Math 2 Unit 5: Comparing Functions and Modeling

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Approximate Time Frame: 2 - 3 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 then considered the parent functions and the general shapes of all of these functions and manipulated expressions and equations to reveal new information about these functions.

Focus of this Unit: This unit focuses on comparing and contrasting linear, exponential, and quadratic functions as models and in applications. Students will also explore the effect of transformations on each of the three types of functions. Students will be modeling volume and surface area of select three-dimensional shapes. They will also manipulate expressions in order to recognize and graph functions. Connections to Subsequent Learning: In Math 3, students will extend their study of functions to logarithmic, exponential, polynomial, and 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. Students will be connecting volume and surface area to density. In Math 3, students will also continue to use models and formulas in real life applications. 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, 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.

Math 2 Unit 5: Comparing Functions and Modeling

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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).

Math 2 Unit 5: Comparing Functions and Modeling

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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. a) Graph linear and quadratic functions and show intercepts, maxima, and minima. e) 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.

Interpret expressions for functions in terms of the situation they model.

F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

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. 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.

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.

a) Determine an explicit expression, a recursive process, or steps for calculation from a context.

Math 2 Unit 5: Comparing Functions and Modeling

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Transfer:

Students will write a function that describes a relationship between two quantities and determine which type of function (linear, quadratic, or exponential) best models a real-world situation. They will also compare properties of functions using different representations.

Ex.

The table shows the breathing rates y (in liters of air per minute) of a runner traveling at different speeds x (in miles per hour). Tell whether the data can be modeled by a linear function, an exponential function, or a quadratic function. Then write an equation for the function.

Speed of runner, x (mi/h)

2 3 4 5 6 7

Breathing rate, y (L/min)

51.4 57.1 63.3 70.3 78.0 86.6

Ex.

Students at a space camp launch rockets during camp. The path of the first group’s rocket is modeled by the equation ℎ = −16𝑡2 + 150𝑡 + 1 where t is the

time in seconds and h is the distance from the ground in feet. The path of the rocket for a second group is graphed below.

Which rocket flew higher?

Which rocket stayed in the air longer?

What is the reasonable domain and range for each model?

Math 2 Unit 5: Comparing Functions and Modeling

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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 that…

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.

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

A plot of residuals reveals additional information about the fit of a mathematical model to bivariate data.

The formulas for circumference and area of a circle and for volume of a sphere 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?

Math 2 Unit 5: Comparing Functions and Modeling

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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. 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:

Students should already be able to:

Solve one-variable equations and recognize equivalent forms.

Model real-world one-variable equations and two-variable equations limited to linear, exponential, and quadratic.

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.

Determine a line of best fit to a linear model of data.

Advanced Skills/Concepts: Some students may be ready to:

Recognize that volume is a linear function of area.

Analyze the sum of the squares of the residuals.

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

*

*

*

Math 2 Unit 5: Comparing Functions and Modeling

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Knowledge: Students will know…

The parent graphs for linear, quadratic, and exponential functions.

Skills: Students will be able to …

Write an appropriate equation and/or function to model a real-life situation.

Use a model of an appropriate function to interpret information about a real-life situation.

Use and compare multiple representations of functions including tables, graphs, equations, and real-life situations.

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

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

Write a function that describes a relationship between two quantities.

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).

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

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.

Academic Vocabulary:

Critical Terms: Quadratic Parabola Translation Transformation Residual

Supplemental Terms: Tables Graphs Equations Correlation Coefficient

Math 2 Unit 5: Comparing Functions and Modeling

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Assessment

Pre-Assessments Formative Assessments Summative Assessments Self-Assessments

Line of Best Fit Tortoise, the Hare, and the Aardvark Comparing Functions Practice Truffle Tin

Tortoise, the Hare, and the Aardvark Truffle Tin Functionville

Sample Lesson Sequence 1. F.IF. 6,7,8,9 F.BF.3, F.LE.3 Comparing Linear, quadratic, and exponential functions. (Model Lesson)

a. Compare the graphs

b. Compare the equations

c. Compare what happens when they transform the equation and the graph

2. F.IF.8a Rearrange functions to a friendly version to graph.

3. S.ID.6 & F.IF.9 Line of Best Fit Revisited