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Quantitative Literacy and the Humanities

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Student Learning Objective: Estimate using Arithmetic

The habit of making a quick, rough estimate can be invaluable in understanding the magnitude of numbers in everyday life, past and present.

Activity or Assignment:
Make a rough estimate in order to understand the magnitude of an issue, even if you lack a precise number.
  • How many trees were needed to provide energy to a village in 1200?
  • How many people lived in ancient Rome?
  • How many orphans were there after World War I?
  • How long would it take to travel 100 miles in 1789? in 1870? in 1940?

In order to estimate, you probably will need to...

Get used to uncertainty. 
You will probably not be able to pin down all the pieces needed for your estimation.  That's ok--the goal is to get within an order of magnitude.  You can continually refine your estimate later.

Expect that there will be multiple ways to do the job.  There is not one right way to derive an estimate, but some ways are more reasonable than others.  Fortunately, humanities folks are used to seeing multiple paths.

Distinguish between precision and accuracy.  Some measurements are precise, without being accurate, and vice versa. 

  • Precise and accurate statement: The population of France in 1914 was 39.6 million.
  • Accurate statement, but less precise: The population of France in 1914 was about 40 million.
  • Precise statement, but less accurate: The population of France in 1914 was 54.3 million.
When estimating, an accurate but less precise number may be the best.  When calculating, a precise number is important. And it's crucial not to be fooled into thinking that a precise number is necessarily accurate.

Memorize a few common numbers and percentages.  Identify a few benchmark numbers or percentages that will be commonly used in your course for estimation.
  • The population of the U.S. in 2014 is about 315 million. 
  • In 2012, the poverty rate in the U.S. was about 15 percent.
  • 1.4 million French men were killed in World War I. 

Think about the dimensions of an issue. 
Are you determining an area, a cost, a human relationship?  Think step by step--number by number--to build your estimate.  Scale up your estimates using multiplication, scale down using division.  Sometimes you'll need to use addition and subtraction.

State your assumptions.  State all assumptions about the numbers, even if that assumption is "this is just a guess."  List the sources for each step so that you'll know which pieces are reasonable guesses, and which are based in evidence.  Refine the estimate with better data later.

Convert from one unit to another, through multiplication and division.
  Unit conversion requires an understanding of the relationship between two measures. 
  • The phrase "30 miles per gallon" means that for every one gallon, a car can travel 30 miles.  It is expressed mathematically as 30 miles/1 gallon. 
  • If we want to convert gallons to liters, we need to multiply so that the gallon units cancel out.  In this case, multiple 30 miles/1 gallon x 1 gallon/3.785 liters = 7.92 miles per liter. 
  • But in estimation mode, you might divide 30 by 4 to get 7.5.  Depending on how rough you are allowing your estimations to be, you might simply divide 30 by 5 to get 6.  It's not very precise, but it is accurate to an order of magnitude.

Multiply using percentages or simple ratios.  If you are estimating that half of the total population is male, then multiply the total population by .5 (50%) or divide by 2 (a ratio of 1/2) to obtain an estimate.

Convert between trillions, billions, millions, and thousands.  Multiply by 1,000 to convert to the next smaller unit, for example from millions to thousands (3 million = 3,000 thousands). 

For instance, if you want to quickly compare 63 million with 2.4 billion, you can turn 2.4 billion into 2,400 million to make a more intuitive comparison (63 vs. 2,400) at the correct order of magnitude.

Refine estimates and provide context with research.  See the next page in this path, "Use Evidence to Make an Argument."

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