Math for APES Calculations Without Calculators

Pamela J. Shlachtman and Kathryn Weatherhead NSTA Boston 2008

Solutions:

1. Use exponents whenever numbers are especially large or small.

Scientific notation is a way to express, numbers the form of exponents as the product of a number (between 1 and 10) and raised to a power of 10. For 650000 use 6.5 x 10 5 For 0.000543 use 5.43 x 10 -4

In scientific notation remember to have one number to the left of the decimal and to use correct significant figures.

2. Practice math manipulations

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with exponents

When adding or subtracting numbers with exponents the exponents of each number must be the same before you can do the operation.

Example: (1.9 x 10 -3 ) – (1.5 x 10 -4 ) = (19 x 10 -4 ) - (1.5 x 10 -4 ) = 17.5 x 10 -4

When multiplying numbers with base 10 exponents, multiply the first factors, and then add the exponents.

Example, (3.1 x 10 5 ) (4.5 x 10 5 ) = 13.95 x 10 10 or 1.4 x 10 11 When dividing numbers, the exponents are subtracted, numerator exponent minus denominator exponent.

Example: 9 x 10 5 3 x 10 3 = 3 x 10 2

3. Use Dimensional analysis or factor/label method for calculations The following formula based on the cancellation of units is useful: Given Value x Conversion factor =Answer 1 OR old unit x new unit = new unit 1 old unit Example: Convert 12 km into mm. Report your answer using scientific notation. 12 km x 1000m x 1000 mm = 12000000mm = 1.2 X 10 7 mm 1 km 1 m

Units – Area and Volume • Area – m 2 , cm 2 , mm 2 , etc • Volume – m 3 , cm 3 , mm 3 , ml, L • Volume Conversions – 1000 ml = 1000 cm 3 – 1 L = 1000 ml – Example – Convert 500 cm 3 into m 3 • 100 cm = 1 m  (100cm)(100cm)(100cm) = 10 6 cm 3 • (500 cm 3 )(1m 3 / 10 6 cm 3 ) = 1.0 X 10 -4 m 3 = 1 m 3

4. Be sure to know how to convert numbers to percentages and percent change.

Example: If 200 households in a town of 10000 have solar power, what percent does this represent?

200/10000 x 100 = ? answer = 2.0% Example: If a city of population 10,000 experiences 100 births, 40 deaths, 10 immigrants, and 30 emigrants in the course of a year, what is its net annual percentage growth rate? answer = (100 + 10) – (40 + 30) X 100 = .40% 10,000

5. Keep it simple. They don’t expect you to do calculus without a calculator!

Try reducing the fraction from the previous problem 200/10000 to 20/100= 1/50 Then solve: 1/50 x 100%= 2.0%

6. Remember that the numbers will likely be simple to manipulate.

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The APES folks know you only have limited time to do 100 multiple choice and 4 essays

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If you are getting answers like 1.3657, then it is likely wrong

7. Show ALL of your work and steps of calculations, even if they are so simple you think they are implied.

NO WORK – NO CREDIT !

8. Show all of your units, too!

Numbers given without units are often not counted even if correct.

9. Answers should make sense! LOOK them over before you finish

Example: No one is going to spend 1 billion dollars per gallon of water or $10,000 per kWh electrical energy!

10. Know some basic metric prefixes for simple conversions

Giga G MegaM Kilo k Hecto h 10 10 10 10 9 6 = 1 000 000 3 = 1 000 2 = 1 000 000 000 = 100 1 = 10 Deka dk Nanon 10 Base Unit (m, l, g) Deci d 10 -1 = .1

Centic 10 -2 = .01

Milli m Micro μ 10 10 -3 -6 = .001

10 0 = .000 001 =1 10 -9 = .000 000 01

Conversions from US to metric will probably be given and do not need to be memorized. They should be practiced, however.

Gallons to Liters Liters to Gallons Meters to Yards Yards to Meters Grams to Ounces Ounces to Grams Kilograms to Pounds Pounds to Kilograms Miles to Kilometers Kilometers to Miles 1 gal= 3.8 L 1 L, l= .264 gal 1 m= 1.094 yd 1 yd= .914 m 1 g= .035 oz 1 oz= 28.35 g 1 kg= 2.2 lb 1 lb= 454 g 1 mi= 1.609km

1 km= .621 mi

11. Know some simple energy calculations.

2004 Exam: West Freemont is a community consisting of 3000 homes. The capacity of the power plant is 12 megawatts (MW) and the average household consumes 8,000 kilowatt hours (kWh) of electrical energy each year. The price paid for this energy is $0.10 per kWh.

(a) Assuming that the existing power plant can operate at full capacity for 8,000 hours per year, how many kWh of electricity can be produced by the plant in one year?

12 MW X 1000 kW X 8000 hours = 96000000 kWh/year 1 MW Year or 9.6 X 10 7 kWh/year (b) How many kWh of electricity does the community use in one year?

3000 houses X 8000 kWh = 24000000 kWh/yr or yr 2.4 X 10 7 kWh/yr

12. Rule of 70

• Based on exponential growth • Doubling Time = 70/annual growth rate For example, if a population is growing at an annual rate of 2%, the number of years it will take for that population to double can be found by dividing 70 by 2, i.e., DT = 70/2 = 35 years.

Calculate the doubling time for a population growing at 1.4%.

Answer = 70/1.4 = 50 years

13. Be able to calculate half-life Example: A sample of radioactive waste has a half life of 10 years and an activity level of 2 curies. After how many years will the activity level of this sample be 0.25 curie? Answer = 30 years [(1/2 x )original quantity = amount remaining]

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14. Know how to graph data

Title the graph

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Set up the independent variable along the X axis

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Set up the dependent variable along the Y axis

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Label each axis and give the appropriate units

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Make proportional increments along each axis so the graph is spread out over the entire graph area

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Plot points and sketch a curve if needed. Use a straight edge to connect points unless told to extrapolate a line.

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Label EACH curve if more than one is plotted.

100 90 40 30 20 10 0 80 70 60 50 1 2

Study Time

3 4

Hours per Week

5 6

Be able to interpolate and

extrapolate data

Year 1855 1875 1895 1950 1977 1990

Example – Question 2 – 2003 Exam

CBR 43 43 37 22 CDR 41 20 12 12 10 10 CBR vs CDR for Industria 10

50 45 40 35 CBR CDR 30 25 20 15 10 5 0 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990

Year

15. Know what is meant by “per capita” when solving a problem or interpreting a graph

From Question 2 – 2005 Exam Between 1950 and 2000, global meat production increased from 52 billion kg to 240 billion kg. During this period, global human population increased from 2.6 billion to 6.0 billion. Calculate the per capita meat production in 1950 and 2000.

1950 : 5.2 X 10 10 kg / 2.6 X 10 9 people = 2.0 X10 1 kg/person or 20 kg/person 2000: 2.40 X 10 11 kg / 6.0 X 10 9 people = 4.0 X10 1 kg/person or 40 kg/person

16. pH and Logs Logarithm Calculations (base 10) Log 10 (x) = Y 10 Y = x pH = -log[H + ] If the [H + ] = 10 -3 , the pH = -log[10 -3 ] = 3

17. Population Calculations CBR and CDR  represents the number of individuals born or dies per 1000.

CBR – CDR X 100 = 1000 Rate of Natural Increase (%) [(CBR + Immigration) – (CDR + Emigration)] X 100 = Growth Rate (%)

Example – data from the US Census Bureau Country Year United States United States 2000 2005 CBR 14.39

14.14

CDR 8.52

8.25

Net number of migrants per 1000 3.71

Rate of Natural Increase (%) 0.587

Growth Rate (%) 0.957

3.31

0.589

0.92

Rate of Natural Increase = 14.39 – 8.52 X 100 = 0.587% 1000 Doubling Time based on RNI = 70 / 0.587 = 119 years Doubling Time based on GR = 70 / 0.957 = 73 years

18. Miscellaneous Students should know : • Number of days in a year = 365 • Number of hours in a day = 24 • Population of the US ~300 million • Global Population • How to work with negative numbers – Calculate the change in temperature between 140000 years ago and 125000 if the temperatures were -8 o C and +2 o C, respectively.

Answer = 2 o C –(-8 o C) = 10 o C

19. Practice – practice – practice!!

APES released multiple choice exams and all free response questions are available at AP Central. http://apcentral.collegeboard.com

ANY QUESTIONS??

Kathryn Weatherhead’s website: http://web.beaufort.k12.sc.us/education/staff/ staff.php?sectiondetailid=5121 Pam Shlachtman’s website : www.yourclasspage.com

(schoolcode = 3052351360 – scroll down to Shlachtman - APES) E-mail address: [email protected]