| Problem Solving |
- Tools and strategies are strategically selected and used to solve particular applications.
- Reflection on the process and reasonableness of the solution moves students from the symbolic to the practical.
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- How do I know where to begin when solving a problem?
- What are the characteristics of a successful problem solver?
- What is the best method or technique for working toward a solution?
- What is the meaning of the solution and does it make sense?
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| Connections |
- Connections exist within mathematical concepts and can broaden understanding of the world.
- Mathematical ideas interconnect and build on one another to produce a coherent whole.
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- Where do mathematical ideas surface in the world?
- How can I use mathematical knowledge in different contexts?
- How is mathematics a coherent discipline that contains internal structure and interconnectedness?
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| Reasoning |
- Abstract and quantitative reasoning are required to process mathematical information and solve problems.
- Mathematical conjectures are developed and investigated through observing patterns.
- Sound reasoning requires the ability to distinguish between valid and invalid arguments and to critique the reasoning of others.
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- How do I select appropriate reasoning tools to progress mathematically?
- What is a possible approach? What happens if I follow that approach?
- Is my / their reasoning sound? What does a well-reasoned argument entail?
- What constitutes proof?
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| Communication |
- Mathematical ideas must be communicated clearly in written, visual, or oral form.
- Communication of mathematical thinking should demonstrate clear and concise organization.
- Mathematical language can be used to express ideas symbolically, numerically, and graphically.
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- How do I communicate mathematical ideas clearly?
- How can I organize and consolidate my thinking through communication?
- How can mathematics be perceived as a language? What terms, notations, and representations most accurately and succinctly convey my ideas?
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| Representation |
- Symbols, graphs, pictures, and tables can be used to represent real situations.
- Flexibility in one’s ability to read and interpret various forms is important in understanding problems and solutions.
- Various mathematical representations are useful for problem solving and communicating a solution.
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- What representation best illuminates this relationship?
- How do I move between mathematical abstraction and physical reality?
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| Number and Operations |
- Understanding numbers, their representations, properties, and relationships assist in higher level thinking.
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- How does the context of a problem dictate the number system that should be used?
- What operation is most appropriate given the context of the problem?
- When is a rough estimate, an approximation, or an exact answer suitable for a solution to a problem?
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| Algebraic Understanding |
- Patterns, relations, and functions are mathematical ways to describe connectedness and dependence.
- Mathematical situations and structures can be represented and analyzed using symbols to advance algebraic thinking.
- Mathematical models can be used to represent and understand quantitative relationships.
- Change can be modeled in a variety of mathematically ways.
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- How can I generalize patterns, describe relationships, and analyze functions?
- How do I use the tools of symbolic algebra to judge the reasonableness of mathematical representations?
- How is the idea of change analyzed?
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Geometric
Understanding and
Measurement |
- Two and three dimensional shapes have properties and relationships similar to each other.
- Coordinate geometry can be used to describe spatial relationships and location.
- The study of transformations and symmetry provides a deeper understanding of physical change.
- Visualization, spatial reasoning, and geometric modeling are strategies to enhance problem solving.
- In order to assign numerical values to spatial and physical attributes, objects can be measured using appropriate systems, units, and processes.
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- How do the tools of geometry such as definitions, theorems, and properties foster an increasing ability to spatially visualize and logically deduce conclusions?
- What is the relationship between units and physical quantity?
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| Data Analysis and Probability |
- Data collection and its organization helps formulate relevant questions that can be answered using mathematical tools.
- Selection of the appropriate statistical method to
- analyze data will progress students toward solutions and subsequent inferences.
- A study of probability helps illuminate the randomness of our everyday world.
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- What types of questions can be answered by analyzing data?
- When is data analysis valid and what is its purpose?
- How can I become a critical interpreter of data?
- How does probability relate to the real world?
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