Unit 6: Linear Modeling

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1 Approximate Time Frame: 3-4 weeks Connections to Previous Learning: Students synthesize and apply their knowledge of linear equations and inequalities to model data. By-hand graphing skills will quickly be supplemented by technology, allowing for inspection and discussion of several data sets. Focus of the Unit: Students generate and interpret linear equations and inequalities to model data in real-world contexts. Students interpret key features, such as intercepts, slope, correlation, causation, and linear fit. Connections to Subsequent Learning: Later units will call on students' understanding of modeling and various functions, including linear, exponential, piecewise, and absolute value to compare and choose appropriate models. From the Grade 8, High School, Functions Progression Document p. 11: Build a function that models a relationship between two quantities: This cluster of standards is very closely related to the algebra standard on writing equations in tow variables. Indeed, that algebra standard might well be met by a curriculum in the same unit as this cluster. Although students will eventually study various families of functions, it is useful for them to have experiences of building functions from scratch, without the aid of a host of special recipes, by grappling with concrete context for clues. For example, in the Lake Algae task in the margin, parts (a) (c)lead students through reasoning that allows them to construct the function in part (d) directly. Students who try a more conventional approach in part (d) of fitting the general function f(t) = ab t to the situation might well get confused or replicate work already done. The Algebra Progression discussed the difference between a function and an expression. Not all functions are given by expressions, and in many situations it is natural to use a function defined recursively. Calculating mortgage payment and drug dosages are typical cases where recursively defined functions are useful (see example). 1/30/2014 7:06:03 PM Adapted from UbD framework Page 1

2 Desired Outcomes Standard(s): Interpret the structure of expressions. A.SSE.1 Interpret expressions that represent a quantity in terms of its context. a) Interpret parts of expressions, such as terms, factors, and coefficients. Create equations that describe numbers or relationships. A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear 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.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reason as in solving equations. For example, rearrange Ohm s law v = IR to highlight resistance R. Interpret functions that arise in Applications in terms of the context. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Build a function that models a relationship between two quantities. F.BF.1 Write a function that describes a relationship between two quantities. a.) Determine an explicit expression, a recursive process, or steps for calculation from a context. Represent and solve equations and inequalities graphically. A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Analyze functions using different representations. F-IF 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a) Graph linear and quadratic functions and show intercepts, maxima, and minima. b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 1/30/2014 7:06:03 PM Adapted from UbD framework Page 2

3 N-Q 2. Define appropriate quantities for the purpose of descriptive modeling. N-Q 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. *Elements of this standard that are crossed out are not addressed in this unit and will be addressed at a different time. Transfer: Students will be able to develop and apply procedures of representations of linear relations to model real-world situations. Ex. Bubba fell out of a tree into a swimming pool. If it takes 10 seconds. How far did Bubba fall? WIDA Standard: (English Language Learners) English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from: discussion regarding the connections between visual, tabular, symbolic and real-life representations of data. explicit vocabulary instruction regarding the difference between correlation and causation. Understandings: Students will understand Linear models can be created, used, and interpreted for real-life situations. Essential Questions: What real world situations can be modeled by a linear relationship? How can technology help to determine whether a linear model is appropriate in a given situation? 1/30/2014 7:06:03 PM Adapted from UbD framework Page 3

4 Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) 1. Make sense of problems and persevere in solving them. * 2. Reason abstractly and quantitatively. Students interpret the contextual meaning of slope and y-intercept of a linear model for a given data set. They make predictions based on a linear model and reason whether their predictions are likely to be realistic. * 3. Construct viable arguments and critique the reasoning of others. Students determine and make predictions based on a linear model. They consider appropriateness of the model and predictions made using it. 4. Model with mathematics. Students use linear relationships to model real-world situations. 5. Use appropriate tools strategically. * 6. Attend to precision. Students will use appropriate scales and levels of accuracy in their models and predictions, as determined by the units in the data or situation. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Prerequisite Skills/Concepts: Students should already be able to: Graph a line, given its equation or a list of ordered pairs. Write a linear equation or inequality, given two ordered pairs or slope and one ordered pair. Use linear equations/inequalities to solve problems. Translate among representations of linear functions including tables, graphs, equations and real-life situations. Use technology to graph lines, changing the viewing window as necessary to show and determine intercepts. Advanced Skills/Concepts: Some students may be ready to: Calculate residuals, squares of residuals, and the sum of least squares to determine a line of best fit and the appropriateness of a linear model. Use residuals and their graph to determine appropriateness of a linear model. Write an original optimization problem that requires a system of linear inequalities, use technology to represent the system graphically, and solve the system. 1/30/2014 7:06:03 PM Adapted from UbD framework Page 4

5 Knowledge: Students will know Correlation can be positive or negative. Skills: Students will be able to Interpret parts of expressions, such as terms, factors, and coefficients. Determine for what range of values a linear model might be appropriate (restrictions on domain/range) for a given situation. Estimate the rate of change over a specified interval. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Define appropriate quantities for the purpose of descriptive modeling. Choose a level of accuracy appropriate to limitations of measurement when reporting quantities. Graph linear and quadratic functions and show intercepts, maxima, and minima. Academic Vocabulary: Critical Terms: Linear regression Supplemental Terms: Terms Factors Coefficients Interpolate Extrapolate Significant digits Outlier 1/30/2014 7:06:03 PM Adapted from UbD framework Page 5

6 Assessment Pre-Assessments Formative Assessments Summative Assessments Self-Assessments #0 Pre-assessment #2 Hand-Span Activity #3 Station Experimentation #4 Observational Checklist #5 Discussion Questions #6 Tie the Knot #7 Summative Assessment #1 Radar 1. Applying one-variable equations 2. Station Experimentation (Model Lesson) 3. Applying graphing of inequalities Sample Lesson Sequence 1/30/2014 7:06:03 PM Adapted from UbD framework Page 6