Connections

__**Connections Standard:**__

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

The benchmarks for **mathematical processes** articulate what students should demonstrate in problem solving, representation, communication, reasoning and connections at key points in their mathematics program. Specific grade-level indicators have not been included for the mathematical processes standard because content and processes should be interconnected at the indicator level. Therefore, mathematical processes have been embedded within the grade-level indicators for the five content standards.

**By the end of the K-2 program:** A. Use a variety of strategies to understand problem situations; e.g., discussing with peers, stating problems in own words, modeling problems with diagrams or physical materials, identifying a pattern. B. Identify and restate in own words the question or problem and the information needed to solve the problem. C. Generate alternative strategies to solve problems. D. Evaluate the reasonableness of predictions, estimations and solutions. E. Explain to others how a problem was solved. F. Draw pictures and use physical models to represent problem situations and solutions. G. Use invented and conventional symbols and common language to describe a problem situation and solution. H. Recognize the mathematical meaning of common words and phrases, and relate everyday language to mathematical language and symbols. I. Communicate mathematical thinking by using everyday language and appropriate mathematical language.

**By the end of the 3-4 program:** A. Apply and justify the use of a variety of problem-solving strategies; e.g., make an organized list, guess and check. B. Use an organized approach and appropriate strategies to solve multi-step problems. C. Interpret results in the context of the problem being solved; e.g., the solution must be a whole number of buses when determining the number of buses necessary to transport students. D. Use mathematical strategies to solve problems that relate to other curriculum areas and the real world; e.g., use a timeline to sequence events; use symmetry in artwork. E. Link concepts to procedures and to symbolic notation; e.g., model 3 x 4 with a geometric array, represent one-third by dividing an object into three equal parts. F. Recognize relationships among different topics within mathematics; e.g., the length of an object can be represented by a number. G. Use reasoning skills to determine and explain the reasonableness of a solution with respect to the problem situation. H. Recognize basic valid and invalid arguments, and use examples and counter examples, models, number relationships, and logic to support or refute. I. Represent problem situations in a variety of forms (physical model, diagram, in words or symbols), and recognize when some ways of representing a problem may be more helpful than others. J. Read, interpret, discuss and write about mathematical ideas and concepts using both everyday and mathematical language. K. Use mathematical language to explain and justify mathematical ideas, strategies and solutions.

A. Clarify problem-solving situation and identify potential solution processes; e.g., consider different strategies and approaches to a problem, restate problem from various perspectives. B. Apply and adapt problem-solving strategies to solve a variety of problems, including unfamiliar and non-routine problem situations. C. Use more than one strategy to solve a problem, and recognize there are advantages associated with various methods. D. Recognize whether an estimate or an exact solution is appropriate for a given problem situation. E. Use deductive thinking to construct informal arguments to support reasoning and to justify solutions to problems. F. Use inductive thinking to generalize a pattern of observations for particular cases, make conjectures, and provide supporting arguments for conjectures. G. Relate mathematical ideas to one another and to other content areas; e.g., use area models for adding fractions, interpret graphs in reading, science and social studies. H. Use representations to organize and communicate mathematical thinking and problem solutions. I. Select, apply, and translate among mathematical representations to solve problems; e.g., representing a number as a fraction, decimal or percent as appropriate for a problem. J. Communicate mathematical thinking to others and analyze the mathematical thinking and strategies of others. K. Recognize and use mathematical language and symbols when reading, writing and conversing with others.
 * By the end of the 5-7 program:**

A. Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution. B. Apply mathematical knowledge and skills routinely in other content areas and practical situations. C. Recognize and use connections between equivalent representations and related procedures for a mathematical concept; e.g., zero of a function and the x-intercept of the graph of the function, apply proportional thinking when measuring, describing functions, and comparing probabilities. D. Apply reasoning processes and skills to construct logical verifications or counter-examples to test conjectures and to justify and defend algorithms and solutions. E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas. F. Use precise mathematical language and notations to represent problem situations and mathematical ideas. G. Write clearly and coherently about mathematical thinking and ideas. H. Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner.
 * By the end of the 8-10 program:**

A. Construct algorithms for multi-step and non-routine problems. B. Construct logical verifications or counterexamples to test conjectures and to justify or refute algorithms and solutions to problems. C. Assess the adequacy and reliability of information available to solve a problem. D. Select and use various types of reasoning and methods of proof. E. Evaluate a mathematical argument and use reasoning and logic to judge its validity. F. Present complete and convincing arguments and justifications, using inductive and deductive reasoning, adapted to be effective for various audiences. G. Understand the difference between a statement that is verified by mathematical proof, such as a theorem, and one that is verified empirically using examples or data. H. Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations. I. Communicate mathematical ideas orally and in writing with a clear purpose and appropriate for a specific audience. J. Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation. media type="youtube" key="TMs0FTb1XPg" height="315" width="560"
 * By the end of the 11-12 program:**