SI Measurement System Introduction to Engineering Design © 2012 Project Lead The Way, Inc.

The International System of Units (SI) Presentation Name Course Name Unit # – Lesson #.# – Lesson Name The International System of Units (SI) The International System of Units (SI) is a system of units of measurement consisting of seven base units Mostly widely used system of measurement The United States is the only industrialized nation that has not adopted the SI system Unit Name Symbol Measurement meter m length kilogram* kg mass second s time ampere A electric current kelvin K thermodynamic temperature candela cd luminous intensity mole mol amount of substance The abbreviation SI is from the French Systeme Internationale d’unites. Note that even though kilogram has the kilo- prefix, it is defined as a base unit and is used in definitions of derived units.

The U S Customary System Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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

The International System of Units Presentation Name Course Name Unit # – Lesson #.# – Lesson Name The International System of Units Often referred to as the metric scale Prefixes indicate an integer power of 10 Power of 10 Prefix Abbreviation 101 deca- da 102 hecto- h 103 kilo- k 106 Mega- M 109 Giga- G 1012 Tera- T Power of 10 Prefix Abbreviation 10-1 deci- d 10-2 centi- c 10-3 milli- m 10-6 micro- µ 10-9 nano- n 10-12 pico- p Note that the kilo- prefix in kilogram indicates that a kilogram is 10^3 = 1000 grams. The fact that the kilogram is a base unit does not affect the meaning of the prefix but allows for the use of the kilogram as a unit in the definition of derived units. [These and additional prefixes are shown on the PLTW Engineering Formula Sheet. Students do not need to write these prefixes in their notes.]

Common Items: Size Comparison Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Common Items: Size Comparison Students can understand more when you relate to common objects.

Common Items: Size Comparison Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Common Items: Size Comparison Students can understand more when you relate to common objects.

Recording Measurements Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Recording Measurements A measurement always includes a value A measurement always includes units A measurement always involves uncertainty A measurement is the best estimate of a quantity Be sure to always include units when recording measurements. An important part of a measurement is the numerical value of the measurement; however, the value is meaningless with units. A measurement is never certain. There are always errors in measurements, even if they are very small. It is important to know the level of error that may be inherent in a measurement. A measurement is only useful if a value is associated with units and the uncertainty of the value is understood.

Recording Measurements Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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?

Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Significant Digits Scientists and engineers often use significant digits to indicate the uncertainty of a measurement Significant digits are digits in a decimal number that carry meaning indicating the certainty of the value All digits you record for a measurement are considered significant Include all certain digits in a measurement and one uncertain or estimated digit

Significant Digits General Rules Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Significant Digits General Rules Digital Instruments – Read and record all digits, 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 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.

Metric Scale A typical metric scale often includes a 30+ centimeter graduated scale Each centimeter is graduated into 10 millimeters

The Millimeter The millimeter is the smallest increment found on a typical metric scale 1 mm

The Millimeter The next larger marking on a metric scale shows 5 millimeters 5 mm

The Millimeter Largest markings on a metric scale represent centimeters (cm) These are the only marks that are actually numbered 1 cm = 10 mm

Measurement: Using a Decimal Scale Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Measurement: Using a Decimal Scale How long is the rectangle? Let’s look a little closer Let’s look a little closer. [click]

Measurement: Using a Decimal Scale Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Measurement: Using a Decimal Scale How long is the rectangle? You can tell that the length of the rectangle is between 3 and 4 centimeters. [click] Because the scale is incremented in millimeters, you can also be certain that the measurement is between 3.8 and 3.9 centimeters (assuming the scale is accurate). So you are certain that the first digit after the decimal, the tenths place, is 8. [click] Because there are no tick marks between millimeter marks, you can only estimate the hundredths place of the measurement. Perhaps you would estimate 3.83 or 3.84 cm. The last digit is an estimate – your best guess as to where, within the millimeter distance, the measurement falls.

Recording a Measurement Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Recording a Measurement How long is the rectangle? Remember the general rule Decimal Scaled Instruments – Record all digits that you can certainly determine from the scale markings and estimate one more digit Best Estimate = 3.84 cm

Recording a Measurement Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Recording a Measurement How long is the rectangle? Remember the general rule Decimal Scaled Instruments – Record all digits that you can certainly determine from the scale markings and estimate one more digit Best Estimate = 3.84 cm Since the measurement is certainly between 3.8 and 3.9, you can be certain that the 3 and the 8 are correct. However, the 4 is an estimate based on your best guess. This number has three significant digits : 3, 8, and 4. Certain

6.33 cm 3 Your Turn How would you record the length of this rectangle? Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Your Turn How would you record the length of this rectangle? How many significant digits? 6.33 cm 3 [click to zoom in on scale. Allow student to estimate the measurement.] [click to reveal estimate of 6.33 cm.] Answers may vary, but a good estimate that reflects the appropriate precision is 6.33 cm. Other good estimates are 6.32 or 6.34 cm. [Click and allow students to answer question, then click again to reveal answer of 3.]

Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Your Turn [The teacher will pass out Activity 3.1a Linear Measurement with Metric Units and have students complete the metric linear measurements.] Record each measurement in centimeters using the appropriate number of significant digits.

Fractional Length Measurement Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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.

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.

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

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

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

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

Measurement: Using a Fractional Scale Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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.]

Measurement: Using a Fractional Scale Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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

Measurement: Using a Fractional Scale Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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.

Measurement: Using a Fractional Scale Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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  

Recording a Measurement: Using a Fractional Scale Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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 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 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. = .0625 in.

Recording a Measurement: Using a Fractional Scale Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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.

Your Turn Record each measurement in fractional and decimal inches. Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Your Turn Have students complete the U S Customary linear measurements required in Activity 1.3.1 Linear Measurement. Record each measurement in fractional and decimal inches.

Precision and Accuracy Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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

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

Recording Measurements Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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.