1. US Customary Measurement System

2. The U S Customary System
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The U S Customary System
System of measurement used in the United States
Similar to the British Imperial System of Measurement, but not identical
Common U S Customary Units
Measurement
Symbol
Unit
length
in.
inch ft
foot mi
mile
mass
slug
force lb
pound
time s
second
thermodynamic temperature F
Fahrenheit degree

3. Common Items: Size Comparison
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Common Items: Size Comparison
Students can understand more when you relate to common objects.

4. Recording Measurements
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Recording Measurements
A measurement always includes units
A measurement always includes error
A measurement is the best estimate of a quantity
Scientists and engineers often use significant digits to indicate the uncertainty of a measurement
Indicate the accuracy and precision of your measurement
Be sure to always include units when recording measurements.
There are always errors in measurements, even if the errors are very small. It is important to know the level of error that may be inherent in a measurement.
It is important to understand how accurate the recorded measurement is. For instance, if you know an object measures 3 inches in length, you can’t really be sure if the object is actually somewhat longer or shorter than 3 inches. Perhaps the object is 3 1/16 inches long, or 2 15/16 inches long. If the object must fit into a 3 inch space – which again may be somewhat larger or somewhat smaller than the recorded measurement – how can you be sure the part will fit?

5. Precision and Accuracy
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Precision and Accuracy
Precision (repeatability) = The degree to which repeated measurements show the same result
Accuracy = The degree of closeness of measurements of a quantity to the actual (or accepted) value
Although precision and accuracy are often confused, there is a difference between the meanings of the two terms in the fields of science and engineering.
Precision indicates how close together repeated measurements of the same quantity are to each other. So a precise bathroom scale would give the same weight each time you stepped on the scale within a short time (even if it did not report your true weight).
Accuracy indicates how close measurements are to the actual quantity being measured. For example, if you put a 5 pound weight on a scale, we would consider the scale accurate if it reported a weight of 5 pounds.
A target analogy is sometimes used to differentiate between the two terms. Consider the “arrows” or dots on the targets to be repeated measurements of a quantity.
[click] The first target shows that the arrows (or repeated measurements) are “centered” around the center of the target, so on the whole, the measurements are fairly close to the target (actual) measurement, making the measuring device accurate. But the repeated measurements are not close to each other, so the precision of the measuring device is low.
[click] The second target show that the arrows (or repeated measurements) are close together, so the precision is high. But the “center” of the measurements is not close to the target (actual) value of the quantity.
What should the target look like if the measurement is both highly accurate and highly precise? [allow student to answer then click]. The third target shows both precision (because the measurements are close together) and accuracy (because the “center” of the measurements is close to the target value).
High Accuracy
Low Precision
Low Accuracy
High Precision
High Accuracy
High Precision

6. Recording Measurements
Ideally, a measurement device is both accurate and precise
Accuracy is dependent on calibration to a standard
Precision is dependent on the characteristics and/or capabilities of the measuring device and its use
Record only to the precision to which you and your measuring device can measure

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Significant Digits
Accepted practice in science is to indicate uncertainty of measurement
Significant digits are digits in a decimal number that carry meaning contributing to the uncertainty of the quantity
The digits you record for a measurement are considered significant
Include all certain digits in a measurement and one uncertain digit
Note: Fractions are “fuzzy” numbers in which significant digits are not directly indicated
Laying tile involves accuracy, so significant figures are useful. Let's say you want to know how wide 10 tiles would go. You measure one tile and you get 11 7/8 inches on one side of the tape measure and 30.2 centimeters on the other side. If you convert 11 7/8 inches to a decimal fraction, you get inches. That implies accuracy down to a thousandth of an inch. That isn't true because the tape can't measure to the nearest thousandths of an inch, only to the nearest 16th of an inch. So significant numbers are easier to determine when a measurement is done with decimal fractions.

8. Recording Measurements
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Recording Measurements
General Rules
Digital Instruments: Read and record all the numbers, including zeros after the decimal point, exactly as displayed
Decimal Scaled Instruments: Record all digits that you can certainly determine from the scale markings and estimate one more digit
Preferred over fractional scaled instruments
Fractional Scaled Instruments: Need special consideration
We will concentrate on measuring and recording linear length measurements in this presentation, but the techniques discussed apply to all types of measurements.
We’ll look at an example of a decimal scaled instrument first – a metric scale. Later we’ll talk about a fractional scale – a ruler divided into fractions of inches.

9. Fractional Length Measurement
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Fractional Length Measurement
A typical ruler provides
A 12 inch graduated scale in US Customary units
Each inch is graduated into smaller divisions, typically 1/16” increments
In this presentation we will concentrate on linear measurements of length.

10. The Inch
The divisions on the U S Customary units scale are easily identified by different sized markings. The largest markings on the scale identify the inch.

11. The Inch
Each subsequently shorter tick mark indicates half of the distance between next longer tick marks.
For example the next smaller tick mark indicates half of an inch = ½ inch
1/2

12. The Inch
Half of a half = ¼ inch. An English scale shows ¼ inch and ¾ inch marks.
All fractions must be reduced to lowest terms.
1/4
3/4

13. The Inch
Half of a quarter = 1/8 inch
1/8
3/8
5/8
7/8

14. The Inch Half of an eighth = 1/16 inch 1/16 5/16 9/16 13/16 3/16 7/16
11/16
15/16

15. Measurement: Using a Fractional Scale
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Measurement: Using a Fractional Scale
How long is the rectangle?
Let’s look a little closer
[click to zoom in on scale. Allow student to estimate the distance then click again.]

16. Measurement: Using a Fractional Scale
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Measurement: Using a Fractional Scale
How long is the rectangle?
What fraction of an inch does this mark represent?
3/16
1/4
1/2
You can tell that the length of the rectangle is between 2 and 3 inches. So the first inch digit of the number is certainly 2.
[slowly click through ½, ¼, and 1/8 indicators. Then click to reveal the question. Allow students to answer, then click again.
[click] Because the scale is incremented in 16ths of an inch, you can also be certain that the measurement is between 2 1/8 in. and 2 3/16 in. (assuming the scale is accurate).
1/8

17. Measurement: Using a Fractional Scale
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Measurement: Using a Fractional Scale
How long is the rectangle?
1/8
3/16
What is the midpoint of 2 1/8 and 2 3/16?
5/32
[click] We may be tempted to estimate the length to be right in the middle. What is the midpoint between these two tick marks? [Allow students to answer, then click]. 2 5/32.

18. Measurement: Using a Fractional Scale
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Measurement: Using a Fractional Scale
How do we determine that 5/32 is midway between 1/8 and 3/16?
Convert each fraction to a common denominator: 32
To determine the midpoint on the scale, convert both endpoint fractions to 32nds of an inch. This is done by multiplying the fraction by another fraction that is equal to one. In other words, multiply by a fraction with the same factor in the numerator and the denominator. In order to convert a fraction in eighths to a fraction in terms of 32nds, multiply by 4 / 4. [click]
To convert a fraction in terms of eights of an inch to 32nds of an inch, multiply by 2/2. [click]
The midpoint is represented by the average of these two numbers. [click] 5
Find the average of the two measurements

19. Recording a Measurement: Using a Fractional Scale
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Recording a Measurement: Using a Fractional Scale
How long is the rectangle?
Remember the General Rule
Fractional Scaled Instruments require special consideration
Since the measurement is certainly between 2 and 3 inches, you can be certain that the 2 is correct.
You are also certain the measurement is between 2 1/8 = 2 2/16 and 2 3/16. What does that mean with respect to significant figures? Significant digits don’t really apply to fractions, so let’s convert the fraction to a decimal.
The decimal equivalent of 2 5/32 is inches. If we assume that all of these figures are significant, it would suggest that we are certain of the measurement to the nearest ten thousandth of an inch and that we estimated the one hundred thousandth of an inch. There is NO WAY we can be that accurate with a standard ruler that shows 1/16 inch increments.
Since we can be certain of the measurement to the nearest 1/16 inch, which is equivalent to approximately 0.06 inches, estimates to the nearest 0.01 in would be an estimate.
Are 6 significant digits appropriate???
1/16 in. = in.

20. Recording a Measurement: Using a Fractional Scale
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Recording a Measurement: Using a Fractional Scale
For the standard ruler marked in 1/16 inch increments (least count = 1/16 in.)
Record fraction measurements to the nearest 1/32 inch
Record decimal equivalent to the nearest hundredths of an inch
Record with your data
The least count of the scale (1/16 in.)
The increment to which measurements are estimated (nearest 1/32 in.) 2 5 32
in.
2.16 in.
For our purposes here, we will record measurements made on a fractional scale (incremented to the 1/16 inch) to the nearest 1/32 inch. You may record the fraction in lowest terms (e.g., 6/32 = 3/16 in lowest terms).
When converting the number to a decimal, we will record the number to the hundredths place such that the tenths place is certain and the hundredths place is estimated.
Because you can not use fractions to indicate the precision to which you are measuring, record the least count of the scale and the increment to which you are estimating your measurements. In the case of a standard ruler marked in 1/16 in. increments, the least count is 1/16 inch. You will then estimate to the nearest 1/32 in.

21. Your Turn Record each measurement in fractional and decimal inches.
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Your Turn
Have students complete the U S Customary linear measurements required in Activity Linear Measurement.
Record each measurement in fractional and decimal inches.