  # Number and Operations

N&0 – 20 Relative magnitude: Relative magnitude is the property of relative size. It
refers to the relative size of numbers, objects, distances, brightness, and other things that
can be quantified. The GLEs specifically require students to demonstrate understanding
of the relative magnitude of different types of numbers by comparing, ordering, and
identifying equivalent forms of numbers within and across number formats using models,
number lines, equality (=) and inequality symbols (≠, ≤, ≥, >, <), and explanations. (See

Table 1: Number Formats by Grade for Demonstrating Relative Magnitude

 Grade Number Formats Within Number Formats Across Number Formats 2 o Whole numbers 0 to 199 3 o Whole numbers 0 to 999 o Positive fractions (halves, thirds, or fourths) 4 o Whole numbers 0 to 999,999 o Positive fractions (halves, thirds, fourths, fifths, sixths, eighths, or tenths) o Decimals (to hundredths place) 5 o Whole numbers 0 to 9,999,999 o Positive fractions (halves, fourths, eighths, thirds, fourths, sixths, twelfths, fifths, or powers of 10) o Decimals (to thousandths) o Benchmark percents (10%, 25%, 50%, 75%, or 100%) o Integers in context 6 o Numbers with whole number bases and whole number exponents (e.g., 34 compared to 43) o Integers o Rational numbers (fractions, decimals, whole number percents from 1 – 100%)  7 o Numbers with whole number bases and whole number exponents (e.g., 34 compared to 43) o Integers o Rational numbers o Absolute values o Numbers in scientific notation  8 o Numbers with whole number of fractional bases and whole number exponents (e.g., 34 compared to ) o Integers o Rational numbers o Absolute values o Square roots o Numbers in scientific notation o Common irrational numbers (e.g., , π )  N&0 – 21 Within number formats: To compare numbers within number formats means
to compare whole numbers to whole numbers, fractions to fractions, decimals to
decimals
, and so on.

N&0 – 22 Across number formats: To compare numbers across number formats means
to compare whole numbers to fractions, fractions to decimals, decimals to percents, and
so on.

Rational numbers: (See N&0 – 1.)

Whole Numbers: (See N&0 – 2.)

Positive fractions: (See N&0 – 3, N&0 – 4, N&0 – 5, N&0 – 6.)

Decimals: (See N&0 – 7.)

Percents: (See N&0 – 8.)

Integers: (See N&0 – 9.)

Real Numbers: (See N&0 – 11.)

Irrational Numbers: (See N&0 – 10.)

N&0 – 23 Absolute value: The absolute value of a real number is the distance
between 0 and the number on the number line. The absolute value of a real number x is
written as . The absolute value of a non- negative number (a number greater than or
equal to zero ) is the number (e.g., since the number 5 is a distance of five units
from on the number line). The absolute value of a negative number is the opposite of the
number (e.g., since the number – 5 is five units from 0 on the number line).

N&0 – 24 Whole number bases and whole number exponents, and fractional bases
with whole number exponents:
A whole number exponent is a whole number (See
N&O – 2) that indicates repeated multiplication of the same number. The number being
multiplied is called the base. The exponent is typically written to the right of the base and
slightly raised. The exponent indicates how many times the base is used as a factor (See
N&O – 38).

Example 24.1 – Whole number base and whole number exponent: Example 24.2 – Fractional base with whole number exponent: N&0 – 25 Scientific notation: Scientific notation is a way of representing very large
or very small numbers as the product of a number , n, and a power of 10, where the
absolute value of n is greater than or equal to 1 and less than 10.

Examples 25.1: N&0 – 26 Ordering: Ordering numbers means placing the numbers in numerical order
from the least to the greatest or from the greatest to the least.

Example 26.1:

Order the following numbers from the least to the greatest: Note: students may be asked to provide an explanation or place the numbers on a number line.

Answer: N&0 – 27 Comparing: Comparing numbers, a and b, means to determine if a is less
than b, if a is greater than b, or if a is equal to b.

Example 27.1: Alisa is placing on the number line below . Between which two
numbers should she place ? Answer: should be placed between and 1 because is greater than and less than 1.

N&0 – 28 Number line: A number line is a line where every real number corresponds
to a unique point on the line. Thus, if two numbers correspond to the same point on the
line the numbers are equivalent (See N&O – 14).

Examples 28.1 – Horizontal number lines: Example 28.2 – Using a vertical number line to locate integers in context:

At what temperature is the arrow pointing? Answer: The arrow is pointing at – 5°F.

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