Precalculus, Quarter 4, Unit 4.1 Conic Sections Overview Number of instructional days: 12 (1 day = 45 minutes) Content to be learned 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). 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. 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? What is the intersection of a cone and a plane parallel to a line along the side of a cone? Mathematical practices to be integrated 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. 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? Cumberland, Lincoln, and Woonsocket Public Schools C-39
Precalculus, Quarter 4, Unit 4.1 Conic Sections (12 days) Written Curriculum Grade Span Expectations M(G&M) AM 9 Solves problems using analytic geometry (including three-dimensions) and circular trigonometry (e.g., 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. (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. 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). 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 Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, general quadratic equations in two variables give rise to equations whose graphs are circles, ellipses, parabolas, and hyperbolas. 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). C-40 Cumberland, Lincoln, and Woonsocket Public Schools
Precalculus, Quarter 4, Unit 4.2 Data Analysis Overview Number of instructional days: 10 (1 day = 45 50 minutes) Content to be learned 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). Calculate, analyze, and interpret measures of central tendency for the normal curve. Mathematical practices to be integrated 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). 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? Cumberland, Lincoln, and Woonsocket Public Schools C-41
Precalculus, Quarter 4, Unit 4.2 Data Analysis (10 days) Written Curriculum Grade-Level Expectations/Grade-Span Expectations M(DSP)-12-1 Interprets a given representation(s) (e.g., regression function including linear, quadratic, and exponential) to analyze the data to make inferences and to formulate, justify, and critique conclusions. (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. Students also analyzed trends using more, less, or equal. In grade 1, representations using pictographs with one-to-one correspondence and tables were added. Grade 2 students explored representations of line plots, and in grade 3, students also interpreted bar graphs, made predictions, and analyzed trends using most frequent (mode), least frequent, largest, or smallest. In grade 4, pictographs and circle graphs were interpreted, and students analyzed data and justified conclusions to make predictions and solve problems. Students also used measures of central tendency (median or mode) or range. In grade 5, line graphs were also interpreted. Students used measures of central tendency (mean, median, or mode) or range to analyze situations and solve problems. In grade 6, stem-and-leaf plots were added, and dispersion (range) was added to analyze situations. In grade 7, students interpreted scatter plots representing discrete linear relationships or histograms. Students identified outliers in data and analyzed situations to determine their effect on mean, median, or mode, and evaluated the sample from which the statistics were developed (bias). In grade 8, scatterplots and box-and-whisker plots (quartile values) were analyzed, and an estimated line of best fit was determined. Students evaluated samples to determine if a survey was random, nonrandom, or biased. In grades 9 10, students analyzed data to formulate and justify conclusions, make predictions, and solve problems. Students estimated a regression line and found correlation (strong positive, strong negative, or no correlation. They also solved problems involving conceptual understanding of the sample. C-42 Cumberland, Lincoln, and Woonsocket Public Schools
Data Analysis (10 days) Precalculus, Quarter 4, Unit 4.2 Current Learning Students interpret linear regression functions to analyze data, make inferences, and formulate, justify, and critique conclusions. Students analyze measures of dispersion including standard deviation, variance, percentiles, and central tendency for the normal distribution. They also interpret the correlation coefficient and coefficient of determination in the context of data. Future Learning Students will use and encounter statistics in the field of economics, business, research, science, medicine, engineering, sports, and information technology. Additional Research Findings Through statistics, we develop useful ways of organizing data. The research references listed below validate that knowledge of statistics is essential to many careers while explaining which statistics concepts should be taught at each grade level. Research supports theories of normal distribution and the central limit theorem while reinforcing how important it is for students to make valid statistical comparisons. Beyond Numeracy, (pp. 227 230) Science for All Americans, (pp. 137 139) A Research Companion to Principles and Standards for School Mathematics, (pp. 193 199) Benchmarks for Science Literacy, (pp. 226 230). Principles and Standards for School Mathematics, (pp. 48 50, 324 330) Cumberland, Lincoln, and Woonsocket Public Schools C-43
Precalculus, Quarter 4, Unit 4.2 Data Analysis (10 days) C-44 Cumberland, Lincoln, and Woonsocket Public Schools
Precalculus, Quarter 4, Unit 4.3 Sampling Overview Number of instructional days: 8 (1 day = 45 50 minutes) Content to be learned Collect, organize, and display data using appropriate techniques; draw conclusions from data. Make hypotheses and answer questions to interpret data. Justify conclusions about sample and sampling used in data collection. Essential questions What are the advantages of using sampling techniques in society? Provide an example utilizing an appropriate sampling method. What are the societal implications (historically) of using sampling techniques? Provide an example (e.g., biomedicine, sports). Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Ask appropriate questions regarding situations requiring sampling. Communicate conclusions regarding sampling problem situations. Attend to precision. Use precise mathematical vocabulary and symbols to communicate efficiently and effectively in problems involving sampling. Model with mathematics. Relate what has been learned in mathematics to everyday life. Improve the model if it needs modification. Does the sampling method affect the results of the study/survey? Explain. Cumberland, Lincoln, and Woonsocket Public Schools C-45
Precalculus, Quarter 4, Unit 4.3 Sampling (8 days) Written Curriculum Grade-Level Expectations/Grade-Span Expectations M(DSP)-12-6 In response to a teacher or student generated question or hypothesis decides the most effective method (e.g., survey, observation, research, experimentation) and sampling techniques (e.g., random sample, stratified random sample) to collect the data necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the questions or hypotheses being tested while considering the limitations of the data that could effect interpretations; and when appropriate makes predications, asks new questions, or makes connections to real-world situations. (Local) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 10 2.) Clarifying the Standards Prior Learning In grades 2 3, groups of students chose the most effective method among survey, observation, or experimentation to collect data. At this level, data was categorical or numerical. Students collected, organized, and displayed the data appropriately. They analyzed data to draw conclusions about the question or hypothesis being tested and made required predictions when appropriate. In grades 4 6, the above learning continued, but students had the ability to ask new questions and make connections to the real-world situations in response to the given hypothesis or question. In grades 5 6, individuals, in addition to groups, made decisions involving the study. In grades 7 8, students also considered limitations that could affect interpretations of the data under study. In grade 9, research was added to the statistical study in addition to survey, observation, and experimentation. Sampling techniques, such as random sampling and stratified random sampling, were used to collect data. Current Learning Students employ the full definition of a statistical study. They collect, organize, describe, display, calculate, analyze, synthesize, and interpret data to make predictions and conclusions. Students use a variety of sampling techniques to accomplish this task, which can be either teacher- or student-generated. Future Learning Data, statistics, and probability are widely used in the fields of economics, business, research, science, medicine, engineering, sports, information technology, politics, etc., to name a few. Numerous facets of life are based upon current, prior, or future statistical analysis. Additional Research Findings Research information on this content can be found in the following references: Science for All Americans (pp. 139 140) A Research Companion to Principles and Standards for School Mathematics (pp. 193 215) Benchmarks for Science Literacy (pp. 226 230) Principles and Standards for School Mathematics (pp. 48 50 and p. 230) C-46 Cumberland, Lincoln, and Woonsocket Public Schools
Sampling (8 days) Precalculus, Quarter 4, Unit 4.3 This research indicates that if sampling is done without bias in the method, then the larger the sample is, the more likely it is to represent the whole accurately. Central elements of statistical analysis is discussed defining an appropriate sample, collecting data from that sample, and making reasonable inferences relating the sample and the population. People can be alert to possible bias in choosing samples that others take but maybe are unable to take adequate precautions against bias in designing a study of their own. An understanding of how samples are taken is crucial to effectively participate in political debates about the environment, health care, quality of education, and equity. Cumberland, Lincoln, and Woonsocket Public Schools C-47
Precalculus, Quarter 4, Unit 4.3 Sampling (8 days) C-48 Cumberland, Lincoln, and Woonsocket Public Schools
Precalculus, Quarter 4, Unit 4.4 Limits, Proofs, and Non-Euclidean Geometry Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Use geometric models to represent and distinguish between Euclidean and non- Euclidean systems. Use informal concepts of successive approximation, upper and lower bounds, and limits in measurement situations. Use measurement conversion strategies. Extend and deepen knowledge and usage of proofs and proof techniques. Essential questions What roles does the concept of limit play in everyday life? How does this relate to mathematics use of limits? What implications does the study of non- Euclidean geometry have in various segments of society? How is the process of proving used in various disciplines? What is the connection between finding the partial sum of a sequence and finding its limit? Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Use proofs to make logical arguments and conclusions. Make conjectures and prove them, including using counter examples. Look for and make use of structure. Identify patterns and structures. Use appropriate tools strategically. Use various tools, including technology, to help solve problems. Use technology to visualize results. How is the process of finding the limits of partial sums useful in determining the area and volume of 2-D and 3-D figures, respectively? How does finding the upper and lower limit of a graphical region relate to the boundaries of the function? What is the significance of expressing dimensions as units? Cumberland, Lincoln, and Woonsocket Public Schools C-49
Precalculus, Quarter 4, Unit 4.4 2010-2011 Limits, Proofs, and Non-Euclidean Geometry (10 days) Written Curriculum Grade Span Expectations M(G&M) 12 7 Uses informal concepts of successive approximation, upper and lower bounds, and limits in measurement situations (e.g., use successive approximation to find the area of a pond); uses measurement conversion strategies (e.g., unit/dimensional analysis). (Local) M(G&M) AM 2 Extends and deepens knowledge and usage of proofs and proof techniques; and uses geometric models to represent and distinguish between Euclidean and non-euclidean Systems. (Local) Clarifying the Standards Prior Learning In geometry, students made conjectures, constructed geometric arguments, and used geometric properties or theorems to solve problems (proofs). Students studied Euclidean geometry in the high school geometry course. In unit 2.2 of this course, students proved trigonometric identities. In unit 3.2, they studied infinite sequences and series, which directly relates to limits. Current Learning Students use geometric models to represent and distinguish between Euclidean and non-euclidean systems. Students use informal concepts of successive approximation, upper and lower bounds, and limits in measurement situations. Students also use measurement conversion strategies. Students extend and deepen their knowledge and usage of proofs and proof techniques. Future Learning In calculus, the derivative and integral will be introduced using the definition of limit. This will lead to further investigation of rates of change of functions, net accumulation of functions, and applications of the derivative and integral. Inductive proof will be further analyzed in higher mathematics courses. Students will explore Euclidean and non-euclidean geometry in greater detail in college mathematics and science courses. Additional Research Findings According to Beyond Numeracy, mathematical induction can be used to prove any proposition that involves an arbitrary integer n (pp. 120 123). C-50 Cumberland, Lincoln, and Woonsocket Public Schools