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Big Idea:
Numeric reasoning involves fluency and facility with numbers. |
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4.1 A.
Number Sense |
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Descriptive Statement: Number sense is an intuitive feel
for numbers and a common sense approach to using them. It is a
comfort with what numbers represent that comes from investigating
their characteristics and using them in diverse situations. It
involves an understanding of how different types of numbers, such as
fractions and decimals, are related to each other, and how each can
best be used to describe a particular situation. It subsumes the
more traditional category of school mathematics curriculum called
numeration and thus includes the important concepts of place value,
number base, magnitude, and approximation and estimation. |
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Essential Questions |
Enduring Understandings |
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- How do mathematical ideas interconnect and
build on one another to produce a coherent whole? (4.5C1; 4.5C6)
- How can we compare and contrast numbers? (4.5A4)
- How can counting, measuring, or labeling help to make sense of the
world around us? |
- One representation may sometimes be more helpful than another;
and, used together, multiple representations give a fuller
understanding of a problem.
- A quantity can be represented numerically in various ways. Problem
solving depends upon choosing wise ways.
- Numeric fluency includes both the understanding of and the ability
to appropriately use numbers. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Extend
understanding of the number system to all real numbers. |
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2.
Compare and order rational and irrational numbers. |
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3.
Develop
conjectures and informal proofs of properties of number systems and sets of
numbers. |
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4.1 B. Numerical Operations |
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Descriptive Statement: Numerical Operations are an
essential part of the mathematics curriculum, especially in the
elementary grades. Students must be able to select and apply various
computational methods, including mental math, pencil-and-paper
techniques, and the use of calculators. Students must understand how
to add, subtract, multiply, and divide whole numbers, fractions,
decimals, and other kinds of numbers. With the availability of
calculators that perform these operations quickly and accurately,
the instructional emphasis now is on understanding the meanings and
uses of these operations, and on estimation and mental skills,
rather than solely on the development of paper-and-pencil
proficiency. |
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Essential Questions |
Enduring Understandings |
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- What makes a computational strategy both
effective and efficient? (4.5D1)
- How do operations affect numbers?
- How do mathematical representations reflect the needs of society
across cultures? (An essential question with broad applicability
across multiple standards) (4.5C5) |
- Computational fluency includes
understanding not only the meaning, but also the appropriate use of
numerical operations.
- The magnitude of numbers affects the outcome of operations on
them.
- In many cases, there are multiple algorithms for finding a
mathematical solution, and those algorithms are frequently
associated with different cultures. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Extend understanding and use of operations to real numbers and
algebraic procedures. |
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2.
Develop, apply, and explain methods for solving problems involving
rational and negative exponents. |
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3.
Perform operations
on matrices.
·
Addition and
subtraction
·
Scalar
multiplication
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4.
Understand and
apply the laws of exponents to simplify expressions involving numbers raised to
powers. |
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4.1 C. Estimation |
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Descriptive Statement: Estimation is a process that is
used constantly by mathematically capable adults, and one that can
be easily mastered by children. It involves an educated guess about
a quantity or an intelligent prediction of the outcome of a
computation. The growing use of calculators makes it more important
than ever that students know when a computed answer is reasonable;
the best way to make that determination is through the use of strong
estimation skills. Equally important is an awareness of the many
situations in which an approximate answer is as good as, or even
preferable to, an exact one. Students can learn to make these
judgments and use mathematics more powerfully as a result. |
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Essential Questions |
Enduring Understandings |
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- How can we decide when to use an exact
answer and when to use an estimate? |
- Context is critical when using
estimation. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Recognize the
limitations of estimation, assess the amount of error resulting from estimation,
and determine whether the error is within acceptable tolerance limits. |
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