1. Math For Environmental Science
Point out to the students that it is not nearly as complex as the picture on the left!
Really it is not much more than simple arithmetic, the key is figuring out what the question is asking.

2. Dimensional Analysis Dimensional Analysis (Factor-Label)
It is critical for success on the exam that you have a thorough understanding of dimensional analysis.

3. Sample Question Heating a house in Virginia. Assumptions:
50,000 BTUs of heat per square foot are required to heat the house for the winter.
One cubic foot of natural gas supplies 1,000 BTUs of heat energy.
Natural gas is available at a cost of $4.00 per thousand cubic feet.
The house has 3,000 square feet of living space.
Natural Gas Facts:
Top 3 producers – Russia, USA, Canada
Contributes the least amount of pollution of the fossil fuels

4. Sample Question for Dimensional Analysis
The number of cubic feet of natural gas required to heat the house for one winter
This problem wants to know cubic feet of natural gas needed each winter to heat the house. In math terms, it looks like this:
1 ft3 natural gas

5. Sample Question for Dimensional Analysis
We know the house is 3,000 square ft. We also know that it takes 50,000 BTUs of heat per square foot to heat the home. Finally, we know that one cubic foot of natural gas supplies 1,000 BTUs of heat.
50,000 BTUs 1 ft or ft2 50,000 BTUs
and
  ft3 1,000 BTUs or 1,000 BTUs 1 ft3 and
 3000 ft2 home

6. Sample Question for Dimensional Analysis
The next step is to pick the statements that will cancel the units you do not want. You want ft3 in your answer. You need to eliminate BTUs and ft2.
50,000 BTUs 1 ft or ft2 50,000 BTUs
and
  ft3 1,000 BTUs or 1,000 BTUs 1 ft3 and
 3000 ft2 home

7. Sample Question for Dimensional Analysis
Now you need to eliminate BTUs. To do this you will need to use the equivalency that has BTUs on the bottom and cubic feet on the top.
3000 ft2 × 50,000 BTUs 1 ft2 ×
1 ft3 1,000 BTUs =
150,000 ft3 natural gas

8. Sample Question for Dimensional Analysis
1,000 ft3 $ or $4.00 1,000 ft3
3000 ft2 × 50,000 BTUs 1 ft2 ×
1 ft3 1,000 BTUs × $4.00 1,000 ft3 =
$600
For part ii, you need an additional statement.

9. Energy Efficiency or Win = Wout Efficiency Efficiency = Wout Win
Systems will never be 100 % efficient (unless stated), so pay careful attention.
Efficiency = Wout Win or
Win = Wout Efficiency

10. Energy Efficiency 150,000 ft3 natural gas 0.50 =
Now, if we add a third part to the question stating the system is 50 % efficient, then the output must be divided by the efficiency to determine how much input energy is needed. REMEMBER: A percentage efficiency must be changed to its decimal equivalent.
150,000 ft3 natural gas =
300,000 ft3 natural gas

11. Scientific Notation
A number written in scientific notation consists of a coefficient and an exponent. Coefficients need to be between 1 and 9. The coefficient is then multiplied by ten raised to an exponent, 10exponent.
Determine the exponent on the “10” by counting the number of places you move the decimal point. If you move the decimal to the right, the exponent will be negative. If you move the decimal point to the left, the exponent will be positive.

12. Converting to Scientific Notation1
575,000 =
5.75 × 105

13. Adding and Subtracting in Scientific Notation2
First, make sure both numbers have the same exponent. Next, add the coefficients.
2.4 × × 105=
4.24 × 105
.24 × 105 (moving the decimal to the left made the exponents equivalent)
Subtracting numbers written in scientific notation works the same way.

14. Multiplying in Scientific Notation3
This is much easier. Multiply the coefficients and then add the exponents.
(4.0 × 105) × (2.2 × 104) =
8.8 × 109

15. Dividing in Scientific Notation4
Divide the two coefficients. Then, subtract their exponents.
(8.8 × 109) / (2.2 × 104) =
4.0 × 105

16. Sample Question
Estimate the potential reduction in petroleum consumption (gallons of gasoline per year) that could be achieved in the United States by introducing electric vehicles under the following assumptions:
The mileage rate for the average car is 20 miles per gallon of gasoline.
The average car is driven 15,000 miles per year.
The United States has 200 million cars.
Twenty percent of the US cars could be replaced with electric cars.
Number of electric vehicles in U.S. – since ,000 plugin electrics have been sold

17. Sample Question
Start with simplifying and converting the numbers into scientific notation. Then follow the steps outlined in the dimensional analysis section.
2.0 × 108 cars× 1.5 × 104 miles car ∙year
× 1 gallon 2.0× 101 miles =
1.5 × 1011 gallons per year

18. Sample Question Now take 20% of your answer
1.5 × 1011 gallons per year
× 0.20 =
3.0 × 1010 gallons per year
Now take 20% of your answer

19. Population Growth Rate1
Population Growth Rate
The most common way to express population growth is as a percentage.
In the equation for population growth rate we subtract deaths from births and divide by the total population then multiply by 100
Births−Deaths Total Population × 100
=PGR (%)

20. Population Growth Rate2
Population Growth Rate
The crude birth rate (CBR) is the total number of births per year per 1,000 people and the crude death rate (CDR) is the total number of deaths per year per 1,000 people. When given the CBR and the CDR use the following equation.
CBR−CDR 10 =PGR (%)
we wouldn’t need to divide by 10 if the CBR & CDR were expressed per “100” instead of per 1,000
CBR of U.S – 13.7
CBR of World – 18.9
CDR of U.S – 8.4
CDR of World – 7.9
Have students calculate current us and world growth.

21. Sample Question
Electra is tracking its population data. In 1955, the population was 6000, with a crude birth rate of 55. At that time the population of Electra was growing rapidly, because of the low crude death rate of 10. In 1975 the population growth began to slow. The number of deaths totaled 50 and births numbered The total population at the beginning of 1975 was 7000.

22. Sample Question a)
What was the population growth rate of Electra in 1955? What was the population growth rate in Electra in 1975?
1955:
55−10 10 =4.5%
1975:
120− × 100=1%

23. 70
Rule of
To calculate the time required for a doubling of a population based upon population growth rate, PGR, expressed as a percentage. Leave growth percentage as a percentage!
70 PGR(as decimal ) =
Doubling Time
Give several other quick examples

24. b) Sample Question 70 4.5 = 15 ½ years
If Electra had maintained the 1955 growth rate how many years would it have taken for the population to double
= 15 ½ years

25. Per Capita
Per capita is a Latin term that translates into "by head.” It is determined by dividing the total resource by the population.
 Map is GDP ($) per capita.
Dark blue = 30,000
Teal = 6,000 – 12,000
Yellow = 2,000 – 3,500
Orange = 500 – 1,000
Red = 0 – 500

26. a) Sample Question 84 × 109 kg 7.0 × 109 = 12 × 100
Between 1966 and 2012, the global human population increased from 3.5 billion to 7.0 billion. Global poultry production increased from 35 billion kilograms to 84 billion kilograms during this period. a)
1966
35 × 109 kg 3.5 × 109 =10 × 100
or 10 kg per capita
2012
84 × 109 kg 7.0 × 109 = 12 × 100
or 12 kg per capita
Calculate the per capita poultry production in 1966 and in 2012.

27. Data, Data Tables and Graphs
When analyzing data, contemplate the independent vs. dependent variables. It may help to remember “DRY MIX” to determine which axis to place the variables as you design your graph.
Do not forget that slope is calculated most simply by change in y (rise) / change in x (run)
Dependent, Responding on the Y-axis
Manipulated, Independent on the X- axis

28. a)
Sample Question
Identify the ten-year period during which the greatest increase of the world population growth rate took place.
The world population growth rate rose from about 1.5 percent per year from to a peak of over 2 percent in the early 1960s due to reductions in mortality. Growth rates thereafter started to decline due to rising age at marriage as well as increasing availability and use of effective contraceptive methods. Note that changes in population growth have not always been steady. A dip in the growth rate from , for instance, was due to the Great Leap Forward in China. During that time, both natural disasters and decreased agricultural output in the wake of massive social reorganization caused China's death rate to rise sharply and its fertility rate to fall by almost half.

29. b)
Sample Question
The world population increased from 3 billion in 1959 to 6 billion by 1999, a doubling that occurred over 40 years. The Census Bureau's latest projections imply that population growth will continue into the 21st century, although more slowly. The world population is projected to grow from 6 billion in 1999 to 9 billion by 2044, an increase of 50 percent that is expected to require 45 years.
How many years did it take for the population in 1960 to double?
40 Years

30. Sample Question
The graphs estimate the Earth’s changing carbon dioxide (CO2) concentration (top) and Antarctic temperature (bottom), based on analysis of ice core data extending back 800,000 years.
Until the past century, natural factors caused atmospheric CO2 concentrations to vary within a range of about 180 to 300 parts per million by volume (ppmv). Warmer periods coincide with periods of relatively high CO2 concentrations. NOTE: The past century’s temperature changes and rapid CO2 rise (to 390 ppmv in 2010) are not shown here. Increases over the past half century are shown in the {Recent Role section} [internal link to The Recent Role of the Greenhouse Effect section] Source: Based on data appearing in NRC (2010)

31. a)
Sample Question
Calculate the net change in atmospheric CO2 concentration between 50,000 years ago and today.
80 ppm

32. b)
Sample Question
Calculate the temperature difference between 550,000 years ago and today.
20 °F

33. Logarithmic Scale
A scale of measurement that displays the value of a physical quantity using intervals corresponding to orders of magnitude, rather than a standard linear scale. A simple example is a chart whose vertical or horizontal axis has equally spaced increments that are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4. Examples of logarithmic scales:
Richter (earthquakes)
pH (acids & bases)
decibels (human hearing)

34. a) b) c) Sample Question
Determine the threshold concentration of pyrethrum.
Determine the LD50 for pyrethrum in respect to Daphnia
In toxicology, the median lethal dose, LD50 (abbreviation for lethal dose, 50%”), LC50 (lethal concentration, 50%) or LCt50 (lethal concentration & time) of a toxin, radiation, pathogen, is the dose required to kill half the members of a tested population after a specified test duration.
threshold dose  the minimum dose of radiation, a chemical, or a drug that will produce a detectable degree of any given effect
1.0 mg/L
  400 mg/L