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Chapter 2 A Mathematical Toolkit

Chapter 2 A Mathematical Toolkit. Measurement Système Internationale d̀Unité́s/Metric System Accuracy and Precision Significant Figures Visualizing Data/Graphing. Objectives. 2.1 The Measure of Science Define the SI standards of measurement Use common metric prefixes

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Chapter 2 A Mathematical Toolkit

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  1. Chapter 2 A Mathematical Toolkit Measurement Système Internationale d̀Unité́s/Metric System Accuracy and Precision Significant Figures Visualizing Data/Graphing

  2. Objectives • 2.1 The Measure of Science • Define the SI standards of measurement • Use common metric prefixes • Estimate measurements and solutions to problems • Perform arithmetic operations using scientific notation

  3. Objectives • 2.2 Measurement Uncertainty • Distinguish between accuracy and precision • Indicate the precision of measured quantities with significant digits • Perform arithmetic operations with significant digits

  4. Objectives • 2.3 Visualizing Data • Graph the relationship between independent and dependent variables • Recognize linear and direct relationships and interpret the slope of a curve • Recognize quadratic and inverse relationships

  5. What is measurement? • Defined as a comparison of an unknown quantity to a known Standard. The measurement instrument must be standardized against the known standard • Every measurement has a value and a unit Standard kilogram of mass, officially known as the “International prototype of the kilogram” composed of platinum-iridium alloy, stored under glass in a vacuum since 1889

  6. Standards of Measurement When we measure, we use a measuring tool to compare some dimension of an object to a standard. Historical standard platinum iridium meter bar The meter now is defined as the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. The speed of light is c = 299,792,458 m/s For example, at one time the standard for length was the king’s foot. What are some problems with this standard?

  7. SI measurement • Le Système international d'unités • The only countries that have not officially adopted SI are Liberia (in western Africa) and Myanmar (Burma), but may now may using metric regularly • Metrication is a process that does not happen all at once, but is rather a process that happens over time. Why???? • Among countries with non-metric usage, the U.S. is the only country significantly holding out.The U.S. officially adopted SI in 1866. Information from U.S. Metric Association

  8. Measurement In Action On 9/23/99, $125,000,000 Mars Climate Orbiter entered Mar’s atmosphere 100 km lower than planned and was destroyed by heat. 1 lb = 1 N 1 lb = 4.45 N “This is going to be the cautionary tale that will be embedded into introduction to the metric system in elementary school, high school, and college science courses till the end of time.”

  9. Base Units of Measurement in SI Length Mass Time Temperature _______________ _________________ _______________ _____________

  10. 7 total SI Base Units • Electrical charge Coulomb • Amount of substance Mole • Luminous intensity Candela

  11. Derived Units • Units which are “calculated” using base units or other derived units Frequency Hertz S-1 Area square meter m2 Pressure Pascal N/m2 Energy/WorkJoule kg·m2/s2 Speed meter/sec m/s And many others…..

  12. Other Derived Units

  13. Some Tools for Measurement Which tool(s) would you use to measure: A. temperature B. volume C. time D. weight

  14. Learning Check Match L) length M) mass V) volume ____ A. A bag of tomatoes is 4.6 kg. ____ B. A person is 2.0 m tall. ____ C. A medication contains 0.50g aspirin. ____ D. A bottle contains 1.5 L of water.

  15. Learning Check What are some U.S. units that are used to measure each of the following? A. length B. time C. weight D. temperature

  16. Solution Some possible answers are A. length-- inch, foot, yard, mile B. volume-- cup, teaspoon, gallon, pint, quart C. weight-- ounce, pound (lb), ton D. temperature-- °F, Rankine

  17. Metric Prefixes • Kilo- means 1000 of that unit • 1 kilometer (km) = 1000 meters (m) • Centi- means 1/100 of that unit • 1 meter (m) = 100 centimeters (cm) • 1 dollar = 100 cents • Milli- means 1/1000 of that unit • 1 Liter (L) = 1000 milliliters (mL)

  18. You are responsible for “Giga” through “Nano”

  19. Learning Check Select the unit you would use to measure 1. Your height a) millimeters b) meters c) kilometers 2. Your mass a) milligrams b) grams c) kilograms 3. The distance between two cities a) millimeters b) meters c) kilometers 4. The width of an artery a) millimeters b) meters c) kilometers

  20. Scientific Notation

  21. Scientific notation consists of two parts: • A number between 1 and 10 • A power of 10 N x 10x

  22. Scientific Notation • “Writing a number as a power of 10.” • Why? It makes very large and very small numbers more manageable to write and use. • Also, all digits in scientific notation (1-10) are considered to be significant and are clearly shown (no question about significance) • Example 1200 mg has 2 sig.figs, written as 1.2 X 103 .00000230g has 3 sig. figs written as 2.30 X 10-6 • Rule of thumb: Use when number is greater than 1000 or smaller than 0.001 Or, you may always use it!

  23. Writing in scientific notation • Move the decimal point in the original number so that it is located to the right of the first nonzero digit. • Multiply the new number by 10 raised to the proper power that is equal to the number of places the decimal moved. • If the decimal point moves: • To the left, the power of 10 is positive. • To the right, the power of 10 is negative.

  24. Examples • Given: 289,800,000 • Use: 2.898 (moved 8 places) • Answer:2.898 x 108 • Given: 0.000567 • Use: 5.67 (moved 4 places) • Answer:5.67 x 10-4

  25. Learning Check • Express these numbers in Scientific Notation: • 405789 • 0.003872 • 3000000000 • 2 • 0.478260

  26. To change scientific notation to standard form… • Simply move the decimal point to the right for positive exponent 10. • Move the decimal point to the left for negative exponent 10. (Use zeros to fill in places.)

  27. Example • Given: 5.093 x 106 • Answer:5,093,000(moved 6 places to the right) • Given: 1.976 x 10-4 • Answer:0.0001976(moved 4 places to the left)

  28. 2.2 Measurement Uncertainty

  29. Measurement needs to be precise and accurate • Precision: • How closely multiple measurements of the same quantity come to each other. • This will depend on the measuring device. For example, a thermometer that shows degrees in tenths is more precise than one that only shows single degrees.

  30. Measurement and numbers • Measurements consist of two parts • The number itself (the quantity) • The units (the nature of the quantity measured) • There are two kinds of numbers • Counted or defined - exact, not subject to estimate. Ex: number of eggs in a carton. • Measured - always carries some amount of uncertainty because measurement involvesestimation. The size of uncertainty depends on the precision of the measuring device AND the skill of the person using the device

  31. 1 cm 2 cm 3 cm 4 cm 5 cm Estimation in measurement • When we measure, the quantity rarely falls exactly on the calibration marks of the scale we are using. • Because of this we are estimating the last digit of the measurement. • For instance, we could measure “A” above as about 2.3 cm. We are certain of the digit “2”, but the “.3” part is a guess - an estimate. • What is your estimate for B and C? A B C

  32. 1 cm 2 cm 3 cm Higher Precision • A measuring device with more marks on the scale is more precise. I.e., we are estimating less, and get a more accurate reading. Here we are estimating the hundredths place instead of the tenths. • Here, we can measure A as 1.25 cm. Only the last digit is uncertain. • Usually we assume the last digit is accurate ± 1. • How would you read B, C, and D? A B C D

  33. Reading a Meter stick . l2. . . . I . . . . I3 . . . .I . . . . I4. . cm First digit (known) = 2 2.?? cm Second digit (known) = 0.7 2.7? cm Third digit (estimated) between 0.05- 0.07 Length reported = 2.75 cm or 2.74 cm or 2.76 cm

  34. Zero as a Measured Number . l3. . . . I . . . . I4 . . . . I . . . . I5. . cm What is the length of the line? First digit 5.?? cm Second digit 5.0? cm Last (estimated) digit is 5.00 cm

  35. Accuracy • Refers to how close a measurement comes to the true or accepted value. • This depends on both the measuring device and the skill of the person using the measuring device. • This can be determined by comparing the measured value to the known or accepted value.

  36. A graduated cylinder: 41.2 mL (3 sig figs = very precise) 41.0 50 mL Graduated cylinder 100 mL Beaker A beaker: 40. mL (2 sig figs = not as precise) 50

  37. Accurate or Precise? • What is the temperature at which water boils? • Measurements: 95.0°C, 95.1°C, 95.3°C • True value: 100°C Precise! (but not accurate)

  38. Accurate or Precise? • What is the temperature at which water freezes? • Measurements: 1.0°C, 1.2°C, -5.0°C • True value: 0.0°C Accurate! (it’s hard to be accurate without being precise)

  39. Accurate or Precise? • What is the atmospheric pressure at sea level? • Measurements: 10.01 atm, 0.25 atm, 234.5 atm • True value: 1.00 atm Not Accurate & Not Precise (don’t quit your day job)

  40. Accurate or Precise? • What is the mass of one Liter of water? • Measurements: 1.000 kg, 0.999 kg, 1.002 kg • True value: 1.000 kg Accurate & Precise (it’s time to go pro)

  41. Can you hit the bull's-eye? Three targets with three arrows each to shoot. How do they compare? Both accurate and precise Precise but not accurate Neither accurate nor precise Can you define accuracy and precision?

  42. Accuracy or Precision? When deciding on accuracy, precision, both, or neither….it is quantitative data (numerical), not qualitative (descriptive) The recipe calls for 25 chocolate chips per cookie. The cookies analyzed have 34, 35, and 32 respectively. The percent NaCl is 99%, 99%, and 98%. The number of grams of KF required is 0.04 g. The amounts used were 0.038, 0.039, 0.041, and 0.040. To win, Henry must earn 500 points. In his three trials, he earned 400, 480, and 395 points.

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