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MATHEMATICS : A JOY RIDE Mathematical Crossword Across 1 an exterior angle of any angle <180 2 perimeter of a circle 3 Indian genius ,a student of Prof. Hardy Down 2 Lines intersecting in the same point 4 figure formed by two rays originating at same point 5 this theorem was first used by Maharshi Bodhayan 5 closed rectilinear figure 6 first counting machine 9 Equation represents a straight line 7 centroid is point of intersection of – 10 graph representing the data 8 amount of space taken up by a 3D object 11 triangle with all sides unequal . 12 mean of a statistical data 13 Father of the coordinate geometry 15 points on the same line are ---- 14 shape of a box 17 Point of intersection of altitudes of a triangle 16 another name for indices 18 3X4 = 4X3 1 MATHS IS FUN & JOY TRY TRY, DON’T CRY 2 THREE R’S Teaching math is about providing an atmosphere of playful engagement with mathematical problems, where students feel confident in failing, in order to try again; a place where students become transformed by exercising their own mathematical powers of reasoning. TIME TO GETUP 3 HAPPY & WISE EARLY TO BED, EARLY TO RISE MAKES THE PERSON HAPPY & WISE TIME TO SLEEP 4 RESULT 5 R/MENTAL BLOCK/DISLIKE? Who is responsible for creating MATHSPHOBIA in the child’s mind? MOTHER ? Dull teaching causes most people to shy away from maths. Understanding how children learn best is an important step towards improving maths learning.By providing conducive atmosphere. TEACHER ? 6 Lack of practice? “I hear and I forget. I see and I remember. I do and I understand.” 1. Recognize you have an aversion to math, whether it's full-blown math phobia or just a few math blocks here and there. 2. Make a conscious decision to do something about it. 3. Give yourself a regular math workout, however small to start with. You'll find it all gets easier, and you'll soon enjoy math once again. 7 INDIAN GENIUS WHY GO SO FAR? STORY OF THE SON OF OUR OWN SOIL SIR RAMANUJAN Who was Srinivasa Ramanujan? A famous Indian mathematician 1729? who lived from 1887 to 1920. The theory of numbers brought worldwide fame to Ramanujan. Some of us here know Sir Ramanujan worked at Cambridge University with the great mathematician, G.H. Hardy.His birth centenary was celebrated in1987. 8 Story time • Once the inspector visited the school. He entered the 4th std. class where his favorite subject was being tought.He posed a small question to the children. He asked them the sum of first 100 counting numbers.. All the children got busy to find the answer. Some started writing in the notebook,some started counting fingers. One little boy on the last bench was sitting very quietly watching the rest of the children. Inspectors always have a bad habit of catching the back benchers as during inspection teachers make the dull children sit at the back. So he asked the child ,”sweetheart, why don’t you want to give It a try?” Pat came the reply, sir, it's not a big deal. Answer is 5050. Inspector was very impressed with the child & asked him to explain. Child confidently replied ,sir ,if one adds two numbers at the extreme, every time one gets a total of 101. • (as 100+1;99+2,------50+51) One gets 50 such pairs. Hence the answer is 101x50=5050. Inspector knew that one day this child prodigy is going to be a high achiever in life. Yes, his prophecy was true. Till the date we know him as Sir Ramanujan. 9 Internet blogs Teaching math is about being a physician who with care & affection ,above all patience finds the remedy for the patient (student).Patient too cooperates & follows the treatment religiously. A place where students become transformed by exercising their own mathematical powers of reasoning Math inquiry lessons are student-focused. Teachers give students materials and minimal direction; students then explore the topic and construct their own meaning . Movies with inquiry bases ,hands on math activities applications on the futureschannel.com Visit the sites: mathtopper.com;videomathtutor.com;articlesbase.com; mathworks.in,futureschannel.com & many more. Just type mathphobia or remedial teaching in math in Google search 11 From a strictly mathematical viewpoint: What Equals 100%? What does it mean to give MORE than 100%? If: ABCDEFGHIJKLMNOPQRST UVWXYZ Is represented as: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Therefore, one can conclude 18 19 20 21 22 23 24 25 26. with mathematical certainty that: If: While Hard Work and Knowledge H-A-R-D-W-O-R- K will get you close, and Attitude 8+1+18+4+23+15+18+11 = 98% And: will get you there, K-N-O-W-L-E-D-G-E It's the Love of God /Faith that 11+14+15+23+12+5+4+7+5 = 96% will put you over the top! But: A-T-T-I-T-U-D-E 1+20+20+9+20+21+4+5 = 100% L-O-V-E -O-F- G-O-D / FAITH 12+15+22+5+15+6+7+15+4 = 101% 12 LEARN WHILE YOU PLAY PAPER FOLDING CRAFT WORK TEACHING AIDS ADDITIONAL INFORMATION FALLACIES/PUZZLES 13 LEARN TO ANSWER WHY & HOW ? A C A B B C B A C B A C 14 SIMPLE RESULTS Some Algebraic Facts (a+b)2 = a2 + 2ab+ b2 a ab a a2 b b2 b Pascal triangle = (10+1)0 = (10 + 1)1 1 1 = (10 + 1)2 1 2 1 1 3 3 1 = (10 + 1)3 1 4 6 4 1 = (10 + 1)4 1 5 10 10 5 1 = (10 + 1)5 1 b ab (a+b)2 geometrically gives area of a square whose sides are (a+b) units. Sequence of the numbers of the Pascal’s triangle represent the binomial coefficients in the expansion of (x+y)n 15 AL-G-BAR Some Algebraic Facts (a+b+c)2 = a2+b2+c2+2ab+2bc+2ac a b c a2 ab ac ab b2 bc ac bc c2 16 SUM OF THE ANGLES OF REGULAR POLYGONS Shape of the regular polygon No. of sides & angles 3 Rule of Sum of the angles 180 (3-2) Sum of the angles 180 Rule for measure of each angle Degree measure of each angle 180(3-2) 3 60 180(4-2) 4 4 180 (4-2) 360 90 5 180 (5 -2) 540 180(5-2) 5 108 6 180(6-2) 720 180(6-2) 6 120 n 180(n -2) 180(n-2) 180(n -2) n 180(n-2) n 17 MATHS CRAFT 1 Helpful website: www.scribd.com/doc/2726617/chapter-11 Mathematics stimulates the imagination, anchors speculation, and promotes an awareness of 18 reality. To derive the formula for area of a circle Recall circumference of a circle is 2∏r Area of a triangle is = ½ base X height h=r 2∏r Area of a ∆ = ½ base X height =1/2 X 2∏r X r = ∏ r2 = Area of a circle 19 l= length b= breadth Area= l X b sq.units r b=h, height Total surface area of a solid cylinder = l =2 ∏r 2 ∏rh + 2 ∏r2 sq.units CSA =l Xb =2 ∏ r h 20 Volume of a cylinder = Volume of a cuboid = lXbXh=∏r . r . h= ∏ r2 h= volume of a cylinder ∏r h h r ∏r 21 Making a Cube Making a Triangular Prism 4 5 3 22 CRAF T 12cms To make a cone and find its surface area and volume Materials required: o 6 o 1. A square piece of thin cardboard of side 12 cm 2. A thick square cardboard of side 36 cm each 1. `36 18 18 120 18 120 O 120 2. Bring the edges OA & OB together .Stick them.Attach the circular piece above to the bottom of the cone formed. 3. Length of the arc=circumference of the circle as 4. 2 ∏ x6 = 2 ∏ x18 x120/360 =12 ∏cm 18 36cms. Scissors, Adhesive, Compasses 5. l = slant height =18cms. ∏ r l = ∏ x6x18=108∏ sq.cms. 6. CSA = 7. TSA = ∏ x6x6 + 108 ∏ =144 ∏ sq.cms. UNDERSTAND BETTER • SQUARE PYRAMID • HEXAGONAL PYRAMID 24 MATHS THROUGH CRAFT ACTIVITY HEXAGONAL PRISM 25 Sharpen your reasoning with puzzles Change the direction of the fish moving 3 sticks. Can you divide no.of 17 cows between three brothers so that elder one gets ½,middle one gets 1/3 & the youngest get 1/9 th of the total cows? 26 FATHER OF ALGEBRA Linear equations The riddle begins, “Diophantus ” youth lasted 1/6 of his life. He grew a beard after 1/12 more after 1/7 more of his life he married 5 years later he had a son ALGEBRA SOLVES A RIDDLE Little is known about the life of Diophantus the Greek father of algebra, except his age at death, which has been preserved in the famous 1,500-year-old riddle shown here. If we assume ‘x’ as his age at the time of death then we get the equation The son lived exactly ½ as long as his father. And Diophantus died just 4 years after his son. All this adds up to the years Diophantus lived x = x/6 + x/12 + x/7 + 5 + x/2 + 4, which reduces to 3x/28 = 9 telling us 27 Diophantus was 84 years old when he died. ACHILLES FOOT Achilles and tortoise Speed of Achilles is 10 times that of tortoise. However tortoise gets a head start of 100 meters. When will Achilles catch on with the tortoise? 100 0 (A = 100, T = 110); (A = 110, T = 111); (A = 111, T = 111.1) (A=111.1,T=111.11 tortoise will be always ahead of the 28 Achilles, even if by a mere eyelash. Scratch ur head • Let x=2 ; x(x – 1) = 2 ( x – 1) ;x2 - x = 2x -2 ; x2 – x –x = 2x – 2 – x ; x2 – 2x = x -2 ; x (x – 2 ) = x – 2 ; x =1 But x= 2 hence 2 = 1 o If you jog half way from A to B at a steady rate of 2miles/hr ; how fast would you have to run the rest of the way in order to average 4 miles /hr for the entire trip. BEWARE IT MIGHT BE A FALLACY • Have you noticed? 11 x 11 =121; 111 x111=12321,1111x1111=1234321 what next? • 371= 3 3 + 7 3 + 1 3; 407= 4 3 +o 3 + 7 3 • Palindromes: 56765 both ways read same • e.g.57+75=132+231=363 29 Numeral & NUMBER WHICH NUMERAL IS SMALLER? WHICH NUMBER IS SMALLER? 30 FIBONACCI Fibonacci Numbers Sequence is 1, 1, 2, 3, 5, 8, 13, ………if you’ve ever thought maths wasn’t “natural,” think again. The numbers of many flowerpetals are Fibonacci numbers. The numbers of spirals in a pine cone, pineapple, and sunflower seed heads also tend to be Every ratio of the Fibonacci numbers starting from 3/2, 5/3, 8/5, ….. is called golden ratio more about it Fibonacci numbers. in the next slide. New borne One month old 31 GOLDEN RECTANGLE Calling someone a “SQUARE” is an insult but calling them a “GOLDEN RECTANGLE” isn’t so bad. This old man portrait of Leonardo da Vinci shows a picture with a square subdivided into rectangles having golden ratio. The ratio AG:AB represent the golden ratio and is donated by B A G 1+ 5 = 2 =1.6 :1 This construction which is used in many temples, mosques fits into golden rectangles. In each of the square if you put a quarter circle then it represents the pattern which we see in some seashell This rectangle is the most harmonious & pleasing to the eye hence we have sheets of paper, book of pages, standard photo frame, monitor, credit cards, windows and so on in the shape of rectangle. Usually ratio of all rectangular things is between 1.4:1 and 1.8:1,credit cards,TV ,monitors32etc. RELATION & FUNCTION 33 THEORY OF CONVERGENCE • (using a shrinking ruler to measure the unmeasurable) • A fundamental concept of calculus is ‘convergence of limit’. The idea that an unknown value can be measured by ”closing in “through approximations that are made finer & finer until they are refined, in effect to a precise value.1) the tracks converging on the horizon appear to join at a particular point, though they actually never meet.2)Images of a boy holding a mirror photographed in another mirror, although actually never shrink, but they appear to be converging on such a small area that it is considered to be a point.3)The lines AE,AD,AC &AB show average growth rates for successively smaller periods of time. But for the instant A ,the growth rate is shown by the tangent at A. 34 CONVERGENCE 35 Vector analysis V2 R V1 • VECTOR ANALYSIS Shooting at a target on a windy day is a problem illustrating one of Carl Gauss’s realm of mathematics known as ‘vector analysis’ The velocity of the wind blowing from west to east is represented by an arrow i.e. vector V1.The rifleman compensates by moving his gun slightly left of the target as represented by vector V2.The bullet flies in a compromise pathway to the bull’s eye along the line R. 36 The great Galileo & Isac newton : Gravitational force g=32ft/sec A parachuter in free fall drops faster every monent.Calculus finds his rate at any instant by in effect, measuring shorter & shorter time segments.In the first bracketed period he falls at an average speed of 88ft/sec for half a sec.In the next equal period 104 feet.In two shorter periods he drops 94.4 ft per sec & 97.6 ft.The ever narrowing range finally converge to 96 ft/sec at exactly 3 sec. Timing an object as it falls from a given height is the most straight forward method of gauging the efects of gravity.It was this technique which Galileo used about 1585 to arrive at his free fall eqn.y=16t2 ;y representing the distance fallen in ft. and t the elapsed time I sec. after the first fall. Newton further proved that it is law of nature that every free falling object falls to earth with a constant acceleration of 32ft /sec every sec. 37 THEODOLITE & SEXTANT A theodolite is a surveying instrument used for measuring horizontal and vertical angles. A Sextant is an instrument used to measure the angle of elevation of the sun above the 38 horizon. MOBIUS STRIP Not even Picasso could paint this ring in two different colours. It proves the strip has only one side 39 KLEIN BOTTLE Three diagrams at left illustrate how a stretchable glass tube can be transformed in to A Klein bottle. One end becomes the neck, the other the base. The neck goes through the side of the bottle& the neck & the base join, making inside continuous with the outside. 40 Beauty of Math! 1x8+1=9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 +10= 1111111111 look at this symmetry: 9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888 1x1=1 11 x 11 = 121 111 x 111 = 12321 1111 x 1111 = 1234321 11111 x 11111 = 123454321 111111 x 111111 = 12345654321 1111111 x 1111111 = 1234567654321 11111111 x 11111111 = 123456787654321 111111111 x 111111111=12345678987654321 Brilliant, isn't it? 41 U KNOW HIM IN 1882 A GERMAN COUPLE WORRIED THAT THEIR THREE YEAR OLD CHILD HAD NOT LEARNT TO SPEAK A WORD . HOWEVER HE GREW UP WITH THE SIDE INTEREST OF OBSCURE MATHS. WHICH EARNED HIM A NOBLE PRIZE. TILL THE DATE THE WORLD REMEMBER S HIM AS E = mc2 ALBERT EINSTEIN 42 HELLO,HOPE U R WITH ME. THEN JUST READ IF THESE INTEREST U. John Napier a Scottish mathematician invention of log table. He also is known for an invention of a slide rule. 1550-1617 43 580-500a.c. Pythagoras, Greek mathematician, formulated the Pythagoras, theorem. 44 1642-1727 Sir Isaac Newton’s greatest contribution to mathematics was the invention of calculus. The lines show average growth rate for successive periods. But for the instant it is shown by a gradient of the tangent. A picket fence is a simple key to integration. Calculus solves the problem by dividing the area in to small intervals so that the top becomes negligible. 45 1596-1650 Rene Descartes, a French mathematician and philosopher, invented analytic(co-ordinate) geometry. Cartesian plane is named after him. y (3,5) x 46 1777-1855 Carl Friedrich Gauss along with Archimedes and Newton, Carl Friedrich Gauss has been called the greatest mathematician ever. He contributed in the field of astronomy, surveying & electromagnetism. 47 Charles Babbage British Mathematician & Engineer develop an early computer. 1791-1871 48 INDIA IS PROUD OF U & INDEBTED TO U FOR EVER • Aryabhatta: gave the value of Pi. For his astronomical contributions India’s first satellite was named after him. • Brahmagupta: Developed a decimal system by giving Zero. • Bhaskara: developed Trigonometry. • Jayant Naralikar : Theory of relativity. • S.N.Bose : an eminent statistician 49 INTERESTING !!! If U are AWAKE. • 1729? As I told you earlier at schol Ramanujan was a studnt star in Maths.He went beyond what was tought in class.Fascination for the beauties in maths overpowered him.1729 is the famous taxi no.which is often mentioned in narrating his love for nos.While in the U.K. Prof. Hardy visited him in the hospital as Ramanujan was lying ill.Hardy mentioned the no.of the taxi in which he came.At once Ramanujan gave out the property of 1729 as the smallest no.that can be expressed as a sum of two cubes in two different ways.This theory later helped immensly in solving indeterminate eqns. Who was Leelavati? This unortunate daughter of Bhaskarachrya became a first woman mathematician as the going got tough for her. As we all know Bhaskaracharya was a great astronomer & had developed a science of astronomical calculations. He had calculated an auspicious muhurtam for his daughter to get married. However he also knew something which made him worry. What was that U want me to tell? THIS IS JUST TO SEE HOW MANY OF U ARE STILL AWAKE!!Ok so the story50 goes. This was a humble effort to demonstrate the power and sophistication of these ideas, and explore how mathematics teaching can be structured to resonate with children's sophisticated thinking. • I HOPE THIS TALK PROMOTES THE LOVE FOR MATHEMATICS & DEVELOPES BETTER UNDERSTANDING. • I AM GRATEFUL FOR YOUR PRESENCE AND INTERACTION. • Hope this orientation helps in redefining maths • I REQUEST YOU TO GIVE CANDID OPINION FOR FURTHER IMPROVEMENT ON THIS EFFORT TO PROMOTE LOVE FOR 51 MATH AND HELP REDUCE THE FAIL %. MATHEMATICS : A JOY RIDE Mathematical Crossword Across 2. The result in multiplication (7) 5. Approximately equal to 3.1415 (2) 7. Number added to another in addition (6) 9. The bottom number in division (7) 10. A positive or negative whole number (7) 12. A sign used in subtraction (5) 13. Amount of space taken up by a 3D object (6) 18. 1/2 or 3/4, for example (8) 20. This shape has all points at the same distance from its center (6) 21. The 3 or the 2 in 3 X 2 = 6 (6) 22. Is identical in value (6) 23. Figure formed by two lines extending from the same point (5) 24. Take away (8) Down 1. Rectlinear closed figure with three sides 3. Angle greater than 90 degrees and less than 180 degrees is this (6) 4. Longer dimension of a rectangle (6) 5. ____ sign is used in addition (4) 6. Sharing a pizza between friends requires this kind of operation (8) 8. For finding total you need to this operation 11. To determine the product (8) 14. A gram, a foot or 87 degrees (7) 15. A three-sided figure having two equal sides (9) 16. The answer in a division problem (8) 17. A quadrilateral with four sides equal (6) 19. An angle measuring less than 90 degrees (5) Time 5 mts. Made by Alka Damle 54 Across 2. The result in multiplication (7) 5. Approximately equal to 3.1415 (2) 7. Number added to another in addition (6) 9. The bottom number in division (7) 10. A positive or negative whole number (7) 12. A sign used in subtraction (5) 13. Amount of space taken up by a 3D object (6) 18. 1/2 or 3/4, for example (8) 20. This shape has all points the same distance from its center (6) 21. The 3 or the 2 in 3 X 2 = 6 (6) 22. Is identical in value (6) 23. Figure formed by two lines extending from the same point (5) 24. Take away (8 Down 1. Rectlinear closed figure with three sides 3. Angle greater than 90 degrees and less than 180 degrees is this (6) 4. Longer dimension of a rectangle (6) 5. ____ sign is used in addition (4) 6. Sharing a pizza between friends requires this kind of operation (8) 8. For finding total you need to this operation 11. To determine the product (8) 14. A gram, a foot or 87 degrees (7) 15. A three-sided figure having two equal sides (9) 16. The answer in a division problem (8) 17. A quadrilateral with four sides equal (6) 19. An angle measuring less than 90 degrees 55