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The Georgia Department of Juvenile Justice 7th Grade MathematicsUnits of Instruction Resource Manual Table of Contents7th 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 InstructionChapter6: Units of InstructionUnit 1: Algebraic Reasoning Task 1Task 2Task 3Task 4Task 5Focus CAPsUnit 2: Integers & Rational Numbers, Applying Rational Numbers, Patterns & FunctionsTask 1Task 2Task 3Task 4Task 5Task 6Task 7 Task 8Task 9 Focus CAPsUnit 3: Proportional RelationshipsTask 1Task 2Task 3Task 4Task 5Task 6 Task 7Task 8Task 9Task 10Task 11 Focus CAPsUnit 4: PercentsTask 1Task 2Task 3Task 4 Focus CAPsUnit 5: Collecting Displaying and Analyzing DataTask 1Task 2Task 3Task 4Task 5 Focus CAPsUnit 6: Geometric FiguresTask 1Task 2Task 3Task 4Task 5Task 6Task 7 Focus CAPsUnit 7: Measurements Two Dimensional FiguresTask 1Task 2Task 3Task 4Task 5Task 6 Focus CAPsUnit 8: Three Dimensional Figures, Probability & Multi-Step Equations & InequalitiesTask 1- Culminating TaskTask 2 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 7th 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 7th 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 7th 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 StandardsM7A1 Students will represent and evaluate quantities using algebraic expressions.a. Translate verbal phrases to algebraic expressions.b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.c. Add and subtract linear expressions. M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations. M7A3 Students will understand relationships between two variables.a. Plot points on a coordinate plane.b. Represent, describe, and analyze relations from tables, graphs, and formulas.c. Describe how change in one variable affects the other variable.d. Describe patterns in the graphs of proportional relationships, both direct (y = kx) and inverse (y = k/x). M7D1 Students will pose questions, collect data, represent and analyze the data, and interpret results.a. Formulate questions and collect data from a census of at least 30 objects and from samples of varying sizes.b. Construct frequency distributions.c. Analyze data using measures of central tendency (mean, median, and mode), including recognition of outliers.d. Analyze data with respect to measures of variation (range, quartiles, interquartile range).e. Compare measures of central tendency and variation from samples to those from a census. Observe that sample statistics are more likely to approximate the population parameters as sample size increases.f. Analyze data using appropriate graphs, including pictographs, histograms, bar graphs, line graphs, circle graphs, and line plots introduced earlier, and using box-and-whisker plots and scatter plots.g. Analyze and draw conclusions about data, including a description of the relationship between two variables. M7G1 Students will construct plane figures that meet given conditions.a. Perform basic constructions using both compass and straight edge, and appropriate technology. Constructions should include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.b. Recognize that many constructions are based on the creation of congruent triangles. M7G2 Students will demonstrate understanding of transformations.a. Demonstrate understanding of translations, dilations, rotations, reflections, and relate symmetry to appropriate transformations.b. Given a figure in the coordinate plane, determine the coordinates resulting from a translation, dilation, rotation, or reflection. M7G3 Students will use the properties of similarity and apply these concepts to geometric figures.a. Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts.b. Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures.c. Understand congruence of geometric figures as a special case of similarity: The figures have the same size and shape. M7G4 Students will further develop their understanding of three-dimensional figures.a. Describe three-dimensional figures formed by translations and rotations of plane figures through space.b. Sketch, model, and describe cross-sections of cones, cylinders, pyramids, and prisms. M7N1 Students will understand the meaning of positive and negative rational numbers and use them in computation.a. Find the absolute value of a number and understand it as the distance from zero on a number line.b. Compare and order rational numbers, including repeating decimals.c. Add, subtract, multiply, and divide positive and negative rational numbers.d. Solve problems using rational numbers. M7P1 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.M7P2 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.M7P3 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.M7P4 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.M7P5 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.M7RC1 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. HYPERLINK ""?DJJ 7th Grade MathematicsGeorgia Performance Standards:? Curriculum Map??????????????????????????????????????????? 1st Semester2nd SemesterAlgebraic ReasoningIntegers & Rational NumbersApplying Rational NumbersPatterns & FunctionsProportional RelationshipsPercentsCollecting, Displaying & Analyzing DataGeometric FiguresMeasurements Two Dimensional FiguresThree Dimensional Figures, Probability & Multi-Step Equations& InequalitiesChapter 1CAPs1-7Chapter 2CAPs8-14Chapter 5CAPs 26- 31Chapter 6CAPs 32-36Chapter 7CAPs 37-42Chapter 8CAPs 43-49Chapter 9CAPs 50-54Chapter 10CAPs 55-58315-211159-63422-251264-68GPS:M7P2.c,dM7P5 a,b,cM7P3 a,c,dM7P1a,b,cM7P4 a,b,cM7A1a,b,cM7A2a,bGPS:M7N1.a,b,c,dM7P5.a,b,cM7P3 a,b,c,dM7P4a,b,cM7P2c,dM7A1 a,bM7P1 a,b,c,dM7A2a,bM7A3a,b,cGPS:M7P1a,b,c,dM7P4a,b, cM7P5a,b,c,M7P3a,cM7A3a,bM7N1b,dM7P2.cM7A2a,bM7G3a,bGPS:M7P4a,b,cM7P5a,b,cM7P1b,c,dM7P3a,b,cM7A2a,bM7A1.bGPS:M7D1a,b,c,f,gM7P4cM7P5a,bM7P1a,bM7P3a,cM7A2a,bM7A3a,b cM7P2.cGPS:M7P1a,b,c,dM7P3a,b,c,dM7P5a,bM7P4a,b,cM7P2a,b,c,dM7A1aM7A2a,bM7G1M7D1.fM7G3.cM7G2a,bGPS:M7P1a,b,c,dM7P4a,b,cM7P2a,b,c,dM7A3aM7P5a,bM7A1a,bM7A2a,bM7P3a,cM7A3.bGPS:M7P3a,b,c,dM7P5a,b,cM7P1a,b,c,dM7P4 cM7P2c,dM7A1a,bM7G3a,bM7N1.dM7A2a,bM7D1.gFocus CAPs:1&7Focus CAPs:Chapter 28&14Chapter 315 & 21Chapter 422 & 25Focus CAPs:26 & 31Focus CAPs:32 & 36Focus CAPs:37 & 42Focus CAPs:43 & 49Focus CAPs:50 & 54Focus CAPs:55 & 58Enduring Understandings & Essential QuestionsAlgebraic ReasoningEnduring Understandings:In mathematics, letters are used to represent quantities that vary. 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. The properties of real numbers are true for algebraic as well as numeric expressions. Algebraic expressions may be used to represent and generalize real situations. Inverse operations are helpful in understanding and solving equations. 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 a formula? What strategies can I use to help me understand and represent real situations using algebraic expressions and equations? What properties and conventions do I need to understand in order to simplify and evaluate algebraic expressions? How is an equation like a balance? How can the idea of balance help me solve an equation? Integers & Rational Numbers, Applying Rational Numbers,Patterns & FunctionsEnduring Understanding:Negative numbers are used to represent quantities that are less than zero such as temperatures, scores in games or sports, and loss of income in business. Absolute value is useful in ordering and graphing positive and negative numbers. Computation with positive and negative numbers is often necessary to determine relationships between quantities. Models, diagrams, manipulatives and patterns are useful in developing and remembering algorithms for computing with positive and negative numbers. Properties of real numbers hold for all rational numbers. Positive and negative numbers are often used to solve problems in everyday life. Essential Questions:When are negative numbers used and why are they important? Why is it useful for me to know the absolute value of a number? What strategies are most useful in helping me develop algorithms for adding, subtracting, multiplying, and dividing positive and negative numbers? What properties and conventions do I need to understand in order to simplify and evaluate algebraic expressions? Why is graphing on a coordinate plane helpful? Proportional Relationships Enduring Understandings: A dilation is a transformation that changes the size of a figure, but not the shape. The notation used to describe a dilation includes a scale factor and a center of dilation. A dilation of scale factor k with the center of dilation at the origin may be described by the notation (kx, ky). Two similar figures are related by a scale factor, which is the ratio of the lengths of the corresponding sides. The sides and perimeters of similar figures are related by a scale factor and the areas are related by the square of the scale factor. Scale factors, length ratios, and area ratios may be used to determine missing side lengths and areas in similar geometric figures. Three-dimensional objects can be created from two-dimensional plane figures through transformations such as translations and rotations. Cross-sections of three-dimensional objects can be formed in a variety of ways, depending on the angle of the cut with the base of the object. All three-dimensional objects can be built by stacking congruent or similar plane figures. Parallel cross sections of the same solid may be similar or congruent plane figures. Essential Questions:What is dilation and how does this transformation affect a figure in the coordinate plane? How can I tell if two figures are similar? In what ways can I represent the relationships that exist between similar figures using the scale factors, length ratios, and area ratios? What strategies can I use to determine missing side lengths and areas of similar figures? Under what conditions are similar figures congruent? What plane figures can I make by slicing a cube by planes? What about when I use cones, prisms, cylinders, and pyramids instead of cubes? How can I be sure I have found all possible cross-sections of a solid? When rotating or translating plane figures through space, what solids can I form? If I stack congruent or similar plane figures, what kinds of solids are formed?PercentsEnduring Understandings:Percent means “per hundred.” The symbol % also means percent or “per hundred.” Percent is a business term going back to the Roman times which is still used to measure interest rates, commission rates, tips, mark-ups, discounts, and tax rates.Percent is another way of writing a fraction with a denominator of 100. A percentage represents a part of a whole. When all fractional parts are included the result is equal to the whole and to one.Decimals, fractions, and percents are three equivalent ways of saying the same amount:0.30 is the same as saying “thirty hundredths”30/100 is also the same as saying “thirty hundredths”30% so when you say “thirty percent,” it means “thirty per hundred”When money is borrowed, you must pay to use it because someone else is losing an opportunity to use it while you have it. What you pay to use the money is called interest. The rate of interest is a percent. The money you borrow is called the principal. The formula used for interest isInterest = Principal x Rate of Interest x Time (I = P x R x T)Models, diagrams, manipulatives and patterns are useful in understanding, developing and remembering percentages. Essential Question:What are the advantages of using percents as a standardized means of making comparisons? Mental math lends itself to "solutions at the ready." What benefit might this provide for one when dealing withpercents in a store, restaurant, bank, or school? How does understanding percents make one a "smart consumer"? Collecting Displaying and Analyzing DataEnduring Understandings:Data can be represented graphically in a variety of ways. The type of graph is selected to best represent a particular data set. Measures of center (mean, median, mode) and measures of variation (range, quartiles, interquartile range) can be used to analyze data. Larger samples are more likely to be representative of a population. Conclusions can be drawn about data sets based on graphs, measures of center, and measures of variation. We can use graphs to investigate the relationship between data sets.Essential Questions:What is meant by the center of a data set, how is it found and how is it useful when analyzing data? How can I describe variation within a data set? In what ways are sample statistics related to the corresponding population parameters? How do I choose and create appropriate graphs to represent data? What conclusions can be drawn from data? Geometric FiguresEnduring Understandings:Coordinate geometry can be a useful tool for understanding geometric shapes and transformations. Reflections, translations and rotations are actions that produce congruent geometric objects. Many geometric constructions are based upon congruent triangles. Essential Questions:How can the coordinate plane help me understand properties of reflections, translations and rotations? What is the relationship between reflections, translations and rotations? Why is the definition of a circle a foundation for geometric constructions? In what ways can I use congruent triangles to justify many geometric constructions? Measurements Two Dimensional FiguresEnduring Understandings:Double number lines, models and manipulatives are helpful in recognizing and describing proportional relationships. The equation y = kx describes a proportional relationship in which y varies directly as x. The equation y = k/x describes a proportional relationship in which y varies inversely as x. Proportional relationships can be represented using words, rules, tables and graphs. Many problems encountered in everyday life can be solved using proportions. Essential Questions:How can I tell the difference between an inverse proportion and a direct proportion? In an inverse proportion, how do quantities vary in relation to each other? How can I decide if data varies directly or inversely? In what real world situations can I find direct and inverse variation? How can I determine the constant of proportionality in a proportional relationship by looking at a table, graph, or equation? Three Dimensional Figures, Probability & Multi-Step Equations & InequalitiesEnduring Understandings: Proficiency in computing with positive and negative rational numbers is vital to success in real life. Algebraic expressions may be used to represent relationships between variables when solving problems. Proportional relationships can be represented using words, rules, tables and graphs. Inverse operations and the basic properties of our number system are essential to understanding and solving linear equations. Constructions and transformations can be used to verify congruence. Essential Question:How do I add, subtract, multiply and divide using positive and negative rational numbers? In what ways are natural numbers, whole numbers, integers, and rational numbers alike?In what ways are they different? How can I use congruent triangles to justify geometric constructions? How could using the coordinate plane help me understand properties of reflections, translations and rotations? How can I be sure I have found all possible cross-sections of a solid? What information do I need to know in order to be assured that two figures are similar and/or congruent? How does an inverse proportion differ from a direct proportion? What role does solving equations play in solving real world problems? How can I formulate questions, gather data, display and decipher the results? Unit 1: Algebraic ReasoningGeorgia Performance Standards:M7P2 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.M7P5 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.M7P3 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.M7P1 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.M7P4 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.M7A1 Students will represent and evaluate quantities using algebraic expressions.a. Translate verbal phrases to algebraic expressions.b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.c. Add and subtract linear expressions.M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.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 HYPERLINK "" . 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: paperColored pencilsActivity Have you ever tried to walk on the beach when the sand was so hot it burned your feet? The following table contains data collected by heating dry sand with a heat lamp simulating what happens when the sun begins to heat the sand on a beach.Time (seconds)Temperature of Dry Sand (?C)0233023.960249024.912026.415027.218027.62102924029.827030.930031.933032.8What variable did you put on the x-axis? Explain your reasoning.Would it make sense to connect the points on this graph? Why or why not?Discuss how the temperature of the sand changed over time.The table below contains data collected by heating wet sand with the same heat lamp over the same period of time.Time (seconds)Temperature of Wet Sand (?C)0233023.36023.49023.912024.015024.618024.72102524025.227025.330025.733026Graph this data on the same coordinate plane you used for the dry pare the graphs. Discuss how they are alike and how they are different. What does this tell you?Can you give a scientific explanation for the differences in the heating of wet and dry sand?What does this data suggest about taking a walk on the beach?Discussion, Suggestions, Possible Solutions1.SolutionTime: Time is the independent variable in this case. Temperature depends on time.Make sure that students have labeled their axes and used appropriate scales.2.SolutionYes. It would make sense to connect the points. Time passes between points and temperature continues to increase between points. If we connect the points, it is easier to approximate what the temperature might be between the 30 second intervals. It may also be easier to see the trends in the graph. If we connect the points with a line segment, we are assuming a constant rate of change in the temperature over the time interval. Students may or may not be ready for this idea but it is worth discussing.3.SolutionThe temperature of the sand increased over time. The total amount of increase was 32.8-23 = 9.8 degrees centigrade. The relationship between time and temperature in this case is fairly linear. This is not the place to be formal about linear functions but it is good for students to notice that the graph has a somewhat linear shape.Questions to access knowledge:Explain the difference in temperature between 60 and 90 seconds.Explain the difference in temperature between 120 and 180 seconds.How long did it take for the temperature to increase one degree? Explain how this is illustrated in the table and on the graph.Possible Graph5.SolutionBoth graphs start at the point (0, 23) which means that the temperatures of the wet and dry sand were the same when the heating began. Both graphs are increasing which means that in both cases the temperatures increase over time. The graph for the dry sand lies above the graph for the wet sand. At each plotted point after time is 0, the temperature of the dry sand is greater than the temperature of the wet sand. This means that the dry sand is heating faster than the wet sand. Students may say that the graph for the dry sand is steeper than the graph for the wet sand. This is not the place for a formal discussion of slope but it can be mentioned.6.Possible solutionEvaporation occurs when water changes from a liquid to a gas. This requires an input of energy, usually heat energy. When evaporation occurs, the matter from which the heat energy is derived is cooled. Seventh grade students have not yet studied physical science but they may be able to relate this idea to the life science idea of sweating to cool the body or of a dog panting. Dogs pant to evaporate water from their tongues thereby lowering their body temperature.7.Possible SolutionThis suggests that sand closer to the ocean still wet, or more recently wet, from the tide coming in and out, will be cooler than sand further from the shore than has not been wet by the tide. So, if you don’t want your feet burned, walk by the water!Questions to access knowledge:Explain the difference in temperature between 60 and 90 seconds for both dry sand and wet pare the difference in temperature between 60 and 90 seconds for both dry sand and wet sand. How is this difference illustrated in the table and on the graph?Explain the difference in temperature between 120 and 180 seconds for both dry sand and wet sand. How is this difference illustrated in the table and on the graph?Compare the difference in temperature between 120 and 180 seconds for both dry sand and wet sand.How long did it take for the temperature to increase one degree? Explain how this is illustrated in the table and on the graph.(Time, Temp)(120, 24)(Time, Temp)(120, 26.4)CommentStudents may benefit from drawing vertical and horizontal lines to connect data points to the corresponding x-value and y-value on a graph. Students may also benefit from labeling the data points as a means of interpreting the data. Students should select, apply, and translate among mathematical representations. In this case, students should make a connection between the data in the graph and the data in the table.Time (seconds)Temperature of Dry Sand (?C)Temperature of Wet Sand (?C)023233023.923.3602423.49024.923.912026.424.015027.224.618027.624.7210292524029.825.227030.925.330031.925.733032.826Task: 2Resources: 3a + 4 4 6 2aA corner has been removed from this rectangle. Find an expression for the perimeter of the rectangle. Write your expression as simply as possible. What is the perimeter of the rectangle if a = ? inch? Show your calculations step-by-step.Find an expression for the area of the rectangle. Write your expression as simply as possible. What is the area of the rectangle if a = 1.8 feet? Show your calculations step-by-step.Discussion, Suggestions, Possible Solutions 1.SolutionThe perimeter of the rectangle is found by adding all of the sides. The sum of the sides is 4 + 3a + 4 + 6 + 2a + 2 + (3a+ 4 -2a). In simplest form this expression becomes: 3a +2a + a + 4 + 4 + 6 + 2 + 4 subtraction and commutative property of addition (3a +2a + a )+( 4 + 4 + 6 + 2 + 4) associative property of addition 6a + 20If a = ?, then 6(?) + 20 = 24.5 inches.2.SolutionThere are two ways to find this area. One is to treat the shape as the composition of two rectangles. The area of the smaller rectangle is 2 x 2a or 4a. The area of the larger rectangle is 4(3a + 4). Thus the area is:4a + 4(3a + 4)4a + 12a + 16distributive property16a + 16If a = 1.8 feet, then the area is 16(1.8) + 16 = 28.8 + 16 = 44.8 square feet.Task: 3 Resources: (see example 3) tiles or cubesActivity Jamal wants to buy a Sony PlayStation 3 with accessories. The entire package costs $234.10. Jamal has already saved $39. Every Saturday night Jamal’s Aunt Eunice comes over for dinner. Aunt Eunice has no children and is always interested in what Jamal is doing. He told her about the PlayStation and she made him a deal. Since she believes that saving money for the things that you want is a virtue, she will match every dollar Jamal saves 3 to 1 beginning at that moment. She will not match the $39 he has already saved. When Jamal has saved all of the money he needs, Aunt Eunice will take him, after Saturday night dinner, to buy the PlayStation.Jamal figures that he can save $4 per week. The Saturday that Aunt Eunice came for dinner was April 1st. On what day can Jamal pick up his play station? How much more do you need? If you save {1,2,3}, how much would she give you? At what point did he figure out it was April Fool’s Day?Show how you figured it out. You may use models, pictures, tables, etc. but you must also write and solve an equation, labeling your variables. Give a written explanation of your work.Discussion, Suggestions, Possible SolutionsCommentFor every dollar Jamal saves, Aunt Eunice will give him $3. Students might want to draw a model or use manipulatives to represent this situation.For example:JamalAunt EuniceTotal 1 $42 $4 + $4 = $8$4(2) = $83 $4 + $4 + $4 = $12$4(3) = $124 $4 + $4 + $4 + $4 = $16$4(4) = $16x$4(x) = 4xThis means that for every dollar Jamal saves, he actually saves four dollars. Students should be able to recognize that if x represents Jamal’s money, 3x will represent the amount of money that Aunt Eunice contributes. Jamal’s money plus Aunt Eunice’s money = x + 3x = 4xJamal has already saved $39. So, the $39 he has already saved plus the money he gets by saving and matching (4x) needs to equal $234.10. Possible solutionJamal’s money plus Aunt Eunice’s money plus $39 must add up to $234.10.Our equation is:Since x represents the amount of money that Jamal must save, Jamal must save $48.78. If he saves $4 per week, as he thinks he can, it will take him QUOTE or just a little more than 12 weeks to save the money. Decimals here should lead to some good discussion. What do we do about 12.195 weeks? Since we know that Jamal must have all of his money and he will buy the PlayStation on a Saturday night, he and Aunt Eunice can buy the PlayStation on a Saturday night 13 weeks from April 1st. This means he will buy the PlayStation on July 1st. Task: 4Resources: Reasoning Styrofoam cups with lips approximately ? of an inchGraph paperActivity You will be given a stack of identical paper cups similar to those shown in the picture below. The paper cups shown here are identical. By making appropriate measurements, you are to represent the relationship between the number of cups in a stack and the height of the stack using a table, a coordinate graph, a formula and a written description. In the case of each representation, discuss the advantage of that representation over the other three (e.g. What does the table tell you that the graph does not?) Height of stackLip of cupBase of cupUse your representations to answer each of the following questions.What is the height of 60 cups? Show how you know.How many cups will fit on a shelf that is 18 inches tall? Show how you know.Discussion, Suggestions, Possible SolutionsCommentsTeachers should get real cups for students to measure. The answers in this problem will depend on the size of the paper cups. Large styrofoam cups with lips approximately ? of an inch work well.For purposes of discussion, we will use cups for which the base is approximately 7 inches and the lip is ? of an inch.Students should measure and then begin working on their representations. Students may choose to measure the cups in centimeters.Number of CupsHeight of Lips (in)Height of Base (in)Height of Stack (in)1.7577.7521.578.532.2579.254371053.75710.7564.5711.575.25712.25The table needs enough values to show a pattern. Students may or may not include 0 cups but should understand that if the number of cups is 0, the height of the stack is 0. Some students may include a variable representation for the height of the stack given n cups. Advantages of the table over other representations include that the table shows exactly how much the height of the stack increases with each successive cup.A coordinate graph of the relationship should have the number of cups on the x-axis and the height of the stack on the y-axis since height depends on the number of cups. Make sure that students do not connect the points on the line since having a piece of a cup makes no sense in this case. Students should recognize that all the points lie on the same line. The graph shows that the height of the stack increases by a constant amount with each additional cup. (cups, height)(4, 10) The formula for this relationship is h = 7 + .75n where h represents the height of the stack and n represents the number of cups in the stack. One of the advantages of the formula over other representations is that it allows you to find the height of the stack for any number of cups or the number of cups given any height of the stack.One verbal description of this relationship is as follows:The base of the bottom cup is 7 inches high. Each lip is .75 or ? of an inch. Only one base is part of the height of the stack because all other cups are stacked inside of it. The lip of each cup contributes ? of an inch to the height, including the lip of the first cup. Therefore, the total height of the stack is the measure of the base of the bottom cup (7 inches) plus .75 inch (or ?) times the number of cups in the stack. One advantage of a verbal explanation is that it helps clarify other representations.Number of CupsHeight of Stack (in)10.75 in7.7520.75 in8.530.75 in9.25410510.75611.5712.25The formula for this relationship is h = 7 + .75n where h represents the height of the stack and n represents the number of cups in the stack.The height of 60 cups is QUOTE inches.The number of cups that can fit on a shelf that is 18 inches high is found by substituting 18 in our equation for h.Since there is no such thing as QUOTE or .67 parts of a cup in this situation, 14 cups can be stacked on a shelf that is 18 inches high. Discuss the remainder in the context of the problem. You may want to ask students why you would not round up in this situation. Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 1 Georgia Performance Standard(s): ??M7P2.c Develop and evaluate mathematical arguments and proofs. ??M7P2.d Select and use various types of reasoning and methods of proof. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. Objective(s): ??The student identifies and extends patterns. Students will represent numbers by using exponents. Instructional Resources: Holt Mathematics Course 2 Pgs. 6-13 Chapter 1 Resource Book (CRB) One-Stop Planner Activities: ??Complete Are You Ready? In textbook pg. 3 ??Read textbook pgs. 6-9. ??Complete Think and Discuss, pg.7 in textbook. ??Complete Practice and Problem Solving, Problems 1-4, 8-11, and 30-41 on pgs. 8-9 in textbook. ??Complete Practice A 1-1 CRB, pg. 3. ??Complete Reading Strategies 1-1 CRB, pg. 9. ??Read textbook pgs. 10-13. ??Complete Think and Discuss, pg. 11 in textbook. ??Complete Practice and Problem Solving, Problems 1-5. 11-20, 36-43, 53, and 60-69 on pgs. 12-1 3 in textbook. ??Complete Practice A 1-2 CRB, pg. 11. ??Complete Reading Strategies 1-2 CRB, pg. 17. Evaluation: Complete Power Presentations Lesson Quiz 1-1 and 1-2. Modifications: Performance Tasks: IDEA works CD Course Title: 7th Mathematics GaDJJ: State Code: 27.0220000 CAP: 7 Georgia Performance Standard(s): ??M7P2.d Select and use various types of reasoning and methods of proof. ????M7P3.a Organize and consolidate their mathematical thinking through communication. ????M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. ??M7P2.c Develop and evaluate mathematical arguments and proofs. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P1.c Apply and adapt a variety of appropriate strategies to solve problems. ????M7P3.d Use the language of mathematics to express mathematical ideas precisely. ????M7P5.b Select, apply, and translate among mathematical representations to solve problems. Objective(s): ????The student organizes and reviews key concepts and skills presented in Chapter One. ??The student assesses mastery of concepts and skills in Chapter One. Instructional Resources: Holt Mathematics Course Two Textbook Chapter 1 Resource Book (CRB) One Stop Planner Activities: ??Complete Study Guide: Review 1-51 on pgs. 64-66 in textbook. ????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ??Complete Chapter Test 1-45 on pg. 67 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 in textbook on pg. 67 with 80% accuracy. Modifications: Performance Tasks: IDEA Works Unit 2: Integers & Rational Numbers, Applying Rational Numbers, Georgia Performance Standards: M7N1 Students will understand the meaning of positive and negative rational numbers and use them in computation.a. Find the absolute value of a number and understand it as the distance from zero on a number line.b. Compare and order rational numbers, including repeating decimals.c. Add, subtract, multiply, and divide positive and negative rational numbers.d. Solve problems using rational numbers. M7P5 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.M7P3 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.M7P4 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.M7P2 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.M7A1 Students will represent and evaluate quantities using algebraic expressions.a. Translate verbal phrases to algebraic expressions.b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.M7P1 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.M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.M7A3 Students will understand relationships between two variables.a. Plot points on a coordinate plane.b. Represent, describe, and analyze relations from tables, graphs, and formulas.c. Describe how change in one variable affects the other variable.Selected Terms and Symbols:Variable: A symbol (often a letter) that represents a number.Proportion: An equation that states two ratios are equal.Ratio: A comparison of two quantities that share a fixed, multiplicative relationship.Rational Number: A number that can be written as a/b where a and b are integers, but b is not equal to 0.Equation: A mathematical sentence that contains an equal sign.Algebraic Expression: A mathematical phrase involving at least one variable and sometimes numbers and operation symbols. 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. -10051015-5 Task: 1 Resources: Which temperature is colder, -10 (ten below zero) or 0 (zero)?Plot both numbers on the number line. Plot both numbers on the number line.Which temperature is colder, -5 (five below zero) or 0 (zero)?Plot both numbers on the number line.Which temperature is warmer, -10 (ten below zero) or -5 (five below zero)? 4. Which temperature is warmer, -10 (ten below zero) or 15 (fifteen degrees)?Plot both numbers on the number line. 5. What do you notice about negative numbers?Discussion, Suggestions, Possible Solutions1.-10 degrees is colder than 0 degrees2.-5 degrees is colder than 0 degrees.3-5 degrees is warmer than -10 degrees4.15 degrees is warmer than -10 degrees5.Students should recognize all numbers below zero are located to the left of zero and all numbers above zero are located to the right of zero on the number line. Students should understand that when reading the number line from left to right, the numbers are ordered least to greatest; therefore, negative numbers are less than positive numbers and any positive number is greater than any negative number. Students should also understand the number to the left of any number is smaller in value. Consequently, the number to the right of any number is greater in value.Part IIGeorgia AvenuePeach RoadPulaski StreetBroad StreetPiedmont AvenueIf the park is located at zero on the number line, plot the location of the house and school if they are located one unit from the park. What do you notice about the placement of your plots on the number line? Plot the location of the house and school if they are two units from the park. What do you notice about the placement of your plots on the number line?Plot the location of the house and school if they are five units from the park. What do you notice about the placement of your plots on the number line?Plot the location of the house and school if they are nine units from the park. What do you notice about the placement of your plots on the number line?The distance between a number and zero on the number line is called absolute value. The symbol for absolute value is shown in this equation |8| = 8 and |-8| = 8.Explain |4|. Explain |-7|. Explain |8|.Explain |-21|. Explain |d|.Discussion, Suggestions, Possible SolutionsPart II.1.Both plots are one unit from zero. The distance between 0 and 1 is 1. The distance between -1 and 0 is also 1. There is one plot on positive one and one plot on negative one. 2.Both plots are two units from zero. The distance between 0 and 2 is 2. The distance between -2 and 0 is also 2. There is one plot on positive two and one plot on negative two. 3.Both plots are five units from zero. The distance between 0 and 5 is 5. The distance between -5 and 0 is also 5. There is one plot on positive five and one plot on negative five. 4.Both plots are nine units from zero. The distance between 0 and 9 is 9. The distance between -9 and 0 is also 9. There is one plot on positive nine and one plot on negative nine. 5.The absolute value of 4 is 4 because the distance between 0 and 4 is 4.6.The absolute value of -7 is 7 because the distance between 0 and -7 is 7.7.The absolute value of 8 is 8 because the distance between 0 and 8 is 8.8.The absolute value of -21 is 21 because the distance between 0 and -21 is 21.9.The absolute value of d is d because the distance between 0 and d is d, where d is any value.Task: 2 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 feetTask: 3Resources: Graph these numbers on the number line and then answer questions 1 – 4.How did you scale your number line? Explain why you chose this increment.Which number has the larger absolute value 1.8 or QUOTE ? How do you know? Look at the fractions and mixed numbers in this list. Which of these numbers, when written as decimals, are repeating decimals? Which form terminating decimals? Can you tell, without dividing, which fractions will repeat and which will terminate? How do you know?Compare your number line with a partner. Did you both use the same increment? Is one choice better than the other? Why or why not?Explain how you placed your numbers. Are your numbers in the same order? If not, decide who is correct and why.Discussion, Suggestions, Possible SolutionsFrom least to greatest:1.Students may scale their number lines using different increments. Tenths work well when plotting these numbers but scaling in tenths is not the only correct possibility. 2.The best way to compare the two numbers is to write them both in the same form, QUOTE is -1.75 in decimal form; therefore, 1.8 has the larger absolute value. Students should notice that 1.8 is farther away from 0 on the number line than is QUOTE . They should also be able to compare the decimal forms 1.8 (1.80) and 1.75 realizing that 1.8 is larger3.The fractions QUOTE and QUOTE form repeating decimals. The fractions QUOTE and QUOTE for terminating decimals.Students need to see that if the denominator of a fraction, in lowest terms, is a factor of (or divides evenly into) 10, 100, 1000, etc., (powers of 10) then the fraction will terminate. If not, it will repeat. Numbers whose only prime factors are 2 and/or 5 will divide evenly into powers of 10.4.The fractions QUOTE and QUOTE form repeating decimals. The fractions QUOTE and QUOTE for terminating decimals.Students need to see that if the denominator of a fraction, in lowest terms, is a factor of (or divides evenly into) 10, 100, 1000, etc., (powers of 10) then the fraction will terminate. If not, it will repeat. Numbers whose only prime factors are 2 and/or 5 will divide evenly into powers of 10.Task: 4Resources:education. Do you remember the activity books from elementary school? There were dots on a page numbered one to a larger number and you connected the dots to form a picture. Your task is to create a series of points on the coordinate plane that, when connected, will form a picture. You must then create a script that tells someone how to connect the dots. Your picture must have a minimum of eight points and there must be at least one point in each of the four quadrants.The example below is a simple version of what you could create.Script:Start at (-1,1) and draw a line segment to (-4, 1). Connect (-4, 1) to (-3,-1). Connect this dot to (3, -1) and continue with a line segment to (4,1). Connect this dot to (-1, 1).Continuing from (-1, 1), connect to (-1, 6). This will then be connected to (-4, 2). Connect this to (0,2). Finish the drawing by drawing a line segment from here to (-1,6).Discussion, Suggestions, Possible SolutionsThis task helps students apply their knowledge of points on a coordinate plane. It also can be used as a partner task. Each student could create a script and then have the partner create the image from this script. Teachers may find it beneficial to have students draw their pictures first and then identify the points to use in the script. For students who need extra practice, there are many commercially prepared activity books that have series of points listed for graphing that when completed, create a picture. Another activity would have teacher-created scripts for the vertices of basic geometric plane figures for students to plot and identify.A common game used in classrooms is a version of Hangman. Twenty-six points are located on the coordinate plane and are labeled with the letters of the alphabet. Students then ask their partners to identify a word clue by its coordinates. Teachers can also make up practice problems by posing silly riddles with the letters to the answers spelled out in point coordinates.This task can also be used in conjunction with most graphing calculators. Many calculators can be programmed to create the students’ pictures once the coordinate points have been listed into the program. Texas Instruments has published several activities using this skill. Going to the website education. may yield other activities suited to extension of this concept.By going to the site , students can demonstrate their knowledge of coordinate graphing by navigating a robot through a minefield. Also allows students to identify points by their coordinates interactively. The site also has many other games that students can use to increase their skill level in identifying and placing points on the coordinate plane.Task: 5Resources: the table below, you are given ten statements. You are to choose whether each statement is always true (A), sometimes true (S), or never true (N).If the statement is always true, you should be able to give a rule or a property that justifies your claim.If the statement is sometimes true, you should be able to give an example showing that it can be true and a counterexample showing that it can be false.If the statement is never true, you should again be able to give a rule or a property that is contradicted by the statement.1. 3a > 32. a + 2 = 2 + a3. -2b < 04. a + -a = 05. a – 6 > a6. 5a + 5b = 5(a + b)7. a - 2 = 2 - a8. a + 10 > 109. If two different numbers have the same absolute value, their sum is zero.10. If the sum of two numbers is negative, their product is negative. Discussion, Suggestions, Possible Solutions1.Sometimes true. If a is negative or a fractional value less than 1, 3 times a will be less than 3. Any example showing that the statement is sometimes true and a counterexample showing that the statement is sometimes false is a sufficient student response. However, you may want to discuss exactly when the statement is true and when it is false as a group.2.Always true. This is the commutative property for addition.3.Sometimes true. -2b < 0 if b is a positive number and -2b > 0 if b is a negative number. If b = 0, the statement is also false because -2 x 0 = 0 which is not less than zero. Some students may come up with this example. If not, it is worth asking the students to investigate what happens when b = 0.4.Always true. This is the property of additive inverses.5.Never true. If we take 6 away from a no matter what number a might be, the result will be smaller. If a is -2, -2 - 6 = -8. -8 is smaller than -2. If a is 0, 0 – 6 = -6. -6 is less than -2, etc.6.Always true. The distributive property.7.Sometimes true but only when a = 2. Then we would get 0 = 0. For all other values of a, this statement is false. It is important to bring out here that subtraction is not commutative. 8.Sometimes true. If a is any positive number, a + 10 >10. If a is 0 or a negative number a + 10 < 10. Any example showing that the statement is sometimes true and a counterexample showing that the statement is sometimes false is a sufficient student response. However, you may want to discuss exactly when the statement is true and when it is false as a group.9.Always true. The key word here is different. If two different numbers have the same absolute value, the numbers will always be opposites and therefore by the property of additive inverses their sum will be 0.10.Sometimes true. If the numbers have opposite signs, the product will be negative. -6 + 3 = -3. -6 x 3 = -18. However, if both numbers are negative, the sum will be positive. If one of the numbers is 0, the sum may be negative but the product will be 0 which is neither positive nor negative. 0 + -5 = -5. 0 x -5 = 0.Task: 6Resources: this task, your job is to create a CAP multiple choice test. You are given ten problems. For each problem, you are to write four possible “answer” choices. Your choices should include exactly one correct answer and at least one choice that contain common errors or misconceptions.On a separate sheet of paper, for each problem, show how you got the correct answer; and explain the choices you made based on common errors or misconceptions.So, here is the test!Evaluating and Simplifying Algebraic Expressions1. Which of the following is the value of 3a – 2b when a = 2 and b = -6?2. Which of the following is the value of 3a2 – 4 when a = -5?3. Which of the following is the value of 2(4 – b)2 – b when b = 5?4. Which of the following is the value of QUOTE when QUOTE ? 5. Which of the following is the value of (a + 7) – 3b when a = -3 and b = 2?6. Which of the following shows the expression -3m – 2 + 5m – 8 in simplest possible form with no parentheses and no like terms?7. Which of the following shows the expression 4(3a – 5) + 2(-3a + 7) in simplest possible form with no parentheses and no like terms?8. Which of the following shows the expression QUOTE in simplest possible form with no parentheses and no like terms?9. Which of the following shows the expression QUOTE in simplest possible form with no parentheses and no like terms?10. Which of the following shows the expression 2ab + a – ab - 7 in simplest possible form with no parentheses and no like terms?Discussion, Suggestions, Possible Solutions1. The correct answer is 3?2 - 2?(-6) = 6 + 12 = 18. Students may select any other choices they choose. One of the most common mistakes is to misunderstand that you are subtracting -12. Missing the sign, a student might get 6 – 12 or -6 rather than 18. 2. The correct answer is 3? (-5)2 -4 = 3?25 – 4 = 75 – 4 = 71.One mistake student often make is to square the product of 3 and -5. This would give an answer of 221.3. The correct answer is 2(4 – 5)2-5 = 2(-1)2 – 5 = 2?1 – 5 = 2 – 5 = -3.4. The correct answer is (10?2/5 – 6)/4 = (20/5 – 6)/4 = (4 – 6)/4 = -1/2. A common mistake is to divide part of the numerator by the denominator and not the whole numerator. For example, if a student calculated QUOTE and divided only the first term in the numerator (4) by 4, the result would be QUOTE – 6 = 1 – 6 = -5. Of course, there are also lots of mistakes that students might make in using the fractional value QUOTE for a.5. The correct answer is (-3 + 7) – 3?2 = 4 – 3?2 = 4 – 6 = -2. One common mistake at the step 4 – 3?2 is to subtract 3 from 4 before multiplying 3 times 2. This would yield an answer of 2.6. The correct answer is: -3m – 2 + 5m – 8 = 2m – 10. The commutative and associative properties of addition are used to simplify this expression. The signs when collecting the constants in a problem of this nature are often missed.7. The correct answer is 4(3a – 5) + 2(-3a + 7) = 12a – 20 + -6a + 14 = 6a – 6. The distributive property and the commutative and associative properties of addition are used to simplify this expression.8. The correct answer is QUOTE . Common mistakes include using the distributive property incorrectly and multiplying fractions incorrectly. 9. The correct answer is QUOTE . Mistakes with fractions are common.10 The correct answer is 2ab + a – ab – 7 = ab + a – 7. Common mistakes include combining the unlike term a with the terms 2ab and – ab and not recognizing – ab as having the same result as adding -1ab.Task: 7Resources: and Odd NumbersThe algebraic expression 2n is often used to represent an even number. Why do you think this is true? Illustrate your explanation with pictures and/or models.If 2n represents an even number, could n represent any number? Why or why not.Write an algebraic expression that could represent an odd number. Explain your thinking and illustrate with a picture or a model.What kind of number do you get when you add two even numbers? Justify your answer two different ways, by using models and by using algebraic expressions.What kind of number do you get when you add two odd numbers? Justify your answer two different ways, by using models and by using algebraic expressions.Consecutive Integers IThe lengths of the sides of the triangle below are consecutive integers.What are consecutive integers? Give examples.How can you represent the lengths of the sides of the triangle in terms of one variable? Explain your thinking.Consecutive Integers IISuppose three consecutive integers have a sum of -195. If x denotes the middle integer, how can you represent the other two in terms of x?Write an equation in x that can be used to find the integers.Show that the sum of any three consecutive integers is always a multiple of three. Task: 8Resources: are to make and present a poster showing what you have learned from your study of positive and negative rational numbers. Choose a theme for your poster. Be creative!Choose four rational numbers. At least two of your numbers should be between –1 and 1, one of which should be written as a decimal and the other should be written as a fraction. Two of the numbers should be positive and two of the numbers should be negative. Make your poster using the rubric below:ComparingUse the >, <, and = to compare your negative numbers. Graph all four numbers on a number line.Absolute valueWrite the absolute value of each number and explain what is meant by absolute value.Number problemsCreate two addition problems; one using numbers with like signs and the other using numbers with different signs.Create two subtraction problems; one using numbers with like signs and the other using numbers with different signs.Create two multiplication problems; one using numbers with like signs and the other using numbers with different signs.Create two division problems; one using numbers with like signs and the other using numbers with different signs.Model three of your problems with different operational signs.Three real-life problems with solutionsWrite three real-life problems involving rational numbers and solve to show their solutions. Use a different operation in each problem.Properties of real numbersUse two of your numbers to illustrate the commutative property of additionUse two of your numbers to illustrate the associative property of multiplication.Use your numbers to illustrate the distributive property.Give the additive inverse of one of your numbers.Give the multiplicative inverse of one of your numbers.Rules and common misconceptionsList any rules you have found for computing with positive and negative numbersGive examples of common misconceptions students have when working with positive and negative numbers Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 8 Georgia Performance Standard(s): ??M7N1.a Find the absolute value of a number and understand it as the distance from zero on a number line. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P5.c Use representations to model and interpret physical, social, and mathematical phenomena. ??M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P2.c Develop and evaluate mathematical arguments and proofs. ??M7A1.b Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate. ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P3.a Organize and consolidate their mathematical thinking through communication. Objective(s): ??The student compares and orders integers and determines absolute value. Students will add integers. Instructional Resources: Holt Mathematics Course 2, Pgs. 76-85. Chapter 2 Resource Book (CRB) One-Stop Planner Activities: ??Complete Are You Ready, in textbook pg. 73. ??Read textbook pgs. 73-81. ??Complete Think and Discuss, pg. 77 in textbook. ??Complete Practice and Problem Solving, Problems 1-8, 16-23, 31-38, and 50-59 on pgs. 78-79 in textbook. ??Complete Practice A 2-1 CRB, pg. 3. ??Complete Reading Strategies 2-1 CRB, pg. 9. ??Read textbook pgs. 82-87. ??Complete Think and Discuss, pg. 83 in textbook. ??Complete Practice and Problem Solving, Problems 1-11, 13-31, and 56-65 on pgs. 84-85 in textbook. ??Complete Practice A 2-2 CRB, pg. 11. ??Complete Reading Strategies A 2-2 CRB, pg. 17. Evaluation: Complete Power Presentations Lesson Quiz 2-1 and 2-2. Modifications: Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 14 Georgia Performance Standard(s): ??M7P2.d Select and use various types of reasoning and methods of proof. ????M7P3.a Organize and consolidate their mathematical thinking through communication. ????M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. ??M7P2.c Develop and evaluate mathematical arguments and proofs. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P1.c Apply and adapt a variety of appropriate strategies to solve problems. ????M7P3.d Use the language of mathematics to express mathematical ideas precisely. ????M7P5.b Select, apply, and translate among mathematical representations to solve problems. ????M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ????M7P1.a Build new mathematical knowledge through problem solving. Objective(s): ????The student organizes and reviews key concepts and skills presented in Chapter Two. ??The student assesses mastery of concepts and skills in Chapter Two. Instructional Resources: Holt Mathematics Course Two Textbook Chapter 2 Resource Book (CRB) One Stop Planner Activities: ??Complete Study Guide: Review 1-69 on pgs. 138-140 in textbook. ????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ??Complete Chapter Test 1-44 on pg. 141 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 in textbook on pg. 141 with 80% accuracy. Modifications: Performance Tasks: IDEA Works Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 15 Georgia Performance Standard(s): ??M7N1.c Add, subtract, multiply, and divide positive and negative rational numbers. ??M7N1.d Solve problems using rational numbers. ??M7P1.c Apply and adapt a variety of appropriate strategies to solve problems. ??M7P1.d Monitor and reflect on the process of mathematical problem solving. ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7P1.b Solve problems that arise in mathematics and in other contexts. Objective(s): ??The student estimates decimal sums, differences, products, and quotients. The student will add and subtract decimals. Instructional Resources: Holt Mathematics Course 2, Pgs. 150-157 Chapter 3 Resource Book (CRB) One-Stop Planner Activities: ??Complete Are You Ready, in textbook pg. 147. ??Read textbook pgs. 147-153. ??Complete Think and Discuss, pg. 151 in textbook. ??Complete Practice and Problem Solving, Problems 1-3, 8-13, 21-24, and 46-54 on pgs. 152-153 in textbook. ??Complete Practice A 3-1 CRB, pg. 3. ??Complete Reading Strategies 3-1 CRB, pg. 9. ??Read textbook pgs. 154-159. ??Complete Think and Discuss, pg. 155 in textbook. ??Complete Practice and Problem Solving, Problems 1-4, 10-17, and 48-57 on pgs. 156-157 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 and 3-2. Modifications: Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 21 Georgia Performance Standard(s): ?M7N1.c Add, subtract, multiply, and divide positive and negative rational numbers. ?M7N1.d Solve problems using rational numbers. ?M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ?M7P5.c Use representations to model and interpret physical, social, and mathematical phenomena. ?M7P1.a Build new mathematical knowledge through problem solving. ?M7P1.b Solve problems that arise in mathematics and in other contexts. ?M7A2.a Given a problem, define a variable, write an equation, solve the equation, and interpret the solution. ?M7A2.b Use the addition and multiplication properties of equality to solve one- and two-step linear equations. ?M7P3.a Organize and consolidate their mathematical thinking through communication. ?M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. Objective(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 Textbook Chapter 3 Resource Book (CRB) One Stop Planner Activities: Complete the odd number problems on the Study Guide, pgs. 212-213 in textbook. Complete the CRCT Review for the average learner, pgs. 218-219 in textbook. ????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ??Complete Chapter Test pg. 215 in textbook. ??H 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. 215 in textbook with 80% accuracy. Modifications: IDEA Works CD Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 22Georgia Performance Standard(s): ??M7A3.a Plot points on a coordinate plane. ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7A3.b Represent, describe, and analyze relations from tables, graphs, and formulas. ??M7P4.a Recognize and use connections among mathematical ideas. ??M7P4.b Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. ??M7P5.c Use representations to model and interpret physical, social, and mathematical phenomena. ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. Objective(s): ??The student plots and identifies ordered pairs on a coordinate plane. The student will identify and graph ordered pairs from a table of values. Instructional Resources: Holt Mathematics Course 2, Pgs. 224-231 Chapter 4 Resource Book (CRB) One-Stop Planner Activities: ??Complete Are You Ready, in textbook pg. 221. ??Read textbook pgs. 221-227. ??Complete Think and Discuss, pg. 225 in textbook. ??Complete Practice and Problem Solving, Problems 1-4, 13-16, and 39-48 on pgs. 226-227 in textbook. ??Complete Practice A 4-1 CRB, pg. 3. ??Complete Reading Strategies 4-1 CRB, pg. 9. ??Read textbook pgs. 228-231. ??Complete Think and Discuss, pg. 229 in textbook. ??Complete Practice and Problem Solving, Problems 1-3, 7-9, and 18-31 on pgs. 230-231 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 and 4-2. Modifications: Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 25 Georgia Performance Standard(s): ??M7A1.a Translate verbal phrases to algebraic expressions. ??M7A1.b Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate. ??M7A3.b Represent, describe, and analyze relations from tables, graphs, and formulas. ??M7A3.c Describe how change in one variable affects the other variables. ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P5.c Use representations to model and interpret physical, social, and mathematical phenomena. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7A3.a Plot points on a coordinate plane. M M7P3.a Organize and consolidate their mathematical thinking through communication Objective(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 Textbook Chapter 4 Resource Book (CRB) One Stop Planner Activities: Complete the odd number problems on the Study Guide, pgs. 258-259 in textbook. Complete the CRCT Review for the average learner, pgs. 262-263 in textbook. ????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ??Complete Chapter Test pg. 261 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. 261 in textbook with 80% accuracy. Modifications: IDEA Works CD Unit 3: Proportional RelationshipsGeorgia Performance Standards:M7P1 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.M7P4 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.M7P5 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.M7P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.M7A3 Students will understand relationships between two variables.a. Plot points on a coordinate plane.b. Represent, describe, and analyze relations from tables, graphs, and formulas.M7N1 Students will understand the meaning of positive and negative rational numbers and use them in computation.b. Compare and order rational numbers, including repeating decimals.d. Solve problems using rational numbers. M7P2 Students will reason and evaluate mathematical arguments.c. Develop and evaluate mathematical arguments and proofs.M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.M7G3 Students will use the properties of similarity and apply these concepts to geometric figures.a. Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts.b. Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures.Selected Terms and Symbols:Absolute value: The distance between a number and zero on the number line. The symbol for absolute value is shown in this equation |-8| = 8.Associative property: In addition or multiplication, the result of the expression will remain the same regardless of grouping. Examples: a + (b + c) = (a + b) + c; a(bc) = (ab)cCommutative property: The sum or product of numbers is the same no matter how the numbers are arranged. Examples: a + b = b + a; ab = baDistributive property: The sum of two addends multiplied by a number will be the sum of the product of each addend and the number. Example: a(b + c) = ab + acIntegers: The set of whole numbers and their opposites {… -3, -2, -1, 0, 1, 2, 3, …}Natural numbers: The set of numbers {1, 2, 3, 4, …}. Natural numbers can also be called counting numbers. Negative Numbers: The set of numbers less than zero.Opposite Numbers: Two different numbers that have the same absolute value. Example: 4 and –4 are opposite numbers because both have an absolute value of 4.Positive Numbers: The set of numbers greater than zero.Rational Numbers: The set of numbers that can be written in the form QUOTE where a and b are integers and b ≠ 0.Sign: a symbol that indicates whether a number is positive or negative. Example: in –4, the (–) sign shows this number is read “negative four”.Whole numbers: The set of all natural numbers and the number zero.Variation and proportion are defined to be the relationship between two or more variables with regard to a constant of proportionality.?x and y are directly proportional, if y = kx where k denotes a constant of proportionality and k ≠ 0.?x and y are inversely proportional, if xy = k where k denotes a constant of proportionality and k ≠ 0.?Variation problems describe the proportional relationship between two (or more) different values. In direct variation problems, we usually see the phrase “is directly proportional to” or “varies directly as”. For example, we can state a scale drawing problem two ways:Real size is directly proportional to scale size. Real size varies directly as scale size.We can use either one of these statements to write a general formula about real size and scale size. The real size (r) is equal to some number (k) times scale size (s). That is r = ks.Adapted from quantities, A and B, are in inverse proportion if by whatever factor A changes, B changes by the multiplicative inverse, or reciprocal, of that factor. Example: If A changes by a factor of 3, then B changes by a factor of 1/3.Adapted from : A mathematical sentence that contains an equal sign.Proportion: An equation which states that two ratios are equal.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: Activity In this task, you will use an equation that relates distance, rate and time to investigate how two quantities vary in relation to each other. The formula d = rt represents the fact that distance traveled is equal to the product of the rate of travel and the time traveled.You are to work with two or three other people to conduct an experiment and record your data. You will mark off a distance of 20 yards. Each person in your group will cover the twenty yards at four different speeds by walking, skipping, jogging, and running. You are to record distance, rate and time for at least one person in your group using the table you have created. When you return to the classroom, you should copy the data for everyone in your group into your table.Making the TableBefore going to the site of the experiment, you should create a table for recording your data. Remember that each person in your group will have four sets of data. You should record distance, rate and time (to the nearest second) for each person.Think about the equation, d = rt (distance = rate x time). Solve the equation for r (rate). Show your work. How will you determine each person’s rate? What units should you use for rate?Conducting the ExperimentYou will need to take the following items with you to the site of the experiment: a measuring tape, a stopwatch, a pencil, a calculator and the table you have created for recording data.To conduct the experiment, your group will need a walker, a timer and a recorder. Take turns performing these tasks. Make sure that each person in your group travels the 20 yards 4 times using different rates of speed (walking, skipping, jogging, and running) and that each person’s data is recorded. Time should be recorded to the nearest second.Analyzing the DataWhen you return to the classroom copy the data for everyone in your group into your table and use the data to complete the following.1. What do you notice about the distance?2. Which two quantities vary in this experiment? How do think these two quantities vary in relation to each other?3. In the coordinate plane, graph the ordered pair (time, rate) for each trial in your table.4. What does the graph tell you about the relationship between rate and time?5. Write the equation that is represented by your graph. How do we describe this type of equation? What is the constant of proportionality in this relationship and what does it represent?Discussion, Suggestions, Possible SolutionsEach group of students will need a measuring tape, a stopwatch and a calculator. Every student should have a pencil and their own chart, created ahead of time for recording data.Begin this task in the classroom by discussing the experiment and the equation d = rt. Students can be reminded of this equation using very simple examples (i.e. If a train travels 60 miles an hour for 6 hours, how far does it travel?).Ask students how they might find rate (speed) if they know how far the train has traveled and the time it took? Ask them to write the equation d = rt in terms of r. Applying this information to their experiment, make sure students understand how they will find the rate they are to record, if they know distance and time, and the units that should be used.The table might look something like this. It is important to let students think about the kind of table needed to organize their data.WalkingSkippingJoggingRunningPersonDTRDTRDTRDTRStudents should take turns performing the experiment, timing and recording data. When every student has completed four trials, return to the classroom. Give students time to copy the data for everyone in their group into their table before discussing the questions given.1. What do you notice about the distance? Distance is always constant. (20 yards)2. Which two quantities vary in this experiment? Rate and time are the quantities that vary.How do think these two quantities vary in relation to each other? As one increases, the other decreases.3. In the coordinate plane, graph the ordered pair (time, rate) for each trial in your table. Graphs will vary based on data but all should show an inverse proportion. Time (in seconds) should be graphed on the horizontal axis and rate (in yards per second) should be represented on the vertical axis based on the order of the ordered pair given (time, rate).4. What does the graph tell you about the relationship between rate and time? As time increases, rate decreases.5. Write the equation that is represented by your graph. How do we describe this type of equation? What is the constant of proportionality in this relationship and what does it represent?20 = rt or r = 20/t is an inverse variation or proportion. 20 is the constant of proportionality and represents the distance traveled in this case.Task: 2Resources: Activity If a grown man and a small child sit on opposite ends of a seesaw, what happens? Would changing or moving the weight on one end of the seesaw affect the balance? You’ll find out as you do the experiment in this investigation.Step 1: On a flat desk or table, try to balance a ruler across a pencil that is taped to the desk, near the ruler’s 6-inch mark.Step 2: Stack two nickels on the ruler so that they’re centered 3 inches to the right of the pencil. You may need to tape them in place.Step 3: Place one nickel on the ruler to the left of the pencil so that it balances the two right-side nickels. Be sure the ruler stays centered over the pencil. How far from the pencil is this one nickel centered?Step 4: Repeat Step 3 for two, three, four, and six nickels on the ruler to the left of the pencil. Measure to the nearest ?inch. Copy and complete the following table.Left sideRight sideNumber ofnickelsDistancefrom pencilNumber ofnickelsDistancefrom pencil122232Step 5: As you increase the number of nickels on the left side, how does the distance from the balance point change? What relationships do you notice?Step 6: Make a new table and repeat the investigation with three nickels stacked 3 inches to the right of center. Does the same relationship seem to hold true?Step 7: Review the data in your tables. How does the number of nickels on the left and their distance from the pencil compare to the number of nickels on the right and their distance from the pencil? Write a sentence using the words left nickels, right nickels, left distance, and right distance to explain the relationship between the quantities in this investigation. Define variables and rewrite your sentence as an equation.Step 8: Graph the equation you wrote in Step 7. How is this graph similar to or different from the graph of a direct variation?Discussion, Suggestions, Possible SolutionsStudents will need a pencil, a ruler, and 8 nickels.Step 3: The one nickel should balance at 6 inches from the pencil.Left sideRight sideNumber ofnickelsDistancefrom pencilNumber ofnickelsDistancefrom pencil16232323322341.5236123Step 4:Step 5: As the number of nickels on the left side increases, the distance from the pencil decreases. The number of nickels multiplied by the distance equals 6.Step 6: The same relationship holds—as the number of nickels on the left side increases, the distance from the pencil decreases. This time, the product of the number of nickels and the distance is 9.Left sideRight sideNumberof nickelsDistance frompencilNumber ofnickelsDistancefrom pencil1Can’t be done3324.533333342.253361.533Step 7: The product of the number of nickels on the left side and the distance from the pencil on the left side is equal to the product of the number of nickels on the right side and the distance from the pencil on the right side. The number of nickels and distance from the pencil are constant on the right side.(left nickels) · (left distance) = (right nickels) · (right distance) OR left nickels_ = right nickels right distanceleft distanceLet x = the number of nickels on the left side and y = the distance from the pencil on the left side. Let k, a constant, equal the product of the number of nickels and distance from the pencil on the right side. Then, xy = k.Step 8: As one quantity increases, the other decreases. The right side columns remained constant, and the product of the left side values remained constant. The graph of a direct variation is a straight line. The graph of the equation in this investigation is curved. As x increases, y decreases. The graph approaches the x axis but does not cross it.Graph of y = 6 / xGraph of y = 9 / x10108 86 64 42 2Task: 3Resources: Activity JDC Jones decided to give the students in Y cottage a party for receiving the Director’s Leadership Award this month. However, if a student receives a DR this week they will not be allowed to attend the party. JDC Jones ordered a large ice cream cake for the party, but is not sure how many students will attend. How will the number of students attending the party affect the portion each person gets? Does this represent a direct or inverse proportion? Explain your answer and illustrate with pictures, a table, and a graph. What is the constant of proportionality and what does it represent?Discussion, Suggestions, Possible SolutionsThe Number of People123456789Portion of the Cake for Each11/21/31/41/51/61/71/81/9As the number of people attending the party increases by a factor of n, the portion of cake each person receives decreases by a factor of 1/n.# peoplePortion of the cake for each person11 whole cake2??31/31/31/34????51/51/51/51/51/561/61/61/61/61/61/6This represents an inverse proportion because the portion of cake (c) is related to the number of people (p) by the equation c=1/p. As the number of people increases by a factor of n, the portion for each person decreases by a factor of1/n. The graph is a curve that does not cross the x or y axes. The constant of proportionality is 1 because cp = 1. In this case, the constant of proportionality represents 1 whole cakeTask: 4Resources: Activity The 7th Grade class is responsible for planning the Valentine’s Dance at Aaron Cohn YDC. Sarah is the chairperson of the decorating committee and knows that if she had to decorate the gym alone it will take her 40 hours. Sarah said, “The more people that help, the quicker we will get the job done.” Help her estimate how much time it will take to decorate the gym.1. Describe the relationship between the amount of time it will take to decorate the gym and the number of people working.2. How much time would it take for 2 people to decorate the gym? Four people? Make a table that includes 1-8 people working and decorating times for each number. How did you determine the amount of time each group would work?3. Create a graph to represent the situation. What do you notice?4. Could the point (2.5, 16) be on this graph? Explain why or why not.5. Would this graph ever cross either of the axes? Explain why or why not.6. Is this graph directly or inversely proportional? Justify your answer.7. Write an equation that shows the relationship between the time it will take to decorate the gym and the number of people working.8. How is the factor of change in the number of students working related to the factor of change in the total number of hours to decorate?9. What is the constant of proportionality and what does it represent?Discussion, Suggestions, Possible Solutions1. As the number of people increased by a factor of n, the number of hours to decorate changed by a factor of 1/n. The morepeople that work, the less total time it will take to decorate the gym.2. When the number of people working is 2, the 40 man-hours will be divided among the 2 people. So each person will work 20# PeopleTime140220313.34105866.775.785hours. When the number of people working is 4, the 40 man-hours will be divided among the 4 people. So each person will work 10 hours.To determine the number of hours needed, multiply 40 by 1/p, where p represents the number of people working. It is also possible to divide 40 by p.3. The graph is a curve that does not cross the vertical or horizontal axes and it shows that as the number of people increases by a factor of n, 40 hours decreases by a factor of 1/n.4. The point (2.5,16) cannot be on the graph because the x coordinate represents the number of people working and you cannot have 2.5 people.5. The graph would not cross the x-axis because this would mean that it took 0 hours to decorate which is not possible.The graph would not cross the y-axis because this would mean that 0 people decorated the gym and that would not be possible.6. The graph is inversely proportional because the product of the two quantities, number of people times the total number of hours to decorate, always equals 40.7. T = 1/n · 40 where T represents the total time to decorate and n represents the number of students working.8. The factor of change in the students working, n, is the reciprocal of the factor of change in the amount of time todecorate, 1/n. n and 1/n are also multiplicative inverses. For example, when the number of people changes by a factor of 2, the number of ours to decorate changes by a factor of ?.9. The constant of proportionality is 40 because nT always equals 40 where n is the number of people working and T is the total number of hours to decorate the gym. The number 40 represents the total number of man-hours needed to decorate the gym.Task: 5Resources: Activity How long do you think it would take you to place 100 post-it notes on the wall of your classroom? What if someone helped you? What if the entire class helped? Let’s investigate what happens.Your teacher will start by asking only two students to place the post-it notes and time how long it takes. By the end of the experiment, everyone in the class will be placing post-it notes on the classroom walls.1. What should happen to time as more students participate? Explain your answer.2. Complete the table below using your class data.number ofstudentstime23. Create a scatter plot for the data.4. Describe the pattern you see.5. What happens to the amount of time as the number of students increases? What mathematical terminology describes this situation?Discussion, Suggestions, Possible SolutionsYou will need 100 post-it notes for this task. They can be of any size. However, to accommodate less skillful students, you might want to use a larger size that is easily managed. Usually post-it notes come in packages of 50, so two packages will be sufficient. After the first two students perform the task, the teacher should remove the post it notes from the walls and put them into however many stacks to be used for the next group (5 students = 5 stacks).For each successive trial, you should (delete this) add 2-3 students to complete the task and record the time. The last trial should involve the entire class. The table the students fill in does not have the “number of students” column completed. This is to allow for differing class sizes. Ideally, the experiment should consist of 7-9 trials.The scatter plot, consisting of data for the 7-9 trials, will show the trend that as the number of students increases, the amount of time needed to complete the task will decrease. Students should recall creating scatter plots from Unit 1.Students may want to describe this as direct variation where k is negative. However, because of the variability in students and times needed there is no guarantee that the experiment will yield a constant value. This is a good opportunity to lead a discussion that not all variation is direct or inverse. This was discussed earlier in the “Name That Graph” task and can be explored graphically here. Sometimes there are trends in data that provide valuable information yet do not fit direct or inverse variation.Task: 6Resources: can I make with a rectangle??Construct a rectangle and label its vertices ABCD in order.?Construct a congruent rectangle A’B’C’D’ by translating ABCD in the plane in some direction other than vertical or horizontal.?Construct line segments connecting the vertices of ABCD with the corresponding vertices of A’B’C’D’.?This picture represents a shadow of what kind of solid? Why? (How many vertices does the solid have? How many edges?)?Make a three-dimensional model of the solid by translating your rectangle through space in a direction that is perpendicular to the plane containing ABCD. Explain how you made the solid. What kind of solid is it? What kinds of plane figures make up the faces of the solid??What is the volume and surface area of the solid? What information did you need to know to answer this question?Justify your answers.?What if you translate the rectangle through space in a direction that is not perpendicular to the plane containingABCD? What kind of solid would you have? Why?What else could I make??What solids could you make by translating a parallelogram through space??What solids could you make by translating a circle through space? Explain how you know.?What solids could you make by translating a square through space so that the distance of the translation is the same measure as the length of a side of the square? Explain how you know.Discussion, Suggestions, Possible SolutionsWhat can I make with a rectangle?A two-dimensional drawing of a rectangular prism (the shadow of a rectangular prism) is obtained by translating a rectangle through two-dimensions and connecting the vertices. The teacher might want to encourage the students to make both squares and rectangles that are not squares and have them compare their results.By translating a rectangle through space in a direction that is perpendicular to the plane containing the rectangle, the students form a right rectangular prism. By translating a rectangle through space in a direction that is not perpendicular to the plan containing the rectangle, the students form an oblique prism whose bases are rectangles but lateral faces are parallelograms. Task: 7Resources: to make each of the following cross sections by slicing a cube:a. Square.b. Equilateral triangle.c. Rectangle, not a square.d. Triangle, not equilateral. e. Pentagon.f.Regular hexagon.g. Hexagon, not regular. h. Octagon.i.Trapezoid, not a parallelogram. j.Parallelogram, not a rectangle. k. Circle.Record which of the shapes you were able to create and how you did it. If you can’t make the shape, explain why not. Describe, name, and sketch any additional cross-sections that are possible and explain why they are possible.Discussion, Suggestions, Possible Solutionsa. A square cross section can be created by slicing the cube by a plane parallel to one of its square faces.b. An equilateral triangle cross-section can be obtained by slicing off a corner of the cube so that the three vertices of the triangle are at the same distance from the corner.c. One way to obtain a rectangle that is not a square is by slicing the cube with a plane parallel to one of its edges, but not parallel to one of its square faces.d. If we slice off a corner of a cube so that the three vertices of the triangle are not at the same distance from the corner, the resulting triangle will not be equilateral.e. To get a pentagon, slice with a plane going through five of the six faces of the cube.f. To get a regular hexagon, slice with a plane going through the center of the cube and perpendicular to an interior diagonal.g. Any other slice that goes through all six square faces of the cube gives a non-regular hexagon.h. It is not possible to create an octagonal cross-section of a cube because a cube has only six faces.i. To create a trapezoid that is not a parallelogram, slice with a plane going through one face near a vertex through the opposite face at a different distance from the opposite vertex.j. To create a non-rectangular parallelogram, slice the cube by any plane that goes through two opposite corners of the cube but not containing any other vertex of the cube.k. It is not possible to create a circular cross-section of a cube because all slices are polygons with sides formed by slicing the square faces of the cube.Task: 8Resources: the possible cross-sections for these solids. Explain how you know that these are possible cross-sections. a. Cylinder. b. Cone. c. Sphere.Use models of the above solids to confirm your predictions. Sketch and describe the cross-sections.Name three plane figures that cannot be formed from cross-sections of the above figures and explain why they cannot be formed.Discussion, Suggestions, Possible Solutionsa. Possible cross sections for the cylinder are circles (cut parallel to the base), rectangles (perpendicular to the base), and ellipses.b. Possible cross sections for the cone are circles (cut parallel to the circular base), ellipses (cut at an angle, not parallel to the circular base and not intersecting the base of the cone), parabolas (cut parallel to the edge of the cone, not intersecting the vertex but intersecting the base), and hyperbolas (cut perpendicular to the base, but not intersecting the vertex).c. The only possible cross sections for a sphere are circles.Task: 9Resources: pyramidsActivityThe Great Pyramid of Giza in Egypt is often called one of the Seven Ancient Wonders of the world. The monument wasbuilt by the Egyptian pharaoh Khufu of the Fourth Dynasty around the year 2560 BC to serve as a tomb when he died. When it was built, the Great pyramid was 145.75 m high. The base is square with each side 231 m in length.Construct a model of the pyramid, with a base that is 1 foot on each side. Be sure to make the height proportional to the base just as in the real pyramid.Suppose the pyramid is sliced by a plane parallel to the base and halfway down from the top. What will be the shape of the top? What will the dimensions of the slice be? Justify your answer. What if the slice is 15% of the way down from the top? What if the slicing plane is not parallel to the base? What will the shape of the slice be under those conditions?Discussion, Suggestions, Possible SolutionsThe model of the pyramid should have a base that is 1 ft by 1 ft and .63 feet or about 7.5 inches high.If you slice the pyramid halfway down from the top, you’ll have a cross-section that is square and half the dimension of the base of the pyramid. If the slice is 15% of the way down from the top, you’ll have a square that is reduced in size from the base by 85%.Task: 10Resources: Someone gives you a flower vase.a) Suppose the vase was created by taking a rectangular prism and slicing one end of the prism by a plane, not necessarily parallel to the base. What plane figure is the shape of the top of the vase? How do you know?b) Suppose the vase was created by taking a right circular cylinder and slicing one end of the cylinder by a plane, not necessarily parallel to the base. What plane figure is the shape of the top of the vase? How do you know? c) Suppose the vase was created by taking a cylinder with an oval base. Is it possible to slice the top so that the slice is a circle? Why or why not?Discussion, Suggestions, Possible Solutionsa) If the base is a rectangle and the slicing plane is parallel to the base, then the top of the vase will be a rectangle. If the slicing plane is not parallel to the base, then the top of the vase will be a non-rectangular parallelogram. (Additional challenge: When will the slice be a rhombus?)b) If the base is a circle and the slicing plane is parallel to the base, then the top of the vase will be a circle. If the slicing plane is not parallel to the base, then the top of the vase would be a non-circular ellipsec) Yes, it is possible to slice the top so that the cross section is a circle. Using a toilet paper roll, squeezed to form an oval-based cylinder, angle the cut at the top so that you form a circle.Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 26 Georgia Performance Standard(s): ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P4.a Recognize and use connections among mathematical ideas. ??M7P4.b Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7P5.c Use representations to model and interpret physical, social, and mathematical phenomena. Objective(s): ??The student identifies, writes, and compares ratios. The student will find unit and compare unit rates, such as average speed and unit price. Instructional Resources: Holt Mathematics Course 2, Pgs. 270-277 Chapter 5 Resource Book (CRB) One-Stop Planner Activities: ??Complete Are You Ready, in textbook pg. 267. ??Read textbook pgs. 267-273. ??Complete Think and Discuss, pg. 271 in textbook. ??Complete Practice and Problem Solving, Problems 1, 2, 5-8, and 21-26 on pgs. 272-273 in textbook. ??Complete Practice A 5-1 CRB, pg. 3. ??Complete Reading Strategies 5-1 CRB, pg. 10. ??Read textbook pgs. 274-277. ??Complete Think and Discuss, pg. 275 in textbook. ??Complete Practice and Problem Solving, Problems 1, 2, 5, 6, 9-16, and 31-36 on pgs. 276-277 in textbook. ??Complete Practice A 5-2 CRB, pg. 12. ??Complete Reading Strategies 5-2 CRB, pg. 18. Evaluation: Complete Power Presentations Lesson Quiz 5-1 and 5-2. Modifications: Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 31 Georgia Performance Standard(s): ?M7G3.b Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures. ?M7P1.b Solve problems that arise in mathematics and in other contexts. ?M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ?M7P3.a Organize and consolidate their mathematical thinking through communication. M 7P5.c Use representations to model and interpret physical, social, and mathematical phenomena. Objective(s): ????The student organizes and reviews key concepts and skills presented in Chapter Five. ??The student assesses mastery of concepts and skills in Chapter Five. Instructional Resources: Holt Mathematics Course Two Textbook Chapter5 Resource Book (CRB) One Stop Planner Activities: Complete the odd number problems on the Study Guide, pgs. 318-319 in textbook. Complete the CRCT Review for the average learner, pgs. 324-325 in textbook. ?????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ???Complete Chapter Test pg 321 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. 321 in textbook with 80% accuracy. Modifications: IDEA Works CD Unit 4: PercentsGeorgia Performance Standards:M7P4 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.M7P5 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.M7P1 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 solve problems.d. Monitor and reflect on the process of mathematical problem solving.M7P3 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.M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.M7A1 Students will represent and evaluate quantities using algebraic expressions.Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.Selected Terms and Symbols:Percent- A ratio comparing a number to 100. The symbol % is used to indicate that a number is a percent.Percent of Change- The amount, stated as a percent, that a number increases or decreases.Percent of Change= Amount of Change Original AmountPercent of Decrease- A percent change describing a decrease in a quantity.Percent of Increase- A percent change describing an increase in a quantity.Interest- The amount that is collected or paid for the use of money.Simple Interest- The amount of money paid only on the principal.Principal- The amount of money deposited or borrowedTeacher’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, justifying 8. 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: favorite clothing store at the mall is having a sale. All items are 30% off. The tax rate is 6%. You buy an outfit including a shirt (orig.$21.30), shorts (orig.$29.50), and a pair of shoes orig. $49.60). Make a chart to show the discount on each item, the price after discount and the total price with tax.Discussion, Suggestions, Possible Solutions:In this problem the student will need to find out the percent of a number and create a chart to demonstrate their understanding of how to compute discounts, taxes, and total price after discount. The student’s chart may be set up in the following manner:Item PurchasedPrice before discount30%DiscountPrice after discountShirt$21.30Shorts$29.50Shoes$49.606%Tax Total PriceIn order to solve the problem the student will first need to convert 30% to a decimal which will be .30. the student will then need to multiply each item by .30 to figure out the amount of discount. The student will then need to subtract the discount from the original price to get the price of each item after discount. In order to get the total price with tax the student will need to add the price after discount of all items and find out the tax on the cost of the items. To do this they will convert 6% to a decimal or .06 and multiply. They will then add the tax to the price to arrive at the total price after discount.Answers:Item PurchasedPrice before discount30%DiscountPrice after discountShirt$21.30-6.39$14.91Shorts$29.50-8.85$20.65Shoes$49.60-14.88$34.726%Tax $3.32Total Price$58.69Task 2Resources: want to open a savings account so you decide to talk to your supervisor, Tim, to learn more about it. Tim says that putting away money in a savings account is a smart idea. He says he’s planning on making the down payment on a house with the money he earns on his account. He advises you to speak with a representative at the bank. He states that his bank’s rate of interest is 6.4% and that he earns $640 per year on the principal he deposited into his account. How much principal did Tim put into the account to earn that much in interest?Write the steps that you would use to solve this problem.Discussion, Suggestions, Possible Solutions: This is a simple interest problem in which the student will need to find the original amount or the principal. The students will use the formula I=prt to solve the problem. They will first need to rearrange the equation to solve for the principal. The rearranged formula will be : I =P RT (r)ate = 6.4% = 0.064(t)ime = 1 year(i)nterest = $640P = principal__$640 = P (1)(0.064) The principal is $10,000Task 3Resources: linen closet at Claxton RYDC was packed with 46 t-shirts, 39 shorts, 44 jumpsuits, 106 wash cloths, and 72 towels but still wouldn’t hold enough for the new shipment that was coming in. Officer Melvin decided to rearrange the closet to make more room. When she was done the closet had enough room to hold. 50 t-shirts, 50 shorts, 60 jumpsuits, 150 washcloths, and 150 towels. What was the percent of increase for each item? What was the overall percent of increase in storage capacity of the linen closet after Officer Melvin cleaned up and reorganized? Round your answers to the nearest whole number.Discussion, Suggestions, Possible Solutions: This is a percent of change problem which will require the student to demonstrate knowledge of the terms, percent of change, percent of increase, and percent of decrease. In order to solve the problem the student will need to know how to convert decimals to percent. The formula for figuring out the percent of change is: Percent of Change = Amount of Change Original AmountAs an example, for T-shirts the student would need to subtract the original number of T-shirts from the number of T-shirts that would fit in the closet after it was rearranged. This will give the amount of change. 50 t-shirts – 46 t-shirts = 4 (the amount of change)Then plug the information into the equation:Percent of Change = 4 t-shirts = .086 (round up to .09) or 9% 46 t-shirtsAnswer: The percent of change in space available for t-shirts is 9%Remaining Answers: T-shirts: .09 or 9%Shorts: .28 or 28%Jumpsuits: .36 or 36%Washcloths: .42 or 42%Towels: 1.08 or 108%Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 32 Georgia Performance Standard(s): ??M7P4.a Recognize and use connections among mathematical ideas. ??M7P4.b Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P5.c Use representations to model and interpret physical, social, and mathematical phenomena. ??M7P1.c Apply and adapt a variety of appropriate strategies to solve problems. ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7P1.d Monitor and reflect on the process of mathematical problem solving. ??M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. Objective(s): ??The student models and writes percents as equivalent fractions and decimals. The student will write decimals and fractions as percents. Instructional Resources: Holt Mathematics Course 2, Pgs. 330-335 Chapter 6 Resource Book (CRB) One-Stop Planner Activities: ??Complete Are You Ready, in textbook pg. 327. ??Read textbook pgs. 327-332. ??Complete Think and Discuss, pg. 331 in textbook. ??Complete Practice and Problem Solving, Problems1-3, 14-16, and 44-53 on pgs. 331-332 in textbook. ??Complete Practice A 6-1 CRB, pg. 3. ??Complete Reading Strategies 6-1 CRB, pg. 9. ??Read textbook pgs. 333-335. ??Complete Think and Discuss, pg. 334 in textbook. ??Complete Practice and Problem Solving, Problems 1-5, 12-16, and 32-37 on pgs. 334-335 in textbook. ??Complete Practice A 6-2 CRB, pg. 11. ??Complete Reading Strategies 6-2 CRB, pg. 17. Evaluation: Complete Power Presentations Lesson Quiz 6-1 and 6-2. Modifications: Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 36 Georgia Performance Standard(s): M7A1.b Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate. M7A2.b Use the addition and multiplication properties of equality to solve one- and two-step linear equations. M7P1.a Build new mathematical knowledge through problem solving. M7P1.c Apply and adapt a variety of appropriate strategies to solve problems. M7P1.d Monitor and reflect on the process of mathematical problem solving. M7P1.b Solve problems that arise in mathematics and in other contexts. M7P3.a Organize and consolidate their mathematical thinking through communication. M M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. Objective(s): ????The student organizes and reviews key concepts and skills presented in Chapter Six. ??The student assesses mastery of concepts and skills in Chapter Six. Instructional Resources: Holt Mathematics Course Two Textbook Chapter 6 Resource Book (CRB) One Stop Planner Activities: Complete the odd number problems on the Study Guide, pgs. 364-365 in textbook. Complete the CRCT Review for the average learner, pgs. 368-369 in textbook. ????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ??Complete Chapter Test pg. 367 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. 367 in textbook with 80% accuracy. Modifications: IDEA Works CD Unit 5: Collecting & Displaying Data Georgia Performance Standards:M7D1 Students will pose questions, collect data, represent and analyze the data, and interpret results.a. Formulate questions and collect data from a census of at least 30 objects and from samples of varying sizes.b. Construct frequency distributions.c. Analyze data using measures of central tendency (mean, median, and mode), including recognition of outliers.f. Analyze data using appropriate graphs, including pictographs, histograms, bar graphs, line graphs, circle graphs, and line plots introduced earlier, and using box-and-whisker plots and scatter plots.g. Analyze and draw conclusions about data, including a description of the relationship between two variables.M7P4 Students will make connections among mathematical ideas and to other disciplines.c. Recognize and apply mathematics in contexts outside of mathematics.M7P5 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.M7P1 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.M7P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.M7A3 Students will understand relationships between two variables.a. Plot points on a coordinate plane.b. Represent, describe, and analyze relations from tables, graphs, and formulas.c. Describe how change in one variable affects the other variable.M7P2 Students will reason and evaluate mathematical arguments.c. Develop and evaluate mathematical arguments and proofs. Selected Terms and Symbols:Dilation : Transformation that changes the size of a figure, but not the shape.Proportion: An equation which states that two ratios are equal.Ratio : Comparison of two quantities by division and may be written as r/s, r:s, or r to s.Scale Factor : The ratio of any two corresponding lengths of the sides of two similar figures. Similar Figures : Figures that have the same shape but not necessarily the same size. Congruent Figures : Figures that have the same size and shape.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, justifying 8. 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: Have you ever thought about your birthday or the birthdays of your classmates?Today we will take a survey of your class and find out the month of each person’s birth to see if we can draw any conclusions from this information.1) Describe how you collected your data. Which method did you choose for organizing your data and why?2) Is this data numerical or categorical? Tell how you know.3) Use an appropriate graph to display this data. Explain why you chose that type of graph.4) What interesting information can you determine from analyzing this data?Discussion, Suggestions, Possible Solutions Students may want to use a frequency table or line plot to keep their data organized as they gather it. Both of these methods will help make it easy to analyze the results and determine a good type of graphical display.1) Collect your data in an organized manner. Which method did you choose for organizing your data and why? Students may want to use a frequency table or line plot to keep their data organized as they gather it. Both othese methods will help make it easy to analyze the results and determine a good type of graphical display.2) Is this data numerical or categorical? Tell how you know.The information discovered here is called categorical information. This is because its value is a name, not a number. Information in the form of a number is called numerical. Line plots are one of the easiest ways to graph categorical information. The ability to quickly recognize and count values through the “x” representation helps many readers in evaluating a lone plot.3) Use an appropriate graph to display this data. Explain why you choose that type of graph.Line plots are one of the easiest ways to graph categorical information. The ability to quickly recognize and count values through the “x” representation helps many readers in evaluating a lone plot. However, there are other appropriate ways of displaying this information such as a histogram or circle graph.4) What interesting information can you determine from analyzing this data?Some interesting points may be that the information is not evenly distributed among the twelve months. The data should have at least one mode and more than likely, there will be students with the same birthday.Task: 2Resources: The principal at Macon RYDC was interested in finding out how long 7th graders spent on their CAP work each day. She randomly selected 12 students and asked them how much time they spent on CAP work the previous day. The responses of those 12 students are shown below.0 minutes20 minutes15 minutes1 hour30 minutes45 minutes1 hour1 hour and 10 minutes 15 minutes0 minutes15 minutes30 minutesYour job is to help the principal at Macon RYDC find out the average amount of time 7th graders spend on CAP work .1. What is the average amount of time these seventh graders spend on CAP work ?2. Explain how you determined your answer.3. Does your answer reflect the mean, the median or the mode? Explain how you know.4. If you had found a different measure of central tendency, would your answer be the same or different?Explain why or why not. Discussion, Suggestions, Possible SolutionsAdditional tasks in this unit will explore the relationship between mean, median, and mode and investigate when one measure of center is better than another. The purpose of this task is not to begin an in-depth analysis of statistical concepts, but instead to help teachers have a better understanding of the performance level of their students.1 and 2.a) Students may place the number of minutes over the number of minutes in an hour and work with fractions.0, 20 , 15 , 1,30 ,45 , 1, 110 , 15 , 0, 15 , 3060 60606060 606060Notice these fractions have not been simplified. Students may have simplified the fractions, however, in the nextproblem students will be adding each of the fractions, so keeping terms with a denominator of 60 may be more useful.To find the mean, add each of the fractions and divide by 12. Notice that division by 12 is necessary even though there are two values of 0 in the data. The mean must account for ALL data points.The sum of the data is 6. When divided by 12, this gives minutes.1 which students should be able to recognize as 302b) Students may have converted all the data into minutes, added the number of minutes (360) and divided by 12 (data points). This answer also results in 30 minutes.c) To find the median, the data must be put in order. Since there are 12 data points, the median is found by adding the 6th and 7th order data points and dividing by 2.If students chose the fraction method, this represents the data points in order.20 + 3050Median = 60 60= 60 = 50 × 1 = 50 which is 25 minutes. Again, students can work with simplified forms for 2 60 2 120this problem.The mode of the data is 15 minutes.1. Students should explain the method that they used and justify their answer. Result should vary.2. The mean=30 minutes, the mode = 15 minutes, the median = 25 minutes. 3. If students use the mean value of 30 minutes, then the percent of school time spent on CAP work is approximately 3% of class time is spent on CAP work. Task: 3 Resources: The data below represent grades of 7th grade math students at Augusta RYDC:84 71 100 88 23 100 81 93 92 100 87 751. Find the mean: 2. Find the median: 3. Find the mode: 4. Which of these numbers do you think best describes the middle of the data? Explain your reasoning.Remove the outlier from the data, and then do the following: 5. Find the mean: _____________________ 6. Find the median: ____________________ 7. Find the mode: _____________________Compare these results to the original measures. Keep your observations in mind for some questions later.Now go back to the original data set (with the outlier included). One of the scores listed is 92. Suppose that had been recorded incorrectly and that it should have been 99 rather than 92. How would that have affected these measures of center?Replace the score of 92 with a score of 99, and then do the following:8. Find the mean: 9. Find the median: 10. Find the mode: Compare these results to the original measures. Keep your observations in mind for the following questions.11. Which measure(s) of center take the value of every item of data into account? Explain your thinking.12. Which measure(s) of center are affected by outliers?Explain.13. Which of these numbers in questions 1-3 do you think best describes the middle of the data? Explain your reasoning.Options for differentiation:Suppose your math grade for this grading period is to be determined using 10 CAP grades, group work, and class participation scores. All of the scores are equally important. You get to decide which measure of central tendency, the mode, median, or mean, will be used for your grade.14. Would you ever prefer to use the median rather than the mean? If so, what would have to be true about the scores? If not, explain why you think using the median wouldn’t ever help your grade.15. Would you ever prefer to use the mean rather than the median? If so, what would have to be true about the scores? If not, explain why you think using the mean wouldn’t ever help your grade.16. Is it possible that you could prefer the mode rather than the median or mean? If so, what would have to be true about the scores? If not, explain why you think using the mode wouldn’t ever help your grade.Discussion, Suggestions, Possible Solutions84 71 100 88 23 100 81 93 92 100 87 751. Find the mean:Finding the mean is a concept/skill to maintain. The sum of the above numbers is 994 and dividing this by the number of tests (12) gives a mean of 82 5/6 or approx. 82.83.2. Find the median:Students should be familiar with median prior to seventh grade. Ordering these test scores yields 23, 71, 75, 81,84, 87, 88, 92, 93, 100, 100, 100. Because there is an even amount of data, the median is the mean of 87 and 88 or 87.5.3. Find the mode:This is another concept/skill that should be maintained from previous grades. Because 100 is listed more frequently than any of the other test scores, it is the mode.4. Which of these numbers do you think best describes the middle of the data?Explain your reasoning.At this point, students’ reasoning is extremely important. The explanation and justification are more important than a particular answer.Note: It is appropriate in this unit to treat the concept of outliers informally. Teachers may wish to point out, however, that there is a mathematical process for determining which data, if any, are outliers. Formally, outliers are often defined as data that lie more than 1.5 times the interquartile range beyond the quartiles. The computations below provide justification for considering 23 as an outlier.Q1 (first or lower quartile) = 78Q3 (third or upper quartile) = 96.5So the interquartile range (IQR) = 18.5. Multiply 18.5 by 1.5 to get 27.75.Now subtract 27.75 from Q1 and add it to Q3.78-27.75 = 50.2598+27.75 = 125.75The boundaries for the outliers have now been established. Data below 50.25 or above 125.75 are considered outliers. The only data point that meets those criteria is 23. Thus, 23 is indeed an outlier.Remove the outlier from the data, and then do the following:5. Find the mean:The new sum of the data is 971and dividing this by the new number of tests (11) gives a mean of 88 3/11 or approx. 88.276. Find the median:71 75 81 84 87 88 92 93 100 100 100Because we now have an odd number of grades, the middle number is 88.7. Find the mode:As in number 3, the test score that appeared the most is 100.Replace the score of 92 with a score of 99, and then do the following:8. Find the mean:The new sum of the data is 1001 and dividing this by the number of tests (12) gives a mean of 83 5/12 or approx.83.429. Find the median:The new data in order would be 23 71 75 81 84 87 88 93 99 100 100 100This does not affect the answer to #2 above, so the median is the same 87.5.10. Find the mode:As in number 3, the test score that appeared the most is 100.11. Which measure(s) of center take the value of every item of data into account?Explain your thinking.In this example, the mean changed but the median and mode were unaffected. The mean will always be affected by a change in the value of a single score. Depending on which score is changed, the median and/or mode could be changed as well.12. Which measure(s) of center are affected by outliers?Explain.The mean. Removing the outlier had no effect on the mode, and only a small effect on the median. However, the mean changed by almost 6 points.13. Which of these numbers in questions 1-3 do you think best describes the middle of the data? Explain your reasoning.Because 23 is an outlier, the median, 87.5, could be considered to be more representative of the center than the mean. The mode is clearly not a good representation of the center of the data because it is one of the extreme values (maximum). However, the measure that best represents the center actually depends on the intended use of that measure, and this question is intended to spark discussion of that. The mode might be used if the intent were to describe how “most” students scored. Task: 4Resources: The following table lists four of the greatest DJJ YDCs Baseball teams with the number of home runs they made each year.Macon WaycrossAugustaMetroYearHomerunsYearHomerunsYearHomerunsYearHomeruns19935419831198613199039199459198401988231991611995351985201989211992331996411986161990271993231997461987471991371994261998251988271992521995819994719893519933419961320006019904119944220015419914619953120024619923419964020034919933219975420044619944919983020055411995301999152006341996492000352007722199737200119199829200223199902003222004181. Study these records. Which center appears to be the greatest home run hitters? Why did you choose this center? Share your predictions with the class. 2. a. Make a stem and leaf plot for each center. b. Using your stem and leaf plot, create a grouped frequency table for each center. c. Using your frequency table, create a histogram for each center. d. Use your graphical displays to discuss the home runs of each center. 3. Find the mean, median and mode for each center. Compare the three values you computed for each center. Which value do you think best describes the performance of each center? Why? 4. Find the range and interquartile range for each center. Which center has the greatest range for home runs per year? Which center has the greatest interquartile range? In trying to determine which center perform consistently from year to year, would you compare ranges or interquartile ranges? Explain why you chose the measure you did. 5. Construct a box and whisker plot for each center and graph them together so that you are able to make comparisons. Write a paragraph describing what you can tell by comparing the four box and whisker plots.6. Based on your graphs and computations in problems 2 - 5, is your prediction still the same as it was in problem 1? Why or why not?Discussion, Suggestions, Possible Solutions 1. Share your predictions with the class.2. Make a stem and leaf plot for each center.Shown below are ordered stem and leaf plots for each center. An ordered stem and leaf plot makes finding the mean and quartiles much easier. Having the data ordered also makes it easier to identify the mode of the data.It is important to emphasize several points about these plots.?Make sure students include a key with the plots. ?Two of the centers, Waycross and Augusta, had years in which their home run totals were single digits. Be sure to point out how single digits are handled in a plot where everything else is a two-digit number.With the Metro plot, be sure students know that although there are no data values in the 40’s or 50’s, the 4 and 5 MUST be included in the plot. Part of the purpose of a stem and leaf plot is to be able to see the distribution of the data and to compare it to a histogram. If the 4 and 5 are not included, then the stem and leaf plot does not have the same shape as the corresponding histogram, and would be an incorrect representation of the data.Macon2 | 5 represents 25 home runsWaycross 1| 6 represents 16stemleaf00 0 11620 7 930 2 4 5 741 6 7 9 9 Stem leaf22534541166 6 7 95449 60Augusta 1/3 reprsents 13 home runsMetro 1/3 represents 13 home runsstemleaf081323 633 94561stemleaf135891233 730145 7402524b. Using your stem and leaf plot, create a grouped frequency table for each center.home runsfrequency0-9310-19120-29330-39540-495home runsfrequency20-29230-39240-49750-59360-691MaconWaycrossAugustaMetrohome runsfrequency 10-19420-29530-39540-49250-592d. Use your graphical displays to discuss the home run hitting of each player.The teacher should monitor discussion.3.MaconMean = 44Mode = 46Median = 46WaycrossMean = 29Mode = 0 and 49Median = 32AugustaMean = 30Mode = 23Median = 28.5MetroMean = 29Mode = noneMedian = 264.MaconRange = 38Interquartile Range = 19WaycrossRange = 49Interquartile Range = 25.5AugustaRange = 41Interquartile Range = 16MetroRange = 53Interquartile Range = 26Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 37 Georgia Performance Standard(s): ??M7D1.b Construct frequency distributions. ??M7D1.f Analyze data using appropriate graphs, including pictographs, histograms, bar graphs, line graphs, circle graphs, and line plots introduced earlier, and using box-and-whisker plots and scatter plots. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P1.a Build new mathematical knowledge through problem solving. ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7D1.c Analyze data using measures of control tendency (mean, median, and mode), including recognition of outliers. ??M7D1.g Analyze and draw conclusions about data, including a description of the relationship between two variables. Objective(s): ??The student organizes and interprets data in frequency tables, stem-and-leaf plots, and line plots. The student will find the mean, median, mode, and range of a data sheet. Instructional Resources: Holt Mathematics Course 2, Pgs. 376-385 Chapter 7 Resource Book (CRB) One-Stop Planner Activities: ??Complete Are You Ready, in textbook pg. 373. ??Read textbook pgs. 373-380. ??Complete Think and Discuss, pg. 378 in textbook. ??Complete Practice and Problem Solving, Problems 1, 4, and 16-21 on pgs. 378-380 in textbook. ??Complete Practice A 7-1 CRB, 3. ??Complete Reading Strategies 7-1 CRB, pg. 10. ??Read textbook pgs. 381-385. ??Complete Think and Discuss pg. 383. ??Complete Practice and Problem Solving, Problems 1, 2, 5, 6, 7, 16, and 20-23 on pgs. 384-385 in textbook. ??Complete Practice A 7-2 CRB, pg. 12. ??Complete Reading Strategies 7-2 CRB, pg. 19. Evaluation: Complete Power Presentations Lesson Quiz 7-1 and 7-2. Modifications: Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 42 Georgia Performance Standard(s): ??M7A3.a Plot points on a coordinate plane. ??M7A3.b Represent, describe, and analyze relations from tables, graphs, and formulas. ??M7A3.c Describe how change in one variable affects the other variable. ??M7d1.f Analyze data using appropriate graphs. ??M7D1.g Analyze and draw conclusions about data, including a description of the relationship between two variables. ??M7P1.a Build new mathematical knowledge through problem solving. ??M7P2.c Develop and evaluate mathematical arguments and proofs. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7d1.a Formulate questions and collect data from a census of at least 30 objects and from samples of varying sizes. ??M7D1.e Compare measures of control tendency and variation from samples to those from a census. Objective(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 Two Textbook Chapter7 Resource Book (CRB) One Stop Planner Activities: Complete the odd number problems on the Study Guide, pgs. 430-431 in textbook. Complete the CRCT Review for the average learner, pgs. 434-435 in textbook. ?????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ???Complete Chapter Test pg. 433 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. 433 in textbook with 80% accuracy. Modifications: IDEA Works CD Unit 6: Geometrical FiguresGeorgia Performance Standards:M7P1 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.M7P3 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.M7P5 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.M7P4 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.M7P2 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.M7A1 Students will represent and evaluate quantities using algebraic expressions.a. Translate verbal phrases to algebraic expressions.M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.M7G1 Students will construct plane figures that meet given conditions.M7D1 Students will pose questions, collect data, represent and analyze the data, and interpret results.f. Analyze data using appropriate graphs, including pictographs, histograms, bar graphs, line graphs, circle graphs, and line plots introduced earlier, and using box-and-whisker plots and scatter plots.M7G3 Students will use the properties of similarity and apply these concepts to geometric figures.c. Understand congruence of geometric figures as a special case of similarity: The figures have the same size and shape.M7G2 Students will demonstrate understanding of transformations.a. Demonstrate understanding of translations, dilations, rotations, reflections, and relate symmetry to appropriate transformations.b. Given a figure in the coordinate plane, determine the coordinates resulting fromSelected Terms and Symbols:Transformation: The mapping, or movement, of all the points of a figure in a plane according to a common operation.Reflection: A transformation that "flips" a figure over a line of reflectionReflection Line: A line that acts as a mirror or perpendicular bisector so that corresponding points are the same distance from the mirror.Translation: A transformation that "slides" each point of a figure the same distance in the same direction Rotation: A transformation that turns a figure about a fixed point through a given angle and a given direction. Bisector: A bisector divides a segment or angle into two equal parts.Parallel lines: Two lines are parallel if they lie in the same plane and they do not intersect. AB || CD denotes that AB is parallel to CD . Perpendicular lines: Two lines are perpendicular if they intersect to form right angles.AB ⊥ CD denotes that AB is perpendicular to CD .Congruent: Having the same size, shape and measure. ∠A ?∠ B denotes that ∠A is congruent to ∠B.Point: One of the basic undefined terms of geometry. Traditionally thought of as having no length, width, or thickness, and often a dotis used to represent it.Line: One of the basic undefined terms of geometry. Traditionally thought of as a set of points that has no thickness but its length goes on forever in two opposite directions. AB denotes a line that passes through point A and B.Plane: One of the basic undefined terms of geometry. Traditionally thought of as going on forever in all directions (in two-dimensions) and is flat (i.e., it has no thickness).Line segment or segment: The part of a line between two points on the line. AB denotes a line segment between the points A and B.Endpoints: The points at an end of a line segmentIntersection: The point at which two or more lines intersect or cross.Ray: A ray begins at a point and goes on forever in one direction.Corresponding sides: Sides that have the same relative positions in geometric figuresCorresponding angles: Angles that have the same relative positions in geometric figuresAngle: Angles are created by two distinct rays that share a common endpoint (also known as a vertex). ∠ABC or ∠B denote angles with vertex B.Angle of Rotation: The amount of rotation about a fixed point.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: study of Geometry was born in Ancient Greece, where mathematics was thought to be embedded in everything from music to art to the governing of the universe. Plato, an ancient philosopher and teacher, had the statement, “Let no man ignorant of geometry enter here,” placed at the entrance of his school. This illustrates the importance of the study of shapes and logic during that era. Everyone who learned geometry was challenged to construct geometric objects using two simple tools, known as Euclidean tools:A straight edge without any markingsA compassThe straight edge could be used to construct lines; the compass to construct circles. As geometry grew in popularity, math students and mathematicians would challenge each other to create constructions using only these two tools. Some constructions were fairly easy (Can you construct a square?), some more challenging, (Can you construct a regular pentagon?), and some impossible even for the greatest geometers (Can you trisect an angle? In other words, can you divide an angle into three equal angles?). Archimedes (287-212 B.C.E.) came close to solving the trisection problem, but his solution used a marked straight edge. What constructions can you create? Your First Challenge: Can you construct a Target?A very simple target consists of three circles. The largest circle would have a radius that is three times the length of the radius of the smallest circle, and the middle circle would have a radius two times the length of the radius of the smallest circle. On a separate sheet of paper, construct a target with a straight edge and compass then write a set of instructions that another student could use to create your target.Discussion, Suggestions, Possible SolutionsThe first day that students touch a compass should be spent “playing” by making pretty, colorful, and neat pictures or designs with the requirement of using many circles. This task is just one of many that may be used and provides a historical context for the activity. In doing any initial construction activity, students should “discover” these tips and if not by themselves, then through student sharing and teacher questioning:Pencil points must be very sharp.Using a different color of pencil for each change in radius helps understand congruences.Bearing down lightly makes it easier to manipulate the compass more accurately. (Some students may prefer to turn the paper than the compass.)Placing cardboard or several layers of paper under the paper helps prevent slips and tears.Teachers should have additional sheets of paper available for students who struggle with initial constructions and need to perform the constructions multiple times to do so with precision.With the target construction, students should learn how to construct circles proficiently and use the compass to construct congruent segments. Teachers may need to question students to ensure that they understand what a target is and what is required for the task. One strategy a student might use is to construct the center circle and a line that passes through the center and extends to both edges of the paper. The radius of this circle can then be copied by placing the point of the compass on the intersection of the line and the circle (point B), and the pencil at the center of the circle. As a new circle is constructed, it will intersect the line at the center of the small circle (point A) and the endpoint of the radius of what will be the middle circle (point C). The process can be extended to create the outer circle of the target. Another strategy may begin by using the diameter of the first circle constructed as the radius for the second circle.A B CWriting instructions for the constructions is a critical component of the task since it is a precursor to writing proofs. Just as students learn to write papers by creating and revising drafts, students need ample time to write, critique and revise their instructions. By using peers to proofread instructions, students learn both how to write clear instructions and how to critique and provide feedback on how to refine the instructions. Teachers should allot sufficient instructional time for this writing process.Possible Solution: A line segment is constructed through point C that is visually longer than AB . A circle with center C and radius AB is constructed. (Typically students will first construct a circle with center A and radius AB and use that measure to construct the second circle). Label one of the intersections of the circle with center C and the first lineThe following is one possible method thata student could choose to develop this construction.A circle with a radius ofAPwas A circle was draw with a center at Bdrawn with a center at P.and a radius ofBC.A second circle was drawn with aA congruent circle was drawn with aradius of BP and a center at P.center atB′.Congruent circles with radiiAC were drawn with centers at A and .A′The construction is completed upon drawing the radiusB′C′ and the radiusA′C′.Task: 2Resources: ActivityCan you imagine living in a two-dimensional world? What would it look like? What would a triangle look like? A square? A circle? Edwin A. Abbott envisioned this world when he wrote a book called Flatland: A Romance of Many Dimensions, a tale of shapes and their lives on a flat surface. In his vision, shapes with more sides (such as hexagons and octagons) were thought to be more intelligent than those with fewer sides (such as the two triangles shown below).Below are two triangles that could have been part of Flatland. They are complete opposites, true reflections of each other. Since they are opposites, they cannot tolerate anything about each other and fight constantly. As a superior being, can you construct a wall between them that separates them fairly? After you complete the task, write the triangles a letter explaining how you determined the fair line and why you think it divides their space evenly.BECADFlDiscussion, Suggestions, Possible SolutionsThis activity provides a literature connection for mathematics through the popular book Flatland. The book is appropriate for 9-12 grade readers, however students may find more information about the story at or by using an alternative version written at the 5th grade level, Life in Flatland (MH Edition). Teachers may choose to use the text for extensions by exploring ideas of what lines, fences, and squares would look like in Flatland and lookingat pattern blocks and spaghetti from different angles on a flat desk.Congruent circles were drawn with radii of more than half the distance of AD . The intersection of the two circles indicates where the “wall” should be drawn.This is the perpendicular bisector ofADRadii of the circles have been drawn to show the congruent triangles that verify the concept of perpendicular bisector.Students can construct the reflection by constructing a perpendicular bisector of AD . While they will not understand this terminology, from the previous exercise they should recognize if they construct circles with centers A and D and radii greater than half of AD, those circles will intersect on the reflection line. After students have constructed thereflection line teachers should ask guiding questions to help students to notice the features of a perpendicular bisector. Such questions include:?Is the midpoint of AD on the reflection line? (Yes.) How can you be certain? (One way to check this is to fold the paper with the endpoints matching. The crease should lie on the reflection line.)?How do you know your reflection line bisects AD ? (When the paper is folded as in the above question, the two segments are the same length.)?What other line segments are bisected by the reflection line? ( FD is one of the others.)?What can you conjecture about the angles created by the reflection line and AD ? (They are right angles.)Teachers should also introduce the terminology bisect, perpendicular lines, and perpendicular bisector. As students produce their letters or explanations of their work, teachers should continue to monitor the use of correct notation and terminology. Task: 3Resources: Activity You’ve already completed many challenging constructions, so you are ready for a final challenge: The Construction of a Regular Octagon. How would you construct a regular octagon? Discuss this with a partner and come up with a strategy. Think about what constructions might be needed and how they might be completed. Be prepared to share your ideaswith the class.Experiment to see if your strategy works and justify why your strategy works. Discussion, Suggestions, Possible SolutionsStudents may need guiding questions or prompts to assist them in this construction. A common approach to this construction requires the following constructions:?Constructing perpendicular lines?Bisecting AnglesTeachers should allow students to develop their own strategy to bisect an angle. One such approach is:?Create a circle (or arc) with center A that intersects both of the visible sides of the angle.?Connecting the intersection of the circle with the angle creates a line segment.?The construction method for bisecting a segment (or creating perpendicular bisector of a segment) follows naturally.Radii are drawn for this to show the congruent triangles that justify the angle bisection. Because of the vertex A, students may notice that it is not necessary to draw the complete circles for the perpendicular bisector. However, they certainly may do so.Seventh graders should know that each pair of segments are congruent because they are radii of the same circle. The concept of congruent circles is started in 6th grade when they are finding the circumference and area of circles with the same radii.Seventh graders should note that in any congruent figures, all corresponding angles are congruent and all corresponding sides are congruent. It would be very beneficial to students to begin to make these observations informally in the seventh grade to support later geometric understandings.Possible solutions to the octagon construction may include constructing a circle with a diameter, constructing a perpendicular bisector of the diameter, then bisecting each right angle. Alternatively, students may choose to begin with a line and construct a perpendicular bisector, bisect each of the right angles, then construct a circle to determine the vertices of the octagon.Task: 4Resources: Your task is to plot any creative polygon you want on the coordinate plane, and then create polygons congruent to the one you designed using the three translations described below.1. For each vertex of your original polygon in the form (x, y), create its image at the coordinates (x+4, y). 2. For each vertex of your original polygon in the form (x, y), create its image at the coordinates (x, y – 3).3. For each vertex of your original polygon in the form (x, y), create its image at the coordinates (x – 4, y+1).The vertices of your original polygon combined with their images must be mapped to points in all four quadrants of the coordinate plane to receive full credit.Options for differentiation:Provide a description of each of the following translations, where c can represent any number.1. (x + c, y)2. (x, y – c)3. (x – c, y)4. (x , y+ c)Discussion, Suggestions, Possible SolutionsAnswers will vary. Teachers should encourage students to make fairly simple polygons at first, but then move to more complicated designs. Students should also recognize through class discussion that all points, not merely integer coordinates would be translated using the notation (x+h, y+k). As an extension, teachers can use a variety of rational number coordinates.Options for differentiation:1. (x + c, y)If c is positive, this moves each point c units to the right (parallel to the x-axis).2. (x, y – c)If c is positive, this moves each point c units down (parallel to the y-axis).3. (x – c, y)If c is positive, this moves each point c units to the left.4. (x , y+ c)If c is positive, this moves each point c units up.Task: 5Resources: As you have discovered, parallel lines are present in any translation and in order to construct translations with a compass and straight edge, you first need to know how to construct parallel lines and why those constructions work.Let’s start by exploring features of parallel lines. In the figure below, lines m and n are parallel and the line t intersects both.?Label a new point C anywhere you choose on the line m. Connect B and C to form ?ABC.?Construct a point D on line n so that points D and C are on opposite sides of line t and AC = BD.?Verify that ?ABC is congruent to ?ABD.tmAnB1. Name all corresponding and congruent parts of this construction.2. What can you conclude about ∠CAB and ∠DBA? Will this always be true, regardless of where you choose C to be?Does it matter how line t is drawn? (In other words could line t be perpendicular to both lines? Or slanted the other way?)3. What type of quadrilateral is CADB? Why do you think this is true?Drawing a line that intersects two parallel lines creates two sets of four congruent angles. Use this observation to construct a parallel line to AB through a given point P.PABm4. Construct a perpendicular line to AB that passes through P. Label the intersection with line m as Q.5. Construct a perpendicular line to PQ that passes through P. Is this new line parallel to AB ? If so, how can you be certain?Discussion, Suggestions, Possible SolutionsSome teachers may choose to have students discover this construction, if time allows, however the focus of the activity is to observe properties of parallel lines which will be extended next year. These properties include that parallel lines are everywhere equidistant and that two lines perpendicular to the same line are parallel. It is important to prompt students to provide informal explanations as to why the lines they construct are parallel that move beyond statements such as “they just look parallel.”The construction of parallel lines using two perpendicular line constructions is presented here although there are other possible options. In the eighth grade, students will be introduced to the construction of parallel lines using a random line as the transversal. This will not only reinforce the constructions that they are doing in the seventh grade, but will also help their understanding of the special angles that are formed by a transversal (a line that crosses two or more lines) intersecting parallel lines.One way for seventh graders to verify that the two triangles are congruent is to cut them out hold them together. This is an easy way for them to notice the corresponding parts are congruent. The teacher may suggest that they label the parts in a way that the labels are visible after they are cut out.1. Name all corresponding and congruent parts of this construction.Students may find it difficult to recognize congruent triangles that share a side or are not a reflection at this point, so they are led first to examine the three sets of corresponding congruent sides.AC ? BD; AB ? AB; BC ? AD and ∨ CAB ? ∨ ABD; ∨ ACB ? ∨ ADB; ∨ ABC ? ∨ BAD2. What can you conclude about ∠CAB and ∠DBA? Will this always be true, regardless of where you choose C to be?Does it matter how line t is drawn?The angles will always be congruent because of the congruent triangles which were constructed. Students may wish to create additional constructions to verify this, but this is a key observation of the construction. It does not matter how line t is drawn.3. What type of quadrilateral is CADB? Why do you think this is true?The quadrilateral is a parallelogram. In the 5th grade, students develop the area formulas of rectangles, parallelograms and triangles based on dividing a parallelogram by its diagonal into two congruent triangles. Therefore, informal explanations that students may provide could include relationships between triangles and parallelograms, and each set of parallel lines appears to be the same distance apart because of the congruent segments they identified. Teachers may use guiding questions to reinforce properties of parallelograms:What makes a quadrilateral a parallelogram? (Opposite sides are both congruent and parallel.)When you divide a parallelogram in half along its diagonal, what is created? (Two congruent triangles are created.)What parts are congruent in a parallelogram? (Both opposite sides are congruent and opposite angles are congruent.)4. Construct a perpendicular line toAB that passes through P. Label the intersection with line m as Q.Drawing a circle with center P that intersects line m twicegives us a line segment.Construct the perpendicular bisector of this line segment by drawing two congruent circles with radii that are more than half the length of this segment using the intersections shown in the previous step as the centers.Connecting the last two circles’ intersections gives a perpendicular line toAB .6. Construct a perpendicular line to PQ that passes through P. Is this new line parallel to AB ? If so, how can you be certain?Constructing a circle with center P gives a new line segmentConstructing the perpendicular bisector of this segment formed by the intersection of the new circle and line PQ. using the methods mentioned earlier in this unit helps tocomplete the construction.This new perpendicular bisector is parallel to line m through point P. Radii have been drawn to help students to verify this relationship.Yes, this new line is parallel to the line segment AB. Students may fold the paper in half, lining up the perpendicular to show a reflection of the parallel lines; may argue that perpendicular lines create right angles that are congruent and from the previous exercise, they know they must be parallel; or may argue that they are forming half a rectangle or square with opposite sides parallel.Task: 6Resources: Have students visit the National Library of Virtual Manipulatives to explore and describe properties of rotation. Use the manipulative named, 9-12 Geometry “Transformations- Rotations” and click on “Activities” to access the following:1. Playing with Rotations2. Hitting a Target3. Describing Rotations Discussion, Suggestions, Possible SolutionsThe introduction to rotations using this web site may be done as a whole group with a projector or in a computer lab individually or in pairs. If the students work on the tasks as individuals or in pairs, prepare a list of questions for them to answer while exploring the web site. For example, “What determines the location of the image of a rotation?” or “If a rectangle is rotated 90 degrees counterclockwise, what happens to the coordinates of its vertices?” At the end of thissession, whether the activity is done as a whole group, individually, or in pairs, students should report to the whole class what they have learned.Teachers familiar with Geometer’s Sketchpad may also choose to utilize many published activities that will introduce translations to their students.Teachers may also choose to investigate the TransmoGrapher 2 at Interactivate Activities: Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 43 Georgia Performance Standard(s): ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P3.d Use the language of mathematics to express mathematical ideas precisely. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P1.a Build new mathematical knowledge through problem solving. ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P2.b Make and investigate mathematical conjectures. ??M7P3.b Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. ??M7A1.a Translate verbal phrases to algebraic expressions. ??M7A2.a Given a problem, define a variable, write an equation, solve the equation, and interpret the solution. ??M7A2.b Use the addition and multiplication properties of equality to solve one- and two-step linear equations. ??M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. Objective(s): The student identifies and describes geometric figures. The student will identify angles and angle pairs. Instructional Resources: Holt Mathematics Course 2, Pgs. 442-451 Chapter 8 Resource Book (CRB) One-Stop Planner Activities: Complete Are You Ready, pg. 439. Read textbook pgs. 439-447. ???Complete Think and Discuss pg. 443. ???Complete Practice and Problem Solving, Problems 1-3, 7-9, and 21-29 on pgs. 444-445. Problems 1-3, 10-12, and 27-32 on pgs. 450-451. ???Complete Practice A 8-1 CRB, pg. 3. ???Complete Reading Strategies 8-1 CRB, pg. 9. ?Read textbook pgs. 448-451. ????Complete Think and Discuss, pg. 449 in textbook. ????Complete Practice and Problem Solving, Problems 1-3, 10-12, and 27-32 on pgs. 450-451 in textbook. ????Complete Practice A 8-2 CRB, pg. 11. ????Complete Reading Strategies 8-2 CRB, pg. 17. Evaluation: Complete Power Presentations Lesson Quiz 8-1 and 8-2. Modifications: Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 49 Georgia Performance Standard(s): M7G2.a Demonstrate understanding of translations, dilations, rotations, reflections, and relate symmetry to appropriate transformations. M7P4.c Recognize and apply mathematics in contexts outside of mathematics. M7P5.b Select, apply, and translate among mathematical representations to solve problems. M7P3.a Organize and consolidate their mathematical thinking through communication. M7P4.a Recognize and use connections among mathematical ideas. M7P4.b Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. M M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. Objective(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 Textbook Chapter 8 Resource Book (CRB) One Stop Planner Activities: Complete the odd number problems on the Study Guide, pgs. 506-507 in textbook. Complete the CRCT Review for the average learner, pgs. 510-511 in textbook. ?????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ???Complete Chapter Test pg. 509 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. 509 in textbook with 80% accuracy. Modifications: IDEA Works CD Unit 7: Measurement Two Dimensional FiguresGeorgia Performance Standards:M7P1 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.M7P4 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.M7P2 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.M7A3 Students will understand relationships between two variables.a. Plot points on a coordinate plane.M7P5 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.M7A1 Students will represent and evaluate quantities using algebraic expressions.a. Translate verbal phrases to algebraic expressions.b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations. M7P3 Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.c. Analyze and evaluate the mathematical thinking and strategies of others.M7A3 Students will understand relationships between two variables.b. Represent, describe, and analyze relations from tables, graphs, and formulas.Selected Terms and Symbols:Cone – A three-dimensional object with a circular or elliptical base and one vertex.Base of a cone – The flat circular or elliptical portion of the cone.Oblique cone – A cone with a vertex that is not aligned directly above the center of the base.Right circular cone – A cone with a circular base and with a vertex that is aligned directly above the center of the base.Cross section - A plane figure obtained by slicing a solid with a plane.Cube – A regular polyhedron whose six faces are congruent squares.Cylinder – A three-dimensional object with two parallel congruent circular bases.Bases of a cylinder – The two congruent and parallel circular regions that form the ends of the cylinder.Oblique cylinder – A cylinder with bases that are not aligned one directly above the other.Right circular cylinder – A cylinder with circular bases that are aligned one directly above the other. Polyhedron – A collection of polygons joined at their edges. Each of these polygons is called a “face.” Prism – A polyhedron with two parallel and congruent faces and all other faces that are parallelograms.Bases of a prism – The two faces formed by congruent polygons that lie in parallel planes, all of the other faces being parallelograms.Lateral faces of a prism – A face that is not the base of the solid.Rectangular prism – A prism whose bases are rectangles.Right rectangular prism – A prism whose faces and bases are rectangles.Pyramid – A pyramid is a polyhedron with one face (the “base”) a polygon and all the other faces triangles meeting at a common point called the vertex.Base of a pyramid – The face that does not intersect the other faces at the vertex. The base is a polygonal region.Lateral faces of a pyramid – Faces that intersect at the vertex.Right pyramid – A pyramid that has its vertex aligned directly above the center of its base. 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: to the following website for this investigation: on “Show Values”.a. Change the scale by moving the red point on the segment in the top left corner. What do you observe when the scale is less than 1? Equal to 1? Greater than 1? As you are changing the scale, observe what is happening to the area of the red triangle and the ratio of the areas of the triangles. Describe what you observe. Why do you think this happens?b. Move the point X to different locations outside, inside, and on the triangle. What changes in the values do you notice as you move X? Explain why you think this happens.c. As you moved X in part b, other than the values, describe all the changes you noticed. Why do you think these changes occurred?d. What are some real-world situations in which this might be used?Discussion, Suggestions, Possible Solutions:Another web site where students can play with dilations is the National Library of Virtual Manipulatives (NLVM) site. Below is a direct link to the activities involving dilations. the NLVM site, students can explore dilations with different shapes. Suggested questions are provided on the NLVMweb site. In this investigation, students will observe the effects of a dilation with a given center and scale factor. Students should observe that the three lines passing through corresponding vertices of the two triangles intersect at a common point. This point is called the center of dilation. The distance from the center of dilation to the original triangle is reduced or enlarged according to the scale factor.a. The shape of the triangle remains the same. When the scale is less than 1, the red triangle is a reduction of ΔABC and is closer to the center of dilation. When the scale is equal to 1, the red triangle is congruent to ΔABC. When the scale is greater than 1, the red triangle is an enlargement of ΔABC and is further away from the center of dilation. The distance from X to a vertex of the red triangle depends on the scale factor (e.g., if the scale factor is0.5, the distance from X to a vertex of the red triangle is ? the distance from X to the corresponding vertex on the blue triangle).As the scale increases, the area of the red triangle increases because increasing the scale makes the red triangle larger. The ratio of the area of the red triangle to the blue triangle also increases because the numerator of the fraction representing the ratio is getting larger.[Note to teachers: Some students might observe that the ratio of the areas of the triangles is the square of the scale factor. This might not be obvious unless students look at the ratio for scale factors such as 0.5 or 2. This concept will be encountered in other activities, so it is not essential that students make this observation at this point.]b. The values do not change because the scale and the areas of the triangles are not changing. The only thing that changes is the location of X and the red triangle in relationship to the blue triangle.c. The lines through the corresponding vertices of the red triangle and ΔABC always intersect at a common point, X. [Teachers may want to tell students that this common point is called the center of dilation.]d. Projecting something on a screen (e.g., using an overhead projector), reducing or enlarging pictures, scale drawings (e.g., maps, building plans). Task: 2Resources: the ordered pairs given in the table to make six different figures. Draw each figure on a separate sheet of graph paper. Connect the points with line segments as follows:?For Set 1, connect the points in order. Connect the last point in the set to the first point in the set.?For Set 2, connect the points in order. Connect the last point in the set to the first point in the set.? For Set 3, connect the points in order. Do not connect the last point in the set to the first point in the set?For Set 4, make a dot at each point (don’t connect the dots).After drawing the six figures, compare Figure 1 to each of the other figures and answer the following questions.1. Which figures are similar? Explain your thinking.2. Describe any similarities and/or differences between Figure 1 and each of the other figures.?Describe how corresponding sides compare.?Describe how corresponding angles compare.3. How do the coordinates of each figure compare to the coordinates of Figure 1? If possible, write general rules for making Figures 2-6.4. Is having the same angle measures enough to make two figures similar? Why or why not?5. What would be the effect of multiplying each of the coordinates in Figure 1 by ??6. Translate, reflect, rotate (between 0 and 90°), and dilate Figure 1 so that it lies entirely in Quadrant III on the coordinate plane. You may perform the transformations in any order that you choose. Draw a picture of the new figure at each step and explain the procedures you followed to get the new figure. Use coordinates to describe the transformations and give the scale factor you used. Describe the similarities and differences between your new figures and Figure 1.Figure 1Figure 2Figure 3Figure 4Figure 5Figure 6Set 1Set 1Set 1Set 1Set 1Set 1(6, 4)(12, 8)(18, 4)(18, 12)(6, 12)(8, 6)(6, -4)(12, -8)(18, -4)(18, -12)(6, -12)(8, -2)(-6, -4)(-12, -8)(-18, -4)(-18, -12)(-6, -12)(-4, -2)(-6, 4)(-12, 8)(-18, 4)(-18, 12)(-6, 12)(-4, 6)Set 2Set 2Set 2Set 2Set 2Set 2(1, 1)(2, 2)(3, 1)(3, 3)(1, 3)(3, 3)(1, -1)(2, -2)(3, -1)(3, -3)(1, -3)(3, 1)(-1, -1)(-2, -2)(-3, -1)(-3, -3)(-1, -3)(1, 1)(-1, 1)(-2, 2)(-3, 1)(-3, 3)(-1, 3)(1, 3)Set 3Set 3Set 3Set 3Set 3Set 3(4, -2)(8, -4)(12, -2)(12, -6)(4, -6)(6, 0)(3, -3)(6, -6)(9, -3)(9, -9)(3, -9)(5, -1)Discussion, Suggestions, Possible SolutionsSource: Adapted from Stretching and Shrinking: Similarity, Connected Mathematics, Dale Seymour PublicationsStudents will find rules to describe transformations in the coordinate plane. Rules of the form (nx, ny) transform a figure in the plane into a similar figure in the plane. This transformation is called a dilation with the center of dilation at the origin. The coefficient of x and y is the scale factor. Adding a number to x or y results in a translation of the original figure but does not affect the size. Thus, a more general rule for dilations centered at the origin is (nx + a, ny + b).Students will also observe that congruence is a special case of similarity (n=1). Congruent figures have the same size and shape. As students learned in the previous unit, transformations that preserve congruence are translations, reflections, and rotations.Possible solutions:Note: The scale used on the x- and y-axes in the figures below is 2 units. Each square is 4 square units (2 x 2).Figure 1:Figure 2:Figure 3:Figure 4:Figure 5:Figure 6:1. Figures 1, 2, 4 and 6 are similar. Students may observe visually that these figures have the same shape but are different sizes (except for Figure 6). Figure 6 is congruent to Figure 1. Note that congruence is a special case of similarity – figures have the same size and shape. Figures 3 and 5 are longer (or taller) and skinnier. Students may also notice that corresponding angles are equal for all figures. The scale factor from Figure 1 to Figure 2 is2. The scale factor from Figure 1 to Figure 4 is 3. The side lengths of Figure 2 are twice the side lengths of Figure1 and the side lengths of Figure 4 are three times the side lengths of Figure 1. The scale factor from Figure 1 to Figure 6 is 1 because it is congruent to Figure 1. In Figures 3 and 5, one dimension increases by a factor of 3 and the other does not.2. Figure 2 is an enlargement of Figure 1. The figures have the same shape but different sizes.The ratio of the lengths of the corresponding sides is 1 to 2. The corresponding angles are equal in measure.Figure 3 is wider or longer than Figure 1. The figures are different shapes and sizes.The ratio of the lengths of the corresponding sides is not constant. For one dimension, the ratio is 1 to 3; for the other dimension, the ratio is 1 to 1.The corresponding angles are equal in measure.Figure 4 is an enlargement of Figure 1. The figures have the same shape but different sizes. The ratio of the lengths of the corresponding sides is 1 to 3.The corresponding angles are equal in measure.Figure 5 is taller than Figure 1. The figures have different shapes and sizes.The ratio of the lengths of the corresponding sides is not constant. For one dimension, the ratio is 1 to 3; for the other dimension, the ratio is 1 to 1.The corresponding angles are equal in measure.Figure 6 is the same shape and size as Figure 1. Figure 1 is shifted (i.e., translated) up and to the right to getFigure 6.The ratio of the lengths of the corresponding sides is 1 to 1. The corresponding angles are equal in measure.3. Figure 2: Both the x and y coordinates are multiplied by 2. (2x, 2y)Figure 3: The x coordinates in Figure 3 are three times the corresponding x coordinates in Figure 1; the y coordinates are the same. (3x, y)Figure 4: Both the x and y coordinates are multiplied by 3. (3x, 3y)Figure 5: The x coordinates in Figure 5 are the same as the corresponding x coordinates in Figure 1. The y coordinates are three times the corresponding y coordinates in Figure 1. (x, 3y)Figure 6: Two is added to both the x and y coordinates. (x + 2, y + 2)4.No. All angles of the figures (except angles of the smiles) have the same angle measures, but the figures are notsimilar. Figures 3 and 5 are long (or tall) and skinny, unlike Figure 1.5. The figure would shrink and the lengths of the sides would be half as long. [Note to teachers: Students may say that the new figure is “? the size” of the original figure which might imply that the area of the new figure is ? the area of the original. In actuality, the area of the new figure is ?? or ? the size of the original figure. Be sure that students understand that the side lengths are reduced by a factor of ?.]6. Answers will vary depending on the transformations that students use. Students must recall what they learned in the transformations unit about translations, reflections, and rotations. The translation, reflection, and rotation do not change the size or shape of the figure. The final figure is a reduction or enlargement of Figure 1 and it has a different orientation in the coordinate plane because of the reflection and rotation.Task: 3Resources: in this unit, you are going to be designing a logo for a club at your school. To prepare for this project, draw a non- rectangular shape in the coordinate plane so that portions of the shape are in each of the four quadrants. Explain what would happen to your shape if you transformed it using each of the given rules with the center of dilation at the origin.a. (4x, 4y)b. (0.25x, 0.25y)c. (2x, y)d. (3x, 3y + 5)e. (x + 5, y - 5)f.(? x - 1, ? y)g. Will any of the transformed figures be similar to the original figure? Explain.h. If you make a new figure by adding 2 units to the length of each side of your shape, will the two figures be similar? Why or why not? i.Write a general rule for transformations in the plane that produce similar figures.Discussion, Suggestions, Possible SolutionsThis task assesses students’ ability to identify the effects of transforming a figure according to a rule involving dilations and/or translations.Possible solutionsa. The figure would grow by a scale factor of 4. The distance from the origin to the object would increase by a scale factor of 4.b. The figure would shrink by a scale factor of 0.25. The distance from the origin to the object would decrease by a scale factor of 0.25.c. The figure would increase on one dimension by a scale factor of 2; the other dimension would stay the same.d. The figure would grow by a scale factor of 3 and move up 5 units.e. The figure would move right five units and down five units.f.The figure would shrink by a scale factor of ? and move left 1 unit.g. Figures a, b, d, e, and f will be similar to the original figure. Both dimensions increase by the same scale factor.Figure e will be congruent to the original figure because the side lengths and shape do not change. The ratio of the lengths of the corresponding sides will be 1:1 and the measures of the corresponding angles will be equal. Notethat congruence is a special case of similarity. [Figure e is congruent to the original figure.]h. The figures would not be similar. Adding a constant amount to each side will distort the figure. The ratio of the lengths of the corresponding sides will not be constant.i.(nx + a, ny + b) Task: 4Resources: students from dorm A are shooting hoops using the basketball goal outside their dorm. They think that the goal seems lower than the 10 ft. goal in the gym. They wonder how far the goal is from the ground. The students can’t reach the goal to measure the distance to the ground, but they remember something from math class that may help. First, they need to estimate the distance from the bottom of the goal post to the top of the backboard. To do this, they measure the length of the shadow cast by the goal post and backboard. They then stand a yardstick on the ground so that it is perpendicular to the ground, and measures the length of the shadow cast by the yardstick. Here are their measurements:Length of shadow cast by goal post and backboard: 5 ft. 9 in. Length of yardstick’s shadow: 1 ft. 6 in.Draw and label a picture to illustrate the student’s experiment. Using their measurements, determine the height from the bottom of the goal post to the top of the backboard.If the goal is approximately 24 inches from the top of the backboard, how does the height of the basketball goal outsideA dorm compare to the one in the gym? Justify your answer.Discussion, Suggestions, Possible Solutions:x3 ft.5.75 ft.1.5 ft.Find height of backboard, x, by solving the following proportion: x_ = _3_5.751.51.5x = 17.25x = 17.25 ÷ 1.5 x = 11.5 ft.If the goal is approximately 24 inches or 2 ft. from the top of the backboard, then the height of the goal is approximately9.5 ft. so A Dorm’s goal is about 6 inches (or ? ft.) lower than the goal in the gym.Solution Method 2: Students might choose to convert all the measurements to inches. x_ = 36691818x = 2484x = 2484 ÷ 18x = 138 in. or 11.5 ft.If the goal is approximately 24 in. or 2 ft. from the top of the backboard, then the height of the goal is approximately 114 in. or 9.5 ft., so A dorm’s goal is about 6 inches (or ? ft.) lower than the goal in the gym..Task: 5Resources: director of your center wants a new center logo and is sponsoring a contest to get ideas from students. The winner of the contest will receive a $100.00 gift certificate to be used to decorate their cottage. Your cottage really could use a make over, so you and your cottage mates have decided to enter the contest. Your task is to design a logo for the center that meets the following requirements:1. You must use constructions to create the logo.2. The design must include at least one translation, reflection, or rotation.3. Design your logo on an 8.5 inch by 11 inch coordinate plane so that it will fit on the front or back of a t-shirt.4. Dilate your design so that it will fit on a sign (no larger than 5 feet by 5 feet).Discussion, Suggestions, Possible Solutions:This task is designed to reinforce concepts learned in this unit and the previous unit on transformations. If students did the summative task from Unit 4, Analyzing Quilts, they should be able to use the coordinate plane to describe transformations. You may want to review that task before engaging students in the activity of designing a logo. Be sure that students are thinking about the concepts of similarity learned in this unit as they compare the logo for the t-shirt and for the sign. They should discuss relationships between corresponding sides and angles (or other parts of the logo), perimeters, and areas. They may verify scale factors by measuring or estimating (e.g., estimate area of dilated logo by observing approximately how many t-shirt logos will fit in the sign logo).One way to dilate the t-shirt design is to trace the design on a transparency and project it onto a sheet of poster board using an overhead projector. Students could then trace the design on the poster board. To find the scale factor for the dilation, students could measure various parts of the logos and find the ratio of corresponding parts.Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 50 Georgia Performance Standard(s): ??M7P1.c Apply and adapt a variety of appropriate strategies to solve problems. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P1.a Build new mathematical knowledge through problem solving. ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P2.c Develop and evaluate mathematical arguments and proofs. ??M7A3.a Plot points on a coordinate plane. ??M7A3.b Represent, describe, and analyze relations from tables, graphs, and formulas. ??M7P5.b Create and use representations to organize, record, and communicate mathematical ideas. ??M7A1.b Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate. ??M7A2.a Given a problem, define a variable, write an equation, solve the equation, and interpret the solution. ??M7A2.b Use the addition and multiplication properties of equality to solve one- and two-step linear equations. ??M7P3.a Organize and consolidate their mathematical thinking through communication. Objective(s): ??The student compares the precision of measurements and determines acceptable levels of accuracy. The student will find the perimeter of a polygon and the circumference of a circle. Instructional Resources: Holt Mathematics Course 2, Pgs. 518-527 Chapter 9 Resource Book (CRB) One-Stop Planner Activities: Complete Are You Ready, in textbook pg. 515. Read textbook pgs. 515-523. ???Complete Think and Discuss, pg. 519 in textbook. ???Complete Practice and Problem Solving, Problems 1-3, 13-18, 37-40, and 52-59 on pgs. 520-521 in textbook. ???Complete Practice A 9-1 CRB, pg. 3. ???Complete Reading Strategies 9-1 CRB, pg. 9. Read textbook pgs. 524-529. ???Complete Think and Discuss, pg. 525 in textbook. ???Complete Practice and Problem Solving, Problems 1-3, 11-13, 26, and 32-38 on pgs. 526-527 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 and 9-2. Modifications: Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 54 Georgia Performance Standard(s): ??M7A3.b Represent, describe, and analyze relations from tables, graphs, and formulas. ??M7P2.c Develop and evaluate mathematical arguments and proofs. ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P1.c Apply and adapt a variety of appropriate strategies to solve problems. ??M7P4.a Recognize and use connections among mathematical ideas. ??M7P4.b Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P2.a Recognize reasoning and proof as fundamental aspects of mathematics. ??M7P2.b Make and investigate mathematical conjectures. ??M7P2.d Select and use various types of reasoning and methods of proof. ??M7A1.a Translate verbal phrases to algebraic expressions. ??M7A1.b Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate. Objective(s): ????The student organizes and reviews key concepts and skills presented in Chapter Nine. ??The student assesses mastery of concepts and skills in Chapter Nine. Instructional Resources: Holt Mathematics Course Two Textbook Chapter 9 Resource Book (CRB) One Stop Planner Activities: Complete the odd number problems on the Study Guide, pgs. 566-567 in textbook. Complete the CRCT Review for the average learner, pgs. 572-573 in textbook. ????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB to cover concepts not understood. ???Complete Chapter Test pg. 569 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. 569 in textbook with 80% accuracy. Modifications: IDEA Works CD Unit 8: Three Dimensional Figures, Probability & Multi-Step Equations& InequalitiesGeorgia Performance Standards:M7P3 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.M7P5 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 phenomenaM7P1 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.M7P4 Students will make connections among mathematical ideas and to other disciplines.c. Recognize and apply mathematics in contexts outside of mathematics.M7P2 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.M7A1 Students will represent and evaluate quantities using algebraic expressions.a. Translate verbal phrases to algebraic expressions.b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.M7G3 Students will use the properties of similarity and apply these concepts to geometric figures.a. Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts.b. Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures.M7N1 Students will understand the meaning of positive and negative rational numbers and use them in computation.d. Solve problems using rational numbers.M7A2 Students will understand and apply linear equations in one variable.a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.M7D1 Students will pose questions, collect data, represent and analyze the data, and interpret results.g. Analyze and draw conclusions about data, including a description of the relationship between 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 bell ringers (web-based bell ringer links are located in the resource(s) section above). 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:, , federal government has established a grant that is to be awarded to one seventh grade student. This grant wasdesigned as a research experiment to determine the effect of joining the work force while still in middle school. The result of this experiment will be considered when Congress votes on possibly lowering the minimum age requirement for work permits. A maximum amount of $1,000,000 will be awarded for the selected student to start his/her own business. You have decided to apply for this grant.Part 1:You Are UniqueOne of the conditions of the grant is that the business must be unlike any other that is currently in existence.a) Write a description of your business. Include its purpose and why you believe that there will be a demand for this service or product. Give reasons for your belief that this business will eventually make a profit. In other words, explain how the world will be a better place because of this new idea and why people will be willing to spend money for it.b) Give your business a unique name and slogan that will be easy for people to remember and easy for them to associate with its purpose. Explain your reasons for choosing this name and this slogan.c) Design a creative logo for your business. Use at least three of the basic geometric constructions in the design of your logo. Show the development of this logo with your construction markings and label all endpoints and circle centers. Report how this logo will help people remember the name of your business and its objective. A printing company will be reproducing the logo and requires that it be positioned on a Cartesian plane. Neatly transfer the final version of your logo onto an 8 in. by 11 in. Cartesian plane (portrait orientation) so that portions of it are in all four quadrants. Remember that it will be displayed on all of your stationary, employee uniforms, advertisements, etc.Give this portion of the application to your teacher so that he/she may verify the originality of your idea before you proceed with the next part.Part 2: Being a BossA second requirement of the grant is for you to demonstrate your budgeting and staffing skills. You will be given information about potential employees to assess your skills in these areas.a) Setting SalariesAlthough your business will start off small, it will be large enough to require that you hire three employees.1. Using a website similar to , , or other types of resources (i.e. newspapers, periodicals, etc.) as a guide, collect at least thirty samples of salaries that would be appropriate for you to offer your three employees based on their responsibilities and requirements to assure that your business is a successful venture. Be sure to record this information in an organized fashion. Explain how and why you choose that particular group of thirty salaries.2. Find the measures of central tendency, measures of variation, quartiles and possible outliers for your salary data. Justify your work for determining each of the above measures.3. Make the most appropriate display of the data.From this information, determine the salary to be offered to your employees and explain why this is the salary you have chosen. Make certain to reference your statistics and any displays you create in your explanation.b) Keeping RecordsIndicate to the grant review committee that you are able to organize information, record this information in a logical manner, complete accurate computations and make wise decisions concerning employees. You are to use the given information provided by a company that began last year to show that you are capable of handling large amounts of money appropriately.Make a table or chart that indicates the amount of loss or profit for each of three regions of business. For the sake of this assignment, these regions will be called A, B, and C.1. The first month, the business had a total loss of $493,827.12.oRegion A was responsible for 50% of the total loss. Regions B and C equally shared the remaining loss for the month.oDefine a variable, write and solve an equation to determine the amount of loss for each of the three regions.2. The second month the business broke even, meaning profits equal losses.oRegion A had a profit of $201,609.33. Region B’s loss was half that of Region C’s.oDefine a variable, write and solve an equation to determine the amount of loss for Regions C and B.oShow how you know that your result is correct.3. As with the previous month, the third month the business broke even.oThis time, although Region B had a profit, the mean of the incomes from Regions A and B was a loss of$117,200.00.oExplain how you determined each region’s amount of profit or loss for this month.4. Make an appropriate display of the data for regions A, B, and C during the three months.Analyze the results of this display.5. Company policy is to give an award each quarter to the best region. You are asked to recommend one of the regions for this award. Explain which one you believe should receive the award and how you came to your decision.Part 3: Water in the officeA third requirement of the grant is that you demonstrate that you are able to take care of office details.1. You have decided to have bottled water available in the office. The water bottle company charges $0.75 for each bottle of water that is delivered.a. Express the cost (y) in terms of the number of bottles of water delivered (x).b. Find the range of dollar amounts (y values) when the number of bottles (x values) ranges from 24 bottles to216 bottles.c. Make a table/chart that shows the relationship of x and y.d. Graph this relationship, name the constant of proportionality; and tell whether the relationship is a direct or inverse proportion. Explain how you know.2. A special offer of a substantial discount has been presented to new customers when the first order is 12 boxes with 24 bottles of water each.The water company will continue to offer this price for all other orders throughout the first year of business, regardless of the size of the orders.a. If the discounted price for this special offer is 2/3 of the everyday price, express the cost (y) in terms of the number of bottles of water delivered (x).b. Find the range of dollar amounts (y values) when the number of bottles (x values) ranges from 24 bottles to the number of bottles in the special first order offer.c. Make a table/chart that shows the relationship of x and y.d. Graph this relationship, name the constant of proportionality; and tell whether the relationship is a direct or inverse proportion. Explain how you know.e. What would the total cost (without tax) be to have this special offer delivered to your office? Show how you know.3. The water bottle company has told you that the average person drinks 1.6 bottles of water each day. Let x represent the number of people drinking bottled water. Let y represent the number of days those people are drinking water.a. Using an estimate of 1.6 bottles per day, write an equation that represents the relationshipbetween x, y and the number of bottles of water in the special first year offer. Solve your equation for y. b. Make a table/chart that shows the relationship between x and y.c. Graph this relationship, name the constant of proportionality; and tell whether the relationship is a direct or inverse proportion. Explain how you know.d. How long will it take to use up all of the water that was ordered with the special discount when you have yourself and three employees in the office each day? Justify your answer.Turn in your grant application to your teacher so that he/she may proofread it for any errors prior to submitting it to the federal government.Part 4: The Final PhaseWow! You have just received a letter notifying that you are one of the top 52 finalists! The federal government has picked one from each of the 50 states, the District of Columbia and Puerto Rico. Your grant application was the best one from Georgia.To help the review committee make a decision as to which student will be awarded the grant, they are asking that each finalist determine how they will publicize their service or product.a) Building a Billboard1. You have decided to use the 8 in. by 11 in. logo design that you did on the coordinate plane in Part 1 as part of a design for a billboard advertisement to tell the world about your new company. The billboard will be 10 ft. by7 ft. in a landscape orientation. Use another piece of four-quadrant graph paper to show your billboard design. This design must include a dilation of the original logo.State the scale factor of the dilation. Explain how you know.Give the new dimensions of the logo on the billboard and justify your answer.2. The committee has requested that you tell them the distance from the top of the billboard to the ground..To do this, you have decided to use similar triangles formed by using shadows.Draw an illustration of the billboard and its shadow. Label the length of the shadow as 12 ft.Next to this, draw an illustration of yourself and your shadow. Label the length of your shadow as 1 ft. 6 in. and the illustration of your self as your actual height to the nearest inch.Define a variable, write and solve an equation that uses the information shown in your illustration to determine the height of the billboard. Justify your work.b) The InterviewA member of the committee that will determine the recipient of the grant is meeting with you, one of your parents, and your math teacher today to interview you and review the billboard information. Your parent picked you up from school to take you to the appointment, but your teacher could not leave until 5 minutes later. Your parent drove 65 mph for the entire trip, but your teacher drove 70 mph using the same directions. If you and your teacher reached the meeting place at the same time, how long did it take for your math teacher to drive to the appointment location? Use pictures, models, tables and/or graphs and explain your thinking. Define your variable, make up an equation that describes the situation and solve it. How far away was this meeting place? Show how you know.c) A Special GiftThe art teacher at your school and the students in the seventh grade art class are excited about the fact that you have progressed this far in your quest for the grant. They want your help in designing a statue to be donated to the school in your honor. The statue will be created by dilating your logo and sweeping it through space to make a three-dimensional figure.Help your classmates with this design by clearly writing out the directions for the statue. Be sure to include the scale factor used and how the logo should be swept through space.Make a model of the statue. State the dimensions of the actual statue and explain why you decided to use these dimensions.Oh dear! You were taking the model to the art room so that the art teacher could see what you have in mind when you accidentally dropped it. When it hit the floor, a piece of the model broke off in a neat and smooth break. Sketch the two pieces of your three-dimensional model so that you are showing the congruent cross sections that were formed by the break.Part 5:Congratulations!Discussion, Suggestions, Possible SolutionsThroughout this culminating activity, or one similar in rigor and depth, the teacher should monitor students for understanding and help students correct any obvious weaknesses.Each “Entrepreneur” should enjoy developing an innovative and creative business adventure. Therefore, due to the fact that each product should be original and unique, all solutions will vary.Part 1: You Are UniqueAll sections of Part 1 will be different for each student.a) Setting SalariesAll sections of Part 2a “Setting Salaries” will be different for each student.b) Keeping RecordsMake a table or chart that indicates the amount of loss or profit for each of three regions of business. For the sake of this assignment, these regions will be called A, B, and C.Part 2: Being a Boss1. The first month, the business had a total loss of $493,827.12.oRegion A was responsible for 50% of the total loss. Regions B and C equally shared the remaining loss for the month.oDefine a variable, write and solve an equation to determine the amount of loss for each of the three regions.Let A = income from Region A. Let B = income from Region B. Let C = income from Region C.One of several possible solutions for the first month’s record is shown below:A + B + C = -493,827.12A = .5(A + B + C) to find 50%.A = .5(-493,827.12)by substitution.A = -246,913.56 using multiplication.B = C because B and C equally share the remaining loss. So, A + 2B = -493,827.12 by substituting B for C.2B = -493,827.12 – A by subtracting from both sides of the equation.2B = -493,827.12 – (-246,913.56) by substituting the above value for A.2B = -493,827.12 + 246,913.56 because the inverse of a negative is a positive.2B = -246,913.56 by using additionB = -246,913.56 by dividing both sides of the equation by 2.2B = -123,456.78 by division.So Region A has a loss of $246,913.56. Region B and Region C each have a loss of $123,456.78 in the first month.2. The second month the business broke even, meaning profits equal losses.oRegion A had a profit of $201,609.33. Region B’s loss was half that of Region C’s.oDefine a variable, write and solve an equation to determine the amount of loss for Regions C and B.oShow how you know that your result is correct.During the second month they break even, so A + B + C = 0. A = $201,609.33B and C both have a loss.Because B’s loss is half C’s loss, we know C = 2B.A + B + C = 0A + B + 2B = 0 by substitution. A + 3B = 0 by addition.201,609.33 + 3B = 0 by substitution3B = 0 – 201,609.33 by subtracting 201,609.33 from both sides.3B = -201,609.33 because of subtraction.B = -201,609.33 by dividing both sides by 3.3B = -$67,203.11 because of division. Because C = 2B, C = 2(-67,203.11)C = -$134,406.22So Region A has a profit of $201,609.33, Region B has a loss of $67,203.11, and Region C has a loss of$134,406.22 in the second month.Checking this we have:$201,609.33 - $67,203.11 - $134,406.22 = 0SoA+B+C= 0 and the business did break even.3. As with the previous month, the third month the business broke even.oThis time, although Region B had a profit, the mean of the incomes from Regions A and B was a loss of$117,200.00.oExplain how you determined each region’s amount of profit or loss for this month.Once again, the business broke even during the third month. So A + B + C = 0B has a profit.The mean o incomes from A and B is a loss of $117,200. So A + B= -$117,200 using the definition of mean2A + B = -$234,400 by multiplying both sides by 2.At this point, students have choices and answers will vary. Suppose B has a profit of $100,000.Then A + 100,000 = -234,400A = -$334,400 by subtracting both sides by 100,000. Since A + B + C = 0,-334,400 + 100,000 + C = 0 by substitution.-234,400 +C = 0 using additionC = $234,400 by subtracting -234,400 from both sides. Hence, if B has a profit of $100,000, then A has a loss of $334,400 and C has a profit of $234,400.4. Make an appropriate display of the data for regions A, B, and C during the three months.Analyze the results of this display.Students have a choice in determining their display of the data.One possible choice could be a chart or table such as the one shown below.(Note that the 3rd month’s results are flexible. The results of this month affect the additional columns to its right.)RegionFirst month resultsSecond month results*Third month resultTotalA-$246,913.56+$201,609.33-$334,400.00-$379,704.23B-$123,456.78-$67,203.11+100,000.00-$90,659.89C-$123,456.78-$134,406.22+$234,400.00-$23,463.00Totals-$493,827.1200-$493,827.12*Note: The third month’s results are based on supposing that Region B has a profit of $100,000.Region3 month total results3 month mean3 month rangeMediansMode(s)A-$379,704.23-$126,568.08$536,009.33-$246,913.56B-$90,659.89-$30,219.96$223,456.78-$67,203.11C-$23,463.00-$7,821.00$368,806.22-$123,456.78Totals-$493,827.12-$123,456.785. Company policy is to give an award each quarter to the best region. You are asked to recommend one of the regions for this award. Explain which one you believe should receive the award and how you came to your decision.There are many different reasons that a student may use to determine which area is rewarded at the end of the third month.Region A had the largest improvement from one month to the other (first to second month gain of $448,522.89). Region B had the best median (-$67,203.11) and was the most consistent with the smallest range ($223,456.78).Region C had the highest profitable month (+$234,400.00) and the best three month total (-$23,463.00).Part 3: Water in the office1. You have decided to have bottled water available in the office. The water bottle company charges $0.75 for each bottle of water that is delivered.a. Express the cost (y) in terms of the number of bottles of water delivered (x).y = cost of x bottles of water delivered.$0.75 = cost of one bottle of water deliveredSo y = .75xb. Find the range of dollar amounts (y values) when the number of bottles (x values) ranges from 24 bottles to216 bottles.For x = 24 y = (.75)(24)For x = 216y = (.75)(216)y = 18y = 162So for 24 bottles of water delivered, the cost is $18. For 216 bottles, the cost is $162.The range of the values of y would be (24)(0.75) ″ x ″ (216)(0.75) or $18 ″ x ″ $162 because it would cost $18 for24 bottles of water and $162 for 216 bottles of water.c. Make a table/chart that shows the relationship of x and y.An example of a possible table or chart might look something like the one below.x$0.75xy0.75(0).002.75(2)1.504.75(4)3.006.75(6)4.50d. Graph this relationship, name the constant of proportionality; and tell whether the relationship is a direct or inverse proportion. Explain how you know.Graphing this relationship might look like the graph below. Students should know by this time of the year that the points should not be connected because they may not purchase part of a bottle of water.Number of bottles of waterThe constant of proportionality is .75. We know that y = .75x is an example of a direct proportion because it is a it may be written in the form of y = kx with k = .75.2. A special offer of a substantial discount has been presented to new customers when the first order is 12 boxes with 24 bottles of water each.The water company will continue to offer this price for all other orders throughout the first year of business, regardless of the size of the orders.a. If the discounted price for this special offer is 2/3 of the everyday price, express the cost (y) in terms of the number of bottles of water delivered (x).To determine the discounted price, we could use the equation y = (2/3)(.75)x which simplifies to y = .5x.b. Find the range of dollar amounts (y values) when the number of bottles (x values) ranges from 24 bottles to the number of bottles in the special first order offer.The range of the values of y would be (24)(0.50) ″ x ″ (288)(0.50) or $12 ″ x ″ $144 because it would cost $28 for24 bottles of water and $144 for 288 bottles of water.c. Make a table/chart that shows the relationship of x and y.An example of a possible table or chart of the special offer might look something like the one below.x$0.50xy0(0.50)(0)$02(0.50)(2)$1.004(0.50)(4)$2.006(0.50)(6)$3.00d. Graph this relationship, name the constant of proportionality; and tell whether the relationship is a direct or inverse proportion. Explain how you know.Graphing this relationship might look like the graph below.Number of bottles of waterThe constant of proportionality is the price of each bottle of water or $0.50.This is an example of direct proportionality because it is of the form 7 = ax when a = 0.50.Not only does this mean that the company will save a lot of money because all of the costs have dropped, but also the graph looks different because the line is not so steep.e. What would the total cost (without tax) be to have this special offer delivered to your office? Show how you know.To determine the discounted price, we could use the equation y = (2/3)(.75)x which simplifies to y = .5x. In order to find the number of bottles of water, we have x = (12)(24) = 288.So we have y = .5x for x = 288. y = .5(288)y = 144Therefore, the discounted price for 288 bottles (12 boxes of 24 each) is $144 and this would be the cost to have the special offer delivered to the office.3. The water bottle company has told you that the average person drinks 1.6 bottles of water each day. Let x represent the number of people drinking bottled water. Let y represent the number of days those people are drinking water.a. Using an estimate of 1.6 bottles per day, write an equation that represents the relationshipbetween x, y and the number of bottles of water in the special first year offer. Solve your equation for y.x = number of people drinking bottled water and y = number of days those people drinking water1.6 bottles/day per person (estimated average)288 = number of bottles of water in the special offer(1.6 bottles/day)(x people)(y days/person) = 2881.6 xy = 288 by substitutionxy = 288 dividing both sides by 1.61.6xy = 180 using divisiony = 180 dividing both sides by x xb. Make a table/chart that shows the relationship between x and y.An example of a possible table or chart of the special offer might look something like the one below.x180 xy11801180101801018100180100951801801801c. Graph this relationship, name the constant of proportionality; and tell whether the relationship is a direct or inverse proportion. Explain how you know.Graphing this relationship would look like the graph below.The constant of proportionality is 180.y = 180 is an inverse proportion because it may be written in the form y = k when k = 180. xxd. How long will it take to use up all of the water that was ordered with the special discount when you have yourself and three employees in the office each day? Justify your answer.To determine how long it will take to use up all of the ordered water, 4 would be substituted for the value of x because there is a total of four people in the office each day.For 4 people, y = 180 xy = 1804y = 45So 4 people will drink the 288 bottles in about 45 days or 9 work weeks or 2 ? months.Part 4: The Final Phasea) Building a Billboard1. You have decided to use the 8 in. by 11 in. logo design that you did on the coordinate plane in Part 1 as part of a design for a billboard advertisement to tell the world about your new company. The billboard will be 10 ft. by7 ft. in a landscape orientation. Use another piece of four-quadrant graph paper to show your billboard design. This design must include a dilation of the original logo.State the scale factor of the dilation. Explain how you know.Give the new dimensions of the logo on the billboard and justify your answer.Student work should vary due to originality.2. The committee has requested that you tell them the distance from the top of the billboard to the ground..To do this, you have decided to use similar triangles formed by using shadows.Draw an illustration of the billboard and its shadow. Label the length of the shadow as 12 ft.Next to this, draw an illustration of yourself and your shadow. Label the length of your shadow as 1 ft. 6 in. and the illustration of your self as your actual height to the nearest inch.Define a variable, write and solve an equation that uses the information shown in your illustration to determine the height of the billboard. Justify your work.The teacher should guide the students to understand that although all students’ shadows for this task are the same because they are not actually going outside to measure the shadows, but are setting the problem up according to the instructions typed. In reality, the shadows would be different lengths for students with different heights. This would also be true based on the different times of day that the shadows were measured because of the sun’s angle. Because of the variety of student heights, their billboards will have different heights.To find the height of the billboard, they should illustrate the situation by a drawing similar to the one below.The student should noticeHe/she may choose to set up athat the figures form proportion using the corresponding similar right triangles.sides of these triangles.x ft5 ? ft1 ? ft12 ftIf a student is 5 feet 6 inches tall, the proportion isHeight of billboard=Height of studentLength of billboard’s shadowLength of student’s shadow orx = 5.5 121.5 One method that a student could use to solve this proportion might is to find the cross products. This would yield(1.5)x = (5.5)(12)Multiplication produces(1.5)x = 66Dividing both sides by 1.5 leaves a solution of x = 41.25.Note that 41.25 feet means 41 feet and ? of a foot. This is the same thing as 41 feet and ? of 12 inches, or 41 feet and 3inches.The height of the billboard is 41 feet 3inches tall if the student is 5 feet 6 inches tall.b) The InterviewA member of the committee that will determine the recipient of the grant is meeting with you, one of your parents, and your math teacher today to interview you and review the billboard information. Your parent picked you up from school to take you to the appointment, but your teacher could not leave until 5 minutes later. Your parent drove 65 mph for the entire trip, but your teacher drove 70 mph using the same directions. If you and your teacher reached the meeting place at the same time, how long did it take for your math teacher to drive to the appointment location? Use pictures, models, tables and/or graphs and explain your thinking. Define your variable, make up an equation that describes the situation and solve it. How far away was this meeting place? Show how you know.There are several different approaches that students may wish to use for this situation. One approach is shown below:SchoolParent’s car at 65mphMeeting placeDistance covered in5 minutesDistance covered in x minutesThis could represent a distance of 65(5 + x).SchoolTeacher’s car at 70mphMeeting placeDistance covered in x minutesThis represents a distance of 70x.A sample solution for defining a variable, making up and solving an equation follows:Let x = the time spent driving.In the case of your teacher, the distance is 70x mph.In the case you’re your parent, the distance is 65(5 + x) mph.Because the distance is the same for both vehicles, you may combine them with an equal sign to form an equation.70x = 65(5 + x). Using the distributive property, this may be simplified to70x = 325 + 65x. By subtracting 65x from both sides of the equation, you now have5x = 325. Dividing 5 into both sides of the equation will leave you withx = 65 minutes.This means that your teacher drove for 65 minutes or 1 hour and 5 minutes to reach the meeting place.To determine the distance from school to the meeting place, a student may use the following approach. Substitute the 65 minutes in for the value of x in either of the expressions 70x or 65(5 + x).Because the expressions use miles per hour, 65 minutes must changed to 1 5/60 hour or 65/60 hr. Substituting this into the expression 70x for the distance produces 70(1 5/60) which would be 75 5/6 miles. Therefore, the meeting place is a little less than 76 miles from school.c) A Special GiftHelp your classmates with this design by clearly writing out the directions for the statue. Be sure to include the scale factor used and how the logo should be swept through space.Make a model of the statue. State the dimensions of the actual statue and explain why you decided to use these dimensions.Oh dear! You were taking the model to the art room so that the art teacher could see what you have in mind when you accidentally dropped it. When it hit the floor, a piece of the model broke off in a neat and smooth break. Sketch the two pieces of your three-dimensional model so that you are showing the congruent cross sections that were formed by the break.Each student’s work should be original, creative and unique. The teacher should carefully observe student progress throughout this section.Part 5:Students should be very proud of their accomplishment in the completion of this task. Let them really “Show WhatThey Know” by celebrating their achievement in an appropriate and fun manner.The teacher may actually want to have some way of having the class determine “winners” with certificates or some other means of rewarding them.Course Title: 7th GradeGaDJJ State Code: 27.0220000CAP: 55 Georgia Performance Standard(s): ??M7P3.a Organize and consolidate their mathematical thinking through communication. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P3.d Use the language of mathematics to express mathematical ideas precisely. ??M7P1.a Build new mathematical knowledge through problem solving. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P2.c Develop and evaluate mathematical arguments and proofs. ??M7P2.d Select and use various types of reasoning and methods of proof. ??M7A1.b Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate. ??M7P1.b Solve problems that arise in mathematics and in other contexts. ??M7P3.c Analyze and evaluate the mathematical thinking and strategies of others. Objective(s): The student identifies various three-dimensional figures. The student will find the volume of prisms and cylinders. Instructional Resources: Holt Mathematics Course 2, Pgs. 578-589 Chapter 10 Resource Book (CRB) One-Stop Planner Activities: Complete Are You Ready, in textbook pg. 575. Read textbook pgs. 575-585. ???Complete Think and Discuss, pg. 581 in textbook. ???Complete Practice and Problem Solving, Problems 1-3, 7-9, 13, 19, and 23-29 on pgs. 582-583. in textbook. ???Complete Practice A 10-1 CRB, pg. 3. ???Complete Reading Strategies 10-1 CRB, pg. 9. Read textbook pgs. 586-589. ??Complete Think and Discuss, pg. 587 in textbook ??Complete Practice and Problem Solving, Problems 1-3, 8-10, and 21-25 on pgs. 588-589.in textbook ??Complete Practice A 10-2 CRB, pg. 11. ??Complete Reading Strategies 10-2 CRB, pg. 17. Evaluation: Complete Power Presentations Lesson Quiz 10-1 and 10-2. Modifications: P Performance Tasks: IDEA works CD Course Title: Seventh Grade Mathematics GaDJJ: State Code: 27.0220000 CAP: 58 Georgia Performance Standard(s): ??M7G3.a understands the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts. ??M7G3.b Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures. ??M7P5.a Create and use representations to organize, record, and communicate mathematical ideas. ??M7P5.b Select, apply, and translate among mathematical representations to solve problems. ??M7P1.a Build new mathematical knowledge through problem solving. ??M7P4.c Recognize and apply mathematics in contexts outside of mathematics. ??M7P1.c Apply and adapt a variety of appropriate strategies to solve problems. ??M7P2.c Develop and evaluate mathematical arguments and proofs Objective(s): ????The student organizes and reviews key concepts and skills presented in Chapter Ten ??The student assesses mastery of concepts and skills in Chapter Ten. Instructional Resources: Holt Mathematics Course Two Textbook Chapter10 Resource Book (CRB) One Stop Planner Activities: Complete the odd number problems on the Study Guide, pgs. 616-617 in textbook. Complete the CRCT Review for the average learner, pgs. 620-621 in textbook. ?????Have teacher check your work on review. If score is at least 80%, then go on to Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in CRB 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 concepts still not understood. Evaluation: Complete Chapter Test pg. 619 in textbook with 80% accuracy. Modifications: IDEA Works CD stemleaf081323 633 94561home runsfrequency0-9310-19120-29330-39540-495home runsfrequency20-29230-39240-49750-59360-691stemleaf00 0 11620 7 930 2 4 5 741 6 7 9 9Task Websites 1 (see example 3) tiles or cubesUnit 2 3 4 5 6 7 8, , ................
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