Standard 4.1 Number and Numerical Operations

All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.

Big Idea: Numeric reasoning involves fluency and facility with numbers.

4.1.8 A. Number Sense

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.

Essential Questions

Enduring Understandings

- 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.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

1.         Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well):

·        Rational numbers

·        Percents

·        Exponents

·        Roots

·        Absolute values

·        Numbers represented in scientific notation

3.         Understand and use ratios, rates,  proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.

• The word “rates” was added to this CPI by the State Board of Education on January 9, 2008. This is an area of focus in grade 8 and should be linked to the concept of slope (4.3.8B1).

6.         Recognize that repeating decimals correspond to fractions and determine their fractional equivalents.

·        5/7 = 0. 714285714285…  =  0.

Sample ECR Item: With only a DVD and a piece of string, explain how you could approximate the value of π.

4.1.8 B. Numerical Operations

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.

Essential Questions

Enduring Understandings

-  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.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

1.         Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

·        Pencil-and-paper

·        Mental math

·        Calculator

Pat has 1296 one-inch square tiles. Which of the following are the dimensions of the largest square table top Pat could cover with the tiles? a. 324 in. x 324 in.

b. 9 ft x 9 ft

* c. 1 yd x 1 yd

d. 36 m x 36 m

4.1.8 C. Estimation

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.

Essential Questions

Enduring Understandings

-  How can we decide when to use an exact answer and when to use an estimate?

-  Context is critical when using estimation.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples