1st Six Weeks Unit 2
--
6th grade Math
Ordering Fractions, Decimals and Integers
5 Days
TEKS for the Unit
(Readiness TEKs in Bold)
6.2A Number and operations--classify whole numbers, integers, and rational numbers using a visual representation
such as a Venn diagram to describe relationships between sets of numbers.
6.2B Number and operations--identify a number, its opposite, and its absolute value.
6.2C Number and operations--locate, compare, and order integers and rational numbers using a number line.
6.2D Number and operations--order a set of rational numbers arising from mathematical and real-world
contexts.
6.4G Proportionality--generate equivalent forms of fractions, decimals, and percents using real-world
problems, including problems that involve money. (repeated from Unit 1, used when comparing numbers
to find equivalent forms)
The following sets of numbers are used in this unit:
o Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of
one each time {1, 2, 3, ..., n}
o Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
o Integers – the set of counting (natural) numbers, their opposites, and zero
{–n, …, –3, –2, –1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.
a
o Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and
b
b ≠ 0, which includes the subsets of integers, whole numbers, and counting (natural) numbers
1 11 4
(e.g., –3, 0, 2, 1.35, 15%, - ,
,1 , 0 .23 , etc.). The set of rational numbers is denoted by the
2 7
7
symbol Q.
Note:
 Gr 4 represented fractions and decimals to the tenths or hundredths as distances from zero on a number line.
 Gr 3 orders only whole numbers, Gr 4 compares 2 positive fractions, Gr 5 compares 2 positive decimals.
 Gr 6 needs to teach how to compare mixed positive rationals, then integers, then all rationals.
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Supporting TEKS 6.2A
6.2A Number and operations. The student applies
mathematical process standards to represent and use
rational numbers in a variety of forms.
The student is expected to classify whole
numbers, integers, and rational numbers using a
visual representation such as a Venn diagram to
describe relationships between sets of numbers.

A Venn diagram is an applicable visual
representation as the SE focuses on classification of
numbers.
As there is no unified definition for these terms, the
natural numbers will be taken to mean {1, 2, 3 . . .},
and the whole numbers will be taken to mean {0, 1,
2, 3 . . .}.
Visual representations of the relationships between sets and subsets of rational numbers
 RELATIONSHIPS BETWEEN SETS OF NUMBERS, including, but not limited to:
o All counting (natural) numbers are a subset of whole numbers, integers, and rational numbers.
 Ex: Two is a counting (natural) number, whole number, integer, and rational number.
o All whole numbers are a subset of integers and rational numbers.
 Ex: Zero is a whole number, integer, and rational number, but not a counting (natural) number.
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o All integers are a subset of rational numbers.
 Ex: Negative two is an integer and rational number, but neither a whole number nor a counting (natural)
number.
o All counting (natural) numbers, whole numbers, and integers are a subset of rational numbers.
 Ex: Four is a counting (natural) number, whole number, integer, and rational number.
o Not all rational numbers are an integer, whole number, or counting (natural) number.
 Ex: One-half is a rational number but not an integer, whole number, or counting (natural) number.
o Terminating and repeating decimals are rational numbers but not integers, whole numbers, or counting
(natural) numbers.
 Ex: 1 .3 is a repeating decimal; therefore, it is a rational number.
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Supporting TEKS 6.2B
6.2B Number and operations.
The student applies mathematical
process standards to represent and
use rational numbers in a variety of
forms.
The student is expected to
identify a number, its opposite,
and its absolute value.
This SE may be used to introduce the concept of integers with the
identification of a number, its opposite, and its absolute value.
When 6(2)(B) is paired with 6(1)(A), students may be expected to apply
the skill of identifying integers in everyday life.
The SE includes the use of the absolute value symbol and the formal
mathematics vocabulary as students identify a number and its opposite as
being the same distance from zero, or having the same absolute value.
The term “opposite” refers to the additive inverse of a number.
 Numbers
o Positive numbers are to the right of zero on a horizontal number line and above zero on a vertical number
line.
 Represented with a (+) symbol or no symbol at all
o Negative numbers are to the left of zero on a horizontal number line and below zero on a vertical number
line.
 Represented with a (–) symbol
o Zero is neither positive nor negative.
 Quantities from mathematical and real-world problem situations are represented with positive and negative
numbers.
o Ex: above/below, ascend/descend, credit/debit, deposit/withdrawal, forward/backward, gain/loss,
increase/decrease, positive/negative, profit/loss, up/down, warmer/colder, etc.
 Relationships between a number and its opposite
o All numbers have an opposite. Zero is its own opposite.
o The term “opposite” refers to the additive inverse of a number, meaning the number that when added to a
given number results in zero.
o Opposite numbers are equidistant from zero on a number line.
 Ex: Positive 25 and negative 25 are opposite numbers; therefore, 25 + (–25) = 0.
 Ex: –3.5 and 3.5 are opposite numbers; therefore, (–3.5) + 3.5 = 0.
o The opposite of the opposite of a number is the number itself.
 Ex: –(–3.5) = 3.5
 Relationships between a number and its absolute value
o Absolute value – the distance of a value from zero on a number line
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 Notation for absolute value is |x|, where x is any number
 Distance is always a positive value or zero.
o The distance of a number from zero is the same as the distance of its opposite from zero.
o As a positive number decreases, the absolute value of the positive number decreases.

Ex: |7| = 7 and |6| = 6; 7 > 6
o As a positive number increases, the absolute value of the positive number increases.

Ex: |7| = 7 and |8| = 8; 7 < 8
o As a negative number decreases, the absolute value of the negative number increases.

Ex: |–7| = 7 and |–8| = 8; 7 < 8
o As a negative number increases, the absolute value of the negative number decreases.

Ex: |–7| = 7 and |–6| = 6; 7 > 6
o The absolute value of zero is zero.

Relationship between a number, its opposite, and its absolute value
o A number and its opposite are equidistant from zero.
o The absolute value of a number and the absolute value of its opposite are equivalent.
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Supporting TEKS 6.2C
6.2C Number and operations. The student applies
Comparing and ordering of rational numbers
mathematical process standards to represent and use
includes integers and negative rational numbers.
rational numbers in a variety of forms.
The SE includes the number line as a tool for
The student is expected to locate, compare, and order
locating, comparing, and ordering integers and
integers and rational numbers using a number line.
rational numbers.
 Equivalence of various forms of rational numbers
125
1
o Ex: 1.25 is represented as
and 1
100
4
 All integers and rational numbers can be located as a specified point on a number line.
o Characteristics of a scaled number line:
 A number line begins as a line with predetermined intervals (or tick marks), with a minimum of two
positions/numbers labeled.
 Numbers on a number line represent the distance from zero.
 The distance between the tick marks is counted rather than the tick marks themselves.
 The placement of the labeled positions/numbers on a number line determines the scale of the number
line, and the intervals between position/numbers are proportional.
 When reasoning on a number line, the position of zero may or may not be placed. When working with
larger numbers, a number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical
number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the
left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
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o Characteristics of an open number line (
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)
An open number line begins as a line with no intervals (or tick marks) and no positions/numbers labeled.
Numbers/positions are placed on the empty number line only as they are needed.
When reasoning on an open number line, the position of zero is often not placed. When working with
larger numbers, an open number line without the constraint of distance from zero allows the ability to
“zoom-in” on the relevant section of the number line.
The placement of the first two numbers on an open number line determines the scale of the number line.
 Once the scale of the number line has been established by the placement of the first two numbers,
intervals between additional numbers placed are approximately proportional.
The differences between numbers are approximated by the distance between the positions on the
number line.
Open number lines extend infinitely in both directions (arows indicate infinitey)
Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical
number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the
left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
Landmark (or anchor) numbers may be placed on the open number line to help locate other numbers.
Relative magnitude of a number describes the size of a number and its relationship to another number.
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

 Ex: 1 is to the left of 5 on a number line, so 1 < 5; or 5 is to the right of 1 on a number line, so
5 > 1.
 Ex: –10 is to the left of –2 on a number line, so –10 < –2; or –2 is to the right of –10 on a number
line, so –2 > –10.
Comparison words and symbols
 Inequality words and symbols
 Greater than (>)
 Less than (<)
 Equality words and symbol
 Equal to (=)
Quantifying descriptor in mathematical and real-world problem situations (e.g., between two given
numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest,
longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)
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Readiness TEKS 6.2D
6.2D Number and operations. The student applies
mathematical process standards to represent and use rational
numbers in a variety of forms.
The student is expected to order a set of rational numbers
arising from mathematical and real-world contexts.
The SE continues the ordering of rational
numbers.
The SE extends the ordering of rational
numbers to include integers and negative
rational numbers.
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one
thousands, ten thousands, etc.
 Order numbers – to arrange a set of numbers based on their numerical value
 Number lines (horizontal/vertical)
o Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number
line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Quantifying descriptor in mathematical and real-world problem situations (e.g., between two given numbers,
greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest,
heaviest/lightest, closest/farthest, oldest/youngest, etc.)
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Readiness TEKS 6.4G—repeated from Unit 1
6.4G Proportionality. The student applies mathematical
process standards to develop an understanding of proportional
relationships in problem situations.
The student is expected to generate equivalent forms of
fractions, decimals, and percents using real-world
problems, including problems that involve money.

Ideas related to percent have been grouped
together under the Proportionality strand.
When the SE is paired with the 6(1)(A), the
expectation is that students order numbers
arising from mathematical and real-world
contexts, including those involving money.
Equivalent forms of positive rational numbers in real-world problem situations, including money
o Given a fraction, generate a decimal and percent
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o Given a decimal, generate a fraction and percent
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o Given a percent, generate a fraction and decimal
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