Math 2 Unit 3 Comparing Functions Modeling & …



Approximate Time Frame: 5 – 6 Weeks

Connections to Previous Learning:

In Math 1, students studied linear and exponential functions and compared them in a modeling context. Students looked at multiple representations of linear and exponential functions. Students were exposed to square root, cube root, piecewise, absolute value and step functions and represented these on graphs. Students considered the parent functions and the general shapes of all of these functions. Students manipulated expressions and equations to reveal new information about these functions.

Focus of this Unit:

This unit focuses on comparing all the functions in Math 1 and Math 2. Additionally, students engage in working with quadratic functions. Students look at the parent function of a quadratic and the general shape of any quadratic function. Students will use quadratic functions to create models of real-life situations and compare these to previously learned functions. Students will use technology to fit linear, quadratic, and exponential models to data and analyze the fit of the model using residuals. Students will also explore transformations in this unit. In subsequent units, students will use transformations to consider congruence and similarity.

Connections to Subsequent Learning:

In Math 3, students will extend their study of functions to logarithmic, exponential, polynomial, rational and trigonometric functions. Students will continue to study the general shape of the functions, the parent functions, how the function can be transformed and how equivalent equations can reveal new information about the function. After studying this collection of functions, students will again be asked to choose a model that best fits a real-world situation.

From the Grade 8, High School Functions Progression Document p. 3:

In modeling situations, knowledge of the context and statistics are sometimes used together to find algebraic expressions that best fit an observed relationship between quantities. Then the algebraic expressions can be used to interpolate (i.e., approximate or predict function values between and among the collected data values) and to extrapolate (i.e., to approximate or predict function values beyond the collected data values). One must always ask whether such approximations are reasonable in the context.

In school mathematics, functional relationships are often given by algebraic expressions. For example, [pic] gives the nth square number. But in many modeling situations, such as the temperature at Boston’s Logan Airport as a function of time, algebraic expressions are generally not suitable.

From the Grade 8, High School Functions Progression Document p. 7:

The high school standards on functions are organized into four groups:

Interpreting Functions (F-IF); Building Functions (F-BF); Linear, Quadratic and Exponential Models (F LE); and Trigonometric Functions (F-TF). The organization of the first two groups under mathematical practices rather than types of function is an important aspect of the Standards: students should develop ways of thinking that are general and allow them to approach any type of function, work with it, and understand how it behaves, rather than see each function as a completely different animal in the bestiary. For example, they should see linear and exponential functions as arising out of structurally similar growth principles; they should see quadratic, polynomial, and rational functions as belonging to the same system (helped along by the unified study in the Algebra category of Arithmetic with Polynomials and Rational Expressions).

|Desired Outcomes |

|Standard(s): |

|Interpret functions that arise in applications in terms of the context. |

|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. |

|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. |

|Graph linear and quadratic functions and show intercepts, maxima, and minima. |

|Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. |

|F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. |

|F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions). For example, given a graph of one quadratic function and |

|an algebraic expression for another, say which has the larger maximum. |

|Summarize, represent, and interpret data on two categorical and quantitative variables. |

|S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. |

|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. |

|Informally assess the fit of a function by plotting and analyzing residuals. |

|Explain volume formulas and use them to solve problems. |

|G.GMD.1 Give an informal argument for the formula for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments. Calvalieri’s principal and informal limit |

|arguments. |

|G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. |

|Write expressions in equivalent forms to solve problems. |

|A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. |

|Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as |

|(1.151/12) 12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. |

|Build new functions from existing functions. |

|F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and|

|illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. |

|Build a function that models a relationship between two quantities. |

|F.BF.1 Write a function that describes a relationship between two quantities. |

|Write a function that describes a relationship between two quantities. |

|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: |

|explicit vocabulary instruction with regard to components of functions, graphs, geometric figures, expressions and equations. |

|guided discussions about the relationship between various representations of the same concepts and contexts. |

|Understandings: Students will understand… |

|Quadratic functions have key features that can be represented on a graph and can be interpreted to provide information to describe relationships of two quantities. These graphs can be compared to linear and |

|exponential functions to model a situation. |

|The meaning of average rate of change of a quadratic model is interpreted based upon the context. |

|Quadratic expressions have equivalent forms that can reveal new information to aid in solving problems. |

|Data can be represented on and interpreted from a scatter plot. |

|Equations are affected by transformations of a graph and vice versa. |

|The formulas for circumference, area, and volume of a circle can be seen as linear, quadratic, and other functions of the radius. |

|Essential Questions: |

|What do the key features of a graph represent in a modeling situation? |

|What new information will be revealed if this equation is written in a different but equivalent form? |

|How do you create an appropriate function to model data or situations given within context? |

|When changes are made to an equation, what changes are made to the graph? |

|How do you create or interpret a scatter plot from data? |

|How can familiar formulas for circles be viewed as linear and quadratic models? |

|Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) |

|*1. Make sense of problems and persevere in solving them. Students will consider a real-life situation, determine if it is best modeled by a linear, exponential or quadratic function, and answer questions and solve |

|problems based on this model. |

|2. Reason abstractly and quantitatively. |

|3. Construct viable arguments and critique the reasoning of others. |

|*4. Model with mathematics. Students will use graphs, tables, and equations to model linear, exponential and quadratic equations. Students will interpret the appropriateness of linear, exponential or quadratic |

|models for a given data set or situation. |

|*5. Use appropriate tools strategically. Students will use technology to fit linear, quadratic, and exponential models to data. |

|6. Attend to precision. |

|*7. Look for and make use of structure. Students will see how the changes made in the structure of a function have similar graphical effects as those made on a different function. The students will see many |

|functions as having a parent function that can be transformed by slightly changing the equation. |

|8. Look for and express regularity in repeated reasoning. |

|Prerequisite Skills/Concepts: |Advanced Skills/Concepts: |

|Students should already be able to: |Some students may be ready to: |

|Solve one-variable equations and recognize equivalent forms. |Recognize that volume is a linear function of area. |

|Model real-world one-variable equations and two-variable equations limited to linear and |Analyze the sum of the squares of the residuals. |

|exponential. |Explain the relationship between the sum of the squares of the residuals and the correlation coefficient. |

|Define a function and distinguish between coefficients, factors and terms. | |

|Recognize how functions can be represented on a graph and in a table and how scale and labels can| |

|modify the appearance of the representation. | |

|Represent real-world situations with a linear or exponential model and make decisions about the | |

|appropriateness of each. | |

|Represent data on a scatter plot both by hand and with technology. | |

|Knowledge: Students will know… |Skills: Students will be able to … |

|The parent graphs for linear, quadratic, and exponential functions. |Write a quadratic equation and/or function to model a real-life situation. |

| |Use a model of a quadratic function to interpret information about a real-life situation. |

| |Use and compare multiple representations of quadratic functions including tables, graphs, equations and real-life |

| |situations. |

| |Distinguish between linear, exponential and quadratic functions from multiple representations. |

| |Rewrite quadratic and exponential functions in different forms to reveal new information. |

| |Estimate, calculate and interpret average rate of change over a specified interval. |

| |Compare two functions represented in different ways (such as an equation compared to a table or graph). |

| |Fit a linear, quadratic, or exponential model to data. |

| |Assess the fit of a model to data by analyzing residuals and residual plots. |

| |Describe an informal proof for and use geometric formulas. |

| |Transform graphs based on changes in equations and write equations based on a transformed parent graph. |

|Academic Vocabulary: |

| | |

|Critical Terms: |Supplemental Terms: |

|Quadratic |Tables |

|Parabola |Graphs |

|Translation |Equations |

|Transformation |Correlation Coefficient |

|Residual | |

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