Number System

UNIT 1

NUMBER SYSTEM

A recapitulation of Hindu – Arabic system of numeration should involve the Comparison of numbers, stating the place of a digit in a number, writing numbers in place – value charts, and in expanded form. The student should be able of differentiate between natural numbers and whole numbers and represent whole numbers on the number-line. The concept of integers, introduced in the primary level, is reinforced with special emphasis on abstract concept of negative integers, comparing it with real-life situations, especially while using them to describe distances in specific directions.

Therecapitulation of factors and multiples needs to cover the subsets of natural numbers viz. Even and odd numbers, prime, composite and co-prime numbers. Finding H.C.F and L.C.M of sets of numbers need to relate more to real-life situations, so that students can judge for themselves whether to find L.C.M or H.C.F in given situation.

“Tell me, and I’ll forget

Show me, and I may remember

Involve me, and I’ll understand”

Confucius

The fascinating world of Mathematics can inculcate in the learner the ability to think, reason, explore and conjecture. In addition to grasping the basic concepts, we believe that a comprehensive study of mathematics must enable the learner to appreciate the applicability of mathematics in real – life situations.

LESSON PLAN FORMAT

Period No: 1 of Lesson No: 1/Unit

Date&Time:Class:VISubject: MathematicsMedium: English

Sita how old are you?

Can you tell me how many days are there in a year?

When do you all wake-up?

If cost of 1pencil is `2. What is the cost of 2 pencils?

In this way we can say that numbers form a part of our daily life.

Consider the given number

If 3 is placed before the number what do you get?

/ I am 12 years
There are 365 days in a year
We wake-up at 5AM every day.
2x2 =`4 or 2+2=`4
We have 2 tens and 5 ones / Sita is 12 years old.
There are 365 days in a year
They wake-up at 5AM
Cost of 1 pencil = `2
Cost of 2 pencils = 2x2 =`4 or 2+2 = `4
NUMBER SYSTEM
Let the given number be 25, 325
325 / ANALYTIC - SYNTHETIC METHOD / How many tens and ones do we have in 25?
How many hundreds do we have 325?
NDIAN SYSTEM OF NUMERATION
It uses 10 digits from 0 to 9 describe all numbers
PLACE VALUE OF A NUMBER
The value of a digit owing to its place in a numeral is known as its place value
EXPANDED FORM /

Now read the number325
If you add 4 before it how do you read
How are you able to read them?

This system is Hindu-Arabic system of numeration in which we have “lakhs”place
Observe the given number
In 36 what is the place value of ‘3’
In 32, 40, 129 what is the place value of ‘3’
Now that we know the place values we can write the numbers in the expanded form.
Write the place value of all the digits in the given numbers. / It is 3 hundred and twenty five
It is four thousand three hundred and twenty five
From their places
‘3’ is in Ten’s place
Its place value id 30
Its place value is 30,00,000
‘8’ is in one’s place.
Place value of 8:
(8X1)=8
In the same ways pupils give the place values of other digits in the given number. / 4325
Indian place value chart
T.C / C / T.L / L / T.Th / Th / H / T / O
Place value in a number
what is the place value of 3 in given numbers
Numbers / Place value of 3
36 / 30 or Thirty
32,40,129 / 30,00,0000 or Thirty lakhs
Problem: Write the expanded form of 2,63,118
Place value of 8: (8X1)=8
Place value of 1: (1x10)=10
Place value of 1: (1x100)=100
Place value of 3: (3x1000)=3000
Place value of 6: (6x10000)=60000
Place value of 2: (2x1,00,000)=2,00,000
Expanded form of 2,63,118 =2,00,000+60000+3000+100+10+8 / AID : PLACE VALUE CHART / 1.Write in figures Twenty two lakhs and twenty two
2.Write 93,42,318 in words
3. Where do you put commas when you write a number in Indian system?
4. Find the difference in place values of ‘7’ in 1,70,071
Write 13,747 in expanded form
Remember!
To move up to larger place values in a place value chart
Successive place values
are multiplied by 10.
To move down to smaller value, successive place values are divided by 10.

At the end of the lesson: Behavioural changes expected and achieved

a)To make them understand concept of place-value

b)He develops ability to perform calculations conversions.

c)He is able to solve both oral and written mathematical problems independently.

d)The pupils are able to make use of place value chart in solving elementary mathematical problems.

Assignments

Complete the following conversions:-

1 hundred =______Tens =______ones

1Thousand =______Hundreds =______ones

1 Lakh = ______Thousands = ______Hundreds = ______ones

Write place value of 1 in the given numbers:

81, 32,604 ii. 63,91,754

Write in expanded form:

6, 86,794 ii. 38,47,612

LESSON PLAN FORMAT

Period No:2 of Lesson No:1/Unit1

Date&Time:Class:VISubject: MathematicsMedium: English

Content Analysis/Concepts/Sub-Concepts / Activities including behavioural/Learning outcome / Aid/Equipments to be done/Demonstrations etc., including writing Board work / Method / Evaluation
By Teacher / By Pupils
MOTIVATION
SUCCESSOR AND PREDECESSOR
The Successor of a given number is ‘1’ more than the given number
The Predecessor of a given number is 1 less than the given number
COMPARING NUMBERS

The number that comes later in number line is greater
The

number
with more
digits is greater number

The number with greater digit in the same place is greater number

ASCENDING ORDER
Writing the given numerals from small to big is called Ascending order
DESCENDING ORDER
Writing the given numerals from big to small is called descending order
INTERNATIONAL SYSTEMS
Indian system to read & write a large number the digits are broken into
periods separated by commas(,)
1st Period – Ones, Tens, Hundreds.
The other Periods have two place
Values each.
But in International System of Numeration, all the periods have three place values each. /

Give a largest two digit numbers?

If ‘1’is added what do you get?

Successor of a number

What is 99-1=?

This is called Predecessor

What should be done to find successor?
So, find the successor of given number?

Similarly find predecessor of given number
What is the smallest
number in the given numbers?
What is the next smallest number in the remaining numbers Complete the process for all the numbers.
In this way writing the given numbers from smallest to biggest is called Ascending order.
Similarly if we write
from bigger number to smaller number we get descending order
SOL: So, in this way we can write the given numbers in Ascending order
For writing a 6 digited number we use lakhs in our country. But in other countries we write in different way. This system is International System.
Divide the periods according to International system.
Read the number in International system. / ‘99’
‘100’
99-1=98

‘1’should be added to the given number to find successor
Successor of 37,99,13,799+1=37,99,13

We get predecessor by subtracting ‘1’from the given number.
‘24’
‘98’
We have 3,6 digited number, one – 7 digited number & one – 8 digited number
First write the three 6 digited numbers then 7 digited numbers & then 8 digited numbers to write the given numbers in Ascending order
We have three place values in each period.
They read aloud / 99’is the largest 2 digit number
If ‘1’ is added =99+1=100
If ‘1’ is subtracted 99-1=98
Problem: What is the succeeding number of 37, 99, and 13,799? & Preceding number of 1, 00, 00,006?
800 successor of 37,99,13,799 is 37,99.13,800
1,00,00,006-1=100,00,005
The Predecessor of 1,00,00,006 is 1,00,00,005
Consider the numbers:
(1) (2) (3) (4) (5)
24, 346, 98, 467, 3684
(small to big)
Ascending order
24,98, 346, 467, 3684
Descending Order
(big to small)
3684,467,346,98,24
Problem: Arrange the numbers in ascending and descending order
(6) (6) (7)
5,45,352; 5,45,253; 25,45,253;
(6) (8)
3,53,451; 5,43,73,181
Consider the number of digits in each number.
We have 3,6 digited numbers
In these 3,53,451 is the smallest
In the next two numbers 5,45,352 and 5,45,253
The smallest is 5, 45,253.
The next smallest is 5,45,352
Next we have a 7 digited number 25,45,253 & then a 8 digited number 5,43,73,181
Ascending order:(Small to Big)
3,53,451; 5,45,253; 5,45,352; 25,45,253; 5,43,73,181
Descending order:(Big to Small)
5,43,73,181; 25,45,253; 5,45,352; 5,45,253; 3,53,451
Write 2,34,137 in words
Two lakhs, Thirty four Thousand, one hundred & Thirty Seven
INTERNATIONL PLACE VALUE CHART
Millions / Thousands / Ones
H.M / T.M / M / H.Th / T.Th / Th / H / T / o
Now we can write the same number in International system as follows:-
234,137
Two Hundred and Thirty four Thousands, One Hundred and Thirty Seven / A
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D / write the Predecessor and Successor of the following number

1,000
99,999
8,922

How many digited numbers you have in the given series?
How to arrange them in Ascending order?
How are the Periods taken in International System?
Write 2,076,211 in words in International System.

At the end of the Lesson:

Behavioural changes expected and achieved.

a)They are able to write successor, predecessor of a given number

b)They are able to compare numbers and write the given numerals in Ascending and Descending order.

c)They acquire the knowledge of International System.

ASSIGNMENTS:

Write the Successors & Predecessors

Predecessor(-1) / Number / Successor(+1)
38,26,940
3,00,00,000

Observe the periods and then write relevant number system and number – word

Numbers / System / Number - Word
4,278,445,106
1,58,63,572

Write 91,89,697; 8,17,31,867; 23,61,731; 1,89,32,714; 1,07,187 in Ascending and Descending order

LESSON PLAN FORMAT

Period No: 3of Lesson No:2/Unit1

Date&Time:Class:VISubject: MathematicsMedium: English

Content Analysis/Concepts/Sub-Concepts / Activities including behavioural/Learning outcome / Aid/Equipments to be done/Demonstrations etc., including writing Board work / Method / Evaluation
By Teacher / By Pupils
FACTORS:
The numbers that are multiplied to get a product are called factors of that product
REMEMBER:

1 is a factor of every number and is the smallest factor.
Every number is a factor of itself and is the largest factor.
The factor of a number is less than or equal than

Or equal to that number.
MULTIPLES OF A NUMBER:
The product of a number and counting numbers(1,2,3,- - - ) are known as Multiples of that Number
Remember:

Every number is a multiple of itself.
There are infinite number of multiples of a natural number.
To find multiples multiply the number with all natural numbers

PRIME AND COMPOSITE NUMBERS
SIEVE OF ERATOSTHENES IS
USED TO FIND PRIME NUMBERS
Remember:
‘1’ is neither prime or composite /

Write 56 as product of two numbers.

Let us see the other ways.

Write factors of 12 & 32.

What is the smallest factor of 12?
What is the largest factor largest factor of 12?
What is the smallest & largest factor of 32?
How much is 36x1=?
36x2=?
36x3=?
36x4=?
From the table what are the numbers which are having only 2 factors(1&itself)
Such numbers are called Prime Numbers .Many factors do the other numbers have? Such numbers are called Composite Numbers.
All even numbers are multiples of 2 i.e. they have 2 as a factor
i.e. they are not prime numbers
So cross every multiple of 2
Similarly cross out multiples of 3
Similarly cross multiples of 5 and number on till every number is circled or crossed / 8x7
12 = 12 x1
= 2 x 6
= 3 x 4
1,2,3,4,6,12 are factors of 12
32 = 32 x 1
= 16x2
= 4 x 8
1,2,4,8,16,32 are factors of 32
‘1’
‘12’
‘1’ is the smallest & ‘32’ is the largest factor.
36x1= 36
They are 2,3,5&7
They have 3 or more factors.
Sol:
22,24,26,28,30,32,34,36,38,40,42,
44, 46, 48 are multiples of ‘2’.
21, 27,33,39,45 are multiples of 3 in the remaining numbers. / 56 = 8x7
product factors
56 can also be written as product of two numbers in the following ways also
56 = 4x14
= 2 x28
= 1 x 56
Here 1,2,4,7,8,14,28.56 are all factors of 56 & 56 is multiple of all these numbers
12 = 12 x 1
= 2 x 6
= 3 x 4
1,2,3,4,6,12 are factors of 12
1 is the smallest & 12 is the largest
32 = 32 x 1
= 16 x 2
= 4 x 8
1,2,4,8,16,32 are factors of 32.
Problem: Write Multiples of 36.
36x1=36
36x2=72
36x3=108
36x4=144
Here 36, 72,108,144 are first four multiples of 36.
Numbers / Factors / No. of Factors
1
2
3
4
5
6
7
8 / 1
1,2
1,3
1,2,4
1,5
1,2,3,6
1,7
1,2,4,8 / 1
2
2
3
2
4
2
4
Prime Number: A Number which has two factors 1 and itself is called a Prime Number(Number/1)
Composite Number: Numbers having more than two factors are called composite Numbers
Problem: Write all Prime Numbers between 20 and 50.
The numbers between 20 and 50 are:-

21 22 23 24 25 26 27

28 29 30 31 32 33 34

35 36 37 38 39 40 41

42 43 44 45 46 47 48

49

Multiples of 2 are crossed out similarly for 3,5 and number on
Thus Primes between 20 and 50 are 23, 29,31,37,41, 43, and 47. / A
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D / Complete the factor tree

Write 3 multiples for 2391,51,76
Give some examples of Prime & Composite Numbers
Classify as Prime & Composite and state the reason
7, 29, 57, 89, 48.

At the end of the Lesson:

Behavioural change expected and achieved.

a)He develops the ability to use mathematical tables in finding factors & multiples

b)Pupil is able to find out Prime & Composite Numbers.

c)The Student solves mathematical Problems.

d)The Student develops the habit of logical thinking & reasoning.

Assignments:

Fill up the blanks.

______is the number which is neither prime nor composite.
______is the only even prime number.
The first five multiples of 7 are ______

Encircle the prime numbers.

49 64 17 87 23 99 143 191 51 6 1

Is 221 a composite number?

LESSON PLAN FORMAT

Period No: 4of Lesson No: 2/Unit1

Date&Time:Class:VISubject: MathematicsMedium: English

Content Analysis/Concepts/Sub-Concepts / Activities including behavioural/Learning outcome / Aid/Equipments to be done/Demonstrations etc., including writing Board work / Method / Evaluation
By Teacher / By Pupils
PROBLEM:
Twin Primes: / To write 12 as sum of two prime numbers give the possible ways of getting 12 as sum
What is the sum in which we have both prime numbers?
What is the difference between them
Such primes are called Twin Primes
From the pairs of twin primes discussed write the given numbers as sum of twin primes / 1+11
2+10
3+9
4+
5+7
6+6
5+7
7-5=’2’

‘2’ / Problem: Write 12 as sum of two prime numbers
Sol: Possible ways of getting 12 as sum
1+11=12
2+10=12
3+9=12
4+8=12
5+7=12(both 5 & 7 are prime numbers)
6+6=12
Among these ways 5+7=12 is the possible way to write 12 as sum of two prime numbers.
Twin Primes: If the difference between a pair of prime numbers is ‘2’, they are called Twin Primes.
Problem: Write 8, 24, 60 and 84 as sum of twin prime numbers.
8=3+5
24=11+13
60=29+31 / ANALYTICAL - SYNTHETIMC METHOD / Write 39 as sum of three odd prime numbers?
Give some pairs of Twin Primes
What is the difference between twin Primes?
CO-PRIME OR RELATIVELY PRIME NUMBERS
Numbers having only ‘1’as their common factor / Write the factors of 12, 13
Now write down the factors of 13
What is their common factor? Such numbers are said to be relatively prime (co-prime) numbers. / The factors if 12 are1, 2, 3, 4, 6, 12
1, 13 are factors of 13
‘1’ / 84=41+43
12
12 x 1
3 x 4
2 x 6
1, 2, 3, 4, 6, 12 are factors of 12
13
13 x 1
1,13 are factors of 13
‘1’is their common factor / Write ‘2’examples relatively prime numbers

At the end of the Lesson:

Behavioral changes expected and achieved:

a)The Pupil is able to write the given Numbers as sum of twin prime numbers.

b)The Pupil solves relevant problems with confidence.

c)The Pupil tries to collect enough examples for co-prime & twin primes.

d)The Pupil takes active part in finding out prime numbers.

Assignments:

Which of the following are co-primes

6, 8 ii. 9, 10 iii. 1 3, 39

Write 60 as sum of twin prime numbers.
The difference between twin primes is ______
Co-Prime numbers have______as common factor.

LESSON PLAN FORMAT

Period No: 5of Lesson No: 3/Unit1

Date&Time:Class:VISubject: MathematicsMedium: English

Content Analysis/Concepts/Sub-Concepts / Activities including behavioural/Learning outcome / Aid/Equipments to be done/Demonstrations etc., including writing Board work / Method / Evaluation
By Teacher / By Pupils
PRIME FACTORISATION:
Prime factorisation is the expression of a number of only prime numbers
Method 1:

By finding factors

By factor tree

/ Write the factors of the given numbers.
In 35=7x5
7 & 5 are prime numbers
In 70 = 7x10
7 is a prime number and 10 is a composite number.
So ‘10’ can again be written as 5x2 which is a product of two prime numbers .Now 70 is expressed as product of prime numbers
Write the ways in which 36 can be written as product of two numbers
Again factorise the composite numbers so that finally you get all factors as Prime. / 35=7x5
70=7x10
=7x5x2
36=2x18
=3x12
=4x9
=6x6
4x9=2x2xx3x3
6x6=2x3x2x3 / Observe the following numbers:-
35, 70, 132, 196
Let us write their factors
35=7x5 or35x1
70=7x10 or7x5x2 or 70x1
132= 1x132
=2x66x1
=2x2x33x1
=2x2x3x11x1
196=2x98
=2x2x49
=2x2x7x7
Express 36 as product of prime factors / ANALYTIC - SYNTHETIC METHOD / Split 132 & 196 as product of Prime factors

By Successive

Division / Divide the numbers successively by smallest prime number possible, till the quotient is 1
980 is divisible by 2. So first divide by 2 then by 2, 5, 7 & 7 / Students give the prime numbers with which the given number is divisible /

Write 980 as product of prime factors by division
Method
2 980

2 490

5 245

7 49

7 7

1
Hence product of prime factor of 980 = 2x2x5x7x7 / Give the smallest prime numbers with which the given number is divisible.

At the end of the Lesson:

Behavioural changes expected and achieved:

While solving a problem, the Pupil analyses it, collects all the known evidences and then draws proper inferences
To develop the Pupils habit of systematic and neat- working.
To develop the concept of Prime factorization

Assignment:

Write 231, 429 as product of prime factors
Complete the factor tree

Write 20570 as product of prime numbers by Division method.
Write factor tree for 90 and 136

LESSON PLAN FORMAT

Period No: 6of Lesson No: 4/Unit1

Date&Time:Class:VISubject: MathematicsMedium: English

Content Analysis/Concepts/Sub-Concepts / Activities including behavioural/Learning outcome / Aid/Equipments to be done/Demonstrations etc., including writing Board work / Method / Evaluation
By Teacher / By Pupils
GREATEST COMMON DIVISOR OR HIGHEST COMMON FACTOR
(G.C.D) OR(H.C.F)
From the common factor of two given numbers, the largest number is said to be the Greatest Common Divisor(G.C.D) or Highest Common Factor(H.C.F) / Express 34 as product of prime factors
Similarly write 102 as product of prime factors of the two sets what are the common factors The product of common prime factors = 2x17
=34
This is G.C.D or H.C.F of given number
What are the factors of 25 similarly find factors of 17 & 21
What is their common factor? /
34= 2 x 17

102 = 2 x 3 x 17
2, 17
Factors of 25
1, 5, 25
Factors of 17 are 1, 17
Factors of 21 are 1, 3, 7, 21
Their common factor is ‘1’ / Take two numbers 34, 102
Step 1: Write prime 2 34
Factors of given numbers

34 = 2 x 17 2 102
3 51
17
102 = 2 x 3 x 17
Step 2: Write common factors common factors = 2, 17
Step 3: Multiply common factors
G.C.D or = 2 x 17
H.C.F = 34
Find G.C.D of 2, 17 and 21
25 = 5x5 = 1 x 25
17 = 17 x 1
21 = 3 x 7 = 21 x 1
Common Factor of 25, 17, 21 = 1
G.C.D = 1 / ANALYTIC - SYNTHETIC METHOD / Find G.C.D of 126, 216 by Prime factorisation Method
Write some numbers whose G.C.D is ‘1’
What is G.C.D of twin primes?
G.C.D by Division method

First divide larger number by smaller number
Divide the quotients with remainders
Continue the division till we get zero as remainder.
The Last divisor with zero as remainder is the G.C.D

FINDING G.C.D OF TWO OR MORE NUMBERS
First find G.C.D of any two numbers, then find G.C.D of the next number and G.C.D we got previously continue the process till all numbers are completed / Step 1: Divide the larger number 198 by 144 what is the remainder
Step 2: Take 54 as new divisor and old divisor as new dividend and divide
Step 3: Continue step 2 till remainder is ‘o’
To find G.C.D of the given numbers first find the G.C.D of first two numbers
What is the G.C.D of 3120, 5200
Now again find the G.C.D of 1040 and third number which will be G.C.D of the given three numbers / ‘54’
The Students continue the division as per the directions given
3120)5200(1
3120
2080)3120(1
2080
1040)2080(2
2080
0
1040
1040)3640(3
3120
520)1040(2
1040
0 / Find G.C.D of 144, 198 by Division Method.
144) 198(1
144
54)144(2
108
36)54(1
36
18)36(2
36
0
Problem: Find G.C.D of 3120, 5200 and 3640
First find G.C.D of First two numbers
3620)5200(1
3120
2080)3120(1
2080
1040)2080(2
2080
0
Now find G.C.D of 1040 and 3640.
1040)3640(3
3120
520)1040(2
1040
0
G.C.D of 3120, 5200 and 3640 = 520

At the end of the Lesson: