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Mathematics Fundamental Process

Doing and understanding mathematics fosters growth in the following fundamental processes:

Big Ideas Enduring Understandings Essential Questions
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.
  • 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? 
  • 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. 
  • 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? 
  • 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. 
  • 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? 
  • 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. 
  • 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? 
  • 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. 
  • What representation best illuminates this relationship?
  • How do I move between mathematical abstraction and physical reality? 
Number and Operations
  • Understanding numbers, their representations, properties, and relationships assist in higher level thinking. 
  • 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? 
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. 
  • 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?
Understanding and
  • 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. 
  • 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? 
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. 
  • 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?