Mississippi College and Career Readiness Standards for
Mathematics Scaffolding Document
Grade 6
September 2016 Page 1 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems
6.RP.1
Understand the concept
of a ratio and use ratio
language to describe a
ratio relationship
between two quantities.
For example, “The ratio of
wings to beaks in the bird
house at the zoo was 2:1,
because for every 2 wings
there was 1 beak.” “For
every vote candidate A
received, candidate C
received nearly three
votes.”
Desired Student Performance
A student should know
• A ratio is a pair of
nonnegative numbers, A:B,
where both are not zero, and
are used to indicate a
relationship between two
quantities.
• For the ratio A:B, the value is
the quotient of A/B.
• The order of the numbers is
important to the meaning of
the ratio. Switching the
numbers changes the
relationship.
• Descriptions of a ratio
relationship include words
such as to, for each, for
every.
• How to reason abstractly and
quantitatively.
A student should understand
• Solving problems involving
multiplicative comparisons.
• Interpreting a fraction as
division of the numerator by
the denominator
(a/b = a ÷ b).
• How to find and use the
Greatest Common Factor to
simplify fractions.
• Changing the order of the
numbers represents a
different relationship.
A student should be able to do
• Write a ratio that describes a
relationship between two
quantities.
• Use ratio reasoning to solve
real-world and mathematical
problems.
• Compare data from bar
diagrams and frequency
tables using ratios.
• Use ratios to describe a
simple set of data in different
ways: girls to boys, boys to
girls, boys to total, total to
girls.
• May use a four-function
calculator for computations.
September 2016 Page 2 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems
6.RP.2
Understand the concept
of a unit rate a/b
associated with a ratio
a:b with b
≠
0, and use
rate language in the
context of a ratio
relationship. For
example, “This recipe has
a ratio of 3 cups of flour to
4 cups of sugar, so there
is ¾ cup of flour for each
cup of sugar.” “We paid
$75 for 15 hamburgers,
which is a rate of $5 per
hamburger.”
Desired Student Performance
A student should know
• A rate indicates, for a
proportional relationship
between two quantities, how
many units of one quantity
there are for every one unit of
the second quantity.
• A unit rate is a ratio, a:b,
where b = 1.
• The unit price is the cost per
unit.
• Dividing the numerator by the
denominator will find the unit
rate.
A student should understand
• Equivalent fractions as
equivalent ratios.
• Interpreting a fraction as
division of the numerator by
the denominator
(a/b = a ÷ b).
• Descriptions of a unit rate
include words such as per,
in, and for every.
A student should be able to do
• Convert a given ratio to a
unit rate.
• Use ratio and rate reasoning
to solve real-world and
mathematical problems.
• Compare unit rates.
• Calculate and justify the best
buy using unit price.
• May use a four-function
calculator for computations.
September 2016 Page 3 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems
6.RP.3
Use ratio and rate
reasoning to solve real-
world and mathematical
problems, e.g., by
reasoning about tables
of equivalent ratios, tape
diagrams, double
number line diagrams,
or equations.
Desired Student Performance
A student should know
• Rate tables show a collection
of equivalent ratios.
• An equivalent ratio can be
found my multiplying both
quantities by the same
amount (a:b = 2a:2b).
• In an equation, the constant
value represents the rate
(y = 3x; 3 is the unit rate).
• A double number line has
one set of numbers running
along the top representing
one quantity and a second
set of numbers running along
the bottom representing the
second quantity.
A student should understand
• Ratios and proportional
relationships are used to
express how quantities are
related and how quantities
change in relation to each
other.
• Equivalent fractions.
A student should be able to do
• Use ratio and rate reasoning
to solve real-world and
mathematical problems.
• Use a variety of tools: tape
diagrams, double number
lines, or equations to
demonstrate equivalent
ratios.
• Use a four-function calculator
for computations.
September 2016 Page 4 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems
6.RP.3a
Use ratio and rate
reasoning to solve real-
world and mathematical
problems, e.g., by
reasoning about tables
of equivalent ratios, tape
diagrams, double
number line diagrams,
or equations.
a. Make tables of
equivalent ratios
relating quantities with
whole-number
measurements, find
missing values in the
tables, and plot the pairs
of values on the
coordinate plane. Use
tables to compare
ratios.
Desired Student Performance
A student should know
• Equivalent fractions are
equivalent ratios.
• The “rule” for a rate table is
the unit rate.
• The equation y = mx, relates
independent variables,
dependent variables, and
rates.
• How to plot points in all four
quadrants of the coordinate
plane.
A student should understand
• The relationship between
dependent and
independent variables.
• The unit rate for y is the
point located at (1,y).
• Pairs of numbers that have
the same ratio can be
organized into a ratio table.
• Scaled ratios (equivalent
fractions) can be created by
multiplying or dividing the
two related quantities by the
same number.
• Ratios can be scaled up or
down.
A student should be able to do
• Make a table of equivalent
ratios.
• Use tables to compare ratios.
• Find missing values in
tables.
• Plot values on the coordinate
plane.
• Determine that the steeper
line represents the greater
ratio.
• Use ratio and rate reasoning
to solve real-world and
mathematical problems, such
as increasing a recipe to
serve more people.
• Use a four-function calculator
for computing.
September 2016 Page 5 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems
6.RP.3b
Use ratio and rate
reasoning to solve real-
world and mathematical
problems, e.g., by
reasoning about tables
of equivalent ratios, tape
diagrams, double
number line diagrams,
or equations.
b. Solve unit rate
problems including
those involving unit
pricing and constant
speed. For example, if it
took 7 hours to mow 4
lawns, then at that rate,
how many lawns could be
mowed in 35 hours? At
what rate were lawns
being mowed?
Desired Student Performance
A student should know
• A unit rate is a ratio, a:b,
where b = 1.
• Unit price is the cost per unit.
• Dividing the numerator by the
denominator will find the unit
rate.
• The distance formula is d = rt,
where d is distance, r is the
unit rate, and t is time.
• Dependent variables can be
found by multiplying the
independent variable by the
unit rate.
• Descriptions of a unit rate
include words such as per, in,
and, for every.
A student should understand
• Knowing two values in an
equation leads to
calculation of the third.
• The inverse relationship
between multiplication and
division.
• Division of whole numbers
with decimals quotients.
A student should be able to do
• Calculate speed, if distance
and time are known.
• Calculate unit price, if total
cost and quantity are known.
• Find and justify the “best
buy.”
• Use ratio and rate reasoning
to solve real-world and
mathematical problems.
• Use a four-function calculator
for computing.
September 2016 Page 6 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems
6.RP.3c
Use ratio and rate
reasoning to solve real-
world and mathematical
problems, e.g., by
reasoning about tables
of equivalent ratios, tape
diagrams, double
number line diagrams,
or equations.
c. Find a percent of a
quantity as a rate per
100 (e.g., 30% of a
quantity means 30/100
times the quantity.);
solve problems
involving finding the
whole, given a part and
the percent.
Desired Student Performance
A student should know
• Percents are rates per 100.
• The percentage of a number
is the product of the percent
in fraction or decimal form
and the original number,
50% of 10 = 50/100 times 10.
• How to fluently write decimals
as fractions and percents.
• How to fluently write fractions
as decimals and percents.
• How to fluently write percents
as decimals and fractions.
• How to represent percents
greater than 100% and less
than 1%.
• Equivalent fractions.
• How to solve for the unknown
in an equation.
• How to reason abstractly and
quantitatively.
A student should understand
• Fraction and percent
equivalents.
• Fractions demonstrate the
relationships between
parts and wholes.
• How to compare and order
decimals, fractions, and
percents.
• How to use the percent
proportion, part/whole = %
/100.
• Proportional reasoning.
A student should be able to do
• Write percents as a rate per
one hundred.
• Find a percent of a quantity.
• Solve problems involving
finding the whole when given
a part and the percent.
• Use ratio and rate reasoning
to solve real-world and
mathematical problems.
• Use a visual representation
to model percents.
September 2016 Page 7 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems
6.RP.3d
Use ratio and rate
reasoning to solve real-
world and mathematical
problems, e.g., by
reasoning about tables
of equivalent ratios, tape
diagrams, double
number line diagrams,
or equations.
d. Use ratio reasoning to
convert measurement
units; manipulate and
transform units
appropriately when
multiplying or dividing
quantities.
Desired Student Performance
A student should know
• How to convert among
different-sized standard
measurement units.
• Common conversion factors
in the customary system:
inches, feet, miles, ounces,
pounds, tons, fluid ounces,
cups, quarts, and gallons.
• Common conversion factors
in the metric system: kilo-,
centi-, milli-, meters, liters,
and grams.
• How to reason abstractly and
quantitatively.
A student should understand
• Dividing by the conversion
factor when transforming
from smaller units to larger
units (inches to feet).
• Multiplying by the conversion
factor when transforming
from larger units to smaller
units (gallons to cups).
• Equivalent ratios are used as
conversion factors: 12 in. / 1
ft. = 1 ft. / 12 in.1 lb. / 16 oz.
= 16 oz. / 1 lb.
A student should be able to do
• Use a ratio as a conversion
factor when working with
measurements of different
units.
• Use ratio and rate reasoning
to solve real-world and
mathematical problems.
• Use a four-function calculator
for computing.
September 2016 Page 8 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of multiplication and division to divide fractions by fractions
6.NS.1
Interpret and compute
quotients of fractions, and
solve word problems
involving division of
fractions by fractions, e.g.,
by using visual fraction
models and equations to
represent the problem. For
example, create a story
context for (2/3) ÷ (3/4) and
use a visual fraction model to
show the quotient; use the
relationship between
multiplication and division to
explain that (2/3) ÷ (3/4) = 8/9
because 3/4 of 8/9 is 2/3. (In
general, (a/b) ÷ (c/d) = ad/bc.)
How much chocolate will each
person get if 3 people share
1/2 lb of chocolate equally?
How many 3/4-cup servings
are in 2/3 of a cup of yogurt?
How wide is a rectangular strip
of land with length 3/4 mi and
area 1/2 square mi?
Desired Student Performance
A student should know
• This standard completes the
extension of operations to
fractions.
• Fractions should be seen and
treated as regular numbers.
• The meaning of multiplication.
• The codependent relationship
between multiplication and
division.
• How to solve for the unknown in
an equation.
• How to reason abstractly and
quantitatively.
• Interpretation means to
communicate symbolically,
numerically, abstractly, and/or
with a model.
• How to create a story context
from a set of given information.
A student should understand
• A fraction 1/b as the quantity
formed by 1 part when a whole
is partitioned into b equal parts
(unit fraction).
• A fraction a/b as the quantity
formed by a parts of size 1/b.
• All fractions are rational.
• Fractions allow us to solve word
problems that may not be
possible to solve with whole
numbers or integers.
• Three uses of division are for
equal sharing, measuring, and
finding unknown factors.
• Fractions have multiple
interpretations, and making
sense of them depends on
identifying the unit.
• Equivalent fractions can be used
as a strategy for solving various
word problems.
• The close relationship between
fractions and ratios.
A student should be able to do
• Plot, label, and identify fractions
on a number line.
• Evaluate the reasonableness of
a solution based on the
benchmark fractions of 0, ½,
and 1.
• Perform +, -, and
⋅
with fractions,
and with whole numbers and
fractions (with like and unlike
denominators).
• Make comparisons between
fractions given in multiple
representations.
• Perform operations with mixed
numbers.
• Use a variety of visual fraction
models (tape diagram, number
line diagram, or area model).
• Must demonstrate use of the
standard algorithm to convert
between fractions and decimals.
September 2016 Page 9 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Compute fluently with multi-digit numbers and find common factors and multiples
6.NS.2
Fluently divide multi-digit
numbers using the standard
algorithm.
Desired Student Performance
A student should know
• Multiplication facts (0–12).
• The difference between
dividend, divisor, and quotient.
• This is the culminating
standard for several years’
worth of work with division of
whole numbers.
• How to attend to precision.
A student should understand
• Divisibility rules for numbers 2
through 10.
• Division is repeated subtraction.
• Rational numbers can be
represented in multiple ways
and are useful when examining
situations involving numbers
that are not whole.
• Estimation as a tool for
checking reasonableness of a
quotient.
A student should be able to do
• Divide multi-digit numbers using
the standard algorithm.
• Check quotients for
reasonableness.
September 2016 Page 10 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Compute fluently with multi-digit numbers and find common factors and multiples
6.NS.3
Fluently add, subtract,
multiply, and divide multi-
digit decimals using the
standard algorithm for each
operation.
Desired Student Performance
A student should know
• Place value to the left and right
of the decimal point.
• Vocabulary for adding,
subtracting, multiplying, and
dividing: addends, sum,
subtrahend, minuend,
difference, factors, product,
dividend, divisor, and quotient.
• When adding and/or
subtracting decimals, decimal
points must be lined up.
• Commutative Properties of
addition and multiplication.
• How to attend to precision.
A student should understand
• Addition and subtraction are
inverse operations.
• Multiplication and division are
inverse operations.
• Subtraction problems may be
interpreted as missing
addend problems.
• Division problems may be
interpreted as missing factor
problems.
• The difference between a
terminating and repeating
decimal.
• Multiplying a whole number
by a decimal less than 1
results in a product less than
the original factor.
A student should be able to do
• Add and subtract multi-digit
decimals using the standard
algorithm.
• Multiply and divide multi-digit
decimals using the standard
algorithm.
• Use estimation to check
answers for reasonableness.
September 2016 Page 11 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Compute fluently with multi-digit numbers and find common factors and multiples
6.NS.4
Find the greatest common
factor of two whole numbers
less than or equal to 100 and
the least common multiple of
two whole numbers less
than or equal to 12. Use the
distributive property to
express a sum of two whole
numbers 1-100 with a
common factor as a multiple
of a sum of two whole
numbers with no common
factor. For example, express
36 + 8 as 4(9 + 2).
Desired Student Performance
A student should know
• The greatest common factor of
any two numbers is the largest
number that will divide evenly
into both numbers.
• The least common multiple of
any two numbers is the first
consecutive number divisible
by the two numbers.
• A prime number is a number
with exactly two factors: 1 and
itself.
• Composite numbers are
numbers with more than two
factors.
• How to look for and make use
of structure.
A student should understand
• 1 is neither prime nor composite.
• Prime factorization of a number is
a multiplication expression
composed of only prime numbers.
• Distributive property of
multiplication over addition means
you can multiply a sum by a
number and get the same result
as multiplying each addend
separately, a(b + c) = ab + ac.
A student should be able to do
• Find the greatest common
factor of two whole numbers
less than or equal to 100.
• Find the least common multiple
of two whole numbers less than
or equal to 12.
• Use the distributive property to
express a sum of two whole
numbers 1–100 with a common
factor as a multiple of a sum of
two whole numbers with no
common factor.
September 2016 Page 12 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.5
Understand that positive and
negative numbers are used
together to describe
quantities having opposite
directions or values (e.g.,
temperature above/below
zero, elevation above/below
sea level, credits/debits,
positive/negative electric
charge); use positive and
negative numbers to
represent quantities in the
real-world contexts,
explaining the meaning of 0
in each situation.
Desired Student Performance
A student should know
• Numbers greater than 0,
located to the right of 0 on the
number line, are called positive
numbers.
• Numbers less than 0, located
to the left of 0 on the number
line, are called negative
numbers.
• Any negative number is less
than any positive number.
• 0 is neither positive nor
negative.
• How to reason abstractly and
quantitatively.
• How to use appropriate tools
strategically.
A student should understand
• Two numbers with opposite signs,
such as 5, and -5, are equidistant
from 0 on the number line.
• Number lines may be displayed
horizontally or vertically. This
does not affect a number’s value.
• 0 is the point at which direction or
value changes.
A student should be able to do
• Explain the relationship
between positive and negative
numbers in real-world context:
temperature, money, sea level,
and electric charge.
• Explain the meaning of zero in
any real-world context.
September 2016 Page 13 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.6
Understand a rational
number as a point on the
number line. Extend number
line diagrams and
coordinate axes familiar
from previous grades to
represent points on the line
and in the plane with
negative number
coordinates.
Desired Student Performance
A student should know
• A rational number is a number
that may be displayed as a
fraction as long as the
denominator is not 0.
• The number line extends
infinitely in both positive and
negative directions.
• An integer is any positive or
negative whole number.
• Points on the number line may
be integers, fractions, or
decimals.
• How to attend to precision.
• How to look for and make use
of structure.
A student should understand
• every rational number can be
represented by a point on a
number line.
• The coordinate plane is 2
number lines intersecting at 0,
effectively called the x- and
y-axes.
• As such, the coordinate plane
extends infinitely.
A student should be able to do
• Plot a rational number as a
point on the number line.
• Extend number lines as needed
to display data.
• Extend coordinate axes learned
in previous grades.
• Plot ordered pairs that may
include negative coordinates.
September 2016 Page 14 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.6a
Understand a rational
number as a point on the
number line. Extend number
line diagrams and
coordinate axes familiar
from previous grades to
represent points on the line
and in the plane with
negative number
coordinates.
a. Recognize opposite signs
of numbers as indicating
locations on opposite sides
of 0 on the number line;
recognize that the opposite
of the opposite of a number
is the number itself, e.g.,
-(-3) = 3, and that 0 is its own
opposite.
Desired Student Performance
A student should know
• The opposite of a positive
number is a negative number,
and the reverse is true.
• Two numbers with opposite
signs, such as 5 and -5,
represent numbers equidistant
from 0 on the number line.
• Zero is its own opposite.
• Look for and express regularity
in repeated reasoning.
A student should understand
• The opposite of an opposite of a
number is the original number.
A student should be able to do
• Recognize opposite signs of
numbers as indicating locations
on opposite sides of 0 on the
number line.
• Find the opposite of any
number.
• Read numbers accurately
plotted on the number line.
• Plot numbers accurately on the
number line.
September 2016 Page 15 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.6b
Understand a rational
number as a point on the
number line. Extend number
line diagrams and
coordinate axes familiar
from previous grades to
represent points on the line
and in the plane with
negative number
coordinates.
b. Understand signs of
numbers in ordered pairs as
indicating locations in
quadrants of the coordinate
plane; recognize that when
two ordered pairs differ only
by signs, the locations of the
points are related by
reflections across one or
both axes.
Desired Student Performance
A student should know
• The coordinate plane is made
up of four quadrants that
extend infinitely.
• Numbers with opposite signs
represent locations opposite to
one another.
• A reflection refers to the exact
opposite position as to create a
mirror image.
• How to attend to precision.
• How to look for and make use
of structure.
A student should understand
• The signs of numbers in an
ordered pair relate to their
location on the coordinate plane.
• (5, 5) and (-5, -5) would be
points reflected about the origin.
You move in exact opposite
directions when plotting.
A student should be able to do
• Use the signs of the coordinates
to determine the location of an
ordered pair in the coordinate
plane.
• Plot a point on a coordinate
plane.
• Read a point plotted on the
coordinate plane.
September 2016 Page 16 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.6c
Understand a rational
number as a point on the
number line. Extend number
line diagrams and
coordinate axes familiar
from previous grades to
represent points on the line
and in the plane with
negative number
coordinates.
c. Find and position integers
and other rational numbers
on a horizontal or vertical
number line diagram; find
and position pairs of
integers and other rational
numbers on a coordinate
plane.
Desired Student Performance
A student should know
• Integers are positive and
negative whole numbers.
• Number lines may run
horizontally or vertically.
• Ordered pairs are coordinates
that are positive, negative, or
one of each.
• Coordinates are not limited to
integers.
A student should understand
• The coordinate plane is two
number lines intersecting at 0,
effectively called the x- and
y- axes.
• As such, the coordinate plane
extends infinitely.
A student should be able to do
• Find and position integers and
other rational numbers on a
horizontal or vertical number
line.
• Find and position pairs of
integers and other rational
numbers on a coordinate plane.
September 2016 Page 17 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.7
Understand ordering and
absolute value of rational
numbers.
Desired Student Performance
A student should know
• Rational numbers may include
whole numbers, integers,
fractions, and decimals.
• Ordering refers to placing
numbers greatest to least or
least to greatest.
• Absolute value is the distance
a number is from 0 on the
number line.
• The symbol for absolute value
is
| |
.
A student should understand
• Since absolute value refers to
distance from zero, it is always
represented by a positive
number.
A student should be able to do
• Order rational numbers least to
greatest or greatest to least.
• Find the absolute value of a
rational number.
September 2016 Page 18 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.7a
Understand ordering and
absolute value of rational
numbers.
a. Interpret statements of
inequality as statements
about the relative position of
two numbers on a number
line diagram. For example,
interpret -3 > -7 as a statement
that -3 is located to the right of
-7 on a number line oriented
from left to right.
Desired Student Performance
A student should know
• Numbers to the right of zero
are positive on a number line
diagram.
• Numbers to the left of zero are
negative on a number line
diagram.
• Relation symbols are <, >, <,
>, and ≠.
• How to reason abstractly and
quantitatively.
• How to use appropriate tools
strategically.
A student should understand
• The further left a number is
from zero, the more negative
(smaller) it is.
• The further right a number is
from zero, the more positive
(larger) it is.
• When comparing any two
numbers on a number line
diagram, the number to the left
is always smaller if the number
line is oriented from left to right.
A student should be able to do
• Describe the relative position of
two numbers on a number line
when given an inequality.
• Interpret statements of
inequality as statements about
the relative position of two
numbers on a number line
diagram.
September 2016 Page 19 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.7b
Understand ordering and
absolute value of rational
numbers.
b. Write, interpret, and
explain statements of order
for rational numbers in real-
world contexts. For example,
write -3 °C> -7°C to express
the fact that -3°C is warmer
than -7°C.
Desired Student Performance
A student should know
• Relation symbols are <, >, <,
>, and ≠.
• The position of positive and
negative numbers in relation to
zero on a number line diagram.
• How to reason abstractly and
quantitatively.
• How to use appropriate tools
strategically.
A student should understand
• The further a negative number is
from zero, the smaller the value.
• The further a positive number is
from zero, the greater the value.
A student should be able to do
• Write and interpret statements
of inequality in terms of a real-
world situation.
• Explain what the numbers in an
inequality represent.
September 2016 Page 20 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.7c
Understand ordering and
absolute value of rational
numbers.
c. Understand the absolute
value of a rational number
as its distance from 0 on
the number line; interpret
absolute value as
magnitude for a positive or
negative quantity in a real-
world situation. For
example, for an account
balance of -30 dollars, write
|
−30
|
= 30 to describe the
size of the debt in dollars.
Desired Student Performance
A student should know
• Absolute value is the distance
a number is from zero on the
number line.
• The symbol for absolute value
is
| |
.
• Magnitude refers to the
amount or size relative to the
context.
• How to reason abstractly and
quantitatively.
• How to use appropriate tools
strategically.
A student should understand
• The absolute value of any
number, positive or negative, is
positive.
• Absolute value is positive
because it represents distance
from zero.
• The difference between a
signed number, such as -5,
and the absolute value of a
signed number,
|
−5
|
, in real-
world context.
A student should be able to do
• Explain absolute value.
• Relate absolute value to real-
world situations such as sea
level, temperature, and debt.
September 2016 Page 21 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.7d
Understand ordering and
absolute value of rational
numbers.
d. Distinguish comparisons
of absolute value from
statements about order. For
example, recognize that an
account balance less than -30
dollars represents a debt
greater than 30 dollars.
Desired Student Performance
A student should know
• How to compare rational
numbers.
• Absolute value is the distance
from zero.
• How to reason abstractly and
quantitatively.
• How to use appropriate tools
strategically.
A student should understand
• Using a number line diagram as
a tool for comparison.
• Absolute value may not be
limited to integers.
A student should be able to do
• Distinguish comparisons of
absolute value from statements
about order.
September 2016 Page 22 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.8
Solve real-world and
mathematical problems by
graphing points in all four
quadrants of the coordinate
plane. Include use of
coordinates and absolute
value to find distance
between points with the
same first coordinate or the
same second coordinate.
Desired Student Performance
A student should know
• How to plot points in all four
quadrants.
• One can use coordinates or
absolute value to find distance
between plotted points.
• How to reason abstractly and
quantitatively.
• How to use appropriate tools
strategically.
A student should understand
• Distance between two plotted
points with the same
x-coordinate is found by
subtracting the y-coordinates
using standard rules for
subtraction.
• Distance between two plotted
points with the same
y-coordinate is found by
subtracting the x-coordinates
using standard rules for
subtraction.
• One may also consider the
absolute value of the difference
between coordinates. For
example, (2,4) and (2,7): 7 – 4
= 3 and 4 – 7 = -3. Either
solution is an absolute value of
3. As such, the points are 3
units apart.
A student should be able to do
• Solve real-world and
mathematical problems by
graphing points in all four
quadrants of the coordinate
plane.
• Use coordinates and absolute
value to find the distance
between points.
September 2016 Page 23 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.9a
Apply and extend previous
understanding of addition
and subtraction to add and
subtract integers; represent
addition and subtraction on
a horizontal or vertical
number line diagram.
a. Describe situations in
which opposite quantities
combine to make 0. For
example, a hydrogen atom has
0 charge because its two
constituents are oppositely
charged.
Desired Student Performance
A student should know
• Numbers found to the left of
zero on the number line are
negative; meaning their value
is less than zero.
• Two numbers that are
equidistant from zero on the
number line are called
opposites.
• Integers are all whole numbers
and their opposites.
• Integers are included in the
larger group of rational
numbers.
• Procedures for adding and
subtracting both positive and
negative integers with or
without a number line.
A student should understand
• Two numbers whose sum is
zero are opposites. These are
also called additive inverses.
• How to find the opposite of a
number.
• How to use appropriate tools
strategically. (number line)
• How to look for and make use
of structure.
A student should be able to do
• Use a horizontal or vertical
number line to add any
combination of positive and/or
negative numbers.
• Use a horizontal or vertical
number line to subtract any
combination of positive and/or
negative numbers.
• Apply strategies to solve
integers problems in a real-
world context.
September 2016 Page 24 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.9b
Apply and extend previous
understanding of addition
and subtraction to add and
subtract integers; represent
addition and subtraction on
a horizontal or vertical
number line diagram.
b. Understand p + q as the
number located a distance
|
|
from p, in the positive or
negative direction
depending on whether q is
positive or negative. Show
that a number and its
opposite have a sum of 0
(are additive inverses).
Interpret sums of integers by
describing real-world
contexts.
Desired Student Performance
A student should know
• The absolute value of a
number is its distance from
zero.
• Since absolute value is a
measure of distance, it always
has a positive value.
• The difference between the
absolute value of a number
and its opposite.
• Opposites are additive
inverses.
• The commutative property for
addition.
A student should understand
• How to find absolute value of a
number vs. finding its opposite.
• A positive integer added to a
positive integer results in a
positive integer.
• A negative integer added to a
negative integer results in a
negative integer.
• How to reason abstractly and
quantitatively.
A student should be able to do
• Use a horizontal or vertical
number line to add p + q,
regardless of whether either
number is positive or negative.
• Use a horizontal or vertical
number line to show that a
number and its opposite have a
sum of zero.
• Interpret sums of integers in
real-world contexts.
September 2016 Page 25 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.9c
Apply and extend previous
understanding of addition
and subtraction to add and
subtract integers; represent
addition and subtraction on
a horizontal or vertical
number line diagram.
c. Understand subtraction of
integers as adding the
additive inverse,
p – q = p + (-q). Show that
the distance between two
integers on the number line
is the absolute value of their
difference, and apply this
principle in real-world
contexts.
Desired Student Performance
A student should know
• The absolute value of a
number is its distance from
zero.
• The opposite of a number is its
additive inverse.
• The procedures for adding
integers with or without a
number line.
• The procedures for subtracting
integers with or without a
number line.
A student should understand
• Subtracting a number is equal
to adding its additive inverse
(opposite). For example 5 – 6 =
5 + (-6)
A student should be able to do
• Use a horizontal or vertical
number line to find p – q.
• Use a horizontal or vertical
number line to find p + (-q).
• Solve subtraction of integers
without the context of a real-
world situation.
September 2016 Page 26 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers
6.NS.9d
Apply and extend previous
understanding of addition
and subtraction to add and
subtract integers; represent
addition and subtraction on
a horizontal or vertical
number line diagram.
d. Apply properties of
operations as strategies to
add and subtract integers.
Desired Student Performance
A student should know
• The associative property of
addition:
(a + b) + c = a + (b + c).
• The commutative property of
addition: a + b = b + a.
• The additive identity property
of zero: a + 0 = 0 + a = a.
• The distributive property of
multiplication over addition:
a x (b + c) = a x b + a x c.
• The distributive property of
multiplication over subtraction:
a x (b - c) = a x b - a x c.
A student should understand
• Applying appropriate properties
to add or subtract integers.
• How to reason abstractly and
quantitatively.
• How to use appropriate tools
strategically. (properties)
A student should be able to do
• Add or subtract integers with or
without a number line.
• Add or subtract integers with or
without a four-function
calculator.
• Demonstrate understanding of
adding and subtracting by
recognizing equivalent
expressions.
September 2016 Page 27 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions
6.EE.1
Write and evaluate
numerical expressions
involving whole-number
exponents.
Desired Student Performance
A student should know
• A base is the number being
raised to an exponent, or power.
• A number raised to a power
represents repeated
multiplication of the base.
• How to evaluate the means to
solve for the product of the base.
• How to read and identify the
parts of a numerical and
algebraic expression using
mathematical terms.
A student should understand
• A number raised to a power
represents an algebraic
expression.
• Using exponents to make
sense of quantitative
relationships.
• Any number, whole or
fractional, may be raised to
an exponent.
• The function of each part of a
numerical expression.
A student should be able to do
• Write an expression using
exponents to illustrate repeated
multiplication.
• Multiply fluently whole numbers
and fractions.
• Evaluate expressions that
consist of whole numbers,
exponents, fractions and
decimals.
• Use a four-function calculator to
evaluate expressions.
September 2016 Page 28 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions
6.EE.2
Write, read, and evaluate
expressions in which
letters stand for numbers.
Desired Student Performance
A student should know
• A variable is a symbol that
stands in the place of an
unknown value.
• An expression is a mathematical
phrase containing numbers,
variables, and operation
symbols.
• A term, such as 3x, is read 3
times x.
• How to evaluate the means to
substitute a value for a variable
and solve the expression.
• How to attend to precision.
A student should understand
• Common algebraic
expressions such as less
than, more than, times,
shared equally.
• The difference between an
expression and an equation.
• Variables may represent any
whole number, fraction,
decimal, or exponent.
• Variables may also represent
positive or negative values.
• Order of operations.
A student should be able to do
• Read accurately an algebraic
expression containing variables
and exponents (reading).
• Translate an expression from
words to symbols (writing).
• Substitute in a value for the
given variable and complete the
calculations (evaluating).
• Add, subtract, multiply, and
divide fluently with whole
numbers, fractions, and
decimals.
• Apply order of operations.
• Use a four-function calculator
for computations.
September 2016 Page 29 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions
6.EE.2a
Write, read, and evaluate
expressions in which
letters stand for numbers.
a. Write expressions that
record operations with
numbers and with letters
standing for numbers.
For example, express the
calculation “Subtract y from
5” as 5 – y.
Desired Student Performance
A student should know
• Mathematical terms such as
sum, quotient, product,
difference, coefficient.
• A variable is a symbol that
stands in the place of an
unknown value.
• When a term is representing
addition/subtraction versus
multiplication/division.
• How to look for and express
regularity in repeated reasoning.
• How to read an expression.
A student should understand
• Common algebraic
expressions such as less
than, more than, times,
shared equally.
• An expression can be a final
answer: 5 – y is the answer
until you have a value to
substitute in for y.
• The difference between the
expressions 5 – y and y – 5.
• Order of Operations.
A student should be able to do
• Write an expression when using
whole numbers, fractions, and
decimals.
September 2016 Page 30 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions
6.EE.2b
Write, read, and evaluate
expressions in which
letters stand for numbers.
b. Identify parts of an
expression using
mathematical terms (sum,
term, product, factor,
quotient, coefficient); view
one or more parts of an
expression as a single
entity. For example,
describe the expression
2(8 + 7) as a product of two
factors; view (8 + 7) as both
a single entity and a sum of
two terms.
Desired Student Performance
A student should know
• Parts of an expression.
• Which operations apply to the
mathematical terms sum,
difference, product, quotients,
less than, more than, less, etc.
A student should understand
• 2(8 + 7) is a product of two
factors: 2 and the sum of 8
and 7.
• An expression has a single
value but may also be viewed
as multiple terms that
operations are performed on.
• That variables in an
expression represent a value.
A student should be able to do
• Identify accurately the parts of
an expression.
September 2016 Page 31 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions
6.EE.2c
Write, read, and evaluate
expressions in which
letters stand for numbers.
c. Evaluate expressions at
specific values of their
variables. Include
expressions that arise
from formulas used in real-
world problems. Perform
arithmetic operations,
including those involving
whole-number exponents,
in the conventional order
when there are no
parentheses to specify a
particular order (Order of
Operations). For example,
use the formulas V = s
3
and
A = 6s
2
to find the volume
and surface area of a cube
with sides of lengths s = ½.
Desired Student Performance
A student should know
• A variable is a symbol that
stands in the place of an
unknown value.
• How to evaluate a means to
substitute a value for the variable
and solve the expression.
A student should understand
• Substituting a value for a
variable.
• Order of operations.
A student should be able to do
• Evaluate an expression for a
given value.
• Substitute values in formulas to
solve real-world problems.
• Apply order of operations with
or without parentheses.
• Evaluate expressions that arise
from formulas; however,
students are not required to
manipulate the formulas.
September 2016 Page 32 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions
6.EE.3
Apply the properties of
operations to generate
equivalent expressions.
For example, apply the
distributive property to the
expression 3(2 + x) to
produce the equivalent
expression 6 + 3x; apply the
distributive property to the
expression 24x + 18y to
produce the equivalent
expression 6(4x + 3y); apply
properties of operations to
y + y + y to produce the
equivalent expression 3y.
Desired Student Performance
A student should know
• Commutative properties of
addition and multiplication,
associative properties of addition
and multiplication, and
distributive properties of
multiplication over addition and
subtraction.
• Repeated addition is
multiplication.
• Equivalent means expressions
have the same value.
• How to look for and make use of
structure.
A student should understand
• 2 * 2 has the same value as 4.
As such, 3(2 + x) is the same as 6
+ 3x.
• Distributing is multiplying a term
outside of parentheses times
every term inside the
parentheses.
• The importance of fluently adding,
subtracting, multiplying and
dividing whole numbers, fractions,
and decimals.
A student should be able to do
• Generate two or more
equivalent expressions using
the properties.
• Compose and decompose
expressions using the
properties.
September 2016 Page 33 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions
6.EE.4
Identify when two
expressions are equivalent
(i.e., when the two
expressions name the
same number regardless
of which value is
substituted into them). For
example, the expressions y
+ y + y and 3y are equivalent
because they name the
same number regardless of
which number y stands for.
Desired Student Performance
A student should know
• A variable is a symbol that
stands in the place of an
unknown value (number).
• When two or more expressions
are equivalent, that means the
value of each expression is the
same.
A student should understand
• Variables represent numerical
values.
• Expressions may have to be
simplified to determine
equivalency.
A student should be able to do
• Determine whether two
expressions are equivalent by
using the same value to
evaluate both expressions.
• Identify equivalent expressions.
• Use properties of operations to
justify that two expressions are
equivalent.
September 2016 Page 34 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Reason about and solve one-variable equations and inequalities
6.EE.5
Understand solving an
equation or inequality as a
process of answering a
question: which values
from a specified set, if any,
make the equation or
inequality true? Use
substitution to determine
whether a given number in
a specified set makes an
equation or inequality true.
Desired Student Performance
A student should know
• An expression is a mathematical
phrase containing numbers,
variables, and operation
symbols.
• An equation is a number
sentence that’s equal to a
specific value.
• An inequality is a number
sentence that utilizes relation
symbols other than the equal
sign (i.e., <, >, ≤, ≥, or ≠).
• How to reason abstractly or
quantitatively.
A student should understand
• The difference between
expressions, equations and
inequalities.
• An inequality may contain a
variable that can represent
more than one value. For
example, x < 5; x = all real
numbers less than 5.
• A solution is the number or set
of numbers that makes an
inequality true.
A student should be able to do
• Utilize substitution to decide if
an equation or inequality is true.
• Solve an equation or inequality
to find the value of the variable.
• Use a four-function calculator to
solve equations and
inequalities.
September 2016 Page 35 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Reason about and solve one-variable equations and inequalities
6.EE.6
Use variables to represent
numbers and write
expressions when solving
a real-world or
mathematical problem;
understand that a variable
can represent an unknown
number, or, depending on
the purpose at hand, any
number in a specified set.
Desired Student Performance
A student should know
• The difference between an
equation, expression, and
inequality.
• Variables and their purpose.
A student should understand
• A variable may represent one
number or more than one
number (equation versus
inequality).
• A variable may represent any
whole number, fraction, or
decimal.
• A variable may represent
positive or negative numbers.
A student should be able to do
• Use variables to represent
numbers to solve real-world
problems.
• Determine the function of the
variable in a real-world or
mathematical problem.
• Write expressions when solving
real-world or mathematical
problems.
• Identify the relationship of the
variable in real-world or
mathematical problems.
September 2016 Page 36 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Reason about and solve one-variable equations and inequalities
6.EE.7
Solve real-world and
mathematical problems by
writing and solving
equations of the form
x + p = q and px = q for
cases in which p, q, and x
are all nonnegative rational
numbers.
Desired Student Performance
A student should know
• Nonnegative rational numbers
include positive numbers that
may be written as a quotient of
two integers where the
denominator is not zero.
• Decimals that are rational
numbers either terminate or
repeat.
• How to reason abstractly and
quantitatively.
• How to look for and make use of
structure.
A student should understand
• Substitution of values as they
pertain to a real-world problem.
A student should be able to do
• Solve equations when the
values for the variables are
given.
• Write and solve equations that
represent real-world problems.
• Fluently add, subtract, multiply
and divide whole numbers,
fractions, and decimals.
• Evaluate reasonableness of
solutions.
September 2016 Page 37 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Reason about and solve one-variable equations and inequalities
6.EE.8
Write an inequality of the
form x > c or x < c to
represent a constraint or
condition in a real-world or
mathematical problem.
Recognize that inequalities
of the form x > c or x < c
have infinitely many
solutions; represent
solutions of such
inequalities on number line
diagrams
.
Desired Student Performance
A student should know
• A simple inequality is x > 4 or
x < 4.
• X can represent any value that
proves the inequality true.
• How to reason abstractly and
quantitatively.
• How to look for and make use of
structure.
A student should understand
• C, the constraint value, is not
limited to integers.
• A constraint value is the
value that x is greater than or
less than.
• X in an inequality, x < c, has
an infinite number of
solutions.
• That there are an infinite
number of solutions for an
inequality.
A student should be able to do
• Write an inequality to represent
constraints or conditions in a
real-world or mathematical
problem.
• Graph a solution set of an
inequality on a number line.
• Explain what the solution set of
an inequality represents.
September 2016 Page 38 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Expressions and Equations
Represent and analyze quantitative relationships between dependent and independent variables
6.EE.9
Use variables to represent
two quantities in a real-
world problem that change
in relationship to one
another; write an equation
to express one quantity,
thought of as the
dependent variable, in
terms of the other quantity,
thought of as the
independent variable.
Analyze the relationship
between the dependent
and independent variables
using graphs and tables,
and relate these to the
equation. For example, in a
problem involving motion at
constant speed, list and
graph ordered pairs of
distances and times, and
write the equation d = 65t to
represent the relationship
between distance and time.
Desired Student Performance
A student should know
• Dependent variable is a variable
whose value is determined by
another variable in the
expression. For example, the
distance you travel is determined
by how long you drive.
• Independent variable is a
variable whose value decides the
value of the other variable. For
example, in a fundraiser, how
many items you sell determines
the amount of money you make.
• Plotting points in all four
quadrants of the coordinate
plane.
• How to model with mathematics.
• How to attend to precision.
• How to look for and express
regularity in repeated reasoning.
A student should understand
• The relationship between the
dependent and independent
variable in a real-world
relationship.
• The function of the dependent
and independent variable.
• The pattern y=mx, to show that
the dependent variable is the
product of a rate times the
independent variable.
• The y and x in y=mx refer to the
x and y axis on the coordinate
plane.
• The effect x has on y
corresponds to the rate in the
equation. For example, d= 65t,
means for every hour (t), d will
increase by 65 miles.
A student should be able to do
• Analyze tables and graphs to
determine the dependent and
independent variable.
• Analyze tables and graphs to
determine the relationship
between dependent and
independent variables.
• Write an equation with variables
that represent the relationship
between the dependent and
independent variables.
• Create a table of two variables
that represent a real-world
situation in which one quantity
will change in relation to the
other.
• Use data to plot points on the
coordinate plane.
• Interpret patterns in the table
and graph and relate them back
to the equation.
• Use a four-function calculator to
determine either variable.
September 2016 Page 39 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Geometry
Solve real-world and mathematical problems involving area, surface area, and volume
6.G.1
Find the area of right
triangles, other
triangles, special
quadrilaterals, and
polygons by composing
into rectangles or
decomposing into
triangles and other
shapes; apply these
techniques in the
context of solving
real-world and
mathematical problems.
Desired Student Performance
A student should know
• A quadrilateral is defined as a
four-sided shape.
• Recognize squares,
rectangles, rhombuses, and
trapezoids as quadrilaterals.
• Simple shapes compose to
form larger shapes.
• Shapes can be partitioned
into parts of equal areas.
• Multiplication and division are
inverse operations.
• How to reason abstractly and
quantitatively.
• How to look for and make use
of structure.
• Area as defined as the inside
shape or space measured in
square units.
• How to identify a right
triangle.
A student should understand
• A trapezoid is defined as a
quadrilateral with at least
one pair of parallel sides.
• Area formulas for rectangles,
parallelograms, triangles,
and trapezoids are related.
• The relationship between
area of a rectangle and area
of a triangle.
• Area of a right triangle
equals one-half base times
height.
• Shapes with more than three
sides can be decomposed
into triangles (1 square = 2
triangles).
• Triangles can be used to
compose larger polygons.
A student should be able to do
• Calculate area of triangles
and quadrilaterals when
given base and height.
• Calculate base or height
when given area.
• Compose polygons from
triangles.
• Decompose polygons into
triangles.
• Solve real-world and
mathematical problems.
• Use a four-function calculator
to solve for area.
September 2016 Page 40 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Geometry
Solve real-world and mathematical problems involving area, surface area, and volume
6.G.2
Find the volume of a
right rectangular prism
with fractional edge
lengths by packing it
with unit cubes of the
appropriate unit fraction
edge lengths, and show
that the volume is the
same as would be found
by multiplying the edge
lengths of the prism.
Apply the formulas
V=lwh and V=bh to find
volumes of right
rectangular prisms with
fractional edge lengths
in the context of solving
real-world and
mathematical problems.
Desired Student Performance
A student should know
• Volume of a right rectangular
prism equals the product of
the length times the width
times the height of the figure.
• A right rectangular prism is
defined as a prism whose
lateral faces are rectangles.
• A unit cube is a cube whose
side lengths are 1 unit long.
• Multiplication of fractions and
mixed numbers.
• What volume means and
what volume represents.
• How to reason abstractly and
quantitatively.
A student should understand
• The connection between
computing volume and
packing the solid figure with
cubes of varying sizes.
• The formula V = bh refers to
multiplying the area of the
base times the height.
• Units that measure volume
are cubic (cm
3
, m
3
, ft
3
).
A student should be able to do
• Compute volume after
packing a rectangular prism
with unit cubes.
• Apply formulas to solve
problems with real-world
contexts.
• Calculate volume with and
without a four-function
calculator.
• Evaluate reasonableness of
the volume of a prism in
regard to its length, width,
and height.
September 2016 Page 41 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Geometry
Solve real-world and mathematical problems involving area, surface area, and volume
6.G.3
Draw polygons in the
coordinate plane given
coordinates for the
vertices; use
coordinates to find the
length of a side joining
points with the same
first coordinate or the
same second
coordinate. Apply these
techniques in the
context of solving real-
world and mathematical
problems.
Desired Student Performance
A student should know
• Polygons are closed figures
with no curved sides.
• Vertices is plural for vertex;
vertex being the point where
two segments meet.
• Coordinates (ordered pairs)
are two numbers that
describe the location of a
point on the coordinate grid.
• How to plot points on the
coordinate grid in all four
quadrants.
• How to find, name, and label
coordinates.
• How to identify and label a
line segment.
• How to use appropriate tools
strategically.
A student should understand
• Length of a segment with
joining points is also the
distance between said
points.
• The distance between two
points with the same first
coordinate is found my
subtracting the points’
second coordinates.
• Conversely, the distance
between two points with the
same second coordinate is
found by subtracting the
points’ first coordinates.
A student should be able to do
• Draw polygons in the
coordinate plane given
coordinates for the vertices.
• Use coordinates to find the
length of a side joining points
with the same first coordinate
or the same second
coordinate.
• Subtract positive and
negative numbers.
• Find the perimeter and area
of polygons.
• Solve real-world and
mathematical problems.
September 2016 Page 42 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Geometry
Solve real-world and mathematical problems involving area, surface area, and volume
6.G.4
Represent three-
dimensional figures
using nets made up of
rectangles and triangles,
and use the nets to find
the surface area of
these figures. Apply
these techniques in the
context of solving
real-world and
mathematical problems.
Desired Student Performance
A student should know
• The difference between right
rectangular prisms, right
triangular prisms, right square
prisms (cube), and right
tetrahedrons.
• A right tetrahedron is also
called a triangular pyramid.
• Area formulas for triangles
and rectangles.
• How to model with
mathematics.
A student should understand
• A net is a two-dimensional
pattern for a three-
dimensional figure.
• Nets may be rearranged to
form the same three-
dimensional figure. For
example, there are 11
different nets for a right
square prism (cube).
• Surface area is the sum of
the areas of each face of a
three-dimensional figure.
A student should be able to do
• Match nets with
corresponding three-
dimensional figures.
• Draw nets when given the
name of a three-dimensional
figure.
• Calculate surface area with
and without a four-function
calculator.
• Evaluate reasonableness of
the surface area considering
the lengths and widths of the
faces of the figure.
• Solve real-world and
mathematical problems.
September 2016 Page 43 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Statistics and Probability
Develop understanding of statistical variability
6.SP.1
Recognize a statistical
question as one that
anticipates variability in
the data related to the
question and accounts
for it in the answers.
For example, “How old am
I?” is not a statistical
question, but “How old are
the students in my
school?” is a statistical
question because one
anticipates variability in
students’ ages.
Desired Student Performance
A student should know
• Variability means that not all
the data values will have the
same value.
• A statistical question poses a
question where data must be
collected to answer the
question.
• How to reason abstractly and
quantitatively.
A student should understand
• What makes a good
statistical question.
• The difference between
numerical data and
categorical data. For
example, heights of
basketball players versus
their favorite colors.
A student should be able to do
• Recognize a statistical
question.
• Develop a question that can
be used to collect statistical
information.
• Collect data to demonstrate
the variability of the answers
to the question.
September 2016 Page 44 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Statistics and Probability
Develop understanding of statistical variability
6.SP.2
Understand that a set of
data collected to answer
a statistical question
has a distribution which
can be described by its
center, spread, and
overall shape.
Desired Student Performance
A student should know
• Distribution refers to the entire
data set as a whole.
• Distribution can be described
in terms of center, spread,
and shape.
• Mean is the average of all the
numerical data.
• Median is the exact middle
value of the data.
• Mode is the most frequently
occurring data value.
• How to model with
mathematics.
A student should understand
• The center of a distribution
can be described in terms of
mean, median, and mode.
• The spread of a distribution
can be described in terms of
clusters, gaps, and outliers.
• Shape can be described as
symmetric or skewed.
• A box plot is a method of
displaying a distribution of
data values by using the
median, quartiles, and
extremes of the data set.
• A box shows the middle 50%
of the data.
A student should be able to do
• Describe a distribution of
data in terms of center,
spread, and overall shape.
• Construct a box plot to show
the distribution of a set of
data.
• Interpret data from a box
plot.
• Compare multiple
distributions looking for
similar centers, spreads, and
overall shapes.
September 2016 Page 45 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Statistics and Probability
Develop understanding of statistical variability
6.SP.3
Recognize that a
measure of center for a
numerical data set
summarizes all of its
values with a single
number, while a
measure of variation
describes how its
values vary with a single
number.
Desired Student Performance
A student should know
• Measure of center refers to
mean, median, and mode.
• Any of these three represent
the entire data set with just
one number.
• Measure of variation refers to
range, interquartile range, or
mean absolute deviation.
• Range is the difference
between maximum and
minimum data values.
• Mean absolute deviation is
the average of the absolute
value of the distances of the
data values from the mean.
• How to attend to precision.
A student should understand
• When mean is used as the
measure of center, that
number may not itself be a
value from the data set.
• When median is used as the
measure of center, it may or
may not be a value from the
data set. That is dependent
on whether the data set is
made up of an even or odd
set of data.
• When mode is used as the
measure of center, it is an
actual value from the data
set.
A student should be able to do
• Calculate measures of center
(mean, median, and mode)
of a set of numerical data.
• Calculate measures of
variation by calculating
range, interquartile range, or
mean absolute deviation of a
set of numerical data.
• Cannot use a calculator but
must be able to use the
standard algorithm for
calculating.
September 2016 Page 46 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Statistics and Probability
Summarize and describe distributions
6.SP.4
Display numerical data
in plots on a number
line, including dot plots,
histograms, and box
plots.
Desired Student Performance
A student should know
• Dot plots are also called line
plots. Data are represented
by Xs or dots on a number
line.
• Histograms are bar graphs
where data are grouped and
displayed within intervals.
• Intervals must be continuous
(bars must be touching).
• Box plots, also called box-
and-whisker plots, graph five
summary measures. They
show the center as well as the
variability of the data.
• How to model with
mathematics.
• How to attend to precision.
A student should understand
• The difference between how
data are represented by dot
plots, histograms, and box
plots.
• Graphical representations
should be chosen based on
what information needs to be
communicated about the data
set.
A student should be able to do
• Organize and display data as
a line plot or dot plot.
• Organize and display data in
a histogram.
• Organize and display data in
a box plot.
• Calculate extremes, range,
median, and mean to be able
to display data in a box plot.
• Identify a graphical
representation that is
representative of a given
data set.
September 2016 Page 47 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Statistics and Probability
Summarize and describe distributions
6.SP.5a
Summarize numerical
data sets in relation to
their context such as by:
a. Reporting the number
of observations.
Desired Student Performance
A student should know
• Observations refer to the
number of data values in the
set.
• Measures of center are mean,
median, and mode.
• Measures of variability are
range, interquartile range, and
mean absolute deviation.
A student should understand
• Reporting the number of
observations does not by
itself lend any information to
measures of center or
variability, only how many
data values were collected.
A student should be able to do
• Use a four-function calculator
for rapid calculation of
measures of center or
variability.
• Report number of
observations.
September 2016 Page 48 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Statistics and Probability
Summarize and describe distributions
6.SP.5b
Summarize numerical
data sets in relation to
their context such as by:
b. Describing the nature
of the attribute under
investigation, including
how it was measured
and its units of
measurement.
Desired Student Performance
A student should know
• An attribute is a particular
characteristic or feature being
investigated: typical age of 6
th
graders, typical numbers of
pets, or how many states
have most students visited in
their lifetime.
• How to model with
mathematics.
A student should understand
• The difference between an
attribute and the units used
to measure that attribute.
A student should be able to do
• Identify the attribute being
investigated.
• Identify how the attribute was
measured and by what units.
September 2016 Page 49 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Statistics and Probability
Summarize and describe distributions
6.SP.5c
Summarize numerical
data sets in relation to
their context such as by:
c. Giving quantitative
measures of center
(median, and/or mean)
and variability
(interquartile range), as
well as describing any
overall pattern and any
striking deviations from
the overall pattern with
reference to the context
in which the data were
gathered.
Desired Student Performance
A student should know
• Measures of center are mean,
median, and mode.
• Mean is the average of all the
data sets.
• Median is the exact middle
data value.
• Measures of variability are
range, interquartile range, and
mean absolute deviation.
• Range is the difference in the
two extremes.
• Interquartile range (IQR) is
the difference between the
upper and lower quartiles.
• Mean absolute deviation
(MAD) is the average of the
absolute values of the
distances of the data values
from the mean.
A student should understand
• When the median and the
mean are the same data
value, or almost the same,
the distribution is said to
be symmetric.
• If the data does not
resemble a mirror image
due to clusters, gaps, or
outliers, the distribution is
said to be skewed.
• Outliers, extreme high or
low data values, have a
direct effect on the mean
of the data set.
A student should be able to do
• Calculate measures of
center: mean, median, and
mode.
• Calculate measures of
variability: range,
interquartile range, and
mean absolute deviation.
• Identify clusters, gaps,
extremes, and outliers in the
data set.
• Describe overall patterns and
how those patterns relate to
the context of the data.
• Describe any deviations from
the overall pattern and how
they relate to the context of
the data.
September 2016 Page 50 of 50
College- and Career-Readiness Standards for Mathematics
GRADE 6
Statistics and Probability
Summarize and describe distributions
6.SP.5d
Summarize numerical
data sets in relation to
their context such as by:
d. Relating the choice
of measures of center
and variability to the
shape of the data
distribution and the
context in which the
data were gathered.
Desired Student Performance
A student should know
• Measures of center: mean,
median, and mode.
• Measures of variability: range
and interquartile range
• Shape is described as
symmetrical or skewed.
• How to construct viable
arguments and critique the
reasoning of others.
A student should understand
• There is no wrong choice of
measure of the center—only
a wrong interpretation of it.
• The shape of the data should
be considered before
deciding on which measure
of center or variability should
be used to summarize the
data.
• The effect adding or
removing data values will
have on measures of center
or variability.
A student should be able to do
• Calculate measures of
center.
• Calculate measures of
variability.
• Draw inferences about the
shape of the distribution
using measures of center
and/or variability.
• Justify the use of a particular
measure of center or
variability based on the
shape of the data.
