Gazillions

Thinking up an activity for the Common Core State Standards.

• 8.EE.3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. 
• 8.EE.4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

I've been thinking about this recently because I seriously impressed some school kids by multiplying in my head a couple of numbers in the billions. Then when Char Beckmann needed an activity for the  Adventures in Mathematics 8th grade book - opportunity! (Or rationalization...) (These books are from the Michigan Council of Teachers of Mathematics.)

My first couple of ideas were: something based on the brilliant scale of the universe applet, or a game looking at different representations of these numbers (my love for rummy games), or an activity based on Fermi problems.

Walking the kids to school this morning I was thinking about the rummy idea, and came up with a new game mechanic variation on rummy:  instead of collecting sets, each turn you have to play a card out in front of you. Then opponent can capture that card with a match. Then you could capture the pair with another matching card... kind of a slow run building mechanic.  Don't think it will fit for the book, but I will definitely try it in a game later.

Thinking about the matching puzzle, we have:

Number Names	Measurement Prefixes	Things	Power of Ten
One		Humans (meters)	10^0
Ten	deca-	Orcas, Anacondas (meters)	10^1
Hundred	hecto-	Redwood (meters)	10^2
Thousand	kilo-	Mountains' height (meters), Number of visible stars	10^3, 10*10*10
Million	mega-	Width of USA (meters)	10^6
Billion	giga-	Diameter of the Sun (meters), Age of the universe (years)	10^9
Trillion	tera-	Diameter of the Solar System (meters), US national debt (dollars)	10^12
Quadrillion	peta-	One light year (meters)	10^15
Quintillion	exa-	Number of grains of sand on earth	10^18
Sextillion	zetta-	Diameter of the Milky Way, Number of water molecules in a drop	10^21
Septillion	yotta-	Diameter of the Universe (meters), Number of stars in the universe	10^24
Octillion	hella- (petitioned)	Diameter of Universe (mm) Mass of the earth (grams)	10^27
Nonillion		Number of bacteria on earth	10^30
Decillion		Mass of the sun (grams)	10^33
		Number of atoms in the universe, Volume of the observable universe (m^3)	10^80
Googol		Possible volume of whole universe (m^3)	10^100
Centillion			10^303
Googolplex			10^10^100

Why aren't millions called unillions? Or just an Illion? Mil- means 1000! I've always thought it must be because it should be 1000 thousands. Would numbers be more comprehensible without the -illions? The US national debt is 15 thousand thousand thousand thousands!

In grad school we proposed (probably it was Richard) a number system where there would be big numbers (since everyone knows what a big number is), and then a really big number would be a number that the number of digits was a big number. A really, really big number, then, is a number whose number of digits has a big number of digits. Quite sensible.


So the activity for the book could be matching quantities in different columns, though that doesn't give any opportunities for computation. Maybe a bit of a matching puzzle with some clues that require computation and comparison.

I cut things out of my table until it felt a little challenging, with enough structure to serve as an example for deduction and learning. I then put together some clues to help students fill in most, but leave a few for research, deduction or guessing.

The chart on the next page needs to be completed. The researcher has the data to fill in but no idea where to put it. Solve the puzzle of where to put the extra information. There are some blanks in the table, and those are shaded in. However some of the open spaces must get two comparisons, because there are too many for just one in each open space. 

Unfortunately, these are NOT in order. 

Names to fill in: Trillion, Quintillion, Centillion, Decillion, Octillion, Nonillion, Quadrillion, and Googol. 

Prefixes to fill in: yotta, hecto, peta, zetta, mega, and tera. 

Comparisons to fill in: Possible volume of whole universe (m3), Age of the universe (years), Mountains' height (meters), Width of USA (meters), Anacondas, Diameter of observable universe (mm), Mass of the earth (grams), Number of water molecules in a drop, One light year (meters), Number of bacteria on earth, Number of grains of sand on earth, Diameter of the Solar System (meters), Number of stars in the universe, and Redwood Trees’ height (meters) 

There were some weird facts the researcher remembered – maybe it will help you fill in the missing information! 

1. A googolplex has a googol zeroes. 

2. Thinking about word connections like tricycle and quadrilateral might help. 

3. The researcher remembers thinking that the number of grains of sand was exallent. 

4. Number of bacteria on earth is so big that there is about a sextillion for each human. (And there’s billions of humans!) 

5. A weird science measure is a mole. One mole of water is about 18 g, and has about 602 sextillion atoms. 

6. It would take about a million earths to have the same mass as the sun, even though the sun is made out of hydrogen and helium, mostly. 

7. It’s about 2000 km from Michigan to Florida. 

8. An average grain of sand is about 1mm wide. If you made a line out of all the sand on earth it would stretch for a light year! (The distance light can travel in a year.) 

9. The biggest official distance measurement is a yottameter, which is a billion times bigger than a petameter. 

10. In computers, a terabyte (TB) is 1000 GB, and a gigabyte is 1000 MB.

My favorite scientific notation/order of magnitude problems are Fermi problems, so I did put in a few of these for extensions.

Extensions
The brilliant physicist Enrico Fermi used to love posing crazy questions to his students and colleagues, so that now sometimes people call crazy estimation questions ‘Fermi Problems’ in his honor.

For example, he’d ask how many piano tuners there are in Chicago. He’d make a guess as to how many people, how many pianos, how many times they needed tuning and how many pianos one tuner could tune.

Try these Fermi problems and then make up your own! A tip is to think mostly about the powers of ten.

1. How many jars of peanut butter to fill up the Empire State Building?
2. How many photographs are in all the houses in your town?
3. How many middle schools are there be in the United States?
4. How many songs are downloaded in Michigan each day? 

Dr. Fermi said if you make enough guesses, some are over and some are under, and you would be surprised how accurate you might end up!