By the end of grade seven, students are adept at manipulating numbers and equations and understand the general principles at work. Students understand and use factoring of numerators and denominators and properties of exponents. They know the Pythagorean theorem and solve problems in which they compute the length of an unknown side. Students know how to compute the surface area and volume of basic three-dimensional objects and understand how area and volume change with a change in scale. Students make conversions between different units of measurement. They know and use different representations of fractional numbers (fractions, decimals, and percents) and are proficient at changing from one to another. They increase their facility with ratio and proportion, compute percents of increase and decrease, and compute simple and compound interest. They graph linear functions and understand the idea of slope and its relation to ratio.

• 1.0 Students make decisions about how to approach problems:

  1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.

  1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.

  1.3 Determine when and how to break a problem into simpler parts.

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• 2.0 Students use strategies, skills, and concepts in finding solutions:

  2.1 Use estimation to verify the reasonableness of calculated results.

  2.2 Apply strategies and results from simpler problems to more complex problems.
  

  2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.

  2.4 Make and test conjectures by using both inductive and deductive reasoning.

  2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

  2.6 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.

  2.7 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.

  2.8 Make precise calculations and check the validity of the results from the context of the problem.

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• 3.0 Students determine a solution is complete and move beyond a particular problem by generalizing to other situations:

  3.1 Evaluate the reasonableness of the solution in the context of the original situation.

  3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.

  3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations.

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