Reporting QuantitiesAccuracy and Limitations

Words to KnowAccuracy – 1 freedom from error : 2 conformity to a standard.Estimate– to judge the approximate value, size, or cost of on the basis of experience or observation rather than actual measurement .Measure –to mark or fix in multiples of a specific unit.Order of Magnitude – the power of ten to which a value is rounded.Power of ten – the place value in a base 10 system.Precision – the quality or state of being precise Rounding –to drop any digits to the right of a given decimal place and increase the last remaining digit by 1 if the first dropped digit is 5 or greater.

EstimatingThe late great physicist Enrico Fermi used to solve problems by making educated guesses or estimates.

In fact, we all use estimation in our daily lives. Like when we estimate how much the items in our grocery cart will cost or how long it will take us to do our homework.

Say you want to estimate the number of cans of soda drunk in San Francisco in one year. You’ll come pretty close if you guess that there are about 800,000 people in S.F. and that one person drinks on average about 100 cans per year. So, around 80,000,000 cans are consumed every year.

Sure, this answer is not exactly right, but it is likely not off by more than a factor of 10 (i.e., an “order of magnitude”). That is, even though we guessed, we’re going to be in the ballpark of the right answer. This is often the first step in working out any math problem.

Estimating Vs. MeasuringSome situations can only be predicted by using estimation, but in other situations it is necessary to actually measure.

The biggest problem with measuring is how accurately we take the measures. The accuracy of any calculations based on those measures depends on how accurately the parts were measured to begin with.

Just like shooting at a target, the smaller the target the more you need to aim.

When accuracy counts you must be as precise as possible in your measurements.

Surveying or land surveying is the technique and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them.

Surveying has been an essential element in the development of the human environment since the beginning of recorded history.

It is required in the planning and execution of nearly every form of construction.

Its most familiar modern uses are in the fields of building, transportation, mapping, communication, and the definition of legal boundaries for land ownership.

Think about place values and measurement: How accurate do you think they could measure

the distance between two cities 5,000 years ago? What about 100 years ago?

What level of accuracy is possible today? Why is measuring different today?

An all-female surveying crew in Idaho, 1918

Putting a limb in 4 different states? Well, maybe you thought you were ...

SALT LAKE CITY — Tourists who think they're putting a hand or foot in each of four U.S. states at the Four Corners area are apparently missing the mark — by about .25 miles.

National Geodetic Survey officials say the Four Corners marker showing the intersection of Arizona, Colorado, New Mexico and Utah is about 2.5 miles west of where it should be, the Deseret News reported.

The only place in the United States where four state boundaries come together was first surveyed by the government in 1868 during the initial survey of Colorado's southern boundary. The survey was inaccurate.

Officials said Monday the accurate location lies to the east of U.S. 160 in Colorado and northeast of the San Juan River as it flows into New Mexico.

Associated PressApril 20, 2010

According to Dave Doyle of the National Geodetic Survey, the monument marking the intersection of Arizona, Colorado, New Mexico and Utah is actually only about 1,807 feet east of where it should have been placed in 1875. Doyle says the monument's location has been legally adopted by all the states as the official corner.

• What level of accuracy did the Associated Press news report?

The news article said the difference was 2.5 miles.

• What level of accuracy did Mr. Doyle from the National Geodesic Survey report?

Mr. Doyle reported the difference as 1870 feet, which is actually only about 1/3 of a mile.

What effect does this new information about four-corners have on calculating the area of the state of Utah?

The area of Utah is a calculation based on the length of its borders. According to the old placement of the four-corners, the southern border of Utah was ____________. The new length is __________________, 1870 feet shorter. The calculations for the new area is smaller than before.

Fortunately for Utah, all the states involved have agreed to keep the old boundaries for simplicity.

Calculations are affected by the precision of the measurements.

Just like the area of Utah, any time we put our measurements into mathematical formulas the accuracy of our answers are dependent upon the precision of our measurements.

So, unless we’re able to measure in molecules, we can’t really say that we have the exact answer. There is always a more precise measurement we could have used.

What level of accuracy would you need?

A veterinarian gives tranquilizers based upon the weight of an animal. If an animal receives too much it will die.

What units would you use to weigh an adult elephant? Why? Since adult elephants easily weigh more that a thousand pounds, the difference of one pound is small enough that the elephant would be fine with just being weighed in pounds (or kilograms.) What units would work for a dog? Why? Dogs are much smaller than elephants so the margin of error for the anesthetic would be smaller. For a large dog like a Great Dane we could use one decimal place, but for a Chihuahua we would need to use at least two decimal places. What about a hamster? Why? Hamsters can weigh between two to six ounces. This is very small! Veterinarians would use the most precise measure available , at least three or four decimal places.

What level of accuracy would you need?

Missile trajectory is based on the measured angles.

Shooting a basket in a basketball game for fun. While we try to do our best we’re just there for fun. We probably would estimate a range of 5-10 degrees. Shooting a free throw in a championship basketball game. A lot more pressure to get the ball in the hoop! We would most likely try to get within 1-3 degrees of the perfect shot. A missile being launched at a practice target 2 miles away. Laser targeting systems have been developed because of the need to only blow up what your aiming at. Even though it’s a practice shot, we need to use hundreths or thousandths of a degree to hit the target. Launching a rocket to Mars. The farther away we are shooting for, the more accurate we need to be. This is because what is a small error aiming at something close, would be a massive error when aiming at Mars. Millionths or even billionths of a degree would be the accuracy needed for this trip.

Fractions vs. Decimals

A general rule is that if a problem is given in fractions, the answer should be in fractions. Same with decimals or radicals.

This is especially true when we look at repeating decimals. If we round a repeating decimal, it is not as accurate as the fraction equivalent.13>0.3

Irrational numbers can’t be written as a fraction. To maintain accuracy we usually leave them in a symbolic or radical form.

So to say the area of a certain circle is 6.28 is not as accurate as saying it is

Actual vs. Rounded