Algebra 1, Quarter 3, Unit 3.1 Representing and Interpreting Data and Using Statistics to Solve Problems Overview Number of instructional days: 7 (1 day = 45 minutes) Content to be learned Represent data using dotplots, histograms, and boxplots (one variable). Construct and analyze a function (line of fit) from a scatterplot (two variables); use the function to solve problems in the context of the data. Use statistics to compare the data distribution of two or more data sets (mean, median, interquartile range, and standard deviation). Recognize possible associations and trends in data. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (joint, marginal, and conditional relative frequencies). Informally assess the fit of a function by plotting and analyzing residuals. Essential questions How can statistics be used to solve real-world problems? What benefits are gained by determining and analyzing trends in scatterplots? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Understand and explain the meaning of a problem. 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. Use appropriate tools strategically. Consider available tools such as graphing calculators. Make sound decisions about when these tools might be helpful. Use technological tools to explore and deepen understanding. How do extreme data points affect the difference in shape, center, and spread in the data set? 33
Algebra 1, Quarter 3, Unit 3.1 Representing and Interpreting Data and Using Statistics to Solve Problems (7 days) Written Curriculum Common Core State Standards for Mathematical Content Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable S-ID.1 S-ID.2 S-ID.3 Represent data with plots on the real number line (dot plots, histograms, and box plots). Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Summarize, represent, and interpret data on two categorical and quantitative variables [Linear focus, discuss general principle] S-ID.5 S-ID.6 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. 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. 34
Algebra 1, Quarter 3, Unit 3.1 Representing and Interpreting Data and Using Statistics to Solve Problems (7 days) 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 Students learned the characteristics of data distribution (e.g., they are able to identify the center, spread, and overall shape/trend of the data). In grade 6, they learned the definitions of mean, median, and range. In grade 8, students constructed and interpreted scatterplots in two variables and investigated patterns of association between two quantities. Current Learning Students represent data using dotplots, histograms, and boxplots. They use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more sets. Students compare and contrast two or more data sets and recognize possible associations and trends in data. They construct and analyze a function from a scatterplot, and they use the function to solve problems in the context of the data. Future Learning In the next unit, students will reinforce their ability to interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. In addition, they will compute (using technology) and interpret the correlation coefficient of a linear fit and develop the ability to distinguish whether the relationship between the variables is correlation or causation. Additional Findings [Need to complete.] 35
Algebra 1, Quarter 3, Unit 3.1 Representing and Interpreting Data and Using Statistics to Solve Problems (7 days) 36
Algebra 1, Quarter 3, Unit 3.2 Interpreting Slope, Computing and Interpreting Correlation Coefficients, and Interpreting and Using Structure of Expressions Overview Number of instructional days: 14 (1 day = 45 minutes) Content to be learned Analyze the slope and y-intercept of a linear model in a problem context. Differentiate between correlation and causation. Compute and interpret the correlation coefficient of the line of best fit. Essential questions What does the slope and y-intercept of a linear model reveal about the data? What does the correlation coefficient mean for a set of data? Mathematical practices to be integrated Attend to precision. Calculate accurately and efficiently. Use clear definitions and state the meaning of symbols. Look for and express regularity in repeated reasoning. Notice if calculations are repeated. Maintain oversight of the process while attending to details. Evaluate the reasonableness of each step in a solution. What is a real-world example that exemplifies causation rather than correlation? 37
Algebra 1, Quarter 3, Unit 3.2 Interpreting Slope, Computing and Interpreting Correlation Coefficients, and Interpreting and Using Structure of Expressions (14 days) Written Curriculum Common Core State Standards for Mathematical Content Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable Interpret linear models S-ID.7 S-ID.8 S-ID.9 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Compute (using technology) and interpret the correlation coefficient of a linear fit. Distinguish between correlation and causation. Seeing Structure in Expressions A-SSE Interpret the structure of expressions [Linear, exponential, quadratic] A-SSE.1 Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Common Core Standards for Mathematical Practice 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. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 38
Algebra 1, Quarter 3, Unit 3.2 Interpreting Slope, Computing and Interpreting Correlation Coefficients, and Interpreting and Using Structure of Expressions (14 days) Clarifying the Standards Prior Learning In grade 8, students interpreted the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. They also graphed proportional relationships, interpreting the unit rate as the slope of the graph. Earlier in Algebra 1, students calculated and interpreted the average rate of change of a function. Current Learning Students reinforce their ability to interpret the slope and the y-intercept of a linear equation in the context of a problem. Students compute and interpret the correlation coefficient of a line, and they also learn to distinguish between correlation and causation. Future Learning In Algebra 2, students will make inferences and justify conclusions from sample surveys, experiments, and observational studies. They will also use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Additional Findings correlation coefficient: a single summary number that gives you a good idea about how closely one variable is related to another. Correlation is a statistical technique that can show whether and how strongly pairs of variables are related. For example, height and weight are related; taller people tend to be heavier than shorter people. The relationship isn't perfect. People of the same height vary in weight, and you can easily think of two people you know, where the shorter one is heavier than the taller one. Nonetheless, the average weight of people 5'5'' is less than the average weight of people 5'6'', and their average weight is less than that of people 5'7'', etc. Correlation can tell you just how much of the variation in peoples weights is related to their heights. (www.surveysystem.com/correlation.htm) There has been a lot of publicity over the purported relationship between autism and vaccinations, for example. As vaccination rates went up across the United States, so did autism. However, this correlation (which has led many to conclude that vaccination causes autism) has been widely dismissed by public health experts. The rise in autism rates is likely to do with increased awareness and diagnosis or one of many other possible factors that have changed over the past 50 years. (http://stats.org/in_depth/faq/causation_correlation.htm) More information on the correlation coefficient is available at www.biddle.com/documents/bcg_comp_chapter2.pdf 39
Algebra 1, Quarter 3, Unit 3.2 Interpreting Slope, Computing and Interpreting Correlation Coefficients, and Interpreting and Using Structure of Expressions (14 days) 40
Algebra 1, Quarter 3, Unit 3.3 Adding, Subtracting, and Multiplying Polynomial Expressions Overview Number of instructional days: 4 (1 day = 45 minutes) Content to be learned Add polynomial expressions. Subtract polynomials expressions. Multiply polynomial expressions. Mathematical practices to be integrated Look for and make use of structure. Look for a pattern or structure when performing operations. See complicated things as being composed of several smaller objects. Look for and express regularity in repeated reasoning. Look for general methods and shortcuts when performing operations. Maintain oversight of the process while attending to details (keep exponents when adding/subtracting; add exponents when multiplying). Essential questions How is multiplying polynomials related to the Distributive Property? How can you use operations of polynomials to find the perimeter or area of objects? 41
Algebra 1, Quarter 3, Unit 3.3 Adding, Subtracting, and Multiplying Polynomial Expressions (4 days) Written Curriculum Common Core State Standards for Mathematical Content Arithmetic with Polynomials and Rational Expressions A-APR Perform arithmetic operations on polynomials [Linear and quadratic] A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Common Core Standards for Mathematical Practice 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In grade 8, students interpreted the structure of expressions. They wrote expressions in equivalent forms to solve problems. Current Learning Students perform operations with polynomials. This includes addition, subtraction, and multiplication. Students understand that operations of polynomials are closed under addition, subtraction, and multiplication. 42
Algebra 1, Quarter 3, Unit 3.3 Adding, Subtracting, and Multiplying Polynomial Expressions (4 days) Future Learning In Unit 3.4, students will master simplifying rational expressions. In Algebra 2, they will interpret the structure of expressions. Students will write expressions in equivalent forms to solve problems. Additionally, they will perform arithmetic operations on polynomials beyond quadratic relationships. Additional Findings Teachers may find that using the FOIL method may be restrictive in the sense that it is only useful when multiplying two binomials. Students may find that thinking of multiplying two polynomials as distributing one monomial at a time to the polynomial. Students seem to find it easier to multiply polynomials using the box method. A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set. If an element outside the set is produced, the operation is not closed. (www.regentsprep.org/regents/math/algebra/an1/closure.htm) 43
Algebra 1, Quarter 3, Unit 3.3 Adding, Subtracting, and Multiplying Polynomial Expressions (4 days) 44
Algebra 1, Quarter 3, Unit 3.4 Factoring Expressions and Creating Equations and Inequalities to Solve Problems Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Interpret expressions that represent a quantity in terms of its context. Factor quadratic expressions to find zeroes of the function. Complete the square to find maximum or minimum value of the function. Create and graph equations and inequalities in one and two variables to represent relationships between two quantities. Isolate a squared variable within a formula. Use properties of exponents to transform expressions for exponential functions. Mathematical practices to be integrated Reason abstractly and quantitatively. Make sense of quantities and relationships within a problem. Attend to the meaning of the quantities, not just the computation. Construct viable arguments and critique the reasoning of others. Use stated assumptions, definitions, and previously established results to develop arguments. Reason inductively when factoring a quadratic expression. Check for the reasonableness of the answer. Look for and express regularity in repeated reasoning. Maintain oversight of the process while attending to details. Look for general methods and shortcuts when factoring. Essential questions What are significant points on the graph of a quadratic function? What information is found by factoring a quadratic equation? What can be learned from completing the square of a quadratic function? What are methods of graphing that can be used to graph a quadratic equation or inequality from a given equation or inequality? What are the properties of exponents? How can they be used to transform expressions for exponential functions? 45
Algebra 1, Quarter 3, Unit 3.4 Factoring Expressions and Creating Equations and Inequalities to Solve Problems (10 days) Written Curriculum Common Core State Standards for Mathematical Content Seeing Structure in Expressions A-SSE Interpret the structure of expressions [Linear, exponential, quadratic] A-SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. Write expressions in equivalent forms to solve problems [Quadratic and exponential] A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Creating Equations A-CED Create equations that describe numbers or relationships [Linear, quadratic, and exponential (integer inputs only); for A.CED.3 linear only] A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. Common Core Standards for Mathematical Practice 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 quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 46
Algebra 1, Quarter 3, Unit 3.4 Factoring Expressions and Creating Equations and Inequalities to Solve Problems (10 days) 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. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In Unit 3.2, students interpreted linear expressions. In Unit 3.3, they learned to multiply binomials. In addition, in previous units in Algebra 1, students created and graphed linear equations and inequalities in one and two variables. In Unit 2.4, they isolated a variable within a formula. Current Learning Students connect the interpretation of linear expressions to quadratic expressions. They factor quadratic expressions to reveal the zeroes of the function and learn to use completing the square to identify the maximum or minimum value of the function. Students understand that factoring a trinomial produces the product of two binomials. They master the ability to isolate a variable within a formula (e.g., solving for r in the formula A = pi r 2 ). Future Learning In Unit 4.1, students will master multiple techniques for solving quadratic equations, including factoring, finding square roots, completing the square, and the quadratic formula. In Algebra 2, students will construct and compare linear, quadratic, and exponential models to solve problems. They will also write expressions in equivalent forms to solve problems. 47
Algebra 1, Quarter 3, Unit 3.4 Factoring Expressions and Creating Equations and Inequalities to Solve Problems (10 days) Additional Findings The completing-the-square method converts the function into vertex form to reveal the maximum of minimum values of the function. 48