Algebra 1, Quarter 1, Unit 1.1 Investigating Data Overview Number of instructional days: 7 (1 day assessment) (1 day = 45 minutes) Content to be learned Select and describe a data set using a variety of representations or data. (1 day) Organize, describe, categorize, analyze, and interpret data, using a variety of representations. (1 day) Analyzing data to formulate conclusions. (1 day) Calculate and interpret measures of central tendency. (1 day) Represent data in a variety of representations (list, graph, table, plots, etc.). (1 day) Analyze graphs to determine if their visual impressions are accurate or misleading. (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 two-way tables, graphs, and flowcharts to determine if results make sense. Interpret results in the context of a problem. Essential questions How can collecting and analyzing data help you make decisions or predictions? How can you make and interpret different representations of data? What real-world applications use statistics to analyze data? How can you determine which data display is the best for your data? Why is a particular representation of data helpful when analyzing data? What effect do outliers have on measures of central tendency? What characteristics on a graph can be altered to make the graph misleading? Cumberland, Lincoln, and Woonsocket Public Schools C-1
Investigating Data (7 days) Algebra 1, Quarter 1, Unit 1.1 Grade-Span Expectations Written Curriculum M(DSP) 10 3 Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP) 10 1. (State) M(DSP) 10 1 Interprets a given representation (e.g., box-and-whisker plots, scatter plots, bar graphs, line graphs, circle graphs, histograms, frequency charts) to make observations, to answer questions, to analyze the data to formulate or justify conclusions, critique conclusions, make predictions, or to solve problems within mathematics or across disciplines or contexts (e.g. media, workplace, social and environmental situations). (State) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 10 2.) 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) M(N&O) 10 8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems. (Local) Clarifying the Standards Prior Learning In earlier grades, students developed a basic understanding of data representations by analyzing and creating pictographs, line plots, circle graphs, bar graphs, stem and leaf plots, histograms, scatter plots and box-and-whisker plots. Students used these representations to solve problems and make predictions. Students have previously analyzed patterns, trends, or distributions in data in a variety of contexts. They have used measures of central tendency to analyze situations and solve problems. Current Learning This unit is at the reinforcement level. Students determine and create the best display for a given set of data, formulate and justify conclusions, and make predictions with teacher-supplied or student-collected data. They use a variety of representations to solve problems within mathematics and/or across disciplines. Analyzing data is studied again in unit of study 3.2. At this time, it is recommended that teachers prepare for future units by brainstorming ideas for surveys, observations, research, or experiments to be conducted later in the year. Future Learning These GSEs are revisited in grades 11, 12, and advanced math, where the students will learn about standard deviation and variance. Cumberland, Lincoln, and Woonsocket Public Schools C-2
Investigating Data (7 days) Algebra 1, Quarter 1, Unit 1.1 Additional Research Findings According to Benchmarks for Science Literacy, by the end of grade 12 students should understand that the way data are displayed can make a big difference in how they are interpreted (p. 230). A Research Companion to the Principles and Standards for School Mathematics states that students need to learn how to turn observations into data and probe the data to answer questions. Understanding data representation and analysis involves many complex issues, from sorting through what different numbers on a graph mean, to choosing appropriate measures, to identifying relationships between variables (p. 212). Science for All Americans emphasizes that it is important for students to understand the different sampling techniques and how to accurately summarize data (pp. 130 139). Cumberland, Lincoln, and Woonsocket Public Schools C-3
Investigating Data (7 days) Algebra 1, Quarter 1, Unit 1.1 Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-4
Algebra I Quarter 1, Unit 1.2 Expressions (Simplifying and Magnitude) Overview Number of instructional days: 10 (1 day assessment) (1 day = 45 minutes) Content to be learned Order and compare rational numbers, square roots, integers, numbers with rational bases and integral exponents, the absolute value of numbers, and numbers represented in scientific notation. (1 day) Solve problems involving rational numbers, common irrational numbers, numbers with rational bases and integer exponents, square roots, absolute value, integers, and numbers represented in scientific notation. (2 days) Use the rules for exponents to solve problems. (2 days) Add, subtract, multiply, and divide numbers written in scientific notation form. Use scientific notation in problem situations. (2 days) Express numerical and algebraic expressions in written form and vice versa. (2 days) Mathematical practices to be integrated Model with mathematics. Identify the important quantities and their relationships using mathematical models. Look for and make use of structure. Apply prior learning to new situations. Identify patterns and structures. Essential questions How can the properties of numbers be used when working with algebraic expressions? What are the similarities and differences between rational and irrational numbers? How can mathematical phrases and realworld relationships be represented using symbols and operations? How can absolute value be applied to realworld situations? Cumberland, Lincoln, and Woonsocket Public Schools C-5
Algebra 1, Quarter 1, Unit 1.2 Expressions (Simplifying and Magnitude) (10 days) Written Curriculum Grade-Span Expectations M(N&O) 10 2 Demonstrates understanding of the relative magnitude of real numbers by solving problems involving ordering or comparing rational numbers, common irrational numbers (e.g., 2 ), rational bases with integer exponents, square roots, absolute values, integers, or numbers represented in scientific notation using number lines or equality and inequality symbols. (State) M(N&O) 10 6 Uses a variety of mental computation strategies to solve problems. Calculates benchmark perfect squares and related square roots (e.g., 1 2, 2 2,, 12 2, 15 2, 20 2, 25 2, 100 2, 1000 2 ). Determines any whole number percentage of a number or any multiples of 100% up to 500%. Determines benchmark fractions of a number. (Local) (IMPORTANT: The intent of this GSE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) M(N&O) 10 7 Makes appropriate estimates in a given situation by determining the level of accuracy needed and analyzing the accuracy of results. Estimates tips, discounts, and tax and estimates the value of a non-perfect square root or cube root. (Local) (IMPORTANT: The intent of this GSE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) M(N&O) 10 8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems. (Local) 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) M(N&O) 12 4 Accurately solves problems involving scientific notation or uses significant digits to assess the precision of an answer. Interprets rational exponents and their relation to radicals; computes by hand in simple cases (e.g. ), and using a calculator when appropriate. Interprets numbers given in scientific notation and carries out computations of them with and without a calculator. Solves problems involving compound interest. (Local) Clarifying the Standards Prior Learning Beginning in grade 3, students ordered and compared whole numbers, fractions, decimals, and benchmark percents with number formations (fractions to fractions, decimals to decimals, and percents to percents). Starting in sixth grade, students compared numbers with exponents, rational numbers, and absolute value. They also compared numbers in scientific notation, irrational numbers, and square roots of numbers. Also, beginning in the third grade, students mentally added and subtracted whole numbers, working with two- and three-digit whole numbers (multiples of 10). Students mentally calculated change from $1.00, $5.00, and $10.00. By grade 6, students were calculating change (denominations up to $100.00) and using Cumberland, Lincoln, and Woonsocket Public Schools C-6
Algebra 1, Quarter 1, Unit 1.2 Expressions (Simplifying and Magnitude) (10 days) benchmark percents (1%, 10%, 25%, 50%, and 75%). In seventh grade, students mentally calculated benchmark perfect squares and related square roots. Students began estimating in kindergarten; however, they began to make estimates and use various techniques for estimating numbers in grade 3. In fifth grade, students determined accuracy and analyzed the effects of estimation. In later grades, students used estimation to determine tips, discounts, and tax, as well as estimating values of non-perfect square roots. In grade 4, students began using symbols to represent unknown quantities to write simple linear algebraic expressions. In fifth grade, students began evaluating linear algebraic expressions by using whole numbers. In sixth grade, students worked with four operations, writing and evaluating linear algebraic expressions with more than one variable. By eighth grade, students wrote and evaluated linear algebraic expressions with rational numbers and exponents. Current Learning Students solve problems involving ordering or comparing rational numbers, common irrational numbers (e.g. 2 ), rational bases with integer exponents, square roots, absolute values, integers, and numbers represented in scientific notation, using number lines or equality and inequality symbols. Students solve problems involving algebraic expressions by simplifying expressions (e.g., expressions involving integer exponents, square roots, or absolute values), by evaluating expressions, or by translating problem situations into algebraic expressions. Future Learning In future units and courses, students will solve problems that involve ordering or comparing any subset of real numbers. 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 higher degree polynomials, including difference of squares; and apply properties of logarithms. Additional Research Findings According to Principles and Standards for School Mathematics, it is important to understand the meanings of operations and how they are related to one another; understand numbers, ways of representing numbers, relationships among numbers and number systems; and to compute fluently and make reasonable estimates (pp. 148, 214, 222, 290). Additionally, Benchmarks for Science Literacy states that, by eighth grade, students should know that there is no one right way to solve a math problem and that different methods have different advantages and disadvantages (p. 28). Cumberland, Lincoln, and Woonsocket Public Schools C-7
Algebra 1, Quarter 1, Unit 1.2 Expressions (Simplifying and Magnitude) (10 days) Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-8
Algebra I Quarter 1, Unit 1.3 Solving Linear Equations Using Multiple Representations Overview Number of instructional days: 13 (1 day assessment) (1 day = 45 minutes) Content to be learned Create problem situations from equations. (3 days) Translate problem situations into equations. (3 days) Solve linear equations symbolically (algebraically and graphically). (3 days) Calculate the solution set of an algebraic equation or graph and state the meaning of the solution. (3 days) Mathematical practices to be integrated Make sense of problems and persevere in solving them. Work among different representations. Reason abstractly and quantitatively. Represent mathematically what is read and do something with it. Move back and forth between the true meaning of symbols and the abstract manipulation of symbols. Model with mathematics. Relate what has been learned in mathematics to everyday life. Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Essential questions Can equations that appear to be different be equivalent? What are the ways that a problem scenario can be represented? How are multiple representations useful for solving linear equations? Cumberland, Lincoln, and Woonsocket Public Schools C-9
Algebra 1, Quarter 1, Unit 1.3 Solving Linear Equations Using Multiple Representations (13 days) Written Curriculum Grade-Span Expectations M (F&A) 10 4 Demonstrates conceptual understanding of equality by solving problems involving algebraic reasoning about equality; by translating problem situations into equations; by solving linear equations (symbolically and graphically) and expressing the solution set symbolically or graphically, or provides the meaning of the graphical interpretations of solution(s) in problem-solving situations; or by solving problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. (State) Clarifying the Standards Prior Learning Students worked with equivalence between two expressions in third grade. Thereafter, they have worked with one-step linear equations and multi-step linear equations. In grades 7 and 8, students translated problem-solving situations into equations. Students have also manipulated equations, solving for different variables (literal equations). Current Learning Students solve real-world problems symbolically and graphically and interpret the meaning of the results. Future Learning In later units and courses, students will solve equations and systems of equations or inequalities and interpret the solutions algebraically and graphically by factoring, completing the square, using the quadratic formula, and graphing quadratic functions to solve quadratic equations; by solving and interpreting solutions of equations involving polynomial, rational, and radical expressions; and by analyzing the effect of simplifying radical or rational expressions on the solution set of equations involving such expressions. Additional Research Findings According to Principles and Standards for School Mathematics, students at this level should represent and analyze mathematical situation and structure using algebraic symbols, and they should use mathematical models to represent and understand quantitative relationships (pp. 394 395). For further background, see Stasis and Change: Integrating Patterns, Functions, and Algebra Through the K-12 Curriculum in A Research Companion to Principles and Standards for School Mathematics (pp. 136 149). Also, reference the following sections in Science for All Americans: Mathematical Inquiry (p. 19), Symbolic Relationships (p. 132). Cumberland, Lincoln, and Woonsocket Public Schools C-10
Algebra 1, Quarter 1, Unit 1.4 Proportional Relationships, Percents, Ratios, and Rates Overview Number of instructional days: 6 (1 day assessment) (1 day = 45 minutes) Content to be learned Solve multi-step problems involving proportional relationships, percents, ratios, and rates in a variety of contexts. (2 days) Makes appropriate estimates in a given situation such as tips, discounts, and tax. (3 days) Mathematical practices to be integrated Reason abstractly and quantitatively. Rewrite the problem in simpler terms. Construct viable arguments and critique the reasoning of others. Think logically, determine if there are errors, and explain reasoning. Communicate conclusions. Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Use units of measure correctly. Essential questions What kinds of relationships can proportions, percents, ratios, and rates represent? In what real-life situations would an estimate be used over an exact computation? How can you determine if a result is reasonable by using estimation? Cumberland, Lincoln, and Woonsocket Public Schools C-11
Algebra 1, Quarter 1, Unit 1.4 Proportional Relationships, Percents, Ratios, and Rates (6 days) Written Curriculum Grade-Span Expectations M(N&O) 10 6 Uses a variety of mental computation strategies to solve problems. Calculates benchmark perfect squares and related square roots (e.g., 1 2, 2 2,, 12 2, 15 2, 20 2, 25 2, 100 2, 1000 2 ). Determines any whole number percentage of a number or any multiples of 100% up to 500%. Determines benchmark fractions of a number. (Local) (IMPORTANT: The intent of this GSE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) M(N&O) 10 7 Makes appropriate estimates in a given situation by determining the level of accuracy needed and analyzing the accuracy of results. Estimates tips, discounts, and tax and estimates the value of a non-perfect square root or cube root. (Local) (IMPORTANT: The intent of this GSE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) M(N&O) 10 4 Accurately solves problems that involve but are not limited to proportional relationships, percents, ratios, and rates. (The problems might be drawn from contexts outside of and within mathematics including those that cut across content strands or disciplines.) (State) Clarifying the Standards Prior Learning In grade 3, students mentally added and subtracted whole numbers working with two- and three-digit whole numbers (multiples of 10). Student also mentally calculated change from $1.00, $5.00, and $10.00. By sixth grade, students calculated change back (denominations up to $100.00) and used benchmark percents (1%, 10%, 25%, 50%, and 75%). In seventh grade, students mentally calculated benchmark perfect squares and related square roots. Students began estimating in kindergarten; however they began to make estimates and use various techniques for estimating numbers in the third grade. In the fifth grade, students determined accuracy and analyzed the effects of estimation. In later grades, students used estimation to determine tips, discounts, and tax, as well as estimated values of non-perfect square roots. Starting in third grade, students used arithmetic to solve problems, progressing up to grade 8, where they solved problems using proportional reasoning, multiplication, and division of integers. Students also calculated squares, cubes, square roots, and cube roots. Current Learning Students solve situational real-world problems that include proportional relationships, percents, ratios, and rates across mathematics and other disciplines. Cumberland, Lincoln, and Woonsocket Public Schools C-12
Algebra 1, Quarter 1, Unit 1.4 Proportional Relationships, Percents, Ratios, and Rates (6 days) Future Learning Students will accurately solve problems involving scientific notation and use significant digits to assess the precision of an answer. Students will interpret rational exponents and their relation to radicals. Additional Research Findings According to Principles and Standards for School Mathematics, it is important to understand the meanings of operations and how they related to one another; to understand numbers, ways of representing numbers, the relationships among numbers and number systems; and to compute fluently and make reasonable estimates (pp. 148, 214, 222, 290). Benchmarks for Science Literacy states that, by the end of eighth grade, students should be able to find what percentage one number is of another number and be able to figure any percentage of any number (p. 291). See Building Fluency with Computational Method (p. 71) in A Research Companion to Principles and Standards for School Mathematics. Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-13
Algebra 1, Quarter 1, Unit 1.4 Proportional Relationships, Percents, Ratios, and Rates (6 days) Cumberland, Lincoln, and Woonsocket Public Schools C-14
Algebra 1, Quarter 1, Unit 1.5 Patterns Overview Number of instructional days: 4 (1 day assessment) (1 day = 45 minutes) Content to be learned Demonstrate conceptual understanding of linear patterns through identifying (in words), extending (n th term), and generalizing (using an equation) those patterns. (1 day) Use models, tables, graphs, and sequences to display a linear pattern. (1 day) Develop a pattern based on models, tables, graphs, and sequences. (1 day) Mathematical practices to be integrated Make sense of problems and persevere in solving them. Work among different representations. Reason abstractly and quantitatively. Move back and forth between the true meaning of symbols and the abstract manipulation of symbols. Use symbolic representation effectively. Essential questions How are patterns used in the real world? How can patterns be used to make generalizations? What does the equation tell you about the pattern that it represents? Cumberland, Lincoln, and Woonsocket Public Schools C-15
Algebra 1, Quarter 1, Unit 1.5 Patterns (4 days) Written Curriculum Grade-Span Expectations M(F&A) 10 1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems. (State) Clarifying the Standards Prior Learning Students have identified patterns since kindergarten. They have found missing elements in a pattern and written rules using words or symbols. They have generalized linear and nonlinear relationships using words and symbols. Current Learning Students solve problems through identifying, extending, and generalizing patterns. Future Learning In grades 11 and 12, students will identify arithmetic and geometric sequences and generalize these patterns in order to find a specific term. In advanced math, they will work with infinite sequences and make connections to linear and exponential functions. Additional Research Findings According to Science For All Americans, mathematics is the science of patterns and relationships (Chapter 2). Benchmarks for Science Literacy asserts that getting a grasp on the nature of today s mathematics, which is still engaged in eliciting new patterns and relationships, is likely to be a challenge for all students (Chapter 2a). Principles and Standards for School Mathematics states that high school students algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with more sophistication than in the middle grades (pp. 296 300). Cumberland, Lincoln, and Woonsocket Public Schools C-16