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# 3.1b

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• Students can understand more when you relate to common objects.
• 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?
• 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).
• Laying tile involves accuracy, so significant figures are useful. Let&apos;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 11.875 inches. That implies accuracy down to a thousandth of an inch. That isn&apos;t true because the tape can&apos;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.
• 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.
• In this presentation we will concentrate on linear measurements of length.
• [click to zoom in on scale. Allow student to estimate the distance then click again.]
• 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).
• [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.
• 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]
• 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 2.15625 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 theone 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.
• 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.
• Have students complete the U S Customary linear measurements required in Activity 1.3.1 Linear Measurement.
• ### 3.1b

1. 1. US Customary Measurement System
2. 2. 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 slug force lb pound time s second thermodynamic temperature F Fahrenheit degree
3. 3. Common Items: Size Comparison
4. 4. 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
5. 5. 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 High Accuracy Low Precision Low Accuracy High Precision High Accuracy High Precision
6. 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
7. 7. 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
8. 8. 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
9. 9. 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
10. 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. 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. 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. 13. The Inch • Half of a quarter = 1/8 inch 1/8 3/8 7/85/8
14. 14. The Inch • Half of an eighth = 1/16 inch 1/16 3/16 5/16 13/16 7/16 11/16 9/16 15/16
15. 15. Measurement: Using a Fractional Scale • How long is the rectangle? • Let’s look a little closer
16. 16. Measurement: Using a Fractional Scale • How long is the rectangle? • What fraction of an inch does this mark represent? 1/21/4 1/8 3/16
17. 17. 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
18. 18. 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 5Find the average of the two measurements
19. 19. Recording a Measurement: Using a Fractional Scale • How long is the rectangle? • Remember the General Rule – Fractional Scaled Instruments require special consideration Are 6 significant digits appropriate??? • 1/16 in. = .0625 in.
20. 20. 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.
21. 21. Your Turn Record each measurement in fractional and decimal inches.