Scientific Numeracy

What is the current temperature of the room?

A. 22

B. 72

C. 295

What is the current temperature of the room?

22°F = 72°F = 295°K

The temperature is 72 feet in here.

Mars Climate Orbiter

Mars Climate Orbiter $327.6 million Gimli Glider

Space errors handout

Activity• Have a large footed and small

footed student measure the length of the room.

• How many feet long is the room?

• Who tells a ruler how long to be?

• http://en.wikipedia.org/wiki/Foot_(unit)

7 BASE SI UnitsUnit Name and

Symbol Dimension Symbol What does it measure?

Meter (m) L

Kilogram (kg) M

Second (s) T

Ampere (A) I

Kelvin (K) Θ

Mole (mol) N

Candela (cd) J

7 BASE SI UnitsUnit Name and

Symbol Dimension Symbol What does it measure?

Meter (m) L Length

Kilogram (kg) M Mass

Second (s) T Time

Ampere (A) I Electric Current

Kelvin (K) Θ Temperature

Mole (mol) N Amount of substance

Candela (cd) J Luminous Intensity

Other units are DERIVED units

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D =m

v

€

S =d

t

€

P =F

A

€

F = ma

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E = mc2

Some units get special names!

• A NEWTON is the force required to accelerate a one kilogram object at a rate of one meter per second squared.

• Instead of writing (kg)(m)/(s2) physicists just use Newton (N) instead.

• Thus 1N = 1(kg)(m)/(s2)

Pressure = Force/Area

• Force = (kg)(m)/(s2) • Area = m2

• Units for pressure

€

kg∗ms2

m2

€

N

m2= A Pascal=

Concept Check

• What is the difference between base, derived and special units?

• http://www.physics.nist.gov/cuu/Units/units.html

Metric Prefixes

• Metric Madness

• Metric is SI

• US is only industrialized country not using it.

• We use an inferior system.

• yotta- (Y) 1024 1 septillion units• zetta- (Z) 1021 1 sextillion units• exa- (E) 1018 1 quintillion units• peta- (P) 1015 1 quadrillion units• tera- (T) 1012 1 trillion units• giga- (G) 109 1 billion units• mega- (M) 106 1 million

units• kilo- (k) 103 1 thousand units• hecto- (h) 102 1 hundred units• deka- (da) 10 1 ten units

• BASIC UNIT• deci- (d) 10-1 1 tenth units• centi- (c) 10-2 1 hundredth units• milli- (m) 10-3 1 thousandth units• micro- (µ) 10-6 1 millionth units• nano- (n) 10-9 1 billionth units• pico- (p) 10-12 1 trillionth units• femto- (f) 10-15 1 quadrillionth• atto- (a) 10-18 1 quintillionth• zepto- (z) 10-21 1 sextillionth• yocto- (y) 10-24 1 septillionth

Conversion is EasierThree Yards into Feet

• 3 goes to 36• We have a new number.

Three Meters into Centimeters

• 3 goes to 300• Decimal moves two places

Just moves the decimal to the right two places. Much more convenient especially when dealing with the very large and small numbers of science.

Metric System

Handouts metric mania

Scientific Notation

• How long would it take to count to a million? How do you know?

• Big and Small numbers in science.

Mass of Electron Mass of EarthMass of Proton Mass of SunDiameter of Atom Size of Galaxy

Handout

H2O is a Compound

• One glass of water has 8.36 x 1024 molecules of H2O inside it.

8,360,000,000,000,000,000,000,00018

Exponential Numbers: Powers of 10

• Any Number written as A x 10B • A is usually between 1 and 10• B is usually an integer.

• Example I: 93,450,000 9.345 x 107 • 7th power means move the decimal place right 7 places.

• Example II: 0.0001728 1.728 x 10-4

• The - 4th power means to move the decimal place to the left 4 places.

Exponent Rules• Rule 1

• Example

Exponent Rules• Rule 2

• Example

Exponent Rules• Rule 3

• Proof from Rule 1:

Exponent Rules• Rule 4

• Example

Why Scientific Notation is Easier

Example: 6,350,000,000 x 424,000,000 = 6.35 x10^9 x 4.24x10^8.

• Using rule one we add the exponents (8+9) and multiply the leading numbers (6.35 x 4.24). This is an easier calculation to perform.

• Answer = (6.35 x 4.24) x 10^17

Addition and Subtraction

• Convert to same power.• Keep the exponent.• Add the leading numbers

Two Videos

• Scale of the Universe • Yakkos Univese

Dimensional AnalysisRules for Converting Units

1) Units combine when multiplied just as an x does.

7cm · 5cm = 35cm2 (7x · 5x = 35x2) 7cm · 2cm · 3cm = 42cm3 (7x ·2x · 3x = 42x3) 2) Units cancel in division just as x or unknown

number would. 10cm / 5cm = 2 (10x / 5x = 2) 8cm2 / 2cm = 4cm (8x2 / 2x = 4x)

3) Units stay the same in addition and subtraction. 10cm + 5cm = 15 (10x + 5x = 15x) 8cm - 2cm = 6cm (8x - 2x = 6x) 4) You cannot subtract or add units with different

powers 10cm2 + 5cm = 10cm2 + 5cm

just as 10x2 + 5x = 10x2 + 5x

Unit Conversion Factor: essentially a ratio that is equal to one.

• To convert between different units we use a conversion factor like the two above. When you multiple a number by a conversion factor, you are not changing its value as you are essentially multiplying it by 1 and any number times 1 is equal to itself (245 · 1 = 245).

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12in

1ft

Problem 1 (Level 2)

• Problem 1: Convert 442 seconds into minutes.

• Solution: First we must find a Conversion Factor and set it up so the units cancel. A conversion factor is essentially equal to One.

• Conversion Factor: 60 seconds = 1 minute:

•The seconds cancel and we are left with 442 x 60 (minutes)

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442s∗60min

1s=

Problem 2 (Level 2)

• Problem 1: Convert 10 miles into centimeters.

• Solution: First we must find a Conversion Factor and set it up so the units cancel. A conversion factor is essentially equal to One.

• Conversion Factor: 1 Mile = 1.609 Kilometers so we first convert the 10 miles into kilometers:

•1.609 kilometers = 1 mile is the conversion factor•The units of miles cancel in division.•Now we convert kilometers to meters and then meters to centimeters

Problem 3 (level II)

• Problem 2: Convert 60 milers per hour into meters per second.

• Solution: We must convert both miles into meters and hours into seconds.

• Conversion Factor: 1 Mile = 1.609 Kilometers so we first convert the 60 miles into kilometers:

Handout on Unit Conversions

How Does Temperature Affect the Height of a Basketball’s bounce?

• Importance of Control• Independent vs

Dependent Variable• How are they graphed?

• IV: X-axis • DV:Y-axis

Accuracy vs Precision

Accuracy: how close a set of measurements is to the actual value.Precision: how close a set of measurements are to one another.

Percent Error

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Theoretical Value − Experimental

Theoretical Value∗100

A percent error tells you how far off an experimental value is from the currently accepted or theoretical value.

Percent Error Samples

• The accepted value for the density of gold is 19.3g/cm3. A student working in the lab measured it to be 18.7g/cm3. What was the student’s percent error?

• A student measures the acceleration due to gravity on earth to be 9.86m/s2 whereas the currently accepted value is 9.8m/s2. How far off was this student?

1) Clyde Clumsy was directed to determine the mass of a 500g piece of metal. After diligently goofing off for ten minutes, he quickly weighed the object and reported 458g.

2) Willomina Witty was assigned to determine the density of a sample of nickel metal. When she finished, she reported the density of nickel as 5.59 g/ml. However, her textbook reported the density of nickel to be 6.44 g/ml.

3) An experiment to determine the volume of a "mole" of a gas was assigned to Barry Bungleditup. He didn't read the experiment carefully and concluded the volume was 18.7 liters when he should have obtained 22.4 liters.

4) A student should have received a 93 for a grade but Mr. Sapone accidentally (Scouts honor!) put in an 83 instead. Calculate Mr. Sapone’s error.

Percent Error Problems

Correlation ≠ Causation

• All we have to do to stop global warming is become pirates!

Measurements & Uncertainty

• Suppose I ask you to measure the length of a desk with a meter stick.

• You tell me it is 0.756563874 meters.

• Should I applaud you for your high level of precision and accuracy?

Yo mama so stupid she tried to climb Mountain Dew

• Did you really read the meter stick with your naked eye to a billionth of a meter?

• Your naked eye is not that precise and your value is suspect.

Significant Figures• Important because of Measured vs. known numbers.

• Measured: the length of a desk.• Known Number: the number of desks in this room. (counting

or by definition)

• Think of some other examples.

Significant Figures

• Sigfigs tell us that the result of any experiment cannot be more accurate than the data used.

• Sigfigs let readers know the accuracy you used in an experiment.

Reading a Meter Stick / Ruler

• The Significant Figures of a measured value include those numbers directly readable from a measuring device plus one doubtful figure.

• Calculators make errors since they assume all numbers are known.

• You must stick with what is known and then include ONE doubtful number.

• Just because a device works does not mean it is accurate. It must be CALIBRATED.

Calibrate: (1) mark an instrument with a standard scale of readings.(2) correlate the readings of an instrument with those of a standard in order to check the instrument's accuracy.

How can you check to see if a thermometer is accurate before you use it?

• Why would you want to?

Significant Figures Rules

1. All non-zero numbers are always significant (e.g. 123456789)2. All zeroes between non-zero numbers are significant (7007). 3. All zeroes both simultaneously to the right of a decimal point and

at the end of a number are significant. (.007 has 1 sf and 700 = 1 but 7.00 = 3)

4. All zeroes left of a decimal point in a number > or = to 10 with a decimal point are significant. (700.4 = 4 sf)

To check #3 and #4 write the number in scientific notation. If you can get rid of the zeroes they are not significant.

Number # of SigFigs Rule(s)

48,923

3.967

900.06

0.0004

8.1000

501.040

3,000,000

10.0

Number # of SigFigs Rule(s)

48,923 5 1

3.967 4 1

900.06 5 1,2,4

0.0004 1 1,4

8.1000 5 1,3

501.040 6 1,2,3,4

3,000,000 1 1

10.0 3 1,3,4

Sigfig Products and Quotients

• When multiplying or dividing, the answer cannot have more significant figures than the term with the least number of significant figures.

• For example 25.2 x 2.543 = 64.0836 in a calculator.

• The answer is 64.1 however.

Is this accurate?

• 16.235 × 0.217 × 5 = 17.614975

Is this accurate?

• 16.235 × 0.217 × 5 = 17.614975

• There can only be ONE significant digit:

• 16.235 × 0.217 × 5 = 20

Is this accurate?

• 0.00435 × 4.6 = 0.02001

Is this accurate?

• 0.00435 × 4.6 = 0.02001

• 0.00435 × 4.6 = 0.020

• The answer is NOT 0.02 because there must be 2 significant digits.

Sigfigs: Addition and Subtraction

• In addition and subtraction the number of decimal places is what is important.

• The answer cannot have more decimal places than the term with the least number.

• 25.331 + 1.33 = 26.66 not 26.661• 13.214 + 234.6 + 7.0350 + 6.38 = 261.2 WHY?

Is this correct?

• 1247 + 134.5 + 450 + 78 = 1909.5

Is this correct?

• 1247 + 134.5 + 450 + 78 = 1909.5

• Round the result to the tens place:

• 1247 + 134.5 + 450 + 78 = 1910

Sigfigs’s

• HANDOUT

Statical Analysis > Sigfigs

• In science there is no such thing as a measurement.

• ALL MEASUREMENTS HAVE ERROR.

• Any result given without a consideration of error is useless!

• Standard Deviation is a measure of how spread out numbers are.

• Its symbol is σ (the Greek letter sigma)

• The formula is easy: it is the square root of the Variance which is the average of the squared differences from the Mean.

Standard Deviation (sample)1. Calculate the mean or average of each

data set. To do this, add up all the numbers in a data set and divide by the total number of pieces of data.

2. Get the deviance of each piece of data by subtracting the mean from each number.

3. Square each of the deviations. 4. Add up all of the squared deviations.5. Divide this number by one less than the

number of items in the data set.6. Calculate the square root of the

resulting value. This is the sample standard deviation.

Standard Deviation Curve

• The average height for adult men in the United States is about 70 inches, with a standard deviation of around 3 inches.

• 68% have a height within 3 inches of the mean (67–73 inches)

• 95% have a height within 6 inches of the mean (64–76 inches

Standard Deviation Problem