This student activity is designed to explore the concept of work. Using inclined planes with different slopes, students are lead to discover the invariance of force times distance for objects starting at the same height. Included are notes for instructors wishing to use this material.
9-12: 1B/H1. Investigations are conducted for different reasons, including to explore new phenomena, to check on previous results, to test how well a theory predicts, and to compare theories.
9-12: 1B/H6b. In the long run, theories are judged by the range of observations they explain, how well they explain observations, and how useful they are in making accurate predictions.
2. The Nature of Mathematics
2A. Patterns and Relationships
9-12: 2A/H1. Mathematics is the study of quantities and shapes, the patterns and relationships between quantities or shapes, and operations on either quantities or shapes. Some of these relationships involve natural phenomena, while others deal with abstractions not tied to the physical world.
2B. Mathematics, Science, and Technology
9-12: 2B/H3. Mathematics provides a precise language to describe objects and events and the relationships among them. In addition, mathematics provides tools for solving problems, analyzing data, and making logical arguments.
3. The Nature of Technology
3A. Technology and Science
9-12: 3A/H2. Mathematics, creativity, logic, and originality are all needed to improve technology.
9-12: 3A/H4. Engineers use knowledge of science and technology, together with strategies of design, to solve practical problems. Scientific knowledge provides a means of estimating what the behavior of things will be even before they are made. Moreover, science often suggests new kinds of behavior that had not even been imagined before, and so leads to new technologies.
4. The Physical Setting
4E. Energy Transformations
6-8: 4E/M4. Energy appears in different forms and can be transformed within a system. Motion energy is associated with the speed of an object. Thermal energy is associated with the temperature of an object. Gravitational energy is associated with the height of an object above a reference point. Elastic energy is associated with the stretching or compressing of an elastic object. Chemical energy is associated with the composition of a substance. Electrical energy is associated with an electric current in a circuit. Light energy is associated with the frequency of electromagnetic waves.
9-12: 4E/H9. Many forms of energy can be considered to be either kinetic energy, which is the energy of motion, or potential energy, which depends on the separation between mutually attracting or repelling objects.
9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.
9. The Mathematical World
9B. Symbolic Relationships
6-8: 9B/M3. Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease, increase or decrease in steps, or do something different from any of these.
9-12: 9B/H4. Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.
9-12: 9B/H5. When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes by more than one, and sometimes not at all.
11. Common Themes
9-12: 11A/H2. Understanding how things work and designing solutions to problems of almost any kind can be facilitated by systems analysis. In defining a system, it is important to specify its boundaries and subsystems, indicate its relation to other systems, and identify what its input and output are expected to be.
9-12: 11B/H1a. A mathematical model uses rules and relationships to describe and predict objects and events in the real world.
9-12: 11B/H5. The behavior of a physical model cannot ever be expected to represent the full-scale phenomenon with complete accuracy, not even in the limited set of characteristics being studied. The inappropriateness of a model may be related to differences between the model and what is being modeled.
Next Generation Science Standards
Motion and Stability: Forces and Interactions (HS-PS2)
Students who demonstrate understanding can: (9-12)
Analyze data to support the claim that Newton's second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. (HS-PS2-1)
Disciplinary Core Ideas (K-12)
Definitions of Energy (PS3.A)
A system of objects may also contain stored (potential) energy, depending on their relative positions. (6-8)
Conservation of Energy and Energy Transfer (PS3.B)
Mathematical expressions, which quantify how the stored energy in a system depends on its configuration (e.g. relative positions of charged particles, compression of a spring) and how kinetic energy depends on mass and speed, allow the concept of conservation of energy to be used to predict and describe system behavior. (9-12)
Crosscutting Concepts (K-12)
Graphs, charts, and images can be used to identify patterns in data. (6-8)
Empirical evidence is needed to identify patterns. (9-12)
Cause and Effect (K-12)
Systems can be designed to cause a desired effect. (9-12)
Scale, Proportion, and Quantity (3-12)
Algebraic thinking is used to examine scientific data and predict the effect of a change in one variable on another (e.g., linear growth vs. exponential growth). (9-12)
Science and Engineering Practices (K-12)
Analyzing and Interpreting Data (K-12)
Analyzing data in 9–12 builds on K–8 and progresses to introducing more detailed statistical analysis, the comparison of data sets for consistency, and the use of models to generate and analyze data. (9-12)
Analyze data using tools, technologies, and/or models (e.g., computational, mathematical) in order to make valid and reliable scientific claims or determine an optimal design solution. (9-12)
Planning and Carrying Out Investigations (K-12)
Planning and carrying out investigations in 9-12 builds on K-8 experiences and progresses to include investigations that provide evidence for and test conceptual, mathematical, physical, and empirical models. (9-12)
Plan and conduct an investigation individually and collaboratively to produce data to serve as the basis for evidence, and in the design: decide on types, how much, and accuracy of data needed to produce reliable measurements and consider limitations on the precision of the data (e.g., number of trials, cost, risk, time), and refine the design accordingly. (9-12)
Science Models, Laws, Mechanisms, and Theories Explain Natural Phenomena (2-12)
A scientific theory is a substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment and the science community validates each theory before it is accepted. If new evidence is discovered that the theory does not accommodate, the theory is generally modified in light of this new evidence. (9-12)
Using Mathematics and Computational Thinking (5-12)
Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. (9-12)
Use mathematical representations of phenomena to describe explanations. (9-12)
Common Core State Standards for Mathematics Alignments
High School — Algebra (9-12)
Reasoning with Equations and Inequalities (9-12)
A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.?
High School — Functions (9-12)
Interpreting Functions (9-12)
F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.?
F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Building Functions (9-12)
F-BF.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Linear, Quadratic, and Exponential Models? (9-12)
F-LE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
%0 Book Section %A Roeder, John %D 2003 %T Teaching About Energy: Designing a Roller Coaster %B Teaching about Energy %U http://www.compadre.org/Repository/document/ServeFile.cfm?ID=3375&DocID=69
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