To the Moon and Beyond-Algebra II Unit
By Jason Avent, Ellen Lukasik, Michael West
|
Target Audience Although this project was contains units for Biology and Physics, this version focuses on applications for Algebra II, and is adaptable to regular or honors curriculum. Project Description This project is a collaborative effort between math and science classes that allows students to apply traditional curriculum with individual ideas, talents, and perspectives over the course of a long-term unit. Integrating concepts and principles learned in the classroom, students in small groups will design a flight plan to the moon. They will identify key concerns, analyzing options, and justifying decisions with evidence, experimentation, and research. They will organize and present their results in an unspecified visual manner (examples include a poster, diorama, power point presentation, or newspaper). Final oral presentations will incorporate all classes, emphasizing the various positions and considerations of different disciplines and permitting peer reviews. Local officials, professionals of interest, and media will be encouraged to attend. Driving Question (modified; original by Ryan Odom) Overall Goals of the Project The overall mission for “To the Moon and Beyond” is to develop a strong, interdisciplinary curriculum in a project-based approach that develops skill, perspective, and understanding while modeling real world applications. This project was designed for every student, so that each one can achieve success with autonomy and due challenge. Most importantly, “To the Moon and Beyond” enables students to make connections, communicate mathematically, and integrate classroom lessons into actual practice, giving math meaning, context, and depth. Project Objectives Algebra II Students will be able to:
Rationale Although many different theories exist on how students learn, those entrusted to teach view constructivism as the actual way students absorb and retain knowledge. Constructivism is an approach to teaching based on the idea that learning occurs as mental construction, linking and building on knowledge, beliefs, and attitudes that students possess. It stresses active participation, teaching in appropriate contexts, exploratory and generative learning, self evaluation, and problem and inquiry based lessons. Research has supported this view of pedagogy, and my experience has demonstrated and supported the effectiveness of this approach. Using student inquiry to answer and explore possibilities to solve problems fosters interaction and allows students to develop relationships, create connections, and establish metaphors throughout content. Tools, symbols, and manipulatives further enable students to play, struggle, and learn. Without the ability to express detailed problems and meaningful processes to find solutions, mathematics would be an isolated process remanded to individuals. Collaboration and group discussions give students the opportunity to practice and refine these skills. “To the Moon and Beyond” creates a learning environment based and rooted in these principles. It was designed from the curriculum up, ensuring the inclusion of collaboration and active inquiry to establish the ultimate learning experience for students. Background “One important outcome for this unit is helping students develop an appreciation of the power and limitations of mathematical modeling. They should realize that the two most basic expectations of models are (1) the ability to account for or represent known phenomena and (2) the ability to predict future results. Thus, with the models that students develop in this unit, they should continually be asking such questions as, What will happen if this trend continues? Or What if this element is change? Other outcomes that we anticipate from this unit are a better understanding of the difference among linear, quadratic, and exponential functions (models) and the patterns of growth that arise from them; a more concrete realization of the vastness of space and the seeming paradox of having very large quantities of debris in orbit yet, at the same time, quite small probabilities of encountering any - although with potentially lethal results should an encounter happen; and an appreciation of the power of mathematics to help us "get our arms around " seemingly unmanageable problems. Data included in this unit are taken from various NASA sources, but you will want to be alert to updated information and current events related to the problem. For example, three events in early 1996 received extensive coverage in the media. One was the Space Shuttle Endeavour's maneuver to steer clear of a defunct satellite. A second was the breaking loose of a satellite and its 12-mile-long tether during an attempt to deploy it from the Shuttle. A third was the return to Earth of a Chinese satellite, which problem landed somewhere in the “In January 1996, two days into their mission, the crew of the Space Shuttle Endeavour performed a maneuver to slow its speed by 4 feet per second, thereby steering clear of a 350-pound piece of "space junk," a defunct Air Force satellite that would have passed within 4/5 of a mile of Endeavour. Although the Shuttle was in no danger of collision, NASA flight safety rules call for at least 1.3 miles of separation between the Shuttle and any other orbiting object. The maneuver widened the distance to 6 miles.
The incident, however, called attention to the growing problem of space debris. In seventy-four Shuttle missions, only about six have had to perform maneuvers to avoid orbiting objects; yet the Air Force was catalogs nearly 8000 orbiting manufactured objects of grapefruit size and larger. As humans and their vehicles and probes prepare to spend longer and longer times in space, they must also prepare to confront the growing problem of space debris. NASA scientists are just beginning to realize the enormity of this space-age form of pollution. And just as pollution is creating terrible problems for us on Earth, so, too, are we experiencing hazards with space pollution. Estimates made in 1990 stated that more than 4 million pounds of manufactured materials were in Earth orbit. Of that amount, only 5 percent represented operating payloads; the other 95 percent consisted of human-made debris: old rocket parts, nonfunctioning satellites, discarded tools, the by-products of explosions and collisions, and other odds and ends, as well as countless, numbers of smaller objects, such as paint chips, and dust-sized particles. On the basis of the rate at which launches were occurring in 1990, we expected the nations in the space business to contribute nearly 2.7 million pounds per year by the year 2000. The prediction warns that if we don't change our ways, we will have 9.5 million pounds of human-made materials circling the Earth. Rings are already forming around Earth - not the rings of rock and dust and ice that encircle other planets, but rings of human-made orbital debris - and their density is increasing. According to Don Kessler, a NASA scientist who has made a career of studying space debris, "Rings are nature's way of saying it doesn't like things in non-circular orbit out of Earth's equatorial plane. Nature wants to tear these objects apart and reform them into either a ring or a single object-it is just a question of when." The North American Aerospace Defense Command (NORAD) reported in 1986 that 4488 of 6194 radar-trackable objects were orbital debris. The rest included 1582 payloads, 68 interplanetary probes, and 56 items of interplanetary probe debris. A radar-trackable object in space is baseball size or larger, but during hypervelocity, which begins at 3 kilometers per second, particles as small as a paint flake can be damaging or even lethal. By the mid-1980's, ground based telescopes made it possible for scientists to see marble-sized pieces of orbital debris. From these observations, they concluded that the number of debris objects was many times the number that NORAD had cataloged. The first loss of a spacecraft part directly attributable to human-made orbital debris occurred during the Shuttle mission of STS-7 in 1983. The crew of the Challenger reported an impact crater on one of the orbiter's windows significant enough to require replacement despite the fact that the window was 5/8 inch thick and built to withstand pressures of 8600 pounds per square inch and temperatures up to 482°C. By studying the traces found in the pitted window, NASA determined that the damage was caused by white paint specks about 0.2 millimeters in diameter traveling between 3 and 5 kilometers per second. Cosmonauts on the Soviet spacecraft Salyut 7 reported a similar window incident just weeks later and even reported hearing the impact. The Solar Maximum Mission satellite had been in space for fifty months when the crew of STS-41C repaired it in space and returned to Earth with 15 square feet of the insulation blanket and 10 square feet of aluminum louvers showing thousands of pits and excessive wear and tear. The blanket showed thirty-two holes per square foot and the louvers, six holes per square foot, many more than NASA scientists expected; analysis revealed that most of the pits were caused by paint flakes. Collisions and breakups significantly increase the number of particles orbiting Earth, but they differ in fragmentation and the resulting hazards. Collisions produce smaller fragments and increased hazard. If a 10-pound mass hits a 1000 pound stage at orbital speeds, a tremendous amount of energy is released that could result in 4 million particles and 10000 larger pieces. Breakups caused by explosions produce fewer small fragments, which give scientists a greater likelihood that they can learn about the causes of explosions and thus prevent them. For example, when seven second-stage Delta rocket breakups occurred three years after launch, scientists were able to determine that they had been caused by unspent hypergolic fuels; this knowledge resulted in launch changes that apparently corrected that problem.” From Modeling Orbital Debris Problems in Mission Mathematics, Linking Aerospace and the NCTM Standards, 9-12, a NASA/NCTM project, NCTM 1997. Standards C1 (A) The student identifies and sketches graphs of parent functions, including linear (y = x), quadratic (y = x2), square root (y = Ö x), inverse (y = 1/x), exponential (y = ax), and logarithmic (y = logax) functions. C1 (B) The student extends parent functions with parameters such as m in y = mx and describes parameter changes on the graph of parent functions. D1 (A) For given contexts, the student determines the reasonable domain and range values of quadratic functions, as well as interprets and determines the reasonableness of solutions to quadratic equations and inequalities. D1 (B) The student relates representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions. D1 (C) The student determines a quadratic function from its roots or a graph. E(1) The student uses quotients to describe the graphs of rational functions, describes limitations on the domains and ranges, and examines asymptotic behavior. E (2) The student analyzes various representations of rational functions with respect to problem situations. E (3) For given contexts, the student determines the reasonable domain and range values of rational functions, as well as interprets and determines the reasonableness of solutions to rational equations and inequalities. E (4) The student solves rational equations and inequalities using graphs, tables, and algebraic methods. E (5) The student analyzes a situation modeled by a rational function, formulates an equation or inequality composed of a linear or quadratic function, and solves the problem. E (6) The student uses direct and inverse variation functions as models to make predictions in problem situations. F (2) The student uses the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describes limitations on the domains and ranges, and examines asymptotic behavior. F (3) For given contexts, the student determines the reasonable domain and range values of exponential and logarithmic functions, as well as interprets and determines the reasonableness of solutions to exponential and logarithmic equations and inequalities. F (4) The student solves exponential and logarithmic equations and inequalities using graphs, tables, and algebraic methods. F (5) The student analyzes a situation modeled by an exponential function, formulates an equation or inequality, and solves the problem. National Council of Teachers of Mathematics Algebra Standards Represent and analyze mathematical situations and structures using algebraic symbols
Use mathematical models to represent and understand quantitative relationships
Description of assessments, including final product Assessments are formative and summative, occurring throughout the six week period and concluding with a formal presentation session. Assessments over the length of the unit include Concept Maps, Application Cards, Documented Problem Solutions, Categorizing Grids, Productive Study Time Logs, and Course Related Self-Confidence Surveys. Complete descriptions of these assessments can be found in Classroom Assessment Techniques by Thomas A. Angelo and K. Patricia Cross. We will conclude our project with a group presentation session with an audience of peers, media, and related professionals. Students will attend all presentations, introducing concerns and issues from a number of perspectives and disciplines. They will provide peer reviews with constructive feedback, using critical thinking skills to evaluate the strength of others’ creativity, arguments, and visual media. |
