Year long indiana_standards

Kindergarten Pacing Guide
Quarter 1 Quarter 2 Quarter 3 Quarter 4
Oral Counting:
By 1s * to 30. * from any # to 60. * from any # to 100. * from any # to 100.
By 10s * up to 30. * up to 60. * up to 100. * up to 100.
Backward from 20. from any # up to 100.
Object Counting:
Counting out Count 20 ordered items. * up to 20 items. up to 60, forming 10s. up to 100, forming 10s.
Subitizing /
Reasoning
* # talks with dots.
Creatively count up to
10 dots.
Develop # talks. Include
rekenreks, number lines,
and other manipulatives.
Develop # talks. Determine
which of two sets has more
objects using strategies.
Fluent # talks. Use words
to compare quantities
and explain reasoning.
Fingers * Show up to 5 fingers
with counting.
* Show up to 5 fingers
without counting.
with counting.
without counting.
Read & Write
Numbers
* Up to 10. * Up to 20.
Read # words up to 10.
Up to 50.
Read # words up to 10.
Up to 100 (or higher!).
Compare #s (to 20).
Addition &
Subtraction:
Modeling
Guided +/− within 10 using
fingers, objects, drawings,
acting, sounds.
Independent +/− within 10
using fingers, objects,
drawings, acting, sounds.
Independent +/− within 20
using choice of tools.
Problem
Writing
Problems with +/−, equal
groups, and patterns.
Guided problem writing for
equations.
Independent small group
problem writing.
Problem
Solving
Guided problem solving
Acting, Pictures, #s, Words.
Develop independent small
group solving, writing,
presenting.
Fluent independent small
presenting.
Geometry,
Measurement,
and Data
Describe and Identify:
triangle, square, circle,
rectangle, hexagon
above, behind, in front of,
below, beside, next to.
Understand concepts of time.
State attributes of:
triangle, square, circle
rectangle, hexagon.
Compare length, capacity,
weight, and temperature.
Understand concepts of time.
Describe and Identify:
cone, cube, cylinder, sphere
Classify 2-D and 3-D objects.
Describe scenes in terms of
2-D/3-D shapes and their
relative positions.
Create, extend, and give
rules for repeating and
growing patterns with
numbers and shapes
Sort objects by attributes.
* Indicates recommended formative assessment content.
Text in purple shows benchmarks not explicitly included in the Indiana Math Standards. See explanation section for rationales.

1st Grade Pacing Guide
Count * Read, write #s to 120.
Count forward or
backward by 1s, 5s, 10s.
* Count out up to 120
items, grouping in 10s.
Use ordinal # words.
Find the # that is
10 more or less than
any # up to 100.
Fluently count to find the
value of a collection of
dimes, nickels, and pennies.
Compare/Order Compare and arrange given
#s up to 20 in order.
Compare and arrange given
Represent #s Create representations
of a # up to 20
using objects, pictures,
and number sentences.
Create representations
of a # up to 50
* Create representations
of a # up to 100
Create equations where
each side totals a # up
to 100. Determine if a
given equation is true.
+/− Models +/− up to 20 using fingers,
objects & pictures.
+/− up to 50 using
objects & pictures.
* +/− up to 100 using
objects & pictures.
* +/− up to 100 using
objects & pictures.
+/− Strategies Develop strategies for +/−
within 20.
Explain strategies for +/−
within 20.
Develop strategies for +/−
within 100.
Explain strategies for +/−
within 100.
Fact Mastery Fluent mastery of +/−
facts within 5.
* Fluent mastery of +/−
facts within 10.
facts within 10.
Fluent use of strategies for
+/− facts within 20.
Problem Solving Guided problem solving
Develop independent / small
presenting.
Fluent independent / small
presenting.
Fluent independent / small
group problem writing
to match given equation.
Measurement Compare and order objects
by length, area, capacity,
Compare and order objects
by length, area, capacity,
Use non-standard units to
compare lengths, areas,
capacities, and weights.
Use non-standard units to
compare lengths, areas,
capacities, and weights.
Time Read hours and minutes
using digital clocks.
Read and write analog clock
times to the nearest hour.
Read and write times to the
nearest half hour.
Relate time to events: be-
fore/after, shorter/longer.
Geometry Name/draw/build 2-D and
3-D shapes.
Sort shapes by attribute.
Distinguish defining and
non-defining attributes.
Compose and decompose
shapes.
Partition circles and
rectangles into halves
and fourths/quarters.

2nd Grade Pacing Guide
Numbers:
Models
* Count by 1s, 2s, 5s, 10s,
and 100s forward or
backward up to 1000.
Use ordinal # words.
Represent #s to 1000:
models, drawings, #s
words, equations.
Explore even and odd #s.
* Use models, drawings,
numbers, to +/−
#s up to 1000.
Write equations for arrays.
* Use models, drawings,
numbers, to +/−
#s up to 1000.
Create/extend # patterns.
# line * Place given numbers
up to 100 on a # line.
* Place given numbers
* Place given numbers
Use <, =, and >
to compare numbers.
Addition &
Subtraction:
Strategies
Use / articulate
strategies/properties
for +/− within 20.
* Use / articulate
Mentally +/− 10 or 100
to/from a given #
within 1000.
Use / articulate
Fact mastery Fluent mastery of +/−
facts within 10.
facts within 20.
facts within 20.
facts within 20.
Problem Solving Guided problem solving
Develop individual/small
presenting, problem
creating for equations.
Fluent small group
solving, writing,
presenting, problem
creating for equations.
Fluent individual
solving, writing,
problem creating.
Include:
+/− situations
Count coins within $1
Count money within $100
Lengths up to 100 units
Picture and bar graphs.
Include:
+/− situations
Count coins within $5
Count money within $1000
Lengths up to 1000 units
Include:
+/− situations
+/− coins with exchanging
+/− $ with exchanging
Elapsed time
Include:
+/− situations
+/− coins with exchanging
+/− $ with exchanging
Elapsed time
Measurement:
Length /
Capacity
Measure length/capacity
using copies of a unit:
cm/m/in/ft/yd/cup/pint.
Measure/construct lengths
using rulers, yard/meter
sticks, tape measures, etc.
Measure same object with
diﬀerent units. Predict/
compare measurements.
Solve +/− length problems.
Draw pictures of rulers to
show solutions.
Time Tell time to the half hour. Tell time to the quarter hour. Tell time to nearest 5 min. Describe time relationships.
Geometry Name/draw/build/classify
2-D and 3-D shapes.
Compose and decompose
shapes.
Decompose rectangles into
unit squares.
Partition rectangles and
circles into 1
2 s, 1
3 s, 1
4 s.

3rd Grade Pacing Guide
Addition &
Subtraction:
Represent #s
Represent/compare/round
#s up to 10,000 with words
money, pictures, # lines,
expanded form, equations.
Represent/compare/round
#s up to 10,000 with words
money, pictures, # lines,
expanded form, equations.
to 10,000.
to 10,000.
Strategies Use manipulatives to
+/− within 1000.
* Students ﬂuently use /
articulate strategies for
+/− within 1000.
+/− within 1000.
+/− within 1000.
Fact mastery Fluent mastery of +/−
facts within 20.
Fluent mastery of +/−
facts within 20.
facts within 20.
facts within 20.
Multiplication &
Division:
Models &
strategies
Interpret / model ×/÷ to
100 using objects, pictures,
# lines, money, arrays,
grids. For ÷, include
equal groups, repeated
subtraction, inverse of ×.
Use models and
articulate strategies
for ×/÷ within 100.
* Write equal groups /
repeated subtraction
word problems for ÷.
* Fluently use and
articulate strategies
for ×/÷.
Understand properties of
×/÷ (including properties
of 0 and 1).
Create, extend, and ﬁnd the
rule for number patterns
involving multiplication.
Fact mastery Practice counting by 2s, 3s,
4s, 5s, and 6s
Practice counting by 7s, 8s,
9s, and 10s
×/÷ fact families
with factors up to 5.
×/÷ fact families
with factors up to 10.
Problem Solving Transition from guided to
independent small group
Small group solving, writing,
presenting, problem
creating given an equation.
* Fluent individual
solving, writing,
problem creating.
* Fluent small group
solving, writing,
problem creating.
Include:
+/ − / × /÷ interpretation
Unknown quantities
Fraction problems
Time, Money, Area
Lengths with mixed units
Line plots, tables, graphs
Scaled picture graphs.
Include:
Unknown quantities
Fraction problems
Time, Money, Area
Include:
Unknown quantities
Fraction problems
Time, Money, Area
Include:
Unknown quantities
Fraction problems
Time, Money, Area
Text in green shows benchmarks from a prior grade level in the Indiana Math Standards.

3rd Grade Pacing Guide
Fractions:
Models
Introduce area, set, # line
models with halves, thirds,
fourths. Include money,
time, rulers, hundred grids,
cup measures, tokens.
Use ratio and equal group
interpretations to show
fractions with area and set
models.
* Use area, set, # line
models, and assorted
manipulatives to show
equivalent fractions.
Use concrete models and
reasoning to compare
fraction sizes.
Diagrams Flexibly and accurately
partition circles,
rectangles, and other
shapes into halves,
thirds, and fourths.
Flexibly and accurately
partition circles,
rectangles, and other
shapes into halves,
thirds, . . . eighths.
Use grid paper to create
pictures of 1/2s, 1/3s,
. . . 1/20s.
Use hundred grids to show
1/2s, 1/4s, 1/5s, 1/10s,
1/20s, 1/25s, 1/50s,
and 1/100s.
Measurement &
Geometry:
Length
Measure/construct
lengths using rulers,
yard/meter sticks, tape
measures.
Measure/construct
lengths to 1/4 inch or
1/2 cm. Include 1/2, 1/4,
1/3 foot, yard, meter.
Make line plots of
measurements to the
nearest fraction of an inch,
foot, or yard.
Measure the same object
with different units.
Predict measurements
Include simple fractions.
Perimeter /
Area
Use physical square inches,
feet, yards, centimeters,
and meters to find areas
and perimeters.
Use strategies
(including distributive
property) to find areas
and perimeters of
rectangles without
counting every square.
Construct rectangles with
given areas or perimeters.
Find rectangles with the
same perimeter and
different areas or vice
versa.
Use strategies to find areas
and perimeters of large
rectangles or of shapes
composed of rectangles.
Include figures with
unknown side lengths.
Time Tell time to nearest minute. Strategies for elapsed time. * Elapsed time problems. * Elapsed time problems.
Other Use appropriate tools/units
to estimate/measure
temperature, weight/mass,
capacity. Include simple
fractions.
Use appropriate tools/units
to estimate/measure
temperature, weight/mass,
capacity. Include simple
fractions.
Identify/describe/categorize
2-D and 3-D shapes.
Identify/describe/draw lines
and line segments using
rulers and technology.
Identify/describe/categorize
2-D and 3-D shapes.
Identify/describe/draw lines
and line segments using
rulers and technology.

4th Grade Pacing Guide
Operations and
Algebra:
Numerical
expressions
* Represent/compare/
round #s to 1,000,000.
Use money, equations,
words, models, # lines.
* Represent/compare/
round #s to 1,000,000.
Use money, equations,
words, models, # lines.
to 1,000,000.
Use equations to describe
relationships between two
variables.
+/− Fluently use strategies
for multi-digit +/−.
Relate +/− algorithms
to a concrete model.
Fluently use the standard
algorithm to +/−
multi-digit #s.
Fluently use the standard
algorithm to +/−
multi-digit #s.
×/÷ Understand × as repeated +,
as area, as an operator.
Understand ÷ as sharing,
repeated −, inverse of ×.
Use manipulatives to
×/÷ multi-digit #s.
* Use properties of
operations and other
strategies for
multi-digit ×/÷.
* Use properties of
operations and other
strategies for
multi-digit ×/÷.
Fact mastery * Demonstrate mastery
of basic +/−/×/÷ facts.
* Demonstrate mastery
Find all factor pairs for #s
up to 100.
Problem Solving * Fluent individual/
presenting, problem
creating for multi-step
expressions.
* Fluent individual/
presenting, problem
expressions.
presenting, problem
equations.
presenting, problem
equations.
Include:
Multiplicative and additive
comparisons
Mixed operations
Unknown quantities
Fraction problems
Time, length, mass/weight
capacity with mixed units
Perimeter and area.
Include:
comparisons
Mixed operations
Unknown quantities
Fraction problems
Time, length, mass/weight,
Perimeter and area.
Include:
comparisons
Mixed operations
Unknown quantities
Fraction problems
Perimeter and area.
Include:
comparisons
Mixed operations
Unknown quantities
Fraction problems
Perimeter and area.

Fractions:
Equivalence
Use ratio/equal group
interpretations of
fractions/decimals to
show equivalence with
area, set, # line
models. Use hundred
grids, money, rulers,
time, tokens, packages.
* Use pictures,
equations, sentences to
explain fraction/
decimal equivalence.
Include word forms
and the grammar of
phrases involving
fractions/decimals.
Use models to explain why
a/b = (n × a)/(n × b).
Compare fractions/decimals
by finding common
numerators/denominators
or by comparing to a
benchmark.
Use models to +/− simple
fractions/decimals.
+/− fractions, decimals,
and mixed numbers with
common denominators.
Create equations with multi-
step expressions involving
fractions and decimals.
Measurement
Mixed units
Measure to find equivalent
quantities using different
units for length,
weight/mass, capacity, and
time. Include simple
Use diagrams, tables, and
other strategies to solve
measurement conversion
problems. Include simple
Use diagrams, tables, and
other strategies to solve
measurement conversion
problems. Include simple
Solve and write explanations
for real-world “how many”
questions involving large
#s, fractions, estimates,
measurements, surveys and
experiments.
Length Measure/construct
mm. Include fractions/
decimals of a
foot, yard, meter.
Measure/construct
decimals of a foot,
yard, meter.
Measure/construct
decimals of a foot,
yard, meter.
Measure/construct
decimals of a foot,
yard, meter.
Area Use physical square
inches / square cm /
square feet to find
perimeters and areas
of shapes composed of
rectangles
Use strategies to find
perimeters/areas of shapes
Discover area/perimeter
formulas for rectangles
and squares.
of large shapes composed
of rectangles. Include
problems with unknown
side lengths.
of large shapes
Include problems with
unknown side lengths.
Geometry Construct/measure/describe
angles, rays, perpendicular,
parallel lines using tools.
Construct/measure/describe
angles, rays, perpendicular,
parallel lines using tools.
Identify/construct/classify
triangles & quadrilaterals.
Recognize and draw lines
of symmetry in two-
dimensional figures.

Fractions:
+/− models
Use ratio and equal groups
approaches to demonstrate
fraction/decimal/percent
equivalence with area, set,
# line models.
* Use area, set, # line
models and strategies
to compare/+/−
decimals and fractions
with unlike
denominators.
Round fractions/
decimals. Use
benchmark fractions to
mentally estimate sums
and diﬀerences.
* Use models to explain
algorithms for adding
and subtracting
decimals and fractions
with unlike
denominators.
×/÷ models Use area, set, and
# line models to
explore interpretations
of fractions as division
of the numerator by the
denominator.
* Use area, set, and
# line models to
× fractions/decimals.
Interpret × as area,
repeated addition,
and scaling.
Use area, set, and # line
models to divide fractions
by whole #s and to divide
whole #s by unit fractions.
* Develop ﬂuency with
strategies for ×/÷ of
Operations and
Algebra:
Numerical
expressions
Generate equivalent
multi-step expressions
correctly using parentheses
and evaluate such
expressions.
Generate equivalent
multi-step expressions
correctly using parentheses
and evaluate such
expressions.
Express geometric and
scenario-based patterns
as equations, tables,
and graphs of ordered
pairs.
Express geometric and
scenario-based patterns
as equations, tables,
and graphs of ordered
pairs.
Strategies &
algorithms
for ×/÷
Understand how ×/÷ by
powers of 10 changes #s.
Use exponents to denote
powers of 10.
Use a concrete model to
explain the algorithm for ×
of multi-digit #s.
Use strategies to divide.
Strategies/algorithms for ×.
Find quotients with 4-digit
dividends and 2-digit
divisors using strategies.
Use properties as strategies
to estimate/compute
+/ − / × ÷ of whole #s,
fractions, and decimals.
Algorithms
for +/− and
×/÷ facts
Demonstrate mastery
of algorithms for multi-
digit +/− and basic
×/÷ facts.
Demonstrate mastery
×/÷ facts.
Demonstrate mastery
×/÷ facts.
Demonstrate mastery
×/÷ facts.

Problem Solving * Solve, write, present
solutions. Create
problems for multi-step
expressions.
* Solve, write, present
solutions. Create
problems for multi-step
expressions.
solutions. Problems
for expressions with
fractions/decimals.
solutions. Problems
for expressions with
fractions/decimals.
Include:
×/÷ including remainders
× and + comparisons
Fraction/decimal problems
Fermi-style problems
Means/medians/modes.
Include:
Include:
Include:
Measurement:
Mixed units
Measure with different units
(time, length, weight/mass,
capacity, temperature).
Include fractions/decimals.
Use diagrams/tables/graphs
to solve measurement
conversion problems.
Use diagrams/tables/graphs
to solve measurement
conversion problems.
Fluently solve real world
problems involving
measurement conversion.
Length * Measure/construct
lengths to 1/16 inch
or mm. Include
fractions/decimals of a
foot, yard, meter.
* Measure/construct
or mm. Include
foot, yard, meter.
* Measure/construct
or mm. Include
foot, yard, meter.
* Measure/construct
or mm. Include
foot, yard, meter.
Area Use diagrams to find areas
of rectangles with
fractional side lengths.
Find areas of shapes
Include fractional lengths.
Develop and use formulas for
areas of triangles,
parallelograms, trapezoids.
Solve real-world problems
involving perimeters/areas.
Volume Use physical cubic in./
cubic cm/cubic ft. to find
rectangular solid volumes.
Develop volume formulas
V = l × w × h, V = B × h
for rectangular prisms.
* Find volumes of rooms/
objects composed of
rectangular solids.
* Find volumes of rooms/
objects composed of
rectangular solids.
Geometry Construct triangles with
specified angle attributes
and/or length attributes.
Use compasses or technology
to draw circles. Identify
radius and diameter.
Construct quadrilaterals,
pentagons, hexagons with
specified attributes.
Classify polygons in a
hierarchy based on
properties.

Ratios and
Rates:
Models &
algorithms
* Use equal groups and
ratios to show fraction/
decimal/% equivalence
with area, set, # line
models. Compare/
+/−/× fractions.
* Interpret fractions as
quotients. Use models
to interpret fraction ÷.
Write and solve
scenarios for fraction ÷
expressions.
Use area, set, # line models
and strategies to
+/−/×/÷ fractions. Know
from memory commonly
used fractions and their
decimal and % equivalents.
* Fluently +/−/×/÷
fractions, decimals, %s
using strategies and
algorithms. Explain
why algorithms work
using models.
Ratios &
Rates
* Use ratio language and
notation to describe
relationships between
quantities.
* Solve ratio problems.
Use tables of equivalent
ratios, double # lines,
and tape diagrams.
Solve problems involving
unit rates, percents, and
unit conversions.
Make tables/graphs of
equivalent ratios. Note
that points form straight
lines through the origin.
Algebraic
Thinking:
Algebraic
expressions
Use properties of operations
as exceptions to order of
operations rules to
simplify computations.
Test calculators to see which
use order of operations.
Use different calculators to
compute expressions.
Use variable expressions
to create spreadsheets.
Evaluate expressions.
* Create a formula for
a scenario by replacing
#s with variables. Use
the formula in
similar situations.
Evaluate expressions.
* Create a formula for
#s with variables. Use
the formula in
similar situations.
* Translate verbal
calculation instructions
into algebraic
expressions and
recognize equivalent
ways of writing those
expressions.
Algebraic
Equations
Create expressions, using
parentheses and exponents.
Pair equal expressions to
form equations.
View equations as quantities
in balance. Guess and
check to find solutions. Log
guesses in tables/graphs.
Write inequalities to show
constraints. Use guess and
check to solve inequalities.
Graph inequality solutions.
Develop strategies to solve
linear equations. Use
inverse operations to
isolate unknown quantities.
Exploring
Relationships
Represent relationships
between variables as
equations, tables, graphs.
* Analyze patterns from
multiple perspectives
to create equations
relating different
algebraic expressions.
Guess and check and use
properties of operations to
see if expressions are equal.
Use variables to express
general properties of #s.
Write equations that express
a dependent variable in
terms of an independent
variable. Use graphs and
tables to find patterns.

Numbers and
Operations:
Algorithms
of algorithms for
multi-digit +/ − /×
* Use concrete models to
explain the long
division algorithm.
of algorithms for
multi-digit +/ − / × /÷
of algorithms for
multi-digit +/ − / × /÷
Factors and
multiples
List all factors and several
multiples of a given #.
Identify primes/composites.
Solve problems featuring
greatest common factors /
least common multiples.
Discover and use divisibility
tests. Use place value to
prove why each test works.
Given a sum of #s, use the
distributive property to
factor out the GCF.
Positive and
negative
rational #s
Represent ± #s with models
and explore real-world
contexts.
* Plot ± rational #s
on # lines and
coordinate planes.
* Plot ± rational #s
on # lines and
coordinate planes.
Order rational #s. Interpret
absolute value as distance
between #s on the # line.
Problem Solving * Fluent small group
solving, writing,
presenting, problem
expressions.
solving, writing,
presenting, problem
expressions.
* Fluent individual
solving, writing,
presenting, problem
equations.
solving, writing,
presenting, problem
equations.
Include:
Ratio and rate problems
Fraction/decimal/%s
Perimeter, area, volume
Ask statistical questions
Line plots, tables, graphs,
histograms, and box plots
Incorporate spreadsheets.
Include:
Fraction/decimal/%s
Include:
Fraction/decimal/%s
Include:
Fraction/decimal/%s
Geometry and
Measurement
Convert between English and
metric systems given
conversion factors.
Find areas by decomposing
complex shapes. Know
sums of interior ∠s for
triangles/quadrilaterals.
Construct prisms from nets
and compute volumes and
surface areas. Include
fractional edge lengths.
Draw polygons in the
coordinate plane. Find
lengths of horizontal/
vertical edges.

Proportions:
Models &
algorithms
Fluently +/ − / × /÷
rational #s using
strategies / algorithms.
Write and solve scenarios
involving numerical
expressions involving
fractions, decimals, %s.
Solve ratio, rate, conversion,
% problems. Use tables of
equivalent ratios, double #
lines, tape diagrams.
Set up and solve algebraic
proportions.
Solve multi-step ratio and
percent problems.
Linear
relationships
Explore a variety of real-
world linear relationships
and examine equations,
tables, and graphs that
model the data.
Interpret slope as a unit rate
in a variety of real-world
situations. Interpret
features of graphs including
x- and y-intercepts.
Understand features of
proportional relationships
presented as lines through
the origin, as tables, and as
equations.
Identify linear relationships
from graphs, tables,
equations, scenarios.
Convert among these
formats.
Algebraic
Thinking:
Writing
expressions
& equations
Evaluate multi-step
operations rules to simplify
computations.
Create a formula for
#s with variables. Use the
formula in similar
situations.
Write inequalities to show
constraints.
Analyze patterns from
different perspectives to
create equivalent
expressions. Use properties
to prove equivalence. Use
graphs and tables to find
patterns.
Translate verbal instructions
into algebraic expressions.
Write equations that express
a dependent variable in
terms of an independent
variable.
Technology Test calculators to see which
Fluently use spreadsheets to
solve real-world problems.
Use equations/inequalities as
spreadsheet conditionals.
Solving
equations
View equations as quantities
in balance. Guess and
check to solve equations
and inequalities. Log
When solving a multi-step
equation or inequality,
undo operations in the
reverse order they would be
performed according to
order of operations.
Record solution steps
using a formal algebraic
approach.
Use graph paper, a
spreadsheet, or a graphing
calculator to solve
equations and inequalities
in one variable by graphing
the left and right sides and
finding the intersection.
Understand why this
method works.
Fluently solve linear
using a formal algebraic
approach, retaining the
habit of checking solutions.

Numbers and
Operations:
Rationals &
irrationals
Convert fractions to decimals
using long ÷. Explore
patterns in remainders.
Plot rationals on # lines.
Know that irrational #s can
not be represented as the
ratio of whole #s.
Know some irrational #s.
Use pictures to explore
repeating decimals.
Plot real #s on # lines.
Given two real #s, find
rational and irrational
#s between them.
Positives &
negatives
Interpret addition and
subtraction of integers
using a # line model.
Discover rules for addition
and subtraction of positives
and negatives.
Use the distributive property
to derive rules for ×/÷ of
positives and negatives.
Explore real-world contexts
involving operations with
positive and negative #s.
Factors and
multiples
Solve problems featuring
GCF, LCM, square #s,
and square roots.
Use prime factorization to
find factors and multiples.
Use factors to find
√
s.
Discover and use divisibility
tests. Use place value to
prove why each test works.
Use the distributive property
to factor out the GCF from
Problem Solving Fluent small group solving
writing, presenting,
creating problems.
Fluent small group solving
creating problems.
creating problems.
creating problems.
Include:
Unknown quantities
Scaled diagrams
Proportional relationships
Fraction/decimal/%s
Positive/negative numbers
Angles, perimeter, area
Surface area, volume
Statistics and probability.
Include:
Unknown quantities
Scaled diagrams
Fraction/decimal/%s
Include:
Unknown quantities
Scaled diagrams
Fraction/decimal/%s
Include:
Unknown quantities
Scaled diagrams
Fraction/decimal/%s
Geometry and
Measurement
Construct all possible
triangles satisfying given
conditions.
Work with scale drawings,
similar polygons, and angle
relationships.
Understand and use formulas
for area and circumference
of a circle.
Find volumes of cylinders
and prisms. Construct nets
to compute surface area.

Functions: Identify independent and
dependent variables in
real-world scenarios.
Explore functions using
equations, tables, graphs.
Sketch graphs for scenarios.
Analyze features of
equations, tables, graphs
to describe relationships.
Contrast linear functions
presented in any format.
Analyze patterns to create
equivalent expressions.
Determine if a pattern is
linear using graphs/tables.
Construct equations, tables,
graphs for linear patterns.
Translate verbal instructions
and real-world situations
into algebraic expressions,
equations, inequalities,
and systems of
equations/inequalities.
Plotting
Data
Use spreadsheets and
graphing calculators to
create scatter plots of data.
Describe patterns in scatter
plots (clustering, outliers,
correlation, linearity).
Fit and interpret lines for
scatter plots informally,
and using technology.
Use equations modeling
linear relationships
to make predictions.
Expressions &
Equations:
operations rules to evaluate
multi-step expressions.
Explore properties of
exponents, square roots,
and inequalities.
Interpret and calculate with
numbers presented in
scientific notation.
Use systems of equations
in two variables to
represent scenarios. Use
guess and check to solve
systems of this type. Log
Technology Test calculators to see which
Fluently use spreadsheets to
solve real-world problems.
Use equations/inequalities as
spreadsheet conditionals.
Solving
equations/
inequalities
View equations as balanced
quantities. Guess to solve
equations/inequalities.
Log guesses in tables/graphs.
When solving a multi-step
equation or inequality,
undo operations in the
reverse order they would be
performed according to
order of operations.
Record solution steps using a
formal algebraic approach.
Use graphs to solve
equations/inequalities in
one variable by graphing
the left and right sides and
finding the intersection.
Explain why linear equations
may have one, zero, or
infinitely many solutions.
Solve systems of linear
with rational coefficients
by graphing by hand and
using technology.
Retain the habit of checking
solutions.

Numbers and
Operations:
Rationals &
irrationals
Convert fractions to decimals
using long ÷. Explore
patterns in remainders,
and understand why the
decimal form of a rational
# terminates or repeats.
Plot rational #s precisely on
a number line. Convert
repeating/terminating
decimals to fractions.
Understand and apply the
Pythagorean Theorem.
Know that irrational #s can
not be represented as the
ratio of whole #s.
Know some irrational #s.
Work through the proof
that
√
2 is irrational.
Given two real #s, find
#s between them.
Round irrational #s from the
Pythagorean Theorem to
the nearest 1/8 inch and
construct the length.
Fluent
computation
rational #s using
rational #s using
rational #s using
rational #s using
Problem Solving Fluent small group solving
creating problems.
creating problems.
creating problems.
creating problems.
Include:
Unknown quantities
Scaled diagrams
Fraction/decimal/%s
Pythagorean Theorem
Include:
Unknown quantities
Scaled diagrams
Fraction/decimal/%s
Pythagorean Theorem
Include:
Unknown quantities
Scaled diagrams
Fraction/decimal/%s
Pythagorean Theorem
Include:
Unknown quantities
Scaled diagrams
Fraction/decimal/%s
Pythagorean Theorem
Geometry and
Measurement
Identify attributes, slices,
volumes, and surface areas
of prisms, cylinders, cones,
spheres, and pyramids.
Inductively discover and use
the Pythagorean Theorem.
Understand the effect of
transformations of figures
in the coordinate plane and
demonstrate congruence.
Understand the effect of
transformations of figures
in the coordinate plane and
demonstrate similarity.

Kindergarten Indiana Math Standards By Strand
Kindergarten – Oral Counting
Related Indiana Math Standards include:
K.NS.1 Count to 100 by ones and by tens and count on by one from any number.
K.NS.3 Find the number that is one more than or one less than any whole number up to 20.
Items not explicitly mentioned in the Indiana Math Standards:
Counting backwards. Many computational strategies that need to be developed in 1st grade would be
greatly aided if students had greater ﬂuency with counting backwards coming out of Kindergarten.
Incorporating backwards counting as a classroom transition routine during the second half of the
year can give students this foundation without adding explicit instructional time.
Kindergarten – Object Counting
K.NS.4 Say the number names in standard order when counting objects, pairing each object with one
and only one number name and each number name with one and only one object. Understand that
the last number name said describes the number of objects counted and that the number of objects
is the same regardless of their arrangement or the order in which they were counted.
K.NS.5 Count up to 20 objects arranged in a line, a rectangular array, or a circle. Count up to 10 objects
in a scattered conﬁguration. Count out the number of objects, given a number from 1 to 20.
K.NS.6 Recognize sets of 1 to 10 objects in patterned arrangements and tell how many without counting.
K.NS.7 Identify whether the number of objects in one group is greater than, less than, or equal to the
number of objects in another group (e.g., by using matching and counting strategies).
K.NS.9 Use correctly the words for comparison, including: one and many; none, some, and all; more and
less; most and least; and equal to, more than, and less than.
K.NS.11 Develop initial understandings of place value and the base 10 number system by showing equiv-
alent forms of whole numbers from 10 to 20 as groups of tens and ones using objects and drawings.
K.CA.3 Use objects, drawings, etc., to decompose numbers less than or equal to 10 into pairs in more than
one way, and record each decomposition with a drawing or an equation (e.g., 5=2+3 and 5=4+1).
[In Kindergarten, students should see equations and be encouraged to trace them, however, writing
equations is not required.]
K.CA.4 Find the number that makes 10 when added to the given number for any number from 1 to 9
(e.g., by using objects or drawings), and record the answer with a drawing or an equation.
Counting out beyond 20 items, forming 10s. The Indiana Math Standards only ask Kindergarten
students to master counting out up to 20 objects, and they are only required to form 10s and 1s with
numbers in the teens. However, the pattern of making 10s and 1s is more readily apparent when
working with larger numbers. Also, the Indiana Math Standards do require students to count by 10s
to 100, and so pairing that skill with physical objects is natural and instructionally useful.

Subitizing and Number Talks. “Subitizing” means visually grouping objects so that they are easier
to notice and count. A “number talk” is a group conversation guided by the teacher that allows
students to verbalize critical thinking skills about math concepts. Number talks with dots involve
showing students a wide variety of scattered dot configurations – some can be similar to dot patterns
on dice or dominoes, but other configurations should also be included. Students discuss how many
dots they think there are and different ways they could mentally group the dots. Number talks also
commonly involve other kinds of manipulatives such as rekenreks, five and ten frames, number lines,
et cetera.
These activities support Indiana Math Standards relating to counting, cardinality, patterned arrange-
ments, addition, subtraction, and even geometry (when students “see” squares or triangles formed
by the dots). These activities also teach students to verbalize mathematical ideas, and to develop
higher level thinking skills. Practice with number talks in Kindergarten also lays a strong foundation
for number sense and critical 1st grade concepts.
Two resources supporting number talks are:
• Kara Kolson, Suzanne Mole, and Manuel Silva. “Dot Card and Ten Frame Activities.” Numer-
acy Project, Winnipeg School Division, 2005 – 2006.
• Sherry Parrish. Number Talks: Helping Children Build Mental Math and Computation Strate-
gies. Math Solutions, Sausalito, CA, 2010.
Finger Cardinality. The Indiana Math Standards do imply the use of fingers as part of addition and
subtraction. However, many children need more practice with quickly showing a given number
of fingers. This is a critical skill that 1st grade students need and do not usually have time to
practice. This skill is closely related to Indiana Math Standards relating to counting, cardinality,
addition, subtraction, and place value. Recent research shows that “finger gnosis” is a predictor of
mathematical ability, and suggests that the area of the brain involving awareness of fingers is closely
connected with areas of the brain governing number sense and computation.
The following article contains more information about finger gnosis and mathematical ability:
• Marcie Penner-Wilger and Michael L. Anderson. “The relation between finger gnosis and math-
ematical ability: why redeployment of neural circuits best explains the finding.” Frontiers in
Psychology. 2013; 4: 877.
Kindergarten – Read & Write Numbers
K.NS.2 Write whole numbers from 0 to 20 and recognize number words from 0 to 10. Represent a number
of objects with a written numeral 0 – 20 (with 0 representing a count of no objects).
K.NS.8 Compare the values of two numbers from 1 to 20 presented as written numerals.
Recognizing numbers beyond 20. The Indiana Math Standards only require mastery of number recog-
nition up to 20, although they require verbal counting up to 100. Although students do not need to
master number recognition beyond 20, they should still see written numbers beyond 20 and practice
writing them when appropriate. Kindergarten students enjoy saying and writing really large num-
bers, so teachers should be prepared to correctly write and say numbers up to a billion (and probably
beyond).

Kindergarten – Addition & Subtraction
K.NS.10 Separate sets of ten or fewer objects into equal groups.
K.CA.1 Use objects, drawings, mental images, sounds, etc., to represent addition and subtraction within
10.
K.CA.2 Solve real-world problems that involve addition and subtraction within 10 (e.g., by using objects
or drawings to represent the problem).
K.CA.5 Create, extend, and give an appropriate rule for simple repeating and growing patterns with
numbers and shapes.
Adding and subtracting for sums from 10 up to 20. The Indiana Math Standards only require ex-
posure to addition and subtraction within 10, though Kindergarten students do decompose numbers
in the teens into a sum of a ten and some ones. Exposure to addition and subtraction within 20 or
beyond during the fourth quarter makes sense if the students are already comfortable working within
10.
Kindergarten – Geometry, Measurement, and Data
K.G.1 Describe the positions of objects and geometric shapes in space using the terms inside, outside,
between, above, below, near, far, under, over, up, down, behind, in front of, behind, in front of, next
to, the the left of and to the right of.
K.G.2 Compare two- and three-dimensional shapes in diﬀerent sizes and orientations, using informal
language to describe their similarities, diﬀerences, parts (e.g., number of sides and vertices/“corners”)
and other attributes (e.g., having sides of equal length).
K.G.3 Model shapes in the world by composing shapes from objects (e.g., sticks and clay balls) and
drawing shapes.
K.G.4 Compose simple shapes to form larger shapes. (e.g., create a rectangle composed of two triangles).
K.M.1 Make direct comparisons of the length, capacity, weight, and temperature of objects, and recognize
which object is shorter, longer, taller, lighter, heavier, warmer, cooler, or holds more.
K.MD.2 Understand concepts of time, including: morning, afternoon, evening, today, yesterday, tomor-
row, day, week, month, and year. Understand that clocks and calendars are tools that measure
time.
K.DA.1 Identify, sort, and classify objects by size, number, and other attributes. Identify objects that
do not belong to a particular group and explain the reasoning used.

1st Grade Indiana Math Standards By Strand
1st Grade – Count, Compare/Order
1.NS.1 Count to at least 120 by ones, fives, and tens from any given number. In this range, read and
write numerals and represent a number of objects with a written numeral.
1.NS.3 Match the ordinal numbers first, second, third, etc., with an ordered set up to 10 items.
1.NS.4 Use place value understanding to compare two two-digit numbers based on meanings of the tens
and ones digits, recording the results of comparisons with the symbols >, =, and <.
1.NS.5 Find mentally 10 more or 10 less than a given two-digit number without having to count, and
explain the thinking process used to get the answer.
1.M.3 Find the value of a collection of pennies, nickels, and dimes.
Counting backward by 1s, 5s, and 10s. The Indiana Math Standards do not explicitly state that stu-
dents must be able to count backward. However, practicing this skill significantly increases student
computational fluency and number sense. It also assists with 1.NS.5 which asks students to determine
the number that is 1 more / 1 less or 10 more / 10 less.
1st Grade – Represent #s / +/− Models
1.NS.2 Understand that 10 can be thought of as a group of ten ones – called a “ten.” Understand that
the numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or
nine ones. Understand that the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three,
four, five, six, seven, eight, or nine tens (and 0 ones).
1.NS.6 Show equivalent forms of whole numbers as groups of tens and ones, and understand that the
individual digits of a two-digit number represent amounts of tens and ones.
1.CA.5 Add within 100, including adding a two-digit number and a one-digit number, and adding a
two-digit number and a multiple of 10, using models or drawings based on place value, properties
of operations, and/or the relationship between addition and subtraction; describe the strategy and
explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens,
ones and ones, and that sometimes it is necessary to compose a ten.
1.CA.6 Understand the meaning of the equal sign, and determine if equations involving addition and
subtraction are true or false (e.g., Which of the following equations are true and which are false?
6 = 6, 7 = 8 − 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2).
1.M.3 Find the value of a collection of pennies, nickels, and dimes.

1st Grade – +/− Strategies / Fact Mastery
1.NS.5 Find mentally 10 more or 10 less than a given two-digit number without having to count, and
explain the thinking process used to get the answer.
1.CA.1 Demonstrate fluency with addition facts and the corresponding subtraction facts within 20. Use
strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a
number leading to a ten (e.g., 13−4 = 13−3−1 = 10−1 = 9); using the relationship between addition
and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 − 8 = 4); and creating equivalent but
easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Understand the role of 0 in addition and subtraction.
1.CA.5 Add within 100, including adding a two-digit number and a one-digit number, and adding a
two-digit number and a multiple of 10, using models or drawings based on place value, properties
of operations, and/or the relationship between addition and subtraction; describe the strategy and
explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens,
ones and ones, and that sometimes it is necessary to compose a ten.
1.CA.7 Create, extend, and given an appropriate rule for number patterns using addition within 100.
1st Grade – Problem Solving
1.CA.2 Solve real-world problems involving addition and subtraction within 20 in situations of adding
to, taking from, putting together, taking apart, and comparing, with unknowns in all parts of the
addition or subtraction problem (e.g., by using objects, drawings, and equations with a symbol for
the unknown number to represent the problem).
1.CA.3 Create a real-world problem to represent a given equation involving addition and subtraction
within 20.
1.CA.4 Solve real-world problems that call for addition of three whole numbers whose sum is within 20
(e.g., by using objeccts, drawings, and equations with a symbol for the unknown number to represent
the problem).
1.DA.1 Organize and interpret data with up to three choices (What is your favorite fruit? apples, bananas,
oranges); ask and answer questions about the total number of data points, how many in each choice,
and how many more or less in once choice compared to another.
1st Grade – Measurement, Time, and Geometry
1.G.1 Identify objects as two-dimensional or three-dimensional. Classify and sort two-dimensional and
three-dimensional objects by shape, size, roundness, and other attributes. Describe how two-dimensional
shapes make up the faces of three-dimensional objects.
1.G.2 Distinguish between defining attributes of two- and three-dimensional shapes (e.g., triangles are
closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size). Create
and draw two-dimensional shapes with defining attributes.

1.G.3 Use two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-
circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right
circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
[In grade 1, students do not need to learn formal names such as “right rectangular prism.”]
1.G.4 Partition circles and rectangles into two and four equal parts; describe the parts using the words
halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of. Describe the
whole as two of, or four of, the parts. Understand for partitioning circles and rectangles into two and
four equal parts that decomposing into more equal parts creates smaller parts.
1.M.1 Use direct comparison or a nonstandard unit to compare and order objects according to length,
area, capacity, weight, and temperature.
1.M.2 Tell and write time to the nearest half-hour and relate time to events (before/after, shorter/longer)
using analog clocks. Understand how to read hours and minutes using digital clocks.

2nd Grade Indiana Math Standards By Strand
2nd Grade – Numbers
2.NS.1 Count by ones, twos, fives, tens, and hundreds up to at least 1,000 from any given number.
2.NS.2 Read and write whole numbers up to 1,000. Use words, models, standard form and expanded
form to represent and show equivalent forms of whole numbers up to 1,000.
2.NS.3 Plot and compare whole numbers up to 1,000 on a number line.
2.NS.4 Match the ordinal numbers first, second, third, etc., with an ordered set up to 30 items.
2.NS.5 Determine whether a group of objects (up to 20) has an odd or even number of members (e.g., by
placing that number of objects in two groups of the same size and recognizing that for even numbers
no object will be left over and for odd numbers one object will be left over, or by pairing objects or
counting them by 2s).
2.NS.6 Understand that the three digits of a three-digit number represent amounts of hundreds, tens,
and ones (e.g., 706 equals 7 hundreds, 0 tens, and 6 ones). Understand that 100 can be thought of
as a group of ten tens – called a “hundred.” Understand that the numbers 100, 200, 300, 400, 500,
600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens
and 0 ones).
2.CA.5 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows
and up to 5 columns; write an equation to express the total as a sum of equal groups.
2.CA.7 Create, extend, and give an appropriate rule for number patterns using addition and subtraction
within 1000.
2.M.7 Find the value of a collection of pennies, nickels, dimes, quarters, and dollars.
2nd Grade – Addition and Subtraction
2.CA.1 Add and subtract fluently within 100.
2.CA.4 Add and subtract within 1000, using models or drawings and strategies based on place value, prop-
erties of operations, and/or the relationship between addition and subtraction; desribe the strategy
and explain the reasoning used. Understand that in adding or subtracting three-digit numbers, one
adds or subtracts hundreds and hundreds, tens and tens, ones and ones, and that sometimes it is
necesary to compose or decompose tens or hundreds.
2.CA.6 Show that the order in which two numbers are added (commutative property) and how the
numbers are grouped in addition (associative property) will not change the sum. These properties
can be used to show that numbers can be added in any order.

2nd Grade – Problem Solving
2.CA.2 Solve real-world problems involving addition and subtraction within 100 in situations of adding
to, taking from, putting together, taking apart, and comparing, with unknowns in all parts of the
addition or subtraction problem (e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem). Use estimation to decide whether answers are reasonable in
addition problems.
2.CA.3 Solve real-world problems involving addition and subtraction within 100 in situations involving
lengths that are given in the same units (e.g., by using drawings, such as drawings of rulers, and
equations with a symbol for the unknown number to represent the problem).
2.M.5 Tell and write time to the nearest five minutes from analog clocks, using a.m. and p.m. Solve
real-world problems involving addition and subtraction of time intervals on the hour or half hour.
2.M.7 Find the value of a collection of pennies, nickels, dimes, quarters, and dollars.
2.DA.1 Draw a picture graph (with single-unit scale) and a bar graph (with single-unit scale) to represent
a data set with up to four choices (What is your favorite color? red, blue, yellow, green). Solve simple
put-together, take-apart, and compare problems using information presented in the graphs.
2nd Grade – Measurement
2.M.1 Describe the relationships among inch, foot, and yard. Describe the relationship between centimeter
and meter.
2.M.2 Estimate and measure the length of an object by selecting and using appropriate tools, such as
rulers, yardsticks, meter sticks, and measuring tapes to the nearest inch, foot, yard, centimeter, and
meter.
2.M.3 Understand that the length of an object does not change regardless of the units used. Measure the
length of an object twice using length units of different lengths for the two measurements. Describe
how the two measurements relate to the size of the unit chosen.
2.M.4 Estimate and measure volume (capacity) using cups and pints.
2.M.5 Tell and write time to the nearest five minutes from analog clocks, using a.m. and p.m. Solve
real-world problems involving addition and subtraction of time intervals on the hour or half hour.
2.M.6 Describe relationships of time, including: seconds in a minute; minutes in an hour; hours in a day;
days in a week; and days, weeks, and months in a year.
Constructing lengths. The Indiana Math Standards do not explicitly ask students to construct lengths,
but I have found this to be a critical skill that needs extensive practice. Many local employers report
that the lack of proficiency of this skill in the local workforce is a major barrier to greater economic
prosperity. Constructing lengths rather than merely measuring develops students motor skills and
attention to precision. Students should be able to draw a line or cut out a strip of paper accurately
(to the nearest inch or centimeter). They should be able to tape together strips of paper to make a
strip measuring multiple feet or yards. It often helps to trace a correct version of the strip so that
students can check their accuracy independently (I like to invite students to play a game pretending

that they are cutting out frog tongues that are just long enough to reach a ﬂy. If their tongue is the
right length, then their frog gets a point.)
2nd Grade – Geometry
2.G.1 Identify, describe, and classify two- and three-dimensional shapes (triangle, square, rectangle, cube,
right rectangular prism) according to the number and shape of faces and the number of sides and/or
vertices. Draw two-dimensional shapes.
2.G.2 Create squares, rectangles, triangles, cubes, and right rectangular prisms using appropriate mate-
rials.
2.G.3 Investigate and predict the result of composing and decomposing two- and three-dimensional shapes.
2.G.4 Partition a rectangle into rows and columns of same-size (unit) squares and count to ﬁnd the total
number of same-size squares.
2.G.5 Partition circles and rectangles into two, three, or four equal parts; describe the parts using the
words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four
fourths. Recognize that equal parts of identical wholes need not have the same shape.

3rd Grade Indiana Math Standards By Strand
3rd Grade – Addition and Subtraction
3.NS.1 Read and write whole numbers up to 10,000. Use words, models, standard form and expanded
form to represent and show equivalent forms of whole numbers up to 10,000.
3.NS.2 Compare two whole numbers up to 10,000 using >, =, < symbols.
3.NS.9 Use place value understanding to round 2- and 3-digit whole numbers to the nearest 10 or 100.
3.C.1 Add and subtract whole numbers ﬂuently within 1000.
Note that “Using manipulatives to add and subtract within 1000” and “Fluent mastery of addition and
subtraction facts within 20” are both from 2nd grade Indiana Math Standards. Third grade students often
need to review and reinforce these skills to achieve mastery.
3rd Grade – Multiplication and Division
3.C.2 Represent the concept of multiplication of whole numbers with the following models: equal-sized
groups, arrays, area models, and equal “jumps” on a number line. Understand the properties of 0
and 1 in multiplication.
3.C.3 Represent the concept of division of whole numbers with the following models: partitioning, sharing,
and an inverse of multiplication. Understand the properties of 0 and 1 in division.
3.C.4 Interpret whole-number quotients of whole numbers (e.g., interpret 56 ÷ 8 as the number of objects
in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when
56 objects are partitioned into equal shares of 8 objects each).
3.C.5 Multiply and divide within 100 using strategies, such as the relationship between multiplication and
division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8), or properties of operations.
3.C.6 Demonstrate ﬂuency with multiplication facts and corresponding division facts of 0 to 10.
3.AT.4 Interpret a multiplication equation as equal groups (e.g., interpret 5 × 7 as the total number of
objects in 5 groups of 7 objects each). Represent verbal statements of equal groups as multiplication
equations.
3.AT.5 Determine the unknown whole number in a multiplication or division equation relating three whole
numbers.
3.AT.6 Create, extend, and give an appropriate rule for number patterns using multiplication within 1000.
3rd Grade – Problem Solving
3.AT.1 Solve real-world problems involving addition and subtraction of whole numbers within 1000 (e.g.,
by using drawings and equations with a symbol for the unknown number to represent the problem).

3.AT.2 Solve real-world problems involving whole number multiplication and division within 100 in sit-
uations involving equal groups, arrays, and measurement quantities (e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem).
3.AT.3 Solve two-step real-world problems using the four operations of addition, subtraction, multiplica-
tion, and division (e.g., by using drawings and equations with a symbol for the unknown number to
represent the problem).
3.M.4 Find the value of any collection of coins and bills. Write amounts less than a dollar using the cents
symbol and write larger amounts using the $ symbol in the form of dollars and cents (e.g., $4.59).
Solve real-world problems to determine whether there is enough money to make a purchase.
3.DA.1 Create scaled picture graphs, scaled bar graphs, and frequency tables to represent a data set –
including data collected through observations, surveys, and experiments – with several categories.
Solve one- and two-step “how many more” and “how many less” problems regarding the data and
make predictions based on the data.
3rd Grade – Fractions
The Indiana Math Standards require that 3rd grade students work concretely with many different models of
fractions including area models, set models, and number line models. Students should also work concretely
with fractions in the contexts of money, time, rulers, hundred grids, cup measures, tokens, items that come
in packs, et cetera.
Students should also frequently work with fractions bigger than 1. Students should be able to count out
23
4
of an inch on a ruler to see that this fraction is equivalent to 5
3
4
inches.
3.NS.3 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b
equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. [In grade 3,
limit denominators of fractions to 2, 3, 4, 6, 8.]
3.NS.4 Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the
whole, and partitioning it into b equal parts. Recognize that each part has size 1/b and that the
endpoint of the part based at 0 locates the number 1/b on the number line.
3.NS.5 Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize
that the resulting interval has size a/b and that its endpoint locates the number a/b on the number
line.
3.NS.6 Understand two fractions as equivalent (equal) if they are the same size, or the same point on a
number line.
3.NS.7 Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the
fractions are equivalent, e.g., by using a visual fraction model.
3.NS.8 Compare two fractions with the same numerator or the same denominator by reasoning about
their size based on the same whole. Record the results of comparisons with the symbols >, =, or <,
and justify the conclusions, e.g., by using a visual fraction model.
3.G.4 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the
whole. (1/2, 1/3, 1/4, 1/6, 1/8).

Use ratio and equal group interpretations to show fractions with area and set models. The In-
diana Math Standards only require equal group interpretations of fractions in 3rd grade, leaving
ratio interpretations until 6th grade. However, I have found that ratio interpretations arise naturally
among 3rd, 4th, and 5th grade students working with area and set models and causes confusion if
not addressed early on in a simple way. To show the fraction
2
3
with 12 two-colored tokens, we could
take a ratio approach and make 2 out of every 3 tokens be red, continuing until we have done this
with all 12 tokens. If we use the equal groups approach, we would divide the 12 tokens into three
equal groups first, and then make two of the groups be red. In either case,
8
12
of the tokens will be
red in the end, showing that
2
3
is equivalent to that fraction.
Use grid paper or hundred grids to show fractions. The Indiana Math Standards only require stu-
dents to master halves through eighths in 3rd grade. The use of grids fits well with ideas of multi-
plication and division also introduced in 3rd grade, and so I recommend introducing them early as a
powerful model for accurately depicting fractions of any size.
3rd Grade – Measurement & Geometry
3.M.1 Estimate and measure the mass of objects in grams (g) and kilograms (kg) and the volume of
objects in quarts (qt), gallons (gal), and liters (l). Add, subtract, multiply, or divide to solve one-
step real-world problems involving masses or volumes that are given in the same units (e.g., by using
drawings, such as a beaker with a measurement scale, to represent the problem).
3.M.2 Choose and use appropriate units and tools to estimate and measure length, weight, and tempera-
ture. Estimate and measure length to a quarter-inch, weight in pounds, and temperature in degrees
Celsius and Fahrenheit.
3.M.3 Tell and write time to the nearest minute from analog clocks, using a.m. and p.m., and measure
time intervals in minutes. Solve real-world problems involving addition and subtraction of time
intervals in minutes.
3.M.5 Find the area of a rectangle with whole-number side lengths by modeling with unit squares, and
show that the area is the same as would be found by multiplying the side lengths. Identify and draw
rectangles with the same perimeter and different areas or with the same area and different perimeters.
3.M.6 Multiply side lengths to find areas of rectangles with whole-number side lengths to solve real-world
problems and other mathematical problems, and represent whole-number products as rectangular
areas in mathematical reasoning.
3.M.7 Find perimeters of polygons given the side lengths or by finding an unknown side length.
3.DA.2 Generate measurement data by measuring lengths with rulers to the nearest quarter of an inch.
Display the data by making a line plot, where the horizontal scale is marked off in appropriate units,
such as whole numbers, halves, or quarters.
3.G.1 Identify and describe the following: cube, sphere, prism, pyramid, cone, and cylinder.
3.G.2 Understand that shapes (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having
four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recog-
nize and draw rhombuses, rectangle, and squares as examples of quadrilaterals. Recognize and draw

examples of quadrilaterals that do not belong to any of these subcategories.
3.G.3 Identify, describe, and draw points, lines and line segments using appropriate tools (e.g., ruler,
straightedge, and technology), and use these terms when describing two-dimensional shapes.
Items not explicitly mentioned in the Indiana Math Standards
but I have found this to be a critical skill that needs extensive practice. Many local employers
report that the lack of proficiency of this skill in the local workforce is a major barrier to greater
economic prosperity. Constructing lengths rather than merely measuring develops students motor
skills and attentiveness to precision. Students should be able to draw a line or cut out a strip of
paper accurately (to the nearest 1/4 inch or 1/2 centimeter). They should be able to tape together
strips of paper to make a strip measuring multiple feet or yards. It often helps to trace a correct
version of the strip so that students can check their accuracy independently (I like to invite students
to play a game pretending that they are cutting out frog tongues that are just long enough to reach
a fly. If their tongue is the right length, then their frog gets a point.)
Constructing 1/2, 1/4, 1/3 foot, yard, meter The Standards do ask students to work with lengths
that are accurate to 1/4 inch or 1/2 cm. The Standards do not explicitly ask students to work with
fractions of a foot, yard, and meter. I added this suggestion because it is consistent with the overall
arc of topics in 3rd grade and gives more practice with length measurement and fractions.
Predicting lengths in one kind of unit given a measurement using another kind of unit. The In-
diana Math Standards for 2nd grade do ask students to measure the same length using many different
kinds of units, and they ask students to predict which measurement will need a larger number. I
think it is helpful to revisit this in 3rd grade, and to take advantage of this activity to ask students to
guess how many units of another type would be required. This practice plants the seeds for estima-
tion and conversion skills. Students should take a moment to draw a diagram to try to predict the
value, and then record their guesses before measuring using the other units to check their answers.
Including simple fractions such as 1/2s, 1/3s, 1/4s, and 1/10s is also helpful for helping students to
contextualize their understanding of fractions.

4th Grade Indiana Math Standards By Strand
4th Grade – Operations and Algebra
4.NS.1 Read and write whole numbers up to 1,000,000. Use words, models, standard form and expanded
form to represent and show equivalent forms of whole numbers up to 1,000,000.
4.NS.2 Compare two whole numbers up to 1,000,000 using >, =, and < symbols.
4.NS.8 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is
a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a
multiple of a given one-digit number.
4.NS.9 Use place value understanding to round multi-digit whole numbers to any given place value.
4.C.1 Add and subtract multi-digit whole numbers fluently using a standard algorithmic approach.
4.C.2 Multiply a whole number of up to four digits by a one-digit whole number and multiply two two-
digit numbers, using strategies based on place value and the properties of operations. Describe the
strategy and explain the reasoning.
4.C.3 Find whole-number quotients and remainders with up to four-digit dividends and one- digit divisors,
using strategies based on place value, the properties of operations, and/or the relationship between
multiplication and division. Describe the strategy and explain the reasoning.
4.C.4 Multiply fluently within 100.
4.C.7 Show how the order in which two numbers are multiplied (commutative property) and how numbers
are grouped in multiplication (associative property) will not change the product. Use these properties
to show that numbers can by multiplied in any order. Understand and use the distributive property.
4.AT.2 Recognize and apply the relationships between addition and multiplication, between subtraction
and division, and the inverse relationship between multiplication and division to solve real-world and
other mathematical problems.
4.AT.6 Understand that an equation, such as y = 3x + 5, is a rule to describe a relationship between two
variables and can be used to find a second number when a first number is given. Generate a number
pattern that follows a given rule.
Fluently use strategies for multi-digit +/−. The Indiana Math Standards ask students to learn the
algorithm for multi-digit +/−, assuming that they have a thorough mastery of using strategies
for multi-digit +/− from their work at prior grade levels. I think it is important to review these
approaches and also to send the message that students should continue to use a variety of strategies
even after they learn the algorithms.
Demonstrate mastery of basic +/ − / × /÷ facts. Students should have learned the basic facts for
the four operations in second and third grade. However, it is likely that they will need continuing
practice on these skills to achieve real fluency.

4th Grade – Problem Solving
4.AT.1 Solve real-world problems involving addition and subtraction of multi-digit whole numbers (e.g.,
by using drawings and equations with a symbol for the unknown number to represent the problem).
4.AT.2 Recognize and apply the relationships between addition and multiplication, between subtraction
and division, and the inverse relationship between multiplication and division to solve real-world and
4.AT.3 Interpret a multiplication equation as a comparison (e.g., interpret 35 = 5×7 as a statement that
35 is 5 times as many as 7, and 7 times as many as 5). Represent verbal statements of multiplicative
comparisons as multiplication equations.
4.AT.4 Solve real-world problems with whole numbers involving multiplicative comparison (e.g., by using
drawings and equations with a symbol for the unknown number to represent the problem), distin-
guishing multiplicative comparison from additive comparison. [In grade 4, division problems should
not include a remainder.]
4.AT.5 Solve real-world problems involving addition and subtraction of fractions referring to the same
whole and having common denominators (e.g., by using visual fraction models and equations to
4.DA.1 Formulate questions that can be addressed with data. Use observations, surveys, and experiments
to collect, represent, and interpret the data using tables (including frequency tables), line plots, and
bar graphs.
4.DA.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve
problems involving addition and subtraction of fractions by using data displayed in line plots.
4.DA.3 Interpret data displayed in a circle graph.
4th Grade – Fractions
4.NS.3 Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers.
Name and write mixed numbers using objects or pictures. Name and write mixed numbers as
improper fractions using objects or pictures.
4.NS.4 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction
models, with attention to how the number and size of the parts differ even though the two fractions
themselves are the same size. Use this principle to recognize and generate equivalent fractions. [In
grade 4, limit denominators of fractions to 2, 3, 4, 5, 6, 8, 10, 25, 100.]
4.NS.5 Compare two fractions with different numerators and different denominators (e.g., by creating
common denominators or numerators, or by comparing to a benchmark such as 0, 1/2, and 1).
Recognize that comparisons are valid only when the two fractions refer to the same whole. Record
the results of comparisons with symbols >, =, or <, and justify the conclusions (e.g., by using a
visual fraction model).
4.NS.6 Write tenths and hundredths in decimal and fraction notations. Use words, models, standard form
and expanded form to represent decimal numbers to hundredths. Know the fraction and decimal
equivalents for halves and fourths (e.g., 1/2 = 0.5 = 0.50, 7/4 = 1 3/4 = 1.75).

4.NS.7 Compare two decimals to hundredths by reasoning about their size based on the same whole.
Record the results of comparisons with the symbols >, =, or <, and justify the conclusions (e.g., by
using a visual model).
4.C.5 Add and subtract fractions with common denominators. Decompose a fraction into a sum of
fractions with common denominators. Understand addition and subtraction of fractions as combining
and separating parts referring to the same whole.
4.C.6 Add and subtract mixed numbers with common denominators (e.g. by replacing each mixed number
with an equivalent fraction and/or by using properties of operations and the relationship between
addition and subtraction).
diana Math Standards only require equal group interpretations of fractions in 4th grade, leaving
2
3
8
12
2
3
The grammar of phrases involving fractions/decimals. Although not explicitly mentioned in the
Indiana Math Standards, students working with fractions benefit from an explicit analysis of the
grammar used in English phrases about fractions. In particular, the prepositional phrase starting
with the word of which follows a fraction or decimal contains important information about the size
of the whole (1), and often implies which kind of fraction model is being used. For example, we
might say “3/10 of the shapes are squares.” This phrase implies that all of the shapes present at this
time constitute the whole. It also implies that we are using a set model rather than an area model
– the size of the shapes does not matter in this case. We might say “0.3 of the package of shapes is
blue.” Notice that the tense of the verb changed from plural to singular because we are now referring
to packages rather than to shapes. Whereas the first phrase implied that all of the shapes together
constitute the whole, this time only the shapes in the package are included. This phrase still implies
that the sizes of the shapes do not matter, so we are still working with a set model. We could say
in another situation “I have 11/4 of a pack of squares.” This would mean that I have packs that
hold a certain number of squares. I have enough of those squares to fill 11/4 of a pack. 4/4 will fill
one pack, and 8/4 will fill two packs. So I have enough to fill two packs completely and I will have
3/4 of another pack. I could describe this situation equivalently as “I have 23
4 packs of squares.”
Notice that the word pack is singular in the example above because I had eleven “fourths of a pack”.
However, when using a mixed number greater than 1, the object described becomes plural and the
preposition “of” is no longer used.
4th Grade – Measurement
4.M.1 Measure length to the nearest quarter-inch, eighth-inch, and millimeter.
4.M.2 Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g;

lb, oz; l, ml; hr, min, sec. Express measurements in a larger unit in terms of a smaller unit within a
single system of measurement. Record measurement equivalents in a two-column table.
4.M.3 Use the four operations (addition, subtraction, multiplication and division) to solve real- world
problems involving distances, intervals of time, volumes, masses of objects, and money. Include
addition and subtraction problems involving simple fractions and problems that require expressing
measurements given in a larger unit in terms of a smaller unit.
4.M.4 Apply the area and perimeter formulas for rectangles to solve real-world problems and other math-
ematical problems. Recognize area as additive and find the area of complex shapes composed of
rectangles by decomposing them into non-overlapping rectangles and adding the areas of the non-
overlapping parts; apply this technique to solve real-world problems and other mathematical problems
involving shapes.
4.M.5 Understand that an angle is measured with reference to a circle, with its center at the common
endpoint of the rays, by considering the fraction of the circular arc between the points where the
two rays intersect the circle. Understand an angle that turns through 1/360 of a circle is called
a ”one-degree angle,” and can be used to measure other angles. Understand an angle that turns
through n one-degree angles is said to have an angle measure of n degrees.
4.M.6 Measure angles in whole-number degrees using appropriate tools. Sketch angles of specified mea-
sure.
but I have found this to be a critical skill that needs extensive practice. Many local employers report
that the lack of proficiency of this skill in the local workforce is a major barrier to greater economic
prosperity. Constructing lengths rather than merely measuring develops students motor skills and
attention to precision. Students should be able to draw a line or cut out a strip of paper accurately
(to the nearest inch or centimeter). They should be able to tape together strips of paper to make a
strip measuring multiple feet or yards. It often helps to trace a correct version of the strip so that
students can check their accuracy independently (I like to invite students to play a game pretending
that they are cutting out frog tongues that are just long enough to reach a fly. If their tongue is the
right length, then their frog gets a point.)
Use physical square units to find areas/perimeters. This is a review of concepts from 3rd grade. I
think it is important to revisit the concrete meaning of area before moving on to discovering formulas
for areas of rectangles.
4th Grade – Geometry
4.G.1 Identify, describe, and draw parallelograms, rhombuses, and trapezoids using appropriate tools
(e.g., ruler, straightedge and technology).
4.G.2 Recognize and draw lines of symmetry in two-dimensional figures. Identify figures that have lines
of symmetry.
4.G.3 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint.
4.G.4 Identify, describe, and draw rays, angles (right, acute, obtuse), and perpendicular and parallel lines
using appropriate tools (e.g., ruler, straightedge and technology). Identify these in two-dimensional
figures.

4.G.5 Classify triangles and quadrilaterals based on the presence or absence of parallel or perpendicular
lines, or the presence or absence of angles (right, acute, obtuse).

5th Grade – Fractions
5.NS.1 Use a number line to compare and order fractions, mixed numbers, and decimals to thousandths.
Write the results using >, =, and < symbols.
5.NS.2 Explain different interpretations of fractions, including: as parts of a whole, parts of a set, and
division of whole numbers by whole numbers.
5.NS.5 Use place value understanding to round decimal numbers up to thousandths to any given place
value.
5.NS.6 Understand, interpret, and model percents as part of a hundred (e.g. by using pictures, diagrams,
and other visual models).
5.C.3 Compare the size of a product to the size of one factor on the basis of the size of the other factor,
without performing the indicated multiplication.
5.C.4 Add and subtract fractions with unlike denominators, including mixed numbers.
5.C.5 Use visual fraction models and numbers to multiply a fraction by a fraction or a whole number.
5.C.6 Explain why multiplying a positive number by a fraction greater than 1 results in a product greater
than the given number. Explain why multiplying a positive number by a fraction less than 1 results
in a product smaller than the given number. Relate the principle of fraction equivalence, a/b =
(n × a)/(n × b), to the effect of multiplying a/b by 1.
5.C.7 Use visual fraction models and numbers to divide a unit fraction by a non-zero whole number and
to divide a whole number by a unit fraction.
5.C.8 Add, subtract, multiply, and divide decimals to hundredths, using models or drawings and strategies
based on place value or the properties of operations. Describe the strategy and explain the reasoning.
5.M.2 Find the area of a rectangle with fractional side lengths by modeling with unit squares of the
appropriate unit fraction side lengths, and show that the area is the same as would be found by
multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent
fraction products as rectangular areas.
diana Math Standards only require equal group interpretations of fractions in 5th grade, leaving
2
3
8
12
2
3

5th Grade – Operations and Algebra
5.NS.3 Recognize the relationship that in a multi-digit number, a digit in one place represents 10 times
as much as it represents in the place to its right, and inversel, a digit in one place represents 1/10 of
what it represents in the place to its left.
5.NS.4 Explain patterns in the number of zeros of the product when multiplying a number by powers
of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or
divided by a power of 10. Use whole-number exponents to denote powers of 10.
5.C.1 Multiply multi-digit whole numbers fluently using a standard algorithmic approach.
5.C.2 Find whole-number quotients and remainders with up to four-digit dividends and two-digit divisors,
using strategies based on place value, the properties of operations, and/or the relationship between
multiplication and division. Describe the strategy and explain the reasoning used.
5.C.9 Evaluate expressions with parentheses or brackets involving whole numbers using the commutative
properties of addition and multiplication, associative properties of addition and multiplication, and
distributive property.
5.AT.6 Graph points with whole number coordinates on a coordinate plane. Explain how the coordinates
relate the point as the distance from the origin on each axis, with the convention that the names of
the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
5.AT.7 Represent real-world problems and equations by graphing ordered pairs in the first quadrant of
the coordinate plane, and interpret coordinate values of points in the context of the situation.
5.AT.8 Define and use up to two variables to write linear expressions that arise from real-world problems,
and evaluate them for given values.
Algorithms for +/− and ×/÷ facts. The Indiana Math Standards presume that students have already
mastered algorithms for addition and subtraction in 4th grade. They also presume that students have
mastered basic addition, subtraction, multiplication, and division facts. 5th grade students typically
need continuing review of these skills.
5.AT.1 Solve real-world problems involving multiplication and division of whole numbers (e.g. by using
equations to represent the problem). In division problems that involve a remainder, explain how the
remainder affects the solution to the problem.
5.AT.2 Solve real-world problems involving addition and subtraction of fractions referring to the same
whole, including cases of unlike denominators (e.g., by using visual fraction models and equations to
represent the problem). Use benchmark fractions and number sense of fractions to estimate mentally
and assess whether the answer is reasonable.
5.AT.3 Solve real-world problems involving multiplication of fractions, including mixed numbers (e.g., by
using visual fraction models and equations to represent the problem).

5.AT.4 Solve real-world problems involving division of unit fractions by non-zero whole numbers, and
division of whole numbers by unit fractions (e.g., by using visual fraction models and equations to
5.AT.5 Solve real-world problems involving addition, subtraction, mutliplication, and division with dec-
imals to hundredths, including problems that involve money in decimal notation (e.g. by using
equations to represent the problem).
5.DS.1 Formulate questions that can be addressed with data and make predictions about the data. Use
observations, surveys, and experiments to collect, represent, and interpret the data using tables
(including frequency tables), line plots, bar graphs, and line graphs. Recognize the differences in
representing categorical and numerical data.
5.DS.2 Understand and use measures of center (mean and median) and frequency (mode) to describe a
data set.
5th Grade – Measurement
5.M.1 Convert among different-sized standard measurement units within a given measurement system,
and use these conversions in solving multi-step real-world problems.
5.M.2 Find the area of a rectangle with fractional side lengths by modeling with unit squares of the
appropriate unit fraction side lengths, and show that the area is the same as would be found by
multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent
fraction products as rectangular areas.
5.M.3 Develop and use formulas for the area of triangles, parallelograms and trapezoids. Solve real-world
and other mathematical problems that involve perimeter and area of triangles, parallelograms and
trapezoids, using appropriate units for measures.
5.M.4 Find the volume of a right rectangular prism with whole-number side lengths by packing it with
unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths
or multiplying the height by the area of the base.
5.M.5 Apply the formulas V = l × w × h and V = B × h for right rectangular prisms to find volumes
of right rectangular prisms with whole-number edge lengths to solve real-world problems and other
mathematical problems.
5.M.6 Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding
the volumes of the non-overlapping parts, applying this technique to solve real-world problems and
Constructing lengths involving fractions and decimals of inches, centimeters, feet, yards, and meters.
The Indiana Math Standards do not explicitly ask students to construct lengths, but I have found
this to be a critical skill that needs extensive practice. Many local employers report that the lack
of proficiency of this skill in the local workforce is a major barrier to greater economic prosperity.
Constructing lengths rather than merely measuring develops students motor skills and attentiveness
to precision. Students should be able to draw a line or cut out a strip of paper with a given length
and width accurately (to the nearest 1/16 inch or millimeter). They should be able to tape together
strips of paper to make a strip measuring multiple feet or yards. It often helps to trace a correct
version of the strip so that students can check their accuracy independently (I like to invite students

to play a game pretending that they are cutting out frog tongues that are just long enough to reach
a ﬂy. If their tongue is the right length, then their frog gets a point.) Target measurements should
also include fractions and decimals of a foot, yard, or meter.
5th Grade – Geometry
5.G.1 Identify, describe, and draw triangles (right, acute, obtuse) and circles using appropriate tools
(e.g., ruler or straightedge, compass and technology). Understand the relationship between radius
and diameter.
5.G.2 Identify and classify polygons including quadrilaterals, pentagons, hexagons, and triangles (equilat-
eral, isosceles, scalene, right, acute and obtuse) based on angle measures and sides. Classify polygons
in a hierarchy based on properties.

6th Grade – Ratios and Rates
6.NS.3 Compare and order rational numbers and plot them on a number line. Write, interpret, and
explain statements of order for rational numbers in real-world contexts.
6.NS.5 Know commonly used fractions (halves, thirds, fourths, fifths, eighths, tenths) and their decimal
and percent equivalents. Convert between any two representations (fractions, decimals, percents) of
positive rational numbers without the use of a calculator.
6.NS.8 Interpret, model, and use ratios to show the relative sizes of two quantities. Describe how a ratio
shows the relationship between two quantities. Use the following notations: a/b, a to b, a : b.
6.NS.9 Understand the concept of a unit rate and use terms related to rate in the context of a ratio
relationship.
6.NS.10 Use reasoning involving rates and ratios to model real-world and other mathematical problems
(e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or
equations).
6.C.2 Compute with positive fractions and positive decimals fluently using a standard algorithmic ap-
proach.
6.C.4 Compute quotients of positive fractions and solve real-world problems involving division of fractions
by fractions. Use a visual fraction model and/or equations to represent these calculations.
6.AF.9 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.
6th Grade – Algebraic Thinking
6.C.5 Evaluate positive rational numbers with whole number exponents.
6.C.6 Apply the order of operations and properties of operations (identity, inverse, commutative properties
of addition and multiplication, associative properties of addition and multiplication, and distributive
property) to evaluate numerical expressions with nonnegative rational numbers, including those using
grouping symbols, such as parentheses, and involving whole number exponents. Justify each step in
the process.
6.AF.1 Evaluate expressions for specific values of their variables, including expressions with whole-number
exponents and those that arise from formulas used in real-world problems.
6.AF.2 Apply the properties of operations (e.g., identity, inverse, commutative, associative, distributive
properties) to create equivalent linear expressions and to justify whether two linear expressions are
equivalent when the two expressions name the same number regardless of which value is substituted
into them.
6.AF.3 Define and use multiple variables when writing expressions to represent real-world and other
mathematical problems, and evaluate them for given values.

6.AF.4 Understand that solving an equation or inequality is the process of answering the following ques-
tion: 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.
6.AF.5 Solve equations of the form x + p = q and px = q fluently for cases in which p, q and x are all
nonnegative rational numbers. Represent real world problems using equations of these forms and
solve such problems.
6.AF.6 Write an inequality of the form x > c, x ≥ c, x < c, or x ≤ c, where c is a rational number,
to represent a constraint or condition in a real-world or other mathematical problem. Recognize
inequalities have infinitely many solutions and represent solutions on a number line diagram.
6.AF.8 Solve real-world and other mathematical problems by graphing points with rational number co-
ordinates on a coordinate plane. Include the use of coordinates and absolute value to find distances
between points with the same first coordinate or the same second coordinate.
6.AF.10 Use variables to represent two quantities in a proportional relationship in a real-world problem;
write an equation to express one quantity, the dependent variable, in terms of the other quantity, the
independent variable. Analyze the relationship between the dependent and independent variables
using graphs and tables, and relate these to the equation.
Testing Calculators and Creating Spreadsheets. The Indiana Math Standards do not explicitly ask
students to explore order of operations via calculators or to create spreadsheets. However, they do
expect students to use technology tools when appropriate, and learning to use these tools in 6th
grade helps to provide important context for creating and manipulating numerical and algebraic
expressions. Students can become less intimidated by multi-step expressions if they understand how
to use calculators and spreadsheets effectively. Ideally, students should experiment with simple four
function calculators, scientific calculators that have parentheses and fraction functions, graphing
calculators, and spreadsheets. Students should learn how to create simple formulas and graphs in
spreadsheets to solve real world and mathematical problems and to summarize data.
6th Grade – Numbers and Operations
6.NS.1 Understand that positive and negative numbers are used to describe quantities having oppo-
site directions or values (e.g., temperature above/below zero, elevation above/below sea level, cred-
its/debits, positive/negative electric charge). Use positive and negative numbers to represent and
compare quantities in real-world contexts, explaining the meaning of 0 in each situation.
6.NS.2 Understand the integer number system. Recognize opposite signs of numbers as indicating lo-
cations 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.
6.NS.4 Understand that the absolute value of a number is the distance from zero on a number line. Find
the absolute value of real numbers and know that the distance between two numbers on the number
line is the absolute value of their difference. Interpret absolute value as magnitude for a positive or
negative quantity in a real-world situation.
6.NS.6 Identify and explain prime and composite numbers.
6.NS.7 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 from 1 to 100, with a common factor as a multiple of a sum of
two whole numbers with no common factor.
6.C.1 Divide multi-digit whole numbers fluently using a standard algorithmic approach.
6.AF.7 Understand that signs of numbers in ordered pairs indicate the quadrant containing the point;
recognize that when two ordered pairs differ only by signs, the locations of the points are related by
reflections across one or both axes. Graph points with rational number coordinates on a coordinate
plane.
Discover, use, and prove divisibility tests. The Indiana Math Standards do not explicitly ask stu-
dents to explore divisibility tests. However, knowing divisibility tests is useful when finding factors
of given numbers. The proofs of divisibility tests are fairly simple and allow students to cement their
understanding of place value while simultaneously being introduced to the idea of proof, which is
critical to the development of algebraic thinking.
6.NS.10 Use reasoning involving rates and ratios to model real-world and other mathematical problems
(e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or
equations).
6.C.3 Solve real-world problems with positive fractions and decimals by using one or two operations.
6.DS.1 Recognize a statistical question as one that anticipates variability in the data related to the
question and accounts for the variability in the answers. 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.
6.DS.2 Select, create, and interpret graphical representations of numerical data, including line plots,
histograms, and box plots.
6.DS.3 Formulate statistical questions; collect and organize the data (e.g., using technology); display and
interpret the data with graphical representations (e.g., using technology).
6.DS.4 Summarize numerical data sets in relation to their context in multiple ways, such as: report the
number of observations; describe the nature of the attribute under investigation, including how it
was measured and its units of measurement; determine quantitative measures of center (mean and/or
median) and spread (range and interquartile range), as well as describe any overall pattern and any
striking deviations from the overall pattern with reference to the context in which the data were
gathered; and relate the choice of measures of center and spread to the shape of the data distribution
and the context in which the data were gathered.
6th Grade – Geometry and Measurement
6.GM.1 Convert between measurement systems (English to metric and metric to English) given conversion
factors, and use these conversions in solving real-world problems.

6.GM.2 Know that the sum of the interior angles of any triangle is 180◦ and that the sum of the inte-
rior angles of any quadrilateral is 360◦. Use this information to solve real-world and mathematical
problems.
6.GM.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 to solve real-world and other mathematical problems.
6.GM.4 Find the area of complex shapes composed of polygons by composing or decomposing into simple
shapes; apply this technique to solve real-world and other mathematical problems.
6.GM.5 Find the volume of a right rectangular prism with fractional edge lengths using unit cubes of
the appropriate unit fraction edge lengths (e.g., using technology or concrete materials), 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 to solve real-world and other mathematical problems.
6.GM.6 Construct right rectangular prisms from nets and use the nets to compute the surface area of
prisms; apply this technique to solve real-world and other mathematical problems.

7th Grade – Proportions
7.C.5 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other
quantities measured in like or different units.
7.C.6 Use proportional relationships to solve ratio and percent problems with multiple operations, such as
the following: simple interest, tax, markups, markdowns, gratuities, commissions, fees, conversions
within and across measurement systems, percent increase and decrease, and percent error.
7.C.7 Compute with rational numbers fluently using a standard algorithmic approach.
7.C.8 Solve real-world problems with rational numbers by using one or two operations.
7.AF.4 Define slope as vertical change for each unit of horizontal change and recognize that a constant
rate of change or constant slope describes a linear function. Identify and describe situations with
constant or varying rates of change.
7.AF.5 Graph a line given its slope and a point on the line. Find the slope of a line given its graph.
7.AF.6 Decide whether two quantities are in a proportional relationship (e.g., by testing for equivalent
ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line
through the origin).
7.AF.7 Identify the unit rate or constant of proportionality in tables, graphs, equations, and verbal
descriptions of proportional relationships.
7.AF.8 Explain what the coordinates of a point on the graph of a proportional relationship mean in terms
of the situation, with special attention to the points (0, 0) and (1, r), where r is the unit rate.
7.AF.9 Identify real-world and other mathematical situations that involve proportional relationships.
Write equations and draw graphs to represent proportional relationships and recognize that these
situations are described by a linear function in the form y = mx, where the unit rate, m, is the slope
of the line.
7th Grade – Algebraic Thinking
7.AF.1 Apply the properties of operations (e.g., identity, inverse, commutative, associative, distributive
properties) to create equivalent linear expressions, including situations that involve factoring (e.g.,
given 2x − 10, create an equivalent expression 2(x − 5)). Justify each step in the process.
7.AF.2 Solve equations of the form px + q = r and p(x + q) = r fluently, where p, q, and r are specific
rational numbers. Represent real-world problems using equations of these forms and solve such
problems.
7.AF.3 Solve inequalities of the form px + q(> or ≥)r or px + q(< or ≤)r, where p, q, and r are specific
rational numbers. Represent real-world problems using inequalities of these forms and solve such
problems. Graph the solution set of the inequality and interpret it in the context of the problem.

7.NS.1 Find the prime factorization of whole numbers and write the results using exponents.
7.NS.2 Understand the inverse relationship between squaring and finding the square root of a perfect
square integer. Find square roots of perfect square integers.
7.NS.3 Know there are rational and irrational numbers. Identify, compare, and order rational and common
irrational numbers (
√
2,
√
3,
√
5, π) and plot them on a number line.
7.C.1 Understand p + q as the number located a distance |q| 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 rational numbers by describing real-world contexts.
7.C.2 Understand subtraction of rational numbers as adding the additive inverse, p − q = p + (−q). Show
that the distance between two rational numbers on the number line is the absolute value of their
difference, and apply this principle in real-world contexts.
7.C.3 Understand that multiplication is extended from fractions to rational numbers by requiring that
operations continue to satisfy the properties of operations, particularly the distributive property,
leading to products such as (−1)(−1) = 1 and the rules for multiplying signed numbers.
7.C.4 Understand that integers can be divided, provided that the divisor is not zero, and that every
quotient of integers (with non-zero divisor) is a rational number. Understand that if p and q are
integers, then −(p/q) = (−p)/q = p/(−q).
7.C.1 Understand p + q as the number located a distance |q| 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 rational numbers by describing real-world contexts.
7.C.2 Understand subtraction of rational numbers as adding the additive inverse, p − q = p + (−q). Show
that the distance between two rational numbers on the number line is the absolute value of their
difference, and apply this principle in real-world contexts.
7.C.8 Solve real-world problems with rational numbers by using one or two operations.
7.AF.9 Identify real-world and other mathematical situations that involve proportional relationships.
Write equations and draw graphs to represent proportional relationships and recognize that these
situations are described by a linear function in the form y = mx, where the unit rate, m, is the slope
of the line.
7.GM.2 Identify and describe similarity relationships of polygons including the angle-angle criterion for
similar triangles, and solve problems involving similarity.
7.GM.3 Solve real-world and other mathematical problems involving scale drawings of geometric figures,
including computing actual lengths and areas from a scale drawing. Create a scale drawing by using
proportional reasoning.
7.GM.4 Solve real-world and other mathematical problems that involve vertical, adjacent, complementary,
and supplementary angles.

7.GM.5 Understand the formulas for area and circumference of a circle and use them to solve real-
world and other mathematical problems; give an informal derivation of the relationship between
circumference and area of a circle.
7.GM.6 Solve real-world and other mathematical problems involving volume of cylinders and three-
dimensional objects composed of right rectangular prisms.
7.GM.7 Construct nets for right rectangular prisms and cylinders and use the nets to compute the surface
area; apply this technique to solve real-world and other mathematical problems.
7.DSP.1 Understand that statistics can be used to gain information about a population by examining
a sample of the population and generalizations about a population from a sample are valid only if
the sample is representative of that population. Understand that random sampling tends to produce
representative samples and support valid inferences.
7.DSP.2 Use data from a random sample to draw inferences about a population. Generate multiple
samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
7.DSP.3 Find, use, and interpret measures of center (mean and median) and measures of spread (range,
interquartile range, and mean absolute deviation) for numerical data from random samples to draw
comparative inferences about two populations.
7.DSP.4 Make observations about the degree of visual overlap of two numerical data distributions repre-
sented in line plots or box plots. Describe how data, particularly outliers, added to a data set may
aﬀect the mean and/or median.
7.DSP.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the
likelihood of the event occurring. Understand that a probability near 0 indicates an unlikely event,
a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near
1 indicates a likely event. Understand that a probability of 1 indicates an event certain to occur and
a probability of 0 indicates an event impossible to occur.
7.DSP.6 Approximate the probability of a chance event by collecting data on the chance process that
produces it and observing its relative frequency from a large sample.
7.DSP.7 Develop probability models that include the sample space and probabilities of outcomes to
represent simple events with equally likely outcomes. Predict the approximate relative frequency
of the event based on the model. Compare probabilities from the model to observed frequencies;
evaluate the level of agreement and explain possible sources of discrepancy.
7.GM.1 Draw triangles (freehand, with ruler and protractor, and using technology) with given conditions
from three measures of angles or sides, and notice when the conditions determine a unique triangle,
more than one triangle, or no triangle.
7.GM.2 Identify and describe similarity relationships of polygons including the angle-angle criterion for
similar triangles, and solve problems involving similarity.
7.GM.3 Solve real-world and other mathematical problems involving scale drawings of geometric ﬁgures,
including computing actual lengths and areas from a scale drawing. Create a scale drawing by using
proportional reasoning.

7.GM.4 Solve real-world and other mathematical problems that involve vertical, adjacent, complementary,
and supplementary angles.
7.GM.5 Understand the formulas for area and circumference of a circle and use them to solve real-
world and other mathematical problems; give an informal derivation of the relationship between
circumference and area of a circle.
7.GM.6 Solve real-world and other mathematical problems involving volume of cylinders and three-
dimensional objects composed of right rectangular prisms.
7.GM.7 Construct nets for right rectangular prisms and cylinders and use the nets to compute the surface
area; apply this technique to solve real-world and other mathematical problems.

8th Grade – Functions
8.AF.3 Understand that a function assigns to each x-value (independent variable) exactly one y-value
(dependent variable), and that the graph of a function is the set of ordered pairs (x, y).
8.AF.4 Describe qualitatively the functional relationship between two quantities by analyzing a graph
(e.g., where the function is increasing or decreasing, linear or nonlinear, has a maximum or minimum
value). Sketch a graph that exhibits the qualitative features of a function that has been verbally
described.
8.AF.5 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line;
give examples of functions that are not linear. Describe similarities and differences between linear
and nonlinear functions from tables, graphs, verbal descriptions, and equations.
8.AF.6 Construct a function to model a linear relationship between two quantities given a verbal descrip-
tion, table of values, or graph. Recognize in y = mx + b that m is the slope (rate of change) and b is
the y-intercept of the graph, and describe the meaning of each in the context of a problem.
8.AF.7 Compare properties of two linear functions given in different forms, such as a table of values,
equation, verbal description, and graph (e.g., compare a distance-time graph to a distance-time
equation to determine which of two moving objects has greater speed).
8.AF.8 Understand that solutions to a system of two linear equations correspond to points of intersection
of their graphs because points of intersection satisfy both equations simultaneously. Approximate
the solution of a system of equations by graphing and interpreting the reasonableness of the approx-
imation.
8.DSP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of
association between two quantitative variables. Describe patterns such as clustering, outliers, positive
or negative association, linear association, and nonlinear association.
8.DSP.2 Know that straight lines are widely used to model relationships between two quantitative vari-
ables. For scatter plots that suggest a linear association, informally fit a straight line, and describe
the model fit by judging the closeness of the data points to the line.
8.DSP.3 Write and use equations that model linear relationships to make predictions, including interpo-
lation and extrapolation, in real-world situations involving bivariate measurement data; interpret the
slope and y-intercept.
8th Grade – Expressions and Equations
8.NS.3 Given a numeric expression with common rational number bases and integer exponents, apply the
properties of exponents to generate equivalent expressions.
8.NS.4 Use square root symbols to represent solutions to equations of the form x2 = p, where p is a
positive rational number.

8.AF.1 Solve linear equations with rational number coefficients fluently, including equations whose solu-
tions require expanding expressions using the distributive property and collecting like terms. Rep-
resent real-world problems using linear equations and inequalities in one variable and solve such
problems.
8.AF.2 Give examples of linear equations in one variable with one solution, infinitely many solutions,
or no solutions. Show which of these possibilities is the case by transforming a given equation into
simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b
are different numbers).
8.NS.1 Give examples of rational and irrational numbers and explain the difference between them. Un-
derstand that every number has a decimal expansion; for rational numbers, show that the decimal
expansion terminates or repeats, and convert a decimal expansion that repeats into a rational number.
8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, plot
them approximately on a number line, and estimate the value of expressions involving irrational
numbers.
8.C.1 Solve real-world problems with rational numbers by using multiple operations.
8.C.2 Solve real-world and other mathematical problems involving numbers expressed in scientific nota-
tion, including problems where both decimal and scientific notation are used. Interpret scientific
notation that has been generated by technology, such as a scientific calculator, graphing calculator,
or excel spreadsheet.
8.AF.6 Construct a function to model a linear relationship between two quantities given a verbal descrip-
tion, table of values, or graph. Recognize in y = mx + b that m is the slope (rate of change) and b is
the y-intercept of the graph, and describe the meaning of each in the context of a problem.
8.DSP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of
association between two quantitative variables. Describe patterns such as clustering, outliers, positive
or negative association, linear association, and nonlinear association.
8.DSP.2 Know that straight lines are widely used to model relationships between two quantitative vari-
ables. For scatter plots that suggest a linear association, informally fit a straight line, and describe
the model fit by judging the closeness of the data points to the line.
8.DSP.3 Write and use equations that model linear relationships to make predictions, including interpo-
lation and extrapolation, in real-world situations involving bivariate measurement data; interpret the
slope and y-intercept.
8.DSP.4 Understand that, just as with simple events, the probability of a compound event is the fraction of
outcomes in the sample space for which the compound event occurs. Understand and use appropriate
terminology to describe independent, dependent, complementary, and mutually exclusive events.

8.DSP.5: Represent sample spaces and find probabilities of compound events (independent and depen-
dent) using methods, such as organized lists, tables, and tree diagrams.
8.DSP.6: For events with a large number of outcomes, understand the use of the multiplication counting
principle. Develop the multiplication counting principle and apply it to situations with a large number
of outcomes.
8.GM.1 Identify, define and describe attributes of three-dimensional geometric objects (right rectangular
prisms, cylinders, cones, spheres, and pyramids). Explore the effects of slicing these objects using
appropriate technology and describe the two-dimensional figure that results.
8.GM.2 Solve real-world and other mathematical problems involving volume of cones, spheres, and pyra-
mids and surface area of spheres.
8.GM.3 Verify experimentally the properties of rotations, reflections, and translations, including: lines
are mapped to lines, and line segments to line segments of the same length; angles are mapped to
angles of the same measure; and parallel lines are mapped to parallel lines.
8.GM.4: Understand that a two-dimensional figure is congruent to another if the second can be obtained
from the first by a sequence of rotations, reflections, and translations. Describe a sequence that
exhibits the congruence between two given congruent figures.
8.GM.5 Understand that a two-dimensional figure is similar to another if the second can be obtained
from the first by a sequence of rotations, reflections, translations, and dilations. Describe a sequence
that exhibits the similarity between two given similar figures.
8.GM.6 Describe the effect of dilations, translations, rotations, and reflections on two- dimensional figures
using coordinates.
8.GM.7 Use inductive reasoning to explain the Pythagorean relationship.
8.GM.8 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-
world and other mathematical problems in two dimensions.
8.GM.9 Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane.

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