Standard 6—Interconnectedness: Common Themes

Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning.

Key ideas are identified by numbers (1).
Performance indicators are identified by bullets.
Sample tasks are identified by triangles (s).

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Elementary Systems Thinking

1. Through systems thinking, people can recognize the commonalities that exist among all systems and how parts of a system interrelate and combine to perform specific functions.

Students:

• observe and describe interactions among components of simple systems.
• identify common things that can be considered to be systems (e.g., a plant population, a subway system, human beings).

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Elementary Models

2. Models are simplified representations of objects, structures, or systems used in analysis, explanation, interpretation, or design.

Students:

• analyze, construct, and operate models in order to discover attributes of the real thing.
• discover that a model of something is different from the real thing but can be used to study the real thing.
• use different types of models, such as graphs, sketches, diagrams, and maps, to represent various aspects of the real world.

This is evident, for example, when students:
s compare toy cars with real automobiles in terms of size and function.
s model structures with building blocks.
s design and construct a working model of the human circulatory system to explore how varying pumping pressure might affect blood flow.
s describe the limitations of model cars, planes, or houses.
s use model vehicles or structures to illustrate how the real object functions.
s use a road map to determine distances between towns and cities.

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Elementary Magnitude and Scale

3. The grouping of magnitudes of size, time, frequency, and pressures or other units of measurement into a series of relative order provides a useful way to deal with the immense range and the changes in scale that affect the behavior and design of systems.

Students:

• provide examples of natural and manufactured things that belong to the same category yet have very different sizes, weights, ages, speeds, and other measurements.
• identify the biggest and the smallest values as well as the average value of a system when given information about its characteristics and behavior.

This is evident, for example, when students:
s compare the weight of small and large animals.
s compare the speed of bicycles, cars, and planes.
s compare the life spans of insects and trees.
s collect and analyze data related to the height of the students in their class, identifying the tallest, the shortest, and the average height.
s compare the annual temperature range of their locality.

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Elementary Equilibrium and Stability

4. Equilibrium is a state of stability due either to a lack of changes (static equilibrium) or a balance between opposing forces (dynamic equilibrium).

Students:

• cite examples of systems in which some features stay the same while other features change.
• distinguish between reasons for stability—from lack of changes to changes that counterbalance one another to changes within cycles.

This is evident, for example, when students:
s record their body temperatures in different weather conditions and observe that the temperature of a healthy human being stays almost constant even though the external temperature changes.
s identify the reasons for the changing amount of fresh water in a reservoir and determine how a constant supply is maintained.

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Elementary Patterns of Change

5. Identifying patterns of change is necessary for making predictions about future behavior and conditions.

Students:

• use simple instruments to measure such quantities as distance, size, and weight and look for patterns in the data.
• analyze data by making tables and graphs and looking for patterns of change.

This is evident, for example, when students:
s compare shoe size with the height of people to determine if there is a trend.
s collect data on the speed of balls rolling down ramps of different slopes and determine the relationship between speed and steepness of the ramp.
s take data they have collected and generate tables and graphs to begin the search for patterns of change.

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Elementary Optimization

6. In order to arrive at the best solution that meets criteria within constraints, it is often necessary to make trade-offs.

Students:

• determine the criteria and constraints of a simple decision making problem.
• use simple quantitative methods, such as ratios, to compare costs to benefits of a decision problem.

This is evident, for example, when students:
s describe the criteria (e.g., size, color, model) and constraints (e.g., budget) used to select the best bicycle to buy.
s compare the cost of cereal to number of servings to figure out the best buy.

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Intermediate Systems Thinking

1. Through systems thinking, people can recognize the commonalities that exist among all systems and how parts of a system interrelate and combine to perform specific functions.

Students:

• describe the differences between dynamic systems and organizational systems.
• describe the differences and similarities between engineering systems, natural systems, and social systems.
• describe the differences between open- and closed-loop systems.
• describe how the output from one part of a system (which can include material, energy, or information) can become the input to other parts.

This is evident, for example, when students:
s compare systems with internal control (e.g., homeostasis in organisms or an ecological system) to systems of related components without internal control (e.g., the Dewey decimal, solar system).

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Intermediate Models

2. Models are simplified representations of objects, structures, or systems used in analysis, explanation, interpretation, or design.

Students:

• select an appropriate model to begin the search for answers or solutions to a question or problem.
• use models to study processes that cannot be studied directly (e.g., when the real process is too slow, too fast, or too dangerous for direct observation).
• demonstrate the effectiveness of different models to represent the same thing and the same model to represent different things.

This is evident, for example, when students:
s choose a mathematical model to predict the distance a car will travel at a given speed in a given time.
s use a computer simulation to observe the process of growing vegetables or to test the performance of cars.
s compare the relative merits of using a flat map or a globe to model where places are situated on Earth.
s use blueprints or scale models to represent room plans.

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Intermediate Magnitude and Scale

3. The grouping of magnitudes of size, time, frequency, and pressures or other units of measurement into a series of relative order provides a useful way to deal with the immense range and the changes in scale that affect the behavior and design of systems.

Students:

• cite examples of how different aspects of natural and designed systems change at different rates with changes in scale.
• use powers of ten notation to represent very small and very large numbers.

This is evident, for example, when students:
s demonstrate that a large container of hot water (more volume) cools off more slowly than a small container (less volume).
s compare the very low frequencies (60 Hertz AC or 6 x 10 Hertz) to the mid-range frequencies (10 Hertz-FM radio) to the higher frequencies (10 15 Hertz) of the electromagnetic spectrum.

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Intermediate Patterns of Change

4. Equilibrium is a state of stability due either to a lack of changes (static equilibrium) or a balance between opposing forces (dynamic equilibrium).

Students:

• describe how feedback mechanisms are used in both designed and natural systems to keep changes within desired limits.
• describe changes within equilibrium cycles in terms of frequency or cycle length and determine the highest and lowest values and when they occur.

This is evident, for example, when students:
s compare the feedback mechanisms used to keep a house at a constant temperature to those used by the human body to maintain a constant temperature.
s analyze the data for the number of hours of sunlight from the shortest day to the longest day of the year.

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Intermediate Equilibrium and Stability

5. Identifying patterns of change is necessary for making predictions about future behavior and conditions.

Students:

• use simple linear equations to represent how a parameter changes with time.
• observe patterns of change in trends or cycles and make predictions on what might happen in the future.

This is evident, for example, when students:
s study how distance changes with time for a car traveling at a constant speed.
s use a graph of a population over time to predict future population levels.

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Intermediate Optimization

6. In order to arrive at the best solution that meets criteria within constraints, it is often necessary to make trade-offs.

Students:

• determine the criteria and constraints and make trade-offs to determine the best decision.
• use graphs of information for a decision making problem to determine the optimum solution.

This is evident, for example, when students:
s choose components for a home stereo system.
s determine the best dimensions for fencing in the maximum area.

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Commencement Systems Thinking

1. Through systems thinking, people can recognize the commonalities that exist among all systems and how parts of a system interrelate and combine to perform specific functions.

Students:

• explain how positive feedback and negative feedback have opposite effects on system outputs.
• use an input-process-output-feedback diagram to model and compare the behavior of natural and engineered systems.
• define boundary conditions when doing systems analysis to determine what influences a system and how it behaves.

This is evident, for example, when students:
s describe how negative feedback is used to control loudness automatically in a stereo system and how positive feedback from loudspeaker to microphone results in louder and louder squeals.

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Commencement Models

2. Models are simplified representations of objects, structures, or systems used in analysis, explanation, interpretation, or design.

Students:

• revise a model to create a more complete or improved representation of the system.
• collect information about the behavior of a system and use modeling tools to represent the operation of the system.
• find and use mathematical models that behave in the same manner as the processes under investigation.
• compare predictions to actual observations using test models.

This is evident, for example, when students:
s add new parameters to an existing spreadsheet model.
s incorporate new design features in a CAD drawing.
s use computer simulation software to create a model of a system under stress, such as a city or an ecosystem.
s design and construct a prototype to test the performance of a temperature control system.
s use mathematical models for scientific laws, such as Hooke’s Law or Newton’s Laws, and relate them to the function of technological systems, such as an automotive suspension system.
s use sinusoidal functions to study systems that exhibit periodic behavior.
s compare actual populations of animals to the numbers predicted by predator/ prey computer simulations.

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Commencement Magnitude and Scale

3. The grouping of magnitudes of size, time, frequency, and pressures or other units of measurement into a series of relative order provides a useful way to deal with the immense range and the changes in scale that affect the behavior and design of systems.

Students:

• describe the effects of changes in scale on the functioning of physical, biological, or designed systems.
• extend their use of powers of ten notation to understanding the exponential function and performing operations with exponential factors.

This is evident, for example, when students:
s explain that an increase in the size of an animal or a structure requires larger supports (legs or columns) because of the greater volume or weight.
s use the relationship that v=f l to determine wave length when given the frequency of an FM radio wave, such as 100.0 megahertz (1.1 x 10 8 Hertz), and velocity of light or EM waves as 3 x 10 8m/sec can.

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Commencement Patterns of Change

4. Equilibrium is a state of stability due either to a lack of changes (static equilibrium) or a balance between opposing forces (dynamic equilibrium).

Students:

• describe specific instances of how disturbances might affect a system’s equilibrium, from small disturbances that do not upset the equilibrium to larger disturbances (threshold level) that cause the system to become unstable.
• cite specific examples of how dynamic equilibrium is achieved by equality of change in opposing directions.

This is evident, for example, when students:
s use mathematical models to predict under what conditions the spread of a disease will become epidemic.
s document the range of external temperatures in which warm-blooded animals can maintain a relatively constant internal temperature and identify the extremes of cold or heat that will cause death.
s experiment with chemical or biological processes when the flow of materials in one way direction is counter-balanced by the flow of materials in the opposite direction.

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Commencement Equilibrium and Stability

5. Identifying patterns of change is necessary for making predictions about future behavior and conditions.

Students:

• use sophisticated mathematical models, such as graphs and equations of various algebraic or trigonometric functions.
• search for multiple trends when analyzing data for patterns, and identify data that do not fit the trends.

This is evident, for example, when students:
s use a sine pattern to model the property of a sound or electromagnetic wave.
s use graphs or equations to model exponential growth of money or populations.
s explore historical data to determine whether the growth of a parameter is linear or exponential or both.

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Commencement Optimization

6. In order to arrive at the best solution that meets criteria within constraints, it is often necessary to make trade-offs.

Students:

• use optimization techniques, such as linear programming, to determine optimum solutions to problems that can be solved using quantitative methods.
• analyze subjective decision making problems to explain the trade-offs that can be made to arrive at the best solution.

This is evident, for example, when students:
s use linear programming to figure the optimum diet for farm animals.
s evaluate alternative proposals for providing people with more access to mass transportation systems.

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