MATHEMAGIC ELLA 2018 INSTRUCTIONS

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1 MATHEMAGIC ELLA 2018 Number I am 63, so 77-63=14 and = 54, the year of my birth! Try your age does it work? How? Let a = your age and b = 2-digit birth year Then 77 a +40 = b Rearranging gives 117 a = b Further 117 = a + b 117 represents the year For all people birth year plus age MUST add to the present year!

2 MATHEMAGIC ELLA 2018 Number When Carl Gauss s class was asked by their grade 4 teacher to find the sum of the numbers from 1 to 100, she wanted to keep them busy for awhile. While the rest of the students began making a column of the numbers 1, 2, 3, and so on, Gauss did a quick calculation and raised his hand with the answer. Here is what he did: He paired off the numbers from opposite ends, seeing that each pair had the same sum, 101. Since there were 50 pairs, the sum would be 50 x 101 = 5050

3 MATHEMAGIC ELLA 2018 INSTRUCTIONS A) CHOOSE ANY NUMBER B) ADD 3 C) MULTIPLY BY 2 D) ADD 8 E) DIVIDE BY 2 F) SUBTRACT THE ORIGINAL NUMBER * I BET THE RESULT IS 7!!

4 MATHEMAGIC ELLA 2018 EG. A) 19 B) 22 C) 44 D) 52 E) 26 F) 7

5 MATHEMAGIC ELLA 2018 HOW DOES IT WORK? WHY IS THE RESULT ALWAYS 7? BECAUSE OF ALGEBRA WATCH! A) X B) X + 3 C) 2X + 6 D) 2X + 14 E) X + 7 F) 7

6 MATHEMAGIC ELLA 2018 Chinese Multiplication Video: The lines are grouped according to ones, tens, hundreds, etc. The intersections of lines, when counted, give the digits for each position even with carrying!

7 MATHEMAGIC 2018 Multiplying by 11 The number 11 has a unique property so that you will never need a calculator to multiply by it. Consider 35 x 11 Now let s try 74 x 11 Now, a larger product 426 x 11 It works every time!

8 MATHEMAGIC ELLA 2018 Mind Reading A Sum Instructions: This relies on knowing the shortcut for multiplying by 11 It works because of the famous Fibonacci sequence The sequence is found in many natural systems, including plants, and the human body. We will explore this later.

9 MATHEMAGIC 2018 a) On a sheet with blanks labeled 1 to 10 and Total, have a student choose and write a 1-digit number in blank 1 and a second student choose a second number for blank 2. b) The next student adds the numbers in the first two blanks and puts the sum in blank 3 c) The next student adds the numbers in the previous two blanks and puts the sum in blank 4. d) Step c) is repeated, always adding the previous two numbers and putting the sum in the next blank, until blank 10 is filled. e) Finally one student will add all of the numbers in blanks 1 to 10 and place the sum in the blank labeled Total. * Now you are going to tell them that total by asking for one number only, the number in blank 7. Multiply that number by 11 in your head and that will be the total!!!

10 MATHEMAGIC 2018 Mind Reading A Sum eg) Total 825 * and the number in blank #7 is 75 and (75)(11) = 825 which matches the total!!

11 MATHEMAGIC 2018 Mind Reading A Sum How does this work?...again it is pure algebra! eg) 1. x 2. y 3. x+y 4. x+2y 5. 2x+3y 6. 3x+5y 7. 5x+8y 8. 8x+13y 9. 13x x+34y Total : 55x+88y Blank #7 has 5x+8y and 11(5x+8y) = 55x+88y which is the total!! P.S. Did you notice the Fibonacci sequence happening in the process above?

12 MATHEMAGIC 2018 Fraction Magic! Consider the following pattern: Pattern: We get the same answer in addition and multiplication!! Can you create one of these yourself? and and and

13 MATHEMAGIC 2018 Fractions with a denominator of 7 have a unique and interesting property: Change 1/7 to a decimal: Change 2/7 to a decimal: Change 3/7 to a decimal: Change 4/7 to a decimal: Change 5/7 to a decimal: Change 6/7 to a decimal: Do you notice a pattern? The six digits are exactly the same; they just start at a different place and follow the rotation!

14 MATHEMAGIC 2018 Math puzzle in video form What happened to the piece of chocolate??? v=dmbspgpu0wc Good explanation: =6Q2S1-buIwU

15 Phone Number Puzzle Instructions: (calculator needed hit Enter after every step) 1. Enter the first 3 digits of your seven digit phone number (not area code) 2. Multiply by Add 1 4. Multiply by Add last 4 digits of your phone number 6. Add last 4 digits of your phone number, again 7. Subtract Divide by 2 * What is the result?...should be your phone number (7 digit)!!

16 Phone Number Puzzle eg) Phone number a) 438 b) c) d) e) f) g) h) which is the original phone number!!

17 Phone Number Puzzle Why does it work? Because of algebra To think of your phone number as an actual number, you must account for place value. To work with the first three digits separately and preserve place value, you must multiply them by so that they are in front of the last 4 digits. (eg is four million three hundred and eighty one thousand five hundred and ninety two) Let the first 3 digits be called m and let the last 4 digits be called n Then the procedure asks us to do the following: (((m*80)+1)*250+n+n-250) / 2 Now, let s look at the first part of that expression alone: In the expression ((m*80)+1)*250 you can distribute the 250 to the 80m and the +1 so: you get 250(80m)+250 which simplifies to 20000m So now adding this to the second part we have: (20000m n + n - 250) / 2 This can be simplified to (20000m + 2n) / 2...and then dividing by the 2 gives us 10000m + n times your first three digits (m) gives you xxx0000, and this, added to the last four digits (n) gives you your phone number back.

18 Buffon s Needle Drop Experiment and Pi Bufffon discovered that if you drop needles that are the width of the floorboards of a hardwood floor, then divide the number of needles dropped by the number of needles that cross a floor line, this number will approximate the value of π. Let s watch a simulation of this: The value of Pi can also be approximated using darts. If the circle is inscribed in a square then the number of darts that hit the circle divided by the number that hit the square (but not in the circle) should be close to Let s watch a simulation of this:

19 Math Forcing Grid The 4 x 4 grid has numbers 1 to 16 in order. Pick any number in the grid and circle it. Cross out all other numbers in that row and that column. Pick any second number that is not crossed out and circle it. Cross out any numbers in that row and that column. Repeat this step for a third number choice. Finally there will be only one number left. Circle it. Find the sum of the 4 circled numbers. Is it 34??!!

20 Math Forcing Grid Try it again on a fresh grid. Do we get the same sum? Why does this happen? For an n x n grid the total, T, will always equal T = n n So, for n = 4, T = = 2(17)=34 For a 5 x 5 grid what sum should result every time?

21 East Indian (Lattice) Multiplication This ancient unique method has actually been taught in certain places in North America as well. Let s start with 42 x 35. Put 42 and 35 on sides of box as shown. Put diagonal lines through all boxes. Start with 5 x 2 = 10. Put 1 and 0 in bottom right box as shown. Move on to 3 x 2 and use 06 as product. Then do 5 x 4 = 20 and 3 x 4 = 12. Finally sum each diagonal from top right to bottom left, giving Try again with 38 x 42. Can you make it work with 658 x 47. Check your work on calculator!

22 Special Cubes This comes from Dr. Ross Hongsberger, a Canadian mathematician. 1. Choose any 3-digit number 2. Raise each digit to the 3 rd power and find the sum of these values. 3. Repeat this with the new number. Continue this process until the same number appears twice in a row guaranteed! What is your final number? Let s try again. Try one of your own. Only 4 possible final numbers exist: 371, 153, 370, and 407

23 Believing in Infinity!! Do you believe that? No, not really close.exactly!! Let s look at two proofs of this: A) Now multiply both sides by So, and1 0.9

24 Believing in infinity!! B) Here is a second proof that Let x Then 10x x So, x x x And, by subtraction x 9 x 1

25 Geometry Here s a nice easy way to prove that the angle sum of any triangle is 180 degrees. Take a triangular piece of paper and fold the top angle over until it meets the opposite side, keeping the fold parallel to the base. Fold the other two angles across until they meet the first fold. The three angles should meet to form a 180 degree angle!

26 Proving Pythagorean Theorem This famous property has been proven in more than 350 different ways! This is a common way that is used with students. One of the more famousproofs was done by U.S. President James Garfield in 1876, using area of a trapezoid. But the most unique proof of the theorem uses water! Watch. kmudeb06o

27 Fractal Geometry Fractals are defined as self-replicating images with a reducing geometric pattern. Examples in nature and broccoli and ferns. In science fractal patterns are used to produce antennas that improve cell phone reception. In moviemaking fractal images are used to create scenes of large crowds.

28 Fractal Geometry, Koch Snowflake, Sierpinski Gasket Koch snowflake is a nice exercise for students learning the connection between geometric sequences and series and fractals. The Sierpinski carpet has an area that, over time, approaches zero, but a perimeter that approaches infinity: AdzgWGU A 3-D Sierpinski fractal produces a hypnotizing image when zoomed in on continuously:

29 Fractal Stairs Construction Many beautiful constructions can be done using fractals. We will create a set of 3-D stairs using a fractal pattern. Instructions: Kro1c&t=5s

30 Paper Folding and Exponential Growth How thick is a sheet of paper?? I will take 100 sheets, measure that thickness, and divide by 100. What happens to the thickness every time I fold a sheet of paper in half?? How many times can I fold it by hand? What if I could fold it 20 times, how thick would it be? 40 times? Here is a nice analogy with a Chinese chef: Hl5-6VdY Prize for invention of game of chess!!

31 Binary Code Puzzle Select a number from one of the six cards and keep it to yourself. I will ask you which of the six cards your number is on, usually several of them. Then I will tell you your number!

32 Binary Code Puzzle How does it work? It works because of binary code. What do all these binary numbers have in common? Again, what do these binary numbers have in common?

33 Binary Code Puzzle What do all the binary numbers from Card 6 have in common? Then when you select a number, say 28, and you tell me it is on cards 3, 4, and 5, I know to just add the first number on each of those cards, ie = 28 In binary that means: from card from card from card 5 Total = which is 28 in binary code.

34 Nautilus and Golden Ratio, Phi (Φ) The nautilus shell has a shape which seems to be related to the golden spiral, which is based on Fibonacci numbers 1, 1, 2, 3, 5, 8, The ratio of these numbers is phi, Φ, This shape results from the nautilus s optimal way of enlarging its exterior armor as it grows.

35 Did you know that there is a mathematical formula for parallel parking?? Rebecca Hoyle of University of Surrey has developed the formula using these parameters: P = initial position of back of your car G = gap between vehicles F = front of your car r = turning radius of your car w = width of your car b = distance between back of car and point midway between axles fg = final gap between vehicles k = distance from curb w P r 2 G w 2r b F w 2r fg

36 Finally, for the perfect park those formulas must satisfy this relationship: , 4 2, 2 k w r r Min b w r f w r Max This mathematics is in the Park Assist software found in many new cars!!

37 Symmetry in Nature The sand dollar has 5-point symmetry. Starfish has 5-point symmetry. Snowflakes have perfect hexagonal (6-point) symmetry. Most molecules have a signature symmetrical bonding pattern of one type or another.

38 Symmetry in Nature The Giant s Causeway on the north coast of Northern Ireland is an area of about 40,000 interlocking hexagonal-shaped basalt stones formed by a volcanic eruption 50 to 60 million years ago. This type of formation occurs in many countries including India, Malaysia, Turkey, Russia, and USA. Is God a mathematician?

39 Birds of Prey Biologists have discovered that birds of prey follow a mathematical pattern when they move downward toward their target. The path is a logarithmic spiral. Why? This path allows the bird to keep its head still and its eyes on the prey as it spins toward the earth!

40 Fibonacci Numbers in Nature Fibonacci first discovered his sequence in predicting the number of pairs of rabbits in any generation. But it has also been found in many places in nature, such as: A) number of bees in any generation B) number of petals on many flowers, such as daisies C) phyllotactic ratio of buds and leaves on plants such as sunflowers and pineapples

41 Card Permutations vs. Grains of Sand Have you ever considered how many different orders of the 52 cards in a deck are possible? The answer is 52 x 51 x 50 x 49 x...x 3 x 2 x 1 This is called 52 factorial, or 52! = How large is this? Watch: Just for comparison sake, Google estimates the number of grains of sand on earth as Hard to believe!!!

42 Mystic Circle of 9 Write down any 3 digit number. Rearrange the digits in any order. Subtract the smaller number from the larger. Find the sum of the digits of your difference. Starting from the # symbol as 1, count around the circle to match your sum. Did you end up at? Try it again with a new 3-digit number. Why does it work?

43 It works because of algebra and the properties of the number 9 A generic 3-digit number is written xyz, but its value is 100x + 10y + z, because of place value. If you scramble this, it could become 100z + 10x + y. We can t tell which is larger so by subtracting 100x + 10y + z - 100z + 10x + y = 90x + 9y 99z Notice that all coefficients are multiples of 9, so the sum of the digits will also be a multiple of 9, every time!

44 Magic of 11 a) Take a 2-digit number and reverse the 2 digits. Add the two numbers. eg) Notice two things: ) The sum of the outside digits equals the middle digit.(1 + 3 = 4) 2) The sum is always divisible by 11. ( 143/11 = 13) b) Try again eg) ) Consider the sum as 088, then = 8 2) 88/11 = 8

45 Drilling Square Holes Do you know what a Venn Diagram looks like? In 1875 a French mathematician named Franz Reuleaux discovered that if you create a drill bit in the shape of the shaded intersection region, it will drill square holes!! Check this out!

46 Pentagon Origami Take an ordinary long rectangular strip of paper and slowly tie a granny knot in the strip. Pull the folds as tightly as possible without tearing the paper. When folds are as tight as possible, hold the shape up to the light. What shape do you see? Can you also see the star (pentagram) inside?

47 Squaring Strategy Consider squaring a number like 27 without a calculator. If you can t do this in your head, consider adding 3 to 27 to get to an easier number, 30. But we must subtract 3 as well, so we will work with 24 and 30 instead of 27. Then, 30 x 24 = 720. Now add 3 2, since 3 was the difference used = 729 = 27 2 Let s try one more. To determine 41 2, use 40 and 42 (ie. +1 and -1) Then 40 x = = 1681 = 41 2 Try one of your own!

48 Why? Check the algebra: Given A and asked to determine the value of A 2, use:

49 Chocolate Craving (2018 version) Instructions: this one is time sensitive and has to be updated every year! Calculator needed! a) Pick the number of times you would like to have chocolate this week (try to keep it under 10!!) b) Multiply this number by 2 c) Add 5 d) Multiply by 50 e) If you have already had your birthday this calendar year, add 1768; if you haven t, add f) Now subtract the four digit number that is your year of birth. * The resulting number has three digits: the first digit is the number of times you want chocolate this week, the other two digits are your age!!

50 Chocolate Craving (2018 version) eg) Samantha was born in January 1988 and loves chocolate: a) 6 b) 12 c) 17 d) 850 e) 2615 f) 630 Samantha wants chocolate 6 times and is 30 years old!!

51 Pascal s Triangle Blaise Pascal made this wonderful connection in the 1600s between Combinatorics and Algebra with his famous triangle that holds many interesting patterns. Can you see how numbers in each row are determined by the numbers in the row above? This pattern determines the coefficients in the expansion of a binomial to any integral degree: ie. p q n

52 Pascal s Triangle Here are some of the patterns in Pascal s triangle: 1) Hockey stick pattern 2) 2 nd diagonal is the set of counting numbers 3) 3 rd diagonal is the set of triangular numbers 4) Every row adds to a power of 2 5) The first 5 rows are the powers of 11 falls apart after that! 6) If the even numbers are shaded, Sierpinski s gasket is produced.

53 Platonic Solids and Viruses There are five solids that can be formed from one and only one regular polygon: tetrahedron, cube (hexahedron), octahedron, icosahedron, and dodecahedron. These are the only fair shapes for dice. Amazingly, many virus cells have the shapes of one of these solids, including HIV, polio, influenza, herpes, rhinovirus, and adenovirus.