Considerations around Math Instruction for Students with
Disabilities1
Text
Chapter Two: Place Value
If anybody doesn’t have the book, there are extra’s in the cabinet
in Room 170
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Outcomes for Session Four
Participants will understand the scope and sequence of the
development of number sense
Participants will deepen their understanding of place value in the
Hindu-Arabic number system
Participants survey the various models for Least Restrictive
Environment to treat number sense issues
Participants will understand the foundation of a systematic lesson
plan for math (30 minute lesson plan)
Participants will explore instructional techniques for the mastery
and application of skills for number sense
Place Value
Place Value: Brahmi Numbers
The Hindu-Arabric number system that we use today can be traced
back to the Brahmi numerals (India) 3rd Century BC
What number is missing from the system?
numbers 1,2,3 are the number of stokes to equal the number (same as
modern Japanese number system
number 4 is marked by the four point in the cross
historians are not sure about the origin of the rest of the
numbers; some think that they were distinct marks that were
combined like a system of cursive writing
Talking about the history of the current number system, helps to
understand the evolution of the number 0 that is used today to
denote place value. Remember the number 0 is not a number that
exists on our internal number line. This was invented to makes
number writing efficient. During this time, each group of tens had
it own symbol. There was a different symbol for 20, 30, 40, etc.,
thus making the number system more cumbersome.
5
1944
1999
MDCCCCLXXXXVIIII
MCMXCIX
MIM
In the Roman Number system, place value was denoted by a symbol.
Zero was not used to denote place value. Letters were used to
denote value. Symbols are placed in order of value, starting with
the largest value first. There are no standard rules to writing in
Roman Numbers. For the number 1999 there are three possible ways.
This was confusing and was gradually replaced by the Arabic
numerals.
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Place Value: Arabic Numerals
Arabic numbers were introduced to Europe in the 10th century by
Arabs of North Africa. There are many numbers systems in the Arab
world. These are the numbers systems introduced from North
Africa.
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Irrational Number
Digit comes from Latin (digita meaning fingers)
10 Symbols of the base 10 number system
0123456789
a number is one or more digits
The Arabic number system used a Zero, which eventually was
developed by Babylonian mathematics but it was typically just a
blank space, then became wedges, to eventually the symbol we used
today 0. The Babylonian system was base-60 and still used today in
telling time.
The Zero is both a number and a digit. A number is a mathematical
object used to count. Numbers include zero, negative numbers,
rational numbers, irrational numbers, real and complex
numbers.
The Zero also is a digit in the a number system. In the base 10
system you have the digits 0-9. In a base 2 system (binary) you
have the digits 0 and 1).
The Zero made writing numbers much easier and efficient. Less
digits were required, especially for large numbers. It became the
preferred method in Europe over the Roman Number system.
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Place Value: Positional Systems and Fraud?
Positional systems or place values systems with the use of an
absolute 0 were controversial. One could simply put a number before
or after the string of digits thus causing fraud. This is why when
writing checks we must write out the words as well as using the
digits to represent the amount.
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themselves.
The location of the digits in relationship to each other, helps us
to determine the value
80 + 4
The number 84 has two digits. The 8 and the 4. The placement of the
digits tells us it value.
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Place Value: Language
The language of place value can be confusing for students with any
type of language processing difficulty. What does ten thousands
really mean? This abstract concept can lead to problems with place
value understanding.
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Solve this problem…
Express 5 hours and 20 minutes as 4 hours and ______ minutes.
Once a student is use to base 10 systems, we throw strange system
at them like telling time. This can create confusion for
students.
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Common Errors for Place Value
Confusion when to use rules to group objects, trade objects or
compute
These are pieces of evidence to indicate an error in place value
understanding, which is a type of number sense error.
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Common Errors for Place Value
Content gaps and language barriers
These are pieces of evidence to indicate an error in place value
understanding, which is a type of number sense error.
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Common Errors for Place Value
Content gaps and language barriers
These are pieces of evidence to indicate an error in place value
understanding, which is a type of number sense error.
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Misapplies the procedure for “counting on”
Counts this as 10, 20, 30, 1, 2
Instead of Counts this as 10, 20, 30, 31, 32
These are pieces of evidence to indicate an error in place value
understanding, which is a type of number sense error.
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Does not identify multi-digit numbers in terms of place value
reads the number correctly
but…
3 + 2
instead of
30 + 2
When asked the values of the numbers the child says “3” and “2”
instead of “30” and “2”
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Does not understand that place determines value
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Misapplies the rule for reading numbers from left to right
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eighteen
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Common Errors for Place Value
Student orders numbers based on the value of the digits, instead of
the place value.
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because 6 and 9 are bigger than 1 and 2
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Lacks understanding of positional systems
Struggle with trading groups (when you reach 10 units then trade
for a 10 bar)
Poor understanding of place value structure
make sure it isn’t due to the following
Poor number sense (instant recognition of number, one-to-one
correspondence, counting forwards and backwards, skip counting ,
poor cardinality)
Place Value: Conceptual Errors
7005000058
4000700208
The child is asked to write out a set of numbers. This child
express is each digit as a separate Cardinal value rather than
multiplying each by its place value. (4 x 1000). For example, he
writes “50” instead of “ 5” to indicate the number of tens in “57”.
This child lacks a conceptual understanding of place value in that
his notation is not corrected to any sense of a positional
system.
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They are equal
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215
They are equal
35
91
Can you figure out why he is making these mistakes? There could be
a number of things going on however the most prominent one is that
he is reading numbers both right to left and left to right. In
question number one he thinks that 13 and 31 are the same number.
This is a strong indication that this child lacks an understanding
of place value.
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Scope and Sequence of Number Sense Development
In a group of 2-4 develop a scope and sequence of Number Sense
Development based on what we have studies in class so far.
Write one skill on a separate post-it using a bold marker.
Arrange the post-it in order from the most basic to the most
advanced skill on the wall.
After all groups have completed the task, then look at each others
scope and sequence.
This basic activity is designed to get a specific sequence to the
development of skills that must be mastered in order to ensure
success in all other mathematic domains. While there is no agreed
upon specific scope and sequence, based upon the work of your
group’s would there be anything that you might change from your
group’s scope and sequence? Lead a discussion around the various
Scopes and sequence is of skill development. Proceed to the next
task where they will sequence the CCSS sequence of number sense
including place value.
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CCSS and Number Sense
The common core state standards provide a very specific scope and
sequence of skill development around numbers and the base 10
system.
At each table is an envelope we have a summary of various skills to
be mastered at each grade level related to number sense.
Rearrange the skills from kindergarten through high school. We will
check your answers in the next upcoming slides.
A copy of this activity is located at the end of the
PowerPoint.
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Kindergarten
Counting and cardinality (know number names and count sequence,
count the number of objects, compare numbers)
Number and operations and base 10 (work with numbers in 11-19 to
gain foundations for place value)
First grade
Number in operations in base 10 (extend the counting sequence,
understand place value, uses place value understanding and
properties of operation to add and subtract)
Second grade
Number in operations in base 10 (extend the counting sequence,
understand place value, uses place value understanding and
properties of operation to add and subtract)
Third grade
Number in operations and base 10 (uses place value understanding
and properties of operation to perform multi-digit
arithmetic)
Number in operations-fractions (develop understanding of fractions
as numbers)
Fourth grade
Number in operations in based 10 (generalize place value
understanding for multi-digit whole numbers, use place value
understanding and properties of operations to perform multi-digit
arithmetic)
Number in operations-fractions(extend understanding of fraction
equivalents and ordering, build fractions from unit fractions by
applying and extending previous understandings of operations on the
whole numbers, understand a small notations for fractions, and
compare decimal fractions)
Fifth grade
Number in operations and base 10 (understand the place value
system, perform operations with multi-digit whole numbers and with
decimals to hundredths)
Number and operations-fractions (use a equivalent fractions as a
strategy to add and subtract fractions, apply and extend previous
understandings of multiplication and division to multiply and
divide fractions)
Sixth grade
Ratios and proportional relationships (understand ratio concepts
and use ratio reasoning to solve problems)
The number system (apply and extend previous understanding of
multiplication and division to divide fractions by fractions,
compute fluently with multi-digit numbers and find common factors
and multiples, apply in extend previous understandings of numbers
to the system of rational numbers)
Seventh grade
Ratios and proportional relationships (analyze proportional
relationships and use them to solve real-world and mathematical
problems)
The number system (apply and extend previous understandings of
operations with fractions to add, subtract, multiply, and divide
all numbers)
Eighth grade
The number system (I know that there are numbers that are not
rational, an approximate them by rational numbers)
High school
The real number system (extend the properties of exponent stew
rational exponents, use properties of rational and irrational
numbers)
Quantities (reason quantitatively and use units to solve
problems)
The complex number system (performs arithmetic operations with
complex numbers, represent complex numbers and their operations on
the complex plane, use complex numbers and polynomial identities
and equations
This slide is a animated. Each click will reveal the different
grade level expectations. Go through each one and the groups can
self check to see if they got it in the correct order.
Provide a hand out that contains a combination of preschool and
school aged skills for a scope and sequence of number sense. This
will be used to develop intervention for students struggling with
this skill.
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Models for LRE for math specially designed instruction in number
sense
Mild number sense disability
Usually has instant recognition of number, carnality, and counting
ability.
Struggles with place value.
-Adaptation of general education materials (Manipulatives )
-Alternative presentations and demonstration of concepts by GEN Ed
teacher (Universal design for learning ) -Additional instruction
and practice in general education in the area of place value
(tutor, peer assistance, small group) -Special education teacher
provides consult
-Adaptation of general education materials (Manipulatives )
-Alternative presentations and demonstration of concepts by GEN Ed
teacher (Universal design for learning ) -Pull out for 10 to 15
minutes to the back of the room for direct instruction in place
value by special education teacher
-Co-teaching the lessons, with special education teacher providing
alternative presentation of the concepts to whole group or small
group -Special education teacher suggesting alternative
presentation and demonstration of learning techniques -Special
education teacher providing small group or one on one instruction
in the back of the room
Go over each one of these possible intervention options for a
student with a mild number sense disability. After going over these
options ask the group to comment. Are you doing any of these
options with your students? Is there something else that you are
doing with your mild students?
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Moderate number sense disability
Knows the name of numbers, has rote counting and 1 to 1
correspondence
Struggles with instant number recognition, inconsistent
cardinality, and struggles with place value to the number 10
Option One
Option Two
Option Three
-Pull out for math instruction -Alternative approach to curriculum
-Analyze each lesson to consider if it meets an essential goal for
the grade level -Conduct lesson using alternative presentation and
demonstration -Eliminate lessons that do not meet an essential goal
for grade level -Include daily systematic routines to develop
automaticity in basic number sense skills
-Pull out for math instruction -Rearrange the curriculum so that
one concept is developed at a time to mastery level -Include daily
systematic routines to develop automaticity in basic number sense
skills
-Co teaching with pull out to the back of the room for certain
topics -Allow student to participate in whole group instruction to
be exposed to the concept -Special education teacher provides
suggestions on alternative presentation and demonstration --Include
daily systematic routines to develop automaticity in basic number
sense skills
Go over each one of these possible intervention options for a
student with a mild number sense disability. After going over these
options ask the group to comment. Are you doing any of these
options with your students? Is there something else that you are
doing with your mild students?
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Has basic number knowledge to five
Has no instant number recognition, cardinality, magnitude of
number, and place value concepts
Option One
Option Two
-Alternative core curriculum designed for students with adaptive
functioning concerns
-Pull out for math instruction -Rearrange the curriculum so that
one concept is developed at a time to mastery level -Include daily
systematic routines to develop automaticity in basic number sense
skills
Go over each one of these possible intervention options for a
student with a mild number sense disability. After going over these
options ask the group to comment. Are you doing any of these
options with your students? Is there something else that you are
doing with your mild students?
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3rd Grade
Struggles with instant recognition of 6-9 on 10 frame
Doesn’t understand the base 10 system with trading 10 ones for 1
ten
Poor number formation through handwriting (reverses 2 and 7)
This is Sam. Sam is in third grade and has a specific math learning
disability in math, that is rooted in weak number sense and recall
of math concepts.
We are going to quickly look at a weeks worth of Everyday Math
lessons for Third Grade. Keep Sam in mind when we look over these
lessons. What do you think he will be successful with? What is he
going to struggle with? What accommodations does he need? What
should his instruction look like?
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In this lesson the students explore numbers that you might find in
a museum. From this discovery based exercise they are to conclude
that there are a variety of a type of numbers and numbers have many
purposes. Then the students will work in their journal doing number
sequences. Then they explore the rooms for evidence of numbers. The
differentiation activity is to play a game called “Number
Squeeze”.
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While a majority of the lesson is to focus on building rituals and
routine, this lesson deals with the number grid. Patterns are
explored through Query type of activities. Then students follow up
previous lessons by doing number sequence sheets and then solve
number grip puzzles. The differentiation activity has the student
solving the number grid puzzles by covering up portions of a number
grid with “sticky notes”
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In this lesson the students are learning how to use their reference
materials to look up information. Then the students play an
addition fact games called “Less Than You!”. The differentiation
task as student’s scooping up a handful of pennies, counting and
then comparing quantity with each other.
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In this lesson students review telling time, measurement and basic
shapes. The purpose is to review the use of tools in mathematical
problem solving. The differentiation task has students making a
paper plate clock.
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This lesson focuses on how to record and display data. It also
introduced the concept of a math box. The differentiation has the
students practicing making tally marks.
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What is he going to struggle with?
What accommodations does he need?
What should his specially designed instruction look like?
Use these guiding questions to lead a discussion regarding Sam.
Also refer to the models of LRE for supporting a child, What model
might be most appropriate for him?
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30 Min Math Lesson: Systematic
We build houses based on blueprints. A well planned out description
of how to build the house. House builders must have blueprints.
Specially designed instruction also needs blueprints. This is
referred to as systematic instruction or well though out sequence
of instruction and well established routines that focus on
understanding, mastery, fluency and application of learning.
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30 Minute Lesson Plan: Systems
The 30 minute lesson plan is adapted from the work of Susan Hall in
the area of reading. Since most specially designed instruction
occurs in blocks of 30 minutes, the 30 minute lesson plan is
designed to maximize the short about of time to get the most out of
instruction and is designed for mastery, understanding, fluency and
application.
Systematic: Each step is done the same way every day, building one
skill onto the next. Use a time to make sure you are not taking too
long on one step. Make sure all materials are ready before the
lesson for seamless transition between the skills.
Students who receive their instruction in a systematic way, are
higher achievers.
This can be adapted for 15 min, 45 min, 60 min and 90 min lesson
plans based on needs.
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Application (10 min)
Fluency of previous over learned skill (5 min)
The 30 Minute Lesson plan
State your goal/purpose
Review Previous Lessons
Accelerated version of previous skills designed to activate their
background knowledge
Might be fluency drills
Pass out the one page 30 min lesson plan. Go over each
section.
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3. New Lesson
1. Sometimes the same lesson must be repeated in this section
because they didn’t get it the first time ( or first 3 times)
2. Use proven strategies for teaching new skills
a. CRA
b. Visualization
c. Multi-Sensory
d. Discovery
4. Practice
1. I do- Model the practice they need to do
2. Do it together
3. Gradually turn over the work to them with supervision
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5. Fluency Practice
1. I do- Model the practice they need to do
2. Do it together
3. Gradually turn over the work to them with supervision
It is very important that only previously taught skills are worked
on for fluency. The purpose is rapid retrieval of already learned
skills. These should be rapid fire drills with the purpose of
increasing speed along with accuracy.
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The 30 Minute Lesson plan
5. Problem Solving: The whole purpose of doing the skill is to
eventually solve problems. This can be an instructional task or a
simple story problem.
1. Teach strategies to address word problems
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All have number sense issues and weak place value concepts
Following the Mathematics Navigator Scope and Sequence supplemented
with Hands on Standards manipulatives and application
Complete daily “Instant Number Recognition” tasks to provide
repetition and rapid recall
All students participate in the large group lesson with
accommodations, does a modified math box (some boxes are eliminated
and others added), participates in the differentiated Everyday Math
lesson
Pulled to the back of the room for the second half of the Math
Block
Thoughts
Written out in images instead of words for the students,
Connected to an IEP goal
Connected to a standard
Thoughts
This is based on previous lessons to make sure they are building
concepts from one lesson to the next
In the previous lesson they built visual representations of models
of numbers
Thoughts
In this new lesson, we are introducing a new concept
Always use visuals and manipulatives to help establish mental
images that students can refer to later
This comes directly from Mathematics Navigator “Beginning Place
Value” page 13
Thoughts
Follow the “I do” , “We do”, and “You do” procedures
Will need to develop your own materials as the curriculum and the
supplements do not necessarily have enough examples and repetition
needed for a student with a disability
Thoughts
Always have a fluency practice stage.
In this case we are practicing a skill that they have learned and
should understand. If they have not fully developed this
understanding then go back to a skill that they know.
We only develop fluency of a skill that we have developed
conceptual knowledge of
In this lesson we are going to use Power Point to work on Instant
recognition of two digit numbers (show the base ten, then the
students use their knowledge of the ten frame to recognize the
numbers)
Switch to the Instant Recognition of two digit numbers ppt to
demonstrate a fluency drill.
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Thoughts
Always have a practical application component to every lesson. This
can be an instructional task, story problem, or a project.
Teach strategies for solving problems
This problem comes from HOS page 43
Instruction for Number Sense
Project Based Learning or Instructional Tasks and Application of
learning
Systematic Instruction and Application Learning
systematic instruction for the acquisition of skills
project work (problems, dilemma, instructional tasks, projects) for
the application of skills acquired earlier
Special Education is often criticized for only focusing on skills
development and not purposely transferring the skills to
application. Both must occur! At the elementary, middle, and high
school levels, there are some parts of the curriculum in which
students are necessarily dependent on the teacher, and others in
which students can work more independently. In particular, two
aspects of the curriculum provide for students’ learning
needs:
1. systematic instruction for the acquisition of
skills 2. project work for the application of
skills acquired earlier
This is why the “30 min” lesson always ends with an Instructional
Task that applies the skills learned.
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Student follows instructions
Intrinsic motivation characterizes the work particularly
Teacher addresses student's deficiencies
Three Interventions
We are going to take a look at these three programs that are used
in DPS and how they address number sense. These programs are not
designed to meet the needs of all learners but provide supplement
to the core curriculum for students who struggle. Each presents one
lesson for a different topic on the scope and sequence. They may
not provide the extra practice, instruct strategies appropriate for
student with a disability, sufficient fluency practice or a
sensible sequence of instruction for remediation.
Mathematics Navigator is grouped by subject which makes it easier
to use for specially designed instruction. There is no grade level
designation. The Beginning Place Value module would be appropriate
as early as Kindergarten for the first lesson but probably most
appropriate for 2nd grade and up.
Hands on Standards is loosely grouped by grade level. You will
probably need multiple grade levels to see the entire scope and
sequence of number sense skills. The manuals really just show you
how to use the manipulatives. You have to create your own practice
items.
Origomath is by grade levels. You will need to find the number
sense skills at each grade level. Essentially you are rearranging
the curriculum to be more like Mathematics Navigator. This does
provide a lot of applicaion or instructional tasks that the other
programs don’t provide.
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Hover mouse over Classroom
If possible go to school net to find these three curriculum. If not
then use these slides to show how to find them in School Net. These
slides are here for folks to refer to if they are struggling on how
to find the.
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Click Search
School Net
23= 20 +3
Lesson 9 in “Beginning Place Value”
Developing Number Sense Using Place Value
1. Review: representing, counting, and recording tens (20, 30, 40,
etc)
3. Practice using CRA
Sample Lesson: ORIGOmath
First Grade Unit 5 Working with Place Value and Relative
Position
Using a two-digit Numeral Expander
tens
ones
fold
tape or laminate
This lesson uses the Numeral Expander. See how to make this
manipulative. Pass out a copy of the Numeral Expander. Show them
how to fold it.
Use this to demonstrate how a number is made with place and value.
Show
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tens
ones
3
2
2
3
How many 10s? Write that number on the tape in the 10s box using a
wet erase markers so you can use it over and over again. How many
1s? Write that number on the tape in the 1s box. Then fold the
expander to reveal the number. You can do this reversed as well.
Put the numbers in the boxes and have them build models with the
base 10 blocks.
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Next time we meet… 2-5-13
Chapter 4-5 Teaching Students who struggle with addition and
subtraction of whole numbers
Counting and cardinality (know number names and count sequence,
count the number of objects, compare numbers)
Number and operations and base 10 (work with numbers in 11-19 to
gain foundations for place value)
Number in operations in base 10 (extend the counting sequence,
understand place value, uses place value understanding and
properties of operation to add and subtract)
Number in operations in base 10 (extend the counting sequence,
understand place value, uses place value understanding and
properties of operation to add and subtract)
Number in operations and base 10 (uses place value understanding
and properties of operation to perform multi-digit
arithmetic)
Number in operations-fractions (develop understanding of fractions
as numbers)
Number in operations in based 10 (generalize place value
understanding for multi-digit whole numbers, use place value
understanding and properties of operations to perform multi-digit
arithmetic)
Number in operations-fractions(extend understanding of fraction
equivalents and ordering, build fractions from unit fractions by
applying and extending previous understandings of operations on the
whole numbers, understand a small notations for fractions, and
compare decimal fractions)
Number in operations and base 10 (understand the place value
system, perform operations with multi-digit whole numbers and with
decimals to hundredths)
Number and operations-fractions (use a equivalent fractions as a
strategy to add and subtract fractions, apply and extend previous
understandings of multiplication and division to multiply and
divide fractions)
Ratios and proportional relationships (understand ratio concepts
and use ratio reasoning to solve problems)
The number system (apply and extend previous understanding of
multiplication and division to divide fractions by fractions,
compute fluently with multi-digit numbers and find common factors
and multiples, apply in extend previous understandings of numbers
to the system of rational numbers)
Ratios and proportional relationships (analyze proportional
relationships and use them to solve real-world and mathematical
problems)
The number system (apply and extend previous understandings of
operations with fractions to add, subtract, multiply, and divide
all numbers)
The number system (I know that there are numbers that are not
rational, an approximate them by rational numbers)
The real number system (extend the properties of exponent stew
rational exponents, use properties of rational and irrational
numbers)
Quantities (reason quantitatively and use units to solve
problems)
The complex number system (performs arithmetic operations with
complex numbers, represent complex numbers and their operations on
the complex plane, use complex numbers and polynomial identities
and equations
Common core state standards sorting activity: Cut out each section
and put into an envelope for each table.
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