Chapter 2: A Mathematical Toolkit Dr. Zalesinsky

SI Units 2.1 Scientific Measurement Conversions Scientific Notation 2.2 Measurements with Uncertainties Sig Figs Operations with Sig Figs 2.3 Visualizing Data

• •

Graphs Relationships

The Metric System

• • • • • • •

Length or distance is measured in meters Mass is measured in grams Volume or capacity is measured in liters Time is measured in seconds Energy is measured in Joules Temperature is measured in Kelvin (not degrees) Quantity is measured in moles

The Liter (L)

The meter is slightly longer than a yard 1 inch = 2.54 cm (this is the only English to metric conversion you must know The Meter (m)

Gram (g) A gram’s mass is approximately the mass of 2 paperclips. Many medicines are massed by their active ingredients in grams or parts of a gram.

The really small and really large SCIENTIFIC NOTATION

Scientific Notation Scientific notation is the idea of writing a very large or small number as a power of 10.

Move the decimal so that there’s only ONE non-zero digit in front of the decimal and the number of places it has been moved is the exponent. Examples to follow

Scientific Notation Examples

•

Change to Scientific Notation: 985,000,000 9.85 x 10 8 0.0000674 6.74 x 10 -5 Change to Standard Notation: 8.27 x 10 6 9.565x10

-2 8,270,000 .09565

Using Scientific Notation with Calculator Find the EE or EXP button on your calculator (not the 10 x nor the SCI buttons)

TI Graphing Calculator

Examples

•

Do this calculation: (don’t type in the “x 10 part —use the EE or EXP button) Multiply:(8.76 x 10 -10 )(7.9 x10 11 ) = 692 or 6.92x10

2 Divide: (9.43x10

43 )/(7.33x10

23 ) = (approximate) 1.3 x10 20 be careful to turn your display into correct scientific notation 1.3

20 is not the same!

Uncertainty in Measurement SIGNIFICANT FIGURES

•

English to Metric Comparisons Which is larger? 1. Meter or yard

•

What is the abbreviation for each unit? 2. Mile or Kilometer 1. Meter = _____ 3. Gallon or liter 2. Gram = _____ 4. Pound or gram 3. Liter = ______ 5. Quart or liter 4. Second = ______ 6. Centimeter or inch 5. Joule = _______ 7. Gram or ounce 6. Mole = _______

Metric Abbreviations

•

The BASE units for the metric system are gram, liter, meter, second, Kelvin, Joule and mole.

•

There are larger and smaller portions of each of these

•

Their abbreviations come in front of the unit’s abbreviation (ex. centimeter = cm)

Larger and Smaller Prefixes

• • • • • •

Larger 1 billion = GIGA (G) 1 million = MEGA (M) 1,000 = kilo (k) 100 = hecto (h) 10 = deca (D, dk, or da)

• • • • • • •

Smaller 1trillion = pico (p) 1 billion = nano (n) 1 million = micro (

m

) 1,000 = milli (m) 100 = centi (c ) 10 = deci (d)

Match the abbreviation with the name 1. Cm = ________ 2. mg = ________ 3. ML = ________ 4. Gg = ________

5. m

s = ________ 6. km = _______ 7. mL = _______ 8. kJ = ________ 9. mm= _______ 10. Dg = _______ 1. Centimeter 2. Milligram 3. Megaliter 4. Gigagram 5. Microsecond 6. Kilometer 7. Milliliter 8. Kilojoule 9. Millimeter 10. Decagram

Kilo 0.000765

Conversions

Hecto Deca UNIT Deci 1.0

100 3.09

Centi Milli 2594 73.60

0.763

84300 730638

G 1x10 9 M k h

Conversions

D unit 1.0

d 3.9

c m m 7x 10 6 n p 403 7.62

27 848 626

Significant Figures

•

Read the correct number of significant figures

Measure the following using significant figures

Use Sig. Figs to find this measurement

Use the correct number of sig. figs in this measurement

What digit would be estimated in using Ruler A? A. Ones B. Tenths C. Hundredths D. Thousandths E. Tens

What decimal place is estimated when using Ruler B? A. Ones B. Tenths C. Hundredths D. Thousandths E. Tens

Measure the width of the rectangle using the correct number of sig figs. A. 3. 75 cm B. 3.6 cm C. 2. 6 cm D. 3.60 cm E. 4.25 cm

Measure the length of the rectangle using the correct number of sig figs.

A. 12.55 cm B. 12. 5 cm C. 12.0 cm D. 13. 5 cm E. 13.50 cm

How many sig figs should be in the correct measurement of the length of this rectangle? A. 2 sig figs B. 3 sig figs C. 4 sig figs D. 5 sig figs E. 1 sig fig

The width of this rectangle is 0.90 cm. How many significant figures are in this measurement?

A. 3 sig figs B. 2 sig figs C. 1 sig fig D. infinite sig figs E. none of the above

Data Table 1.1

Dimension W X Y

Length (cm) Longer side Width (cm) Shorter side

Z

Data Table 1.2

Y Z

Rectangle Measured Length # of sig figs in Length Measured Width Number of Sig Figs in Width

W X

Dimension

Length (cm) Longer side Width (cm) Shorter side

Data Table 1.5

W X Y Z

Data Table 1.6

Y Z

Rectangle Measured Length # of sig figs in Length Measured Width Number of Sig Figs in Width

W X

Calculating with Uncertainty CALCULATIONS WITH SIGNIFICANT FIGURES

Multiplication and Division with Sig Figs

• • •

The least number of sig figs in the input is equal to the number of sig figs in the answer (output). Remember all conversion factors and counted numbers have INFINITE sig figs!

Example: 8.03 g x 4.0 cm 3 /g = ?

Addition and Subtraction with Sig Figs

• • • •

The least number of decimal places in the input is the same number of decimal places in the output (answer). 12.573 m + 3847.9 m - 378 m = ?

3482.473 (unrounded) 3482 rounded to the correct number of decimal places

Calculations with Sig Figs 1. A rectangle has a width of 5.0 cm and a length of 8.40 cm. What is the area of this rectangle in cm 2 ? ___________ 2. A rectangular prism has the following measurements: length 8.54 cm, width 7.80cm, and height 10.00 cm. What is the volume in cm 3 ? ______________

W X Y Z

Data Table 1.2

Rectangle Measured Length (cm)

12.2

13.6

3.3

20.2

# of Sig Figs in Length

3 3 2 3

Measured Width (cm)

10.1

0.7

2.4

2.5

# of Sig Figs in Width

3 1 2 2

Data Table 1.3

Rectangle Justified # of Sig Fig in Area Unrounded Area (cm 2 ) Rounded Area (cm 2 )

W X Y Z

Data Table 1.6

Rectangle Measured Length (cm)

12.19

13.51

3.27

20.19

# of Sig Figs in Length

4 4 3 4

Measured Width (cm)

10.09

.61

2.49

2.38

# of Sig Figs in Width

4 2 3 3

Data Table 1.7

Rectangle Justified # of Sig Fig in Area Unrounded Area (cm 2 ) Rounded Area (cm 2 )

2.3 VISUALIZING DATA

Linear Relationships

• • •

Dependent (y) and independent (x) variables when graphed form a straight line Slope is positive if they are DIRECTLY PROPORTIONAL Slope is negative if they are INVERSELY PROPORTIONAL

4,5 4 3,5 3 2,5 2 1,5 1 0,5 0 1 2 3 4 Proportio nal Inverse No change

Nonlinear Relationships

• •

Quadratic equations relate parabolic relationships Some inverse relationships are hyperbolic

Practice Problem

•

Create a graph from the data chart given

•

Describe the relationship (linear or non)

•

If linear, find slope (in/direct) Volume of oil (cc)

0 30 40 48 49 51

Concentration (ppm)

0 1.3

1.5

2.0

3.0

11.0