Category : 5th Class
LEADING OBJECTIVES
This lesson will help you to:
Amazing Facts
QUICK CONCEPT WIEW
PLACE VALUE
B The value of the place, or position, of a digit in a number or series is called place value. Each place has a value of 10 times the place to its right.
The idea of place value is at the heart of our number
The concept of place value is as follows:
Beginning with the ones place at the right, each place value is multiplied by increasing powers of 10. For example, the value of the first place on the right is "one" the value of the place to the left of it is "ten," which is 10 times 1. The place to the left of the tens place is hundreds, which is 10 times 10, and so forth.
The place value of number goes beyond 1000 with the next place value being 10 times greater. The place values after thousand are ten thousands (10,000), hundred thousand (1,00,000), millions (10,00,000) and so on.
For easier readability, commas are used to separate each group of three digits, which is called a period. When a number is written in this form, it is said to be in "standard form" Example: four hundred sixteen thousand, seven hundred thirtyone can be written as 416,731.
The role of place value in addition, subtraction and multiplication algorithms.
Misconcept/concept
Misconcept: Numeral 1 is a prime number
Concept: 1 is neither prime nor composite. It does not have any factors except the number itself.
Misconcept: All prime numbers are odd numbers.
Concept: This is not true, since the number 2 has only 2 factors, 1 and 2, and is also an even number.
Example: Add 65,000 and 1500
\[\frac{\begin{align}
& 65,000 \\
& +1,500 \\
\end{align}}{66,500}\]
Example: Subtract 7,400 from 74,000.
\[\frac{\begin{align}
& 74,000 \\
& 7,400 \\
\end{align}}{66,500}\]
Role of place value in multiplication algorithms
Let’s us understand the concept of place value in multiplication, with the help of an example.
Example: Multiply the following numbers: 263 and 64
STEP 1: Multiply the multiplicand by ones digit of the multiplier.
\[\left( 263\times 64 \right)\]
\[3\times 4=12\]
(1 tens, 2 ones).
Put the 2 in one’s column and carry over 1 to the tens column.
Historical preview
1 
2 
3 
4 
5 
6 
7 
8 
9 
 
= 

+ 





Brahmi numerals around 1^{st} century A.D.
\[\left( 263\times 64 \right)\]
\[6\times 4=24+1\]
(carry over from step 1)
= 25 (2 hundreds^ tens).
Put the 5 in tens column and carry over 2 to the hundreds column.
\[\left( 263\times 64 \right)\]
\[2\times 4=8+2\]
(carry over from step 2)
= 10 (1 thousands, 0 hundreds).
Put the 0 in hundreds column and 1 (carry over) in thousands column
STEP 2: Multiply the multiplicand by tens digit of the multiplier
\[\left( 263\times 64 \right)\]
\[3\times 6=18\]
(1 hundreds, 8 tens).
Put the 8 in tens column and carry over 1 to the hundreds column.
\[\left( 263\times 64 \right)\]
\[6\times 6=36+1\]
(carry over from step 1)
= 37 (3 thousands, hundreds).
Put the 7 in hundreds column and carry over 3 to the thousands column.
\[\left( 263\times 64 \right)\]
\[2\times 6=12+3\]
(carry over from step 2)
= 15 (1 ten thousands^ thousands).
Put the 5 in thousands column and 1 (carry over) in ten thousands column.
STEP 3: Add result of Multiplier 1's and Multiplier 10's results and put the result in the Product columns.
1052 + 15780 = 16832
Product of 263 and 64 is 16832
To study about informal and standard division algorithm
Standard Division Algorithms
Unlike addition, subtraction and multiplication, division is performed from left to right.
(Highest place value to lowest place value).
FACTOR AND MULTIPLES
A factor is simply a number that is multiplied to get a product. Factoring a number means taking the number apart to find its factors — it's like multiplying in reverse.
Here are lists of all the factors of 16, 20, and 45.
16 \[\to \],2,4,8,16
20 \[\to \] 1, 2,4, 5,10, 20
45 \[\to \] 1,3,5,9,15,45
Factors are either composite numbers or prime numbers.
Prime .number : A prime number has only two factors, one and itself, so it cannot be divided evenly by any other numbers. Here's a list of prime numbers up to 100.
Prime numbers to 100
2, 3, 5, 7,11,13,17,19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
These numbers cannot be factored any further.
Composite numbers: A composite number is any number that has more than two factors. These numbers can all be factored further. For example, 4 equals 2 times 2, 6 equal 3 times 2, 8 equals 4 times 2, and so forth.
Here's a list of composite numbers up to 20.
Composite numbers up to 20
Composite numbers can be written as a product of prime factors. This is called prime factorization. To find the prime factors of a number, you divide the number by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number.
Example: Find the prime factors of 168.
Since 168 is even, we start by dividing it by the smallest prime number, 2. 168 divided by 2 is 84. 84 divided by 2 is 42. 42 divided by 2 is 21. Since 21 is not divisible by 2, we try dividing by 3, the next biggest prime number. 21 divided by 3 equals 7, and 7 is a prime number. Therefore, 168 is now fully factored.
\[168\div 2=84\]
\[84\div 2=42\]
\[42\div 2=21\]
\[21\div 3=7\]
Prime number prime factors =
\[2\times 2\times 2\times 3\times 7\]
To check the answer, multiply these factors and make sure they equal 168.
Greatest Common Factor: The greatest common factor, or GCF, is the greatest factor that divides two numbers. To find the GCF of two numbers:
Example:
Find the GCF of 18 and 24.
Prime factors of
\[18=2\times 3\times 3\]
Prime factors of
\[24=2\times 2\times 2\times 3\]
There is one 2 and one 3 in common. Therefore, the 6CF =
\[2\times 2=6.\]
DIVISIBILITY RULES
The simple divisibility rules will help you to find factors of a number.
The number is divisible by:
(example: 12346);
(example: 1236, because
\[1+2+3+6=12=3\times 4\]
);
\[44=4\times 11\]
(example: 178965 or 40980);
(example: 124987080, because
\[080=8\times 10;\]
(example: 234612, because
\[2+3+4+6+1+2=18=9\times 2);\]
(example: 99990);
(example: 987600);
MULTIPLES
How to find multiples of a given number?
Multiply the given number by 1, 2, 3, etc. The products are multiple of the given number.
Example: Multiple of 3 are 3
\[\left( =3\times 1 \right),6\left( =3\times 2 \right),9\left( =3\times 3 \right),12\left( =3\times 4 \right),\]
etc.
LOWEST COMMON MULTIPLE (LCM)
Some important points about multiples:
How to find LCM of the given numbers.
Least two numbers once.
The LCM of the given numbers.
Example: Find LCM of 10, 15 and 18.
Prime factors of 10: 2, 5
Prime factors of 15: 3, 5
Prime factors of 18: 2, 3, 3
Factors which are common to at least two numbers: 2, 3, 5
Factors which are not common to at least two numbers: 3
LCM =
\[2\times 3\times 5\times 3=90\]
VARIABLES: A variable is a symbol for a number we don't know yet. It is usually an alphabet.
Example: in x + 2 = 6, x is the variable.
If it is not a variable it is called a constant.
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