuring Objective 7.1b-d will be completed without the use of a calculator. Return to Course Outline

o Example 1: The distance between 5 and 2 is 5 2 3− = or 2 5 3− = .

o Example 2: The distance between 3.5 and ( 7.4− ) is ( )3.5 7.4 10.9− − = or ( )7.4 3.5 10.9− − = .

o Example 3: The distance between ( 4− ) and ( 1− ) is ( ) ( )4 1 3− − − = or ( ) ( )1 4 3− − − = .

o Example 4: The distance between 213

and 145

is 2 1 81 4 23 5 15− = or 1 2 84 1 2

5 3 15− = .

¨ Demonstrate absolute value using a number line.

¨ Determine the absolute value of a rational number.

¨ Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle to solve practical problems.

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Interdisciplinary Concept: Change and Ineractions 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 7.3

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Integer Operations and Proportional Reasoning Virginia SOL 7.3 The student will a. model addition, subtraction,

multiplication and division of integers; and

b. add, subtract, multiply, and divide integers.*

*SOL test items measuring Objective 7.3b will be completed without the use of a calculator. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Model addition, subtraction,

multiplication and division of integers using pictorial representations of concrete manipulatives.

• Formulate rules for addition, subtraction, multiplication, and division of integers.

• Add, subtract, multiply and divide integers.

• Simplify numerical expressions involving addition, subtraction, multiplication and division of integers using order of operations.

• Solve practical problems involving addition, subtraction, multiplication, and division with integers.

Key Vocabulary absolute value integers opposites

Essential Questions and Understandings • The sums, differences, products and quotients of integers are either positive, zero, or

negative. How can this be demonstrated? This can be demonstrated through the use of patterns and models.

Teacher Notes and Elaborations ¨ The set of integers is the set of whole numbers and their opposites

o (…-3, -2, -1, 0, 1, 2, 3…) o Integers are used in practical situations such as temperature changes

(above/below zero), balance a checking account (deposits/withdrawals), and changes in altitude (above /below sea level).

¨ The absolute value of an integer is the distance on a number line that a number is from zero.

o It is always written as a positive number. o Students should recognize and be able to read the symbol for absolute value

(e.g., 7 7− = is read as “The absolute value of negative seven equals seven.”).

¨ The order of operations is a convention that defines the computation order to follow in simplifying an expression. In grades 5 and 6, students simplify expressions by using the order of operations in a demonstrated step-by-step approach.

¨ Modeling using concrete materials should be explored BEFORE any other strategies are used.

¨ Concrete experiences must be used to formulate rules for adding, subtracting,

ACPS Curriculum Framework Math 7

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Return to Course Outline

multiplying, and dividing integers o Use number lines to model addition, subtraction, multiplication and division of

integers o Use manipulatives, such as two-color counters or algebra tiles to model

addition, subtraction, multiplication and division of integers o Use pictorial representations to model addition, subtraction, multiplication and

division of integers o Examine patterns using calculators

¨ Use real-life examples such as weather maps to demonstrate positive and negative

temperatures, the stock market to illustrate gains and losses, banking examples involving credits and debits, and problems involving sea level to understand ways in which positives and negatives are used.

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

Curriculum Information

SOL 7.16 Essential Knowledge and Skills

Key Vocabulary Essential Questions and Understandings

Teacher Notes and Elaborations SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.16 The student will apply the following properties of operations with real numbers: a. the commutative and associative

properties for addition and multiplication;

b. the distributive property; c. the additive and multiplicative

identity properties; d. the additive and multiplicative

inverse properties; and e. the multiplicative property of zero. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Identify properties of operations used in

simplifying expressions. • Apply the properties of operations to

simplify expressions. Key Vocabulary additive identity property additive inverse property associative property of addition associative property of multiplication commutative property of addition commutative property of multiplication distributive property identity elements inverses multiplicative identity property multiplicative inverse property multiplicative property of zero reciprocal

Essential Questions and Understandings • Why is it important to apply properties of operations when simplifying expressions?

Using the properties of operations with real numbers helps with understanding mathematical relationships.

Teacher Notes and Elaborations € The commutative property for addition states that changing the order of the addends

does not change the sum (e.g., 5 + 4 = 4 + 5).

€ The commutative property for multiplication states that changing the order of the factors does not change the product (e.g., 5 · 4 = 4 · 5).

€ The associative property of addition states that regrouping the addends does not change the sum [e.g., 5 + (4 + 3) = (5 + 4) + 3].

€ The associative property of multiplication states that regrouping the factors does not change the product [e.g., 5 · (4 · 3) = (5 · 4) · 3].

€ Subtraction and division are neither commutative nor associative.

€ 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 [e.g., 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. The additive identity is zero (0). The multiplicative identity is one (1).

ACPS Curriculum Framework Math 7

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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 (e.g., 5 + 0 = 5).

€ The multiplicative identity property states that the product of any real number and one is equal to the given real number (e.g., 8 · 1 = 8).

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

[e.g., 5 + (–5) = 0; 15 · 5 = 1].

(continued)

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.16 The student will apply the following properties of operations with real numbers: a. the commutative and associative

properties for addition and multiplication;

b. the distributive property; c. the additive and multiplicative

identity properties; d. the additive and multiplicative

inverse properties; and e. the multiplicative property of zero. Return to Course Outline

¨ The additive inverse property states that the sum of a number and its additive inverse always equals zero [e.g., 5 + (–5) = 0].

¨ The multiplicative inverse property states that the product of a number and its multiplicative inverse (or reciprocal) always equals one

(e.g., 4 · 14 = 1).

¨ Zero has no multiplicative inverse.

¨ The multiplicative property of zero states that the product of any real number and zero is zero.

¨ Division by zero is not a possible arithmetic operation. Division by zero is undefined.

¨ Identify properties of operations used in simplifying expressions. o -25(7)( -4) o 7( -25)( -4) Commutative property of multiplication o 7[( -25)( -4)] Associative property of multiplication o 7(100) o 700

¨ Apply the properties of operations to simplify expressions.

ACPS Curriculum Framework Math 7

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

Curriculum Information SOL 7.13

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.13 The student will a. write verbal expressions as

algebraic expressions and sentences as equations and vice versa; and

b. evaluate algebraic expressions for given replacement values of the variables.

Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Write verbal expressions as algebraic

expressions. Expressions will be limited to no more than two operations.

• Write verbal sentences as algebraic equations. Equations will contain no more than one variable term.

• Translate algebraic expressions and equations to verbal expressions and sentences. Expressions will be limited to no more than two operations.

• Identify examples of expressions and equations.

• Apply the order of operations to evaluate expressions for given replacement values (integers, fractions, and decimals) of the variables. Limit the number of replacements to no more than three per expression.

Key Vocabulary algebraic equation algebraic expression coefficient

Essential Questions and Understandings • How can algebraic expressions and equations be written? Word phrases and sentences can be used to represent algebraic expressions and equations.

Teacher Notes and Elaborations ¨ An expression is a name for a number.

¨ An expression that contains a variable is a variable expression.

¨ An expression that contains only numbers is a numerical expression.

¨ A verbal expression is a word phrase (e.g., “the sum of two consecutive integers”).

¨ A verbal sentence is a complete word statement (e.g., “The sum of two consecutive integers is five.”).

¨ An algebraic expression is a variable expression that contains at least one variable (e.g., 2x – 5).

¨ An algebraic equation is a mathematical statement that says that two expressions are equal (e.g., 2x + 1 = 5).

¨ A term is a number, variable, product, or quotient in an expression of sums and/or differences.

ACPS Curriculum Framework Math 7

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constant expression grouping symbols order of operations substitution term variable variable expression verbal expression verbal sentence

The expression 3x + 4y – 7 contains 3 terms (3x, 4y, 7− ).

¨ A coefficient is the numerical factor of a variable in a term. In the term 2x, 2 is the coefficient of x.

¨ A constant is a numerical expression that is part of an algebraic expression. In the expression 4x + 9, 9 is the constant.

¨ Write verbal expressions as algebraic expressions. Expressions will be limited to no more than 2 operations.

(continued)

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.13 The student will a. write verbal expressions as

algebraic expressions and sentences as equations and vice versa; and

b. evaluate algebraic expressions for given replacement values of the variables.

Return to Course Outline

¨ Write verbal sentences as algebraic equations. Equations will contain no more than 1 variable term.

¨ Translate algebraic expressions and equations to verbal expressions and sentences. Expressions will be limited to no more than 2 operations.

¨ Identify examples of expressions and equations.

¨ Apply the order of operations to evaluate expressions for given replacement values of the variables. Limit the number of replacements to no more than 3 per expression.

o To evaluate an algebraic expression, substitute a given replacement value for a variable and apply the order of operations. For example, if a = 3 and b = -2 then 5a + b can be evaluated as: 5(3) + (-2) 15 + (-2) 13

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Interdisciplinary Concept: Properties and Models; Change and Interactions Math Concept: Models; Cause and Effect ACPS Mathematics Enduring Understandings: 4 - Situations and structures can be represented, modeled and analyzed using algebraic symbols. 10 - Change, in various contexts, both quantitative and qualitative, can be identified and analyzed. 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. Use graphs to analyze the nature of changes in quantities in linear relationships. Life Long Learner Standards

Curriculum Information SOL 7.14

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.14 The student will a. solve one- and two-step linear

equations in one variable; and b. solve practical problems requiring

the solution of one- and two-step linear equations.

Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Represent and demonstrate steps for

solving one- and two-step equations in one variable using concrete materials, pictorial representations, and algebraic sentences.

• Translate word problems/practical problems into algebraic equations and solve them.

• Solve one- and two-step linear equations in one variable.

• Solve practical problems that require the solution of a one- or two-step linear equation.

Key Vocabulary inverse operations coefficient

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 ¨ An equation is a mathematical sentence that states that two expressions are equal.

¨ A one-step equation is defined as an equation that requires the use of one operation to solve (e.g., x + 3 = –4).

¨ Inverse operations undo each other. The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.

¨ A two-step equation is defined as an equation that requires the use of two operations to

solve (e.g., 2x + 1 = -5; -5 = 2x + 1; 7 43x −

= ).

¨ Represent and demonstrate steps for solving one- and two-step equations in one variable. It is important to use the concrete and pictorial representations to create the rules for solving the equations. The concrete and pictorial representations should be explored before moving to the abstract concepts.

o Use concrete materials such as Algeblocks or Algebra tiles o Use pictorial representations such as drawing out the concrete representations

ACPS Curriculum Framework Math 7

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Return to Course Outline

o Use algebraic sentences

¨ Solve one- and two-step linear equations in one variable. o The following demonstrates steps for solving a two-step equation

algebraically. 2( 2) 14x+ =

2( 2) 142 2x +

= divide or multiply by the reciprocal (multiplicative inverse)

2 7x + = -2 -2 subtract or add the opposite (additive inverse)

x + 0 = 5 x = 5

¨ Solve practical problems that require the solution of a one- or two-step linear equation. Translate word problems/practical problems into algebraic equations then solve the problem

ACPS Curriculum Framework Math 7

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Interdisciplinary Concept: Properties and Models; Change and Interactions Math Concept: Models; Cause and Effect ACPS Mathematics Enduring Understandings: 4 - Situations and structures can be represented, modeled and analyzed using algebraic symbols. 10 - Change, in various contexts, both quantitative and qualitative, can be identified and analyzed. 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. Use graphs to analyze the nature of changes in quantities in linear relationships. Life Long Learner Standards

Curriculum Information SOL 7.15

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.15 The student will a. solve one-step inequalities in one

variable and b. graph solutions to inequalities on

the number line. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Represent and demonstrate steps in

solving inequalities in one variable, using concrete materials, pictorial representations, and algebraic sentences.

• Graph solutions to inequalities on the number line.

• Identify a numerical value that satisfies the inequality.

Key Vocabulary inequality inverse operations coefficient

Essential Questions and Understandings • How are the procedures for solving equations and inequalities the same?

The procedures are the same except for the case when an inequality is multiplied or divided on both sides by a negative number. Then the inequality sign is changed from less than to greater than, or greater than to less than.

• How is the solution to an inequality different from that of a linear equation? In an inequality, there can be more than one value for the variable that makes the inequality true.

Teacher Notes and Elaborations ¨ An inequality is a mathematical sentence that states that one quantity is less than (or

greater than) another quantity. An inequality is a mathematical sentence that compares two expressions using one of the symbols <, >, ≤, ≥, or ≠.

¨ A one-step inequality is defined as an inequality that requires the use of one operation to solve (e.g., x – 4 > 9).

¨ Inverse operations undo each other. The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.

¨ When both expressions of an inequality are multiplied or divided by a negative number, the inequality symbol reverses (e.g., –3x < 15 is equivalent to x > –5).

¨ Solutions to inequalities can be represented using a number line. o Inequalities using the < or > symbols are represented on a number line with an

ACPS Curriculum Framework Math 7

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Return to Course Outline

open circle on the number and a shaded line over the solution set. o Inequalities using the ≤ or ≥ symbols are represented on a number line with a

closed circle on the number and shaded line in the direction of the solution set.

¨ Represent and demonstrate steps for solving one- and two-step equations in one variable. It is important to use the concrete and pictorial representations to create the rules for solving the equations. The concrete and pictorial representations should be explored before moving to the abstract concepts.

o Use concrete materials such as Algeblocks or Algebra tiles o Use pictorial representations such as drawing out the concrete representations o Use algebraic sentences

¨ Identify a numerical value that satisfies the inequality.

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Interdisciplinary Concept: Systems Math Concept: Relationships ACPS Mathematics Enduring Understandings: 2 - Spatial relationships can be described using coordinate geometry and other representational systems. ACPS Essential Standard in grade band 6-8: Explore and represent the properties of shapes Life Long Learner Standards

Curriculum Information

SOL 7.12 Essential Knowledge and Skills

Key Vocabulary Essential Questions and Understandings

Teacher Notes and Elaborations SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.12 The student will represent relationships with tables, graphs, rules, and words. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Describe and represent relations and

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

Key Vocabulary function relation table of values vertical horizontal input output dependent variable independent variable

Essential Questions and Understandings • What are the different ways to represent the relationship between two sets of numbers?

Rules that relate elements in two sets can be represented by word sentences, equations, tables of values, graphs or illustrated pictorially.

Teacher Notes and Elaborations ¨ Rules that relate elements in two sets can be represented by word sentences, equations,

tables of values, graphs, or illustrated pictorially.

¨ A relation is any set of ordered pairs. For each first member, there may be many second members.

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Return to Course Outline

¨ A function is a relation in which there is one and only one second member for each first

member. o The function that relates earnings to time worked is earnings = rate of pay ×

hours worked. o The function that relates distance traveled to the rate of travel and the time is

distance = rate × time; for example, a student traveling at 30 miles per hour on a motor bike, would produce the following table:

TIME (t) 1 hour 2 hours 3 hours 4 hours

DISTANCE (d) 30 miles 60 miles 90 miles 120 miles

The equation that represents this function is d = 30t.

o A person makes $30 an hour. A function representing this is e = 30h where e represents the earnings and h is the number of hours worked. The following represents a table of values for this function.

TIME (t) 1 hour 2 hours 3 hours 4 hours

EARNINGS (e) $30 $60 $90 $120

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

each value of the first variable.

(continued)

ACPS Curriculum Framework Math 7

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SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.12 The student will represent relationships with tables, graphs, rules, and words. Return to Course Outline

¨ A table of values is the data used to make a graph in the coordinate system. The values are used to graph points.

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

o Graphs may be constructed from ordered pairs represented in a table. The ordered pairs in the following table are ( 2,0), ( 1,1), (0,2), (1,3), (2,4)− − . The equation represented in this table and graph is 2y x= + .

¨ Some relations are functions; all functions are relations. ¨ Describe and represent relations and functions, using tables, graphs, rules, and words. Given one representation, students will be able

to represent the relation in another form.

x + 2

2− 0

1− 1

0 2

1 3

2 4

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-10-9-8

-7-6-5-4-3

-2-1

12

34567

8910

x

y

ACPS Curriculum Framework Math 7

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

Curriculum Information SOL 7.2

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.2 The student will describe and represent, arithmetic and geometric sequences using variable expressions.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Analyze arithmetic and geometric

sequences to discover a variety of patterns.

• Identify the common difference in an arithmetic sequence.

• Identify the common ratio in a geometric sequence.

• Given an arithmetic or geometric sequence, write a variable expression to describe the relationship between two consecutive terms in the sequence.

Key Vocabulary arithmetic sequence common difference common ratio consecutive terms geometric sequence variable expression

Essential Questions and Understandings • What are arithmetic sequences?

In an arithmetic sequence, the numbers are found by using a common difference. • What are geometric sequences?

In a geometric sequence, the numbers are found by using a common ratio. • When are variable expressions used? Variable expressions can express the relationship between two consecutive terms in

a sequence. Teacher Notes and Elaborations ¨ Analyze arithmetic and geometric sequences to discover a variety of patterns.

¨ In the numeric pattern of an arithmetic sequence, students must determine the difference, called the common difference, between each succeeding number in order to determine what is added to each previous number to obtain the next number.

o Sample arithmetic sequences include: 4, 7, 10, 13, … (The common difference is 3) 10, 3, 4− , 11− , … (The common difference is 7− ) 6− , 1− , 4, 9, … (The common difference is 5)

¨ Identify the common difference in an arithmetic sequence.

¨ What are geometric sequences? In a geometric sequence, the numbers are found by using a common ratio.

¨ In geometric sequences, students must determine what each number is multiplied by in order to obtain the next number in the geometric sequence. This multiplier is called the

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Return to Course Outline

common ratio. o Sample geometric sequences include:

2, 4, 8, 16, 32,… (The common ratio is 2) 1, 5, 25, 125, 625,… (The common ratio is 5)

80, 20, 5, 1.25,…. (The common ratio is 14

)

¨ Identify the common ratio in a geometric sequence.

¨ When are variable expressions used? Variable expressions can express the relationship between two consecutive terms in a sequence.

(continued) SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.2 The student will describe and represent, arithmetic and geometric sequences using variable expressions. Return to Course Outline

¨ A variable expression can be written to express the relationship between two consecutive terms of a sequence o If n represents a number in the sequence 3, 6, 9, 12…, the next term in the sequence can be determined using the variable

expression n + 3. o If n represents a number in the sequence 1, 5, 25, 125…, the next term in the sequence can be determined by using the

variable expression 5n.

¨ Given an arithmetic or geometric sequence, write a variable expression to describe the relationship between two consecutive terms in the sequence.

¨ Consecutive terms immediately follow each other in some order. For example 5 and 6 are consecutive whole numbers, 2 and 4 are consecutive even numbers.

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

Curriculum Information SOL 7.4

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Integer Operations and Proportional Reasoning Virginia SOL 7.4 The student will solve single-step and multi-step practical problems, using proportional reasoning. Pacing By 22.5 weeks Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Write proportions that represent

equivalent relationships between two sets.

• Solve a proportion to find a missing term.

• Apply proportions to convert units of measurement between the U.S. Customary System and the metric system. Calculators may be used.

• Apply proportions to solve problems that involve percents.

• Apply proportions to solve practical problems, including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths. Calculators may be used.

• Using 10% as a benchmark, mentally compute 5%, 10%, 15%, or 20% in a practical situation such as tips, tax and discounts.

• Solve problems involving tips, tax, and discounts. Limit problems to only one percent computation per problem.

Essential Questions and Understandings • What makes two quantities proportional?

Two quantities are proportional, when one quantity is a constant multiple of the other.

Teacher Notes and Elaborations ¨ A proportion is a statement of equality between two ratios.

¨ A ratio is a comparison of two numbers or measures using division. Both numbers in a ratio have the same unit of measure. A ratio may be written three ways: as a fraction ab

, using the notation a:b, or in words a to b.

¨ A proportion can be written as ab =

cd , a:b = c:d, or a is to b as c is to d.

¨ A proportion can be solved by finding the product of the means and the product of the extremes. o In the proportion a:b = c:d, a and d are the extremes and b and c are the means.

If values are substituted for a, b, c, and d such as 5:12 = 10:24, then the product of extremes (5 × 24) is equal to the product of the means (12 × 10).

¨ In a proportional situation, both quantities increase or decrease together.

¨ In a proportional situation, two quantities increase multiplicatively. Both are multiplied by the same factor.

¨ A proportion can be solved by finding equivalent fractions.

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Key Vocabulary discount (amount of discount) equivalent extremes means percent proportion rate (discount rate, tax rate, unit rate) ratio sale price (discount price) scale factor tax tip

¨ A rate is a ratio that compares two quantities measured in different units.

¨ A unit rate is a rate with a denominator of 1. Examples of rates include miles/hour and revolutions/minute.

¨ A percent is a special ratio in which the denominator is 100. o Proportions can be used to represent percent problems as follows:

100percent part

whole=

(continued)

SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Integer Operations and Proportional Reasoning Virginia SOL 7.4 The student will solve single-step and multi-step practical problems, using proportional reasoning. Return to Course Outline

¨ A discount rate is the percent off an item (e.g., 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 (e.g., Virginia has a 5% tax rate on most items purchased.) Sales tax is the amount added to the price of an item based on the tax rate.

¨ A tip is a small sum of money given as acknowledgment of services rendered, (a gratuity). It is often times computed as a percent of the bill or service.

¨ How are proportions used? o Proportions are used in everyday contexts, such as speed, recipe conversions, scale drawings, map reading, reducing and

enlarging, comparison shopping, and monetary conversions. o Apply proportions to solve practical problems, including scale drawings. Scale factors shall have denominators no greater than

12 and decimals no less than tenths. Calculators may be used. o Solve problems involving tips, tax, and discounts. Limit problems to only one percent computation per problem. o Using 10% as a benchmark, mentally compute 5%, 10%, 15%, or 20% in a practical situation such as tips, tax and discounts. o Apply proportions to convert units of measurement between the U.S. Customary System and the metric system. Calculators may

be used. o Proportions can be used to convert between measurement systems. For example: if 2 inches is about 5 cm, how many inches are

in 16 cm? 2 516

inches cmx cm

=

o Write proportions that represent equivalent relationships between two sets. o Solve a proportion to find a missing term.

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Interdisciplinary Concept: Systems Math Concept: Relationships; Quantifying Representation ACPS Mathematics Enduring Understandings: 2 - Spatial relationships can be described using coordinate geometry and other representational systems. 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: Explore and represent the properties of shapes 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 7.7 Essential Knowledge and Skills

Key Vocabulary Essential Questions and Understandings

Teacher Notes and Elaborations SOL Reporting Category Measurement and Geometry Focus Relationships between Figures Virginia SOL 7.7 The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Identify the classification(s) to which a

quadrilateral belongs, using deductive reasoning and inference.

• Compare and contrast attributes of the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid.

Key Vocabulary

congruent decagon diagonal hatch marks heptagon hexagon isosceles trapezoid kite nonagon octagon parallel

parallelogram pentagon plane figure polygon quadrilateral rectangle regular polygon rhombus square trapezoid vertex

Essential Questions and Understandings • Why can some quadrilaterals be classified in more than one category?

Every quadrilateral in a subset has all of the defining attributes of the subset. For example, if a quadrilateral is a rhombus, it has all the attributes of a rhombus. However, if that rhombus also has the additional property of 4 right angles, then that rhombus is also a square.

Teacher Notes and Elaborations ¨ A polygon is a simple closed plane figure whose sides are line segments that intersect

only at their endpoints.

¨ Two lines in the same plane are parallel if they do not intersect. They are always the same distance from each other.

¨ Two geometric figures that are the same shape and size are congruent. Two angles are congruent if they have the same measure. Two line segments are congruent if they are the same length.

¨ A quadrilateral is a closed plane figure (two-dimensional) with four sides that are line segments.

¨ A parallelogram is a quadrilateral whose opposite sides are parallel and congruent. Opposite angles are congruent.

o A diagonal divides the parallelogram into two congruent triangles. The

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diagonals of a parallelogram bisect each other. o Denote which angles are congruent with the

same number of curved lines. Congruent sides are denoted with the same number of hatch marks on each congruent side.

o Arrows are used in diagrams to indicate that lines are parallel.

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SOL Reporting Category Measurement and Geometry Focus Relationships between Figures Virginia SOL 7.7 The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid. Return to Course Outline

¨ A rectangle is a parallelogram with four right angles. The diagonals of a rectangle are the same length (congruent) and bisect each other. Since a rectangle is a parallelogram, a rectangle has the same properties as those of a parallelogram.

¨ A square is a rectangle with four congruent sides and a rhombus with four right angles. Squares have special

characteristics that are true for all squares, such as diagonals are perpendicular bisectors and diagonals bisect opposite angles. Since a square is a rectangle, a square has all the properties of a rectangle and of a parallelogram.

¨ A rhombus is a parallelogram with four congruent sides whose diagonals bisect each other and intersect at right angles. Opposite angles are congruent.

¨ A trapezoid is a quadrilateral with exactly one pair of parallel sides. A trapezoid may have none or two right angles. A trapezoid with congruent, non-parallel sides is called an isosceles trapezoid.

¨ A kite is a quadrilateral with two pairs of adjacent congruent sides. One pair of opposite angles is congruent.

¨ The number of sides determines the name of the polygon. A pentagon has 5 sides; a hexagon, 6 sides; a heptagon, 7 sides; an octagon, 8 sides; a nonagon, 9 sides; and a decagon, 10 sides.

¨ Prefixes in the names of polygons tell the number of sides: penta = 5, hexa = 6, hepta = 7, octa = 8, nona = 9, and deca = 10.

¨ In regular polygons all angles are congruent and all sides are congruent.

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Curriculum Information

SOL 7.6 Essential Knowledge and Skills

Key Vocabulary Essential Questions and Understandings

Teacher Notes and Elaborations SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.6 The student will determine whether plane figures (quadrilaterals and triangles) are similar and write proportions to express the relationships between corresponding sides of similar figures. ESS Lesson Similar Figures (PDF) - Determining whether two plane figures are similar, identifying corresponding sides (Word) Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Identify corresponding sides and

corresponding and congruent angles of similar figures using the traditional notation of curved lines for the angles.

• Write proportions to express the relationships between the lengths of corresponding sides of similar figures.

• Determine if quadrilaterals or triangles are similar by examining congruence of corresponding angles and proportionality of corresponding sides.

• Given two similar figures, write similarity statements using symbols such as ΔABC ~ ΔDEF, ∠A corresponds to ∠D, and AB corresponds to DE .

Key Vocabulary corresponding parts congruent hatch mark polygon proportion ratio similar figures

Essential Questions and Understandings • How do polygons that are similar compare to polygons that are congruent?

Congruent polygons have the same size and shape. Similar polygons have the same shape, and corresponding angles between the similar figures are congruent. However, the lengths of the corresponding sides are proportional. All congruent polygons are considered similar with the ratio of the corresponding sides being 1:1.

Teacher Notes and Elaborations ¨ Two polygons are similar if corresponding (matching) angles are congruent and the

lengths of corresponding sides are proportional. o The symbol ~ is used to indicate that two polygons (a closed plane figure

constructed with three or more straight-line segments that intersect only at their vertices) are similar.

¨ Congruent polygons have the same size and shape.

¨ Congruent polygons are similar polygons for which the ratio of the corresponding sides is 1:1.

¨ Given two similar figures, write similarity statements.

¨ Identify corresponding sides and corresponding and congruent angles of similar figures using the traditional notation of curved lines for the angles.

¨ Similarity statements can be used to determine corresponding parts of similar figures such as:

~ABC DEFΔ Δ ∠A corresponds to ∠D AB corresponds to DE

(continued)

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Curriculum Information SOL 7.6

Essential Questions and Understandings Teacher Notes and Elaborations (continued)

SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.6 The student will determine whether plane figures (quadrilaterals and triangles) are similar and write proportions to express the relationships between corresponding sides of similar figures. Return to Course Outline

¨ The traditional notation for marking congruent angles is to use a curve on each angle. Denote which angles are congruent with the same number of curved lines. For example, if ∠A congruent to ∠B, then both angles will be marked with the same number of curved lines.

¨ Write proportions to express the relationships between the lengths of corresponding sides of similar figures.

¨ Congruent sides are denoted with the same number of hatch marks on each

congruent side. For example, a side on a polygon with 2 hatch marks is congruent to the side with 2 hatch marks on a congruent polygon.

¨ Determine if quadrilaterals or triangles are similar by examining congruence of corresponding angles and proportionality of

corresponding sides.

¨ Corresponding parts is a one-to-one mapping between two figures. Similar figures are the same shape, but not always the same size. o ΔABC ~ ΔDEF. Therefore:

A∠ corresponds to D∠ and A∠ ≅ D∠ B∠ corresponds to E∠ and B∠ ≅ E∠ C∠ corresponds to F∠ and C∠ ≅ F∠

AB BC AC= =DE EF DF

o Two polygons are similar if corresponding (matching) angles are

congruent and the lengths of corresponding sides are proportional.

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Interdisciplinary Concept: Systems; Communication Math Concept: Relationships; Reasoning and Justification ACPS Mathematics Enduring Understandings: 2 - Spatial relationships can be described using coordinate geometry and other representational systems. 12 - Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations. ACPS Essential Standard in grade band 6-8: Explore and represent the properties of shapes Create and quantify the results of various transformations, including dilation Life Long Learner Standards

Curriculum Information SOL 7.8

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Measurement and Geometry Focus Relationships between Figures Virginia SOL 7.8 The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Identify the coordinates of the image of

a right triangle or rectangle that has been translated either vertically, horizontally or a combination of a vertical and horizontal translation.

• Identify the coordinates of the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.

• Identify the coordinates of the image of a right triangle or a rectangle that has been reflected over the x- or y-axis.

• Identify the coordinates of a right triangle or rectangle that has been dilated. The center of the dilation will be the origin.

• Sketch the image of a right triangle or rectangle translated vertically or horizontally.

• Sketch the image of a right triangle or rectangle that has been rotated 90° or

Essential Questions and Understandings • How does the transformation of a figure affect the size, shape and position of that

figure? Translations, rotations and reflections do not change the size or shape of a figure. A dilation of a figure and the original figure are similar. Reflections, translations and rotations usually change the position of the figure.

Teacher Notes and Elaborations ¨ A rotation of a geometric figure is a 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.

¨ Identify the coordinates of the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.

¨ Sketch the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.

¨ A translation of a geometric figure is a slide of the figure in which all the points on the figure move the same distance in the same direction.

¨ Identify the coordinates of the image of a right triangle or rectangle that has been translated either vertically, horizontally, or a combination of a vertical and horizontal translation.

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SOL Reporting Category Measurement and Geometry Focus Relationships between Figures Virginia SOL 7.8 The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane. Return to Course Outline

180° about the origin. • Sketch the image of a right triangle or

rectangle that has been reflected over the x- or y-axis.

• Sketch the image of a dilation of a right triangle or rectangle limited to a scale

factor of 14

, 12

, 2, 3, or 4.

Key Vocabulary

center of rotation coordinate plane coordinates (ordered pair) dilation horizontal axis (x-axis) image origin

pre-image quadrant reflection rotation scale factor transformation translation vertical axis (y-axis)

¨ Sketch the image of a right triangle or rectangle translated vertically or horizontally.

¨ A reflection is a transformation that reflects a figure across a line in the plane.

¨ Identify the coordinates of the image of a right triangle or a rectangle that has been reflected over the x- or y-axis.

¨ Sketch the image of a right triangle or rectangle that has been reflected over the x- or y-axis.

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

¨ Identify the coordinates of a right triangle or rectangle that has been dilated. The

center of the dilation will be the origin.

¨ Sketch the image of a dilation of a right triangle or rectangle limited to a scale factor

of14

,12

, 2, 3 or 4.

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

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

¨ When a geometric figure is translated on a coordinate plane, the new vertices are labeled as follows: point A corresponds to Aʹ′ , point B corresponds to Bʹ′ , and so on. Sometimes double prime (Aʹ′ʹ′ ) and triple prime (Aʹ′ʹ′ʹ′ ) notations are used.

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Interdisciplinary Concept: Properties and Models Math Concept: Analysis and Evaluation ACPS Mathematics Enduring Understandings: 6 - Data can be collected, organized, and displayed in purposeful ways. 7 - Various statistical methods can be used to observe, analyze, predict, and make inferences about data ACPS Essential Standard in grade band 6-8: Formulate questions, design studies, collect relevant data, and create and use appropriate graphical representations of data 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 7.11

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.11 The student, given data in a practical situation, will a. construct and analyze histograms;

and b. compare and contrast histograms

with other types of graphs presenting information from the same data set.

Return to Course Outline

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

data set using histograms. For collection and display of raw data, limit the data to 20 items.

• Determine patterns and relationships within data sets (e.g., trends).

• Make inferences, conjectures, and predictions based on analysis of a set of data.

• Compare and contrast histograms with line plots, circle graphs, and stem and leaf plots presenting information from the same data set.

Key Vocabulary circle graph conjecture frequency distribution histogram inference

Essential Questions and Understandings • What type of data is most appropriate to display in a histogram?

Numerical data that can be characterized using consecutive intervals are best displayed in a histogram.

Teacher Notes and Elaborations ¨ All graphs tell a story and include a title and labels that describe the data.

¨ Collect, analyze, display, and interpret a data set using histograms. For collection and display of raw data, limit the data to 20 items.

¨ Determine patterns and relationships within data sets (e.g., trends).

¨ Make inferences, conjectures, and predictions based on analysis of a set of data.

¨ Compare and contrast histograms with line plots, circle graphs, and stem-and-leaf plots presenting information from the same data set.

¨ A line plot shows the frequency of data on a number line. Line plots are used to show the spread of the data and quickly identify the range, mode, and any outliers.

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intervals line plot prediction stem-and-leaf plot tally trends

¨ A stem-and- leaf plot displays data from least to greatest using the digits of the greatest place value to group data.

(continued)

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SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.11 The student, given data in a practical situation, will a. construct and analyze histograms;

and b. compare and contrast histograms

with other types of graphs presenting information from the same data set.

Return to Course Outline

¨ A frequency distribution shows how often an item, a number, or range of numbers occurs. It can be used to construct a histogram. A tally is a mark used to keep count in each interval.

¨ Bar graphs are utilized to compare counts of different categories both categorical and discrete data. A bar graph uses parallel bars; either horizontal or vertical, to represent counts for several categories. One bar is used for each category with the length of the bar representing the count for that category. There is space before, between, and after the bars. The axis displaying the scale representing the count for the categories should extend one increment above the greatest recorded piece of data. The values should represent equal increments. Each axis should be labeled, and the graph should have a title.

¨ A histogram is a form of bar graph in which the categories are consecutive and equal intervals. If no data exists in an interval, that interval must still be labeled in the graph. A histogram uses numerical instead of categorical data. A histogram is constructed from a frequency table. The intervals are shown on the x-axis and the number of elements in each interval is represented by the height of a bar located above the interval. The length or height of each bar is determined by the number of data elements (frequency) falling into a particular interval. Histograms summarize data but do not provide information about specific data points.

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Interdisciplinary Concept: Communication Math Concept: Theory ACPS Mathematics Enduring Understandings: 13 - Probability and data analysis can be used to make predictions ACPS Essential Standard in grade band 6-8: 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 7.9

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.9 The student will investigate and describe the difference between the experimental probability and theoretical probability of an event. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Determine the theoretical probability of

an event. • Determine the experimental probability

of an event. • Describe changes in the experimental

probability as the number of trials increases.

• Investigate and describe the difference between the probability of an event found through experiment or simulation versus the theoretical probability of that same event.

Key Vocabulary event experimental probability Law of Large Numbers outcome probability sample space sampling simulation theoretical probability

Essential Questions and Understandings • What is the difference between the theoretical and experimental probability of an event?

Theoretical probability of an event is the expected probability and can be found with a formula. The experimental probability of an event is determined by carrying out a simulation or an experiment. In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability.

Teacher Notes and Elaborations ¨ The probability of an event occurring is a ratio expressing the chance or likelihood

that a certain event will occur, given the number of possible outcomes (results) of an experiment. An event is a subset of a sample space. The sample space is the set of all possible outcomes of an experiment.

¨ Determine the theoretical probability of an event.

¨ The theoretical probability of an event is the expected probability and can be found with a formula.

Theoretical probability of an event number of possible favorable outcomestotal number of possible outcomes

=

¨ Determine the experimental probability of an event. Describe changes in the experimental probability as the number of trials increases.

¨ The experimental probability of an event is determined by carrying out a simulation or an experiment. The experimental probability is found by repeating an experiment many times and using the ratio.

o Experimental probability is not exact since the results may vary if the experiment is repeated.

o In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability (Law of Large Numbers).

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Return to Course Outline

o An important use of experimental probability is to make predictions about a large group of people based on the results of a poll or survey. This technique, called sampling, is used when it is impossible to question every member of a group.

Experimental probability number of times desired outcomes occurtotal number of trials in the experiment

=

¨ Investigate and describe the difference between the probability of an event found through experiment or, simulation versus the theoretical probability of that same event.

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Interdisciplinary Concept: Communication Math Concept: Theory ACPS Mathematics Enduring Understandings: 13 - Probability and data analysis can be used to make predictions ACPS Essential Standard in grade band 6-8: 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 7.10

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.10 The student will determine the probability of compound events using the Fundamental (Basic) Counting Principle. Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Compute the number of possible

outcomes by using the Fundamental (Basic) Counting Principle.

• Determine the probability of a compound event containing no more than two events.

Key Vocabulary compound event dependent event Fundamental Counting Principle independent event outcomes probability sample space tree diagram

Essential Questions and Understandings • What is the Fundamental (Basic) Counting Principle?

The Fundamental (Basic) Counting Principle is a computational procedure used to determine the number of possible outcomes of several events.

• What is the role of the Fundamental (Basic) Counting Principle in determining the probability of compound events?

It is the product of the number of outcomes for each event that can be chosen individually (e.g., the possible outcomes or outfits of four shirts, two pants, and three shoes is 4 · 2 · 3 or 24).

Teacher Notes and Elaborations ¨ Compute the number of possible outcomes by using the Fundamental (Basic) Counting

Principle.

¨ Tree diagrams are used to illustrate possible outcomes of events. They can be used to support the Fundamental (Basic) Counting Principle.

o The following tree diagram illustrates the possible outcomes (results). Using the Fundamental (Basic) Counting Principle the possible outcomes can be found by multiplying the number of pant choices times the shirt choices (2 · 3 = 6)

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.

¨ Events are independent when the outcome of one has no effect on the outcome of the other. For example, rolling a number cube and flipping a coin are independent events.

SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.10 The student will determine the probability of compound events using the Fundamental (Basic) Counting Principle. Return to Course Outline

¨ Events are dependent when the outcome of one event is influenced by the outcome of the other. For example, when drawing two marbles from a bag, not replacing the first after it is drawn affects the outcome of the second draw.

¨ A compound event combines two or more simple events (independent or dependent). For example, a bag contains 4 red, 3 green and 2

blue marbles. What is the probability of selecting a green and then a blue marble (with or without replacement)?

o With replacement (independent) the probability is: 3 2 69 9 81⋅ = which can be simplified to 2

27.

o Without replacement (dependent) the probability is: 3 2 69 8 72⋅ = which can be simplified to

112

.

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

Curriculum Information

SOL 7.5 Essential Knowledge and Skills

Key Vocabulary Essential Questions and Understandings

Teacher Notes and Elaborations SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.5 The student will a. describe volume and surface area of

cylinders; b. solve practical problems involving

the volume and surface area of rectangular prisms and cylinders; and

c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

Return to Course Outline

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: • Determine if a practical problem

involving a rectangular prism or cylinder represents the application of volume or surface area.

• Find the surface area of a rectangular prism.

• Solve practical problems that require finding the surface area of a rectangular prism.

• Develop a procedure and formula for finding the surface area of a cylinder.

• Find the surface area of a cylinder. • Solve practical problems that require

finding the surface area of a cylinder. • Find the volume of a rectangular prism. • Solve practical problems that require

finding the volume of a rectangular prism.

• Develop a procedure and formula for finding the volume of a cylinder.

• Find the volume of a cylinder. • Solve practical problems that require

finding the volume of a cylinder. • Describe how the volume of a

Essential Questions and Understandings • How are volume and surface area related?

Volume is a measure of the amount a container holds while surface area is the sum of the areas of the surfaces on the container.

• How does the volume of a rectangular prism change when one of the attributes is increased?

There is a direct relationship between the volume of a rectangular prism increasing when the length of one of the attributes of the prism is changed by a scale factor.

Teacher Notes and Elaborations ¨ The ratio of the circumference of any circle to the length of its diameter is (pi). π

is a nonterminating nonrepeating decimal. The most commonly used rational number

approximations for are 3.14 and 227

.

¨ The area of a rectangle is computed by multiplying the lengths of two adjacent sides.

¨ The area of a circle is computed by squaring the radius and multiplying that product

by π (A = πr2, where π ≈ 3.14 or 227

).

¨ The radius of a circle is a segment connecting the center of the circle to a point on the circle.

¨ The diameter of a circle is a segment connecting two points on the circle and passing through the center.

π

π

ACPS Curriculum Framework Math 7

Page 48 of 51

Return to Course Outline

rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors

(e.g., 12

, 2, 3, 5, and 10) only.

(continued)

¨ Nets are two-dimensional drawings (e.g., a drawing of a figure that has length and width) of three-dimensional figures (e.g., a figure that has length, width, and height) that can be used to help students find surface area.

¨ A net of a solid is a two dimensional figure that can be folded into a three dimensional shape.

¨ Determine if a practical problem involving a rectangular prism or cylinder represents the application of volume or surface area.

(continued)

ACPS Curriculum Framework Math 7

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(continued from previous page) SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.5 The student will a. describe volume and surface area of

cylinders; b. solve practical problems involving

the volume and surface area of rectangular prisms and cylinders; and

c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

Return to Course Outline

(continued from previous page) The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:

• Describe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors

(e.g., 12

, 2, 3, 5, and 10) only.

Key Vocabulary base cube cylinder diameter face formula height length net pi (π ) radius rectangular prism scale factor surface area volume width

(continued from previous page) Teacher Notes and Elaborations ¨ Surface area of any solid figure is the total area of the surface of the solid.

¨ The volume of a solid is the total amount of space inside a three-dimensional object. A unit for measuring volume is the cubic unit.

¨ Find the surface area of a rectangular prism.

¨ Find the volume of a rectangular prism.

¨ Solve practical problems that require finding the surface area of a rectangular prism.

¨ Solve practical problems that require finding the volume of a rectangular prism.

¨ A rectangular prism can be represented on a flat surface as a net that contains six rectangles – two that have measures of the length and width of the base, two others that have measures of the length and height, and two others that have measures of the width and height.

o The surface area of a rectangular prism is the sum of the areas of all six faces (SA = 2lw + 2lh + 2wh).

o The volume of a rectangular prism is computed by multiplying the area of the base, B, (length times width) by the height of the prism (V = lwh or V = Bh).

¨ Find the surface area of a cylinder.

¨ Find the volume of a cylinder.

¨ Solve practical problems that require finding the surface area of a cylinder.

(continued)

ACPS Curriculum Framework Math 7

Page 50 of 51

SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.5 The student will a. describe volume and surface area of

cylinders; b. solve practical problems involving

the volume and surface area of rectangular prisms and cylinders; and

c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

Return to Course Outline

¨ Solve practical problems that require finding the volume of a cylinder.

¨ A cylinder can be represented on a flat surface as a net that contains two circles (bases for the cylinder) and one rectangular region whose length is the circumference of the circular base and whose width is the height of the cylinder.

o The surface area of the cylinder is the area of the two circles and the rectangle (SA = 2πr2 + 2πrh). o The volume of a cylinder is computed by multiplying the area of the circular base, B, (πr2) by the height of the cylinder

(V = πr2h or V = Bh).

¨ Describe how the volume of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors only.

¨ Describe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors only.

(continued)

ACPS Curriculum Framework Math 7

Page 51 of 51

SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.5 The student will a. describe volume and surface area of

cylinders; b. solve practical problems involving

the volume and surface area of rectangular prisms and cylinders; and

c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

Return to Course Outline

¨ There is a direct relationship between changing one measured attribute of a rectangular prism by a scale factor and its volume. For example, doubling the length of a prism will double its volume.

o This direct relationship does not hold true for surface area. For example, doubling the length will only double the area of the affected sides. It will not double the total surface area.

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. A scale factor is a ratio that compares the sizes of the parts of the scale drawing of an object with the actual sizes of the corresponding parts of the object (e.g., If the scale drawing is ten times the size of the actual object, the scale factor is 10).

Length Width Height Volume Surface Area

Original Figure 5 4 3 60 m3 94 m2

Using the original figure:

Multiply length by 2 10 4 3 120 m3 164 m2

Multiply width by 2 5 8 3 120 m3 158 m2

Multiply height by 2 5 4 6 120 m3 148 m2

Multiply length by 12

122

4 3 30 m3 59 m2

Multiply width by 12

5 2 3 30 m3 62 m2

Multiply height by 12

5 4 112

30 m3 67 m2