Get in Line!

Author: Kim Zubrzycki
Grade Level: Course I and Course II (Algebra & Geometry)
School Address: Copenhagen Central, PO Box 30, Copenhagen, NY 13626
School Phone/Fax: 315-688-4411/2001

Subject Area: Quadratic Functions

Unit Overview:

This unit was designed to enhance a current quadratics unit by integrating technology. This unit should be done in conjunction with traditional lessons on graphing quadratic equations. The ultimate goal of the students is to discover a situation that would best be represented by a quadratic equation. Students should be able to graph linear equations, and be familiar with the order of operations. The technology used for this unit includes graphing calculators (for the initiating activity) and PASCO equipment, which includes data collection interfaces that can be transported to PowerPoint and Word are also used for the students to complete the written summary of the unit and each student's oral presentation. By using the PASCO equipment or similar probes, students are allowed to see the results of their data collection in table and graph form.

This unit will take approximately two weeks of class time (each session being 40 minutes), depending on the availability of the technology to complete. Students are allowed to work in teams to complete the culminating activity, which allows students with disabilities to work with a peer he/she feels comfortable with. In addition to working with peers, the teacher acts as the facilitator to guide the students "on the right track"; this gives the teacher time to work with each team individually to further assist the students with disabilities.

This unit could easily be adapted for various levels, and may be possible to integrate with the Earth Science curriculum, the students could collect data during the Earth Science lab and compare results during the math class.

Content Knowledge:

Declarative

• Vocabulary
• Identify differences between functions
• Identify "key" characteristics of a given graph

Procedural

• Construct various graphs
• Identify equations that correspond to given graphs
• Collect data by performing experiments
• Analyze graphs

Essential Questions:

Are there real life applications that can best be represented by a quadratic function? Name at least one and be able to defend it with results from an experiment that you performed along with a written summary and an oral presentation to the class.

Initiating Activity:

Students will begin this unit by using graphing calculators (or a computer spreadsheet) to discover the differences between various quadratic functions. They will be able to see the affect of different coefficients and constants, as well as the different graphs as compared to linear functions.

Learning Experiences:

• Session 1: Initiating Activity-see the description above

• Session 2: Introduce the essential question. The teacher's role is to guide students toward discovery. To do this, the instructor needs to be able to answer questions without giving specific answers; rather help students learn how to answer their own questions.

• Session 3: Discuss trajectory experiments and compare to linear equations.

• Session 4: Analyze quadratic functions, using PASCO technology (motion sensor). Using PASCO technology, students are given the opportunity to experience the difference between linear and nonlinear functions. In PASCO's, Follow the Leader activity, students are asked to move at the rate of the line given to them on the computer screen via the motion sensor. This particular activity is included in the Middle School Science Labs manual from PASCO, and should need little assistance from the instructor. The Linear and Nonlinear Relationships activity may require some more guidance. By allowing students to work in small groups or pairs, they are able to hopefully learn from one another, as well as work cooperatively to reach the goal.

• Session 5: Use equations to graph functions with a graphing calculator or spreadsheet, and then manually. Provide students with a worksheet or "book assignment" to practice sketching linear and nonlinear relationships.

• Session 6: Use a parabolic graph to identify the "key" characteristics-axis of symmetry, vertex, and roots.

• Session 7: Match equations of given graphs to equations. Provide students with some practice matching equations to graphs.

• Session 8: Review for written test of "Regents like" questions, as well as critical thinking questions.

• Session 9: Take a culminating test.

Culminating Performance:

Students will work in teams of two or three to discover a real life situation that would best be represented by a quadratic equation. They will then need to perform necessary experiments using their choice of methods and/or technology to collect data, in order to defend their hypothesis. After collecting the data, they will need to graph the data using calculators and/or, PASCO technology, or spreadsheets, in order to see the function. Ton conclude their project, students will need to be able to defend their situation by writing a brief summary including all of their results as well as complete a short oral presentation sharing their graphs.

Connections to NYS Standards:

MST #1: Analysis, Inquiry, and Design-Commencement Level

Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions.

• Key Idea: Mathematical Analysis-abstraction and symbolic representation are used to communicate mathematically. Performance Indicators-students use algebraic and geometric representations to describe and compare data.
• Key Idea: Scientific Inquiry-the observations made while testing proposed explanations, when analyzed using conventional and invented methods, provide new insights into phenomena. Performance Indicators-students use various means of representing and organizing observations (e.g. diagrams, tables, charts, graphs, equations, matrices) and insightfully interpret the organized data. (Students assess correspondence between the predicted result contained in the hypothesis and the actual result and reach a conclusion as to whether or not the explanation on which the prediction was based is supported.)
• Key Idea: Engineering Design-engineering design is an iterative process involving modeling and optimization finding the best solution within given constraints which is used to develop technological solutions to problems within given constraints. Performance Indicators-students engage in the following steps in a design process: 1) Generate creative solutions, break ideas into significant functional elements, and explore possible refinements; functional modeling techniques; choose the optimal solutions to the problem, clearly documenting ideas against design criteria and constraints; and explain how human understands, economics, ergonomics, and environmental considerations have influenced the solution. 2) Devise a test of the solution according to the design criteria and perform the test; record, portray, and logically evaluate performance test results through quantitative, graphic, and verbal means. Use of variety of creative verbal and graphic techniques effectively and persuasively to present conclusions, predict impacts and new problems, and suggest and pursue modifications.

MST #2: Information Systems-Commencement Level

Students will access, generate, process, and transfer information using appropriate technologies.

• Key Idea: Information Systems-information technology is used to retrieve, process, and communicate information and as a tool to enhance learning. Performance Indicators-students model solutions to a range of problems in mathematics, science, and technology using computer simulation software.

MST #3: Mathematics-Commencement Level

Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world setting, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.

• Key Idea: Modeling/Multiple Representation-students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships. Performance Indicators-students choose appropriate representations to facilitate the solving of a problem. (Students use learning technologies to make and verify geometric conjectures. Students use graphing utilities to create and explore geometric and algebraic models.)
• Key Idea: Patterns/Functions-students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently. Performance Indicators-students represent and analyze functions using verbal descriptions, tables, equations, and graphs. (Students model real-world situations with the appropriate function. Students use computers and graphing calculators to analyze mathematical phenomena.)

MST #5: Technology-Commencement

Students will apply technological knowledge and skills to design, construct, use, and evaluate products and systems to satisfy human and environmental needs.

• Key Idea: Engineering Design-Engineering design is an iterative process involving modeling and optimization used to develop technological solutions to problems within given constraints. Performance Indicators-students engage in the following steps in a design process generate creative solution ideas, break ideas into the significant functional elements, and explore possible refinements; predict possible outcomes using mathematical and functional modeling techniques; choose the optimal solution to the problem, clearly documenting ideas against design criteria and constraints; and explain how human values, economics, ergonomics, and environmental considerations have influenced the solution.

ELA #1: Language for Critical Analysis and Evaluation-Commencement

Students will read, write, listen, and speak for information and understanding.

• Key Idea: Speaking and Writing-speaking and writing to acquire and transmit information requires asking probing and clarifying questions, interpreting information in one's own words, applying information from one context to another, and presenting the information and interpretation clearly, concisely, and comprehensibly. Performance Indicators-students support interpretations and decisions about relative significance of information with explicit statement, evidence, and appropriate argument. (Students use standard English skillfully, applying established rules and conventions for presenting information and making use of a wide range of grammatical constructions and vocabulary to achieve an individual style that communicates effectively.)