The provisions of this written curriculum shall
be implemented
(a) Introduction.
(1) Within a well-balanced mathematics curriculum, the
primary focal points at Kindergarten are
developing whole-number concepts and using
patterns and sorting to explore number, data, and
shape.
(2) Throughout mathematics in Kindergarten-Grade 2,
students build a foundation of basic
understandings in number, operation, and
quantitative reasoning; patterns, relationships,
and algebraic thinking; geometry and spatial
reasoning; measurement; and probability and
statistics. Students use numbers in ordering,
labeling, and expressing quantities and
relationships to solve problems and translate
informal language into mathematical symbols.
Students use patterns to describe objects, express
relationships, make predictions, and solve
problems as they build an understanding of number,
operation, shape, and space. Students use informal
language and observation of geometric properties
to describe shapes, solids, and locations in the
physical world and begin to develop measurement
concepts as they identify and compare attributes
of objects and situations. Students collect,
organize, and display data and use information
from graphs to answer questions, make summary
statements, and make informal predictions based on
their experiences.
(3) Problem solving, language and communication,
connections within and outside mathematics, and
formal and informal reasoning underlie all content
areas in mathematics. Throughout mathematics in
Kindergarten-Grade 2, students use these processes
together with technology and other mathematical
tools such as manipulative materials to develop
conceptual understanding and solve problems as
they do mathematics.
(b) Knowledge and skills.
(K.1)Number, operation, and quantitative reasoning. The
student uses numbers to name quantities. The
student is expected to:
(A) use one-to-one correspondence and language
such as more than, same number as, or two less
than to describe relative sizes of sets of
concrete objects;
(B) use sets of concrete objects to represent
quantities given in verbal or written form
(through 9); and
(C) use numbers to describe how many objects are
in a set (through 20).
(K.2)Number, operation, and quantitative reasoning. The
student describes order of events or objects. The
student is expected to:
(A) use language such as before or after to
describe relative position in a sequence of
events or objects; and
(B) name the ordinal positions in a sequence such
as first, second, third, etc.
(K.3)Number, operation, and quantitative reasoning. The
student recognizes that there are quantities less
than a whole. The student is expected to:
(A) share a whole by separating it into equal
parts; and
(B) explain why a given part is half of the whole.
(K.4) Number, operation, and quantitative
reasoning. The student models addition and
subtraction.
The student is expected to model and create addition
and subtraction problems in real situations with
concrete objects.
(K.5) Patterns, relationships, and algebraic
thinking. The student identifies, extends, and
creates patterns.
The student is expected to identify, extend, and create
patterns of sounds, physical movement, and concrete
objects.
(K.6)Patterns, relationships, and algebraic thinking.
The student uses patterns to make predictions. The
student is expected to:
(A) use patterns to predict what comes next,
including cause-and-effect relationships; and
(B) count by ones to 100.
(K.7)Geometry and spatial reasoning. The student
describes the relative positions of objects. The
student is expected to:
(A) describe one object in relation to another
using informal language such as over, under,
above, and below; and
(B) place an object in a specified position.
(K.8)Geometry and spatial reasoning. The student uses
attributes to determine how objects are alike and
different. The student is expected to:
(A) describe and identify an object by its
attributes using informal language;
(B) compare two objects based on their attributes;
and
(C) sort objects according to their attributes and
describe how those groups are formed.
(K.9)Geometry and spatial reasoning. The student
recognizes characteristics of shapes and solids.
The student is expected to:
(A) describe and compare real-life objects or
models of solids;
(B) recognize shapes in real-life objects or
models of solids; and
(C) describe, identify, and compare circles,
triangles, and rectangles including squares.
(K.10) Measurement. The student uses attributes such
as length, weight, or capacity to compare and order
objects. The student is expected to:
(A) compare and order two or three concrete
objects according to length (shorter or
longer), capacity (holds more or holds less),
or weight (lighter or heavier); and
(B) find concrete objects that are about the same
as, less than, or greater than a given object
according to length, capacity, or weight.
(K.11) Measurement. The student uses time and
temperature to compare and order events,
situations, and/or objects. The student is expected
to:
(A) compare situations or objects according to
temperature such as hotter or colder;
(B) compare events according to duration such as
more time than or less time than;
(C) sequence events; and
(D) read a calendar using days, weeks, and months.
(K.12) Probability and statistics. The student
constructs and uses graphs of real objects or
pictures to answer questions. The student is
expected to:
(A) construct graphs using real objects or
pictures in order to answer questions; and
(B) use information from a graph of real objects
or pictures in order to answer questions.
(K.13) Underlying processes and mathematical tools.
The student applies Kindergarten mathematics to
solve problems connected to everyday experiences
and activities in and outside of school. The
student is expected to:
(A) identify mathematics in everyday situations;
(B) use a problem-solving model, with guidance,
that incorporates understanding the problem,
making a plan, carrying out the plan, and
evaluating the solution for reasonableness;
(C) select or develop an appropriate problem-
solving strategy including drawing a picture,
looking for a pattern, systematic guessing and
checking, or acting it out in order to solve a
problem; and
(D) use tools such as real objects, manipulatives,
and technology to solve problems.
(K.14) Underlying processes and mathematical tools.
The student communicates about Kindergarten
mathematics using informal language. The student is
expected to:
(A) explain and record observations using objects,
words, pictures, numbers, and technology; and
(B) relate everyday language to mathematical
language and symbols.
(K.15) Underlying processes and mathematical tools.
The student uses logical reasoning to make sense
of his or her world.
The student is expected to reason and support his or
her thinking using objects, words, pictures,
numbers, and technology.
(a) Introduction.
(1) Within a well-balanced mathematics curriculum, the
primary focal points at Grade 1 are adding and
subtracting whole numbers and organizing and
analyzing data.
(2) Throughout mathematics in Kindergarten-Grade 2,
students build a foundation of basic
understandings in number, operation, and
quantitative reasoning; patterns, relationships,
and algebraic thinking; geometry and spatial
reasoning; measurement; and probability and
statistics. Students use numbers in ordering,
labeling, and expressing quantities and
relationships to solve problems and translate
informal language into mathematical symbols.
Students use patterns to describe objects, express
relationships, make predictions, and solve
problems as they build an understanding of number,
operation, shape, and space. Students use informal
language and observation of geometric properties
to describe shapes, solids, and locations in the
physical world and begin to develop measurement
concepts as they identify and compare attributes
of objects and situations. Students collect,
organize, and display data and use information
from graphs to answer questions, make summary
statements, and make informal predictions based on
their experiences.
(3) Problem solving, language and communication,
connections within and outside mathematics, and
formal and informal reasoning underlie all content
areas in mathematics. Throughout mathematics in
Kindergarten-Grade 2, students use these processes
together with technology and other mathematical
tools such as manipulative materials to develop
conceptual understanding and solve problems as
they do mathematics.
(b) Knowledge and skills.
(1.1)Number, operation, and quantitative reasoning. The
student uses whole numbers to describe and compare
quantities. The student is expected to:
(A) compare and order whole numbers up to 99 (less
than, greater than, or equal to) using sets of
concrete objects and pictorial models;
(B) create sets of tens and ones using concrete
objects to describe, compare, and order whole
numbers;
(C) use words and numbers to describe the values
of individual coins such as penny, nickel,
dime, and quarter and their relationships; and
(D) read and write numbers to 99 to describe sets
of concrete objects.
(1.2)Number, operation, and quantitative reasoning. The
student uses pairs of whole numbers to describe
fractional parts of whole objects or sets of
objects. The student is expected to:
(A) share a whole by separating it into equal
parts and use appropriate language to describe
the parts such as three out of four equal
parts; and
(B) use appropriate language to describe part of a
set such as three out of the eight crayons are
red.
(1.3)Number, operation, and quantitative reasoning. The
student recognizes and solves problems in addition
and subtraction situations. The student is expected
to:
(A) model and create addition and subtraction
problem situations with concrete objects and
write corresponding number sentences; and
(B) learn and apply basic addition facts (sums to
18) using concrete models.
(1.4)Patterns, relationships, and algebraic thinking.
The student uses patterns to make predictions. The
student is expected to:
(A) identify, describe, and extend concrete and
pictorial patterns in order to make
predictions and solve problems; and
(B) use patterns to skip count by twos, fives, and
tens.
(1.5)Patterns, relationships, and algebraic thinking.
The student recognizes patterns in numbers and
operations. The student is expected to:
(A) find patterns in numbers, including odd and
even;
(B) compare and order whole numbers using place
value; and
(C) identify patterns in related addition and
subtraction sentences (fact families for sums
to 18) such as 2 + 3 = 5,
3 + 2 = 5, 5 - 2 = 3, and 5 - 3 = 2.
(1.6)Geometry and spatial reasoning. The student uses
attributes to identify, compare, and contrast
shapes and solids. The student is expected to:
(A) describe and identify objects in order to sort
them according to a given attribute using
informal language;
(B) identify circles, triangles, and rectangles,
including squares, and describe the shape of
balls, boxes, cans, and cones; and
(C) combine geometric shapes to make new geometric
shapes using concrete models.
(1.7)Measurement. The student uses nonstandard units to
describe length, weight, and capacity. The student
is expected to:
(A) estimate and measure length, capacity, and
weight of objects using nonstandard units; and
(B) describe the relationship between the size of
the unit and the number of units needed in a
measurement.
(1.8)Measurement. The student understands that time and
temperature can be measured. The student is
expected to:
(A) recognize temperatures such as a hot day or a
cold day;
(B) describe time on a clock using hours and half
hours; and
(C) order three or more events by how much time
they take.
(1.9)Probability and statistics. The student displays
data in an organized form. The student is expected
to:
(A) collect and sort data; and
(B) use organized data to construct real object
graphs, picture graphs, and bar-type graphs.
(1.10) Probability and statistics. The student uses
information from organized data. The student is
expected to:
(A) draw conclusions and answer questions using
information organized in real-object graphs,
picture graphs, and bar-type graphs; and
(B) identify events as certain or impossible such
as drawing a red crayon from a bag of green
crayons.
(1.11) Underlying processes and mathematical tools.
The student applies Grade 1 mathematics to solve
problems connected to everyday experiences and
activities in and outside of school. The student is
expected to:
(A) identify mathematics in everyday situations;
(B) use a problem-solving model, with guidance as
needed, that incorporates understanding the
problem, making a plan, carrying out the plan,
and evaluating the solution for
reasonableness;
(C) select or develop an appropriate problem-
solving strategy including drawing a picture,
looking for a pattern, systematic guessing and
checking, or acting it out in order to solve a
problem; and
(D) use tools such as real objects, manipulatives,
and technology to solve problems.
(1.12) Underlying processes and mathematical tools.
The student communicates about Grade 1 mathematics
using informal language. The student is expected
to:
(A) explain and record observations using objects,
words, pictures, numbers, and technology; and
(B) relate informal language to mathematical
language and symbols.
(1.13) Underlying processes and mathematical tools.
The student uses logical reasoning to make sense
of his or her world.
The student is expected to reason and support his or
her thinking using objects, words, pictures,
numbers, and technology.
(a) Introduction.
(1) Within a well-balanced mathematics curriculum, the
primary focal points at Grade 2 are comparing and
ordering whole numbers, applying addition and
subtraction, and using measurement processes.
(2) Throughout mathematics in Kindergarten-Grade 2,
students build a foundation of basic
understandings in number, operation, and
quantitative reasoning; patterns, relationships,
and algebraic thinking; geometry and spatial
reasoning; measurement; and probability and
statistics. Students use numbers in ordering,
labeling, and expressing quantities and
relationships to solve problems and translate
informal language into mathematical symbols.
Students use patterns to describe objects, express
relationships, make predictions, and solve
problems as they build an understanding of number,
operation, shape, and space. Students use informal
language and observation of geometric properties
to describe shapes, solids, and locations in the
physical world and begin to develop measurement
concepts as they identify and compare attributes
of objects and situations. Students collect,
organize, and display data and use information
from graphs to answer questions, make summary
statements, and make informal predictions based on
their experiences.
(3) Problem solving, language and communication,
connections within and outside mathematics, and
formal and informal reasoning underlie all content
areas in mathematics. Throughout mathematics in
Kindergarten-Grade 2, students use these processes
together with technology and other mathematical
tools such as manipulative materials to develop
conceptual understanding and solve problems as
they do mathematics.
(b) Knowledge and skills.
(2.1) Number, operation, and quantitative
reasoning. The student understands how place value
is used to represent whole numbers.
The student is expected to use concrete models to
represent, compare, and order whole numbers
(through 999), read the numbers, and record the
comparisons using numbers and symbols (>, <, =).
(2.2)Number, operation, and quantitative reasoning. The
student uses fraction words to name parts of whole
objects or sets of objects. The student is expected
to:
(A) name fractional parts of a whole object (not
to exceed twelfths) when given a concrete
representation; and
(B) name fractional parts of a set of objects (not
to exceed twelfths) when given a concrete
representation.
(2.3)Number, operation, and quantitative reasoning. The
student adds and subtracts whole numbers to solve
problems. The student is expected to:
(A) recall and apply basic addition facts (sums to
18);
(B) select addition or subtraction and solve
problems using two-digit numbers, whether or
not regrouping is necessary; and
(C) determine the value of a collection of coins
less than one dollar.
(2.4)Number, operation, and quantitative reasoning. The
student models multiplication and division. The
student is expected to:
(A) model, create, and describe multiplication
situations in which equivalent sets of
concrete objects are joined; and
(B) model, create, and describe division
situations in which a set of concrete objects
is separated into equivalent sets.
(2.5)Patterns, relationships, and algebraic thinking.
The student uses patterns in numbers and
operations. The student is expected to:
(A) find patterns in numbers such as in a 100s
chart;
(B) use patterns in place value to compare and
order whole numbers through 999;
(C) use patterns to develop strategies to remember
basic addition facts; and
(D) solve subtraction problems related to addition
facts (fact families) such as
8 + 9 = 17, 9 + 8 = 17, 17 - 8 = 9, and
17 - 9 = 8.
(2.6)Patterns, relationships, and algebraic thinking.
The student uses patterns to describe relationships
and make predictions. The student is expected to:
(A) generate a list of paired numbers based on a
real-life situation such as number of
tricycles related to number of wheels;
(B) identify patterns in a list of related number
pairs based on a real-life situation and
extend the list; and
(C) identify, describe, and extend patterns to
make predictions and solve problems.
(2.7)Geometry and spatial reasoning. The student uses
attributes to identify, compare, and contrast
shapes and solids. The student is expected to:
(A) identify attributes of any shape or solid;
(B) use attributes to describe how two shapes or
two solids are alike or different; and
(C) cut geometric shapes apart and identify the
new shapes made.
(2.8) Geometry and spatial reasoning. The student
recognizes that numbers can be represented by
points on a line.
The student is expected to use whole numbers to locate
and name points on a line.
(2.9)Measurement. The student recognizes and uses
models that approximate standard units (metric and
customary) of length, weight, capacity, and time.
The student is expected to:
(A) identify concrete models that approximate
standard units of length, capacity, and
weight;
(B) measure length, capacity, and weight using
concrete models that approximate standard
units; and
(C) describe activities that take approximately
one second, one minute, and one hour.
(2.10) Measurement. The student uses standard tools
to measure time and temperature. The student is
expected to:
(A) read a thermometer to gather data; and
(B) describe time on a clock using hours and
minutes.
(2.11) Probability and statistics. The student
organizes data to make it useful for interpreting
information. The student is expected to:
(A) construct picture graphs and bar-type graphs;
(B) draw conclusions and answer questions based on
picture graphs and bar-type graphs; and
(C) use data to describe events as more likely or
less likely such as drawing a certain color
crayon from a bag of seven red crayons and
three green crayons.
(2.12) Underlying processes and mathematical tools.
The student applies Grade 2 mathematics to solve
problems connected to everyday experiences and
activities in and outside of school. The student is
expected to:
(A) identify the mathematics in everyday
situations;
(B) use a problem-solving model that incorporates
understanding the problem, making a plan,
carrying out the plan, and evaluating the
solution for reasonableness;
(C) select or develop an appropriate problem-
solving strategy including drawing a picture,
looking for a pattern, systematic guessing and
checking, or acting it out in order to solve a
problem; and
(D) use tools such as real objects, manipulatives,
and technology to solve problems.
(2.13) Underlying processes and mathematical tools.
The student communicates about Grade 2 mathematics
using informal language. The student is expected
to:
(A) explain and record observations using objects,
words, pictures, numbers, and technology; and
(B) relate informal language to mathematical
language and symbols.
(2.14) Underlying processes and mathematical tools.
The student uses logical reasoning to make sense
of his or her world.
The student is expected to reason and support his or
her thinking using objects, words, pictures,
numbers, and technology.
(a) Introduction.
(1) Within a well-balanced mathematics curriculum, the
primary focal points at Grade 3 are multiplying
and dividing whole numbers, connecting fraction
symbols to fractional quantities, and
standardizing language and procedures in geometry
and measurement.
(2) Throughout mathematics in Grades 3-5, students
build a foundation of basic understandings in
number, operation, and quantitative reasoning;
patterns, relationships, and algebraic thinking;
geometry and spatial reasoning; measurement; and
probability and statistics. Students use
algorithms for addition, subtraction,
multiplication, and division as generalizations
connected to concrete experiences; and they
concretely develop basic concepts of fractions and
decimals. Students use appropriate language and
organizational structures such as tables and
charts to represent and communicate relationships,
make predictions, and solve problems. Students
select and use formal language to describe their
reasoning as they identify, compare, and classify
shapes and solids; and they use numbers, standard
units, and measurement tools to describe and
compare objects, make estimates, and solve
application problems. Students organize data,
choose an appropriate method to display the data,
and interpret the data to make decisions and
predictions and solve problems.
(3) Problem solving, language and communication,
connections within and outside mathematics, and
formal and informal reasoning underlie all content
areas in mathematics. Throughout mathematics in
Grades 3-5, students use these processes together
with technology and other mathematical tools such
as manipulative materials to develop conceptual
understanding and solve problems as they do
mathematics.
(b) Knowledge and skills.
(3.1)Number, operation, and quantitative reasoning. The
student uses place value to communicate about
increasingly large whole numbers in verbal and
written form, including money. The student is
expected to:
(A) use place value to read, write (in symbols and
words), and describe the value of whole
numbers through 999,999;
(B) use place value to compare and order whole
numbers through 9,999; and
(C) determine the value of a collection of coins
and bills.
(3.2)Number, operation, and quantitative reasoning. The
student uses fraction names and symbols to describe
fractional parts of whole objects or sets of
objects. The student is expected to:
(A) construct concrete models of fractions;
(B) compare fractional parts of whole objects or
sets of objects in a problem situation using
concrete models;
(C) use fraction names and symbols to describe
fractional parts of whole objects or sets of
objects with denominators of 12 or less; and
(D) construct concrete models of equivalent
fractions for fractional parts of whole
objects.
(3.3)Number, operation, and quantitative reasoning. The
student adds and subtracts to solve meaningful
problems involving whole numbers. The student is
expected to:
(A) model addition and subtraction using pictures,
words, and numbers; and
(B) select addition or subtraction and use the
operation to solve problems involving whole
numbers through 999.
(3.4)Number, operation, and quantitative reasoning. The
student recognizes and solves problems in
multiplication and division situations. The student
is expected to:
(A) learn and apply multiplication facts through
the tens using concrete models;
(B) solve and record multiplication problems (one-
digit multiplier); and
(C) use models to solve division problems and use
number sentences to record the solutions.
(3.5)Number, operation, and quantitative reasoning. The
student estimates to determine reasonable results.
The student is expected to:
(A) round two-digit numbers to the nearest ten and
three-digit numbers to the nearest hundred;
and
(B) estimate sums and differences beyond basic
facts.
(3.6)Patterns, relationships, and algebraic thinking.
The student uses patterns to solve problems. The
student is expected to:
(A) identify and extend whole-number and geometric
patterns to make predictions and solve
problems;
(B) identify patterns in multiplication facts
using concrete objects, pictorial models, or
technology; and
(C) identify patterns in related multiplication
and division sentences (fact families) such as
2 x 3 = 6, 3 x 2 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2.
(3.7)Patterns, relationships, and algebraic thinking.
The student uses lists, tables, and charts to
express patterns and relationships. The student is
expected to:
(A) generate a table of paired numbers based on a
real-life situation such as insects and legs;
and
(B) identify patterns in a table of related number
pairs based on a real-life situation and
extend the table.
(3.8) Geometry and spatial reasoning. The student
uses formal geometric vocabulary.
The student is expected to name, describe, and compare
shapes and solids using formal geometric
vocabulary.
(3.9)Geometry and spatial reasoning. The student
recognizes congruence and symmetry. The student is
expected to:
(A) identify congruent shapes;
(B) create shapes with lines of symmetry using
concrete models and technology; and
(C) identify lines of symmetry in shapes.
(3.10) Geometry and spatial reasoning. The student
recognizes that numbers can be represented by
points on a line.
The student is expected to locate and name points on a
line using whole numbers and fractions such as
halves.
(3.11) Measurement. The student selects and uses
appropriate units and procedures to measure length
and area. The student is expected to:
(A) estimate and measure lengths using standard
units such as inch, foot, yard, centimeter,
decimeter, and meter;
(B) use linear measure to find the perimeter of a
shape; and
(C) use concrete models of square units to
determine the area of shapes.
(3.12) Measurement. The student measures time and
temperature. The student is expected to:
(A) tell and write time shown on traditional and
digital clocks; and
(B) use a thermometer to measure temperature.
(3.13) Measurement. The student applies measurement
concepts.
The student is expected to measure to solve problems
involving length, area, temperature, and time.
(3.14) Probability and statistics. The student solves
problems by collecting, organizing, displaying, and
interpreting sets of data. The student is expected
to:
(A) collect, organize, record, and display data in
pictographs and bar graphs where each picture
or cell might represent more than one piece of
data;
(B) interpret information from pictographs and bar
graphs; and
(C) use data to describe events as more likely,
less likely, or equally likely.
(3.15) Underlying processes and mathematical tools.
The student applies Grade 3 mathematics to solve
problems connected to everyday experiences and
activities in and outside of school. The student is
expected to:
(A) identify the mathematics in everyday
situations;
(B) use a problem-solving model that incorporates
understanding the problem, making a plan,
carrying out the plan, and evaluating the
solution for reasonableness;
(C) select or develop an appropriate problem-
solving strategy, including drawing a picture,
looking for a pattern, systematic guessing and
checking, acting it out, making a table,
working a simpler problem, or working
backwards to solve a problem; and
(D) use tools such as real objects, manipulatives,
and technology to solve problems.
(3.16) Underlying processes and mathematical tools.
The student communicates about Grade 3 mathematics
using informal language. The student is expected
to:
(A) explain and record observations using objects,
words, pictures, numbers, and technology; and
(B) relate informal language to mathematical
language and symbols.
(3.17) Underlying processes and mathematical tools.
The student uses logical reasoning to make sense of
his or her world. The student is expected to:
(A) make generalizations from patterns or sets of
examples and nonexamples; and
(B) justify why an answer is reasonable and
explain the solution process.
(a) Introduction.
(1) Within a well-balanced mathematics curriculum, the
primary focal points at Grade 4 are comparing and
ordering fractions and decimals, applying
multiplication and division, and developing ideas
related to congruence and symmetry.
(2) Throughout mathematics in Grades 3-5, students
build a foundation of basic understandings in
number, operation, and quantitative reasoning;
patterns, relationships, and algebraic thinking;
geometry and spatial reasoning; measurement; and
probability and statistics. Students use
algorithms for addition, subtraction,
multiplication, and division as generalizations
connected to concrete experiences; and they
concretely develop basic concepts of fractions and
decimals. Students use appropriate language and
organizational structures such as tables and
charts to represent and communicate relationships,
make predictions, and solve problems. Students
select and use formal language to describe their
reasoning as they identify, compare, and classify
shapes and solids; and they use numbers, standard
units, and measurement tools to describe and
compare objects, make estimates, and solve
application problems. Students organize data,
choose an appropriate method to display the data,
and interpret the data to make decisions and
predictions and solve problems.
(3) Problem solving, language and communication,
connections within and outside mathematics, and
formal and informal reasoning underlie all content
areas in mathematics. Throughout mathematics in
Grades 3-5, students use these processes together
with technology and other mathematical tools such
as manipulative materials to develop conceptual
understanding and solve problems as they do
mathematics.
(b) Knowledge and skills.
(4.1)Number, operation, and quantitative reasoning. The
student uses place value to represent whole numbers
and decimals. The student is expected to:
(A) use place value to read, write, compare, and
order whole numbers through the millions
place; and
(B) use place value to read, write, compare, and
order decimals involving tenths and
hundredths, including money, using concrete
models.
(4.2)Number, operation, and quantitative reasoning. The
student describes and compares fractional parts of
whole objects or sets of objects. The student is
expected to:
(A) generate equivalent fractions using concrete
and pictorial models;
(B) model fraction quantities greater than one
using concrete materials and pictures;
(C) compare and order fractions using concrete and
pictorial models; and
(D) relate decimals to fractions that name tenths
and hundredths using models.
(4.3)Number, operation, and quantitative reasoning. The
student adds and subtracts to solve meaningful
problems involving whole numbers and decimals. The
student is expected to:
(A) use addition and subtraction to solve problems
involving whole numbers; and
(B) add and subtract decimals to the hundredths
place using concrete and pictorial models.
(4.4)Number, operation, and quantitative reasoning. The
student multiplies and divides to solve meaningful
problems involving whole numbers. The student is
expected to:
(A) model factors and products using arrays and
area models;
(B) represent multiplication and division
situations in picture, word, and number form;
(C) recall and apply multiplication facts through
12 x 12;
(D) use multiplication to solve problems involving
two-digit numbers; and
(E) use division to solve problems involving one-
digit divisors.
(4.5)Number, operation, and quantitative reasoning. The
student estimates to determine reasonable results.
The student is expected to:
(A) round whole numbers to the nearest ten,
hundred, or thousand to approximate reasonable
results in problem situations; and
(B) estimate a product or quotient beyond basic
facts.
(4.6)Patterns, relationships, and algebraic thinking.
The student uses patterns in multiplication and
division. The student is expected to:
(A) use patterns to develop strategies to remember
basic multiplication facts;
(B) solve division problems related to
multiplication facts (fact families) such as
9 x 9 = 81 and 81 ÷ 9 = 9; and
(C) use patterns to multiply by 10 and 100.
(4.7) Patterns, relationships, and algebraic
thinking. The student uses organizational
structures to analyze and describe patterns and
relationships.
The student is expected to describe the relationship
between two sets of related data such as ordered
pairs in a table.
(4.8)Geometry and spatial reasoning. The student
identifies and describes lines, shapes, and solids
using formal geometric language. The student is
expected to:
(A) identify right, acute, and obtuse angles;
(B) identify models of parallel and perpendicular
lines; and
(C) describe shapes and solids in terms of
vertices, edges, and faces.
(4.9)Geometry and spatial reasoning. The student
connects transformations to congruence and
symmetry. The student is expected to:
(A) demonstrate translations, reflections, and
rotations using concrete models;
(B) use translations, reflections, and rotations
to verify that two shapes are congruent; and
(C) use reflections to verify that a shape has
symmetry.
(4.10) Geometry and spatial reasoning. The student
recognizes the connection between numbers and
points on a number line.
The student is expected to locate and name points on a
number line using whole numbers, fractions such as
halves and fourths, and decimals such as tenths.
(4.11) Measurement. The student selects and uses
appropriate units and procedures to measure weight
and capacity. The student is expected to:
(A) estimate and measure weight using standard
units including ounces, pounds, grams, and
kilograms; and
(B) estimate and measure capacity using standard
units including milliliters, liters, cups,
pints, quarts, and gallons.
(4.12) Measurement. The student applies measurement
concepts.
The student is expected to measure to solve problems
involving length, including perimeter, time,
temperature, and area.
(4.13) Probability and statistics. The student solves
problems by collecting, organizing, displaying, and
interpreting sets of data. The student is expected
to:
(A) list all possible outcomes of a probability
experiment such as tossing a coin;
(B) use a pair of numbers to compare favorable
outcomes to all possible outcomes such as four
heads out of six tosses of a coin; and
(C) interpret bar graphs.
(4.14) Underlying processes and mathematical tools.
The student applies Grade 4 mathematics to solve
problems connected to everyday experiences and
activities in and outside of school. The student is
expected to:
(A) identify the mathematics in everyday
situations;
(B) use a problem-solving model that incorporates
understanding the problem, making a plan,
carrying out the plan, and evaluating the
solution for reasonableness;
(C) select or develop an appropriate problem-
solving strategy, including drawing a picture,
looking for a pattern, systematic guessing and
checking, acting it out, making a table,
working a simpler problem, or working
backwards to solve a problem; and
(D) use tools such as real objects, manipulatives,
and technology to solve problems.
(4.15) Underlying processes and mathematical tools.
The student communicates about Grade 4 mathematics
using informal language. The student is expected
to:
(A) explain and record observations using objects,
words, pictures, numbers, and technology; and
(B) relate informal language to mathematical
language and symbols.
(4.16) Underlying processes and mathematical tools.
The student uses logical reasoning to make sense of
his or her world. The student is expected to:
(A) make generalizations from patterns or sets of
examples and nonexamples; and
(B) justify why an answer is reasonable and
explain the solution process.
S
(a) Introduction.
(1) Within a well-balanced mathematics curriculum, the
primary focal points at Grade 5 are comparing and
contrasting lengths, area, and volume of geometric
shapes and solids; representing and interpreting
data in graphs, charts, and tables; and applying
whole number operations in a variety of contexts.
(2) Throughout mathematics in Grades 3-5, students
build a foundation of basic understandings in
number, operation, and quantitative reasoning;
patterns, relationships, and algebraic thinking;
geometry and spatial reasoning; measurement; and
probability and statistics. Students use
algorithms for addition, subtraction,
multiplication, and division as generalizations
connected to concrete experiences; and they
concretely develop basic concepts of fractions and
decimals. Students use appropriate language and
organizational structures such as tables and
charts to represent and communicate relationships,
make predictions, and solve problems. Students
select and use formal language to describe their
reasoning as they identify, compare, and classify
shapes and solids; and they use numbers, standard
units, and measurement tools to describe and
compare objects, make estimates, and solve
application problems. Students organize data,
choose an appropriate method to display the data,
and interpret the data to make decisions and
predictions and solve problems.
(3) Problem solving, language and communication,
connections within and outside mathematics, and
formal and informal reasoning underlie all content
areas in mathematics. Throughout mathematics in
Grades 3-5, students use these processes together
with technology and other mathematical tools such
as manipulative materials to develop conceptual
understanding and solve problems as they do
mathematics.
(b) Knowledge and skills.
(5.1)Number, operation, and quantitative reasoning. The
student uses place value to represent whole numbers
and decimals. The student is expected to:
(A) use place value to read, write, compare, and
order whole numbers through the billions
place; and
(B) use place value to read, write, compare, and
order decimals through the thousandths place.
(5.2)Number, operation, and quantitative reasoning. The
student uses fractions in problem-solving
situations. The student is expected to:
(A) generate equivalent fractions;
(B) compare two fractional quantities in problem-
solving situations using a variety of methods,
including common denominators; and
(C) use models to relate decimals to fractions
that name tenths, hundredths, and thousandths.
(5.3)Number, operation, and quantitative reasoning. The
student adds, subtracts, multiplies, and divides to
solve meaningful problems. The student is expected
to:
(A) use addition and subtraction to solve problems
involving whole numbers and decimals;
(B) use multiplication to solve problems involving
whole numbers (no more than three digits times
two digits without technology);
(C) use division to solve problems involving whole
numbers (no more than two-digit divisors and
three-digit dividends without technology);
(D) identify prime factors of a whole number and
common factors of a set of whole numbers; and
(E) model and record addition and subtraction of
fractions with like denominators in problem-
solving situations.
(5.4)Number, operation, and quantitative reasoning. The
student estimates to determine reasonable results.
The student is expected to:
(A) round whole numbers and decimals through
tenths to approximate reasonable results in
problem situations; and
(B) estimate to solve problems where exact answers
are not required.
(5.5)Patterns, relationships, and algebraic thinking.
The student makes generalizations based on observed
patterns and relationships. The student is expected
to:
(A) use concrete objects or pictures to make
generalizations about determining all possible
combinations;
(B) use lists, tables, charts, and diagrams to
find patterns and make generalizations such as
a procedure for determining equivalent
fractions; and
(C) identify prime and composite numbers using
concrete models and patterns in factor pairs.
(5.6) Patterns, relationships, and algebraic
thinking. The student describes relationships
mathematically.
The student is expected to select from and use diagrams
and number sentences to represent real-life
situations.
(5.7)Geometry and spatial reasoning. The student
generates geometric definitions using critical
attributes. The student is expected to:
(A) identify critical attributes including
parallel, perpendicular, and congruent parts
of geometric shapes and solids; and
(B) use critical attributes to define geometric
shapes or solids.
(5.8)Geometry and spatial reasoning. The student models
transformations. The student is expected to:
(A) sketch the results of translations, rotations,
and reflections; and
(B) describe the transformation that generates one
figure from the other when given two congruent
figures.
(5.9) Geometry and spatial reasoning. The student
recognizes the connection between ordered pairs of
numbers and locations of points on a plane.
The student is expected to locate and name points on a
coordinate grid using ordered pairs of whole
numbers.
(5.10) Measurement. The student selects and uses
appropriate units and procedures to measure volume.
The student is expected to:
(A) measure volume using concrete models of cubic
units; and
(B) estimate volume in cubic units.
(5.11) Measurement. The student applies measurement
concepts. The student is expected to:
(A) measure to solve problems involving length
(including perimeter), weight, capacity, time,
temperature, and area; and
(B) describe numerical relationships between units
of measure within the same measurement system
such as an inch is one-twelfth of a foot.
(5.12) Probability and statistics. The student
describes and predicts the results of a probability
experiment. The student is expected to:
(A) use fractions to describe the results of an
experiment; and
(B) use experimental results to make predictions.
(5.13) Probability and statistics. The student solves
problems by collecting, organizing, displaying, and
interpreting sets of data. The student is expected
to:
(A) use tables of related number pairs to make
line graphs;
(B) describe characteristics of data presented in
tables and graphs including the shape and
spread of the data and the middle number; and
(C) graph a given set of data using an appropriate
graphical representation such as a picture or
line.
(5.14) Underlying processes and mathematical tools.
The student applies Grade 5 mathematics to solve
problems connected to everyday experiences and
activities in and outside of school. The student is
expected to:
(A) identify the mathematics in everyday
situations;
(B) use a problem-solving model that incorporates
understanding the problem, making a plan,
carrying out the plan, and evaluating the
solution for reasonableness;
(C) select or develop an appropriate problem-
solving strategy, including drawing a picture,
looking for a pattern, systematic guessing and
checking, acting it out, making a table,
working a simpler problem, or working
backwards to solve a problem; and
(D) use tools such as real objects, manipulatives,
and technology to solve problems.
(5.15) Underlying processes and mathematical tools.
The student communicates about Grade 5 mathematics
using informal language. The student is expected
to:
(A) explain and record observations using objects,
words, pictures, numbers, and technology; and
(B) relate informal language to mathematical
language and symbols.
(5.16) Underlying processes and mathematical tools.
The student uses logical reasoning to make sense of
his or her world. The student is expected to:
(A) make generalizations from patterns or sets of
examples and nonexamples; and
(B) justify why an answer is reasonable and
explain the solution process.