Precalculus, Quarter 4, Unit 4.1 Conic Sections

1 Cumberland, Lincoln, and Woonsocket Public Schools C-39 in collaboration with the Charles A. Dana Center at the University of Texas at Austin precalculus , Quarter 4, unit Conic Sections Overview Number of instructional days: 12 (1 day = 45 minutes) Content to be learned Mathematical practices to be integrated Use analytic geometry to solve problems involving finding the equation of a circle inscribed in a triangle. Use analytic geometry to solve problems involving finding the distance between opposite vertices in a rectangular solid. Use analytic geometry to solve problems involving using the distance formula to obtain the equations for Conic Sections , including circle, parabola, hyperbola, and ellipse centered at (h, k).

2 Explore and interpret characteristics of conics graphically and algebraically including different planar slices of a double cone yielding different Conic Sections . Explore conics as loci of points satisfying certain stipulated conditions. Attend to precision. Use labels of axes and units of measure correctly. Model with mathematics. Use a simpler problem to solve more complex problems. Look and make use of structure. Apply prior learning to new situations. Use appropriate tools strategically. Use technology to visualize results. Essential questions What are the similarities and differences between the four types of curves known as Conic Sections ? What is a Conic section and how is it developed?

3 What is the intersection of a cone and a plane parallel to a line along the side of a cone? What mathematical theorems and postulates are used in finding the equations of Conic Sections ? What is meant by a locus of points, and how is it used in determining an equation of a Conic section? precalculus , Quarter 4, unit Conic Sections (12 days) 2010 2011 C-40 Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin Written Curriculum Grade Span Expectations M(G&M) AM 9 Solves problems using analytic geometry (including three-dimensions) and circular trigonometry ( , find the equation of a circle inscribed in a triangle; find the distance between opposite vertices in a rectangular solid); explores and interprets the characteristics of Conic Sections graphically and algebraically including understanding how different planar slices of a double cone yield different Conic Sections ; knows the characterization of Conic Sections as loci of points in the plane satisfying certain distance requirements, and uses the distance formula to obtain equations for the Conic Sections .

4 (Local) Clarifying the Standards Prior Learning In kindergarten, students demonstrated an understanding of spatial relationships using location and position to find objects in the environment. In grade 1, position and location was extended to positional words with reference to maps and diagrams. In grade 2, there was an extension to 2-D and 3-D situations to interpret relation positions, create and interpret simple maps, and name locations on a simple coordinate grid. In grade 3, students interpreted and gave directions from one location to another between locations or compass directions. In grade 4, there was an extension to plot points in quadrant one in context and finding horizontal and vertical distances between points on a coordinate grid.

5 In grade 5, plotting points is extended to quadrants one through four, and students identified vertices of polygons as they are reflected, rotated, and translated. From grades 6 through 8, there were no additions. In algebra 1 and geometry, students solved problems on and off the coordinate plane involving distance, midpoint, parallel and perpendicular lines, and slope. In algebra 2, students solved problems involving conics as a locus of points in the plane and found the equation of conics centered at (0, 0). Current Learning Students solve problems using analytical geometry (including 3-D) and circular trigonometry, including finding the equation of a circle inscribed in a triangle, finding the distance between opposite vertices in a rectangular solid, and using the distance formula to obtain the equations for Conic Sections to include circle, parabola, hyperbola, and ellipse centered at (h, k).

6 Students explore and interpret characteristics of conics graphically and algebraically, including different planar slices of a double cone yielding different Conic Sections . Students explore conics as loci of points satisfying certain stipulated conditions. Future Learning Students will explore Conic Sections as relations for parametric equation development that will lead into related rates and derivative implications in the real world. Additional Research Findings Beyond Numeracy states that analytical geometry and its offshoots are so seemingly natural and, thus, so taken for granted, that it sometimes requires a special effort to remember that they are inventions of human beings. By examining graphs of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, general quadratic equations in two variables give rise to equations whose graphs are circles, ellipses, parabolas, and hyperbolas.

7 These are the same figures that are formed by the intersection of a cone and a plane, where the angle of the plane determines which one of the Conic Sections results (pp. 11 14, 199 200). Cumberland, Lincoln, and Woonsocket Public Schools C-41 in collaboration with the Charles A. Dana Center at the University of Texas at Austin precalculus , Quarter 4, unit Data Analysis Overview Number of instructional days: 10 (1 day = 45 50 minutes) Content to be learned Mathematical practices to be integrated Collect, organize, analyze, calculate, and interpret data using scatterplots, linear regression, least squares, and median-median graphs and equations. Calculate, analyze, and interpret measures of dispersion (range, variance, percentiles, and standard deviation).

8 Calculate, analyze, and interpret measures of central tendency for the normal curve. Reason abstractly and quantitatively. Check answers to ensure they are quantitatively sound in problems involving data analysis. Interpret answers to problem situations involving data analysis and relate to other scenarios. Model with mathematics. Draw conclusions from data based on relationships and models. Use two-way tables, graphs, and flowcharts to determine if results make sense. Use appropriate tools strategically. Use technology to visualize results. Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Essential questions Give an example of a real-life situation that can be modeled using a scatterplot and linear regression (line of best fit).

9 What real-life examples can be modeled using a normal distribution? What careers use statistics? What are the correlation coefficient and the coefficient of determination? Do they necessarily determine causality? What real-life situations are not modeled by the normal distribution and why? How do measures of dispersion differ from measures of central tendency? Provide examples of where one or both are used? What are the implications of the least squares and median-median equations relative to a data set? precalculus , Quarter 4, unit Data Analysis (10 days) 2010 2011 C-42 Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin Written Curriculum Grade-Level Expectations/Grade-Span Expectations M(DSP)-12-1 Interprets a given representation(s) ( , regression function including linear, quadratic, and exponential) to analyze the data to make inferences and to formulate, justify, and critique conclusions.

10 (Local) (IMPORTANT: Analyze data consistent with concepts and skills in M(DSP)-11-2). M(DSP)-12-2 Analyzes patterns, trends, or distributions in data in a variety of contexts by calculating and analyzing measures of dispersion (standard deviation, variance, and percentiles). (Local) M(DSP)-AM-2 Analyzes and interprets measures of dispersion (standard deviation, variance, and percentiles) and central tendency for the normal distribution; and interprets the correlation coefficient and the coefficient of determination in the context of data. (Local) M(DSP)-AM-3 Uses technology to explore the method of least squares and median-median for linear regression. (Local) Clarifying the Standards Prior Learning In kindergarten, students answered questions relating to data given in words, diagrams, or verbally, and they scribed responses using models and tally charts.