#4 Notes : METRIC AND MEASUREMENTS/ Basic Math Tools Scientific Notation Significant Digits Metric System Dimensional Analysis

FOCUSED LEARNING TARGET In Analyzing data, I will be able to define, explain, interpret the observed measurements through the exploration of Science and Engineering Practices such as the use of : Scientific Notation Significant Figures Rules of Rounding off of Numbers Measurements Dimensional Analysis

SCIENTIFIC NOTATION 1  x < 10 x 10 exponent Makes very large or small numbers easy to use Two parts: 1  x < 10 (including 1 but NOT 10) x 10 exponent

WRITING SCIENTIFIC NOTATION EXAMPLES: 1) 2,000,000,000 = 2 X 10 9 2) 5430 = 5.43 X 10 3 3) 0.000000123 = 1.23 X 10 -7 4) 0.007872 = 7.872 X 10 -3 5) 966,666,000 = 9.66666 X 10 8 6) 0.0000600 = 6.00 X 10 -5 LARGE NUMBERS (>1) POSITIVE EXPONENTS EQUAL TO 1 or itself ZERO EXPONENTS SMALL NUMBERS (<1) NEGATIVE EXPONENTS

WRITING STANDARD FORM POSITIVE EXPONENTS MOVE TO RIGHT MOVE TO LEFT EXAMPLES: 1) 4.32 X 10 7 = 43,200,000 2) 3.45278 X 10 3 = 3452.78 3) 8.45 X 10 -5 = 0.0000845 4) 5.0010 X 10 -9 = 0.0000000050010 5) 7.00 X 10 -1 = 0.700 6) 1.123 X 10 5 = 112,300 POSITIVE EXPONENTS MOVE TO RIGHT MOVE TO LEFT NEGATIVE EXPONENTS

SIGNIFICANT DIGITS Exact numbers are without uncertainty and error Measured numbers are measured using instruments and have some degree of uncertainty and error Degree of accuracy of measured quantity depends on the measuring instrument

RULES 1) All NONZERO digits are significant Examples: a) 543,454,545 = 9 b) 34,000,000 = 2 = 5 c) 65,945 2) Trailing zeros are NOT significant Examples: = 1 a) 1,000 b) 234,500 = 4 c) 34,288,900,000 = 6

RULES CON’T are significant 3) Zero’s surrounded by significant digits are significant Examples: a) 1,000,330,134 = 10 b) 534,001,000 = 6 c) 7,001,000,100 = 8 4) For scientific notations, all the digits in the first part are significant Examples: a) 1.000 x 10 9 = 4 b) 2.34 x 10 -16 = 3 c) 3.4900 x 10 23 = 5

RULES CON’T 5) Zero’s are significant if a) there is a decimal present (anywhere) b) AND a significant digit in front of the zero Zero’s at beginning of a number are not significant (placement holder) Examples: a) 0.00100 = 3 e) 0.0000007 = 1 f) 0.003400 = 4 b) 0.1001232 = 7 c) 1.00100 = 6 g) 0.0700 = 3 d) 8900.00000 = 9 h) 0.040100 = 5

Rules for Rounding in Calculations

Rounding with 5’s: UP ____ 5 greater than zero 10.257 = 10.3 34.3591 = 34.4 ODD 5 zero 99.750 = 99.8 101.15 = 101.2

Rounding with 5’s: DOWN EVEN 5 zero 6.850 = 6.8 = 101.2 101.25

CALCULATIONS Multiply and Divide: Least number of significant digits Examples: a) 0.102 x 0.0821 x 273 = 2.2861566 b) 0.1001232 x 0.14 x 6.022 x 10 12 = 8.4412 x1010 c) 0.500 / 44.02 = 0.011358473 d) 8900.00000 x 4.031 x 0.08206 0.995 = 2958.770205 = 37.5 e) 150 / 4 f) 4.0 x 104 x 5.021 x 10–3 x 7.34993 x 102 = 147615.9941 g) 3.00 x 10 6 / 4.00 x 10 -7 = 7.5 x 1012

CALCULATIONS 2) Add and Subtract: Least precise decimal position Examples: a) 212.2 + 26.7 + 402.09 212.2 26.7 402.09 640.99 212.2 26.7 402.09 640.99 212.2 26.7 402.09 640.99 212.2 26.7 402.09 640.99 = 641.0

ADD AND SUBTRACT CON’T Examples: b) 1.0028 + 0.221 + 0.10337 1.0028 1.32717 1.0028 0.221 0.10337 1.32717 1.0028 0.221 0.10337 1.32717 1.0028 0.221 0.10337 1.32717 = 1.327

ADD AND SUBTRACT CON’T Examples: c) 102.01 + 0.0023 + 0.15 102.01 102.1623 102.01 0.0023 0.15 102.1623 102.01 0.0023 0.15 102.1623 102.01 0.0023 0.15 102.1623 = 102.16

ADD AND SUBTRACT CON’T Examples: d) 1.000 x 104 - 1 10000 - 1 9999 - 1 9999 10000 - 1 9999 10000 - 1 9999 = 1.000 x 104

ADD AND SUBTRACT CON’T Examples: e) 55.0001 + 0.0002 + 0.104 55.0001 55.1043 55.0001 0.0002 0.104 55.1043 55.0001 0.0002 0.104 55.1043 = 55.104

ADD AND SUBTRACT CON’T Examples: f) 1.02 x 103 + 1.02 x 102 + 1.02 x 101 1020 102 10.2 1132.2 1020 102 10.2 1132.2 1020 102 10.2 1132.2 1020 102 10.2 1132.2 = 1130

MIX PRACTICE Examples: a) 52.331 + 26.01 - 0.9981 = 77.3429 = 77.34 = 77.3429 = 77.34 b) 2.0944 + 0.0003233 + 12.22 7.001 = 2.04466 = 2.04 c) 1.42 x 102 + 1.021 x 103 3.1 x 10 -1 = 3751.613 = 3.8 x 102 d) (6.1982 x 10-4) 2 = 3.841768 x 10-7 = 3.8418 x 10-7 e) (2.3232 + 0.2034 - 0.16) x 4.0 x 103 = 9480 = 9500

Why the Metric System? International unit of measurement: SI units Base units Derived units Based on units of 10’s

LENGTH Measure distances or dimensions in space Meter (m) Length traveled by light in a vacuum in 1/299792458 seconds.

MASS Measure of quantity of matter Kilogram (kg) Mass of a prototype platinum-iridium cylinder

TIME Forward flow of events Second (s) Time is the radiation frequency of the cesium-133 atom.

VOLUME Amount of space an object occupies Cubic meter (m3) Derived unit 1 mL = 1 cm3

METRIC PREFIXES PREFIX SYMBOL DEFINITION MEGA- M 106 = 1,000,000 KILO- 103 = 1000 HECTO- h 102 = 100 DECA- da 101 = 10 BASE 100 = 1 DECI- d 10-1 = 0.1 = 1/10 CENTI- c 10-2 = 0.01 = 1/100 MILLI- m 10-3 = 0.001 = 1/1000 MICRO- μ 10-6 = 0.000001 = 1/1,000,000 NANO- n 10-9 = 0.000000001 = 1/1,000,000,000

DIMENSIONAL ANALYSIS Process to solve problems Factor-Label Method Dimensions of equation may be checked

DIMENSIONAL ANALYSIS Examples: 3 years = _______seconds 1 year = 365 days 1 day = 24 hours 1 hour = 60 minutes 1 min = 60 seconds 3 years 365 days 24 hours 60 minutes 60 seconds 1 year 1 day 1 hour 1 minute = 94608000 seconds = 9 x 10 7 seconds

DIMENSIONAL ANALYSIS Examples: b) 300.100 mL = ________kL 1 L = 1000 mL 1 kL = 1000 L 300.100 mL 1 L 1 kL 1000 mL 1000 L = 3.001 x 10-4 kL = 3.00100 x 10 –4 kL

DIMENSIONAL ANALYSIS Examples: c) 9.450 x 109 Mg = _________dg 1 Mg = 10 6 g 1 g = 10 dg 9.450 x 109 Mg 10 6 g 10 dg 1 Mg 1 g = 9.450 x 1016 dg

Experiment 1 : Measure and Convert Using Dimensional Analysis Lab Station 1: Measure the total width( short side) of the 2 tables Together Width =______ centimeter(cm) Show conversion using dimensional analysis Width = _______meters (m)

Lab Station 2: Measure the length (long side) of one table Length = ______ centimeter(cm) Show your conversion using dimensional analysis Length = ______meters(m)

Lab Station 3: Measure the mass of the weight using the triple beam balance : Mass = _________grams(g) Show your conversion using dimensional analysis Mass = _______kilograms(Kg)

Lab Station 4: Measure the Mass of the object using the digital scale Mass = _______ grams (g) Show your conversion using dimensional analysis Mass= _________kilogram (kg)

Lab Station 5 Measure the Volume of water using The Graduated cylinder Volume = _____ milliliter(mL) Show your conversion using dimensional analysis Volume = _____ cubic centimeter(cc or cm3)

Lab Station 6 : Measure the volume of soda using the Graduated Cylinder Volume = ______ milliliter (mL) Show your conversion using dimensional analysis Volume = ______ Liter (L)

Lab Station 7 : Measure the dimensions of the block. Length =____ cm Width =_____ cm Height = ____ cm Calculate the Volume of the Block . Show calculation Volume = ______ cm3 Show your conversion using dimensional analysis Volume = ________ cubic meters (m3)

Lab Station 8: Measure the distance using a meterstick. Distance = ________ centimeter(cm) Show your conversion using dimensional analysis Distance = ________ meter(m) Distance = ________kilometer(km)

Lab Station 9: Measure your walk time in a given distance Time = _______ seconds (sec) Show your conversion using dimensional analysis Time = _____ hour (h)

Lab Station 10 : Calculate your speed :Show your calculations Speed (v) = distance / time Speed (v) = ______ meters/sec or m/s Show your conversion using dimensional analysis Speed ( v) = ______ kilometers/ hour or km/h

Lab Station 11: Measure the Temperature of Water Temperature = _____0F Convert the temperature to Temperature = _____0C Temperature = _____ K

DIMENSIONAL ANALYSIS Examples: d) 2.356 g OH- = __________ molecules OH- 1 mole = 17 g OH- 1 mole = 6.022 x 10 23 molecules 2.356 g OH - 1 mole OH - 6.022 x 1023 molecules 17 g OH - 1 mole OH - = 8.34578 x 1022 molecules = 8.346 x 10 22 molecules

DIMENSIONAL ANALYSIS Examples: e) 45.00 km = __________cm 1 km = 1000 m 1 m = 100 cm 45.00 km 1000 m 100 cm 1 km 1 m = 4500000 cm = 4.500 x 10 6 cm

DIMENSIONAL ANALYSIS Examples: f) 6.7 x 1099 seconds = _______years 1 year = 365 days 1 day = 24 hours 1 hour = 60 minutes 1 min = 60 seconds 6.7 x 1099 seconds 1 hours 1 minute 1 day 1 year 60 seconds 60 minutes 24 hours 365 days = 2.124556 x 1092 years = 2.1 x 10 92 years

DIMENSIONAL ANALYSIS Examples: g) 1.2400 g He = __________ Liters He 1 mole = 4 g He 1 mole = 22.4 L 1.2400 g He 1 mole He 22.4 Liters He 4 g He 1 mole He = 6.944 Liters He = 6.9440 Liters He