Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
I34 Pre IB Geometry Mathematics Curriculum Essentials Document Boulder Valley School District Department of Curriculum and Instruction January 2012 Boulder Valley School District Mathematics – An Introduction to The Curriculum Essentials Document Background The 2009 Common Core State Standards (CCSS) have brought about a much needed move towards consistency in mathematics throughout the state and nation. In December 2010, the Colorado Academic Standards revisions for Mathematics were adopted by the State Board of Education. These standards aligned the previous state standards to the Common Core State Standards to form the Colorado Academic Standards (CAS). The CAS include additions or changes to the CCSS needed to meet state legislative requirements around Personal Financial Literacy. The Colorado Academic Standards Grade Level Expectations (GLE) for math are being adopted in their entirety and without change in the PK-8 curriculum. This decision was made based on the thorough adherence by the state to the CCSS. These new standards are specific, robust and comprehensive. Additionally, the essential linkage between the standards and the proposed 2014 state assessment system, which may include interim, formative and summative assessments, is based specifically on these standards. The overwhelming opinion amongst the mathematics teachers, school and district level administration and district level mathematics coaches clearly indicated a desire to move to the CAS without creating a BVSD version through additions or changes. The High School standards provided to us by the state did not delineate how courses should be created. Based on information regarding the upcoming assessment system, the expertise of our teachers and the writers of the CCSS, the decision was made to follow the recommendations in the Common Core State Standards for Mathematics- Appendix A: Designing High School Math Courses Based on the Common Core State Standards. The writing teams took the High School CAS and carefully and thoughtfully divided them into courses for the creation of the 2012 BVSD Curriculum Essentials Documents (CED). The Critical Foundations of the 2011 Standards The expectations in these documents are based on mastery of the topics at specific grade levels with the understanding that the standards, themes and big ideas reoccur throughout PK-12 at varying degrees of difficulty, requiring different levels of mastery. The Standards are: 1) Number Sense, Properties, and Operations; 2) Patterns, Functions, and Algebraic Structures; 3) Data Analysis, Statistics, and Probability; 4) Shape, Dimension, and Geometric Relationships. The information in the standards progresses from large to fine grain, detailing specific skills and outcomes students must master: Standards to Prepared Graduate Competencies to Grade Level/Course Expectation to Concepts and Skills Students Master to Evidence Outcomes. The specific indicators of these different levels of mastery are defined in the Evidence Outcomes. It is important not to think of these standards in terms of ―introduction, mastery, reinforcement.‖ All of the evidence outcomes in a certain grade level must be mastered in order for the next higher level of mastery to occur. Again, to maintain consistency and coherence throughout the district, across all levels, adherence to this idea of mastery is vital. In creating the documents for the 2012 Boulder Valley Curriculum Essentials Documents in mathematics, the writing teams focused on clarity, focus and understanding essential changes from the BVSD 2009 standards to the new 2011 CAS. To maintain the integrity of these documents, it is important that teachers throughout the district follow the standards precisely so that each child in every classroom can be guaranteed a viable education, regardless of the school they attend or if they move from another school, another district or another state. Consistency, clarity and coherence are essential to excellence in mathematics instruction district wide. Components of the Curriculum Essentials Document The CED for each grade level and course include the following: -A-Glance page containing: o approximately ten key skills or topics that students will master during the year o the general big ideas of the grade/course o the Standards of Mathematical Practices o assessment tools allow teachers to continuously monitor student progress for planning and pacing needs o description of mathematics at that level with additional topics or more in-depth coverage of topics included in bold text. teachers should be familiar and comfortable using during instruction. It is not a comprehensive list of vocabulary for student use. -12 Prepared Graduate Competencies -12 At-A-Glance Guide from the CAS with notes from the CCSS -12 Explanation of Coding In these documents you will find various abbreviations and coding used by the Colorado Department of Education. MP – Mathematical Practices Standard PFL – Personal Financial Literacy CCSS – Common Core State Standards Example: (CCSS: 1.NBT.1) – taken directly from the Common Core State Standards with an reference to the specific CCSS domain, standard and cluster of evidence outcomes. NBT – Number Operations in Base Ten OA – Operations and Algebraic Thinking MD – Measurement and Data G – Geometry 6/21/2012 BVSD Curriculum Essentials 2 Standards for Mathematical Practice from The Common Core State Standards for Mathematics The Standards for Mathematical Practice have been included in the Nature of Mathematics section in each Grade Level Expectation of the Colorado Academic Standards. The following definitions and explanation of the Standards for Mathematical Practice from the Common Core State Standards can be found on pages 6, 7, and 8 in the Common Core State Standards for Mathematics. Each Mathematical Practices statement has been notated with (MP) at the end of the statement. Mathematics | Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ―processes and proficiencies‖ with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). 1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, ―Does this make sense?‖ They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in 6/21/2012 BVSD Curriculum Essentials 3 an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 6/21/2012 BVSD Curriculum Essentials 4 8. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word ―understand‖ are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential ―points of intersection‖ between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics. 6/21/2012 BVSD Curriculum Essentials 5 21st Century Skills and Readiness Competencies in Mathematics Mathematics in Colorado’s description of 21 st century skills is a synthesis of the essential abilities students must apply in our rapidly changing world. Today’s mathematics students need a repertoire of knowledge and skills that are more diverse, complex, and integrated than any previous generation. Mathematics is inherently demonstrated in each of Colorado 21st century skills, as follows: Critical Thinking and Reasoning Mathematics is a discipline grounded in critical thinking and reasoning. Doing mathematics involves recognizing problematic aspects of situations, devising and carrying out strategies, evaluating the reasonableness of solutions, and justifying methods, strategies, and solutions. Mathematics provides the grammar and structure that make it possible to describe patterns that exist in nature and society. Information Literacy The discipline of mathematics equips students with tools and habits of mind to organize and interpret quantitative data. Informationally literate mathematics students effectively use learning tools, including technology, and clearly communicate using mathematical language. Collaboration Mathematics is a social discipline involving the exchange of ideas. In the course of doing mathematics, students offer ideas, strategies, solutions, justifications, and proofs for others to evaluate. In turn, the mathematics student interprets and evaluates the ideas, strategies, solutions, justifications and proofs of others. Self-Direction Doing mathematics requires a productive disposition and self-direction. It involves monitoring and assessing one’s mathematical thinking and persistence in searching for patterns, relationships, and sensible solutions. Invention Mathematics is a dynamic discipline, ever expanding as new ideas are contributed. Invention is the key element as students make and test conjectures, create mathematical models of real-world phenomena, generalize results, and make connections among ideas, strategies and solutions. 6/21/2012 BVSD Curriculum Essentials 6 Colorado Academic Standards Mathematics The Colorado academic standards in mathematics are the topical organization of the concepts and skills every Colorado student should know and be able to do throughout their preschool through twelfth-grade experience. 1. Number Sense, Properties, and Operations Number sense provides students with a firm foundation in mathematics. Students build a deep understanding of quantity, ways of representing numbers, relationships among numbers, and number systems. Students learn that numbers are governed by properties and understanding these properties leads to fluency with operations. 2. Patterns, Functions, and Algebraic Structures Pattern sense gives students a lens with which to understand trends and commonalities. Students recognize and represent mathematical relationships and analyze change. Students learn that the structures of algebra allow complex ideas to be expressed succinctly. 3. Data Analysis, Statistics, and Probability Data and probability sense provides students with tools to understand information and uncertainty. Students ask questions and gather and use data to answer them. Students use a variety of data analysis and statistics strategies to analyze, develop and evaluate inferences based on data. Probability provides the foundation for collecting, describing, and interpreting data. 4. Shape, Dimension, and Geometric Relationships Geometric sense allows students to comprehend space and shape. Students analyze the characteristics and relationships of shapes and structures, engage in logical reasoning, and use tools and techniques to determine measurement. Students learn that geometry and measurement are useful in representing and solving problems in the real world as well as in mathematics. Modeling Across the Standards Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data. Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards, specific modeling standards appear throughout the high school standards indicated by a star symbol (*). 6/21/2012 BVSD Curriculum Essentials 7 I34 Pre-IB Geometry Course Overview Course Description PIB Geometry will cover the concepts of Geometry with more emphasis on problem solving strategies. PIB Geometry presents a thorough study of the structure of the postulate system and development of formal two-column proof. It considers the topics of congruence, parallelism, perpendicularity, properties of polygons, similarity, and the relationships of circles, spheres, lines, and planes with respect to space as well as the plane. Basic principles of probability will be introduced. The use of algebraic skills is expected. As an advanced course, this course goes beyond the curriculum expectations of a standard course and addresses specific prerequisite skills needed for the study of further International Baccalaureate mathematics by increasing the depth and complexity. Students are engaged in dynamic, high‐level learning. Assessments Teacher created assessments Common Assessments and Finals Grade Level Expectations Standard 1. 2. 3. Big Ideas for Pre IB Geometry 1. Number sets and their properties form a basis for algebraic number sense. 2. Patterns, Functions, & Algebraic Structures 1. Properties of the real number system can be applied algebraically. 2. The coordinate plane allows us to apply algebraic understandings to Geometric concepts. 1. Probability models outcomes for situations in which there is inherent randomness. 1. The use of Geometric definitions and deductive logic form the foundation for expanding understanding to applied Geometry. 2. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically. 3. Concepts of similarity are foundational to geometry and its applications. 4. Logic and the study of reason provide the processes for students to formulate a strategic plan for problem-solving. 5. Attributes of two- and threedimensional objects are measurable and can be quantified. 6. Objects in the real world can be modeled using geometric concepts. 6/21/2012 Topics at a Glance Congruence Similarity Angles and Triangles Coordinate Geometry Areas and Volumes Introduction to Trigonometry Parallel and Perpendicular Lines Geometric Inequalities Circles & Spheres Lines, Planes, Separation and Space Logic & The study of reasoning Probability Standards for Mathematical Practice 1. Number Sense, properties, and operations 3. Data Analysis, Statistics, & Probability 4. Shape, Dimension, & Geometric Relationships 4. 5. 6. 7. 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. 7. Look for and make use of structure. Look for and express regularity in repeated reasoning. Notes on this document: Plain text indicates standards which are part of the Colorado Academic Standards as defined by the Common Core State Standards for the general (M41) Geometry course. Evidence Outcomes in italics are items that have been added or changed from the Colorado Academic Standards as needed to fit the specific needs of BVSD students. Evidence outcomes in bold indicate items that were added to (M41) Geometry standards for additional depth and understanding in Advanced Geometry. Items in bold blue italic font indicate specific topics added to meet the prerequisites for the International Baccalaureate program in mathematics. BVSD Curriculum Essentials 8 Course Information: Students enrolling in PIB Geometry should have successfully completed an advanced Algebra I course. Students wishing to take this course should be highly motivated and hard working. PIB Geometry is designed to prepare students for success in both IB and AP classes later on in high school. The Grade Level Expectations and Evidence Outcomes listed in this course for Standard 3: Data Analysis, Statistics and Probability are a part of the Geometry standards for the Colorado Academic Standards and should be addressed as a part of this course because students will not receive exposure to these concepts elsewhere. Pre-International Baccalaureate Geometry Curriculum Map: Semester 1 Semester 2 6/21/2012 BVSD Curriculum Essentials 9 1. Number Sense, Properties, and Operations Number sense provides students with a firm foundation in mathematics. Students build a deep understanding of quantity, ways of representing numbers, relationships among numbers, and number systems. Students learn that numbers are governed by properties, and understanding these properties leads to fluency with operations. Prepared Graduates The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting. Prepared Graduate Competencies in the Number Sense, Properties, and Operations Standard are: 6/21/2012 Understand the structure and properties of our number system. At their most basic level numbers are abstract symbols that represent real-world quantities Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations Apply transformation to numbers, shapes, functional representations, and data BVSD Curriculum Essentials 10 Content Area: Mathematics - Pre-IB Geometry Standard: 1. Number Sense, Properties, and Operations Prepared Graduates: Understand the structure and properties of our number system. At their most basic level numbers are abstract symbols that represent real-world quantities. GRADE LEVEL EXPECTATION: Concepts and skills students master: 1. Number sets and their properties form a basis for algebraic number sense. Evidence Outcomes 21st Century Skills and Readiness Competencies Students can: a. Learn the meaning and correct notations for universal set, null set, subset, member or element of a set. b. Be able to explain the union, intersection, and complement of sets using proper notation. c. Use Venn Diagrams to solve set problems, specifically classification in 2 and 3 set problems and finding the number of members in a set. 6/21/2012 Inquiry Questions: 1. When you extend to a new number systems (e.g., from integers to rational numbers and from rational numbers to real numbers), what properties apply to the extended number system? Relevance and Application: 1. The understanding of the closed system of real, rational and irrational numbers is the foundation for the truth of properties and operations throughout mathematical development. 2. Venn Diagrams are a useful means of modeling set theory and problem solving complex situations with overlapping characteristics. Nature of the Discipline: 1. Mathematicians build a deep understanding of quantity, ways of representing numbers, and relationships among numbers and number systems. 2. Mathematics involves making and testing conjectures, generalizing results, and making connections among ideas, strategies, and solutions. 3. Mathematicians look for and make use of structure. (MP) 4. Mathematicians look for and express regularity in repeated reasoning. (MP) BVSD Curriculum Essentials 11 2. Patterns, Functions, and Algebraic Structures Pattern sense gives students a lens with which to understand trends and commonalities. Being a student of mathematics involves recognizing and representing mathematical relationships and analyzing change. Students learn that the structures of algebra allow complex ideas to be expressed succinctly. Prepared Graduates The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must have to ensure success in a postsecondary and workforce setting. Prepared Graduate Competencies in the 2. Patterns, Functions, and Algebraic Structures Standard are: 6/21/2012 Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions BVSD Curriculum Essentials 12 Content Area: Mathematics - Pre-IB Geometry Standard: 2: Patterns, Functions, and Algebraic Structures Prepared Graduates: Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency GRADE LEVEL EXPECTATION: Concepts and skills students master: 1. Properties of the real number system can be applied algebraically. Evidence Outcomes 21st Century Skills and Readiness Competencies Students Can: a. Review Algebra I Concepts to be Used in Geometry i. Real number system ii. Number line iii. Commutative, associative, distributive properties iv. Properties of equality v. Properties of inequality vi. Trichotomy property vii. Absolute value Inquiry Questions: 1. When you extend to a new number systems (e.g., from integers to rational numbers and from rational numbers to real numbers), what properties apply to the extended number system? 2. How is the number system extended and applied to geometric figures and properties? 3. Why don’t we have a different number system for shapes and angles? Relevance and Application: 1. The understanding of the closed system of real, rational and irrational numbers is the foundation for the truth of properties and operations throughout mathematical development. Nature of the Discipline: Standards for Mathematical Practice. 1. Reason abstractly and quantitatively. 2. Model with mathematics. 3. Use appropriate tools strategically. 4. Attend to precision. 5. Look for and make use of structure. 6. Look for and express regularity in repeated reasoning. 6/21/2012 BVSD Curriculum Essentials 13 Content Area: Mathematics - Pre-IB Geometry Standard: 2: Patterns, Functions, and Algebraic Structures Prepared Graduates: Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data GRADE LEVEL EXPECTATION Concepts and skills students master: 2. The coordinate plane allows us to apply algebraic understandings to Geometric concepts. Evidence Outcomes 21st Century Skills and Readiness Competencies Students Can: a. Explore algebra and geometry relationships concerning coordinate geometry b. Understand the history behind coordinate geometry and its usefulness today c. Cartesian coordinate system (parts and notation) i. Label points on a graph ii. Find projection of a point to the x and y axis d. Understand a three dimensional coordinate system i. Ordered triple, octants, graphing, projections, to XY, YZ, XZ planes e. Be able to discuss fallacies in graphing and improper graphing techniques f. Be able to graph inequalities and absolute value relationships on a coordinate plane to parallel and no parallel lines g. Express and understand of slope; verbally, graphically and computationally h. Know the difference between positive, negative, zero, and undefined slope i. Prove the theorems for the slope of two parallel and two perpendicular lines and apply in problems with quadrilaterals and triangles. j. Prove the distance and midpoint formula and apply them in two and three dimensional problems. k. Prove previously proven theorems in geometry by coordinate geometry using slope, distance and midpoint formulas l. Describe a line by an equation i. Be familiar with slope intercept form, point slope form, and standard form for a linear equation ii. Write the equation of a line given: 1. Slope, Y-intercept Inquiry Questions: 1. How can the 2 and 3-dimensional coordinate planes be used to systematize applications of Geometric properties? Relevance and Application: 1. Visualization and the use of coordinate Geometry is used in professions such as architecture, robotics, animation, film and computer graphics, navigation, manufacturing, engineering, urban planning, interior design, construction management and military design. Nature of the Discipline: Standards for Mathematical Practice. 1. Reason abstractly and quantitatively. 2. Construct viable arguments and critique the reasoning of others. 3. Model with mathematics. 4. Use appropriate tools strategically. 5. Attend to precision. 6. Look for and make use of structure. 6/21/2012 * Indicates a part of the standard connected to the mathematical practice of Modeling. BVSD Curriculum Essentials 14 m. n. o. p. q. r. s. 2. Slope, A point 3. 2 points 4. ║or ┴ another line through a given point Know how to graph using slope and intercept the intercept method or a table of values Demonstrate graphing 3 variable equations Equation of circles in the coordinate plane Be able to write the equation of a circle at the origin given the radius and center (and vice versa) Be able to write the equation of a circle not at the origin given the center and radius (vice versa) Be able to change an equation of a circle in vertex form to standard form (vice versa) Identify whether the graph of the equation X2+Y2+AX+BY+C=0 is a circle a point or the empty set. 6/21/2012 BVSD Curriculum Essentials 15 3. Data Analysis, Statistics, and Probability Data and probability sense provides students with tools to understand information and uncertainty. Students ask questions and gather and use data to answer them. Students use a variety of data analysis and statistics strategies to analyze, develop and evaluate inferences based on data. Probability provides the foundation for collecting, describing, and interpreting data. Prepared Graduates The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting. Prepared Graduate Competencies in the 3. Data Analysis, Statistics, and Probability Standard are: 6/21/2012 Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions BVSD Curriculum Essentials 16 Content Area: Mathematics - Pre-IB Geometry Standard: 3. Data Analysis, Statistics, and Probability Prepared Graduates: Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts. GRADE LEVEL EXPECTATION: Concepts and skills students master: 1. Probability models outcomes for situations in which there is inherent randomness. 21st Century Skills and Readiness Competencies Evidence Outcomes Students can: Inquiry Questions: a. Understand independence and conditional probability and use them 1. Can probability be used to model all types of uncertain to interpret data. (CCSS: S-CP) situations? For example, can the probability that the 50th i. Describe events as subsets of a sample space5 using president of the United States will be female be characteristics (or categories) of the outcomes, or as determined? unions, intersections, or complements of other events.6 2. How and why are simulations used to determine (CCSS: S-CP.1) probability when the theoretical probability is unknown? ii. Explain that two events A and B are independent if the 3. How does probability relate to obtaining insurance? (PFL) probability of A and B occurring together is the product of Relevance and Application: their probabilities, and use this characterization to 1. Comprehension of probability allows informed decisiondetermine if they are independent. (CCSS: S-CP.2) making, such as whether the cost of insurance is less iii. Using the conditional probability of A given B as P(A and than the expected cost of illness, when the deductible on B)/P(B), interpret the independence of A and B as saying car insurance is optimal, whether gambling pays in the that the conditional probability of A given B is the same as long run, or whether an extended warranty justifies the the probability of A, and the conditional probability of B cost. (PFL) given A is the same as the probability of B. (CCSS: S-CP.3) 2. Probability is used in a wide variety of disciplines including iv. Construct and interpret two-way frequency tables of data physics, biology, engineering, finance, and law. For when two categories are associated with each object being example, employment discrimination cases often present classified. Use the two-way table as a sample space to probability calculations to support a claim. decide if events are independent and to approximate Nature of the Discipline: conditional probabilities.7 (CCSS: S-CP.4) 1. Some work in mathematics is much like a game. v. Recognize and explain the concepts of conditional Mathematicians choose an interesting set of rules and probability and independence in everyday language and then play according to those rules to see what can everyday situations.8 (CCSS: S-CP.5) happen. b. Use the rules of probability to compute probabilities of compound 2. Mathematicians explore randomness and chance through events in a uniform probability model. (CCSS: S-CP) probability. i. Find the conditional probability of A given B as the fraction 3. Mathematicians construct viable arguments and critique of B’s outcomes that also belong to A, and interpret the the reasoning of others. (MP) answer in terms of the model. (CCSS: S-CP.6) 4. Mathematicians model with mathematics. (MP) ii. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and 5 B), and interpret the answer in terms of the model. (CCSS: The set of outcomes. (CCSS: S-CP.1) 6/21/2012 BVSD Curriculum Essentials 17 6 S-CP.7) "Or," "and," "Not". (CCSS: S-CP.1) 7 For example, collect data from a random sample of students in 8 6/21/2012 your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. (CCSS: S-CP.4) For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. (CCSS: S-CP.5) BVSD Curriculum Essentials 18 4. Shape, Dimension, and Geometric Relationships Geometric sense allows students to comprehend space and shape. Students analyze the characteristics and relationships of shapes and structures, engage in logical reasoning, and use tools and techniques to determine measurement. Students learn that geometry and measurement are useful in representing and solving problems in the real world as well as in mathematics. Prepared Graduates The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting. Prepared Graduate Competencies in the 4. Shape, Dimension, and Geometric Relationships standard are: 6/21/2012 Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data Apply transformation to numbers, shapes, functional representations, and data Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions BVSD Curriculum Essentials 19 Content Area: Mathematics - Pre-IB Geometry Standard: 4. Shape, Dimension, and Geometric Relationships Prepared Graduates: Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics GRADE LEVEL EXPECTATION: Concepts and skills students master: 1. The use of Geometric definitions and deductive logic to form the foundation for expanding understanding to applied Geometry. Evidence Outcomes 21st Century Skills and Readiness Competencies Students can: Inquiry Questions: Introduction to Geometry 1. Does the postulate system of Geometry lead to more or less a. State precise definitions of angle, circle, perpendicular uniformity of thought? line, parallel line, and line segment, based on the 2. Why do we need to know the formal definitions, properties, undefined notions of point, line, distance along a line, postulates and theorems to be able to apply logic to and distance around a circular arc. (CCSS: G-CO.1) Geometry? i. A Deductive System of Reasoning 3. How can mathematical concepts be ―undefined‖? What does ii. Learn the 3 undefined terms of geometry- point, this mean for our understanding of other concepts that line, plane depend on the undefined? iii. Definitions- be able to write accurate definitions 4. When is deductive reasoning a more appropriate tool than and recognize faulty definitions inductive reasoning? iv. Distinguish between postulates and theorems Relevance and Application: v. Beginning Postulates in Geometry 1. The understanding of foundational definitions and properties vi. Distance postulate allows for the development of strategic problem-solving skills. vii. Ruler postulate Nature of the Discipline: viii. Line postulate b. Betweeness Property (segment addition) c. Formal Definitions i. Segment, length, endpoint ii. Ray, opposite ray, midpoint, bisector d. Point Plotting Theorem e. Midpoint Theorem i. Lines, Planes, Separation and Space f. Describe how Points, Lines and Planes are Related (2 Dimensional and 3 Dimensional) i. Know the definition of space ii. Be able to draw 3 dimensional figures (one point and two point perspective) iii. Line postulate 6/21/2012iv. BVSD Curriculum Essentials 20 Plane-space postulate v. Discuss the intersection of 2 lines and line with a plane vi. vii. viii. g. h. i. j. k. l. Plane postulate Intersection of 2 planes postulate Determination of a plane 1. 3 noncolinear points 2. line and point not on the line 3. two intersecting lines Describe How Planes and Space are Separated by Lines and Planes i. Definition convex set ii. Plane separation postulate iii. Space separation postulate iv. Angles and Triangles Define Basic Terms using Correct Notation i. Angle ii. Triangle iii. Interior iv. Exterior v. Perimeter Use Postulates for Measuring Angles i. Angle measurement postulate ii. Angle addition postulate iii. Angle construction postulate iv. Supplement postulate Discuss the Relationship Between Angles and How They are Related i. Right angles ii. Obtuse angles iii. Acute angles iv. Perpendicular angles v. Vertical angles vi. Adjacent angles vii. Linear pair viii. Supplementary angles ix. Complimentary angles Know the Equivalence Relation for Angles i. Symmetric property ii. Reflexive property iii. Transitive property Use Theorems Concerning Angles in problems i. Supplement theorem ii. Complement theorem iii. Vertical angles theorem 6/21/2012 7 8 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. (CCSS: G-CO.12) Compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc. (CCSS: G-CO.12) BVSD Curriculum Essentials 21 m. Proof- Using Angles i. Hypothesis- conclusion ii. Acceptable forms n. Acceptable reasons used in proofs o. Make geometric constructions. (CCSS: G-CO) p. Make formal geometric constructions7 with a variety of tools and methods.8 (CCSS: G-CO.12) q. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (CCSS: G-CO.13) 6/21/2012 BVSD Curriculum Essentials 22 Content Area: Mathematics - Pre-IB Geometry Standard: 4. Shape, Dimension, and Geometric Relationships Prepared Graduates: Apply transformation to numbers, shapes, functional representations, and data. GRADE LEVEL EXPECTATION: Concepts and skills students master: 2. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically. Evidence Outcomes 21st Century Skills and Readiness Competencies Students can: Transformations a. Experiment with transformations in the plane. (CCSS: G-CO) b. State precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (CCSS: G-CO.1) c. Represent transformations in the plane using1 appropriate tools. (CCSS: G-CO.2) d. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. (CCSS: GCO.2) e. Compare transformations that preserve distance and angle to those that do not.2 (CCSS: G-CO.2) f. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (CCSS: G-CO.3) g. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (CCSS: G-CO.4) h. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools.3 (CCSS: G-CO.5) i. Specify a sequence of transformations that will carry a given figure onto another. (CCSS: G-CO.5) Congruence j. Understand congruence in terms of rigid motions. (CCSS: GCO) k. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. (CCSS: G-CO.6) l. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. (CCSS: G-CO.6) 6/21/2012 Inquiry Questions: 1. What happens to the coordinates of the vertices of shapes when different transformations are applied in the plane? 2. How would the idea of congruency be used outside of mathematics? 3. What does it mean for two things to be the same? Are there different degrees of ―sameness?‖ 4. What makes a good definition of a shape? 5. What conditions create unique polygons? 6. What does it mean for two lines to be parallel? 7. How can slope and distance be used to create and investigate the relationship of the triangles created by joining the mid-segments of a triangle? Relevance and Application: 1. Comprehension of transformations aids with innovation and creation in the areas of computer graphics and animation. 2. Knowledge of right triangle trigonometry allows modeling and application of angle and distance relationships such as surveying land boundaries, shadow problems, angles in a truss, and the design of structures. 3. Points of concurrency are used in fields such as architecture, engineering and physics. Nature of the Discipline: 1. Geometry involves the investigation of invariants. Geometers examine how some things stay the same while other parts change to analyze situations and solve problems. 2. Mathematicians construct viable arguments and critique the reasoning of others. (MP) 3. Mathematicians attend to precision. (MP) 4. Mathematicians look for and make use of structure. (MP) 5. Geometry involves the investigation of invariants. Geometers examine how some things stay the same while other parts BVSD Curriculum Essentials 23 m. Use Congruence Postulates for Triangles in Proofs i.Understand one to one correspondence ii.Know the definition of congruent triangles, segments and angles iii.Definition- included side, included angle iv.Equivalence relation- triangles v.Work with the SAS, ASA, and SSS postulates in triangle proofs vi. Use the Angle Bisector Theorem in Proofs vii. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. (CCSS: G-CO.7) viii. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (CCSS: G-CO.8) n. Name the parts of Isosceles and Equilateral Triangles o. Use Quadrilaterals in Proofs ix.Rectangle, squares p. Use Medians in Proofs x. Know definition- median q. Prove geometric theorems. (CCSS: G-CO) r. Prove theorems about lines and angles.4 (CCSS: G-CO.9) s. Prove theorems about triangles.5 (CCSS: G-CO.10) i. Use the Isosceles Triangle Theorem and its Converse in Proofs ii. Use the Properties of Equilateral and Equiangular Triangle in Proofs 1. Classify triangles by sides and angles 2. Understand the nature of corollaries 3. Work with overlapping triangles in proofs t. Prove theorems about parallelograms (rectangles, rhombuses, squares).6 (CCSS: G-CO.11) u. Use circle properties (involving chords and angles) to prove theorems about triangles. v. Points of concurrency w. Segments of irrational length (e.g. using equilateral triangles and squares) x. Use specific properties of quadrilaterals and triangles, to construct each. Examples: Construct a parallelogram given lengths of diagonals. Given lengths of two sides 6/21/2012 change to analyze situations and solve problems. 6. Mathematicians make sense of problems and persevere in solving them. (MP) 7. Mathematicians construct viable arguments and critique the reasoning of others. (MP) *Indicates a part of the standard connected to the mathematical practice of Modeling 1 e.g., Transparencies and geometry software. (CCSS: G-CO.2) 2 e.g., Translation versus horizontal stretch. (CCSS: G-CO.2) 3 e.g., Graph paper, tracing paper, or geometry software. (CCSS: GCO.5) 4 Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. (CCSS: G-CO.9) 5 Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. (CCSS:G-CO.10) 6 Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (CCSS: G-CO.11) 11 For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). (CCSS: G-GPE.4) 12 e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point. (CCSS: G-GPE.5) BVSD Curriculum Essentials 24 and a non-included angle, construct two triangles. i. Determine uniqueness. Coordinate Geometry a. Express Geometric Properties with Equations. (CCSS: G-GPE) b. Translate between the geometric description and the equation for a conic section. (CCSS: G-GPE) i. Derive the equation of a circle of given center and radius using the Pythagorean Theorem. (CCSS: GGPE.1) ii. Complete the square to find the center and radius of a circle given by an equation. (CCSS: G-GPE.1) iii. Derive the equation of a parabola given a focus and directrix. (CCSS: G-GPE.2) c. Use coordinates to prove simple geometric theorems algebraically. (CCSS: G-GPE) d. Use coordinates to prove simple geometric theorems11 algebraically. (CCSS: G-GPE.4) e. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.12 (CCSS: G-GPE.5) f. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (CCSS: GGPE.6) g. Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.★ (CCSS: G-GPE.7) Extend to solve problems with similar figures (e.g. triangles created by connecting midpoints of sides of a triangle) h. Use distance and slope to further investigate and informally prove properties of points of concurrency in triangles (orthocenter, incenter, circumcenter, centroid) Geometric Inequalities a. Parts Theorem i.Know the definition of less than for angles and segments b. Use the Exterior Angle Theorem in Inequality Proofs i.Know definition of remote interior and exterior angles c. Be able to use AAS and HL Congruence Postulates in Proof d. Work With Inequalities in a Single Triangle i.Larger angle opposite longest side and converse 6/21/2012 BVSD Curriculum Essentials 25 e. Use the First Minimum Theorem in Proofs i.Know the definition of distance between a line and an external point f. Understand the Triangle Inequality Theorem in Problems and Planes g. Work With the Hinge Theorem and its Converse in Proofs h. Know the definition of an altitude i.Use of altitudes in proofs Lines and Planes in Space a. Perpendicular lines and planes in space i. Learn definition – line and plane perpendicular in space ii. Understand the basic theorem on perpendiculars and its corollary iii. Using theorems in Unit 6, be able to discuss the relationships between intersecting and parallel lines to a plane in space iv. Learn the second minimum theorem and the definition of distance from a point to a plane b. Parallel lines and planes in space i. Know the definition of Parallel planes ii. Understand the relationship between parallel planes in space, intersecting other lines or planes iii. Know the definition , parts and notation for a dihedral angle iv. Discuss the projection of a point or line onto a plane c. Parallel Lines in a Plane i. Understand the facts about parallel lines that are not dependent on the parallel postulate ii. Know the definition of parallel and skew line iii. Be able to name the four ways to determine a plane (emphasize 2 lines perpendicular to a third line) iv. Discuss the existence and uniqueness of line through a point parallel to another line v. Identify transversals and angles created by 2 lines and a transversal vi. If 3 parallel lines intercept congruent segments on one transversal; they intersect congruent segments on any other transversal. vii. Know conditions which guarantee two parallel lines 6/21/2012 BVSD Curriculum Essentials 26 and the proofs of those conditions. 1. AIP Theorem 2. CAP Theorem 3. Same side Interior angles supplementary 4. 2 lines ┴ to 3rd line (in a plane only) viii. Be able to solve problems finding angle measures using the above theorems ix. Understand the parallel postulate and the facts about parallel lines that are dependent upon the parallel postulate x. Given the parallel postulate know the conditions and the proofs of those conditions that allow you to find angle relationships and measures 1. PAI Theorem 2. PCA Theorem 3. ║ Lines → same side interior angles supplementary 4. Lines ┴ to one of two parallel lines is ┴ to the other xi. Prove triangle relationships that use the parallel postulate and be able to use these relationships to solve angle measurements 1. Sum of interior angles of a triangle equals 180◦ 2. Two angles of a triangle congruent to corresponding parts of another →triangle 3rd angles congruent 3. Acute angles of a right triangle are complementary 4. Exterior angle theorem (equality) xii. Using the knowledge of Parallel lines learn some fact about quadrilaterals and triangles xiii. Know the definition of a quadrilateral, diagonal, opposite, and consecutive sides and angles. 1. Know the definition of a; a. Parallelogram b. Trapezoid c. Rectangle d. Rhombus e. Square f. Kite xiv. Name the six properties of a parallelogram and be 6/21/2012 BVSD Curriculum Essentials 27 able to prove the ones which are theorems (5!) Proofs with Quadrilaterals a. Given a quadrilateral, name the four ways to prove the quadrilateral is parallelogram. b. Know the midline theorem, its proof and be able to solve segment length and angle measure problems using the theorem. c. Know all the properties of rectangles rhombus squares trapezoids and kites using the family “tree” of quadrilaterals and be able to justify them by proof. d. Use the knowledge of quadrilaterals to prove facts and right triangles and medians of triangles and trapezoids e. Median to the hypotenuse is half as long as the hypotenuse. f. If the acute of a right triangle has a measure of 3o (degrees), the opposite side is half as long as the hypotenuse. g. Medians of a triangle are concurrent. Point of intersection is 2/3 the way from the vertex along the median. h. Median of a trapezoid is parallel to the bases and the length is one half the sums of the bases and the length is one half the sums of the bases i. Line bisects one side of a triangle and is parallel to a second side, and then it bisects the third side 6/21/2012 BVSD Curriculum Essentials 28 Content Area: Mathematics - Pre-IB Geometry Standard: 4. Shape, Dimension, and Geometric Relationships Prepared Graduates: Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions. GRADE LEVEL EXPECTATION Concepts and skills students master: 3. Concepts of similarity are foundational to geometry and its applications. Evidence Outcomes 21st Century Skills and Readiness Competencies Students can: Inquiry Questions: a. Understand similarity in terms of similarity transformations. 1. What happens to the coordinates of the vertices of shapes (CCSS: G-SRT) when different transformations are applied in the plane? i. Verify experimentally the properties of dilations given by a 2. How would the idea of congruency be used outside of center and a scale factor. (CCSS: G-SRT.1) mathematics? 1. Show that a dilation takes a line not passing through the 3. What does it mean for two things to be the same? Are there center of the dilation to a parallel line, and leaves a line different degrees of ―sameness?‖ passing through the center unchanged. (CCSS: G-SRT.1a) Relevance and Application: 2. Show that the dilation of a line segment is longer or 1. Comprehension of transformations aids with innovation and shorter in the ratio given by the scale factor. (CCSS: Gcreation in the areas of computer graphics and animation. SRT.1b) 2. Special triangles are commonly used in engineering, ii. Given two figures, use the definition of similarity in terms of surveying and physics. similarity transformations to decide if they are similar. (CCSS: Nature of the Discipline: G-SRT.2) 1. Geometry involves the investigation of invariants. Geometers iii. Explain using similarity transformations the meaning of examine how some things stay the same while other parts similarity for triangles as the equality of all corresponding change to analyze situations and solve problems. pairs of angles and the proportionality of all corresponding 2. Mathematicians construct viable arguments and critique the pairs of sides. (CCSS: G-SRT.2) reasoning of others. (MP) a. Definition of ratio and proportion 3. Mathematicians attend to precision. (MP) b. Properties for ratio 4. Mathematicians look for and make use of structure. (MP) c. Similar units (exception-rates) d. Notation (fraction, semicolon) *Indicates a part of the standard connected to the mathematical b. Properties for proportions practice of Modeling 1. Cross product 2. Inverting 3. Numerators over denominator 9 Theorems include: a line parallel to one side of a triangle divides the 4. Adding, subtracting numerators to other two proportionally, and conversely; the Pythagorean Theorem denominators (vice versa) proved using triangle similarity. (CCSS: G-SRT.4) iv. Use the properties of similarity transformations to establish 10Include the relationship between central, inscribed, and the AA criterion for two triangles to be similar. (CCSS: Gcircumscribed angles; inscribed angles on a diameter are right SRT.3) angles; the radius of b. Prove theorems involving similarity. (CCSS: G-SRT) a circle is perpendicular to the tangent where the radius intersects c. Prove theorems about triangles.9 (CCSS: G-SRT.4) the circle. (CCSS: G-C.2) d. Prove that all circles are similar. (CCSS: G-C.1) 6/21/2012 BVSD Curriculum Essentials 29 e. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (CCSS: G-SRT.5) f. Use similarity criteria to discover Pythagorean triplet patterns. g. Review the algebraic properties for the geometric mean h. Definition of geometric mean, arithmetic mean i. Similarity for triangles a. Definition similarity for triangles b. Find lengths of sides and angle measurements in triangles using the definition of similar triangles Proofs with Similarity a. Proportionality theorem (in a triangle and parallel lines) b. Prove and apply the “basic proportionality” theorem and its converse triangle c. Prove and apply the “angle bisector” proportional theorem to a triangle d. Prove and apply the “parallels proportional segment” theorem for 3 or more parallel lines e. Methods to prove triangles similar f. AAA, similarity theorem and AA corollary g. Prove; apply proportion properties to solve various problems involved segment length, angle measure, ECT… h. Line parallel to one size of a triangle intersects the other two sides creating a triangle similar to the original triangle i. Similarity of triangles is and equivalence relation → transit for triangle similarity. j. SAS similarity and SSS similarity (prove and apply in problems) k. Similarity in Right Triangles l. Altitude to the hypotenuse separates the triangle into two triangles similar to each other and to the original triangle i. Altitude to the hypotenuse is the geometric mean between the segments it separates on the hypotenuse. ii. leg of a right triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to the legs (projection!!) m. Prove the Pythagorean Theorem using geometric mean 6/21/2012 BVSD Curriculum Essentials 30 n. Know the relationship between are, perimeter and lengths of side, altitudes, and medians for similar triangles and be able to solve problems 6/21/2012 BVSD Curriculum Essentials 31 Content Area: Mathematics - Pre-IB Geometry Standard: 4. Shape, Dimension, and Geometric Relationships Prepared Graduates: Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics. GRADE LEVEL EXPECTATION Concepts and skills students master: 4. Logic and the study of reason provide the processes for students to formulate a strategic plan for problem-solving. Evidence Outcomes 21st Century Skills and Readiness Competencies Students can: Logic- Study of Reasoning a. Identify Mathematical Sentences i.Open ii.Closed iii.Truth value b. Understand the Connectives used in Logic Statements i. And ii. Or iii. If-then iv. If and only if c. Truth Value and Truth Tables i. Negations ii. Conjunctions iii. Disjunctions iv. Conditional v. BI- conditional vi. Compound statements d. Identify Tautologies i. Conjunctive statements- disjunctive statements e. Use the Law of Detachment (Modus Poens) in Logical arguments i.Hidden conditional f. Use the Law of the Contrapositive in Logical arguments i.Inverse, converse ii.Logically equivalent statements g. Use the Law of Modus Tollens in logical arguments h. Recognize Invalid Arguments i. Conditional and converse ii. Conditional and inverse i. Compare the Chain Rule used in Logic with the transitive property in algebra Inquiry Questions: 1. Why is the study of logic important for building a personal life-long problem-solving schematic? Relevance and Application: 1. The study of reason forms the foundation for problem-solving in law, science, computer programming and research. 6/21/2012 Nature of the Discipline: Standards for Mathematical Practice. 1. Reason abstractly and quantitatively. 2. Construct viable arguments and critique the reasoning of others. 3. Look for and make use of structure. 4. Look for and express regularity in repeated reasoning. BVSD Curriculum Essentials 32 j. k. l. m. n. o. p. q. r. s. t. u. v. Negations and DE Morgan’s Laws i. Double negation ii. Negation of a conjunction iii. Negation of a disjunction Law of Conjunction Law of Simplification Law of Disjunctive Addition Quantifiers Used in Logic Statements i.Universal quantifier ii.Existential quantifier iii.Negation of universal and existential quantifier Writing Logic Proofs More Techniques in Proofs Performance Standards and Objectives Learn to Write Indirect Proofs i. Review the deductive system of reasoning ii. Understand the parts of an indirect proof Statement to be proved Assumption or supposition Conclusion from assumption Know contradictory fact to conclusion i. Work with existence and uniqueness proofs ii. Use the perpendicular bisector theorem and its corollary in proofs iii. Understand auxiliary sets and their contribution in proofs 6/21/2012 BVSD Curriculum Essentials 33 Content Area: Mathematics - Pre-IB Geometry Standard: 4. Shape, Dimension, and Geometric Relationships Prepared Graduates: Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics. GRADE LEVEL EXPECTATION: Concepts and skills students master: 5. Attributes of two- and three-dimensional objects are measurable and can be quantified. Evidence Outcomes 21st Century Skills and Readiness Competencies Students can: a. Visualize relationships between two-dimensional and threedimensional objects. (CCSS: G-GMD) b. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. (CCSS: G-GMD.4) Polygonal regions and their areas a. Learn the postulates for areas polygonal regions b. Learn the definition of triangular region, polygonal region and identify polygonal regions i. Area postulates ii. Congruent postulate iii. Area addition postulate iv. Unit postulate (area of a square region) c. Learn the formulas and be able to prove them for rectangles, triangles, parallelograms, rhombuses, and kites d. Be able to solve area problems using the appropriate the appropriate formulas e. Be able to solve area ratio problems involving changes in bases and height f. Use ratios to solve problems within different dimensions (e.g. given the ratio of the surface area of two similar solids, find the ratio of their volumes and corresponding lengths). Circles and Spheres a. Basic definitions b. Explain volume formulas and use them to solve problems. (CCSS: G-GMD) a. Give an informal argument13 for the formulas for the circumference of a circle, area of a circle, volume and Inquiry Questions: 1. How might surface area and volume be used to explain biological differences in animals? 2. How is the area of an irregular shape measured? 3. How can surface area be minimized while maximizing volume? 6/21/2012 Relevance and Application: 1. Understanding areas and volume enables design and building. For example, a container that maximizes volume and minimizes surface area will reduce costs and increase efficiency. Understanding area helps to decorate a room, or create a blueprint for a new building. Nature of the Discipline: 1. Mathematicians use geometry to model the physical world. Studying properties and relationships of geometric objects provides insights in to the physical world that would otherwise be hidden. 2. Mathematicians make sense of problems and persevere in solving them. (MP) 3. Mathematicians construct viable arguments and critique the reasoning of others. (MP) 4. Mathematicians model with mathematics. (MP) *Indicates a part of the standard connected to the mathematical practice of Modeling 13Use dissection arguments, Cavalieri’s principle, and informal limit arguments. (CCSS: G-GMD.1 10 Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (CCSS: G-C.2) BVSD Curriculum Essentials 34 c. d. e. f. g. h. surface area of a cylinder, pyramid, sphere and cone. (CCSS: G-GMD.1) b. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ (CCSS: G-GMD.3) c. Extend prior knowledge of surface area to cylinder, cones, and spheres. Understand and apply theorems about circles. (CCSS: G-C) Identify and describe relationships among inscribed angles, radii, and chords.10 (CCSS: G-C.2) Construct the inscribed and circumscribed circles of a triangle. (CCSS: G-C.3) Prove properties of angles for a quadrilateral inscribed in a circle. (CCSS: G-C.3) Find arc lengths and areas of sectors of circles. (CCSS: G-C) Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality. (CCSS: G-C.5) Derive the formula for the area of a sector. (CCSS: G-C.5) Derive the formula for the surface area of a cone. i. j. k. l. Circle, sphere, diameter, radius, chord, secant, tangent, interior and exterior of a circle, great circle, externally and internally tangent circles, the following theorems in circles problems. a. Line ┴ to the radius at its outer endpoint is tangent to the circle b. Every tangent to the circle is ┴ to the radius at the point of contact c. The perpendicular from the center of a circle to a chord bisects the chord i. The segment from the center of a circle to the midpoint of a chord (not a diameter) is ┴ to the chord. d. In the plane of a circle, the ┴ bisector of a chord pasts through the center of the circle e. In the same circle or congruent circles, congruent chords are equal is that from the center (converse is true) f. Apply the theorems above to spheres and tangent planes m. Measurement of arcs and angles 6/21/2012 BVSD Curriculum Essentials 35 n. Learn definitions: a. Minor arc b. Major arc c. Semicircle d. Central angle e. Inscribed angle f. Tangent -secant angles g. 2 Chord angles h. 2 secant, secant -tangent, 2 tangent angles o. Prove the theorems for the above angles and arcs and apply in problems p. Know the difference between an angle intercepting an arc and being inscribed in an arc q. Understand what it means to inscribe or circumscribe polygons and circles a. define secant segment, tangent segment b. learn and prove the “power” theorems and apply in problems to find the segment length i. 2 Tangent theorem ii. 2 Secant theorem iii. Tangent- secant theorem iv. 2 chord theorem r. Learn common tangents (external and internal) and apply in problems in circles. Areas of circles and sectors (using polygons) a. Learn the definition of a polygon and the angle sum formulas b. Know the names for polygons with 3 to 10 sides (most commonly used!) c. Learn the definition of convex polygon d. Develop the formulas for the following; i. Number of diagonals in a polygon with n sides ii. Sum of the interior angles of a polygon with n sides iii. Sum of exterior angles (one angle at each vertex) of a polygon with n sides e. Work with regular polygons finding areas, lengths of sides, and measurement of angles f. Definition of a regular polygon and the apothem g. Learn the formula for the area of a regular polygon h. Develop the formulas for circumference and area of a 6/21/2012 BVSD Curriculum Essentials 36 circle Prove these formulas using inscribed regular polygons and the concept of a limit j. Apply in problems; i. Find the circumference and area of inscribed and circumscribed circles given regular polygons (especially 3, 4, 6 sides) ii. Find the area of an annulus iii. Understand the relationship between area, circumference, and radii in ratio problems k. Develop the formulas for length of an area of a sector and area of a segment l. Define length of an arc and compare this definition with the measure of an arc m. Define sector and segment n. Apply these formulas in problems (especially the “continuous belt” program Solids and their volumes a. Study the properties of solid figures, base areas, volumes and surface area’s b. Learn the definition of a prism i. Know the terms right prism, base, altitude, and cross section i. 6/21/2012 BVSD Curriculum Essentials 37 Content Area: Mathematics - Pre-IB Geometry Standard: 4. Shape, Dimension, and Geometric Relationships Prepared Graduates: Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions. GRADE LEVEL EXPECTATION: Concepts and skills students master: 6. Objects in the real world can be modeled using geometric concepts. Evidence Outcomes 21st Century Skills and Readiness Competencies Students can: a. Apply geometric concepts in modeling situations. (CCSS: GMG) b. Use geometric shapes, their measures, and their properties to describe objects.14 ★ (CCSS: G-MG.1) c. Apply concepts of density based on area and volume in modeling situations.15 ★ (CCSS: G-MG.2) d. Apply geometric methods to solve design problems.16 ★ (CCSS: G-MG.3) Right Triangle Trigonometry a. Define trigonometric ratios and solve problems involving right triangles. (CCSS: G-SRT) b. Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. (CCSS: G-SRT.6) c. Explain and use the relationship between the sine and cosine of complementary angles. (CCSS: G-SRT.7) d. Use trigonometric ratios to discover the ratio properties of special right triangles. e. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ (CCSS: G-SRT.8) f. Prove and apply trigonometric identities. (CCSS: F-TF) g. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1. (CCSS: F-TF.8) h. Use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. (CCSS: F-TF.8) i. Prove and use the Pythagorean Theorem for right triangles j. Be able to give several different proofs of the Pythagorean Theorem 6/21/2012 Inquiry Questions: 1. How are mathematical objects different from the physical objects they model? 2. What makes a good geometric model of a physical object or situation? 3. How are mathematical triangles different from built triangles in the physical world? How are they the same? Relevance and Application: 1. Geometry is used to create simplified models of complex physical systems. Analyzing the model helps to understand the system and is used for such applications as creating a floor plan for a house, or creating a schematic diagram for an electrical system. Nature of the Discipline: 1. Mathematicians use geometry to model the physical world. Studying properties and relationships of geometric objects provides insights in to the physical world that would otherwise be hidden. 2. Mathematicians make sense of problems and persevere in solving them. (MP) 3. Mathematicians construct viable arguments and critique the reasoning of others. (MP) 4. Mathematicians model with mathematics. (MP) e.g., Modeling a tree trunk or a human torso as a cylinder. (CCSS: G-MG.1) 15 e.g., Persons per square mile, BTUs per cubic foot. (CCSS: GMG.2) 16 e.g., Designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios. (CCSS: G-MG.3) 14 BVSD Curriculum Essentials 38 k. Solve problems using the Pythagorean Theorem involving the length of sides, altitude, and area with quadrilaterals and triangles. l. Prove the relationship between the sides of a 30-60-90 right triangle and a 45-45-90 right triangle using the Pythagorean Theorem. m. Solve area problems using special right triangles (including Isosceles and equilateral triangle problems) n. Develop the basic trigonometric ratios a. Sine, cosine, tangent ratios b. Develop the above ratios for special right triangles o. Develop the ability to use a trigonometry table and a calculator to evaluate trig ratios p. Find sides and angles missing in triangle problems q. Be able to solve angle elevation and depression problems r. (optional) understand the process of interpolation in estimating trig values s. Develop the ratios to the winding function (the unit circle) t. Distinguish between degree measure and radian measure on the unit circle u. Understand directed angles, terminal and initial sides v. Use the definition of the sine, cosine, and tangent of directed angles to find the value of various trigametric functions of numbers. 6/21/2012 *Indicates a part of the standard connected to the mathematical practice of Modeling BVSD Curriculum Essentials 39 Pre- IB Geometry Academic Vocabulary for Students Standard 1: complement of a set, element of a set, intersection, null set, union, universal set, Venn diagram Standard 2: associative property, commutative property, distance formula, distributive property, midpoint formula, parallel lines, perpendicular lines, Real numbers, Trichotomy property, undefined slope Standard 3: bivariate data, box plot, compound events, dot plot, frequency table, first quartile, Independently combined probability models, independent events, inter-quartile range, line plot, mean, mean absolute variation, median, probability distribution, probability, probability model, sample space, scatter plot, third quartile, uniform probability model Standard 4: associative property, inverses, commutative property, Centroid, Circumcenter, congruent, dilation, identity property of 0, Incenter, Median, Midsegment, multiplicative inverses, Kite, Orthocenter, Points of Concurrency, properties of equality, properties of inequality, properties of operations, rectilinear figure, Rhombus, rigid motion, similarity transformations, transitivity principle for indirect measurement, theorem, transformations, trigonometric ratios, Tessellation, Unique Reference Glossary for Teachers * These are words available for your reference. Not all words below are listed above because these are words that you will see above are for students to know and use while the list below includes words that you, as the teacher, may see in the standards and materials. Word Additive inverses Associative property of addition Associative property of multiplication Bivariate data Box plot Centroid Circumcenter Commutative property Complement of a set Compound events Congruent Dilation 6/21/2012 Definition Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and – 3/4 are additive inverses of one another because 3/4 + (– 3/4) = (– 3/4) + 3/4 = 0. The SUM is the same no matter what way you group the addends. In general, the associative property of addition can be written as: (a + b) + c = a + (b + c). Notice that the PRODUCT is the same no matter what way you group the factors. In general, the associative property of multiplication can be written as: (a × b) × c = a × (b × c). Pairs of linked numerical observations. Example: a list of heights and weights for each player on a football team. A method of visually displaying a distribution of data values by using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.1 That point where the medians of a triangle intersect. That point where any two perpendicular bisectors of the sides of a polygon inscribed in the circle intersect. The Commutative Property of Addition states that changing the order of addends does not change the sum, i.e. if a and b are two real numbers, then a + b = b + a. In set theory, a complement of a set A refers to things not in (that is, things outside of), A. A combination of multiple simple events, can be independent or dependent. Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations). A transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distances from the center by a common scale factor. BVSD Curriculum Essentials 40 Distance Formula The distance between (x1) and (x2) is the length of the line segment between them: Distributive property Multiplication of real numbers distributes over addition of real numbers. Ex: 2 × (1 + 3) = (2 × 1) + (2 × 3). A data display method which records frequency using a ―•‖ notation as shown below. Dot plot First quartile For a data set with median M, the first quartile is the median of the data values less than M. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the first quartile is 6.2 See also: median, third quartile, interquartile range. Frequency table A table that lists items and uses tally marks to show the number of times they occur. Identity property of addition states that the sum of zero and any number or variable is the number or variable itself. For example, 4 + 0 = 4, - 11 + 0 = - 11, y + 0 = y are few examples illustrating the identity property of addition. That point where the bisectors of the angles of a triangle or of a regular polygon intersect. The point where the three angle bisectors of a Identity property of 0 Incenter triangle meet. Independent events Independently combined probability models Inter-quartile Range Intersection 6/21/2012 Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Two probability models are said to be combined independently if the probability of each ordered pair in the combined model equals the product of the original probabilities of the two individual outcomes in the ordered pair. A measure of variation in a set of numerical data, the inter-quartile range is the distance between the first and third quartiles of the data set. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the inter-quartile range is 15 – 6 = 9. See also: first quartile, third quartile. A set that contains elements shared by two or more given sets. BVSD Curriculum Essentials 41 Kite A quadrilateral that has two distinct pairs of consecutive equilateral sides. Line plot A method of visually displaying a distribution of data values where each data value is shown as a dot or mark above a number line. Also known as a dot plot A measure of center in a set of numerical data, computed by adding the values in a list and then dividing by the number of values in the list.4 Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is 21. A measure of variation in a set of numerical data, computed by adding the distances between each data value and the mean, then dividing by the number of data values. Example: For the data set {2, 3, 6, 7, 10,12, 14, 15, 22, 120}, the mean absolute deviation is 20. A measure of center in a set of numerical data. The median of a list of values is the value appearing at the center of a sorted version of the list—or the mean of the two central values, if the list contains an even number of values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the median is 11. The segment connecting the vertex of an angle in a triangle to the midpoint of the side opposite it. Mean Mean absolute deviation Median (Statistical) Median of a Triangle Midpoint formula Midpoint of a line segment is the point that is halfway between the endpoints of the In two-dimensional coordinate plane, the midpoint of a line with coordinates of its endpoints as (x1, y1) and line segment. (x2, y2) is given by Midsegment Multiplicative inverses Null set Orthocenter 6/21/2012 A line segment joining the midpoints of two sides of a triangle. Two numbers whose product is 1 are multiplicative inverses of one another. Example: 3/4 and 4/3 are multiplicative inverses of one another because 3/4 × 4/3 = 4/3 × 3/4 = 1. a set that is empty; a set with no members The point where the three altitudes of a triangle intersect. BVSD Curriculum Essentials 42 Parallel lines Parallel Lines are distinct lines lying in the same plane and they never intersect each other. Parallel lines have the same slope. In the figure below, lines PQ and RS are parallel and the lines l and m are parallel. Perpendicular lines Perpendicular lines are lines that intersect at right angles. If two lines are perpendicular to each other, then the product of their slopes is equal to – 1. In the figure shown below, the lines AB and EF are perpendicular to each other. Points of Concurrency Probability Probability distribution Probability model Real number Rectilinear figure Rigid motion Sample space The place where three or more lines, rays, or segments intersect at the same point. See point H in diagram above. A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (such as tossing a coin, selecting a person at random from a group of people, tossing a ball at a target, or testing for a medical condition). The set of possible values of a random variable with a probability assigned to each. A probability model is used to assign probabilities to outcomes of a chance process by examining the nature of the process. The set of all outcomes is called the sample space, and their probabilities sum to 1. See also: uniform probability model. The real numbers are sometimes thought of as points on an infinitely long line named number line or real line. Rectilinear figures are figures bounded by straight lines. A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are here assumed to preserve distances and angle measures. In a probability model for a random process, a list of the individual outcomes that are to be considered Scatter plot A graph of plotted points that show the relationship between two sets of data (bivariate). Similarity transformation A rigid motion followed by a dilation. 6/21/2012 BVSD Curriculum Essentials 43 Tessellation A covering of a plane with congruent copies of the same region with no holes or overlaps. Theorem An important mathematical statement which can be proven by postulates, definitions, and previously proved theorems. For a data set with median M, the third quartile is the median of the data values greater than M. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the third quartile is 15. See also: median, first quartile, interquartile range. Informally, moving a shape so that it is in a different position, but still has the same size, area, angles and line lengths. Formally, a correspondence between two sets of points such that each point in the pre-image has a unique image and that each point in the image has exactly one pre-image. Third quartile Transformations Transitivity principle for indirect measurement Trichotomy property Trigonometric ratios Undefined slope Uniform probability model Union Unique Universal Set 6/21/2012 If the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of object A is greater than the length of object C. This principle applies to measurement of other quantities as well. The property that for natural numbers a and b , either a is less than b , a equals b , or a is greater than b . A ratio that describes the relationship between a side and an angle of a triangle. Sine, Cosine, Tangent The "slope" of a vertical line. A vertical line has undefined slope because all points on the line have the same x-coordinate. As a result the formula used for slope has a denominator of 0, which makes the slope undefined. A probability model which assigns equal probability to all outcomes. See also: probability model A set, every member of which is an element of one or another of two or more given sets. Limited to a single outcome or result; without alternative possibilities. A set containing all elements of a problem under consideration. BVSD Curriculum Essentials 44 Venn Diagram Diagrams that show all hypothetically possible logical relations between a finite collection of sets (aggregation of things). Definitions adapted from: Boulder Valley School District Curriculum Essentials Document, 2009. ―Math Dictionary‖ www.icoachmath.com/math_dictionary/mathdictionarymain.html. Copyright © 1999 - 2011 HighPoints Learning Inc. December 30, 2011. ―The Mathematics Glossary.‖ Common Core Standards for Mathematics. http://www.corestandards.org/the-standards/mathematics/glossary/glossary/ Copyright 2011. June 23, 2011. ―Thesaurus.‖ http://www.thefreedictionary.com/statistical+distribution . Copyright © 2012 Farlex, Inc. January 5, 2012. (n.d.). Retrieved from sas.com. Statistics Glossary (n.d.). Retrieved from University of Glasgow: http://www.stats.gla.ac.uk/steps/glossary/index.html. ―Illustrated Mathematics Dictionary.‖ http://www.mathsisfun.com/definitions/Copyright © 2011 MathsIsFun.com. January 14, 2012. ―Math Open Reference.‖ http://www.mathopenref.com/triangleincenter.html 2009 Copyright Math Open Reference. January 15, 2012. 6/21/2012 BVSD Curriculum Essentials 45