Using Math and Critical Thinking

1. Using Math and Critical Thinking Metric System Scientific Notation

2. APES Mathematics Review The APES exam will require you to do mathematical calculations. You will not be allowed to use a calculator. Occasionally these calculations may be possible for you to do in your head; nonetheless, it is MANDATORY to show all work for the calculations on the free response part of the APES exam.

3. International System of Measurement Metric Conversions

4. The International System of Measurement is used by most people worldwide. (SI) • Only a few countries do not use the metric system and the U.S. is one. • However, the metric system is used by the U.S. scientific community. • In APES most all of the measurements are metric. If in doubt, use the metric system. Metric Conversions

5. No fractions are involved! • No Naked Numbers! That means all answers must include units. There is no meaning to 56.8 Two Rules for the International System

6. King Henry Died Unfortunately Drinking Chocolate Milk • Kilo K • Hecto H • Decka D • Meter/Liter/Gram M/L/G • deci d • centi c • milli m Metric Prefixes Table

7. Conversion from larger to smaller units: • Meters to millimeters • M > mm • Move decimal to right • Three places • 56.8m = _____mm • Conversion from smaller to larger units? • Centimeters to kilometers • cm < Km • Move decimal to the left • 5 places • 4,885 cm = ______km Metric Problems

8. Introduction • Not only will you use the metric system, you will also do some slightly more complicated math requiring scientific notation and critical thinking skills.

9. Scientific Notation • Very large and very small numbers are more easily manipulated when they are converted to logarithms to the base 10. • Basic fact 100 =1,101 =10, 102 =100, 10-1 =.1 • Usually expressed by placing only one digit to the left of the decimal place. • 1,800,000,000,000 = 1.8 X 1012 • .000000178 = 1.78 X 10-7

10. WS Conversions 1 • Express one billion in scientific notation. • Express 2,360,000 in scientific notation. • Express 23,000,000,000,000 (23 trillion) in scientific notation.

11. Scientific Notation • Multiplication • Multiply the bases and add the exponents • (3 X 103) X (4 X 105) • (3) (4) = 12 • 3 + 5 = 8 • 12 X 108 • 1.2 X 109

12. Divide the number in the numerator by the number in the denominator and subtract the exponent of the denominator from the exponent of the numerator. • NUMERATOR/DENOMINATOR • (5.2 x 104) divided by (2.6 X102) • 5.2 divided by 2.6 = 2 • 4-2 = 2 • 2 X 102 Division

13. Multiply the coefficients • Add exponents • Ex. (3 X 104)X(2 X 105) = • 6 X 109 • EX. (6 X 105)X(7 X 106) = • 42 X 1011 • 4.2 X 1012 Multiplication

14. Addition • Convert both numbers to the same exponent. • Add the numbers. • Convert to scientific notation. • 3,000,000,000 + 14,000,000 • 3000 X 106 + 14 X 106 • 3014 X 106 • 3.014 X 109

15. Subtraction Convert both numbers to the same exponent. Subtract the numbers. 3,000,000,000 – 14,000,000 3000 X 106 – 14 X 106 2986 X 106 2.986 X 109

16. It is often necessary to round numbers in environmental problems to get reasonable answers. • If the answer is in dollars and cents, it should be rounded accordingly. • $16. 43 NOT $16. 4321 • If the answer is a person or other living organism, it should be rounded accordingly. • 4,037 people NOT 4,037.345 people Rounding

17. Mean (average) • Add the numbers and divide the number of numbers. • Median (middle) • Add the number of numbers +1 and divide by 2 to find the middle. • If there are an even number of numbers, the median is the mean of the middle two numbers. • Mode • The number that appears most often. If all the numbers appear the same number of times, there is no mode. Mean, Median, Mode