Questions:

1. Why estimate?
2. What do we need to think about to estimate well?
3. What are the characteristics of good estimators?
4. When is estimation useful?  When are exact answers necessary?
5. When are numbers “in the real world” estimates and not exact?

What are the characteristics of good estimators?

Good estimators have:

• Good recall of the basic facts
• Ability to change data to a mentally manageable form
• Tolerance for error/approximation
• Ability to adjust the initial estimate when more information becomes available
• Ability to translate a problem to a more mentally manageable form

(Shoen, et. al. 1984)

Five strategies for computational estimation (Reys, 1986)

1. Front-end strategy: focuses on left-most or highest place value digits (e.g., 267 + 521, add the front digits and get 7 [700])
2. Rounding strategy: 267 + 521; change to 300 and 500; estimate is 800.
3. Clustering strategy:  used when a set of numbers are close to each other in value.
4. Compatible numbers strategy: when using the compatible numbers strategy, children adjust the numbers so that they are easier to work with.  For example, using compatible numbers in division involves altering the divisor, the dividend, or both so that they are easy to work with mentally. (e.g. 300/11 might be thought of as 300/10)
5. Special numbers strategy: The special numbers strategy involves looking for numbers that are close to “special value” (e.g. friendly) that are easy to work with, such as one-half or powers of ten.

Everyday Situations that Involve Computation and/or Estimation

What makes someone a good estimator?

• Mental flexibility
• Good number sense
• Confidence that mathematics makes sense

Why estimate?

Estimates are easier to understand.

• Estimates may be clearer [rather than saying 72,729 students, we would say approximately 73,000]
• Estimates may be used for greater consistency.  [used in tables, charts, and graph to show data uniformly]

Estimates can help in problem solving [What is a reasonable answer?]

Estimates are necessary in many situations because exact values are unobtainable

• A number may simply be unknown [e.g., measurements of economic conditions]
• A quantity may be different each time it is measured [temperatures, populations, air pressures, etc.]
• Physical measurements are never exact [length of a side of paper is not exactly 11 inches long]
• Getting an exact value may be too expensive [taking samples and using statistics to generate results]
• Decimal or fractional results may not make sense [e.g. 3 candy bars for $1, you wouldn’t say 33 1/3 cents for each candy bar; you would about 34¢ for one]
• Some situations require a built-in margin of error, so quantities are overestimated [you overestimate the costs of the items you are buying to make sure you have enough money to pay the cashier]
• Numbers may not be in a form suitable for computation.  [To add irrational numbers like the square root of 2 or pi, you need to replace them with rational approximations.] 

When are numbers estimates and not exact numbers?

• A certain dinosaur bone is 65 million years old.
• The population of the US is 270,000,000.
• The number of hungry children in the US is 12 million.
• The area of the Sahara Desert is 3,320,000 square miles
• The mean July temperature in Tucson, Arizona, is 86 degrees.
• Jonestown lies 55 miles from the nearest airport
• My office is 12 feet by 9 feet.

Numbers are estimates when:

1. The exact value is unknown, for example, predictions and numbers that are too large or difficult to determine.
2. The value is not constant, for example population and barometric air pressure.
3. There are limitations in the measurement, for example, the length of a desk and the area of a pond.

When do we use estimation and when do we do exact computation?

• Making a budget—cost of college, cost of food per month.
• Determining the affordability of a trip or vacation—camping trip, trip to Europe.
• Deciding which to buy—whether to buy a new car or a used car
• Determining time—how long to get to …
• Determining if we have enough money—being at the grocery store when short on cash
• Deciding how much the tip should be at a restaurant.
• Determining how long the paper will take to write or project will take to complete.

When do you over estimate and when do you under estimate?

• Depends on the situation
• Close to actual—budget when short of cash
• Over-estimate—money needed for a purchase
• Under-estimate—the value of a stock’s potential growth when buying stock

Being able to perform mental math and to estimate requires:

• The ability to apply base 10 and place value concepts
• The ability to compose and decompose the numbers
• The ability to apply properties of the operations, especially the commutative, associative and distributive properties