CURR 383: Elementary Mathematics

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1 CURR 383: Elementary Mathematics Class 8: Measurement Tuesday January 26, 2010

2 Agenda Openers: Measurement Referents, One Minute Challenge, Useful Units and Fun Facts How much is a million? (video based on book by David Schwartz) The Megapenny Project How big is a foot? (a book by Rolf Meyer) and some shoe size history Making a Simple Protractor The Aboriginal method for measuring the height of a tree How big is a cubic metre?

3 Referents

4 Other Everyday Referents

5 Useful Units How long would it take to count, out loud, by ones to 1 million? 1 billion? How many grains on sand in 1 cup? 1 bathtub? 1 classroom? Do you know anyone 1 billion seconds old? Who? Why don t we measure age in seconds? What kind of vehicles would you need to transport 1 billion pennies? The Megapenny Project How much is a million? by David Schwartz

6 The MegaPenny Project Understanding the size of numbers using pennies

7 One Penny Most people in North America have seen them and you probably have one in your pocket right now. Using this small metal disk, with a size and weight familiar to almost everyone. value width height thickness weight 1, (one cent) 0.75 inches, (3/4 of an inch) 0.75 inches, (3/4 of an inch) inches, (1/16 of an inch) 0.1 ounces, (1/10 of an ounce) area square inches

8 Ten Pennies Place one penny on each of your ten fingers. When we multiply our one penny by a factor of ten, we get ten pennies. value width (side-by-side) height (stacked) thickness weight area (laid flat) 10, (ten cents) 7.5 inches 5/8 of an inch inches, (1/16 of an inch) 1.0 ounce square inches

9 Sixteen Pennies Lay sixteen pennies in a line and you have one foot. Stack them and you have an inch. value width (side-by-side) height (stacked) thickness weight area (laid flat) 16, (sixteen cents) 12 inches, (one foot) 1 inch inches, (1/16 of an inch) 1.6 ounces 9 square inches

10 One Thousand Pennies One thousand pennies in your pocket would be the same as having 20 rolls of pennies in your pocket from a bank. 5 pennies wide x 5 pennies high x 40 pennies tall value width height thickness weight height stacked area (laid flat) $10.00, (Ten dollars and no cents) 3.75 inches 3.75 inches 2.5 inches 100 ounces, (6.25 pounds) 62.4 inches, (5.2 feet) square inches (3.9 square feet)

11 One Hundred Thousand Pennies Ninety-eight thousand three hundred and four Pennies [ Two cubic feet ] Doubling our 50,000 pennies to 100,000, we now have two one-foot cubes. Given, this wasn't a big jump, but now we're going to start climbing the numerical ladder by powers of ten. value $ (Nine hundred eightythree dollars and four cents) width height thickness weight height stacked 24 inches, (two feet) 12 inches, (one foot) 12 inches, (one foot) pounds 512 feet $98,304 area (laid flat) 384 square feet

12 Fifty Thousand Pennies [16w x 16h x 192 tall] one cubic foot Forty-nine thousand one hundred and fifty-two Pennies $49,152 Remember the stack of 16 pennies? It was one inch tall. Well, take 12 of those and stack them and you'll have onefoot-tall stack. The cube above is made up of 256 of those one-foot stacks, making one cubic foot of pennies. Remember this cube, since it will be the building block for all upcoming penny-structures you see. value $ (Four hundred ninety-one dollars and fifty-two cents) width height thickness weight height stacked area (laid flat) 12 inches, (one foot) 12 inches, (one foot) 12 inches, (one foot) 4,915.2 ounces, (307.2 pounds) 3,072 inches, (256 feet) 192 square feet

13 SCALE REFERENCE 5 Feet, 10 inches G-Money

14 One Million Pennies One million, three thousand, seven hundred and seventy-six Pennies [ A wall five feet by four feet by one feet thick with a 9-inch cube stepstool $1,003,77 Say hello to our friend G-Money. Now that the pennies have really begun to pile up, he'll be standing in for scale. G-Money is about 5 feet, 10 inches tall, and he weighs about 180 pounds, or about 35 times less than the 1 million pennies stacked beside him. value $10, (Ten thousand, thirty-seven dollars and seventy-six cents) width height thickness weight ] Four feet Five feet 12 inches, (one foot) pounds (3.14 tons) height stacked 5,228 feet ( 0.99 Miles ) area (laid flat) 3,921 square feet

15 Ten Million Pennies Ten Million cents. If you laid these all out flat, side-by-side, like a huge carpet of pennies, it would nearly cover one acre. Ten million, seventeen thousand and twentyfour Pennies [ A cube 6 x 6 x 6 feet ] $10,017,024 value $100, (One hundred thousand, one hundred seventy dollars and twenty-four cents) width height thickness weight height stacked area (laid flat) Six feet Six feet Six feet 31.3 tons 9.88 Miles 39,129 square feet (0.9 acres)

16 SCALE REFERENCE Blue Whale largest living animal, the Blue Whale, weighs less than 150 tons as an adult longest ever recorded was a 108 ft female modern blue whales in the Southern Hemisphere reach lengths of feet Northern Hemisphere counterparts are smaller, on average 75 to 80 feet (23 to 24.5 m)

17 One Hundred Million Pennies Since these cubes are mostly dense metal, their weight is impressive. One hundred million pennies weighs over 300 tons. For comparison, the largest living animal, the Blue Whale, weighs less than 150 tons as an adult. One hundred million, sixteen thousand, six hundred and forty Pennies [ Two cubes, one 12x12x12 feet, the other 7x7x7 feet ] $ 100,016,640 height thickness value width total weight height stacked area (laid flat) $1,000, (One million, one hundred sixty-six dollars and forty cents) 12 feet (7 feet) 12 feet (7 feet) 12 feet (7 feet) tons 99 Miles 390,690 square feet (9 acres)

18 SCALE REFERENCE 9 Feet Wide X 11 Feet High X 41 Feet Long

19 One Billion Pennies value If you were to stack all these pennies in a single pile, one atop the other, the stack would reach nearly one thousand miles high. For comparison, note that the Space Shuttle typically orbits only 225 miles above the Earth's surface. $10,000, (Ten million, one hundred eighty-one dollars and seventy-six cents) One billion, eighteen thousand, one hundred and seventy-six Pennies [ Five school buses. ] $1,000,018,176 width height thickness total weight height stacked area (laid flat) 45 feet 11 feet 41 feet 3,125 tons 987 Miles 3,906,321 square feet (89.7 acres)

20 SCALE REFERENCE

21 Ten Billion Pennies Current estimates place the world's population at six billion people. The pile of pennies above would then be nearly enough for two pennies for every person on Earth. The U.S.Mint currently manufactures about this many pennies every year. Also, note G-Money standing in the front. Ten billion, twenty-three thousand, five hundred and fifty-two Pennies [ Fifty school buses. ] $10,000,023,552 value $100,000, Value (One hundred million, two hundred thirty-five dollars and fifty-two cents) thickness width total weight height stacked area (laid flat) height 90 feet 11 feet 205 feet 31,250 tons 9,864 Miles 897 acres

22 One Hundred Billion Pennies If you took these hundred billion pennies and laid them out like a carpet, you could cover 14 square miles. The cube you see is made up of over 4.1 million stacks of 24,330 pennies. One hundred billion, seventeen million, six hundred fifty-nine thousand three hundred and thirty-six Pennies [ One cube measuring 127 x 127 x 127 feet ] $100,017,659,336 value width height thickness total weight height stacked area (laid flat) $1,000,176, (One billion, one hundred seventy-six thousand five hundred and ninety-three dollars and thirty-six cents) feet feet feet 312,555.2 tons 98,660 Miles 8,969 acres

23 Two Hundred Billion Pennies Since the first penny was minted in 1787, until present-day, over 300 billion pennies have been minted in the United States. Two hundred billion, thirty-five million, three hundred eighteen thousand six hundred and seventy-two Pennies [ Two cubes, each measuring 127 x 127 x 127 feet ] $200,035,318,672 height value thickness total weight width height stacked area (laid flat) $2,000,353, (Two billion, three hundred fiftythree thousand one hundred and eighty-six dollars and seventy-two cents) feet feet feet 625,110.4 tons 197,320 Miles 17,938 acres

24 One Trillion Pennies value $10,000,000, (Ten billion, one hundred and sixty-six dollars and forty cents) width height 273 feet 273 feet One trillion, sixteen thousand six hundred and forty Pennies [ One cube measuring 273 x 273 x 273 feet ] $1,000,000,016,640 The same football field as the last two pages, set beside our new cube for scale. Notice our friend G- Money, still barely visible as a speck at lower left. thickness 273 feet total weight height stacked area (laid flat) 3,125,000 tons 986,426 Miles 89,675.2 acres

25 SCALE REFERENCE

26 One Trillion Pennies (part II) From right to left (to scale), we have the same old football field, then the Lincoln Memorial (yes, the one pictured on the back of the penny), then the Washington Monument (555 feet tall), then our cube of one trillion pennies, then the Empire State Building (1,250 feet tall), then the Sears Tower (1,450 feet tall). One trillion, sixteen thousand six hundred and forty Pennies [ One cube measuring 273 x 273 x 273 feet ] $1,000,000,016,640 value $10,000,000, (Ten billion, one hundred and sixty-six dollars and forty cents) width height thickness total weight height stacked area (laid flat) 273 feet 273 feet 273 feet 3,125,000 tons 986,426 Miles 89,675.2 acres

27 The Empire State Building 1.8 Trillion Pennies New York's Empire State Building contains 37 million cubic feet of space (minus the antenna structure). Using our cubic foot of pennies (49,152 total), it's just a simple multiplication problem - 37,000,000 x 49,152 = 1,818,624,000,000 pennies. One trillion, eight hundred eighteen billion, six hundred twenty-four million Pennies $1,818,624,000,000 value total weight height stacked area (laid flat) $18,186,240, (Eighteen billion, one hundred eighty-six million, two hundred forty thousand dollars and zero cents) 5,683,200 tons 1,793,939 Miles 163,085 acres

28 The Sears Tower 2.6 Trillion Pennies Chicago's Sears Tower occupies 53.4 million cubic feet of space. Using our cubic foot of pennies (49,152 total), it's once again just another multiplication problem - 53,379,000 x 49,152 = 2,623,684,608,000 pennies. Two trillion, six hundred twentythree billion, six hundred eightyfour million six hundred and eight thousand Pennies $2,623,684,608,000 value total weight height stacked area (laid flat) $26,236,846, (Twenty-six billion, two hundred thirty-six million, eight hundred forty-six thousand and eighty dollars) 8,199,014 tons 2,588,073 Miles 235,279.3 acres

29 One Quadrillion Pennies Here we have the buildings we used for scale back at a trillion, but they're now a bit dwarfed by our new cube of pennies. This is a quadrillion, or a thousand times one trillion. This cube is roughly a half-mile wide and would weigh an astonishing three billion tons. One quadrillion, sixty-seven billion, eighty-eight million, three hundred and eightyfour thousand Pennies [ One cube measuring 2,730 x 2,730 x 2,730 feet ] $1,000,067,088,384,000 value width height thickness total weight height stacked area (laid flat) $10,000,670,883, (Ten trillion, six hundred seventy million, eight hundred eighty-three thousand, eight hundred and forty dollars and zero cents) 2,730 feet 2,730 feet 2,730 feet 3,125,000,000 tons 986,426,768 Miles 89,675,161 acres

30 One Quintillion Pennies value $10,000,670,883,840, (Ten quadrillion, six hundred seventy billion, eight hundred eighty-three million, eight hundred and forty thousand dollars and zero cents) Here we have the buildings we used for scale back at a trillion, but they're now a bit dwarfed by our new cube of pennies. This is width a quadrillion, 27,300 or a thousand feet times one trillion. This cube is roughly a half-mile wide and would weigh an astonishing three billion tons. One quintillion, sixty-seven trillion, eighty-eight billion, three hundred and eightyfour million Pennies [ One cube measuring 27,300 x 27,300 x 27,300 feet ] $1,000,067,088,384,000,000 height thickness total weight height stacked area (laid flat) 27,300 feet 27,300 feet 3,125,000,000,000 tons 986,426,767,677 Miles 89,675,160,698 acres

31 One Quintillion Pennies Now we've stepped up another factor of 1,000. One quintillion pennies. This many pennies, if laid out flat like a carpet, would cover the surface of the earth - twice. If you look hard, you can still see the Sears Tower and other buildings at lower right. Another way to see it is to realize Here we have the buildings we used for scale back that at Mt. a trillion, Everest but (29,000 they're now ft.) is a only bit dwarfed by our new cube of pennies. This is a quadrillion, 1,700 feet or taller a thousand than this times 27,300-foot one trillion. This cube is roughly a half-mile wide and would weigh cube. an astonishing three billion tons. Three trillion tons of pennies is quite enough. To imagine larger cubes, (stepping by factors of 1,000), just imagine cubes roughly ten times larger than the last one. For instance, one quintillion pennies makes the cube above - about 5 miles on each side. If you step up to one sextillion, imagine a cube about 50 miles wide tall and thick

32 Table of Very Big Numbers (from 1 to ) one 1 ten 10 hundred 100 thousand 1,000 million 1,000,000 billion 1,000,000,000 trillion 1,000,000,000,000 quadrillion 1,000,000,000,000,000 quintillion 1,000,000,000,000,000,000 sextillion 1,000,000,000,000,000,000,000 septillion 1,000,000,000,000,000,000,000,000 octillion 1,000,000,000,000,000,000,000,000,000 nonillion 1,000,000,000,000,000,000,000,000,000,000 decillion 1,000,000,000,000,000,000,000,000,000,000,000 undecillion 1,000,000,000,000,000,000,000,000,000,000,000,000

33 duodecillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000 tredecillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000 quattuordecillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 quindecillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 sexdecillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 septendecillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 octodecillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 novemdecillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 vigintillion unvigintillion duovigintillion trevigintillion quattuorvigintillion quinvigintillion sexvigintillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

34 septenvigintillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000 octovigintillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000 novemvigintillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 trigintillion untrigintillion duotrigintillion googol (google) 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00 0 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00 0,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000

35 tretrigintillion quattuortrigintillion quintrigintillion sextrigintillion septentrigintillion octotrigintillion novemtrigintillion 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000

36 One Penny You Have to Start Somewhere

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54 Shoe Size History

55 Shoe Size History

56 Coffee Filter Protractor Cut a coffee filter in half. Fold the filter in half. Mark the fold with a marker and the angle measure. Fold in half again. Mark and indicate the angle measure. Your coffee filter has creases? What is the angle measure of each crease? How could you use this in class?

57 How to Measure the Height of a Tree Video 5min.com

58 How big is a cubic metre? Building a cubic metre in the classroom Why? Alternatives? Questions?