Algebraic Representations of Conic Sections

Transcription

1 Geometry, Quarter 4, Unit 4.1 Algebraic Representations of Conic Sections Overview Number of instructional days: 22 (1 day = 45 minutes) Content to be learned Construct the inscribed and circumscribed circles of a triangle. Prove properties of angles for a quadrilateral inscribed in a circle. Make geometric constructions of an equilateral triangle, square, and regular hexagon inscribed in a circle (as well as explain the process of construction). (+) Construct a tangent line from a point outside a given circle to the circle. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove whether the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Translate between the geometric description and the equation for a conic section. Derive the equation of a circle of given center and radius using the Pythagorean Theorem. Complete the square to find the center and radius of a circle given by an equation. Derive the equation of a parabola given a focus and directrix. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Identify the shapes of two-dimensional cross sections of three-dimensional objects. Identify three-dimensional objects generated by rotations of two-dimensional objects. Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments regarding circles on and off coordinate planes. Determine whether the arguments make sense; ask useful questions to clarify and improve an argument or proof. Use appropriate tools strategically. Make geometric constructions with a variety of tools, including a compass, protractor, straightedge, and geometric software. Make sense of problems and persevere in solving them. Explain the meaning of a problem and look for entry points to its solution when writing the equations of conics. Analyze givens, constraints, and relationships, and utilize them to derive the equations of circles, parabolas, ellipses, and hyperbolas. Make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt Check answers to problems using a different method, and continually ask, Does this make sense? Understand the approaches of others when solving complex problems and deriving the formulas in conic sections. 35

2 Geometry, Quarter 4, Unit 4.1 Algebraic Representations of Conic Sections (22 days) Identify correspondences between different approaches in problem solving. Attend to precision. Specify units of measure in constructions. Use theorems and definitions to examine and prove or disprove a mathematical claim. Essential questions How do you construct a circle that circumscribes a triangle? How do you construct a polygon inscribed in a circle? What can you conclude about the angles of a quadrilateral inscribed in a circle? How can you prove it? If you are given a point on a circle and the coordinates of its center, how can you prove or disprove that another point is on the circle? Given the center and radius, how can you develop the equation of a circle? Given a focus and directrix, how can you develop the equation of a parabola? How do you identify cross sections of threedimensional figures? How do you use rotations to generate three-dimensional figures seen in everyday life? Written Curriculum Common Core State Standards for Mathematical Content Circles G-C Understand and apply theorems about circles G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Congruence G-CO Make geometric constructions [Formalize and explain processes] G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Circles G-C Understand and apply theorems about circles G-C.4 (+) Construct a tangent line from a point outside a given circle to the circle. 36

3 Geometry, Quarter 4, Unit 4.1 Algebraic Representations of Conic Sections (22 days) Expressing Geometric Properties with Equations G-GPE Use coordinates to prove simple geometric theorems algebraically [Include distance formula; relate to Pythagorean theorem] G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. 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). Translate between the geometric description and the equation for a conic section G-GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G-GPE.2 Derive the equation of a parabola given a focus and directrix. G-GPE.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Geometric Measurement and Dimension G-GMD Visualize relationships between two-dimensional and three-dimensional objects G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Common Core Standards for Mathematical Practice 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 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. 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 37

4 Geometry, Quarter 4, Unit 4.1 Algebraic Representations of Conic Sections (22 days) 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. 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. 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. Clarifying the Standards Prior Learning In earlier years, students recognized shapes and their attributes. In grade 7, students described the 2-D figures that result from slicing 3-D figures. (7.G.3) In grade 8, students proved and applied the Pythagorean Theorem and its converse in 2-D and 3-D applications. In Algebra I, students completed the square in quadratic expressions. (A.SSE.3b) In Unit 1.1 of this course, students made formal constructions using a variety of tools. (G-CO.12) In Unit 3.1, they proved and began using the distance formula. (G-GPE.7) In Unit 3.3, students studied inscribed and circumscribed angles of circles. (G-C.2) 38

5 Geometry, Quarter 4, Unit 4.1 Algebraic Representations of Conic Sections (22 days) Current Learning Students construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle. They construct, formalize, and explain the process of constructing an equilateral triangle, square, and regular hexagon inscribed in a circle. Students construct a tangent line from a point outside a given circle to the circle (+ standard). They algebraically prove whether a point lies on circle given the origin and a point on a circle [e.g., whether the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2)]. Students translate between the geometric description and the equation for a conic section. They derive the equation of a circle of given center and radius using the Pythagorean Theorem. Students complete the square to find the center and radius of a circle given by an equation. They derive the equation of a parabola given a focus and directrix. Students derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Students identify the shapes of two-dimensional cross sections of three-dimensional objects as well as three-dimensional objects generated by rotations of two-dimensional objects. Future Learning Students will continue their study of quadratic functions in Algebra II. In future mathematics courses, they will utilize and build upon their knowledge of the conic sections, rotation of 2-D objects, and cross section of 3-D objects. Student will encounter these topics in careers such as graphic design, architecture, engineering, and art-related fields. Additional Findings According to Principles and Standards for School Mathematics, Working to understand orientation and drawings in a 3-D rectangular coordinate system will afford opportunities for student to think and reason spatially. In addition, Schooling should provide rich mathematical settings in which they [students] can hone their visualization skills. Visualizing a building represented in architectural plans, the shape of cross section formed when a plane slices through a cone (conic section) or another solid object, or the shape of the solid swept out when a plane figure is rotated about an axis become easier when students work with physical models, drawings, and software capable of manipulating 3-D representations. (pp. 315 and 316) Writing Team Notes In G-GPE.2 and G-GPE.3(+), the writing team interprets the word derive to mean that students should be able to write and use the equation for a parabola, ellipse, and hyperbola given specific conditions. It may not be necessary at this level for students to derive these formulas from scratch. 39

6 Geometry, Quarter 4, Unit 4.1 Algebraic Representations of Conic Sections (22 days) 40

7 Geometry, Quarter 4, Unit 4.2 Introduction to and Applications of Conditional Probability Overview Number of instructional days: 18 (1 day = 45 minutes) Content to be learned Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (or, and, not). Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Apply the Addition Rule, P( A or B) = P( A) + P(B) P( A and B), and interpret the answer in terms of the model (using a modeling approach). Understand the conditional probability of A given B as P(A and B)/P(B) and interpret independence of A and B as P(A B) = P(A), and P(B A) = P(B) (using a modeling approach). Understand that two events A and B are independent if P(A and B) = P(A) P(B) and use this characterization to determine if they are independent (using a modeling approach). Find the conditional probability of A given B as the fraction of Bs outcomes that also belong to A, and interpret the answer in terms of the P( A and B) model. P( A B) = P(B) Mathematical practices to be integrated Make sense of problems and persevere in solving them. Explain the meaning of a problem and analyze givens, constraints, relationships, and goals. Explain correspondence between equations, verbal descriptions, tables, and graphs. Reason abstractly and quantitatively. Represent a given situation symbolically and manipulate the symbols when dealing with conditional probabilities. When using the rules of probability, create a coherent representation of the problem at hand, attending to the meaning of the quantities, not just how to compute them. Model with mathematics. Construct and interpret two-way frequency tables of collected data to determine whether events are independent or dependent and calculate their approximate probabilities. Use the rules of probability to compute probabilities of compound events in a uniform probability model. Identify important qualities in a practical situation and map their relationships using tools such as tree diagrams, two-way tables, Venn diagrams, and formulas. 41

8 Geometry, Quarter 4, Unit 4.2 Introduction to and Applications of Conditional Probability (18 days) Use appropriate tools strategically. Use tools such as graphing calculators, a statistical package, or other computer software to collect random samples of data. Utilize technology when making mathematical models to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Essential questions What are the similarities and differences between experimental and theoretical probability? How could you use a frequency table and probability distribution to draw conclusions about a given situation? How could you determine whether two events are dependent or independent? What is an example of mutually exclusive events? How can you find the probability of these events? How can tables, tree diagrams, and formulas be used to find conditional probability? Is it possible to prove one event causes another event if the events are not independent? Written Curriculum Common Core State Standards for Mathematical Content Conditional Probability and the Rules of Probability S-CP Understand independence and conditional probability and use them to interpret data [Link to data from simulations or experiments] S-CP.1 S-CP.4 S-CP.5 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in 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. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. 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. 42

9 Geometry, Quarter 4, Unit 4.2 Introduction to and Applications of Conditional Probability (18 days) Use the rules of probability to compute probabilities of compound events in a uniform probability model S-CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. Understand independence and conditional probability and use them to interpret data [Link to data from simulations or experiments] S-CP.3 S-CP.2 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Use the rules of probability to compute probabilities of compound events in a uniform probability model S-CP.6 Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. Common Core Standards for Mathematical Practice 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 43

10 Geometry, Quarter 4, Unit 4.2 Introduction to and Applications of Conditional Probability (18 days) quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 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. Clarifying the Standards Prior Learning In grade 1, students organized, represented, and interpreted data with up to three categories. They also asked and answered questions about the data. (1.MD.4) In grade 2, students made line plots, picture graphs, and bar graphs (with single-unit scales) to represent a data set with up to four categories. They also solved problems using information presented in a bar graph. (2.MD.10) In grade 3, students drew scaled picture graphs and scaled bar graphs to represent data in several categories. They solved one- and two-step problems using information presented in scaled bar graphs. (3.MD.3) In grade 6, students represented and analyzed quantitative relationships between dependent and independent variables. (6.EE.9) In grade 7, students understood that sample population and random sampling can be used to gain information and make inferences about a population. They also investigated chance processes and developed, used, and evaluated probability models. Students studied unlikely events, predicted relative frequency given probability, and found probabilities of compound events and 44

11 Geometry, Quarter 4, Unit 4.2 Introduction to and Applications of Conditional Probability (18 days) displayed them in lists, tables, and tree diagrams. They designed a simulation to generate frequencies for compound events. (7.SP.1-8) In grade 8, students used relative frequencies to describe possible associations between two variables. (8.SP.4) In Unit 3.2 of Algebra 1, students summarized categorical data for two categories in two-way frequency tables. They also interpreted relative frequencies in the context of the data. (S-ID.5) Current Learning Students describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (or, and, not). They construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Students use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. They recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. They apply the Addition Rule, P( A or B) = P( A) + P(B) P( A and B), and interpret the answer in terms of the model. Students understand the conditional probability of A given B as P(A and B)/P(B) and interpret independence of A and B as P(A B) = P(A) and P(B A) = P(B). They understand that two events A and B are independent if P(A and B) = P(A) P(B) and use this characterization to determine if they are independent. Students find the conditional probability of A given B as the fraction of Bs outcomes P( A and B) P( A B) = P(B) that also belong to A, and interpret the answer in terms of the model. Future Learning In Unit 4.4 of Algebra II, students will build upon the concept of random sampling in probability to explain how it relates to sample surveys, experiments, and observational studies. In Unit 4.1 of Precalculus, students will utilize the rules of probability to compute probabilities of compound events in a uniform probability model. They will use probability to evaluate outcomes of decisions and strategies such as drawing by lots and using a random number event. Students will calculate the expected value and develop a probability distribution of a random variable defined for a sample space, in which probabilities are assigned empirically or calculated theoretically. They will further develop and apply probability concepts in advanced-level statistics courses and actuarial sciences. Most students will encounter a statistics course at the college level, regardless of their major. Additional Findings Benchmarks for Science Literacy states, One of the many misunderstandings of probability that teachers have to deal with is that a wellestablished probability will be changed by the most recent history: [for example] People tend to believe that a coin that has come up heads ten times in a row is more likely on the next flip to come up tails than heads or that the number that won the lottery last week is less likely to win this week. Those and other confusions about probability are purely mathematical and can be addressed as such, but it is also important to take up some of the questions related to how probabilities are established. Examples should come from medicine, natural catastrophes such as floods and earthquakes, weather patterns, sports events, stock market events, elections, and other topical contexts. (p. 226) By the end of 12th grade, students should know that a physical or mathematical model can be used to estimate the probability of real-world events. (p. 230) 45

12 Geometry, Quarter 4, Unit 4.2 Introduction to and Applications of Conditional Probability (18 days) Writing Team Notes Include discussion of empirical (experimental) and theoretical probabilities to help students better understand the basis of probability. Emphasize that association/correlation does not equal causation; in other words, just because two events are associated, it does not mean that one is a result of the other. This is a common misconception students have. Different textbooks have used different notations for unions ( ) and intersections ( ) instead of and and or stated in the Common Core standards. For more specific examples of each standard and how they translate to instructional resources refer to chapter 11 of the On Core Mathematics by Houghton Mifflin Harcourt or chapter 13 of the Geometry Common Core by Pearson. 46