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# Measuring sf-scinot

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• Holt book says you can estimate with more precision by estimating one extra digit than the actual measure – so this could be 4.40 cm – do you teach it this way?
• Holt book says you can estimate with more precision by estimating one extra digit than the actual measure – so this could be 4.40 cm – do you teach it this way?
• Check answers here.
• Check answers here.
• ### Measuring sf-scinot

1. 1. TWO METHODS OF OBSERVATIONQualitative observations describe Quantitative observations measure • Slightly rough surface • Mass = 23.6 grams • Round • Volume = 38.4 ml • Grayish brown • Density = 0.615 g/ml
2. 2. QUANTITATIVE MEASUREMENT Quantitative data requires a standard unit of measure • SI unit: uses meters, kilograms, seconds • Principal system used in scientific work; What we will be using in class • English units: uses feet, pounds, seconds Length Mass TimeSI Unit(System Meters (m) Kilograms (kg) Seconds (s)International)Measuring devices Electric balance, spring Ruler scale, triple beam stopwatch Measuring stick balanceOriginal Standard 1/10,000,000 of distance Mass of 0.001 cubic meters 0.001574 average solar days from equator to North Pole of waterCurrent Standard 9,192,631,770 times the period of a Distance traveled by light in Mass of a specific platinum- radio wave emitted from a cesium- a vacuum in 3.34 x 10-9 s iridium alloy cylinder 133 atom
3. 3. SI UNIT PREFIXES • In the SI units, the larger and smaller units are defined in multiples of 10 from the standard unit. • See table 1-4 (pg. 11) for complete tableprefix mega kilo hecto Deka Deci Centi Milli micro Meter Gram Value 106 103 102 101 Second 10-1 10-2 10-3 10-6 Liter mAbbrev- g M k h da d c m μ (“mew”) iation s l
4. 4. PRECISION & ACCURACY• Based on the definitions of precision and accuracy, describe the following pictures as being: 1) precise or imprecise, 2) accurate or inaccurate
5. 5. UNDERSTANDING SCIENCE• In physics, we typically understand our world through measurement and developing quantitative relationships (equations) between physical quantities• There is uncertainty in all measurements: • Blunders, human error • Limitation in accuracy of measuring device • Inability to read device beyond smallest division • Manufacturing of device• Two factors to consider in quantitative measure: the accuracy & precision
6. 6. MEASUREMENTS IN SCIENCE• Precision: the degree of exactness with which a measurement is made.• Two ways to describe precision: 1. How close measured values are to one another • Ex: Given 2 sets of data: Precise Imprecise Precise 7.35ml 7.35ml 7.35ml 7.33ml 6.94ml 7.33ml 7.34ml 8.23ml 7.34ml 7.33ml 7.37ml 7.33ml • Standard deviation (a measure of how spread out numbers are) 2. The number of decimal places used to express a value • Ex: a measurement of 23.5 cm is less precise than 23.453 cm
7. 7. PRECISION• When measuring, the precision of a measure depends on the measuring device used. • Precision of a measuring device = smallest division on the device • However, we can estimate to an extra decimal past this!• Looking at the ruler below…what is the precision of the ruler in cm? What is the measurement at the arrow? Measurement is 4.34 cm Precision of ruler is 0.1 cm
8. 8. MEASURING IN SCIENCE 8.24 cm Precision: 0.1 cm 52.8 ml Precision: 1 ml (measure from bottom of meniscus)
9. 9. MEASURING IN SCIENCE• Accuracy: How close a measured value is to the true value of the quantity measured.• Accuracy is affected by: • Lack of calibration or error in measuring instrument • Human error in measurement • Ex: reaction time in measuring time, etc. • Minimized by conducting multiple trials• Relative error tells us the accuracy of a measure.
10. 10. SIGNIFICANT FIGURES • Defined as the number of reliably known digits in a number.Rules for determining significant figures: figures: • Rules for determining significant Examples1. Zeros between other nonzero digits are significant 50.3 m 3.0025 s2. Zeros in front of nonzero digits are not significant 0.893 kg 0.00008 ms3. Zeros that are at the end of a number and to the right of the 57.00 gdecimal are significant 2.000000 kg4. Zeros at the end of a number to the left of a decimal are 1000 msignificant if they have been measured, are the first estimated (could have 1 or 4 significantdigit, or are hatted; otherwise, they are not significant. figures. We will say it has 1 sig figs) 100Ō m
11. 11. SIGNIFICANT FIGURES An alternative way of determining significant figures: Atlantic / Pacific rule • Is the decimal point present or absent?Pacific: Decimal is Atlantic / Pacific Rule Atlantic: Decimal ispresent absent• Start counting sig figs • Start counting from from left starting with right with first nonzero first nonzero digit. Keep digit. Keep counting counting until you run until you run out of 1-9 out of 1-9 digits digits0.0006 => 1 sig fig 40,000 => 1 sig fig0.00935 => 3 sig figs 1,040 => 3 sig figs1.020 => 4 sig figs 1,200,100 => 5 sig figs
12. 12. SIGNIFICANT FIGURES • Hatted zeros indicate significant zero digits Digit Explanation 2000 - Start from first nonzero, nonhatted digit from right => 1 sig fig 2Ō00 - Start from first nonzero nonhatted digit from right => 2 sig figs 2000. - Start from first nonzero digit from left - 4 sig figs
13. 13. SIGNIFICANT FIGURES • Try these out: • Answers • 1020 • 3 • 3300 • 2 • .0012 • 2 • 3000 • 4 • 5.0020 • 5 • 1.00 • 3 • 80,000 • 1 • 0.100 • 3
14. 14. LAB ACTIVITY: HOW FAR DID YOU GO?• Throughout time, are system of measuring has evolved a number of times. • The inch evolved from the barleycorn • The mile evolved from the furlong (the distance a plough team could be driven without rest; 8 furlongs = 1 mile)• Today, you will use 1 or 2 objects of your choosing to measuring an unknown distance• You will then convert this to meters as a way to study precision & accuracy.
15. 15. Farm-derived units of measurement:The rod is a historical unit of length equal to 5½ yards. It may have originated from thetypical length of a mediaeval ox-goad.The furlong (meaning furrow length) was the distance a team of oxen could ploughwithout resting. This was standardised to be exactly 40 rods.An acre was the amount of land tillable by one man behind one ox in one day.Traditional acres were long and narrow due to the difficulty in turning the plough.An oxgang was the amount of land tillable by one ox in a ploughing season. This couldvary from village to village, but was typically around 15 acres.A virgate was the amount of land tillable by two oxen in a ploughing season.A carucate was the amount of land tillable by a team of eight oxen in a ploughingseason. This was equal to 8 oxgangs or 4 virgates.Source: "Furlong." Wikipedia. Wikimedia Foundation, 09 June 2012. Web. 06 Sept.2012. <http://en.wikipedia.org/wiki/Furlong>.
16. 16. SIGNIFICANT FIGURES WITH MATH OPERATIONS• When adding or subtracting, the precision matters! • your answer can only have as many decimal positions as the value with the least number of decimal places 1.2003 ml + 23.25 ml = 24.45 ml NOTE: calculators DO NOT give values in the correct number of significant digits!
17. 17. SIGNIFICANT FIGURES WITH MATH OPERATIONS• When multiplying or dividing, significant figures matter! • your answer can only have as many significant figures as the value with the least number of significant figures. 3.6 cm x 0.01345 cm = 0.048 cm 2 • Must determine the number of significant figures in the numbers being multiplied/divided• NOTE: calculators DO NOT give values in the correct number of significant digits!
18. 18. MATH OPERATIONS WITH SIGNIFICANT FIGURES • Try these: • Try these: 16 x 2 20 4001 + 3.8 = 4005 1.35 x 400. 540. 24.38+0.0078 = 24.39 10,002 x 0.034 340 300÷5.0 60 10.9 15.3– 4.38= 350.÷17.7 19.8 100.– 3.2= 97 11.40– 3.8= 7.6
19. 19. SCIENTIFIC NOTATION• When expressing an extremely large number such as the mass of Earth, or a very small number such as the mass of an electron, scientists use the scientific notation. • For example, the mass of Earth is about 6,000,000,000,000,000,000,000,000 kg and can be written as 6 X 1024 kg. • Makes it clear which figures in a number are significant.
20. 20. MEASURING: SCIENTIFIC NOTATION• Scientific notation converts a number from standard form to one digit, a decimal point, and a power of 10• To convert from standard form to scientific notation: • Move decimal point so that number is one decimal form • Power of 10 = number of spaces moved • If moved to left , exponent is + • If moved to right , exponent is - • Examples: 1 = 1x100 10 = 1x101 100 = 1x102 1/10 = 0.1 = 1x10-1 1000 = 1x103 1/100 = 0.01 = 1x10-2
21. 21. SCIENTIFIC NOTATION• Example 1: Convert 3,020 to scientific notation.• Solution: • Significant figures must be conserved. How many significant figures are there? • Move the decimal so that there is one digit before the decimal. • The decimal must be moved by 3 positions to the right • Will the exponent be positive or negative? • The exponent will be positive3since 3,020 is greater than one. 3.02 x 10
22. 22. SCIENTIFIC NOTATION• Example 2: Convert .0003070 to scientific notation.Solution: • Significant figures must be conserved. How many significant figures are there? • Move the decimal so that there is one digit before the decimal. • Will the exponent be positive or negative? • The exponent will be negative since the number is less than one. 3.070x10 -4 Notice how the zero remains at the end to show that it is significant!
23. 23. SCIENTIFIC NOTATION• Convert the following to scientific notation with correct sig figs.1. 346,000 3.46 x 1052. 0.0210 2.10 x 10-23. 0.00000900 9.00 x 10-64. 500,Ō00 5.000 x 105
24. 24. SCIENTIFIC NOTATION Converting from scientific notation to standard form: 1. Identify & preserve all sig figs from scientific notation 2. Move decimal the number of spaces of the exponent • If exponent is POSITIVE +, move decimal to RIGHT  • If exponent is NEGATIVE -, move decimal to LEFT  • Examples: Scientific Operations Standard Notation notation 8.75 x 10-2 - 3 sig figs 0.0875 - Negative exponent - move decimal to left 3.635 x 105 - 4 sig figs 363,500 - Positive exponent – move decimal to right 2.50 x 102 - 3 sig figs 250. OR - Positive exponent – move decimal to right 25Ō
25. 25. Scientific Notation on a calculator• Use the “EE” key to do scientific notation on a calculator
26. 26. MEASURING: SCIENTIFIC NOTATION • Identify the number of significant figures and convert the number to standard notation for the following: Sig Figs Standard Form 1.52 x 103 3 1520 7.30 x 10-3 3 0.00730 6.75 x 105 3 675000 5.3030 x 10-2 5 0.053030 3.670 x 105 4 367Ō00
27. 27. SCIENTIFIC NOTATION Mathematical operations with significant figures for scientific notation Same rules applyOperation (rule) Operation ExamplesMultiplication /Division Multiplication: 10a(10b) = 10a + b (1.5´10 ) (1.2 ´10 ) = (1.5) (1.2) (10 ) (10 ) =1.8´10 3 4 3 4 7(Input with lowest # of sig 2 sf 2sf 2 sffigs limits output to that Division: (3.5´10 ) = æ 3.5 ö 10 10 = 0.83´10 3 ÷( ) (number of sig figs ) a ç ) 3 -4 -1 = 8.3´10-2 10 = 10 a-b ( 4.20 ´10 ) è 4.20 ø 4 10 b 2 sf / 3 sf 2 sf Note: exponent in denominator becomes negativeAddition / Subtraction Must change 7.4 x 10-3 7.4 x 10-3 (2 sig figs)(Output is as precise as exponents to be the -3.5 x 10-4 -0.35 x 10-3 (2 sig figs)least precise input) same before 7.05 x 10-3 7.1x10-3 (2 sf) performing operation 2.35 x 105 2.35 x 105 (3 sig figs) + 3.70 x 103  + 0.0370x105 (3 sig figs) 2.3870x105  2.39x105 (3 sf)
28. 28. MEASURING: SIGNIFICANT FIGURES • Try these: • Try these: 4001 16 x 2 20 + 3.8 4005 1.35 x 400. 540. 24.38 10,002 x 0.034 24.39 +0.0078 340 300 15.3 = 60 10.9 – 4.38 5.0 350. 19.8 = 100. 17.7 – 3.2 97 11.40 – 3.8 7.6
29. 29. MEASURING: SIGNIFICANT FIGURES Mathematical operations with significant figures for scientific notation Same rules applyOperation (rule) Operation ExamplesMultiplication /Division Multiplication: 10a(10b) = 10a + b (1.5´10 ) (1.2 ´10 ) = (1.5) (1.2) (10 ) (10 ) =1.8´10 3 4 3 4 7(Input with lowest # of sig 2 sf 2sf 2 sffigs limits output to that Division: (3.5´10 ) = æ 3.5 ö 10 10 = 0.83´10 3 ÷( ) (number of sig figs ) a ç ) 3 -4 -1 = 8.3´10-2 10 = 10 a-b ( 4.20 ´10 ) è 4.20 ø 4 10 b 2 sf / 3 sf 2 sf Note: exponent in denominator becomes negativeAddition / Subtraction Must change 7.4 x 10-3 7.4 x 10-3 (2 sig figs)(Output is as precise as exponents to be the -3.5 x 10-4 -0.35 x 10-3 (2 sig figs)least precise input) same before 7.05 x 10-3 7.1x10-3 (2 sf) performing operation 2.35 x 105 2.35 x 105 (3 sig figs) + 3.70 x 103  + 0.0370x105 (3 sig figs) 2.3870x105  2.39x105 (3 sf)