Algebra 1, Quarter 3, Unit 3.1. Line of Best Fit. Overview


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1 Algebra 1, Quarter 3, Unit 3.1 Line of Best Fit Overview Number of instructional days 6 (1 day assessment) (1 day = 45 minutes) Content to be learned Analyze scatter plots and construct the line of best fit using a straight edge. (1 day) Calculate the lineoffit algebraically. (2 days) Interpret a scatter plot to determine correlation (positive, negative, or no correlation) and strength of correlation. (1 day) Determine how the size of the sample space affects the accuracy and reliability of the results. (1 day) Essential questions How can graphing ordered pairs determine whether two sets of numerical data are related? How does the sample size affect the accuracy or reliability of the line of best fit/regression equation? Mathematical practices to be integrated Model with mathematics. Use functions to model problem situations. Analyze the relationships of specific characteristics of functions. Attend to precision. Use labels of axes and units of measure correctly. Calculate and compute accurately (including technology). Why is a scatter plot a good way to show and describe the relationship between two sets of data? How do outliers affect the line of best fit and the generalizations you make from that line? Cumberland, Lincoln, and Woonsocket Public Schools C25
2 Algebra 1, Quarter 3, Unit 3.1 Final, July 2011 Line of Best Fit (6 days) Written Curriculum Grade Span Expectations M(DSP) 10 2 Analyzes patterns, trends, or distributions in data in a variety of contexts by determining, using, or analyzing measures of central tendency (mean, median, or mode), dispersion (range or variation), outliers, quartile values, estimated line of best fit, regression line, or correlation (strong positive, strong negative, or no correlation) to solve problems; and solve problems involving conceptual understanding of the sample from which the statistics were developed. (State) Clarifying the Standards Prior Learning In grades K 3, students worked with data in context and used the data to determine more, less, or equal. In grades 4 6, students were introduced to measures of central tendency and range in order to analyze situations and solve problems. In grades 7 and 8, students were introduced to dispersion (range or variation), outliers, quartile values, or estimated line of best fit as ways to analyze situations or solve problems, and they evaluated the sample from which the statistics were developed (bias, random, or nonrandom). Current Learning Students review and reinforce dispersion (which was taught earlier this year), measures of central tendency, quartiles, and outliers. Students are introduced to regression line and correlation (strong positive, strong negative, or no correlation) to solve problems, and they solve problems involving conceptual understanding of the sample from which the statistics were developed. Future Learning Students will continue to analyze patterns, trends, or distributions of data in a variety of contexts by calculating and analyzing measures of dispersion (standard deviation, variance, and percentiles). Students will then interpret the data through using the correlation coefficient (r) and the coefficient of determination (r 2 ). Additional Research Findings A Research Companion to Principles and Standards for School Mathematics states, Through multiple experiences with a variety of data sets, students begin to develop the tools and concepts they need to use data themselves and to interpret the data they will encounter through life (pp ). Cumberland, Lincoln, and Woonsocket Public Schools C26
3 Algebra 1, Quarter 3, Unit 3.2 Collecting and Analyzing Data Overview Number of instructional days: 7 (1 day assessment) (1 day = 45 minutes) Content to be learned Choose the most effective method (survey, observation, research, experimentation) and sample techniques to collect the data necessary to answer a given question or hypothesis. (1 day) Collect, organize, and appropriately display data to answer a given question or hypothesis. (2 day) Analyze data to interpret and draw conclusions about the question or hypothesis being tested, considering limitations of the data that could affect interpretations. (2 day) Use data, when appropriate, to make predictions, ask new questions, or make connections to realworld situations. (1 day) Mathematical practices to be integrated Make sense of problems and persevere in solving them. Justify and explain solutions. Construct viable arguments and critique the reasoning of others. Analyze and evaluate the mathematical thinking and strategies of others. Model with mathematics. Use twoway tables, graphs, and flowcharts to determine if results make sense. Interpret results in the context of a problem. Improve the model if it needs modification. Look for and express regularity in repeated reasoning. Look for patterns to find generalization. Evaluate the reasonableness of intermediate results. Essential questions How can the method and sample techniques chosen for collecting data impact your ability to answer the question or hypothesis being considered? How can collecting and analyzing data help you make decisions or predictions? Why is it important to choose the most appropriate representation of the data collected for a problem situation? What factors should you consider to determine the best representation for your data? Why is it important to consider the limitations of the data collected? How can appropriate data help you make predictions about realworld situations? Cumberland, Lincoln, and Woonsocket Public Schools C27
4 Algebra 1, Quarter 3, Unit 3.2 Final, July 2011 Collecting and Analyzing Data (7 days) Written Curriculum Grade Span Expectations M(DSP) 10 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 affect interpretations; and when appropriate makes predictions, asks new questions, or makes connections to realworld situations. (Local) Clarifying the Standards Prior Learning In kindergarten, students used models and tally charts, and they analyzed trends. In grade 2, students collected, organized, and appropriately displayed data in order to analyze data and draw conclusions. In the upperelementary grades, students decided the most effective methods (survey, observation, experimentation) to collect data (numerical or categorical) necessary to answer a teachergenerated question. In middle school, students were introduced to limitations of data and its interpretation. Current Learning Students are introduced to sampling techniques including, but not limited to, random sample and stratified random sample. Students choose the appropriate method of collecting data and appropriate sampling techniques for a given question or hypothesis. Students choose the best representation of data for a given situation, and they analyze the limitations of the data that could affect interpretations. When appropriate, students make predictions, ask new questions, or make connections to realworld situations. Future Learning In subsequent high school and college courses, as well as careers, students will collect data to answer a question or hypothesis, develop an appropriate hypothesis, choose appropriate sampling techniques, and analyze and make predictions. Additional Research Findings A Research Companion to Principles and Standards for School Mathematics, states, Through multiple experiences with a variety of data sets, students begin to develop the tools and concepts they need to use data themselves and to interpret the data they will encounter through life (pp ). Notes About Resources and Materials In unit 1.1, the focus of the unit was to determine the best representations for given sets of data. In this unit, we are actually collecting, creating, analyzing, displaying the data and drawing conclusions. Cumberland, Lincoln, and Woonsocket Public Schools C28
5 Algebra 1, Quarter 3, Unit 3.3 Polynomial Expressions Overview Number of instructional days: 13 (2 assessments) (1 day = 45 minutes) Content to be learned Simplify polynomial expressions through o Adding (1 day) o Subtracting (1 day) Evaluate polynomial expressions, given values of the variable. (1 day) Multiplying polynomials o Monomial times binomial (1 day) o Binomial times binomial (1 day) o Binomial times trinomial (1 day) o Squaring a binomial (1 day) Finding and Factoring out GCF. (1 day) Factor trinomials where the leading coefficient is 1. (1 day) Translate problem situations into algebraic expressions. (2 days) Essential questions How can an area model be used to represent the multiplication of two polynomials? Why is it important to translate a problem situation into an algebraic expression? How can the properties of real numbers be used to evaluate a polynomial expression? Mathematical practices to be integrated Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Calculate and compute accurately (including technology). Look for and make use of structure. Recognize the importance of a mathmatical procedure and extend its use to other problems. Evaluate work and make modifications or try a new approach, if necessary. How are adding and subtracting polynomial expressions similar, and how is multiplication different from adding and subtracting polynomial expressions? How do you know when a polynomial expression is in its simplest form? How is the factoring of a polynomial expression related to the multiplication of two polynomials? Cumberland, Lincoln, and Woonsocket Public Schools C29
6 Algebra 1, Quarter 3, Unit 3.3 Final, July 2011 Polynomial Expressions (13 days) Written Curriculum Grade Span Expectations M(F&A) 10 3 Demonstrates conceptual understanding of algebraic expressions by solving problems involving algebraic expressions, by simplifying expressions (e.g., simplifying polynomial or rational expressions, or expressions involving integer exponents, square roots, or absolute values), by evaluating expressions, or by translating problem situations into algebraic expressions. (State) Clarifying the Standards Prior Learning In grade 4, students began using symbols to represent unknown quantities to write simple linear algebraic expressions. In grade 5, students began evaluating linear algebraic expressions using whole numbers. In grade 6, students began working with four operations to write and evaluate linear algebraic expressions with more than one variable. By grade 8, students had written and evaluated algebraic expressions with rational numbers and exponents. Current Learning Students solve problems involving algebraic expressions, simplify polynomial expressions, evaluate expressions, and translate problem situations into algebraic expressions. Future Learning In grades 11, 12, and advanced math, students will manipulate, evaluate, and simplify algebraic and numerical expressions; add, subtract, multiply, and divide polynomials; add, subtract, multiply, and divide rational expressions; simplify complex fractions; factor quadratic and higherdegree polynomials, including difference of squares; and apply properties of logarithms. Students will solve polynomial equations graphically and algebraically. Additional Research Findings Benchmarks for Science Literacy states, Students should know that there is no one right way to solve a math problem; different methods have different advantages and disadvantages (p. 28). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C30
7 Algebra 1, Quarter 3, Unit 3.4 Rational Algebraic Expressions Overview Number of instructional days: 12 (2 days assessment) (1 day = 45 minutes) Content to be learned Simplify rational expressions. (1 day) Multiply and divide rational expressions. (1 day) Add and subtract rational expressions with like denominators. (1 day) Add and subtract rational expressions with unlike denominators. (2 days) Evaluate rational expressions, given values of the variable. (1 day) Perform operations with complex rational expressions. (2 days) Translate problem situations into rational algebraic expressions. (2 day) Mathematical practices to be integrated Look for and express regularity in repeated reasoning. Look for general methods, patterns, repeated calculations, and shortcuts. Look for patterns to find generalizations. Reason abstractly and quantitatively. Rewrite the problem in simpler terms. Use abstract reasoning; check answers to ensure they are quantitatively correct. Essential questions How do the operations of rational numbers apply to rational expressions? How do operations with rational expressions compare to operations of polynomials? Why does one need to understand and be able to simplify a rational expression? Why is it important to translate a problem situation into an algebraic expression? Cumberland, Lincoln, and Woonsocket Public Schools C45
8 Algebra 1, Quarter 3, Unit Rational Algebraic Expressions (11 days) Written Curriculum Grade Span Expectations M(F&A) 10 3 Demonstrates conceptual understanding of algebraic expressions by solving problems involving algebraic expressions, by simplifying expressions (e.g., simplifying polynomial or rational expressions, or expressions involving integer exponents, square roots, or absolute values), by evaluating expressions, or by translating problem situations into algebraic expressions. (State) Clarifying the Standards Prior Learning In grade 4, students used symbols to represent unknown quantities to write simple linear algebraic expressions. In grade 5, students evaluated linear algebraic expressions by using whole numbers. In grade 6, students worked with four operations to write and evaluate linear algebraic expressions with more than one variable. By grade 8, students wrote and evaluated linear algebraic expressions with rational numbers and exponents. Current Learning Students solve problems involving rational expressions, simplify rational expressions, evaluate rational expressions, and translate problem situations into algebraic expressions. Future Learning Students will manipulate, evaluate, and simplify algebraic and numerical expressions; add, subtract, multiply, and divide polynomials; add, subtract, multiply, and divide rational expressions; simplify complex fractions; factor quadratic and higherdegree polynomials, including difference of squares; and apply properties of logarithms. Students will solve polynomial equations graphically and algebraically. Additional Research Findings Benchmarks for Science Literacy states: Students should know that there is no one right way to solve a math problem, different methods have different advantages and disadvantages (p. 28). Notes About Resources and Materials C46 Cumberland, Lincoln, and Woonsocket Public Schools
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