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The Georgia Department of Juvenile Justice8th Grade MathematicsUnits of Instruction Resource ManualTable of Contents8th Grade MathematicsAcknowledgmentsSuperintendent’s LetterMission and Vision StatementsChapter 1: IntroductionChapter 2: Teacher’s GuideChapter 3: Instructional RotationChapter 4: Georgia Performance StandardsChapter 5: Curriculum MapChapter 6: Essential Questions and Enduring UnderstandingsChapter 7: Units of InstructionUnit 1: Principles of Algebra & Rational NumbersTask 1Task 2Task 3Task 4 Focus CAPsUnit 2: Graphs, Functions & SequencesTask 1Task 2Task 3Task 4Task 5Task 6 Focus CAPsUnit 3: Exponents& Roots/ Ratios, Proportions & Similarity/ PercentsTask 1Task 2Task 3Task 4Task 5 Focus CAPsUnit 4: Foundation of Geometry & Perimeter, Area & VolumeTask 1Task 2Task 3Task 4 Focus CAPsUnit 5: Data & StatisticsTask 1Task 2Task 3 Focus CAPsUnit 6: ProbabilityTask 1Task 2Task 3Task 4 Focus CAPsUnit 7: Multi-Step Equations & InequalitiesTask 1Task 2Task 3Task 4Task 5 Focus CAPsUnit 8: Graphing Lines, Sequences & Functions & PolynomialsTask 1Task 2Task 3Task 4 Focus CAPsChapter 8: Task websitesAcknowledgements The Georgia Department of Juvenile Justice Department of Education would like to thank the many educators who have helped to create this 8th Grade Math Units of Instruction Resource Manual. The educators have been particularly helpful in sharing their ideas and resources to ensure the completion and usefulness of this manual.Students served by the DJJ require a special effort if they are to become contributing and participating members of their communities. Federal and state laws, regulations, and rules will mean nothing in the absence of professional commitment and dedication by every staff member.The Georgia Department of Juvenile Justice is very proud of its school system. The school system is Georgia’s 181st and is accredited by the Southern Association of Colleges and Schools (SACS). The DJJ School System has been called exemplary by the US Department of Justice. This didn’t just happen by chance; rather it was the hard work of many teachers, clerks, instructors and administrators that earned DJJ these accolades and accreditations. The DJJ education programs operate well because of the dedicated staff. These dedicated professionals are the heart of our system. These Content Area Units of Instruction were designed to serve as a much needed tool for delivering meaningful whole group instruction. In addition, this resource will serve as a supplement to the skills and knowledge provided by the Georgia Department of Juvenile Justice Curriculum Activity Packets (CAPs). I would like to thank all the DJJ Teaching Staff, the Content Area Leadership Teams, Kimberly Harrison, DJJ Special Education/Curriculum Consultant and Martha Patton, Curriculum Director for initiating this project and seeing it through. Thank you all for your hard work and dedication to the youth we serve. Sincerely yours, James “Jack” Catrett, Ed.D. Associate SuperintendentMissionThe mission of Department of Juvenile Justice Math Consortium (DJJMC) is to build a multiparty effort statewide to achieve continuous, systemic and sustainable improvements in the education system serving the Math students of the Department of Juvenile Justice (DJJ).VisionTo achieve the mission of the DJJMC, members work collaboratively in examining the Georgia Performance Standards. These guidelines speak specifically to teachers being able to: deliver meaning content pertaining to the Characteristics of Math and its content standards across the Math Units of Instruction Resource Manual. The DJJMC will master and develop whole-group unit lessons built around Curriculum Activity Packets (CAPs), critique student work, and work as a team to solve the common challenges of teaching within DJJ. Additionally, the DJJMC jointly analyzes student test data in order to: develop strategies to eradicate common academic deficits among students, align curriculum, and create a coherent learning pathway across grade levels. The DJJMC also reviews research articles, attends workshops or courses, and invites consultants to assist in the acquisition of necessary knowledge and skills. Finally, DJJMC members observe one another in the classroom through focus walks.IntroductionThe 8th Grade Math Units of Instruction Resource Manual is a tool that has been created to serve as a much needed tool for delivering meaningful whole group instruction. This manual is a supplement to the skills and knowledge provided by the Georgia Department of Juvenile Justice Curriculum Activity Packets (CAPs). It is imperative that our students learn to reason mathematically, to evaluate mathematical arguments both formally and informally, to use the language of mathematics to communicate ideas and information precisely, and to work in cooperative learning groups. Best practices in education indicate that teachers should first model new skills for students. Next, teachers should provide opportunities for guided practice. Only then should teachers expect students to successfully complete an activity independently. The 8th Grade Math Units of Instruction meets that challenge. The Georgia Department of Juvenile Justice Office of Education Direct Instruction Lesson PlanTeacher:Subject:______________________________Date:_____________to__________________Period □ 1st□ 2nd□ 3rd□ 4th□ 5th□ 6thStudents will engage in: □ Independent activities □ pairing □ Cooperative learning □ hands-on □ Peer tutoring □ Visuals □ technology integration □ Simulations □ a project □ centers □ lecture □ Other Essential Question(s):Standards:CAPs Covered:Grade Level:____ Unit:______RTI Tier for data collection: 2 or 3Tier 2 Students:Tier 3 Students:TimeProcedures Followed:Material/Text _______Minutes Review of Previously Learned Material/Lesson Connections:Recommended Time: 2 Minutes _______Minutes Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at , or print on blackboard) Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard). Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.Recommended Time: 2 Minutes_______MinutesIntroduce task by stating the purpose of today’s lesson. Recommended Time: 2 Minutes_______MinutesEngage students in conversation by asking open ended questions related to the essential question(s). Recommended Time: 2 Minutes_______Minutes Begin whole group instruction with corrective feedback:Recommended Time: 10 Minutes_______Minutes Lesson Review/Reteach:Recommended Time: 2 Minutes_______Minutes Independent Work CAPs:Recommended Time: 30 MinutesTeacher Reflections: The Instructional Rotation Matrix has been designed to assist language arts teachers in providing a balanced approach to utilizing the Math Units of Instruction across all grade levels on a rotating schedule. MondayTuesdayWednesdayThursday6th Grade ContentMiddle School9th Grade ContentHigh School7th Grade ContentMiddle School10th Grade ContentHigh School8th Grade ContentMiddle School11th Grade ContentHigh School6th Grade ContentMiddle School12th Grade ContentHigh School7th Grade ContentMiddle School9th Grade ContentHigh School8th Grade ContentMiddle School10th Grade ContentHigh School6th Grade ContentMiddle School11th Grade ContentHigh School7th Grade ContentMiddle School12th Grade ContentHigh SchoolGeorgia Performance StandardsM8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.b. Simplify and evaluate algebraic expressions.c. Solve algebraic equations in one variable, including equations involving absolute values.d.Solve equations involving several variables for one variable in terms of the others.e.Interpret solutions in problem contexts. M8A2 Students will understand and graph inequalities in one variable.a. Represent a given situation using an inequality in one variable.b. Use the properties of inequality to solve inequalities.c. Graph the solution of an inequality on a number line.d. Interpret solutions in problem contexts. M8A3 Students will understand relations and linear functions.a. Recognize a relation as a correspondence between varying quantities.b. Recognize a function as a correspondence between inputs and outputs where the output for each input must be unique.c. Distinguish between relations that are functions and those that are not functions.d. Recognize functions in a variety of representations and a variety of contexts.e. Use tables to describe sequences recursively and with a formula in closed form.f. Understand and recognize arithmetic sequences as linear functions with whole-number input values.h. Interpret the constant difference in an arithmetic sequence as the slope of the associated linear function.i. Identify relations and functions as linear or nonlinear.j. Translate among verbal, tabular, graphic, and algebraic representations of functions. M8A4 Students will graph and analyze graphs of linear equations and inequalities.a. Interpret slope as a rate of change.b. Determine the meaning of the slope and y-intercept in a given situation.c. Graph equations of the form y = mx + b.d. Graph equations of the form ax + by = c.e.Graph the solution set of a linear inequality, identifying whether the solution set is an open or a closed half-plane. f.Determine the equation of a line given a graph, numerical information that defines the line, or a context involving a linear relationship. g.Solve problems involving linear relationships. M8A5 Students will understand systems of linear equations and inequalities and use them to solve problems.a.Given a problem context, write an appropriate system of linear equations or inequalities. b. Solve systems of equations graphically and algebraically, using technology as appropriate.c.Graph the solution set of a system of linear inequalities in two variables.d.Interpret solutions in problem contexts. M8D1 Students will apply basic concepts of set theory.a. Demonstrate relationships among sets through use of Venn diagrams.b. Determine subsets, complements, intersection, and union of sets.c. Use set notation to denote elements of a set. M8D2 Students will determine the number of outcomes related to a given event.a. Use tree diagrams to find the number of outcomes.b. Apply the addition and multiplication principles of counting. M8D3 Students will use the basic laws of probability.a. Find the probability of simple independent events.b. Find the probability of compound independent events. M8D4 Students will organize, interpret, and make inferences from statistical data.a. Gather data that can be modeled with a linear function.b. Estimate and determine a line of best fit from a scatter plot. M8G1 Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.b. Apply properties of angle pairs formed by parallel lines cut by a transversal.c. Understand the properties of the ratio of segments of parallel lines cut by one or more transversals.d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent. M8G2 Students will understand and use the Pythagorean theorem.a. Apply properties of right triangles, including the Pythagorean theorem.b. Recognize and interpret the Pythagorean theorem as a statement about areas of squares on the sides of a right triangle. M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.a. Find square roots of perfect squares.b. Recognize the (positive) square root of a number as a length of a side of a square with a given area.c. Recognize square roots as points and as lengths on a number line.d. Understand that the square root of 0 is 0 and that every positive number has two square roots that are opposite in sign.e. Recognize and use the radical symbol to denote the positive square root of a positive number.f. Estimate square roots of positive numbers.g. Simplify, add, subtract, multiply, and divide expressions containing square roots.h. Distinguish between rational and irrational numbers.i. Simplify expressions containing integer exponents.j. Express and use numbers in scientific notation.k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation. M8P1 Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving. M8P2 Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof. M8P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely. M8P4 Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics. M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena. M8RC1 Students will enhance reading in all curriculum areas by:a. Reading in All Curriculum Areas Read a minimum of 25 grade-level appropriate books per year from a variety of subject disciplines and participate in discussions related to curricular learning in all areas. Read both informational and fictional texts in a variety of genres and modes of discourse. Read technical texts related to various subject areas.b. Discussing books Discuss messages and themes from books in all subject areas. Respond to a variety of texts in multiple modes of discourse. Relate messages and themes from one subject area to messages and themes in another area. Evaluate the merit of texts in every subject discipline. Examine author’s purpose in writing. Recognize the features of disciplinary texts.c. Building vocabulary knowledge Demonstrate an understanding of contextual vocabulary in various subjects. Use content vocabulary in writing and speaking. Explore understanding of new words found in subject area texts.d. Establishing context Explore life experiences related to subject area content. Discuss in both writing and speaking how certain words are subject area related. Determine strategies for finding content and contextual meaning for unknown words.HYPERLINK ""?DJJ 8th Grade MathematicsGeorgia Performance Standards:? Curriculum Map??????????????????????????????????????????? 1st Semester2nd SemesterPrinciples of Algebra & Rational NumbersGraphs, Function & SequencesExponents & Roots Ratios, Proportions & Similarity& PercentsFoundations of Geometry & Perimeter, Area & VolumeData & StatisticsProbabilityMulti-Step Equations & InequalitiesGraphing Lines, Sequences & Functions & PolynomialsChapter 1CAPs1-6Chapter 3CAPs12-15Chapter 4CAPs16-20Chapter 7CAPs31-36Chapter 9CAPs43-47 Chapter 10CAPs48-52 Chapter 11CAPs54-57Chapter 12CAPs58-62 27-11521-25837-421363-67626-301468-71GPS:M8A1a,b,c,dM8P1a,b,c,dM8P4a,b,cM8P3a,cM8P5a,b,cM8P2cM8A2b,cM8A1a,bM8P3cGPS:M8A1b,dM8P1a,bM8P5a,b,cM8P2c,dM8P3a,cM8A4c,eM8P4a,b,cM8A3b,c,d,e,iGPS:M8N1a,b,c,d,e,f,g,h,i,j,k M8A1a,b,c,d M8P1a,b,c,d M8P2a,b,c,d M8P3a,c,dM8P4a,b,c M8P5a,b,c M8G2a,b M8G1a,b,d M8A4.b M8N1i,kGPS:M8A1a,b,c,dM8P3a,b,c,d M8P1a,b,c M8P2a,c,dM8G1a,b,dM8P4a,b,cM8A1a,b,c,d M8P5a,b,c M8G2a M8N1.kGPS:M8P2.c M8P3a,c,d M8P4a,b,cM8P1a,b,c,d M8P5a,b,c M8D4.aGPS:M8D3a,b M8P1a,b,c,d M8P3a,c,d M8P4a,b ,cM8P2.cM8P5a,b,cM8A1a,c,dM8D2a,bGPS:M8A1a,b,c,d M8P1b,c,d M8P5a,b,c M8P3a,cM8P4cM8A2a,b,c,d M8A5b,cGPS:M8A3h,d,e,f, M8A4a,b,c,d,e,f M8P1a,b,c,d M8P3a,c M8P4a,b,cM8P5a,b,c M8A3d,h,iM8D4b M8A1a,b,c,d M8P2a,b,c,d M8N1i,kFocus CAPs:Chapter 12 & 6Chapter 27 & 11Focus CAPs:12 & 15Focus CAPs:Chapter 416 & 20Chapter 521 Chapter 626 Focus CAPs:Chapter 731 & 36Chapter 837 & 42Focus CAPs:43 Focus CAPs:48 & 52Focus CAPs:54 & 57Focus CAPs:Chapter 1258 Chapter 1363 Chapter 1468 Enduring Understandings & Essential Question Principles of Algebra & Rational Numbers Enduring Understandings:Algebraic expressions, equations and inequalities are used to represent relationships between numbers. Absolute value is used to represent distances between numbers. Graphs can be used to represent all of the possible solutions to a given situation. Many problems encountered in everyday life can be solved using equations or inequalities. Essential Questions:How can I simplify and evaluate an algebraic expression? How can I solve an equation or inequality? How can I tell the difference between an expression, equation and an inequality? How can I determine the absolute value of an expression? How can I represent absolute value on a number line? Graphs, Functions & Sequences Enduring Understandings:Collecting and examining data can sometimes help one discover patterns in the way in which two quantities vary. Changes in varying quantities are often related by patterns which, once discovered, can be used to predict outcomes and solve problems. Written descriptions, tables, graphs and equations are useful in representing and investigating relationships between varying quantities. Different representations (written descriptions, tables, graphs and equations) of the relationships between varying quantities may have different strengths and weaknesses. Linear functions may be used to represent and generalize real situations. Slope and y-intercept are keys to solving real problems involving linear relationships. Essential Questions:What does the data tell me? How does a change in one variable affect the other variable in a given situation? Which tells me more about the relationship I am investigating – a table, a graph or an equation? Why? What strategies can I use to help me understand and represent real situations involving linear relationships? How can the properties of lines help me to understand graphing linear functions? How is a linear inequality like a linear equation? How are they different? Exponents& Roots/ Ratios, Proportions & Similarity/ PercentsEnduring Understandings:An irrational number is a real number that cannot be written as a ratio of two integers. All real numbers can be plotted on a number line. Exponents are useful for representing very large or very small numbers. Square roots can be rational or irrational. Some properties of real numbers hold for all irrational numbers. There are many relationships between the lengths of the sides of a right triangle. Essential Questions:When are exponents used and why are they important? Why is it useful for me to know the square root of a number? How do I simplify and evaluate algebraic expressions involving integer exponents and square roots? What is the Pythagorean Theorem and when does it hold? Foundation of Geometry & Perimeter, Area & VolumeEnduring Understandings:Parallel lines have the same slope and perpendicular lines have opposite, reciprocal slopes. When two lines intersect, vertical angles are congruent and adjacent angles are supplementary. When parallel lines are cut by a transversal, corresponding, alternate interior and alternate exterior angles are congruent. The length of segments formed by two non-parallel transversals cutting parallel lines is proportional to the distances of the parallel lines from the intersection of the transversals. Parallel lines can be constructed using the properties of parallel lines cut by a transversal. In Euclidean Geometry, there is exactly one line through a given point parallel to a second given line. Essential Questions:How can I be certain whether lines are parallel, perpendicular, or skew lines? Why do I always get a special angle relationship when any two lines intersect? When I draw a transversal through parallel lines, what are the special angle and segment relationships that occur? What information is necessary before I can conclude two figures are congruent? How can my knowledge of constructing congruent triangles be used to construct perpendicular and parallel lines? Can I find parallel lines that intersect? Why or why not?Data & StatisticsEnduring Understandings:Relations show any correspondence between sets, while functions are special relations in which each input from a fixed set is associated with a single output.Linear functions are defined by constant slope. Arithmetic sequences are numerical representations of linear functions.Venn diagrams are visual tools for organizing members of related sets.Like functions and relations, visualizing sets in multiple representations often reveals unexpected patterns.Essential Questions:How can I identify a function?How can I tell the difference between a relation and a function?How can I relate arithmetic sequences to linear functions?When working with sets, when do I use a union, and when do I use an intersection?ProbabilityEnduring Understandings:Tree diagrams are useful for describing relatively small sample spaces and computing probabilities, as well as for visualizing why the number of outcomes can be extremely large. Essential Questions:How do I determine a sample space? How can a tree diagram help me to find the number of possible outcomes related to a given event? When and why do I use addition to determine sample space size? When and why do I use multiplication to determine sample space size? When and why do I use addition to determine probabilities? When and why do I use multiplication to determine probabilities? How can I use probability to determine if a game is fair or to figure my chances of winning the lottery? Multi-Step Equations & InequalitiesEnduring Understandings:There are situations that require two or more equations to be satisfied simultaneously. There are several methods for solving systems of equations. Solutions to systems can be interpreted algebraically, geometrically, and in terms of problem contexts. In some problem contexts, the constraints that must be satisfied are modeled by inequalities rather than equations. The number of solutions to a system of equations or inequalities can vary from no solution to an infinite number of solutions.Essential Questions:How can I interpret the meaning of a “system of equations” algebraically and geometrically? How does mathematical notation indicate that equations are to be treated as a system? What does it mean to solve a system of linear equations? How can the solution to a system be interpreted geometrically? How can I recognize how many solutions a system of equations has prior to solving? How do I decide which method would be easier to use to solve a particular system of equations? Why is graphing a system of inequalities a good way to show the solution set? How can I translate a problem situation into a system of equations or inequalities? What does the solution to a system tell me about the answer to a problem situation?Graphing Lines, Sequences & Functions & PolynomialsEnduring Understandings:Relations show any correspondence between sets, while functions are special relations in which each input from a fixed set is associated with a single output.Linear functions are defined by constant slope. Arithmetic sequences are numerical representations of linear functions.Venn diagrams are visual tools for organizing members of related sets.Like functions and relations, visualizing sets in multiple representations often reveals unexpected patterns.Essential Questions:How can I identify a function?How can I tell the difference between a relation and a function?How can I relate arithmetic sequences to linear functions?When working with sets, when do I use a union, and when do I use an intersection?Unit: Principles of Algebra & Rational NumbersGeorgia Performance Standards:M8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.b. Simplify and evaluate algebraic expressions.c. Solve algebraic equations in one variable, including equations involving absolute values.d.Solve equations involving several variables for one variable in terms of the others.M8P1 Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.M8P4 Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.M8P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena.M8P2 Students will reason and evaluate mathematical arguments.c. Develop and evaluate mathematical arguments and proofs.M8A2 Students will understand and graph inequalities in one variable.b. Use the properties of inequality to solve inequalities.c. Graph the solution of an inequality on a number line.M8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.b. Simplify and evaluate algebraic expressions.M8P3 Students will communicate mathematically.c. Analyze and evaluate the mathematical thinking and strategies of others.Selected Terms and Symbols:Independent events: Events for which the occurrence of one has no impact on the occurrence of the other.Relative frequency: The number of times an outcome occurs divided by the total number of trials.Sample space: All possible outcomes of a given experiment.Event: A subset of a sample space.Simple Event: An event consisting of just one outcome. A simple event can be represented by a single branch of a tree pound Event: A sequence of simple plement: The complement of event E, sometimes denoted E′ (E prime), occurs when E doesn’t. The probability of E′ equals 1 minus the probability of E: P(E′) = 1 – P(E).Counting Principle: If an event A can occur in m ways and for each of these m ways, an event B can occur in n ways, then events A and B can occur in ways. This counting principle can be generalized to more than two events that happen in succession. So, if for each of the m and n ways A and B can occur respectively, there is also an event C that can occur in s ways, then events A, B, and C can occur in ways.Tree diagram: A tree-shaped diagram that illustrates sequentially th HYPERLINK e possible outcomes of a given event.Teacher’s Place:Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.1. Explain the activity (activity requirements)2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at . 3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard. 7. Discuss answers with the students using the following questioning techniques as applicable: Questioning Techniques:Memory QuestionsSignal words: who, what, when, where?Cognitive operations: naming, defining, identifying, designatingConvergent Thinking QuestionsSignal words: who, what, when, where?Cognitive operations: explaining, stating relationships, comparing andcontrasting Divergent Thinking QuestionsSignal words: imagine, suppose, predict, if/thenCognitive operations: predicting, hypothesizing, inferring, reconstructingEvaluative Thinking QuestionsSignal words: defend, judge, justify (what do you think)?Cognitive operations: valuing, judging, defending, justifying8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter) 10. At the end of the *whole group learning session, students will transition into independent CAP assignments. *The phrase, “whole group learning session” is utilized “rather than, the end of the activity” because all of the activities may not be completed in one day. Task: 1 Resources: Part 1:a) Do the following sequence of operations in order:b) What did you get for your final number?c) Check with your partner, what did that person get for their final number?d) Everyone should have the same number. What number is that?e) Why did everyone end with the same number?f) How does this trick work?Write down any number. (This is your ‘start’ number.) Add to it the number that comes before it.Add 11. Divide by 2.Subtract your start number.Part 2: Now try this one:Take the number of your birth month. Add 32.Add the difference between your birth month number and 12. Divide by 4.Add 2.This is your Lucky Number!Do you feel lucky? Why or why not? Explain what made this trick work.Part 3:Work with a partner to write your own Number Trick. Be sure to list your instructions in the appropriate order. Swap Number Tricks with another group or share your Tricks one at a time with the entire class. See if your friends can describe how your Number Trick works.Discussion, Suggestions, Possible SolutionsThe challenge is to see if students can discover how the trick works. If students need a hint, suggest that instead of using an actual number, they use a box or a letter to begin with.Part 1:Write down any number. (This is your ‘start’ number.)Start with N. →→→→→→→→→→→→→→→→→→→→ NAdd to it the number that comes before it.The previous number is N-1 →→→→→→→→→→→→→→ N + (N – 1) = 2N – 1Add 11.Add the 11 to the 2N – 1 →→→→→→→→→→→→→→→ 2N – 1 + 11 = 2N + 10Divide by 2.Dividing 2N + 10 by 2 →→→→→→→→→→→→→→→→ 2N + 10 = N + 52Subtract your start number.The start number was N. →→→→→→→→→→→→→→→ N + 5 – N = 5This means that everyone should have ended with a 5 regardless of the start number.Part 2:Take the number of your birth month.The birth month will be a number between 1 and 12.If “x” is the number of a student’s birth, then 1 ≤ x ≤ 12.Add 32.Adding 32 gives x + 32.Add the difference between your birth month number and 12.When you add the difference between your birth month and 12 you get x + 32 + (12 – x) which is equal to 44.Divide by 4.Dividing by 4 leaves 11.Add 2.Adding 2 yields 13; this is usually considered an unlucky number.Part 3:Work with a partner to write your own Number Trick. Be sure to list your instructions in the appropriate order. Swap Number Tricks with another group or share your Tricks one at a time with the entire class. See if your friends can describe how your Number Trick works.Task: 2 Resources: The students at Eastman RYDC and Eastman YDC are participating in a game room survey. Eastman RYDC is located 5 miles from the game room and Eastman YDC 3 miles from the game room. The owner, of the game room wonders how far apart the centers are.On grid paper, pick a point to represent the location of the game room.Illustrate all of the possible points where Eastman RYDC could be located on the grid paper.Using a different color, illustrate all of the possible points where Eastman YDC could be located. What is the smallest distance, d, that could separate the centers? How did you know?What is the largest distance, d, that could separate the centers? How did you know?Write and graph an inequality in terms of d to show the owner of the game room all of the possible distances that could separate the two centers.Discussion, Suggestions, Possible SolutionsStudents should understand that the centers could be located anywhere on the circle with the game room as the center and the radius as the distance that they are from the game room.Eastman RYDCEastman YDCGame RoomTherefore, the closest the centers could be would be 5 – 3 = 2 miles and the farthest apart that they could be would be 5 + 3 = 8 miles. This may be written as 5 ± 3 = d where d represents the distance from EastmanTask: 3 Resources: Part 1:Your science teacher says the grades, g, in your class can be represented by the inequality |g – 85| < 10. What is the lowest grade and what is the highest grade in the class? Explain your thinking.Part 2:Suppose the grades in your Language Arts class range from 68 to 94. Represent this information on a number line. Then write a compound inequality to represent the information. Finally, write an absolute value inequality to represent the same information.Discussion, Suggestions, Possible SolutionsPart 1: This absolute value inequality means that g – 85 ≤ 10 and g – 85 ≥ -10.Another way to write this is known as a compound inequality -10 ≤ g – 85 ≤ 10 and may be graphed on a number line.Using the same thinking process as used when solving equations we have-10 ≤ g – 85 ≤ 10+85+85 +8575 ≤g≤ 95? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96Part 2:Suppose the grades in your Language Arts class range from 68 to 94. Represent this information on a number line. Then write a compound inequality to represent the information. Finally, write an absolute value inequality to represent the same information.? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95This means that the lowest score included 68 and the highest score included 94. That is why the points are shaded along with all points in between.One possible answer could be determined by 68 ∨ x∨ 94-81-81-81-13 ∨ x – 81 ∨ 13Course Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 2Georgia Performance Standard(s):??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P2.c Develop and evaluate mathematical arguments and proofs.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8A1.b Simplify and evaluate algebraic expressions.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.Objective(s):??The student compares and orders integers and evaluates expressions containing absolute values. The student will be able to add integers.Instructional Resources:Holt Mathematics 8th grade Course 3 Pgs. 14-21Chapter 1 Resource Book (CRB)One-Stop PlannerActivities:??Read textbook pgs. 14-17.??Complete Think and Discuss, pg. 15.in textbook??Complete Practice and Problem Solving, Problems 1-5, 15-19, and 29-39 on pg. 16.in textbook??Complete Practice A 1-3 CRB, pg. 19.??Complete Reading Strategies 1-3 CRB, pg. 25.??Read textbook pgs. 18-21.??Complete Think and Discuss, pg. 19.in textbook??Complete Practice and Problem Solving, Problems 1-8, 13-20, 31-38, and 49-56 on pgs. 20-21.in textbook??Complete Practice A 1-4 CRB, pg. 27.??Complete Reading Strategies 1-4 CRB, pg. 33.Evaluation:Complete Power Presentations Lesson Quiz 1-3, pg. 17 and 1-4, pg. 21.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 6Georgia Performance Standard(s):??M8A2.b Use the properties of inequality to solve inequalities.??M8A2.c Graph the solution of an inequality on a number line.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems??M8P5.c Use representations to model and interpret physical, social, andmathematicalphenomena.??M8P3.a Organize and consolidate their mathematical thinking through communicationObjective(s):??The student organizes and reviews key concepts and skills presented in Chapter One.??The student assesses mastery of concepts and skills in Chapter OneInstructional Resources:Holt Mathematics Course Two TextbookChapter 1 Resource Book (CRB)One Stop PlannerActivities:??Complete the odd number problems on the Study Guide, pgs. 53-54 in textbook.Have teacher check your work on review. If score is at least 80%, then go on toChapter Test. If score is less than 80%, then teacher will give Reteach worksheets inCRB to cover concepts not understood.??Complete Chapter Test pg.55 in textbook.??Have teacher check your work on test. If score is at least 80%, then go on to next CAP. If score is less than 80%, then teacher will give worksheets in CRB covering concepts still not understood.Evaluation:Complete Chapter Test pg.55 in textbook with 80% accuracyModifications: IDEA Works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 7Georgia Performance Standard(s):??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P2.c Develop and evaluate mathematical arguments and proofs.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P4.a Recognize and use connections among mathematical ideas.??M8P4.b Understand how mathematical ideas interconnect and build on one another toproduce a coherent whole.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.Objective(s):??The student writes rational numbers in equivalent forms. The student will be able tocompare and order positive and negative rational numbers written as fractions, decimals,and integers.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 60-71Chapter 2 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 61.??Read textbook pgs. 60-67??Complete Think and Discuss, pg. 65.in textbook??Complete Practice and Problem Solving, Problems 1-10, 29-38, and 67-79 on pgs. 66-67.in textbook??Complete Practice A 2-1 CRB, pg. 3.??Complete Reading Strategies 2-1 CRB, pg. 10.??Read textbook pgs. 68-71.??Complete Think and Discuss on pg 69.in textbook??Complete Practice and Problem Solving, Problems 1-4, 10-17, 27, and 42-53 on pgs. 70-71.in textbook??Complete Practice A 2-2 CRB, pg. 12.??Complete Reading Strategies 2-2 CRB, pg. 18.Evaluation:Complete Power Presentations Lesson Quiz 2-1, pg. 67 and 2-2, pg. 71.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 11Georgia Performance Standard(s):??M8A1.a Represent a given situation using algebraic expressions or equations in onevariable.??M8A1.c Solve algebraic equations in one variable, including equations involving absolute values.??M8A1.d Interpret solutions in problem contexts.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P1.a Build new mathematical knowledge through problem solving.??M8P1.b Solve problems that arise in mathematics and in other contextsObjective(s):??The student organizes and reviews key concepts and skills presented in Chapter Two.??The student assesses mastery of concepts and skills in Chapter TwoInstructional Resources:Holt Mathematics Course Two TextbookChapter 2 Resource Book (CRB)One Stop PlannerActivities:??Complete the odd number problems on the Study Guide, pgs. 53-54 in textbook.Have teacher check your work on review. If score is at least 80%, then go on toChapter Test. If score is less than 80%, then teacher will give Reteach worksheets inCRB to cover concepts not understood.??Complete Chapter Test pg.55 in textbook.??Have teacher check your work on test. If score is at least 80%, then go on to next CAP.If score is less than 80%, then teacher will give worksheets in CRB covering conceptsstill not understood.Evaluation:Complete Chapter Test pg.55 in textbook with 80% accuracy.Modifications: IDEA Works CDUnit: Graphs, Functions & SequencesGeorgia Performance Standards:M8A1 Students will use algebra to represent, analyze, and solve problems.b. Simplify and evaluate algebraic expressions.d. Solve equations involving several variables for one variable in terms of the others.M8P1 Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena.M8P2 Students will reason and evaluate mathematical arguments.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.M8A4 Students will graph and analyze graphs of linear equations and inequalities.c. Graph equations of the form y = mx + b.e.Graph the solution set of a linear inequality, identifying whether the solution set is an open or a closed half-plane. M8P4 Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.M8A3 Students will understand relations and linear functions.b. Recognize a function as a correspondence between inputs and outputs where the output for each input must be unique.c. Distinguish between relations that are functions and those that are not functions.d. Recognize functions in a variety of representations and a variety of contexts.e. Use tables to describe sequences recursively and with a formula in closed form.i. Identify relations and functions as linear or nonlinear.Selected Terms and Symbols:Additive Inverse: The sum of a number and its additive inverse is zero. Also called the opposite of a number. Example: 5 and -5 are additive inverses of each other. Exponent: The number of times a base is used as a factor of repeated multiplication. Exponential Notation: See Scientific Notation below.Hypotenuse: ?The side opposite to the right angle in a right triangle.Irrational: A real number whose decimal form is non-terminating and non-repeating that cannot be written as the ratio of two integers.Leg: Either of the two shorter sides of a right triangle. The two legs form the right angle of the triangle.Pythagorean Theorem: A theorem that relates the lengths of the sides of a right triangle: The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse.Radical: A symbol that is used to indicate square roots.Rational: A number that can be written as the ratio of two integers with a nonzero denominator.Scientific Notation: A representation of real numbers as the product of a number between 1 and 10 and a power of 10, used primarily for very large or very small numbers.Significant Digits: A way of describing how precisely a number is written.Square root: One of two equal factors of a nonnegative number. For example, 5 is a square root of 25 because 5?5 = 25. Another square root of 25 is -5 because (-5)?(-5) = 25. The +5 is called the principle square root of 25 and is always assumed when the radical symbol is used.Teacher’s Place:Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.1. Explain the activity (activity requirements)2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at . Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard. 7. Discuss answers with the students using the following questioning techniques as applicable: Questioning Techniques:Memory QuestionsSignal words: who, what, when, where?Cognitive operations: naming, defining, identifying, designatingConvergent Thinking QuestionsSignal words: who, what, when, where?Cognitive operations: explaining, stating relationships, comparing andcontrasting Divergent Thinking QuestionsSignal words: imagine, suppose, predict, if/thenCognitive operations: predicting, hypothesizing, inferring, reconstructingEvaluative Thinking QuestionsSignal words: defend, judge, justify (what do you think)?Cognitive operations: valuing, judging, defending, justifying8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter) 10. At the end of the *whole group learning session, students will transition into independent CAP assignments. *The phrase, “whole group learning session” is utilized “rather than, the end of the activity” because all of the activities may not be completed in one day. Task: 1Resources:Stopwatches or clocks with minute hands Large grid paper or large grid transparencies Activity Part 1:In the sixth and seventh grade, you studied proportional relationships. Do you think that the number of heartbeats you can count is proportional to the number of seconds that you check your pulse? Explain why or why not.Part 2:Work with your partner to take measurements and test your conjecture. One of you will be the timer and the other will count their own number of heart beats per period of time. First, count and record the number of beats in 10 seconds, and repeat the experiment counting the number of beats in 20 seconds and 40 seconds. Heartbeat Data for ___________________________Number of SecondsNumber of Heartbeats102040Predict how many times your heart would beat in 25 seconds, in 60 seconds, and in 120 seconds. Explain how you made your predictions.Part 3:After gathering this data, change jobs. The person who kept time now checks his/her pulse rate for 10 seconds, 20 seconds, and 40 seconds.Heartbeat Data for ___________________________Number of SecondsNumber of Heartbeats102040Predict how many times your heart would beat in 25 seconds, in 60 seconds, and in 120 seconds. Explain how you made your predictions.Part 4:Develop a function rule (equation) to represent your pulse rate. How does your function rule compare with your partner’s function rule? Explain to your partner why your function rule is valid. Draw a graph (scatter plot) to represent your pulse rate. How does your graph compare with your partner’s graph? Explain to your partner why your graph is valid. Task: 2Resources: calculatorGraphing calculator presentation software Activity Part I:Two students at Sumter YDC Matt and Tyler have been studying graphing linear equations. Matt challenged Tyler to a race to see who could graph y = 3x + 5 in the least amount of time. Matt is going to graph the equation by hand, and Tyler is going to use the graphing calculator. Matt says he can graph the equation in less than ten seconds, in less time than Tyler can enter the equation in the calculator and press the “Graph” key. Explain how you think Matt intends to graph the equation. Illustrate his method on two more linear equations that you make up to give to Matt and Tyler to use for their race.Part II:Who do you think will win the race if Matt and Tyler are given the equation 3x + 4y = 12 to graph? Why? How do you predict Matt will graph the equation?Discussion, Suggestions, Possible SolutionsRegardless of whether students have used graphing calculators to graph linear equations, they should be able to explain how an equation in slope-intercept form may be graphed quickly by first locating the y-intercept and then using the rise and run to locate another point on the line. In the example given, the student should explain that Matt will locate the y-intercept (0, 5), and then go to the right one unit and up three units or up three units and over to the right one unit to locate another point on the line. These two points are then used to sketch the line. Students might also respond that Matt could quickly get a table of values. Using x = 0, Matt has the y-intercept of 5. Using x = 1, Matt obtains the point (1, 8). Task: 3Resources:Graph paperStopwatch or classroom timerUncooked spaghettiActivity Part : Data CollectionWork with your partner to take measurements and test your conjecture. One of you will be the timer and the other will count their own number of heart beats per period of time. First, count and record the number of beats in 10 seconds, and repeat the experiment counting the number of beats in 20 seconds, 30 seconds, 40 seconds, 50 seconds, and 60 seconds. Heartbeat Data for ___________________________Time (sec)Number of Heartbeats102030405060708090100110120Part : Data RepresentationGraph your data.Part : Fitting a Line to DataMethod 1: EstimationUse a piece of raw spaghetti to visualize and estimate a line of best fit. Choose two points on the line.Use your two points to write an equation for a line of best fit.Method 2: Lower and Upper QuartilesFind the five-number summary for your x-values (time).Find the five-number summary for your y-values (number of heartbeats).Record your lower and upper quartiles for your x-values and your y-values.Time (sec)Number of HeartbeatsLower QuartileUpper QuartileDraw a horizontal box-and-whisker plot using the five-number summary for your x-values (time.) Plot your box-and whisker plot under the x-axis on your graph. Draw a vertical box-and-whisker plot using the five-number summary for your y-values (number of heartbeats). Plot your box-and whisker plot next to the y-axis on your graph. Draw vertical lines from the lower and upper quartile values on the x-axis box-and whisker plot.Draw horizontal lines from the lower and upper quartile values on the y-axis box-and whisker plot.Find the coordinates of the vertices of the rectangle formed by the intersection of the vertical and horizontal lines. We will refer to these vertices as quartile points. Note: Use only the quartile points (vertices) that follow the direction of the data.Do the quartile points have to be actual data points? Why or why not?Draw a line connecting the two quartile points.Write an equation for this line. This is a line of best fit for your data.Use one of your equations to predict how many times your heart would beat in 25 seconds, in 240 seconds, and in 3 minutes. Explain how you made your predictions.ExampleNo. of heartbeatsTime (sec)Part : Data CollectionAfter gathering this data, change jobs. The person who kept time now checks his/her pulse rate and repeat part : Fitting a Line to Data.Part : Analysis How does your line of best fit compare with your partner’s line of best fit? Explain to your partner why your line of best fit is valid. Discussion, Suggestions, Possible SolutionsStudents could come up with possible explanations for variations in heart-rate, such as the effect of exercise, health conditions (thyroid problems, e.g.), etc. Students should recognize that the variable representing time represents seconds and students will need to convert three minutes to seconds before using substitution.Task: 4Resources: Last summer one of your classmate Brian went to the mountains and panned for gold. Although they didn’t find any gold, they did find some pyrite (fool’s gold) and many other kinds of minerals. Brian’s friend, who happens to be a geologist, took several of the samples and grouped them together. The he told Brian that all of those minerals were the same. Brian had a hard time believing him, because they are many different colors. He suggested Brian analyze some data about the specimens. Brian carefully weighed each specimen in grams (g) and found the volume of each specimen in milliliters (ml).Brian has asked his math class at Savannah RYDC to help him analyze the data. Write your analysis of his data given below:Specimen NumberMass or weight (g)Volume (ml)117721043135416657362410752 Discussion, Suggestions, Possible SolutionsStudents could graph the volume as the independent variable (x) and mass as the dependent variable (y). Their graphs could be produced either by hand on graph paper or as a STAT PLOT on a graphing calculator. Students could choose two points that are on their line of best fit that they determined by eye-balling the data, using the lower and upper quartiles, or they could use the linear regression feature of the calculator to obtain the regression equation y = 2.41x + .40. Students might divide the mass by the volume by hand for each specimen and then find the average of this value. They could also have the calculator divide the values and find the average. The average value (2.49) is close to the coefficient of the x term in the regression equation. As students explore the meaning of this slope in this problem context, they should come to understand that it means every time the volume goes up by one ml the mass goes up by approximately 2.4 or 2.5 grams. This rate of mass in grams per milliliter of volume is the density of the mineral.Research could be done to find lists of densities for particular minerals. While earth science references will list the density (also called “specific gravity”) of quartz as 2.6, the samples used for the data above were quartz. This could lead to a discussion of the precision and accuracy of measurements, as well as to a discussion of impurities and other factors that could influence the results.The relationship between the mass and volume of the specimens might be described by some students as a proportional relationship. This could lead to the conclusion that a theoretical model for this relationship might be written as y = 2.6x. In other words, a hypothetical sample with a volume of zero would have a mass of zero, so the y-intercept should be zero.As an extension, ask students where data points would be for samples of minerals that have a density greater than 2.6. These would be in the half-plane above the regression line for the quartz samples. This can provide a transition to graphing inequalities.Task: 5Resources: paperGraphing calculatorColored pencilsActivity In math class one day Mrs. Smith conducted an experiment. In the first part of the experiment, her students wrote T’s on a sheet of paper and counted how many they were able to make in one minute. Then Mrs. Smith told them to change their pencils to their other hands, and they repeated the experiment. Mrs. Smith then gathered from each student the information about how many T’s were made with his/her right hand and how many were made with his/her left hand in one minute. Using the data for the right hand as the x values and the data for the left hand as the y values, explain what you expect a graph of this data would look like.In this context, a point on the line y = x would represent data for an ambidextrous person, a person that works equally as well with their left hand as they do their right hand. Left-handed persons would be able to make more T’s with their left hands than with their right hands in one minute, so their data would lie above the line y = x. Since most students are right-handed, most of the points would be graphed below the line y = x.Questions to ask students:What is an ambidextrous person?Can you think of an equation that represents data for an ambidextrous person?Why does y = x represent data for an ambidextrous person? Ambidextrous PeopleyLeft HandxRight HandRight Handed People Left Handed PeopleyyxxLeft HandRight HandRight HandDiscussion, Suggestions, Possible SolutionsUsing a graphing calculator is an efficient way to capture the picture of the actual data. Enter the x values in List 1 (L1), the y values in List 2 (L2), and enter y1 = x. Turn the statplot on to view the data in Lists 1 and 2, select Zoom Stat to set the viewing window appropriately, and select y1 to see the boundary line y = x along with the individual data points. Course Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 12Georgia Performance Standard(s):??M8A1.b Simplify and evaluate algebraic expressions.??M8A1.d Interpret solutions in problem contexts.??M8P1.a Build new mathematical knowledge through problem solving.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8P2.c Develop and evaluate mathematical arguments and proofs.??M8P2.d Select and use various types of reasoning and methods of proof.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8A4.c Graph equations of the form y = mx + b.??M8P4.a Recognize and use connections among mathematical ideas.??M8P4.b Understand how mathematical ideas interconnect and build on one another toproduce a coherent whole.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8P3.a Organize and consolidate their mathematical thinking through communication.Objective(s):??The student writes solutions of equations in two variables as ordered pairs. The student will be able to graph points and lines on the coordinate plane.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 115-126Chapter 3 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 115.??Read textbook pgs. 115-121.??Complete Think and Discuss, pg. 119 and 123.??Complete Practice and Problem Solving, Problems 1-4, 8-11, 23, 24, and 36-44 on pgs. 120-121.in textbook??Complete Practice A 3-1 CRB, pg. 3.??Complete Reading Strategies 3-1 CRB, pg. 9.??Read textbook pgs. 122-125.??Complete Think and Discuss, pg. 126.in textbook??Complete Practice and Problem Solving, Problems 1-6, 17-22, and 38-47 on pgs. 124-125.in textbook??Complete Practice A 3-2 CRB, pg. 11.??Complete Reading Strategies 3-2 CRB, pg. 17.Evaluation:Complete Power Presentations Lesson Quiz 3-1, pg. 121 and 3-2, pg. 125.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 15Georgia Performance Standard(s):??M8A1.b Simplify and evaluate algebraic expressions.??M8A1.d Interpret solutions in problem contexts.??M8A3.i Translate among verbal, tabular, graphic, and algebraic representations offunctions.??M8A4.c Graph equations of the form y = mx + b.??M8A4.e Determine the equation of a line given a graph, numerical information that definesthe line, or a context involving a linear relationship.??M8P2.d Select and use various types of reasoning and methods of proof.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8A3.e Use tables to describe sequences recursively and with a formula in closed form.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P4.c Recognize and apply mathematics in contexts outside of mathematicsObjective(s):??The student organizes and reviews key concepts and skills presented in Chapter Three??The student assesses mastery of concepts and skills in Chapter Three.Instructional Resources:Holt Mathematics Course Two TextbookChapter 3 Resource Book (CRB)One Stop PlannerActivities:??Complete the odd number problems on the Study Guide, pgs. 150-151 in textbook.Have teacher check your work on review. If score is at least 80%, then go on toChapter Test. If score is less than 80%, then teacher will give Reteach worksheets inCRB to cover concepts not understood.??Complete Chapter Test pg.153 in textbook.??Have teacher check your work on test. If score is at least 80%, then go on to next CAP.If score is less than 80%, then teacher will give worksheets in CRB covering conceptsstill not understood.Evaluation:Complete Chapter Test pg.153 in textbook with 80% accuracy.Modifications: IDEA Works CDUnit: Exponents& Roots/ Ratios, Proportions & Similarity/ PercentsGeorgia Performance Standards:M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.a. Find square roots of perfect squares.b. Recognize the (positive) square root of a number as a length of a side of a square with a given area.c. Recognize square roots as points and as lengths on a number line.d. Understand that the square root of 0 is 0 and that every positive number has two square roots that are opposite in sign.e. Recognize and use the radical symbol to denote the positive square root of a positive number.f. Estimate square roots of positive numbers.g. Simplify, add, subtract, multiply, and divide expressions containing square roots.h. Distinguish between rational and irrational numbers.i. Simplify expressions containing integer exponents.j. Express and use numbers in scientific notation.k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.M8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.b. Simplify and evaluate algebraic expressions.c. Solve algebraic equations in one variable, including equations involving absolute values.d.Solve equations involving several variables for one variable in terms of the others.M8P1 Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.M8P2 Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.M8P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely. M8P4 Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics. M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena.M8G2 Students will understand and use the Pythagorean theorem.a. Apply properties of right triangles, including the Pythagorean theorem.b. Recognize and interpret the Pythagorean theorem as a statement about areas of squares on the sides of a right triangle.M8G1 Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.b. Apply properties of angle pairs formed by parallel lines cut by a transversal.d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent. M8A4 Students will graph and analyze graphs of linear equations and inequalities.b. Determine the meaning of the slope and y-intercept in a given situation.M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.i. Simplify expressions containing integer exponents.k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.Selected Terms and Symbols:Absolute Value: The distance a number is from zero on the number line. Examples: |-4| = 4 and |3| = 3Addition Property of Equality: For real numbers a, b, and c, if a = b, then a + c = b + c. In other words, adding the same number to each side of an equation produces an equivalent equation.Additive Inverse: Two numbers that when added together equal 0. Example, 3.2 and -3.2AlgebraicExpression: A mathematical phrase involving at least one variable. Expressions can contain numbers and operation symbols.Equation: A mathematical sentence that contains an equals sign.Evaluate an Algebraic Expression: To perform operations to obtain a single number or value.Inequality: A mathematical sentence that contains the symbols >, <, ≥, or ≤.Inverse Operation: Pairs of operations that undo each other. Examples:Addition and subtraction are inverse operations and multiplication anddivision are inverse operations.Like Terms: Monomials that have the same variable raised to the samepower. In other words, only coefficients of terms can be different.Linear Equation in One Variable: an equation that can be written in theform ax + b = c where a, b, and c are real numbers and a ≠ 0Multiplication Property of Equality: For real numbers a, b, and c (c ≠ 0), if a + b, then ac = bc. In other words, multiplying both sides of an equation by the same number produces an equivalent expression. Multiplicative Inverses: Two numbers that when multiplied together equal 1. Example: 4 and ?. Solution: the value or values of a variable that make an equation a true statementSolve: Identify the value that when substituted for the variable makes the equation a true statement.Variable: A letter or symbol used to represent a number.Teacher’s Place:Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.1. Explain the activity (activity requirements)2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at 3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard. 7. Discuss answers with the students using the following questioning techniques as applicable: Questioning Techniques:Memory QuestionsSignal words: who, what, when, where?Cognitive operations: naming, defining, identifying, designatingConvergent Thinking QuestionsSignal words: who, what, when, where?Cognitive operations: explaining, stating relationships, comparing andcontrasting Divergent Thinking QuestionsSignal words: imagine, suppose, predict, if/thenCognitive operations: predicting, hypothesizing, inferring, reconstructingEvaluative Thinking QuestionsSignal words: defend, judge, justify (what do you think)?Cognitive operations: valuing, judging, defending, justifying8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter) 10. At the end of the *whole group learning session, students will transition into independent CAP assignments. *The phrase, “whole group learning session” is utilized “rather than, the end of the activity” because all of the activities may not be completed in one day. Task: 1 Resources: Part 1:Repeatedly fold one piece of paper in half, recording the number of folds and the resulting number of layers of paper. Assuming that you could continue the pattern, how many layers of paper would there be for 10 folds, 100 folds, n folds? How do you know?Part 2:Explain the process that is used to generate the following patterns? Cup, pint, quart, half gallon, and gallon, and so on.Penny, dime, dollar, ten dollars, one hundred dollars, and so on.Discussion, Suggestions, Possible SolutionsPart 1Number of folds12345…10…nNumber of layers of paper2481632…1024…2nNumber of layers of paper written using integer exponents2122232425…210…2nStudents should see that each fold resulted in twice as many layers of paper as the previous fold.Part 2CupPintQuartHalf gallonGallon1 cup2 cups4 cups8 cups16 cups2021222324In terms of cups, 1 cup, 2 cups, 4 cups, 8 cups, 16 cups. Each measure is twice the previous one.PennyDimeDollarTen dollarsOne hundred dollars0.010.101.0010.00100.0010-210-1100101102In the sequence of money, each amount is 10 times the previous one, giving 1 penny, ten pennies, 100 pennies, and so on.Task: 2Resources: Vehicle Viewing, Inc. and Traveling TV Co. have each designed new small HD flat-screen square-shaped televisions to be used in automobiles. They feel that these will be very popular because of the desire for parents to entertain their children during the many hours spent in traffic each day. Television size is measured along the diagonal. The diagonal in the Vehicle Viewing, Inc. television is inches and the Traveling TV Co. television has a diagonal of 8.5 inches.You are the owner of a packaging company and both Vehicle Viewing, Inc. and Traveling TV Co. have hired you to package their new televisions.Who has the larger television? How do you know?To protect the screen, you need to place a protective foam sheet between the screen and the box. Find the area of each television screen so that you know how much sheeting you will need to order. Verify your results. In order to prevent breakage, you will need to put some foam ribbon around the sides of the television. Exactly how much foam ribbon will need to be used for each television? Justify your answer.Discussion, Suggestions, Possible Solutionsa.Students will need to determine which of the two diagonals, one expressed as a rational number and the other as an irrational number, is greater. Due to the fact that and , students should understand that is between 8 and 9. However, 8.5 is also between 8 and 9. Because (8.5)2 = 72.25, . Because is more than , Vehicle Viewing, Inc.’s television must be larger than Traveling TV Co.’s television.b.Students will first need to determine the length of a side for each television. Since the figures are squares, each of the sides has the same length. The given diagonal cuts the square into two equal right triangles; therefore, the sides of the square are the legs of the right triangle. By using the Pythagorean Theorem, and setting both legs equal causes a2 + b2 = c2 to become a2 + a2 = c2 or 2a2 = c2. For Vehicle Viewing, Inc., the length of each side is aaTo determine the amount of foam sheeting Vehicle Viewing, Inc. will need to make the protective covering for the screen, we need to remember that the area of the television screen would be A = s2 where s = the length of the side of the television. Therefore, each of the Vehicle Viewing, Inc. televisions would need 37.5 in2 of foam sheeting to protect the screen.To find the length of each side of Traveling TV Co.’s television use the same thinking as shown with the Vehicle Viewing, Inc. shown earlier.aaThe area of the screen of the Traveling TV Co. televisions would also be A = s2. Each Traveling TV Co. television would need in2 of foam sheeting to protect the screen.c.The exact perimeter or amount of foam ribbon needed for packing each of Vehicle Viewing, Inc.’s televisions is The exact amount of foam ribbon needed to pack Traveling TV Co.’s televisions is Students may realize that the EXACT answer for Vehicle Viewing is not practical for the actual people cutting the foam ribbon. However, they should understand that because the result is not a rational number, the practical results would be approximations instead of exact. Students should understand that the irrational numbers represent exact lengths.Students may find the approximations by using a calculator. Vehicle Viewing, Inc.’s televisions will need about 24.5 inches of foam ribbon to protect the perimeter of each television and Traveling TV Co’s television would need about 24 inches of foam ribbon to protect the perimeter of each television.Task: 3 Resources:\TI-83Activity This activity will help you to learn how to represent very large and very small numbers.Part 1: Very Small Numbers (negative exponents)Complete the following table. What patterns do you see?106 = 105 = 104 = 103 = 102 = 101 = 100 = 10-1 = 10-2 = 10-3 = USING THE TI-83: Exploring Powers of 10It may be appropriate to use a calculator or computer when data may contain numbers that are difficult to work with using paper and pencil. Exploring very large and very small numbers can be done on the TI-83 or TI-84 graphing calculator with the following instructions.Part 1:ONTurn on the calculator .ENTER^Enter the number 10, press the key, enter the number 6, and press . Continue step 2 for the positive exponents and zero.(?)^ ENTEREnter the number 10, press the key, press the negation key, enter the number 1, and press . Continue step 4 for the negative exponents.Part 2: Very Large Numbers (positive exponents)Sometimes it can be cumbersome to say and write numbers in their common form (for example trying to display numbers on a calculator with limited screen space). Another option is to use exponential notation and the base-ten place-value system. Using a calculator (with exponential capabilities), complete each of the following:Explore 10N for various values of N. What patterns do you notice?Enter 45 followed by a string of zeros. How many will your calculator permit? What happens when you press enter? What does 4.5E10 mean? What about 2.3E4? Can you enter this another way?Try sums like (4.5 x 10N) + (2 x 10K) for different values of N and K. Describe any patterns that you may notice.Try products like (4.5 x 10N) ? (2 x 10K). Describe any patterns that you may notice.Try quotients like (4.5 x 10N) ÷ (2 x 10). Describe any patterns that you may notice.Discussion, Suggestions, Possible SolutionsPart 1.106 = 1000000105 = 100000104 = 10000103 = 1000102 = 100101 = 10100 = 110-1 = 0.1 = QUOTE 10-2 = 0.01 = QUOTE 10-3 = 0.001 = QUOTE Part 21.The value of N determines the placement of the decimal point. As N increases by 1, the value increases by 10; this moves the decimal point to the right one place. Students should see that multiplying by powers of 10 results in movement of the decimal point. The decimal point moves one place to the right for each positive power of 10 and one place to the left for each negative power of 10. 2.The number of digits permitted will depend on the calculator. When the screen is full, the calculator will automatically convert to the shorthand notation, 4.5E10 which represents 4.5 x 1010 = 45000000000. Note: 2.3E4 represents 2.3 x 104 = 23000 Answers may include 2.3E4 = 23E3 = 230E2 =2300E1 = 230003.Answers will vary. Students should discover that the only time the answer written in scientific notation could be 6.5 x 10T is if N=K=T. In most cases, there is no major advantage for using scientific notation when adding or subtracting unless the powers of 10 are the same or at least close.4.9 x 10N+K. Students should be encouraged to look at positive, negative, and zero values of N and K.5.2.25 x 10N-K. Students should be encouraged to look at positive, negative, and zero values of N and K.When students use scientific or graphing calculators to display numbers with more digits than the display will hold, the calculator will display the number using scientific notation. Scientific notation is a decimal number between 1 and 10 times a power of 10. Have students move from the calculator to paper and pencil converting from large numbers to scientific notation and vice versa. Also, have students follow up addition/subtraction and multiplication/division in scientific notation mentally. Notice the advantages of scientific notation for multiplication and division. Here the significant digits can be multiplied mentally (4.5 x 2) and the exponents added to produce 9 x 10N+K. Task: 4Resources: paperActivity The director of your center has requested that your 8th grade math class complete an end of the year project to show what you have learned this school year. After, talking it over with your classmates you all have decided that you will show the director what you have learned about the Pythagorean Theorem, rational and irrational numbers, and line segments. Therefore, in this task, you will construct a number line with several rational and irrational numbers plotted and labeled. Start by constructing a right triangle with legs of one unit. Use the Pythagorean Theorem to compute the length of the hypotenuse. Then copy the segment forming the hypotenuse to a line and mark one left endpoint of the segment as 0 and the other endpoint with the irrational number it represents. Construct other right triangles with two sides (either the two legs or a leg and a hypotenuse) that have lengths that are multiples of the unit you used in the first triangle. Then transfer the lengths of each hypotenuse to a common number line, and label the point that it represents. After you have constructed several irrational lengths, list the irrational numbers in order from smallest to largest. Discussion, Suggestions, Possible SolutionsSuggestionsIt is recommended that this task be done on grid paper to help students maintain the same unit of measure for each side of each triangle as well as for the number line.Students will need to use a compass and straightedge to copy the segments representing the rational and irrational lengths formed by the hypotenuse of each triangle to the number line. Be sure that students use 0 as one endpoint and label the other endpoint as the distance represented.SolutionUsing different combinations of sides from 1 unit to 4 units, students can construct a variety of hypotenuse lengths including: . Also, students may construct some special triangles called Pythagorean triples where all sides of the right triangle have integer lengths (as an example, see the 3-4-5 triangle below). Can they find others?Using unit lengths for the sides, students will get some of the following:Students should find that by plotting the irrational numbers on the number line, they are automatically ordered from least to greatest. When irrational numbers are ordered from least to greatest, they get numbers such as with rational numbers like Another method for constructing a variety of irrational lengths is to recursively construct another right triangle from the first one with the hypotenuse as one of the legs and the other leg a unit of one. This method will construct irrational and rational numbers in an increasing order. Leg lengthsPythagorean TheoremLeg lengthsPythagorean TheoremLeg lengthsLeg lengthsPythagorean TheoremTo visualize the location of these irrational numbers on the number line, students could use a compass to measure the length of the hypotenuse and swing an arc that will cross the number line. Course Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 16Georgia Performance Standard(s):??M8N1.i Simplify expressions containing integer exponents.??M8A1.b Simplify and evaluate algebraic expressions.??M8P1.c Apply and adapt a variety of appropriate strategies to solve problems.??M8P2.c Develop and evaluate mathematical arguments and proofs.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P1.a Build new mathematical knowledge through problem solving.??M8P1.b Solve problems that arise in mathematics and in other contexts.Objective(s):??The student evaluates expressions with exponents. The student will be able to evaluateexpressions with negative exponents and evaluate the zero exponent.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 158-168Chapter 4 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 159.??Read textbook pgs. 158-165.??Complete Think and Discuss, pg. 163.in textbook??Complete Practice and Problem Solving, Problems 1-4, 15-20, and 51-61 on pgs. 164-165. in textbook??Complete Practice A 4-1 CRB, pg. 3.??Complete Reading Strategies 4-1 CRB, pg. 9.??Read textbook pgs. 166-169.??Complete Think and Discuss, pg. 167.in textbook??Complete Practice and Problem Solving, Problems 1-4, 13-16, 44, and 49-58 on pgs. 168- 169.in textbook??Complete Practice A 4-2 CRB, pg. 11.??Complete Reading Strategies 4-2 CRB, pg. 17.Evaluation:Complete Power Presentations Lesson Quiz 4-1, pg. 165 and 4-2, pg. 169.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 20Georgia Performance Standard(s):??M8N1.h Distinguish between rational and irrational numbers.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8G2.a Apply properties of right triangles, including the Pythagorean Theorem.??M8G2.b Recognize and interpret the Pythagorean Theorem as a statement about areasof squares on the sides of a right triangle.??M8P4.a Recognize and use connections among mathematical ideas.??M8P4.b Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.??M8P2.c Develop and evaluate mathematical arguments and proofs.??M8P4.c Recognize and apply mathematics in contexts outside of mathematicsObjective(s):??The student organizes and reviews key concepts and skills presented in Chapter Four??The student assesses mastery of concepts and skills in Chapter Four.Instructional Resources:Holt Mathematics Course Two TextbookChapter 4 Resource Book (CRB)One Stop PlannerActivities:??Complete the odd number problems on the Study Guide, pgs. 204-205 in textbook.Have teacher check your work on review. If score is at least 80%, then go on toChapter Test. If score is less than 80%, then teacher will give Reteach worksheets inCRB to cover concepts not understood.??Complete Chapter Test pg.207 in textbook.??Have teacher check your work on test. If score is at least 80%, then go on to nextCAP. If score is less than 80%, then teacher will give worksheets in CRB coveringconcepts still not understood.Evaluation:Complete Chapter Test pg.207 in textbook with 80% accuracy.Modifications: IDEA Works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 21Georgia Performance Standard(s):??M8P4.a Recognize and use connections among mathematical ideas.??M8P4.b Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P1.b Solve problems that arise in mathematics and in other contexts.Objective(s):??The student finds equivalent ratios to create proportions. The student will be able to work with rates and ratios.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 213-223Chapter 5 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 213.??Read textbook pgs. 213-219.??Complete Think and Discuss, pg. 217.in textbook??Complete Practice and Problem Solving, Problems 1-5, 11-15, and 37-47 on pgs. 218-219.in textbook??Complete Practice A 5-1 CRB, pg. 3.??Complete Reading Strategies 5-1 CRB, pg. 9.??Read textbook pgs. 220-223.??Complete Think and Discuss, pg. 221.in textbook??Complete Practice and Problem Solving, Problems 1, 8, 16-19, 28, and 32-41 on pgs.222-223.in textbook??Complete Practice A 5-2 CRB, pg. 11.??Complete Reading Strategies 5-2 CRB, pg. 17.Evaluation:Complete Power Presentations Lesson Quiz 5-1, pg. 219 and 5-2, pg. 223.Modifications: IDEA Works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 26Georgia Performance Standard(s):??M8P4.a Recognize and use connections among mathematical ideas.??M8P4.b Understand how mathematical ideas interconnect and build on one another toproduce a coherent whole.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8P2.d Select and use various types of reasoning and methods of proof.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P1.c Apply and adapt a variety of appropriate strategies to solve problems.??M8P1.d Monitor and reflect on the process of mathematical problem solving.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.Objective(s):??The student compares and orders decimals, fractions, and percents. The student will beable to estimate percents.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 271-282Chapter 6 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 271.??Read textbook pgs. 271-277.??Complete Think and Discuss, pg. 275.in textbook??Complete Practice and Problem Solving, Problems 1-4, 12-15, 23-25, and 32-40 on pgs. 276-277. in textbook??Complete Practice A 6-1 CRB, pg. 3.??Complete Reading Strategies 6-1 CRB, pg. 9.??Read textbook pgs.278-282.??Complete Think and Discuss, pg. 280.in textbook??Complete Practice and Problem Solving, Problems 1-8, 11-18, 21-30, and 47-63 on pgs. 280-282.in textbook??Complete Practice A 6-2 CRB, pg. 11.??Complete Reading Strategies 6-2 CRB, pg. 18.Evaluation:Complete Power Presentations Lesson Quiz 6-1, pg. 277 and 6-2, pg. 282.Modifications:Performance Tasks: IDEA works CDUnit: Foundation of Geometry & Perimeter, Area & Volume Georgia Performance Standards:M8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.b. Simplify and evaluate algebraic expressions.c. Solve algebraic equations in one variable, including equations involving absolute values.d.Solve equations involving several variables for one variable in terms of the others.M8P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely. M8P1 Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems. M8P2 Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.M8G1 Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.b. Apply properties of angle pairs formed by parallel lines cut by a transversal.d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.M8P4 Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.M8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.b. Simplify and evaluate algebraic expressions.c. Solve algebraic equations in one variable, including equations involving absolute values.d.Solve equations involving several variables for one variable in terms of the others.M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena.M8G2 Students will understand and use the Pythagorean theorem.a. Apply properties of right triangles, including the Pythagorean theorem.M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.Teacher’s Place:Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.1. Explain the activity (activity requirements)2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at . Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard. 7. Discuss answers with the students using the following questioning techniques as applicable: Questioning Techniques:Memory QuestionsSignal words: who, what, when, where?Cognitive operations: naming, defining, identifying, designatingConvergent Thinking QuestionsSignal words: who, what, when, where?Cognitive operations: explaining, stating relationships, comparing andcontrasting Divergent Thinking QuestionsSignal words: imagine, suppose, predict, if/thenCognitive operations: predicting, hypothesizing, inferring, reconstructingEvaluative Thinking QuestionsSignal words: defend, judge, justify (what do you think)?Cognitive operations: valuing, judging, defending, justifying8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter) 10. At the end of the *whole group learning session, students will transition into independent CAP assignments. *The phrase, “whole group learning session” is utilized “rather than, the end of the activity” because all of the activities may not be completed in one day. Task: 1Resources: Paul, Jane, Justin, Sarah, and Opal were finished with lunch and began playing with drink straws. Each one was making a line design using either 3 or 4 straws.They had just come from math class where they had been studying special angles.Paul pulled his pencil out of his bookbag and labeled some of the angles and lines. He then challenged himself and the others to find all the labeled angle measurements and to determine whether the lines that appear to be parallel really could be parallel.Below are the designs the five students created.o2C AmzomC B35o(2x+10)oxo 40onxo(3x+30)on yo zo70o(5x-20)oPaul’s straw designJane’s straw designJustin’s straw design140o3xx+20xo2x+1070onBCSarah’s straw designOpal’s straw design?Find all of the labeled angle measurements. Determine whether the lines that appear to be parallel really could be parallel.Determine if ∨ABC really could be a right angle.Explain the reasoning for your results.Discussion, Suggestions, Possible SolutionsPaul’s straw design:This relies entirely on vertical angles and linear pairs. m∠x ? m∠z ? 140? ; m∠y ? 40? . Use the linearpair relationship for the angles involving C to conclude m∠A ? m∠C ? 60? ; and m∠B ? 120? . Therefore, lines m and n are not parallel because corresponding angles do not have the same measure. NOTE: The argument could also be made because neither alternate interior nor alternate exterior angles are congruent. Also, neither same-side interior nor exterior angles are supplementary.Jane’s straw design:Use corresponding, same-side interior, and vertical angles; linear pairs; and the sum of the angles in a triangle.m∠x ? 145? , m∠y ? 75? and m∠z ? 110? .Justin’s straw design:Vertical angles give 5x – 20 = 3x + 30-3x-3x2x – 20 = 30+ 20 +202x = 502x ÷2 = 50 ÷ 2 x = 25If the lines are parallel, then the same-side interior angles must have a sum of 180o. By substitution, (2x + 10) + (3x + 30) = (2?25 + 10) + (3?25 + 30)= (50 + 10) + (75 + 30)= 60 + 105≠ 180The sum of the algebraic expressions must equal 90o, so (3x) +(x + 20) + (2x + 10) = 90. Simplified, this give 6x +30 = 90 and x = 10.By substitution, we find 3x = 3(10) or 30o; x + 20 = 10 + 20 or 30o; and 2x + 10 = 20 + 10 or 30o.Task: 2Resources: Part 1:Your best friend’s newest blog entry on MySpace reads:Last night was the worst night ever! I was playing ball in the street with my buds when, yes, you guessed it, I broke my neighbor’s front window. Every piece of glass in the window broke! Man, my Mom was soooooooooooo mad at me! My neighbor was cool, but Mom is making me replace the window. Bummer!It is a Tudor-style house with windows that look like the picture below. I called the Clearview Window Company to place an order. What was really weird was that the only measurements that the guy wanted were ∨BAD (60o), ∨BCE (60o), and AG (28 inches). I told him it was a standard rectangular window and that I had measured everything, but he told me not to worry because he could figure out the other measurements. It is going to cost me $20 per square foot, so I need to figure out how to make some money real quick. How did the window guy know all of the other measurements and how much is this going to cost me?Because you are such a good best friend, you are going to reply to the blog by emailing the answers to the questions on the blog along with detailed explanations about how to find every angle measurement and the lengths of each edge of the glass pieces. You will also explain how to figure out the amount of money he will need.Part 2:(Two weeks later)You just received a text message from your best friend and were told that the order of glass had been delivered to the house by Package Express. Unfortunately, one of the pieces was broken upon arrival and needed to be reordered by Clearview Window Company. Because you are very curious, you think it would be a good idea to determine the probability of each piece of glass being the one broken. Write another email to your friend that explains the probabilities and how you determined them.Knowing that m∨BAD = 60o and m∨BCE = 60o because they were given in the blog, it is possible to determine the measures of every other angle of every piece of glass in the window using thinking similar to the following comments.m∨AGC = 60o ? 180o in ?ACG.m∨ADB = 60o ? corresponds to ∨DGE. m∨FDG = 60o ? vertical to ∨ADB. m∨ABD = 60o ? 180o in ?ABD.m∨DBE = 60o ? corresponds to ∨FDG.m∨EBC = 60o ? supplementary to ∨ABE.m∨BEC = 60o ? 180o in ?BCE.Task: 3Resources: The siding being installed on a particular house is available in 12 foot lengths and each is 1 foot high. Your supervisor had no problems installing the siding around the sides of the house, but ran into difficulty when he had to cut the siding for the top of the side of the house under the roof as shown in the picture below. He knows the angle that he has to cut the corners of the boards, but can’t remember how to determine the lengths of the boards. He did remember that you are good at mathematics, so he has asked you for help.As you can see, there are 7 rows of angled boards, each one foot high. If the house is 35 feet wide at this point, determine The length of each side of each board,The minimum length of siding required to cover this portion of the house,The total number of 12-foot boards required,The total number of cuts required to cover this part of the house,Draw a picture showing the locations of the boards in their assembled positions, andState how much siding is left over from your answer to the previous question if none of the siding is lost when it is sawed.Course Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 31Georgia Performance Standard(s):??M8A1.a Represent a given situation using algebraic expressions or equations in onevariable.??M8A1.c Solve algebraic equations in one variable, including equations involving absolutevalues.??M8A1.d Interpret solutions in problem contexts.??M8P3.d Use the language of mathematics to express mathematical ideas precisely.??M8P1.a Build new mathematical knowledge through problem solving.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P2.a Recognize reasoning and proof as fundamental aspects of mathematics.??M8P2.c Develop and evaluate mathematical arguments and proofs.??M8P2.d Select and use various types of reasoning and methods of proof.??M8G1.a Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.??M8G1.b Apply properties of angle pairs formed by parallel lines cut by a transversal.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.Objective(s):??The student classifies and names figures. The student will be able to identify parallel andperpendicular lines and the angles formed by a transversal.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 321-335Chapter 7 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 321.??Read textbook pgs. 321-329??Complete Think and Discuss, pg. 326.in textbook??Complete Practice and Problem Solving, Problems 1-5, 13-17, 25, 26, and 40-46 on pgs. 326-328.in textbook??Complete Practice A 7-1 CRB, pg. 3.??Complete Reading Strategies 7-1 CRB, pg. 10.??Read textbook pgs. 330-335.in textbook??Complete Think and Discuss, pg. 331.in textbook??Complete Practice and Problem Solving, Problems 1, 6, and 26-33 on pgs. 332-333.in textbook??Complete Practice A 7-2 CRB, pg. 12.??Complete Reading Strategies 7-2 CRB, pg. 18.Evaluation:Complete Power Presentations Lesson Quiz 7-1, pg. 328 and 7-2, pg. 333.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 36Georgia Performance Standard(s):.??M8P4.a Recognize and use connections among mathematical ideas.??M8P4.b Understand how mathematical ideas interconnect and build on one another toproduce a coherent whole.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8P1.a Build new mathematical knowledge through problem solving.??M8P1.b Solve problems that arise in mathematics and in other contextsObjective(s):??The student organizes and reviews key concepts and skills presented in Chapter Seven.??The student assesses mastery of concepts and skills in Chapter Seven.Instructional Resources:Holt Mathematics Course Three TextbookChapter 7 Resource Book (CRB)One Stop PlannerActivities:??Complete the odd number problems on the Study Guide, pgs. 376-377 in textbook.Have teacher check your work on review. If score is at least 80%, then go on toChapter Test. If score is less than 80%, then teacher will give Reteach worksheets inCRB to cover concepts not understood.??Complete Chapter Test pg.379 in textbook.??Have teacher check your work on test. If score is at least 80%, then go on to next CAP. If score is less than 80%, then teacher will give worksheets in CRB covering concepts still not understood.Evaluation:Complete Chapter Test pg.379 in textbook with 80% accuracy.Modifications: IDEA Works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 37Georgia Performance Standard(s):??M8A1.b Simplify and evaluate algebraic expressions.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P2.b Make and investigate mathematical conjectures.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8G2.a Apply properties of right triangles, including the Pythagorean theorem.??M8A1.a Represent a given situation using algebraic expressions or equations in onevariable.??M8A1.c Solve algebraic equations in one variable, including equations involving absolutevalues.??M8A1.d Interpret solutions in problem contexts.??M8P1.a Build new mathematical knowledge through problem solving.Objective(s):??The student finds the perimeter and area of rectangles and parallelograms. The student will be able to find the perimeter and area of triangles and trapezoids.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 384-398Chapter 8 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 385??Read textbook pgs. 385-393.??Complete Think and Discuss, pg. 390.in textbook??Complete Practice and Problem Solving, Problems 1-3, 9-11, 17, 18, and 27-37 on pgs. 391-392.in textbook??Complete Practice A 8-1 CRB, pg. 3.??Complete Reading Strategies 8-1 CRB, pg. 10.??Read textbook pgs. 394-399.??Complete Think and Discuss, pg. 396.in textbook??Complete Practice and Problem Solving, Problems 1-3, 12-14, 31, 33, and 37-43 on pgs. 396-398.in textbook??Complete Practice A 8-2 CRB, pg. 12.????Complete Reading Strategies 8-2 CRB, pg. 19.Evaluation:Complete Power Presentations Lesson Quiz 8-1, pg. 392 and 8-2, pg. 398.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 42Georgia Performance Standard(s):??M8A1.b Simplify and evaluate algebraic expressions.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P1.a Build new mathematical knowledge through problem solving.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8A1.a Represent a given situation using algebraic expressions or equations in onevariable.??M8A1.c Solve algebraic equations in one variable, including equations involving absolutevalues.??M8A1.d Interpret solutions in problem contexts.??M8P1.c Apply and adapt a variety of appropriate strategies to solve problems.??M8P3.a Organize and consolidate their mathematical thinking through communicationObjective(s):??The student organizes and reviews key concepts and skills presented in Chapter Eight.??The student assesses mastery of concepts and skills in Chapter Eight...Instructional Resources:Holt Mathematics Course Two TextbookChapter 8 Resource Book (CRB)One Stop PlannerActivities:??Complete the odd number problems on the Study Guide, pgs. 450-451 in textbook.Have teacher check your work on review. If score is at least 80%, then go on toChapter Test. If score is less than 80%, then teacher will give Reteach worksheets inCRB to cover concepts not understood.??Complete Chapter Test pg.453 in textbook.??Have teacher check your work on test. If score is at least 80%, then go on to next CAP.If score is less than 80%, then teacher will give worksheets in CRB covering conceptsstill not understood.Evaluation:Complete Chapter Test pg.453 in textbook with 80% accuracy.Modifications: IDEA Works CDUnit: Data & StatisticsGeorgia Performance Standards:M8P2 Students will reason and evaluate mathematical arguments.c. Develop and evaluate mathematical arguments and proofs.M8P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely. M8P4 Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.M8P1 Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena.M8D4 Students will organize, interpret, and make inferences from statistical data.a. Gather data that can be modeled with a linear function.Selected Terms and Symbols:Graph of a linear inequality: The solutions of a linear inequality, forming a half-plane on one side of a line and may or may not also form the line itself.Half-plane: One portion of a plane when the plane is divided into two regions by a line.Line of best fit: The line that best represents the trend established by the points in a particular scatter plot.Scatter plot: The graph of a collection of ordered pairs.Slope: The steepness (the ratio of rise to run) of a line, which may be calculated by finding the ratio of the difference between the y values of two points on the line to the difference between the corresponding x values of those two points on the line.Slope-intercept form: One method used to write an equation of a line; uses the form y = mx + b, where m is the slope and b is the y-intercept.Standard form of a linear equation: One method used to write an equation of a line; uses the form Ax + By = C.Point-slope form: One method used to write an equation of a line; uses the form y – y1= m(x – x1), where m is the slope and (x1, y1) is a point on the line.Teacher’s Place:Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.1. Explain the activity (activity requirements)2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at 3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard. 7. Discuss answers with the students using the following questioning techniques as applicable: Questioning Techniques:Memory QuestionsSignal words: who, what, when, where?Cognitive operations: naming, defining, identifying, designatingConvergent Thinking QuestionsSignal words: who, what, when, where?Cognitive operations: explaining, stating relationships, comparing andcontrasting Divergent Thinking QuestionsSignal words: imagine, suppose, predict, if/thenCognitive operations: predicting, hypothesizing, inferring, reconstructingEvaluative Thinking QuestionsSignal words: defend, judge, justify (what do you think)?Cognitive operations: valuing, judging, defending, justifying8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter) 10. At the end of the *whole group learning session, students will transition into independent CAP assignments. *The phrase, “whole group learning session” is utilized “rather than, the end of the activity” because all of the activities may not be completed in one day. Task: 1Resources: plete a survey of the students in your class. Expand the following table to include a row for every student and gather the requested information from every classmate.Class surveyStudent #First NameLast NameHeightNumber of Pets1232.How many different types of ordered pairs can be created from this survey data? Explain your answer.HINT: One type of ordered pair you could create from the information you collected in your survey is (Student #, First Name) 3.If the first term of each ordered pair is the independent variable and the second is the dependent, then which of the ordered pairs you identified in question 2 are relations? Which are functions? Explain your answers.Discussion, Suggestions, Possible SolutionsStudents should be encouraged to work in small groups for these problems. Explanations are the most critical part of this work.NOTE: The teacher can easily change the categories on the survey, add some, or take some away.1. The survey data could be gathered by the entire class, by groups or individuals, but ultimately everyone should have a copy with which to work.2. This is a straightforward counting question. Students could draw tables or other diagrams or use the counting principle. Discounting ordered pairs in which the first and second variables are identical, there are5*4 = 20 ordered pair combinations.3. The dependent variable of every piece of data must be unique to the independent variable in order for the set of ordered pairs to be a function. For instance, if there are two students with the last name of ‘Smith’ in the class, then any set of ordered pairs such as (Last name, Student #) would not be a function because there would be more than one output value for the input value of the last name ‘Smith’. The Venn diagram below demonstrates the relationship of relations and functions.Also, there is no mention of "one-to-one correspondence". It is alluded to by the word "unique". Shouldn't this be included in the study of functions (vocabulary)?Task: 2Resources: .For each list of numbers below, determine the next three numbers in the list.a) 1, 2, 3, 4, 5, , , b) 7, 9, 11, 13, 15, , , c) 10, 7, 4, 1, -2, , , d) 2, 4, 7, 11, 16, , , e) 1, -1, 2, -2, 3, -3, , , f)0, 1, 4, 9, 16, 25, , , g) 2.0, 3.5, 5.0, 6.5, 8.0, 9.5, , , h)17.5, 13.2, 8.9, 4.6, 0.3, , , i)2, 5, 11, 23, 47, , , j)1, 1, 2, 3, 5, 8, , , 1.For each list of numbers below, determine the next three numbers in the list. a) 1, 2, 3, 4, 5, _6_, _7_, _8_b) 7, 9, 11, 13, 15, _17_, _19_, _21_ c) 10, 7, 4, 1, -2, _-5_, _-8_, _-11_ d) 2, 4, 7, 11, 16, _22_, _29_, _37_ e) 1, -1, 2, -2, 3, -3, _4_, _-4_, _5_f) 0, 1, 4, 9, 16, 25, _36_, _49_, _64_g) 2.0, 3.5, 5.0, 6.5, 8.0, 9.5, _11.0_, _12.5_, _14.0_h) 17.5, 13.2, 8.9, 4.6, 0.3, _-4.0_, _-8.3_, _-12.6_i) 2, 5, 11, 23, 47, _95_, _191_, _383_j) 1, 1, 2, 3, 5, 8, _13_, _21_, _34_Course Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 43Georgia Performance Standard(s):??M8P2.c Develop and evaluate mathematical arguments and proofs.??M8P3.d Use the language of mathematics to express mathematical ideas precisely.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P1.a Build new mathematical knowledge through problem solving.Objective(s):??The student identifies sampling methods and recognizes biased samples. The student will be able to organize data in tables and stem-and-leaf plots.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 459-471Chapter 9 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 459.??Read textbook pgs. 458-466.??Complete Think and Discuss, pg. 463.in textbook??Complete Practice and Problem Solving, Problems 1, 2, 4, 5, 8-12, 18, and 21-29 on pgs. 464-465.in textbook??Complete Practice A 9-1 CRB, pg. 3.??Complete Reading Strategies 9-1 CRB, pg. 9.??Read textbook pgs. 467-471.??Complete Think and Discuss, pg. 468.in textbook??Complete Practice and Problem Solving, Problems 1, 6, 12, 20, and 22-29 on pgs. 469- 471 in textbook.??Complete Practice A 9-2 CRB, pg. 11.??Complete Reading Strategies 9-2 CRB, pg. 18.Evaluation:Complete Power Presentations Lesson Quiz 9-1, pg. 465 and 9-2, pg. 471.Modifications:Performance Tasks: IDEA works CDUnit: ProbabilityGeorgia Performance Standards:M8D3 Students will use the basic laws of probability.a. Find the probability of simple independent events.b. Find the probability of compound independent events.M8P1 Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving. M8P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely. M8P4 Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics. M8P2 Students will reason and evaluate mathematical arguments.c. Develop and evaluate mathematical arguments and proofs.M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena. M8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.c. Solve algebraic equations in one variable, including equations involving absolute values.d. Solve equations involving several variables for one variable in terms of the others.M8D2 Students will determine the number of outcomes related to a given event.a. Use tree diagrams to find the number of outcomes.b. Apply the addition and multiplication principles of counting.Selected Terms And Symbols: Adjacent Angles: Angles in the same plane that have a common vertex and a common side, but no common interior points.Alternate Exterior Angles: Alternate exterior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. When the two other lines are parallel, the alternate exterior angles are equal.Alternate Interior Angles: Alternate interior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are in between the other two lines. When the two other lines are parallel, the alternate interior angles are equal.Coincidental: Two equivalent linear equations overlap when plementary Angles: Two angles whose sum is 90 degrees.Congruent: Having the same size, shape and measure. Two figures are congruent if all of their corresponding measures are equal.Corresponding Angles: Angles that have the same relative positions in geometric figures.Equiangular: The property of a polygon whose angles are all congruent.Equilateral: The property of a polygon whose sides are all congruent.Intersecting Lines: Two lines in a plane that cross each other. Unless two lines are coincidental, parallel, or skew, they will intersect at one point.Linear Pair: Adjacent, supplementary angles. Excluding their common side, a linear pair forms a straight line.Parallel Lines: Two lines are parallel if they lie in the same plane and they do not intersect.Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle.Reflection Line: A line that is the perpendicular bisector of the segment with endpoints at a pre-image point and the image of that point after a reflection.Regular Polygon: A polygon that is both equilateral and equiangular.Same-Side Interior Angles: Pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are between the other two lines. When the two other lines are parallel, same- side interior angles are supplementary.Same-Side Exterior Angles: Pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are outside the other two lines. When the two other lines are parallel, same- side exterior angles are supplementary.Skew Lines: Two lines that do not lie in the same plane (therefore, they cannot be parallel or intersect).Supplementary Angles: Two angles whose sum is 180 degrees.Transversal: A line that crosses two or more lines.Vertical Angles: Two nonadjacent angles formed by intersecting lines or segments. Also called opposite angles.Teacher’s Place:Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.1. Explain the activity (activity requirements)2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at . Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard. 7. Discuss answers with the students using the following questioning techniques as applicable: Questioning Techniques:Memory QuestionsSignal words: who, what, when, where?Cognitive operations: naming, defining, identifying, designatingConvergent Thinking QuestionsSignal words: who, what, when, where?Cognitive operations: explaining, stating relationships, comparing andcontrasting Divergent Thinking QuestionsSignal words: imagine, suppose, predict, if/thenCognitive operations: predicting, hypothesizing, inferring, reconstructingEvaluative Thinking QuestionsSignal words: defend, judge, justify (what do you think)?Cognitive operations: valuing, judging, defending, justifying8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter) 10. At the end of the *whole group learning session, students will transition into independent CAP assignments. *The phrase, “whole group learning session” is utilized “rather than, the end of the activity” because all of the activities may not be completed in one day. Task: 1 Resources: Activity1. Suppose that you are approached by a classmate who invites you to play a game with the following rules: Each of you takes turns flipping a coin.He gives you $1 each time a coin lands on Tails.You give him $1 each time a coin lands on Heads.What are the possible outcomes? Who is likely to win the most? Make a tree diagram for the 4 possible outcomes—your 2 tosses and his 2 tosses. What are your winnings for each outcome?2. Suppose that your classmate suggests instead that you flip a coin.If you get heads, you give him $2.If you get tails, then he flips his coin.If he gets heads, you give him $1.If he gets tails, he gives you $2.Who is likely to win the most in this game? Use a tree diagram to verify your conclusion.3. Construct a similar game in which you are most likely to be the winner and use a tree diagram to illustrate the outcomes, winnings, and probabilities.Discussion, Suggestions, Possible Solutions (This lesson is adapted from and can be done interactively at the following website: .) Task: 2Resources: Part I:Your next door neighbor, Mrs. Love, is expecting fraternal twins.She has picked out names for two girls, but is having difficulty thinking of any boy names. Since she knows that there is a possibility of having at least one boy, she wants you to help her determine some possible names. A. Make a list of your favorite boy names. (Be sure to have more than three names on your list.)B. Using the names that you have chosen, tell how many ways she could name a boy baby using both a first name and a different middle name from your list.C. Write an explanation to convince her of how you know that this is the number of ways to name the baby boy.Part II:Your best friend said that the probability of Mrs. Love having two girls was 1/3 because she could have both girls, both boys, or one of each. Do you agree or disagree? Explain your reasoningDiscussion, Suggestions, Possible SolutionsPart I:Students may come up with various representations for the possible ways of naming a boy. Some may write out all the names, while some may choose to organize them with a tree diagram. Some students may realize that the basic counting principle allows them to multiply the number of choices for the first name times the number of choices for the second name and find the number of ways without listing all the ways.Suppose a student decides she likes the following names: Kevin, Luke, Mark, Matthew, Michael, Stephen, and Noah. Then, by basic counting principle she will have seven choices for the first name and six choices for the second name (since a person would not want to name the child with the same first name and middle name), resulting in 7 x 6, or 42, ways of naming a son. Although some students may list the names without using a systematic approach, they may find it easier to be sure that they are not skipping a name combination or listing one twice if they do it in an organized manner. One way to systematically list all the possible name combinations might be to pair Kevin with all of the remaining six possible middle names, then pair Luke with all the six possible middle names, etc.The list of possible names would be: Kevin LukeKevin MarkKevin MatthewKevin Michael Kevin Stephen Kevin Noah Luke Kevin Luke Mark Luke Matthew Luke Michael Luke Stephen Luke Noah Mark Kevin Mark Luke Mark Matthew Mark Michael Mark Stephen Mark Noah Matthew Kevin Matthew Luke Matthew Mark Matthew Michael Matthew Stephen Matthew Noah Michael Kevin Michael Luke Michael Matthew Michael Stephen Michael Noah Stephen Kevin Stephen Luke Stephen Mark Stephen Michael Stephen Matthew Stephen Noah Noah Kevin Noah Luke Noah Matthew Noah Mark Noah Michael Noah StephenThe simplest type basic counting principle questions involve making choices from two disjoint sets. In these cases the students will not make many mistakes, since all that is required is multiplying the numbers that appear in the question. For example, if a person has nine ways to choose an entrée and six ways to choose a dessert, then there are 9 x 6, or 54, ways to choose an entrée and dessert.The task of naming a boy child represents a situation of drawing more than once from the same set without having repetitions. It is important to give a variety of examples. Suppose a teacher wants to choose two different students to run errands—one to go to the library and one to go to the office. If students are asked to find how many ways two students out of a class of 30 can be sent to run the errands, a common error would be to say 2 x 30 rather than 30 x 29. Because the number of possibilities can be extremely large, visualizing the possible outcomes by beginning a tree diagram can be a powerful tool.The other variety of basic counting principle questions involves drawing repeatedly from the same set with repetitions allowed. For example, if a license plate consisting of three letters followed by three digits is to be made, then there are 26 x 26 x 26 x 10 x 10 x 10 possibilities. Letters and digits may be repeated; there are 26 choices for each letter and 10 choices for each digitPart II:For the second part of “Mrs. Love’s Children,” students must realize that in the sample space the events listed need to be equally likely outcomes. Since the likelihood of getting one boy and one girl is not the same as the likelihood of getting two girls, the probability of getting two girls is not 1/3 as the friend stated.The sample space consisting of equally likely events is: Boy-BoyBoy-Girl Girl-Boy Girl-GirlThus, the probability of having two girls is ?. In addition to listing the sample space to determine the probability of having two girls, students may consider the events to be independent and multiply the probability the first child is a girl times the probability that the second child is a girl: ? x ? = ?.Task: 3 Resources: In 1821, Frenchman Louis Braille developed a method that is used to help blind people read and write. This system was based on a more complicated process of communication that was formed by Charles Barbier due to an order from Napoleon who wanted soldiers to communicate in the dark and without speaking. Braille met with Barbier and decided to simplify the code by using a six-dot cell because the human finger needed to cover the entire symbol without moving so that it could progress quickly from one symbol to the next.Each Braille symbol is formed by raising different combinations of dots. Below is a sample of the first three letters of the alphabet.A. Using the six-dot Braille cell, how many different combinations are possible? Explain how you know. B. Do you think this is enough symbols for sight-impaired people to use? Why or why not?C. What are some reasons that some of the possible combinations might need to be discarded?D. An extension has been added to the Braille code that contains eight-dots with the two additional ones added to the bottom. How does this change the number of possible different combinations? Justify your answer.Discussion, Suggestions, Possible SolutionsaAllow the students to approach this task in a variety of different ways. Some may actually try to form or draw the different possibilities. This is easier if they number the dots 1-6 and can lead them to discover Sample Space.Others may discover that a dot has only two options – raised or not raised. You may find that some of your students will create a tree diagram. This may help them to discover the Counting Principle due to the usefulness of multiplication resulting in 26or 64 possible combinations.Be sure to have them share their results and discuss how they arrived at those results. Take advantage of any opportunities to introduce new vocabulary within that may arise naturally during student presentations and discussions during this task.b A variety of responses should be expected from the students. They must justify their answers.cSome of the possible combinations of the six-dot Braille cell must be omitted because they feel the same. This is due to the fact that the dots have the same pattern in a different position. Thus many of the Braille characters need to represent more than one meaning. Connecting this to mathematical language, the character mapping is not one-to-one.d This would cause an increase to 28 or 256 possible characters. Once again, some of the possible dot- combinations would need to be discarded for the same reasons given in part CCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 48Georgia Performance Standard(s):??M8D3.a Find the probability of simple independent events.??M8P1.c Apply and adapt a variety of appropriate strategies to solve problems.??M8P1.d Monitor and reflect on the process of mathematical problem solving.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P4.a Recognize and use connections among mathematical ideas.??M8P4.b Understand how mathematical ideas interconnect and build on one another toproduce a coherent whole.??M8P1.a Build new mathematical knowledge through problem solving.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.Objective(s):??The student finds the probability of an event by using the definition of probability. Thestudent will be able to estimate probability using experimental methods.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 519-530Chapter 10 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 519.??Read textbook pgs. 519-526.??Complete Think and Discuss, pg. 524.in textbook??Complete Practice and Problem Solving, Problems 1, 5, and 19-28 on pgs. 525-526.intextbook??Complete Practice A 10-1 CRB, pg. 3.??Complete Reading Strategies 10-1 CRB, pg. 10.??Read textbook pgs. 527-531.??Complete Think and Discuss, pg. 528.in textbook??Complete Practice and Problem Solving, Problems 1, 2, 5, 6, and 18-25 on pgs. 529-530.in textbook??Complete Practice A 10-2 CRB, pg. 12.??Complete Reading Strategies 10-2 CRB, pg. 18.Evaluation:Complete Power Presentations Lesson Quiz 10-1, pg. 526 and 10-2, pg. 530.Modifications:Performance Tasks: IDEA works CD Course Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 52Georgia Performance Standard(s):??M8D2.b Apply the addition and multiplication principles of counting.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P1.c Apply and adapt a variety of appropriate strategies to solve problems.??M8P3.a Organize and consolidate their mathematical thinking through communication.Objective(s):??The student finds permutations and combinations.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 563-567Chapter 10 Resource Book (CRB)One-Stop PlannerActivities:??Read textbook pgs. 563-575.??Complete Think and Discuss, pg. 565.in textbook??Complete Practice and Problem Solving, Problems 1-4, 9-12, 17, 18, 23, and 41-45 onpgs. 566-567.in textbook??Complete Practice A 10-9 CRB, pg. 71.??Complete Reading Strategies 10-9 CRB, pg. 78.??Complete the Ready to Go On section, pg. 568.in textbook??Complete the CRCT Review for the average learner, pgs. 576-577.in textbookEvaluation:Complete Power Presentations Lesson Quiz 10-9, pg. 567.Modifications:Performance Tasks: IDEA works CDUnit: Multi-Step Equations & InequalitiesGeorgia Performance Standards: M8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.b. Simplify and evaluate algebraic expressions.c. Solve algebraic equations in one variable, including equations involving absolute values.d.Solve equations involving several variables for one variable in terms of the others.M8P1. Students will solve problems (using appropriate technology).b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solving problems. d. Monitor and reflect on the process of mathematical problem solving.M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena.M8P3. Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.M8P4 Students will make connections among mathematical ideas and to other disciplines.c. Recognize and apply mathematics in contexts outside of mathematics.M8A2 Students will understand and graph inequalities in one variable.a. Represent a given situation using an inequality in one variable.b. Use the properties of inequality to solve inequalities.c. Graph the solution of an inequality on a number line.d. Interpret solutions in problem contexts.M8A5 Students will understand systems of linear equations and inequalities and use them to solve problems.b. Solve systems of equations graphically and algebraically, using technology as appropriate.c.Graph the solution set of a system of linear inequalities in two variables.Teacher’s Place:Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.1. Explain the activity (activity requirements)2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at 3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard. 7. Discuss answers with the students using the following questioning techniques as applicable: Questioning Techniques:Memory QuestionsSignal words: who, what, when, where?Cognitive operations: naming, defining, identifying, designatingConvergent Thinking QuestionsSignal words: who, what, when, where?Cognitive operations: explaining, stating relationships, comparing andcontrasting Divergent Thinking QuestionsSignal words: imagine, suppose, predict, if/thenCognitive operations: predicting, hypothesizing, inferring, reconstructingEvaluative Thinking QuestionsSignal words: defend, judge, justify (what do you think)?Cognitive operations: valuing, judging, defending, justifying8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter) 10. At the end of the *whole group learning session, students will transition into independent CAP assignments. *The phrase, “whole group learning session” is utilized “rather than, the end of the activity” because all of the activities may not be completed in one day. Task: 1Resources: Cara likes candles. She also likes mathematics and was thinking about using algebra to answer a question that she had about two of her candles. Her taller candle is 16 centimeters tall. Each hour which it burns makes the candle lose 2.5 centimeters in height. Her short candle is 12 centimeters tall and loses 1.5 centimeters in height for each hour that it burns.Cara needs your help to determine whether these two candles would ever reach the same height at the same time if allowed to burn the same length of time. She also wants to know what height the two candles would be at that time. If it is not possible, she wants to know why it could not happen and what would need to be true in orderfor them to be able to reach the same height. To help Cara understand what you are doing, be sure to use multiple representations, justify your results and explain your thinking.Discussion, Suggestions, Possible SolutionsIf x represents the number of hours that the candles burn and y represents the height of the candle after burning x hours, then the two equations that model this problem situation are y = 16 – 2.5x and y = 12 – 1.5x. The system is easily solved by substitution, resulting in16 – 2.5x = 12 – 1.5x-12 + 2.5x -12 + 2.5x4=1x or x = 4.Thus, the candles reach the same height after 4 hours. At that time both candles are 6 cm long, since16 – 2.5(4) = 16 – 10 = 6 cm and 12 – 1.5x = 12 – 1.5(4) = 6 cm.Supporting the work with a table encourages the student to take note of the changing values of the heights over time and think about the reasonableness of the results.x16 -2.5x12 – 1.5x113.510.5211938.57.546653.54.56137-1.51.5Task: 2Resources: in sports medicine inform us that a person’s maximum heart rate can be estimated by subtracting his/her age from 220. For example, the maximum heart rate for a sixty-year-old should be approximately220 – 60 = 160 beats per minute.Those experts also inform us that during exercise a person’s lower-limit exercise heart rate should be about 60% of his/her maximum heart rate. Thus, the sixty-year-old should maintain a minimum heart rate of 96 beats per minute (60% of 160) during exercise.The recommended upper-limit exercise rate is 90% of the maximum heart rate. Consequently, a sixty-year-old should keep his/her heart rate no more than 144 beats per minute (90% of 160) during exercise.Make a graph showing the exercise heart rate range as a function of age. Discussion, Suggestions, Possible SolutionsLet x = age and y = heart rate. The lower boundary for the exercise heart rate range is y = .6(220 – x). The upper boundary for the exercise heart rate range is y = .9(220 – x).Task: 3Resources: Imagine that you are sitting in front of a television watching the Miami Heat playing the Dallas Mavericks. Tony Wade drives the lane and is fouled. As he steps to the free throw line, the announcer states that “Wade is hitting 82 percent of his free throws this year.” He misses the first shot, but makes the second. Later in the game, Tony Wade is fouled for the second time. As he approaches the free throw line, the announcer states that “Wade has made 78 percent of his free throws so far this year.”A.How are free throw percentages calculated?B.What algebraic relationships could you write to represent the two situations?C.What kind of numbers must we always be dealing with? Why?D.How many free throws has Tony Wade attempted so far this year? E.How many free throws has he made so far this year?Discussion, Suggestions, Possible SolutionsA.How are free throw percentages calculated?Free throw percentages are calculated by dividing the total number of attempted shots into the number of successful shots.B.What algebraic relationships could you write to represent the two situations?If x = the total number of attempted free throws for the first situation and y = the number of successful The second situation could be represented as either the proportiony = 0.78x + 0.56. Some students may want to let x – 2 = the first situation’s attempted free throws with x = to the second situation’s free throws instead. This would still be mathematically correct.C.What kind of numbers must we always be dealing with? Why?We are dealing with positive rational numbers that are less than or equal to 1 because they cannot miss more shots than they attempt and if they were successful for every attempt the percentage would equal 1.D.How many free throws has Tony Wade attempted so far this year?This may be solved in several different ways. One possible method would be using substitution. Since y = 0.82x and y = 0.78x + 0.56, we may substitute the first equation for y in the second equation. This will give 0.82x = 0.78x + 0.56. Subtracting 0.78x from both sides of the equation will yield 0.04x = 0.56. Dividing both sides of the equation by 0.04 will result in x = 14, which means that Tony Wade has attempted 14 free throws so far this year.E.How many free throws has he made so far this year?Once again, the students may work this in different ways. Most will substitute the value of x into one of the two equations such as, y = 0.82(14) or 11.48.This result will most likely spark some good discussion about percentages in sports. To arrive at this answer must mean that the percentages were rounded to the nearest 0.01 and were not exact figures. Task: 4Resources: Mr. Nelson went to Taco Town to get lunch for the eighth-grade teachers one day when they were having a grade-level meeting. He bought eight tacos and five burritos, and the total cost before tax was $13.27. On the day of the next grade-level meeting he went back to Taco Town and got six tacos and seven burritos for a cost of $14.47 before tax. The teachers now want to pay Mr. Nelson, but Mr. Nelson doesn’t remember how much one taco costs or how much one burrito costs.Ms. Payne tells the teachers that there is enough information to figure out how much a taco costs and how much a burrito costs and that she will get one of her math students to do the calculation. She has chosen you to do the work. Find the cost of one taco and the cost of one burrito, showing your work in arriving at the answerDiscussion, Suggestions, Possible SolutionsWhile mathematics teachers want to honor different solution paths, the intent is for the student to utilize the efficiency of solving a system of equations by elimination. Using substitution introduces fractions that make the calculations less straightforward than elimination. Likewise, to solve by graphing using a calculator requires putting the equations in y form, resulting in fractions.The two linear equations in the system are 8t + 5b = 13.27 and 6t + 7b = 14.47.One of the first possible steps towards eliminating the t’s could be to multiply the first equation by -3 and the second equation by 4. Thus, the original system has been replaced by the equivalent system:-24t - 15b = -39.8124t + 28b = 57.88Adding the left and right sides of these equations yields 13b = 18.07. Dividing each side by 13 yields b = $1.39.Substituting this b value into the first equation produces 8t + 5(1.39) = 13.27. Multiplying gives 8t + 6.95 = 13.27.Subtracting 6.95 from each side results in 8t = 6.32. Dividing by 8 gives t =.79.Thus the cost of each taco is $.79 cents and the cost of each burrito is $1.39.Course Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 54Georgia Performance Standard(s):??M8A1.b Simplify and evaluate algebraic expressions.??M8A1.c Solve algebraic equations in one variable, including equations involving absolute values.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8A1.a Represent a given situation using algebraic expressions or equations in onevariable.??M8A1.d Interpret solutions in problem contexts.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.Objective(s):??The student combines like terms in an expression. The student will be able to solve multistep equations.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 580-591Chapter 11 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 581.??Read textbook pgs. 581-587.??Complete Think and Discuss, pg. 585.in textbook??Complete Practice and Problem Solving, Problems 1-6, 19-27, 52, and 58-65 on pgs. 586- 587.in textbook??Complete Practice A 11-1 CRB, pg. 3.??Complete Reading Strategies 11-1 CRB, pg. 9.??Read textbook pgs. 588-592.??Complete Think and Discuss, pg. 589.in textbook??Complete Practice and Problem Solving, Problems 1-6, 12-17, 26, 27, and 41-47 on pgs. 590-591.in textbook??Complete Practice A 11-2 CRB, pg. 11.??Complete Reading Strategies 11-2 CRB, pg. 18.Evaluation:Complete Power Presentations Lesson Quiz 11-1, pg. 587 and 11-2, pg. 591.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 57Georgia Performance Standard(s):??M8A2.a Represent a given situation using an inequality in one variable.??M8A2.b Use the properties of inequality to solve inequalities.??M8A2.c Graph the solution of an inequality on a number line.??M8A2.d Interpret solutions in problem contexts.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8A5.b Solve systems of equations graphically and algebraically, using technology asappropriate.??M8A5.c Interpret solutions in problem contexts.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P1.c Apply and adapt a variety of appropriate strategies to solve problemsObjective(s):??The student organizes and reviews key concepts and skills presented in Chapter Eleven.??The student assesses mastery of concepts and skills in Chapter Eleven.Instructional Resources:Holt Mathematics Course Two TextbookChapter 11Resource Book (CRB)One Stop PlannerActivities:??Complete the odd number problems on the Study Guide, pgs. 616-617 in textbook.Have teacher check your work on review. If score is at least 80%, then go on toChapter Test. If score is less than 80%, then teacher will give Reteach worksheets inCRB to cover concepts not understood.??Complete Chapter Test pg 619 in textbook.??Have teacher check your work on test. If score is at least 80%, then go on to next CAP.If score is less than 80%, then teacher will give worksheets in CRB covering conceptsstill not understood.Evaluation:Complete Chapter Test pg.619 in textbook with 80% accuracy.Modifications: IDEA Works CDUnit: Graphing Lines, Sequences & Functions & Polynomials Georgia Performance Standards:M8A3 Students will understand relations and linear functions.d. Recognize functions in a variety of representations and a variety of contexts.e. Use tables to describe sequences recursively and with a formula in closed form.f. Understand and recognize arithmetic sequences as linear functions with whole-number input values.h. Interpret the constant difference in an arithmetic sequence as the slope of the associated linear function.i. Identify relations and functions as linear or nonlinear.M8A4 Students will graph and analyze graphs of linear equations and inequalities.a. Interpret slope as a rate of change.b. Determine the meaning of the slope and y-intercept in a given situation.c. Graph equations of the form y = mx + b.d. Graph equations of the form ax + by = c.e.Graph the solution set of a linear inequality, identifying whether the solution set is an open or a closed half-plane. f.Determine the equation of a line given a graph, numerical information that defines the line, or a context involving a linear relationship. M8P1 Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.M8P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.M8P4 Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.M8P5 Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical phenomena.M8A3 Students will understand relations and linear functions.d. Recognize functions in a variety of representations and a variety of contexts.h. Interpret the constant difference in an arithmetic sequence as the slope of the associated linear function.i. Identify relations and functions as linear or nonlinear.M8D4 Students will organize, interpret, and make inferences from statistical data.b. Estimate and determine a line of best fit from a scatter plot.M8A1 Students will use algebra to represent, analyze, and solve problems.a. Represent a given situation using algebraic expressions or equations in one variable.b. Simplify and evaluate algebraic expressions.c. Solve algebraic equations in one variable, including equations involving absolute values.d.Solve equations involving several variables for one variable in terms of the others.M8P2 Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.i. Simplify expressions containing integer exponents.k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.Teacher’s Place:Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.1. Explain the activity (activity requirements)2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at . 3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard. 7. Discuss answers with the students using the following questioning techniques as applicable: Questioning Techniques:Memory QuestionsSignal words: who, what, when, where?Cognitive operations: naming, defining, identifying, designatingConvergent Thinking QuestionsSignal words: who, what, when, where?Cognitive operations: explaining, stating relationships, comparing andcontrasting Divergent Thinking QuestionsSignal words: imagine, suppose, predict, if/thenCognitive operations: predicting, hypothesizing, inferring, reconstructingEvaluative Thinking QuestionsSignal words: defend, judge, justify (what do you think)?Cognitive operations: valuing, judging, defending, justifying8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter) 10. At the end of the *whole group learning session, students will transition into independent CAP assignments. *The phrase, “whole group learning session” is utilized “rather than, the end of the activity” because all of the activities may not be completed in one day. Task: 1Resources: in sports medicine inform us that a person’s maximum heart rate can be estimated by subtracting his/her age from 220. For example, the maximum heart rate for a sixty-year-old should be approximately220 – 60 = 160 beats per minute.Those experts also inform us that during exercise a person’s lower-limit exercise heart rate should be about 60% of his/her maximum heart rate. Thus, the sixty-year-old should maintain a minimum heart rate of 96 beats per minute (60% of 160) during exercise.The recommended upper-limit exercise rate is 90% of the maximum heart rate. Consequently, a sixty-year-old should keep his/her heart rate no more than 144 beats per minute (90% of 160) during exercise.Make a graph showing the exercise heart rate range as a function of age. Discussion, Suggestions, Possible SolutionsLet x = age and y = heart rate. The lower boundary for the exercise heart rate range is y = .6(220 – x). The upper boundary for the exercise heart rate range is y = .9(220 – x). Task: 2Resources: A college holds math contests for middle school students and high school students each semester. The faculty coordinator buys prizes for the students on the top three teams at each contest. She got a new catalogue and thought about how many items she could buy considering the prices quoted in the catalogue.One item she looked at was wristband keytags. The company charges different prices depending on how many items are purchased. The more items that are purchased, the lower the price per item. The table below shows the prices offered by the company and how many keytags the faculty coordinator would be willing to buy at that price. For example, if she has to pay $2.37 for each keytag, then she is only willing to buy 35 (enough for one contest). At the $1.13 price, she is willing to buy 410 keytags. The number that is wanted is called the “demand.”Price Per Keytag$2.37Number to Buy (Demand)35$2.2565$1.57160$1.37235$1.13410The company cannot afford to sell a large number of items at a very low price, because the company is trying to make money (make a “profit”). Suppose that the company is only willing to sell 50 keytags at a price of $1.13.The table below shows how many items the company is willing to sell to a customer at a given price.Price Per KeytagNumber to Sell (Supply)$2.37800$2.25600$1.57250$1.37100$1.1350The math contest coordinator wants you to find the “equilibrium price”—the price at which the supply is equal to the demand. At this price both the customer and the company will be satisfied. Use your knowledge of lines of best fit to model the data and locate the equilibrium price.Discussion, Suggestions, Possible SolutionsAdditional Questions for Thought:First Contest:Mallory found the intersection of two of her boundary lines and guessed that there were 20 nickels in the cup. Was her guess a good one? Explain how you know.What is the minimum number of nickels you can have? How do you know?What is the maximum amount of money the cup could contain? How do you know? Can you show this graphically?Second Contest:What is the minimum amount of money that can be in the mug?Classroom Ideas:Let a student graph the boundary lines on the board. Ask every student to write their final guess on a sticky note, and then let each student come to the board and plot their point on the board by placing the sticky note on the graph that has been drawn. All of their sticky notes create somewhat of a shading of the feasible region. Sticky notes placed outside of the region give rise to discourse about why the guess was not plausible given the constraints.This also helps students see that they can use certain ordered pairs to check the feasible region they shade to see if the ordered pairs make logical sense as a solution within the context.Point of Interest:Because we cannot have a negative number of coins in the cup, x > 0 and y > 0 are also constraints which limit the feasible region to the first quadrant (where x and y represent the number of the respective currencies.) Inthe first contest, students will have to realize they need to use the constraint x > 0. One may also say that x and y are strictly greater than zero instead of greater than or equal to since we are told the cup holds pennies and nickels.Task: 3 Resources: You are an engineer in charge of testing new equipment that can detect underwater submarines from the air. Part 1: The first three hoursDuring this part of the test, you are in a helicopter 250 feet above the surface of the ocean. The helicopter moves horizontally to remain directly above a submarine. The submarine begins the test positioned at 275 feet below sea level.After one hour, the submarine is 325.8 feet below sea level. After two hours, the submarine dives another 23 feet. After three hours, the submarine dives again, descending by an amount equal to the average of the first two dives.Make a table/chart with five columns (Time, Position of Submarine, Position of Helicopter, Distance between Helicopter and Submarine, and a Mathematical Sentence showing how to determine this distance) and four rows (start, one hour, two hours, three hours).Make a graphical display which shows the positions of the submarine and helicopter using the information in your table/graph.Part 2: The next three hoursThe equipment in the helicopter is able to detect the submarine within a total distance of 750 feet.For each scenario, determine the maximum or minimum location for the other vehicle in order for the helicopter to detect the submarine; and write a mathematical sentence to show your thinking. Determine the ordered pairs for these additional hours and include them on your graph.At the end of the fourth hour, the helicopter remains at 250 feet.At the end of the fifth hour, the submarine returns to the same depth that it was at the end of the third hour.At the end of the sixth hour, the submarine descends to three times its second hour position.Discussion, Suggestions, Possible Solutions1.The Mathematical Sentence column shows two different approaches that students might use.Should a student receive a negative answer in this column, they should recognize the distance between the vehicles as absolute value. For students who still have difficulty with absolute value, has a variety of explanations.TimePosition ofSubmarinePosition of HelicopterDistance betweenSubmarine andHelicopterMathematical SentenceStart-275250525 -275 – 250 = -5251 hour-325.8250575.8250 – (-325.8) = 575.82 hours-348.8250598.8 -348.8 – 250 = - 598.83 hours-385.7250635.7250 – (-385.7) = 635.7For the graph, the following information could be useful.The helicopter would be plotted at +250 and the submarine at –275. Points above sea level should be noted as a positive number while positions below sea level would be negative numbers.In line with the first hour, the submarine would be plotted at –325.8 and the helicopter would remain at 250.The new position of the submarine is –348.8 feet. Helicopter remains at 250.The position of the submarine will be –385.7 feet. Helicopter remains at 250.Note: Students can simulate the submarine and helicopter before completing the table.635.7 ft525 ft750 ft2.The mathematical sentence could be 750 – 250 = 500. If the helicopter is at 250 feet, the minimum level of the submarine I where the helicopter can detect the submarine is at -500 feet. Therefore, the ordered pair for the fourth hour is (4, -500).CommentThis would be a good place for teachers to point out that depth refers to distance rather than position. Students should recognize at the end of the fourth hour they need to determine the minimum position of the submarine.At the end of the third hour, the submarine was at -385.7 feet. Therefore, a mathematical sentence could be 750 + (-385.7) = 364.3 mentStudents should recognize at the end of the fifth hour they need to determine the maximum position of the helicopter.This one is impossible because the submarine would be at a depth of 3(348.8) = 1046.4 feet which is out of the range of the equipment. A sample mathematical sentence is 750 – 3(348.8) = 750 – 1046.4 = -296.4 which shows that the helicopter would need to be under water.GraphsIt is important for students to make connections between different representations including vertical and horizontal graphs.250 feet750 feet500 feet750 feet250 feet500 feetCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 58Georgia Performance Standard(s):??M8A3.h Identify relations and functions as linear or nonlinear.??M8A3.i Translate among verbal, tabular, graphic, and algebraic representations offunctions.??M8A4.c Graph equations of the form y = mx + b.??M8A4.f Solve problems involving linear relationships.??M8P1.c Apply and adapt a variety of appropriate strategies to solve problems.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8A4.a Interpret slope as a rate of change.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.Objective(s):??The student identifies and graphs linear equations. The student will be able to find the slopeof a line and use slope to understand and draw graphs.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 624-637Chapter 12 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 625.??Read textbook pgs. 625-631.??Complete Think and Discuss, pg. 630.in textbook??Complete Practice and Problem Solving, Problems 1-3, 5-10, 15-20, and 30-37 on pgs.631-632.in textbook??Complete Practice A 12-1 CRB.??Complete Reading Strategies 12-1 CRB.??Read textbook pgs. 633-637.??Complete Think and Discuss, pg. 635in textbook??Complete Practice and Problem Solving, Problems 1-3, 8-13, and 30-36 on pgs. 635-637.in textbook??Complete Practice A 12-2 CRB.??Complete Reading Strategies 12-2 CRB.Evaluation:Complete Power Presentations Lesson Quiz 12-1, pg. 632 and 12-2, pg. 637.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 63Georgia Performance Standard(s):??M8A1.b Simplify and evaluate algebraic expressions.??M8A1.c Solve algebraic equations in one variable, including equations involving absolutevalues.??M8A3.e Use tables to describe sequences recursively and with a formula in closed form.??M8P2.d Select and use various types of reasoning and methods of proof.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8N1.i Simplify expressions containing integer exponents.??M8P4.a Recognize and use connections among mathematical ideas.??M8P4.b Understand how mathematical ideas interconnect and build on one another toproduce a coherent whole.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.Objective(s):??The student finds terms in arithmetic sequence. The student will be able to find terms in a geometric sequence.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 678-691Chapter 13 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 679.??Read textbook pgs. 679-686.??Complete Think and Discuss, pg. 684.in textbook??Complete Practice and Problem Solving, Problems 1-6, 12-17, 23-25, and 37-44 on pgs. 685-686.in textbook??Complete Practice A 13-1 CRB.??Complete Reading Strategies 13-1 CRB.??Read textbook pgs. 687-692.??Complete Think and Discuss, pg. 689.in textbook??Complete Practice and Problem Solving, Problems 1-6, 12-17, 25-28, and 48-55 on pgs. 689-691.in textbook??Complete Practice A 13-2 CRB.??Complete Reading Strategies 13-2 CRB.Evaluation:Complete Power Presentations Lesson Quiz 13-1, pg. 686 and 13-2, pg. 691.Modifications:Performance Tasks: IDEA works CDCourse Title: Course 3 8th grade Ga DJJState Code: 27.0230000 CAP: 68Georgia Performance Standard(s):??M8N1.i Simplify expressions containing integer exponents.??M8A1.b Simplify and evaluate algebraic expressions.??M8P3.d Use the language of mathematics to express mathematical ideas precisely.??M8P4.c Recognize and apply mathematics in contexts outside of mathematics.??M8P3.a Organize and consolidate their mathematical thinking through communication.??M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.??M8P5.b Select, apply, and translate among mathematical representations to solveproblems.??M8P5.a Create and use representations to organize, record, and communicatemathematical ideas.??M8P5.c Use representations to model and interpret physical, social, and mathematicalphenomena.??M8P1.b Solve problems that arise in mathematics and in other contexts.??M8P1.a Build new mathematical knowledge through problem solving.Objective(s):??The student classifies polynomials by degree and by the number of terms. The student will be able to simplify polynomials.Instructional Resources:Holt Mathematics 8th grade Course 3, Pgs. 734-747Chapter 14 Resource Book (CRB)One-Stop PlannerActivities:??Complete Are You Ready in textbook pg., pg. 735.??Read textbook pgs. 735-743.??Complete Think and Discuss, pg. 739.in textbook??Complete Practice and Problem Solving, Problems 1-4, 13-18, and 50-57 on pgs. 740-741.in textbook??Complete Practice A 14-1 CRB.??Complete Reading Strategies 14-1 CRB.??Read textbook pgs. 744-750.??Complete Think and Discuss, pg. 745.in textbook??Complete Practice and Problem Solving, Problems 1, 2, 9, 10, and 29-34 on pgs. 746-747.in textbook??Complete Practice A 14-2 plete Reading Strategies 14-2 CRB.??Complete the Ready to Go On section, pg. 748.in textbookEvaluation:Complete Power Presentations Lesson Quiz 14-1, pg. 737 and 14-2, pg. 743.Modifications:Performance Tasks: IDEA works CDTask Websites 1 Unit 2 Unit 3 \Unit 4 5 6 7 8 ................
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