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The Quadratic Function, an Introduction
Carla Bissell
Mathematics




Unit created on 6/9/1999 EST.
Last modified 12/13/1999 12:27:59 PM EST.


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Fundamental Understandings (help)


Introductory Unit at beginning algebra level, 1. To recognize the quadratic function rule and have a sense of the appearance of the graph. The developed sense would include the basic parabola shape and a connection between parameter values and graph location and shape. 2. To identify data that might be quadratic, especially as it is distinguished from linear data. 3. To sense what real world situations might be modelled by quadratic function rules, how the parameters of the function rule relate to the situation and how limiting domain may be appropriate.


Technology ISTE Standards (info) 


Select and apply technology tools for research, information analysis, problem-solving, and decision-making in content learning. Identify capabilities and limitations of contemporary and emerging technology resources and assess the potential of these systems and services to address personal, lifelong learning, and workplace needs.

Information Literacy Standards (info) 


Standard 1 Accesses information efficiently and effectively, as described by the following indicators: 1.recognizes the need for information: 2.recognizes that accurate and comprehensive information is the basis for intelligent decision making; Standard 3: Uses information effectively and creatively, as described by the following indicators: 1.organizes information for practical application; 2.integrates new information into one's own knowledge; 3.applies information in critical thinking and problem solving; Standard 9: Participates effectively in groups to pursue and generate information, as described by the following indicators: 1.shares knowledge and information with others; 2.respects others' ideas and backgrounds and acknowledges their contributions;

Relevance (help)


The study of quadratic behavior is fundamental to the academic mathematics sequence. Student understanding is frequently hampered by a curriculum that concentrates on small associated skills without connections to the behavior of the function. The function concept is central to most of the secondary pre-calculus curriculum and the language and characteristics should be developed at every level.

Assessment (help)


Informal checks on understanding through observation and oral probes during class discussion, individual or small group presentations, student explorations. Students provide evidence that they can look at mixed function family rules, tables and graphs and situations (including but not limited to linear and quadratic) and distinguish linear and quadratic functions. Students provide evidence that they can match quadratic function rules with their graphs - based on an understanding of the effect of the parameters on the graph. (Evidence could be a traditional matching pencil-paper test or could be a well organized oral or written presentation describing the characteristics that would distinguish the quadratic function from other families AND how the parameters effect the graph.) Students would identify a parabola-like shape (water-fountain flow, hanging rope, basketball toss, archatechural feature) and take measurements that would allow them to place the parabola on a coordinate system and identify at least three ordered pairs. These ordered pairs will be entered into a curve-fitting tool and the function rule derived. Students can then present a report that includes a description of the data collection, a pencil sketch on graph paper, and a print-out (if possible) of the graph of their fitted function. Students should analyze how well their fitted function models the physical object they measured and what domain is meaningful.

Components (help)


Introductory Activity: You have 100 meters of fencing to build a rectangular enclosure. Students would explore questions involved in building this enclosure, including but not limited to maximum area that can be enclosed? Students would be encouraged to explore, with a large group concluding activity that would elicit a function rule: Area as a function of length or width. Students could then enter this rule in a technology tool to confirm or challenge their earlier conclusions. (Or some other similar situation) Lesson One: Students will use a graphing technology to view the graphs of quadratic functions. The function rules will gradually add and change parameters until students have experience which should enable them to analyze and predict the effect of parameter changes. Students would have an opportunity to discuss their findings with a group or a partner to compare and contrast conclusions. All students familiar with the technology being used will be able to view the graphs. Being engaged with their own technology, each student will determine the length of time to spend exploring, ways to test conjectures. Having an opportunity to defend conjectures, students will have an additional opportunity to reflect on the form of the function rule and appearance of its graph. Lesson Two: Students will examine data tables that mix data that is linear, quadratic, and neither of these. Students will, perhaps in a small group, decide, based on their experience with linear functions earlier and their introduction to quadratics in Lesson One, match data type with function. Students may decide how to make their decision. They might do by-hand point plots and examine the graph, they may narrow their choices by remembering the linear property of constant rate of change. They may have a technology that allows them to plot points there or to enter points and curve fit. A full class discussion should serve as a good review of the characteristics of linear data and help them process the appearance of quadratic data. Lesson Three: Students will generate their own quadratic rule to model a familiar situation. They will start with a familiar linear demand function with attendance as a function of ticket price. Multiplying the attendance function by price will generate a quadratic revenue function. A class discussion could estimate costs associated with the event being modelled, and a quadratic profit function could be developed. Students can enter all three functions in a technology tool and use the tool to table and graph the functions. Students will be presented with some questions to explore using the tables or graphs (change of increment or trace). Students would also be encouraged to initiate questions about the situation that might be important to the promoter of the event. Lesson Four Students will conduct a data collection. Each student will throw a ball straight up in the air and record the time from release to hitting the ground. Each student will also record the height at which the ball is released. Using the quadratic rule for the height of a projectile as a function of time, students will calculate their initial velocity and using that and the initial height, complete the rule that describes their own toss. Entering this rule in a technology tool, they can explore the graph and the table of values to answer questions about their throw, to determine which part(s) of the graph or table actually model the experiment, to experience a quadratic rule whose parameters have physical meaning. <

URLs (help)


http://
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Workforce Competencies (info) (help)


Information Manager Effective communicator Numeric Problem Solver Creative and Critical Thinkers Ethical and Responsible Workers Cooperative Workers
Lessons


Quadratics, #4-Data Collection
Quadratics, #2-Data Tables
Quadratics, #3-Exploring a Situation
Quadratics, #1-The Graph





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