ND Curriculum Initiative

Subject & Standards

Needs Assessment/Rational

Students at the secondary level often inquire about the necessity of learning algebraic concepts. They often want to know, “When am I ever going to see this or use this?”. Finding real world applications to those concepts has become a constant teaching goal for me. FORMAT instruction always stresses beginning with what students know about a topic and getting them to connect. The more connections students can make the more interesting the topic becomes to them and then deeper understanding can occur. Students do not get enough exposure to mathematical modeling and the use of regression techniques involving real world applications. In addition, I discussed the concept of representation and graphing with many of our high school mathematics teachers, and we came to recognize a deficiency in our students making the connection of the algebra with the graph with the real world and vise versa. This only increased the necessity for finding real world connections to algebraic concepts to enhance deeper mathematical understanding. In addition, I viewed the North Dakota State Assessment Content Standards Performance Report for Mathematics and compared our district’s and our school’s senior high students (at this time they were seniors) to those statewide. This test was given in October of 2003. (Our present test results have not yet been returned to our district.) The data showed me that our district rated at the following overall performance levels in mathematics, Advanced, 8.9% (state, 16.7%), Proficient, 20.9% (state, 19.7%), Partially Proficient, 41.3% (state, 41.2%), and Novice, 28.9% (state, 22.3%). In matching that information to Standard 2: Geometry and spatial sense, I found that our district (school) rated lower than the state average with 50% correct (state, 56%)Benchmark 1: Properties of 2, 3-dimensional models; and 67% correct (state, 70%) in Benchmark 3: Understand congruence, similarity, and symmetry. I also found similar lower results in Standard 3: Data analysis, statistics, and probability, with 40% correct (state, 46%) in Benchmark 6: Using regression techniques; and 83% correct(state, 85%) in Benchmark 7: Draw inferences and predict outcomes. Finally our school rated lower in Standard 5: Algebra, functions, and patterns having 64% correct (state, 67%)in Benchmark 2: Solve equations, inequalities, and systems; and 56% correct (state, 60%) in Benchmark 3: Represent relations algebraically, graphically; and 64% correct (state, 68%) in Benchmark 7: Use patterns, functions to model problems. All of this data supports the need for enhancing any unit so that deeper understanding can occur amongst our students.

Understandings & Goals

Enduring Understanding: Students will find several uses for conic sections. Students will recognize each conic algebraically and graphically. Goal(s): Students will successfully create a graphical representation of a real world conic. Students will connect conics to a practical use in the world. Students will use appropriate technological tools to collect and analyze data to make conjectures. Students will use regression techniques and conic definitions to create a “best-fit” curve. Students will interpret any conic equation and evaluate each of its parts.

Questions Answered

Essential questions: 1. What are the conic sections and why are they called conic sections? 2. What are the parts of each conic? How do they relate algebraically and graphically? 3. Why do we study conic sections? How are conic sections applied to the real worldObjectives: Students will collect, organize, and analyze data from real world conical applications. Students will effectively and precisely communicate and represent the definition and creation of each conic section. Students will represent data in lists, then appropriately graph that data with the aid of technologies such as the graphing calculator, Paint, and Geometer’s Sketchpad. Students will evaluate roots and components of each conic section through algebraic and graphical representations Students will compare and contrast the effects of changing the parameters of each conic section. Students will generate a “best-fit” curve through algebraic regression of quadratics for each conic. Students will interpret, analyze, and make inferences using their data, graphs, and curves and effectively communicate their results. Students will solve and interpret conic equations and accurately state which conic is represented and all details about that conic.

Assessment

What quiz and test items (e.g. simple content-focused questions that require a single, best answer) will provide evidence of understanding? Students will quiz on two conic sections at a time using simple content-focused questions that have a single best answer. Students will finalize their evidence of understanding by taking a test that is based on content-focused questions with a single best answers. What academic prompts (e.g. open-ended questions or problems that require students to think critically and then to prepare a response / product / performance) will provide evidence of understanding? Some of the activities that students will be involved in include doing assigned textbook problems and work sheets that have content-focused questions that require a single response. These will show what detail the student can recall. Students will also do computer activities that involve writing paragraphs on their interpretations of their results. This will show that students are able to analyze conics, as well as, compare and contrast each conic. Students will also be able to show an awareness of how conics relate to their world. Open-ended questions will be used throughout the instruction. Open-ended questions allow the students to draw interpretations and translations of the material. The different teaching methodologies that will be used include lecture, demonstration, hands-on activities for the students. Using several teaching methods allows students to synthesize and and build their knowledge base and understandings. Students are to dicuss on a daily basis their complications, solutions, and questions concerning conic sections. Sharing with others illustrates collaborative learning and brings students to a deeper realization about each conic. Students will gather data and learn how to interpret it specifically for each conic. Students will work on a parabola project using Paint and Geometer’s Sketchpad or their graphing calculators to connect conic sections to the real world. Students will research several media to find conic sections within our world. When students find connections and familiarity to a topic they can come to a deeper understanding about it. What performance tasks and projects (e.g. complex challenges that are authentic, mirror the real world and require a performance or product) will you include that will provide evidence of student understanding? Students will form collaborative groups to create a presentation illustrating real world applications, definitions, graphs, and “best-fit” regression curves of each conic section. When students share and discuss information, they find insights beyond simple recall. What other evidence (e.g. observations, work samples, dialogues, student self-assessment) of understanding will you collect?  Rubrics (teacher-designed and student-designed) will be used to further assess students. Students will also be assessed by the accummulation of conic problems within thier notebooks, observations and discussions held throughout class, work samples produced, and student- produced products.

Instructional Strategies

Students will use inquiry and project-based learnings throughout this unit. Students will research several types of media: videos, journals, magazines, internet web sites, to find real-life uses for conics. Through that research they should also discover how each conic section is derived algebraically and graphically. Students will collect a minimum of one photo representing each conic section. They must use ethical and appropriate-use standards for such media. This will allow students to reach my second and third learning objectives of connecting conics to a practical use in the world and using the appropriate technological tools to collect data. Students will then create a presentation using one of the real-life conical applications that their group accumulated and find the “best-fit” conic curve by using algebra. They will then plot that conic curve graphically. Through this presentation students will have reached two more of the unit learning objectives: 1. Students will sucessfully create a graphical representation of a real world conic, and 4. Students will use regression techniques and conic definitions to create a “best-fit” curve. Once students have completed that, they will collaboratively create a model for each conic section and explain how that conic section came to be. Furthermore, students will create a paragraph that reflects on their discoveries about conics in general, as well as, what they learned through this project. They should be able to find reasons why conic sections are used in the world. They should be able to relate their findings to architecture, mechanical structures, astronomy, sound, etc. These activities should allow students to obtain the remaining learning objectives: 4. Studens will use regression techniques and conic definitions to create a “best-fit” curve, and 5. Students will interpret any conic equation and all its parts. Students will also reach the latter part of objective 3 by analyzing their data to make c