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the arithmetic of numbers, and the algebraic properties that the systems enjoy. In particular, it is
useful to understand how progressively more expansive systems of numbers enable the solution of
progressively more complicated equations. This lays the foundation for enabling students to see
more evidence of mathematics in the real world in as they learn more mathematics.
51. To use quantification efficiently, one has to be able to apply not just numbers, but the number
systems. Numbers themselves are of limited relevance; what makes them into a powerful tool are
the operations that we can perform with them. As such, a good understanding of the operations of
numbers is the foundation of mathematical reasoning.
52. It is also important to understand matters of representation (as symbols involving numerals, as
points on a number line, as geometric quantities, and by special symbols such as
π) and how to
move between them; the ways in which these representations are affected by number systems; the
ways in which algebraic properties of these systems are relevant and matter for operating within
the systems; and the significance of the additive and multiplicative identities, associativity,
commutativity, and the distributive property of multiplication over addition. Algebraic principles
undergird the place value system, allowing for economical expression of numbers and efficient
approaches to operations on them. They are also central to number-line based operations with
numbers, including work with additive inverses that are central to addition and subtraction of first
integers, then rationals and finally reals.
53. The centrality of number as a key concept in all the other mathematical areas under
consideration here and to mathematical reasoning itself, is undeniable. Students’ grasp of the
algebraic principles and properties first experienced through work with numbers is fundamental to
their understanding of the concepts of secondary school algebra, along with their ability to become
fluent in the manipulations of algebraic expressions necessary for solving equations, setting up
models, graphing functions, and programing and making spreadsheet
formulas. And in today’s
data-intensive world, facility with interpretation of patterns of numbers, comparison of patterns, and
other numerical skills are evolving in importance.
54.
A broad understanding of quantity and number systems supports reasoning in the real-world
applications of mathematics envisaged by this framework.
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