Difference between revisions of "Number System"

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*[[Portal:Mathematics| '''Back to Mathematics Portal''']]
+
''[http://karnatakaeducation.org.in/KOER/index.php/ಸಂಖ್ಯೆಗಳು ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ]''
*[[Mathematics:_Topics|'''Back to Topics in School Mathematics''']]
+
= Concept Map =
*[[Resource_Creation_Checklist|'''Resource Creation Checklist''']] - for guidelines on how to add resources.
+
[[File:Numbers.mm|Flash]]
<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#ffffff; vertical-align:top; text-align:center; padding:5px;">
 
''[http://karnatakaeducation.org.in/KOER/index.php/ಸಂಖ್ಯೆಗಳು ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ]''</div>
 
  
= Concept Map =
 
<mm>[[Numbers.mm|Flash]]</mm>
 
 
__FORCETOC__
 
__FORCETOC__
 +
 +
= Introduction =
 +
<br> Our daily life is based on numbers. We use it for shopping, reckoning the time, counting distances and so on. Simple calculations seem effortless and trivial for most of our necessities.So we should know about numbers. Numbers help us count concrete objects. They help us to say which collection of objects . In this we are learning about basic operations of numbers - different types of numbers, representation, etc. <br> <br>[[File:Number system -Resource material_html_m3cabb6c3.png|400px]]<br><br>
 +
How can math be so universal? First, human beings didn't invent math concepts; we discovered them. <br>Also, the language of math is numbers, not English or German or Russian. <br>If we are well versed in this language of numbers, it can help us make important decisions and perform everyday tasks. <br>Math can help us to shop wisely,  understand population growth, or even bet on the horse with the best chance of winning the race. <br>
 +
Mathematics expresses itself everywhere, in almost every face of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing our understanding of all that we observe.Mathematics has been around since the beginnings of time and it most probably began with counting. Many, if not all puzzles and games require mathematical logic and deduction. This section uses the fun and excitement of various popular games and puzzles, and the exhilaration of solving them, to attract and engage the students to realise the mathematics in fun and games. <br><br>
 +
'''Descriptive Statement'''
 +
<br>Number sense is defined as an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent, coming from investigating their characteristics and using them in diverse situations, and how best they can be used to describe a particular situation. Number sense is an attribute of all successful users of mathematics. Our students often do not connect what is happening in their mathematics classrooms with their daily lives. It is essential that the mathematics curriculum build on the sense of number that students bring with them to school. Problems and numbers which arise in the context of the students world are more meaningful to  students than traditional textbook exercises and help them develop their sense of how numbers and operations are used. Frequent use of estimation and mental computation are also important ingredients in the development of number sense, as are regular opportunities for student communication. Discussion of their own invented strategies for problem solutions helps students strengthen their intuitive understanding of numbers and the relationships between numbers.<br><br>
 +
In summary, the commitment to develop number sense requires a dramatic shift in the way students learn mathematics.
 +
 +
= Flow Chart =
 +
[[File:Number system -Resource material_html_m65980570.jpg|400px]]
  
 
= Textbook =
 
= Textbook =
Line 17: Line 24:
  
 
Watch the following video on the story of how numbers evolved.  The video called Story of One tells how numbers evolved and the initial questions around number theory.<br>
 
Watch the following video on the story of how numbers evolved.  The video called Story of One tells how numbers evolved and the initial questions around number theory.<br>
{{#widget:YouTube|id=RSpadYjnYl8}}
+
{{#widget:YouTube|id=jrLQW1vQklE}}
  
 
This video is related to number system, helps to know the basic information about number system
 
This video is related to number system, helps to know the basic information about number system
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{{#widget:YouTube|id=bj4EKEfrKOU}}
 
{{#widget:YouTube|id=bj4EKEfrKOU}}
  
 +
This video is related to irrational numbers by Suchitha
 +
 +
{{#widget:YouTube|id=udKD4yxsWe4}}
 +
 +
This video is relating to exploring number patterns in square numbers.
 +
 +
{{Youtube|MoM2jw7W-ms
 +
}}
 
==Reference Books==
 
==Reference Books==
  
 
= Teaching Outlines =
 
= Teaching Outlines =
 +
*[http://karnatakaeducation.org.in/KOER/en/index.php/Number_bases_activity_1 Number based activity]
 +
==Concept #1 - History of Numbers:  Level 0 ==
 +
The following website takes us on a fascinating journey originating from Prehistoric Mathematics, its evolution in various civilizations such as Egyptian, Greek, Indian, Chinese etc. to the increased complexities and abstractions of the modern era mathematics. This story of history of numbers also includes descriptions related to contributions of some of the important men and women to the development of mathematics.
  
==Concept #1 - History of Numbers==
+
'''http://storyofmathematics.com/story.html'''
 
===Learning objectives===
 
===Learning objectives===
 
#What is the story of numbers?   
 
#What is the story of numbers?   
Line 37: Line 55:
  
 
===Activities===
 
===Activities===
# [[Activity Template]]
+
# Series of Activities in one page- [[Series of Activities on Number Systems|Click Here]]
 
#Activity 1
 
#Activity 1
#Activity 2
 
  
==Concept #2 Number Sense and Counting==
+
==Concept #2 Number Sense and Counting :  Level 0==
  
 
=== Objectives ===
 
=== Objectives ===
 
1. Understand that there is an aspect of quantity that we can develop with disparate objects<br>
 
1. Understand that there is an aspect of quantity that we can develop with disparate objects<br>
2. Comparison and mapping of quanties (more or less or equal)<br>
+
2. Comparison and mapping of quantities (more or less or equal)<br>
 
3. Representation of quantity by numbers and learning the abstraction that “2 represents quantity 2 of a given thing”<br>
 
3. Representation of quantity by numbers and learning the abstraction that “2 represents quantity 2 of a given thing”<br>
 
4. Numbers also have an ordinal value – that of ordering and that is different from the representation aspect of numbers<br>
 
4. Numbers also have an ordinal value – that of ordering and that is different from the representation aspect of numbers<br>
Line 56: Line 73:
  
 
===Activities===
 
===Activities===
#Activity 1 - [[Quantity and Numbers|Introduction_to_quantity_and_numbers]]
+
#Activity 1 - [[Quantity and Numbers|Quantity and Numbers]]
#Activity 2
+
#Activity 2 - [[The_eighth_donkey_story_for_number_sense|The Eighth Donkey Story]]
 +
#Activity 3 - [[Cardinal and Ordinal Numbers]]
  
==Concept #3 Negative numbers are the opposite of positive numbers==
+
==Concept #3 The Number Line :Level 1-2 ==
 +
The number line is not just a school object. It is as much a mathematical idea as functions.
 +
The number line is a geometric “model” of all numbers -- including 0 1, 2, 25, 374 trillion, and -5, Unlike counters, which model only counters, the number line models measurement, which is why it must start with zero. (When we count, the first object we touch is called "one." When we measur using a ruler, we line one end of the object we’re measuring against the zero mark on the ruler.
  
=== Objectives ===
+
<br>[[File:Number system -Resource material_html_1de463c.jpg|400px]]<br>
# To extend the understanding and skill of representing symbolically numbers and manipulating them.<br>
+
 
# To understand that negative numbers are numbers that are created to explain situations in such a way that mathematical operations hold<br>
+
Part of the power of addition and subtraction is that these operations work with both counting and measuring. Therefore, to understand basic operations like addition and subtraction, we need a number line model as well as counters.
# Negative numbers are opposite of positive numbers; the rules of working with negative numbers are opposite to that of working with positive numbers<br>
+
===Objectives===
# Together, the negative numbers and positive numbers form one contiuous number line<br>
+
#Numbers can be represented on a continuum called a number line
# Perform manipulations with negative numbers and express symbolically situations involving negative numbers<br>
+
#Number line is a representation; geometric model of all numbers
 +
#Mathematical operations can b explained by moving along the number line
  
 
===Notes for teachers===
 
===Notes for teachers===
Negative numbers are to be introduced as a type of number; they do the opposite of what positive numbers do.<br>
+
Introduce the number line as a concept by itself as well as a method to count, measure and perform arithmetic operations by moving along the number line through different activities.
Read the activity for more detailed description.
 
  
 
===Activities===
 
===Activities===
#Activity 1 -[[What are negative numbers|Introduction_to_negative_numbers]]
+
#Activity 1- Add,Sub,Product,Sum,Hopping - [[Hopping_on_number_line|To introduce Number line]]
 +
#Activity 2 - [[Operations_on_number_lines_1|Sum of numbers]]
 +
#Activity 3 - [[Building_the_number_line|Classroom number line]]
 +
 
 +
==Concept #4 Number Bases==
 +
===Learning objectives===
 +
===Notes for teachers===
 +
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
  
 +
===Activities===
 +
#Activity 1 - [http://karnatakaeducation.org.in/KOER/en/index.php/Number_bases_activity_1 Activity-1]
 +
#Activity 2 - [http://karnatakaeducation.org.in/KOER/en/index.php/Number_bases_activity_2 Activity-2]
  
 +
==Concept #5 Place Value==
  
 +
=== Learning objectives ===
  
==Concept #3 Number Bases==
 
===Learning objectives===
 
 
===Notes for teachers===
 
===Notes for teachers===
 
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
 
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
  
 
===Activities===
 
===Activities===
#Activity 1 - [http://karnatakaeducation.org.in/KOER/en/index.php/Number_bases_activity_1 Activity-1]
+
#Activity 1 [http://karnatakaeducation.org.in/KOER/en/index.php/Place_value_activity_1 Activity-1]
#Activity 2 - [http://karnatakaeducation.org.in/KOER/en/index.php/Number_bases_activity_2 Activity-2]
+
#Activity 2 [http://karnatakaeducation.org.in/KOER/en/index.php/Place_value_activity_2 Activity-2]
 +
<br>
  
==Concept #4 Place Value==
+
==Concept #6 Negative numbers are the opposite of positive numbers - ==
'''I Making hundreds, tens and ones (1 period)'''
 
  
[[File:1..png|500px]]
+
=== Objectives ===
*Print 32 squares of this.
+
# To extend the understanding and skill of representing symbolically numbers and manipulating them.<br>
*Distribute into 8 groups of 4 children each.
+
# To understand that negative numbers are numbers that are created to explain situations in such a way that mathematical operations hold<br>
*Each group will get 4 squares.
+
# To recognize that negative numbers are opposite of positive numbers; the rules of working with negative numbers are opposite to that of working with positive numbers<br>
*The value of 4 squares will be 4x100 = 400
+
# Together, the negative numbers and positive numbers form one continuous number line<br>
*Each group must cut the solid lines; 1 square will have 10 strips.  These are tens.  So each square has 10 “tens”  (They can either *cut, or work without cutting – up to the children)
+
# Perform manipulations with negative numbers and express symbolically situations involving negative numbers<br>
*Each one of those tens will have 10 ones.
 
*Let the children make numbers and write them down
 
*Ask them what is the largest number each group can make?  400 is the answer – but check if children understand this.
 
  
'''II. Abstraction from here (1 period)'''<br>
+
===Notes for teachers===
Now let us assume children have 9 such squares.<br>
+
Negative numbers are to be introduced as a type of number; they do the opposite of what positive numbers do.<br>
In each group, how many hundreds are possible ? – 9 <br>
+
Read the activity for more detailed description.
In each group, how many tens are there ? – 9 x 10 = 90<br>
 
In each group, how many tens are there ? – 90 x 10 = 900<br>
 
  
'''1-9 ones are possible; 10 ones means one ten.  Ten ones is the same as one ten'''<br>
+
===Activities===
'''1-9 tens are possible; 10 tens means one hundred. Ten tens is the same as one hundred'''.<br>
+
#Activity 1 -[[What are negative numbers|What are negative numbers]]
'''What happens when we have 10 hundreds? What is it the same as?'''<br>
 
  
'''What is the importance of ten?  We count in groups of tens'''<br>
+
== Concept #7 : Types of Numbers ==
  
9+1= 10 = 10 x 1 <br>
+
=== Learning objectives ===
99 + 1 = 100 = 10 x 10 <br>
+
[[Types of numbers|Types of Numbers]]
999 + 1 = 1000 = 10 x 100 <br>
 
'''Greatest 1-digit number + 1 = Smallest 2-digit number''' <br>
 
'''Greatest 2-digit number + 1 = Smallest 3-digit number''' <br>
 
'''Greatest 3-digit number + 1 = Smallest 4-digit number''' <br>
 
'''Following the pattern, we can expect that, on adding 1 to the greatest 4-digit number''' <br>
 
'''(9999 – nine thousand nine hundred and ninety nine) we get the smallest 5-digit number''' <br>
 
'''(9999 + 1 = 10,000 or ten thousand). Further we can expect that 10 x 1000 = 10,000 i.e.''' <br>
 
'''9999 + 1 = 10,000 = 10 x 1000.'''  <br>
 
'''Do this only when children are confident – this is for advanced students'''<br>
 
  
'''Write on the board like this:'''<br>
+
=== Notes for teachers ===
[[File:2.png|500px]]<br>
 
  
 +
=== Activities ===
  
'''III.  Fill number line (1 period)'''<br>
+
= Assessment activities=
 +
'''I Fill number line (1 period)'''<br>
 
Draw these one below the other<br>
 
Draw these one below the other<br>
 
1,2,.......<br>
 
1,2,.......<br>
 
10,20,.......<br>
 
10,20,.......<br>
 
100, 200,.....<br>
 
100, 200,.....<br>
                                                                                                                                     
+
 
'''IV Tell stories and Play With Number Systems (1 period - optional)'''<br>
+
'''II Tell stories and Play With Number Systems (1 period - optional)'''<br>
 
http://www.math.wichita.edu/history/topics/num-sys.html#sense<br>
 
http://www.math.wichita.edu/history/topics/num-sys.html#sense<br>
 
(This page is downloaded and given as reading materials – page is called Number Systems)<br>
 
(This page is downloaded and given as reading materials – page is called Number Systems)<br>
  
'''V Questions/ activities for class'''<br>
+
'''III Questions/ activities for class'''<br>
 
#Arrange in order – shortest, tallest, increasing and decreasing order<br>
 
#Arrange in order – shortest, tallest, increasing and decreasing order<br>
[[File:3..png|400px]]<br>
+
[[File:Tallest.png|400px]]
#Making numbers<br>
 
Suppose we have 4 digits 7, 8, 3, 5. We want to make different 4-digit numbers using these digits such that no digits are repeated in each number.  For example, you can make 7835 and 3578 but 7738 is not allowed because 7 is repeated and 5 is not used. <br>
 
a. What is the greatest number you can make? <br>
 
b. What is the smallest number? <br>
 
c. Can you write down how you make the greatest number and the smallest number? <br>
 
Use the given digits without repetition and make the greatest and smallest 4-digit numbers. Note that 0753 is a 3 digit number and is therefore not allowed. <br>
 
[[File:5.png|600px]]<br>
 
#Now make the greatest and smallest 4-digit numbers by using any one digit twice. <br>
 
Hint: Think about which digit you will use twice.<br>
 
[[File:6.png|600px]]<br>
 
#Make the greatest/ smallest 4-digit numbers using any 4 different digits with conditions as given <br>
 
[[File:7.png|600px]]<br>
 
 
 
===Learning objectives===
 
===Notes for teachers===
 
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
 
 
 
===Activities===
 
#Activity 1 [http://karnatakaeducation.org.in/KOER/en/index.php/Place_value_activity_1 Activity-1]
 
#Activity 2 [http://karnatakaeducation.org.in/KOER/en/index.php/Place_value_activity_2 Activity-2]
 
 
 
= Assessment activities=
 
  
 
= Hints for difficult problems =
 
= Hints for difficult problems =
Line 173: Line 167:
  
 
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
 
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
 +
 +
[[Category:Number system]]

Latest revision as of 04:35, 23 July 2021

ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ

Concept Map

[maximize]


Introduction


Our daily life is based on numbers. We use it for shopping, reckoning the time, counting distances and so on. Simple calculations seem effortless and trivial for most of our necessities.So we should know about numbers. Numbers help us count concrete objects. They help us to say which collection of objects . In this we are learning about basic operations of numbers - different types of numbers, representation, etc.

Number system -Resource material html m3cabb6c3.png

How can math be so universal? First, human beings didn't invent math concepts; we discovered them.
Also, the language of math is numbers, not English or German or Russian.
If we are well versed in this language of numbers, it can help us make important decisions and perform everyday tasks.
Math can help us to shop wisely, understand population growth, or even bet on the horse with the best chance of winning the race.
Mathematics expresses itself everywhere, in almost every face of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing our understanding of all that we observe.Mathematics has been around since the beginnings of time and it most probably began with counting. Many, if not all puzzles and games require mathematical logic and deduction. This section uses the fun and excitement of various popular games and puzzles, and the exhilaration of solving them, to attract and engage the students to realise the mathematics in fun and games.

Descriptive Statement
Number sense is defined as an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent, coming from investigating their characteristics and using them in diverse situations, and how best they can be used to describe a particular situation. Number sense is an attribute of all successful users of mathematics. Our students often do not connect what is happening in their mathematics classrooms with their daily lives. It is essential that the mathematics curriculum build on the sense of number that students bring with them to school. Problems and numbers which arise in the context of the students world are more meaningful to students than traditional textbook exercises and help them develop their sense of how numbers and operations are used. Frequent use of estimation and mental computation are also important ingredients in the development of number sense, as are regular opportunities for student communication. Discussion of their own invented strategies for problem solutions helps students strengthen their intuitive understanding of numbers and the relationships between numbers.

In summary, the commitment to develop number sense requires a dramatic shift in the way students learn mathematics.

Flow Chart

Number system -Resource material html m65980570.jpg

Textbook

To add textbook links, please follow these instructions to: (Click to create the subpage)

Additional Information

Useful websites

Watch the following video on the story of how numbers evolved. The video called Story of One tells how numbers evolved and the initial questions around number theory.

This video is related to number system, helps to know the basic information about number system

This video is related to irrational numbers by Suchitha

This video is relating to exploring number patterns in square numbers.


Reference Books

Teaching Outlines

Concept #1 - History of Numbers: Level 0

The following website takes us on a fascinating journey originating from Prehistoric Mathematics, its evolution in various civilizations such as Egyptian, Greek, Indian, Chinese etc. to the increased complexities and abstractions of the modern era mathematics. This story of history of numbers also includes descriptions related to contributions of some of the important men and women to the development of mathematics.

http://storyofmathematics.com/story.html

Learning objectives

  1. What is the story of numbers?
  2. How did counting begin and learning distinguish between the quantity 2 and the number 2.
  3. The number "2" is an abstraction of the quantity

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

  1. Series of Activities in one page- Click Here
  2. Activity 1

Concept #2 Number Sense and Counting : Level 0

Objectives

1. Understand that there is an aspect of quantity that we can develop with disparate objects
2. Comparison and mapping of quantities (more or less or equal)
3. Representation of quantity by numbers and learning the abstraction that “2 represents quantity 2 of a given thing”
4. Numbers also have an ordinal value – that of ordering and that is different from the representation aspect of numbers
5. Expression of quantities and manipulation of quantities (operations) symbolically
6. Recognizing the quantity represented by numerals and discovering how one number is related to another number
7. This number representation is continuous.

Notes for teachers

This is not one period – but a lesson topic. There could be a few more lessons in this section. For example, for representing collections and making a distinction between 1 apple and a dozen apples. This idea could be explained later to develop fractions. Another activity that can also be used to talk of units of measure. Addition and subtraction have been discussed here – extend this to include multiplication and division).

Activities

  1. Activity 1 - Quantity and Numbers
  2. Activity 2 - The Eighth Donkey Story
  3. Activity 3 - Cardinal and Ordinal Numbers

Concept #3 The Number Line :Level 1-2

The number line is not just a school object. It is as much a mathematical idea as functions. The number line is a geometric “model” of all numbers -- including 0 1, 2, 25, 374 trillion, and -5, Unlike counters, which model only counters, the number line models measurement, which is why it must start with zero. (When we count, the first object we touch is called "one." When we measur using a ruler, we line one end of the object we’re measuring against the zero mark on the ruler.


Number system -Resource material html 1de463c.jpg

Part of the power of addition and subtraction is that these operations work with both counting and measuring. Therefore, to understand basic operations like addition and subtraction, we need a number line model as well as counters.

Objectives

  1. Numbers can be represented on a continuum called a number line
  2. Number line is a representation; geometric model of all numbers
  3. Mathematical operations can b explained by moving along the number line

Notes for teachers

Introduce the number line as a concept by itself as well as a method to count, measure and perform arithmetic operations by moving along the number line through different activities.

Activities

  1. Activity 1- Add,Sub,Product,Sum,Hopping - To introduce Number line
  2. Activity 2 - Sum of numbers
  3. Activity 3 - Classroom number line

Concept #4 Number Bases

Learning objectives

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

  1. Activity 1 - Activity-1
  2. Activity 2 - Activity-2

Concept #5 Place Value

Learning objectives

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

  1. Activity 1 Activity-1
  2. Activity 2 Activity-2


Concept #6 Negative numbers are the opposite of positive numbers -

Objectives

  1. To extend the understanding and skill of representing symbolically numbers and manipulating them.
  2. To understand that negative numbers are numbers that are created to explain situations in such a way that mathematical operations hold
  3. To recognize that negative numbers are opposite of positive numbers; the rules of working with negative numbers are opposite to that of working with positive numbers
  4. Together, the negative numbers and positive numbers form one continuous number line
  5. Perform manipulations with negative numbers and express symbolically situations involving negative numbers

Notes for teachers

Negative numbers are to be introduced as a type of number; they do the opposite of what positive numbers do.
Read the activity for more detailed description.

Activities

  1. Activity 1 -What are negative numbers

Concept #7 : Types of Numbers

Learning objectives

Types of Numbers

Notes for teachers

Activities

Assessment activities

I Fill number line (1 period)
Draw these one below the other
1,2,.......
10,20,.......
100, 200,.....

II Tell stories and Play With Number Systems (1 period - optional)
http://www.math.wichita.edu/history/topics/num-sys.html#sense
(This page is downloaded and given as reading materials – page is called Number Systems)

III Questions/ activities for class

  1. Arrange in order – shortest, tallest, increasing and decreasing order

Tallest.png

Hints for difficult problems

Project Ideas

Math Fun

Usage

Create a new page and type {{subst:Math-Content}} to use this template