preparation of a unit "identities"
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1. The document shows methods for calculating the area of rectangles by splitting them into smaller rectangles. 2. It demonstrates that the area of the original rectangle equals the sum of the areas of the smaller rectangles. 3. Algebraic formulas are developed to represent splitting rectangles and multiplying sums and differences.
preparation of a unit "identities"
- 1. 8 = 4 + 3 6 = 4 + 2 4 x 4 3 x 4 4 x 2 3 x 2
- 2. Product of sums Here is a rectangle of length 12 centimeters and breadth 8 centimeters. 12 8 Rinu is trying to find out its area. Can you help Rinu to calculate its area ? What is it ? To find area, we want to multiply the rectangle’s length and breadth. So it is 12 x 8 . Now, Rinu divide the rectangle into two rectangles. He split the length of the square as shown below. 7 5 8 which is the largest rectangle ? What are their length and breadth ? What about the other one ?
- 3. Is there any change in its breadth ? The area of the first rectangle is 7 x 8. What about the second one ? It is 5 x 8. We can calculate the area of the original rectangle as (7 +5 ) x 8 Now , suppose in the original rectangle, we split the breadth. 12 6 2 Now if we calculate the area of the original rectangle, it is 12 x (6 + 2) Suppose we split the length and breadth simultaneously. 6 2 7 5 Then what is the area of the largest rectangle ?
- 4. + It is ( 7 +5 ) x ( 6 + 2 ) Consider all the small rectangles inside the rectangle. Let us compute all the area. Area of the first rectangle is ( 7 x 6 ) What about the other three rectangles ? So the area of the original rectangle is ( 7 + 5 ) x (6 + 2 ) Which is same as ( 7 x 6 ) + (7 x 2 ) + ( 5 x 6 ) + (5 x 2 ) Thus , ( 7 + 5 ) x (6 + 2 ) = ( 7 x 6 ) + ( 7 x 2 ) + (5 x 6 ) x (5 x 2 ) What did we do here ? To get 12 x 8 • 12 x 8 is split as ( 7 + 5 ) x ( 6 + 2 ) • 6 and 2 are multiplied by 7 • 6 and 2 are multiplied by 3 • All these products are added Let us see how we can get 18 x 22 from 17 x 20 (18 x 22 ) = (17 + 1 ) x (20 + 2 ) = (17 x 20 ) + (17 x 2 ) + (1 x 20 ) + (1 x 2 ) Identities The equation 2x+3 = 3x+2 is true only for x=1.what about the equation x+(x+1) = 2x+1. It is true whatever number we take as x. such equations which are true for all numbers are called identities. In both problems, we multiplied a sum by another sum. What is the general method to do this ? To multiply a sum of positive numbers by a sum of positive numbers, multiply each number in the second sum by each number in the first sum and add.
- 5. Let us do 32 x 44 using this. 32 x 44 = ( 30 +2 ) ( 40+ 4 ) = ( 30 x 40 ) + ( 30 x 4 ) + ( 2 x 40 ) + (2 x 4 ) = 1200 + 120 + 80 + 8 = 1408 What is about 1004 x 301 ? 1004 x 301 = (1000 + 4 ) (300 + 1 ) = 300000 + 1000 + 1200 + 4 = 302204 Using algebra we can write, ( x + y ) ( u + v ) = xu + xv + yu + yv ,for any four positive numbers x, y, u, v. If we consider ( y + 1 ) ( x + 1 ) , what is its product ? Can you state this as a general principle in ordinary language ? Can we do some multiplication in head using this ? What if we take 2 instead of 1 in this ?
- 6. : 1. 6½ x 8⅓ 2. Write numbers like this 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 i. Mark four numbers in a square and find the difference of diagonal products. Is it same for all squares of four numbers ? ii. Explain why this is so ? iii. Instead of a square of four numbers, take a square of nine numbers and mark only the numbers at the four corners. what is the difference of diagonal products ? Explain using algebra. 8 9 10 13 14 15 18 19 20
- 7. 3. 3. Look at this : 1 x 4 = ( 2 x 3 ) – 2 2 x 5 = ( 3 x 4 ) – 2 3 x 6 = ( 4 x 5 ) – 2 4 x 7 = ( 5 x 6 ) – 2 i. Write the next two lines in the pattern. ii. If we take four consecutive natural numbers, what is the relation between the products of the first and the last, and the product of the middle two ? iii. Write this as a general principle in algebra and explain it ? Square of a sum we have seen in class 7, that the square of a product is the product of squares. Is the square of a sum equal to the sum of squares ?
- 8. What is 71² ? To see it, let us split 71². 71² = 71 x 71 = (70 + 1 ) (70 + 1 ) We can write this as the sum of four products. ( 70 + 1 ) ( 70 + 1 ) = ( 70 x 70 ) + ( 70 x 1 ) + (1 x 70 ) + ( 1 x 1 ) = 4900 + 70 + 70 + 1 = 5041 We can split any square like this. How do we write this in algebra ? To get ( x + 1 )² from x², we must add x and the next number x + 1 to x². That is, ( x + 1 )² = ( x + 1 ) ( x + 1 ) = x.x + x.1 + 1.x + 1.1 = x² + x + ( x + 1 ) We know that x + ( x + 1 ) = 2x + 1. so, ( x + 1 )² = x² + 2x + 1. Using this calculate 51² ? Now suppose we want to compute 65². If we try to do this by writing it as ( 64 + 1 )², we could have to compute 64². Suppose we write it as ² = ( 60 + 5 )², how we get 65² ? 65² = (60 + 5) ( 60 + 5 ) = 60² + (60 x 5 ) + ( 5 x 60 ) + 5² = 3600 + 300 + 300 + 25 = 4225 How about 203² ? Let us generalize this idea as : The square of sum of two positive numbers is sum of the squares of the two numbers and twice their product.
- 9. We can write it in algebra like this : ( x + y )² = x² + y² + 2xy, for any two positive numbers x, y. In another way , Consider a square of length ( x + y ). We can split this x + y as x and y. So we obtain four rectangles. As we have seen in the beginning of this chapter, area of the square of side length ( x + y ) is equal to the sum of Areas of four rectangles. Thus, ( x + y )² = x² + xy + xy + y² That is, ( x + y )² = x² + y² + 2xy. x y x y x² xy xy y² 1. Compute the squares of these numbers. ( 10½)² , 45² , 20.4 2. Is there a general method to compute the squares of numbers like 1½, 2½, 3½, …? Explain it using algebra. 3. Given below is a method to calculate 37². 3² = 9 9 x 100 900 2 x ( 3 x 7 ) = 42 42 x 10 420 7² 49 37² 1369 i. check this for some more two digit numbers. ii. Explain why this is correct, using algebra. iii. Find an easy method to compute squares of number ending in 5.
- 10. Look at this : 1 x 3 = 3 = 2² - 1 2 x 4 = 8 = 3² - 1 3 x 5 = 15 = 4² - 1 Does the pattern continue like this ? Product of differences We have seen how some products can be split into sums. For example, 204 x 103 = ( 200 + 4 ) ( 100 + 3 ) = 20000 + 600 + 400 + 12 = 21012 Now, suppose we want to calculate 196 x 97 ? We can split it as 196 x 97 = ( 200 – 4 ) ( 100 – 3 ) How do we split this as before, into four products ? First we write 196 x 103 = (200 – 4 ) x 97 = (200 x 97 ) – ( 4 x 97 ) Now let us write 97 = 100 – 3, and split each product. 200 x 97 = 200 x ( 100 – 3 ) = 20000 – 600 = 19400 4 x 97 = 4 x ( 100 – 3 ) = 400 – 12 = 388 Putting all these together gives,
- 11. 5- 196 x 97 = ( 200 – 4 ) x 97 = ( 200 x 97 ) – ( 4 x 97 ) = 19400 – 388 = 19012 Let us try multiplication of 298 x 195 like this : 298 x 195 = ( 300 – 2 ) x 195 = ( 300 x 195 ) – ( 2 x 195 ) 300 x 195 = 300 x ( 200 – 5 ) = ( 300 x 200 ) – ( 300 x 5 ) = 60000 – 1500 = 58500 2 x 195 = 2 x ( 200 – 5 ) = ( 2 x 200 ) – ( 2 x 5 ) = 400 – 10 = 390 What do we get on putting these together ? 298 x 195 = 58500 – 390 = 58110 It is not easy to write the general principle in ordinary language. In algebra : Cutting down Suppose we shorten by 2 centimeters, the length of a rectangle of length 10 centimeters and breadth 6 centimeters. 6 10 6 8 2 By how much the area decreased ? What is the calculation done here ? ( 10 – 2 ) x 6 = ( 10 x 6 ) ( 2 x 6 ) ( x – y ) ( u – v ) = xu – xv – yu + yv for all positive numbers x, y, u, v with x > y and u > v To find ( x – y )², we put x and y respectively instead of u and v in the above principle. ( x – y )² = ( x – y ) ( x - y ) = ( x . x ) – ( x . y ) – ( y . x ) + ( y . y ) = x² - xy – yx + y² = x² - xy – xy + y² xy + xy = 2xy. So, ( x – y )² = x² - 2xy + y² .
- 12. Let us write it as a general principle. ( x – y )² = x² - 2xy + y² For all positive numbers x, y with x > y. Geometrical explanation Look at this picture of reducing a rectangle by shortening both sides. Look at this pictures : If we subtract the areas of both the rectangle on the top and on the right, the rectangle at the top corner would be subtracted twice. To compensate, we have to add this rectangle once. ( x – y ) ( u – v ) = xu – xv –yu + yv v u-v x-y y (x-y)(u-v) v u-v x-y y xv (x-y)(u-v) v u-v x-y y (x-y)(u-v) uy v u-v x-y y (x-y)(u-v) y u yv
- 13. This we can say in ordinary language also : The square of the difference of two positive numbers is twice their product subtracted from the sum of their squares. For example, 199² = ( 200 – 1 )² = 200² - ( 2 x 100 x 1 ) + 1² = 40000 – 400 + 1 = 39601 Look at this pattern : 2 ( 2² + 1² ) = 10 = 3² + 1² 2 ( 3² + 2² ) = 26 = 5² + 1² 2 ( 5² + 1² ) = 52 = 6² + 4² 2 ( 4² + 6² ) = 104 = 10² + 2² Take some pairs of natural numbers and calculate the sum of the squares ; can you write twice this sum again as a sum of a pair of perfect squares ? What is the relation between the starting pair and final pair ? Find the sum and difference of the starting pair What is the reason for this ?
- 14. 1. Compute the squares of these numbers. 89, 48, 5⅓, 8.25, 7⅔ 2. Consider the following : 8 = 4 x 2 x 1 = 3² - 1² 12 = 4 x 3 x 1 = 4² - 2² 16 = 4 x 4 x 1 = 5² - 3² 20 = 4 x 5 x 1 = 6² - 4² Can we written this as a general principle using algebra ? 3. Look at this pattern : ( ½ )² + ( 1 ½ )² = 2 ½ 2 = 2 x 1² ( 1 ½ )² + ( 2 ½ )² = 8 ½ 8 = 2 x 2² ( 2 ½ )² + ( 3 ½ )² = 18 ½ 18 = 2 x 3² Explain the general principle using algebra. Sum and difference We have calculated products by splitting numbers as sums or differences. For example : 304 x 205 = ( 300 + 4 ) ( 200 + 5 ) 198 x 195 = ( 200 – 2 ) ( 200 – 5 ) In this way how we compute 304 x 198 ? 304 x 198 = ( 300 + 4 ) ( 200 – 2 ) To compute this , first we split 304 alone, as before : 304 x 198 = ( 300 + 4 ) x 198 = ( 300 x 198 ) + ( 4 x 198 ) Now we split 198 and compute the two products separately. 300 x 198 = 300 x ( 200 – 2 ) = 60000 – 600 = 59400 4 x 198 = 4 x ( 200 – 2 ) = 800 – 8 = 792 Putting these together gives, 304 x 198 = 59400 + 792 = 60192 To understand the general principle, let us write 304 x 198 = 60000 – 600 + 800 – 8
- 15. That as ( 300 + 4 ) ( 200 – 2 ) = ( 300 x 200 ) – ( 300 x 2 ) + ( 4 x 200 ) – ( 4 x 2 ) Like this find out 197 x 103 ? We can write the algebraic form of this as, ( x + y ) ( u – v ) = xu – xv + yu – yv for all positive numbers x, y, u, v with u > v If three numbers are such that the sum of the squares of two of them is equals to the square of the third, we call them a pythagoras triple. For example, 3, 4, 5 form a pythagoras triple because 3² + 4² = 5². There is a method to find all such pythagorean triples. Take any two natural numbers m, n and compute x, y, z as follows : x = m² - n² y = 2mn z = m² + n² It can be seen that x² + y² = z² Greek mathematicians knew this technique as early as 300 BC.
- 16. We can find a general method to compute the product of the sum and difference of two numbers. ( x + y ) ( x – y ) = ( x . x ) – ( x. y ) + ( y . x ) – ( y . y ) = x² - xy + yx – y² = x² - y² Hence the product of the sum and difference of two positive numbers is the difference of their squares. ( x + y ) ( x – y ) = x² - y² for all positive numbers x, y with x > y For example, 204 x 196 = ( 200 + 4 ) ( 200 – 4 ) = 200² - 4² = 40000 – 16 = 39984 Differenceofsquares x x² x x y² y x - y x x²-y² y x+y (x+y)(x-y) x-y We can also apply this in reverse : The difference of the squares of the two positive numbers is the product of their sum and difference. For example, 234² - 230² = ( 234+230 ) ( 234 - 230 ) = 464 x 4 = 1856 We have seen that some natural numbers can be written as the difference of two perfect squares. The above principle can be used to do it. For example, consider 45. we want to find numbers x, y such that x² - y² = 45. This we can write 45 = ( x+ y ) ( x – y ) This means ( x + y ) and ( x – y ) must be factors of 45. 45 can be written as a product of two factors in various ways.
- 17. 45 = 45 x 1 45 = 15 x 3 45 = 9 x 3 Taking the factors 45 and 1, let us write x + y = 45 and x – y = 1. So x = 23 and y = 22 ( how ? ) So, 45 = 23² - 22² Similarly let us take 45 = 15 x 3. Think about x and y. We can write 45 = 9² - 6² What about 45 = 9 x 5 ? Can we write any natural number as the difference of two squares, using this method ? For example, let us take 10. we have 10 = 10 x 1. By solving we get 10 = ( 5 ½ )² - ( 4 ½ )² But these are not squares of natural numbers ; that is, not perfect squares. How about using 10 = 5 x 2 ? What kind of natural numbers can not be written as the difference of two perfect squares ?
- 18. Sometimes, writing a product as the difference of two squares makes the computation easier. For example, consider 26.5 x 23.5. can we write it as the difference of two squares ? We need only find two numbers whose sum is 26.5 and product is 23.5, right ? And for that we need only find half the sum and half the difference of 26.5 and 23.5. that is 25 and 1.5, so, 26.5 = 25 + 1.5 23.5 = 25 – 1.5 Using this, 26.5 x 23.5 = ( 25 + 1.5 ) ( 25 – 1.5 ) = 25² - 1.5² = 625 – 2.25 = 622.75 1. Compute the following in head i. 68² - 32² ii. 3.6² - 1.4² iii. ( 3 ½ )² - ( 2 ½ )² iv. 10.7 x 9.3 v. 198 x 202 vi. 3 ⅔ x 1 ⅓ 2. Find out the larger product of each pair below, without actual multiplication. i. 25 x 75, 26 x 74 ii. 76 x 24 83 x 17 iii. 20.4 x 19.6 10.5 x0.5 iv. 11.3 x9.7 10.7 x 19.3 v. 126 x 74 124 x 76
- 19. 3. Mark four numbers forming a square in a calender : 6 7 13 14 Add the squares of the diagonal pair and find the difference of these sums : 6² + 14² = 232 13² + 7² = 218 222 – 218 = 14 4. Look at this pattern : ( 1 ½ )² - ( ½ )² = 2 ( 2 ½ )² - ( 1 ½ )² = 4 ( 3 ½ )² - ( 2 ½ )² = 6 Explain the general principle using algebra. 5. Take nine numbers forming a Square in a calendar and mark the four numbers at the corners. Multiply the diagonal pairs and find the difference of these products. 3 x 19 = 57 17 x 5 = 85 85 – 57 = 28 i. Do this for other four numbers. ii. Explain using algebra, why the difference is 14 always. ?Take some pairs of numbers with the same sum and find their products. How does the product change with the difference of the number? What is an easy method to find the largest product ? 3 4 5 10 11 12 17 18 19 i. Do this for other such squares ii. Explain using algebra, why the difference is always 28 ( it is convenient to take the number at the center as x ).
- 20. Lookingback Learning outcomes What I can With teacher’s help Must improve • explaining the method of multiplying a sum by a sum. • explaining the square of a sum geometrically and algebraically. • explaining the square of a difference geometrically and algebraically. • explaining the characteristics of natural numbers which can be expressed as the differences of perfect squares. • finding out the pair with the largest product from pairs of numbers with the same sum. • express number relations in general using algebra.