By the end of 5th grade, students should know that:

1.       In something that consists of many parts, the parts usually influence one another.

2.       Something may not work as well (or at all) if a part of it is missing, broken, worn out, mismatched, or misconnected.

By the end of 5th grade, students should know that:

1.       Seeing how a model works after changes are made to it may suggest how the real thing would work if the same were done to it.

2.       Geometric figures, number sequences, graphs, diagrams, sketches, number lines, maps, and stories can be used to represent objects, events, and processes in the real world, although such representations can never be exact in every detail.

By the end of 5th grade, students should know that:

1.       Some features of things may stay the same even when other features change.

2.       Things change in steady, repetitive, or irregular ways or sometimes in more than one way at the same time.

By the end of 5th grade, students should know that:

1.       Almost anything has limits on how big or small it can be.

2.       Finding out what the biggest and the smallest possible values of something are is often as revealing as knowing what the usual value is.

By the end of 5th grade, students should know that:

1.       Numbers and shapes and operations on them help to describe and predict things about the world around us.

2.       In using mathematics, choices have to be made about what operations will give the best results.

By the end of 5th grade, students should know that:

1.       Throughout all of history, people everywhere have invented and used tools.

2.       In earlier times, the accumulated information and techniques of each generation of workers were taught on the job directly to the next generation of workers. Today, the knowledge base for technology can be found as well in libraries of print and electronic resources and is often taught in the classroom.

3.       Measuring instruments can be used to gather accurate information for making scientific comparisons of objects and events and for designing and constructing things that will work properly.

4.       Technology extends the ability of people to change the world: to cut, shape, or put together materials; to move things from one place to another; and to reach farther with their hands, voices, senses, and minds.

By the end of 8th grade, students should know that:

1.       Usually there is no one right way to solve a mathematical problem; different methods have different advantages and disadvantages.

2.       Logical connections can be found between different parts of mathematics.

By the end of 8th grade, students should know that:

1.       Mathematics is helpful in almost every kind of human endeavor—from laying bricks to prescribing medicine or drawing a face. In particular, mathematics has contributed to progress in science and technology for thousands of years and still continues to do so.

By the end of 8th grade, students should know that:

1.       Mathematicians often represent things with abstract ideas, such as numbers or perfectly straight lines, and then work with those ideas alone.

2.       When mathematicians use logical rules to work with representations of things, the results may or may not be valid for the things themselves.

By the end of 8th grade, students should know that:

1.       A system can include processes as well as things.

2.       Thinking about things as systems means looking for how every part relates to others.

3.       Any system is usually connected to other systems, both internally and externally.

By the end of 8th grade, students should know that:

1.       Models are often used to think about processes that happen too slowly, too quickly, or on too small a scale to observe directly, or that are too vast to be changed deliberately, or that are potentially dangerous.

2.       Mathematical models can be displayed on a computer and then modified to see what happens.

3.       Different models can be used to represent the same thing.

By the end of 8th grade, students should know that:

1.       Physical and biological systems tend to change until they become stable and then remain that way unless their surroundings change.

2.       A system may stay the same because nothing is happening or because things are happening but exactly counterbalance one another.

3.       Many systems contain feedback mechanisms that serve to keep changes within specified limits.

4.       Symbolic equations can be used to summarize how the quantity of something changes over time or in response to other changes.

5.       Symmetry (or the lack of it) may determine properties of many objects, from molecules and crystals to organisms and designed structures.

6.       Cycles, such as the seasons or body temperature, can be described by their cycle length or frequency, what their highest and lowest values are, and when these values occur.

By the end of 8th grade, students should know that:

1.       Properties of systems that depend on volume, such as capacity and weight, change out of proportion to properties that depend on area, such as strength or surface processes.

2.       As the complexity of any system increases, gaining an understanding of it depends increasingly on summaries, such as averages and ranges, and on descriptions of typical examples of that system.

By the end of 8th grade, students should know that:

1.       In earlier times, the accumulated information and techniques of each generation of workers were taught on the job directly to the next generation of workers.

2.       Technology is essential to science for such purposes as access to outer space and other remote locations, sample collection and treatment, measurement, data collection and storage, computation, and communication of information.

3.       Engineers, architects, and others who engage in design and technology use scientific knowledge to solve practical problems.

By the end of 12th grade, students should know that:

1.       A system usually has some properties that are different from those of its parts, but appear because of the interaction of those parts.

2.       Understanding how things work and designing solutions to problems of almost any kind can be facilitated by systems analysis.

3.       The successful operation of a designed system usually involves feedback.

4.       Even in some very simple systems, it may not always be possible to predict accurately the result of changing some part or connection.

By the end of 12th grade, students should know that:

1.       The basic idea of mathematical modeling is to find a mathematical relationship that behaves in the same ways as the objects or processes under investigation.

2.       Computers have greatly improved the power and use of mathematical models by performing computations that are very long, very complicated, or repetitive.

3.       The usefulness of a model can be tested by comparing its predictions to actual observations in the real world.

By the end of 12th grade, students should know that:

1.       A system in equilibrium may return to the same state of equilibrium if the disturbances it experiences are small.

2.       Along with the theory of atoms, the concept of the conservation of matter led to revolutionary advances in chemical science.

3.       Things can change in detail but remain the same in general (the players change, but the team remains; cells are replaced, but the organism remains).

4.       Graphs and equations are useful (and often equivalent) ways for depicting and analyzing patterns of change.

5.       In many physical, biological, and social systems, changes in one direction tend to produce opposing (but somewhat delayed) influences, leading to repetitive cycles of behavior.

6.       In evolutionary change, the present arises from the materials and forms of the past, more or less gradually, and in ways that can be explained.

7.       Most systems above the molecular level involve so many parts and forces and are so sensitive to tiny differences in conditions that their precise behavior is unpredictable, even if all the rules for change are known.

By the end of 12th grade, students should know that:

1.       Representing large numbers in terms of powers of ten makes it easier to think about them and to compare things that are greatly different.

2.       Because different properties are not affected to the same degree by changes in scale, large changes in scale typically change the way that things work in physical, biological, or social systems.

3.       As the number of parts of a system increases, the number of possible interactions between pairs of parts increases much more rapidly.

By the end of 12th grade, students should know that:

1.       Mathematics is the study of any patterns or relationships, whereas natural science is concerned only with those patterns that are relevant to the observable world.

2.       As in other sciences, simplicity is one of the highest values in mathematics.

3.       Theories and applications in mathematical work influence each other.

4.       New mathematics continues to be invented, and connections between different parts of mathematics continue to be found.

By the end of 12th grade, students should know that:

1.       Mathematical modeling aids in technological design by simulating how a proposed system would theoretically behave.

2.       Mathematics and science as enterprises share many values and features: belief in order, ideals of honesty and openness, the importance of criticism by colleagues, and the essential role played by imagination.

3.       Mathematics provides a precise language for science and technology—to describe objects and events, to characterize relationships between variables, and to argue logically.

4.       Developments in science or technology often stimulate innovations in mathematics by presenting new kinds of problems to be solved.

5.       Developments in mathematics often stimulate innovations in science and technology.

By the end of 12th grade, students should know that:

1.       Some work in mathematics is much like a game mathematicians choose an interesting set of rules and then play according to those rules to see what can happen.

2.       Much of the work of mathematicians involves a modeling cycle, which consists of three steps: (1) using abstractions to represent things or ideas, (2) manipulating the abstractions according to some logical rules, and (3) checking how well the results match the original things or ideas.

By the end of 12th grade, students should know that:

1.       Technological problems often create a demand for new scientific knowledge, and new technologies make it possible for scientists to extend their research in new ways or to undertake entirely new lines of research.

2.       Mathematics, creativity, logic and originality are all needed to improve technology.

3.       Technology usually affects society more directly than science because it solves practical problems and serves human needs (and may create new problems and needs).

Instructional programs from prekindergarten through grade 12 should enable all students to--

• formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them;
• select and use appropriate statistical methods to analyze data;
• develop and evaluate inferences and predictions that are based on data;
• understand and apply basic concepts of probability.

Instructional programs from prekindergarten through grade 12 should enable all students to--

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.