Untitled Document

Lesson 2

Title: Hunt for Functions

Author: Shelly Rogers

Grade Level: Pre-Calculus

Concepts: Students will be able to recognize shapes, designs, and structures existing around them in terms of functions. By having an expanded awareness of various types of functions, they will be able to more easily see how functions can be used to represent particular images. Their experience with the manipulation of functions, along with this increased repertoire of functions, will give the students the freedom to more accurately recreate images that are less contrived. Through this activity, students will also gain an understanding of mathematics as a tool to describe the world in which we live.

Objectives:

Students will be able to:

1) recognize different types of functions that can be used to describe real-world

objects

2) formulate equations of functions that best describe particular relationships

3) apply their knowledge of how to manipulate functions in order to create images of these real-world objects

4) develop connections between mathematics and the real-world.

TEKS:

§111.35. Precalculus (One-Half to One Credit).

(b) Introduction.

(1) In Precalculus, students continue to build on the K-8, Algebra I, Algebra II, and Geometry foundations as they expand their understanding through other mathematical experiences. Students use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study mathematical concepts and the relationships among them. Students use functions, equations, and limits as useful tools for expressing generalizations and as means for analyzing and understanding a broad variety of mathematical relationships. Students also use functions as well as symbolic reasoning to represent and connect ideas in geometry, probability, statistics, trigonometry, and calculus and to model physical situations. Students use a variety of representations (concrete, numerical, algori\thmic, graphical), tools, and technology to model functions and equations and solve real-life problems.

(1) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, radical, exponential, logari\thmic, trigonometric, and piecewise-defined functions. The student is expected to:

(A) describe parent functions symbolically and graphically, including y = xⁿ, y = ln x, y = log_ax, y = , y = e^x, y = a^x, y = sin x, etc.;

(B) determine the domain and range of functions using graphs, tables, and symbols;

(D) recognize and use connections among significant points of a function (roots, maximum points, and minimum points), the graph of a function, and the symbolic representation of a function; and

(2) The student interprets the meaning of the symbolic representations of functions and operations on functions within a context. The student is expected to:

(A) apply basic transformations, including a • f(x), f(x) + d, f(x - c), f(b • x), |f(x)|, f(|x|), to the parent functions;

(3) The student uses functions and their properties to model and solve real-life problems. The student is expected to:

(A) use functions such as logari\thmic, exponential, trigonometric, polynomial, etc. to model real-life data;

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Materials List and Advanced Preparations:

· Computer (1/group)

· Graphmatica (already downloaded on computer)

· ExploreLearning Gizmo: Translating and Scaling Functions (already downloaded on computer)

· Printer accessibility and printing paper —OR— graphing paper

· Images predominately consisting of parabolas (different one for each group)

· Digital Camera with memory card (1/group)

Engagement:

Teacher: * holds up images constructed by the students when they were being “introduced” to the different types of functions, such as the umbrella image that one group recreated on the parabola day.

* asks students how this task of recreating images could be more interesting

*Explain that these images are somewhat contrived, she originally made the images by creating functions herself…what would be more interesting would be to see functions existing in real-world objects

* asks the students to brainstorm objects in everyday life that could be represented by one or more of the different types of functions

* records their brainstormed ideas so that the entire class can read the list

Exploration:

What the Teacher Will Do	What the Students Will Do	Ongoing/Formative Evaluation (Questions you will ask the students)*
The teacher explains the “hunt” to the class. She tells them that: 1) they are to look for things that could be represented by some type(s) of function(s) and take a picture of it with their digital camera 2) they will be working in their groups 3) the groups should try to find as many functions as possible because the group that has the highest average number of functions found per specific type will earn extra points (or some other incentive) 4) their search must stay within the boundaries that she established. (REMEMBER TO TELL ESTAB. BOUNDARIES) 5) they only have X minutes to search (REMEMBER TO DECIDED WHAT X IS)
Teacher walks around to each group, checks for bad behavior, keeps students on task, answers student questions, etc.	Students begin “hunt.”
After X minutes, the teacher regathers the groups.	Students stop searching and return to their classroom tables.
Teacher explains how to upload their pictures and that, after uploading their pictures, they should determine functions for their images using the Gizmo and Graphmatica programs.	Students upload images and open Gizmo and Graphmatica to begin their attempt to recreate the images.
The teacher tells the groups to keep a log of all equations used to recreate their images and should know which equation corresponds to which function in their image. Teacher also tells the class how much time they have to work on this before she brings the groups back together for presentation and discussion.	.
Teacher walks around the classroom making sure the students understand their task, answering questions, and ensuring the students’ participation. (The teacher can and should do this by asking the students questions about their project.)	Students should be working as a group to recreate their images and to make their equations log. Students will either draw their images by hand, or print the images from Graphmatica.
The teacher brings the groups back together as a class, so that they have enough time to present their images and discuss what they have learned.	The groups should have completed the recreation of their images. They should have either printed their newly created images or drawn them by hand. They should provide some key to explain which function corresponds to which equation in their log.
Adds names of images to the previously brainstormed list.

Explanation:

The groups present the digital pictures of what they found existing around them and the images that they created with equations in order to describe what they found.
Teacher asks the students to compare two of their functions.
Discuss how they were able to manipulate the basic function equations in order to create functions that would best describe the original images. (y = a(f(b(x-h))) + k, so we can manipulate a, b, h, k…we can also choose the domain of the functions.)

Elaboration:

Discuss how we’re still limited in our ability to create images…For instance, we are only working with 2-diminsional functions.

Evaluation:

The recreation of their digital images through manipulation of basic functions
The extent to which they participated
Their ability to explain the effects of changing certain aspects of a parabola’s equation

Safety and behavior:

Students must conduct themselves in an orderly manner and respect the equipment.
No running regardless; however, students should be especially aware of the computer chords.
Students must stay within the “hunt’s” boundaries designated by the teacher.
Obey all school rules, as always. Particularly, however, if the students are allowed to do the “hunt” outside of the classroom, they are not to disrupt any other class.

What the Teacher Will Do

What the Students Will Do