Data Analysis Chapter 2

Our Goals Learn: Units of measure: Lb, sec, acre, hours…. How to convert units: Seconds to years Very big and very small numbers Which numbers matter

Too many units Mass: Lb, gram, ounce Temp oF,oC,K Length in, cm Time: Sec, hr

The official units: SI units Quantity Unit Time Second (s) Length Meter (m) Mass Kilogram (kg) Temperature Kelvin (K) Amount of substance Mole (mol) Electric current Ampere (A) Luminous intensity Candela (cd)

Close your notebooks Quantity Unit Time Second (s) Length Meter (m) Mass Kilogram (kg) Temperature Kelvin (K) Amount of substance Mole (mol) Electric current Ampere (A) Luminous intensity Candela (cd)

Prefixes Gram 1 g Kilogram 1000 g Milligram 0.001 g

Prefixes Prefix Symbol Factor Sci. Not. Ex. Giga G 109 Gigameter (Gm) 1,000,000,000 109 Gigameter (Gm) Mega M 1,000,000 106 Megagram (Mg) Kilo k 1,000 103 Kilometer (km) Deci d 1/10 10-1 Deciliter (dL) Centi c 1/100 10-2 Centimeter (cm) Milli c 1/1,000 10-3 Milligram (mg) Micro m 1/1,000,000 10-6 Microgram (mg) Nano n 1/1,000,000,000 10-9 Nanometer (nm) Pico p 1/1,000,000,000,000 10-12 Picometer (pm)

Prefixes: Close your notebooks Symbol Factor Sci. Not. Ex. Giga G 1,000,000,000 109 Gig meter (Gm) Mega M 1,000,000 106 Mega gram (Mg) Kilo k 1,000 103 Kilometer (km) Deci d 1/10 10-1 Deciliter (dL) Centi c 1/100 10-2 Centimeter (cm) Milli c 1/1,000 10-3 Milligram (mg) Micro m 1/1,000,000 10-6 Microgram (mg) Nano n 1/1,000,000,000 10-9 Nanometer (nm) Pico p 1/1,000,000,000,000 10-12 Pico meter (pm)

3 Derived Units Speed Meters per second m/s Density g/mL Mass/volume More soon Volume cm3 1 cm3 = 1 mL

How many cubic centimeters are in one liter? 1L = 1000 mL = 1000 cm3 (= 10 cm x 10 cm x 10 cm)

Density: our first formula

13.5 g of aluminum has a volume of 5 mL. Density?

What is the mass of 2 mL of aluminum? d =m/v v = 2 mL Density = 3.7 g/mL Must rearrange formula…

d = m/v m =? Rearranging formulas- 3 ways 1. pure math d = m/v Multiply both sides by v dv =mv/v cancel out like terms dv =m

d = m/v m =? Rearranging formulas- 3 ways 2. replace with numbers d = m/v 4 = 8/2 8 = 4 x 2 m = dv

d = m/v m =? Rearranging formulas- 3 ways 3. the triangle method d =? m/v v = ? m/d m = ? vd divide m v d multiply

What is the mass of 2 mL of aluminum? d =m/v m = vd v = 2 mL d = 2.7 g/mL m = (2 mL)(2.7 g/mL) = 5.4 g

Reality check Each mL weighs 2.7 g We have 2 of them (2 mL) What is the mass of 2 mL of aluminum? Each mL weighs 2.7 g We have 2 of them (2 mL) Weight should be 5.4 g

Solving problems Find a formula Rearrange Plug in Do a reality check.

Kelvin (our second formula) Temp scale that cannot go below zero K = oC +273

25oC = ?K K = oC +273 = 300 K

K, oC, and oF

Scientific Notation

What it’s for: Mass of earth = 5,974,200,000,000,000,000,000,000 kg Mass of electron = 0.000000000000,000000000000000000910939 kg wowowowow!

Positive Exponents 101 = 10 102 = 10X10= 100 103 = 10X10X10 = 1000 104 = 10X10X10X10 = 10,000

Negative Exponents 10-1 = 1/10 = 0.1 10-2 = 1/100 = 0.01 10-3 = 1/1000 = 0.001 10-4 = 1/10000 = 0.0001

Scientific Notation: big numbers 10,000 = 1 X 104 250,000 = 2.5 X 105 Count places to the left until there is one number to the left of the decimal point. Always 10x Always 1-10

Try some 200 = 2.00 x 102 210 = 2.10 x 102 9742= 9.742 x 103 318746= 3.18746 x 105

Scientific Notation: small numbers 0.00006 = 6 X 10-5 0.00045 = 4.5 X 10-4 Count places to the right until there is one number to the right of the decimal point 0.003 = 3 x 10-3 0.0000025 = ? 2.5 x 10-6

Now lets go back: Mass of earth = 5,974,200,000,000,000,000,000,000 kg =? 5.9742 x 1024 kg Mass of electron = 0.000000000000000000000000000000910939 kg = ? 9.10939 x 10-31 kg better!

Multiplying with Scientific Notation Add the Exponents 102 X 103 = 105 100 X 1000 = 100,000

Multiplying with Scientific Notation: no calculators (2 X 102)(3 X 103) Multiply the Coefficients 2 X 3 = 6 Add the Exponents 102 X 103 = 105 6 x 105 60,000

Watch out- (2 x 103)(6 x 105) = ? Answer: = 12 x 108 No! must be 1-10 = 1.2 x 109...”right-down, left up” Add: = 8 Multiply: =12

QUESTIONS IF YOU DON’T ASK I CAN”T HELP YOU!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ASK NOW OR FOREVER HOLD YOUR PEACE>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Using the Exponent Key on a Calculator

EE or EXP means “times 10 to the…” How to type out 6.02 x 1023: 6 EE . 3 2 6 EE . 3 2 Don’t do it like this… wrong! 6 y x . 3 2 wrong! …or like this… x 1 6 . 2 EE 3 …or like this: too much work. y x 3 2 x 1 6 .

Also, know when to hit your (–) sign… …before the number, …after the number, …or either one.

Example: 1.2 x 105 2.8 x 1013 = 1 . 2 EE 5 3 8 Type this calculation in like this: 4.2857143 –09 Calculator gives… 4.2857143 E–09 or… This is NOT written… 4.3–9 4.3 x 10–9 4.3 E –9 or But instead is written…

No calculators 6 x 102 /3 x 101 = ? Answer: 2 x 101 Divide: 6/3 = 2 Reality check: 600/30 = 20 Subtract: 2-1 = 1

Try some:

I have 24 eggs. How many dozen eggs do I have? 2 dozen Unit Conversion I have 24 eggs. How many dozen eggs do I have? 2 dozen Why was that so easy?

Equalities length 10.0 in. 25.4 cm 12 eggs = 1 dozen eggs

Learning Check 1. 1000 m = 1 ___ a) mm b) km c) dm 2. 0.001 g = 1 ___ a) mg b) kg c) dg 3. 0.1 L = 1 ___ a) mL b) cL c) dL 4. 0.01 m = 1 ___ a) mm b) cm c) dm

Conversion Factors are equalities Example: 1 in. = 2.54 cm Factors: 1 in. and 2.54 cm 2.54 cm 1 in.

How many minutes are in 2.5 hours? Find a conversion factor: 60 minutes = 1 hour Set up your equation so units cancel: Hours cancel

Sample Problem You have $7.25 in your pocket in quarters. How many quarters do you have? 7.25 dollars 4 quarters 1 dollar = 29 quarters X

Learning Check Write conversion factors that relate each of the following pairs of units: 1. Liters and mL 2. Hours and minutes 3. Meters and kilometers

Learning Check a) 2440 cm b) 244 cm c) 24.4 cm A rattlesnake is 2.44 m long. How long is the snake in cm? a) 2440 cm b) 244 cm c) 24.4 cm

How many seconds are in 1.4 days? Unit plan: days hr min seconds Learning Check How many seconds are in 1.4 days? Unit plan: days hr min seconds 1.4 days x 24 hr x ?? 1 day

Solution Unit plan: days hr min seconds 1.4 day x 24 hr x 60 min x 60 sec 1 day 1 hr 1 min = 1.2 x 105 sec

Wait a minute! What is wrong with the following setup? 1.4 day x 1 day x 60 min x 60 sec 24 hr 1 hr 1 min Doesn’t cancel!

English and Metric Conversions If you know ONE conversion for each type of measurement, you can convert anything! 513 only: You must memorize and use these conversions: Mass: 454 grams = 1 pound Length: 2.54 cm = 1 inch Volume: 0.946 L = 1 quart

Learning Check An adult human has 4.65 L of blood. How many gallons of blood is that? Unit plan: L qt gallon Equalities: 1 quart = 0.946 L 1 gallon = 4 quarts Your Setup:

Steps to Problem Solving Read problem Identify data Make a unit plan from the initial unit to the desired unit Select conversion factors Change initial unit to desired unit Cancel units and check Do math on calculator Give an answer using significant figures

Dealing with Two Units – Honors Only If your pace on a treadmill is 65 meters per minute, how many seconds will it take for you to walk a distance of 8450 feet?

Solution Initial 8450 ft x 12 in. x 2.54 cm x 1 m 1 ft 1 in. 100 cm x 1 min x 60 sec = 2400 sec 65 m 1 min

What about Square and Cubic units? – Honors Only Use the conversion factors you already know, but when you square or cube the unit, don’t forget to cube the number also! Best way: Square or cube the ENITRE conversion factor Example: Convert 4.3 cm3 to mm3 10 mm = 1 cm; (10 mm)3 = (1 cm)3; 103 mm3 = 13 cm3; 1000 mm3 = 1 cm3 note 10 mm3 is not equal to 1 cm3 4.3 cm3 1000 mm3 1 cm3 = 4300 mm3

Learning Check A Nalgene water bottle holds 1000 cm3 of dihydrogen monoxide (DHMO). How many cubic decimeters is that?

So, a dm3 is the same as a Liter ! A cm3 is the same as a milliliter. Solution ( ) 1000 cm3 1 dm 3 10 cm = 1 dm3 So, a dm3 is the same as a Liter ! A cm3 is the same as a milliliter.

Percent Error

Calculating Percent Error Calculate the percent errors for each trial. Report your answers to two places after the decimal point.

Calculating Percent Error Data Analysis: Additional Concepts Topic 3 Calculating Percent Error Substitute each error into the percent error equation. Ignore the plus and minus signs. Note that the units for density cancel out.

Significant Figures

Significant Figures The numbers reported in a measurement are limited by the measuring tool Significant figures in a measurement include the known digits plus one estimated digit

Counting Significant Figures RULE 1. Non-zero digits are always significant. Number of Significant Figures 38.15 cm 5.6 ft 65.6 lb ___ 122.55 m ___ 4 2 3 5

Leading Zeros RULE 2. Leading zeros in decimal numbers are NOT significant. Number of Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb ____ 0.000262 mL ____ 2 3

Sandwiched Zeros RULE 3. In-between zeros are significant. 50.8 mm 3 Number of Significant Figures 50.8 mm 3 2001 min 4 0.702 lb ____ 0.00405 m ____ 3 3

Trailing Zeros 25,000 in. 2 200. yr 1, 2, 3 48,600 gal ____ RULE 4. Trailing zeros in numbers without decimals are NOT significant. Trailing zeros with decimals ARE significant Number of Significant Figures 25,000 in. 2 200. yr 1, 2, 3 48,600 gal ____ 25,005,0.00 g ____ 3 8

Learning Check A. Which answers contain 3 significant figures? 1) 0.4760 2) 0.00476 3) 4760 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 103 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x 105

Learning Check 2) 400.0 and 40 3) 0.000015 and 150,000 In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 and 22.00 2) 400.0 and 40 3) 0.000015 and 150,000

Learning Check State the number of significant figures in each of the following: A. 0.030 m 1 2 3 B. 4.050 L 2 3 4 C. 0.0008 g 1 2 4 D. 3.00 m 1 2 3 E. 2,080,000 bees 3 5 7

Significant Numbers in Calculations- please don’t write A calculated answer cannot be more precise than the measuring tool. A calculated answer must match the least precise measurement. Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing

Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25.2 one decimal place + 1.34 two decimal places 26.54 answer?? answer 26.5 one decimal place

Learning Check In each calculation, round the answer to the correct number of significant figures. A. 235.05 + 19.6 + 2.1 = 1) 256.75 2) 256.8 3) 257 B. 58.925 - 18.2 = 1) 40.725 2) 40.73 3) 40.7

Multiplying and Dividing maintain the fewest significant figures in your answer.

Learning Check A. 2.19 X 4.2 = 1) 9 2) 9.2 3) 9.198 B. 4.311 ÷ 0.07 = 1) 9 2) 9.2 3) 9.198 B. 4.311 ÷ 0.07 = 1) 61.58 2) 62 3) 60 C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041

Reading a Meterstick . 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

Known + Estimated Digits In 2.76 cm… Known digits 2 and 7 are 100% certain The third digit 6 is estimated (uncertain) In the reported length, all three digits (2.76 cm) are significant including the estimated one

Learning Check . l8. . . . I . . . . I9. . . .I . . . . I10. . cm What is the length of the line? 1) 9.6 cm 2) 9.60 cm 3) 9.62 cm How does your answer compare with your neighbor’s answer? Why or why not?

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

End chapter 2 