Factorization (Learning by Doing )

Puzzle

Try Yourself

How many squares, of any size, can you find on this chess board that do not contain a Rook?

 

Answer: There are 116 squares without a rook.

1 x 1 : 61 squares.

2 x 2 : 37 squares.
3 x 3 : 15 squares.
4 x 4 :   3 squares.

 

Multiplication of 2 two-digit numbers

Multiplication of 2 two-digit numbers where the first digit of both the numbers are same and the last digit of the two numbers sum to 10

Example

 To calculate 56 x 54

Multiply 5 by 5+1. So, 5 x 6 = 30. Write down 30.

 Multiply together the last digits: 6 x 4 = 24. Write down 24.

 The product of 56 and 54 is thus 3024.

 

Multiplication of two numbers that differ by 2

 (This trick only works if you have memorized or can quickly calculate the squares of numbers.

When two numbers differ by 2, their product is always the square of the number in between these numbers minus 1.

Example

 18 x 20 =( 19 x 19) - 1 = 361 - 1 = 360

 25 x 27 = (26 x 26) - 1 = 676 - 1 = 675

 49 x 51 = (50 x 50 ) - 1 = 2500 - 1 = 2499

 

 

Multiplication of two numbers that differ by 4

(This trick only works if you have memorized or can quickly calculate the squares of numbers.

 If two numbers differ by 4, then their product is the square of the number in the middle (the average of the two numbers) minus 4.

 Example

 22 x 26 = (24 x 24 ) - 4 = 572

 98 x 102 = ( 100 x 100) - 4 = 9996

 148 x 152 = (150 x 150) - 4 = 22496

 

Multiplication of two numbers that differ by 6

(This trick only works if you have memorized or can quickly calculate the squares of numbers.

 If the two numbers differ by 6 then their product is the square of their average minus 9.

 Example

 10 x 16 = (13 x 13) - 9 = 160

 22 x 28 = (25 x 25) - 9 = 616

Activity : Sum of all angles of quadrilateral is 360

·         Draw a Quadrilateral ABCD on colour chart Sheet .
·         Cut such four Quadrilaterals on four different sheets.
·         Mark ÐA as Ð1 ,  ÐB as Ð2 , ÐC as Ð3 and ÐD as Ð4 on each quadrilateral as shown in fig .
·         Arrange all four angles of quadrilateral one from each colour at one point.
·         What you observe ?
·         It forms a complete angle i.e 360⁰
·         This shows that sum of all angles of quadrilateral is 360⁰

 

 

Squaring by shortcut

Squaring

 

Let us take advantage of  algebraic identity.

A² = (A - d)(A + d) + d²

Naturally, this formula works for any value of d, but we should choose d to be the distance to a number close to A that is easy to multiply.

Examples. To square the number 23, we let d = 3 to get

23² = 20 x 26 + 3² = 520 + 9 = 529:

 

To square 48, let d = 2 to get

48² = 50 x 46 + 2² = 2300 + 4 = 2304:

 With just a little practice, it's possible to square any two-digit number in a matter of seconds. Once you have mastered those, we can quickly square three-digit numbers by rounding up and down to the nearest hundred.

Examples. 223² = 200 x 246 + 23²= 49;200 + 529 = 49;729

952² = 1000 x 904 + 48² = 904;000 + 2;304 = 906;304:

To do mental calculations of this size, one needs to be quick at multiplying 2-digit and 3-digit numbers by 1-digit numbers, generating the answer from left to right.

 

Tangram

The tangram  is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. The objective of the puzzle is to form a specific shape (given only an outline or silhouette) using all seven pieces, which may not overlap. It was originally invented in China at some unknown point in history, and then carried over to Europe by trading ships in the early 19th century. It became very popular in Europe for a time then, and then again during World War I. It is one of the most popular dissection puzzles in the world.[

Over 6500 different tangram problems have been compiled from 19th century texts alone, and the current number is ever-growing. The number is finite, however. Fu Traing Wang and Chuan-Chin Hsiung proved in 1942 that there are only thirteen convex tangram configurations (configurations such that a line segment drawn between any two points on the configuration's edge always pass through the configuration's interior, i.e., configurations with no recesses in the outline).

 

 

 

You can get more picture from google.

 

Activity to find the area of parallelogram

Multiplication of two numbers that differ by 2

Multiplication of two numbers that differ by 2

Note :This trick only works if you know the squares of numbers.

 

When two numbers differ by 2, their product is always the square of the number in between these numbers minus 1.

12 x 14 =   (13 x 13)  - 1 =  169 - 1 =  168
 

18 x 20 =   (19 x 19)  - 1 =  361 - 1 =  360

25 x 27 =   (26 x26)   - 1 =  676 - 1 =  675  
13 x 15 =   (14 x 14)  - 1 =  196 - 1 =  195

15 x 17 =   (16 x 16)  - 1 =  256 - 1 =  255

16 x 18 =   (17 x 17)  - 1 =  289 - 1 =  288

     

 

Angle sum property of Triangle

 

Ekādhikena Pūrvena,

Ekādhikena Pūrvena, ("By one more than the previous one") This  sūtra means that the prescribed arithmetical operation is either multiplication or division. Both are implied since we may proceed leftward and multiply (and carry-over the excess value to the next leftward column) or we may go rightward and divide (while prefixing the remainder). When dividing by a nine's family denominator, the digit to the left of the nine (previous) is increased by one to obtain the multiplier.

.Examples:

35×35 = ((3×3)+3),25 = 12,25 and 125×125 = ((12×12)+12),25 = 156,25

 

or by the sūtra, multiply "by one more than the previous one."

35×35 = ((3×4),25 = 12,25 and 125×125 = ((12×13),25 = 156,25

 

The latter portion is multiplied by itself (5 by 5) and the previous portion is square of first digit or first two digit (3×3) or (12×12) and adding the same digit in that figure (3or12) resulting in the answer 1225.
(Proof) This is a simple application of


  when and , i.e.

It can also be applied in multiplications when the last digit is not 5 but the sum of the last digits is the base (10) and the previous parts are the same. Examples:

37 × 33 = (3 × 4),7 × 3 = 12,21

29 × 21 = (2 × 3),9 × 1 = 6,09 ?

 

This uses twice combined with the previous result to produce:

.

We illustrate this sūtra by its application to conversion of fractions into their equivalent decimal form. Consider fraction 1/19. Using this formula, this can be converted into a decimal form in a single step. This can be done by applying the formula for either a multiplication or division operation, thus yielding two methods.