Building and Interpreting Linear, Quadratic, and Exponential Functions and Describing the Relationships among the Functions (12 days)
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1 1 of 5 10/31/2013 2:20 PM Math Model Unit 3.3 Algebra 2 East Bay Overview Algebra 2, Quarter 3, Unit 3.3 Building and Interpreting Linear, Quadratic, and Exponential Functions and Describing the Relationships among the Functions (12 days) Content to be Learned Create and solve equations and inequalities in one variable of simple rational and exponential functions with rational or real exponents. Use the properties of exponents to interpret expressions for exponential functions (classify growth or decay). Create a rational or exponential function using real numbers from a relationship between two quantities by a recursive process, or by steps for calculation from a context. Combine standard function types using arithmetic operations to build a function modeling a real-world situation. Mathematical Practices to be Integrated Model with mathematics. Apply knowledge of exponential functions to solve growth and decay problems. Build and combine functions to model real-world situations. Look for and make use of structure. Examine the structure of functions in order to compose, or decompose them to build new functions. Essential Questions What makes a rational function undefined? If the relationship represents an exponential function, how do you determine whether the function represents growth or decay? What characteristics of the data in a given problem indicate whether an exponential equation is to be used or not? Written Curriculum
2 2 of 5 10/31/2013 2:20 PM Common Core State Standards for Mathematical Content Algebra Creating Equations A-CED Create equations that describe numbers or relationships [Equations using all available types of expressions, including simple root functions] 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. Interpreting Functions F-IF Analyze functions using different representations [Focus on using key features to guide selection of appropriate type of model function] F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay. Building Functions F-BF Build a function that models a relationship between two quantities [For F.BF.1, 2, linear, exponential, and quadratic] 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. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Common Core State Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose 7 Look for and make use of structure.
3 3 of 5 10/31/2013 2:20 PM 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 , 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 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. Clarifying the Standards
4 ath Model Unit 3.3 Algebra 2 East Bay of 5 10/31/2013 2:20 PM Prior Learning In grade 6, students began to investigate and write equations to represent relationships, and they encountered exponents for the first time. In grade 8, students learned how to apply the properties of integer exponents to generate equivalent numerical expressions. They used square-root and cube-root symbols to represent solutions to equations, and they also evaluated square and cube roots of small perfect squares and cubes. In algebra 1, students interpreted parts of exponential expressions and used properties of exponents to rewrite exponential expressions in equivalent forms. They also created and solved equations in one or more variables to represent exponential relationships. Finally, students interpreted geometric sequences as exponential functions, rearranged exponential formulas to highlight a quantitative interest, and used function notation to evaluate and reveal different properties of the exponential function (including exponential growth and decay). Current Learning In algebra 2, students create and solve equations and inequalities in one variable of simple rational and exponential functions with rational or real exponents. They use the properties of exponents to interpret expressions for exponential functions. They classify growth or decay problems. Students create a linear, quadratic, or exponential function using real numbers from a relationship between two quantities by a recursive process, or by steps for calculation from a context. Students combine standard function types using arithmetic operations to build a function modeling a real-world situation. Modeling with functions is a critical area for algebra 2. Students identify appropriate types of functions for given data, adjust parameters as needed, and compare the quality of models. In this unit, students synthesize and generalize what they have learned about a variety of function families. Future Learning Students will continue mastering concepts related to exponential equations and functions in precalculus and advanced placement calculus. In calculus, students will need to apply rules for derivatives and integrals of exponential functions. A variety of careers will require understanding of the concepts related to exponential equations and/or functions. For example, scientists use exponential functions to model populations, to carbon date artifacts, and to help coroners determine time of death. Financial analysts use exponential functions to compute investments, bankers use them to compute interest, and the government uses them to compute our national debt. Chemists use exponential functions to graph the rate at which temperatures level off. If students are considering any kind of career in science, they need a solid grasp of exponents. Exponents will be used in many types of scientific experiments. Doctors may use exponents to figure out how quickly a disease is going to spread. Research scientists often use exponential functions when performing statistical calculations and interpreting data. Additional Findings Students struggle with solving real-world problems; they have difficulty making connections between the representations of the problem. How children understand the role of the process of sharing and analyzing representations in learning mathematics is an important aspect of learning. (A Research Companion to Principles and Standards for School Mathematics, p. 273) It is important that students gain experience using multiple representations to deepen their
5 5 of 5 10/31/2013 2:20 PM understanding. Students should recognize connections among different representations, thus enabling them to use these representations flexibly. (Principles and Standards for School Mathematics, p. 309) Resources (0) Standards (6) Other Info
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