PREREQUISITE BASIC MATHEMATICAL SKILLS
The APES exam has a significant amount of math and does not allow the use of calculators! Most students find that with a little practice, the math is not difficult, but as many of us have not had practice with setting up and solving problems without a calculator in a long time, in the beginning it can be daunting.
The following is meant to help you review basic math and science numeracy skills.
Percentage
17% = = 0.17
· Remember that “percent” literally means divided by 100
· Percentage is a measure of the part of the whole, or part divided by whole
For Example: 15 million is what percentage of the US population?
= 0.05 = 5%
For Example: What is 20% of this $15 bill so that I can give a good tip?
$15 x 0.20 = $15 x = $3
Rates
Rise
=
Y2 – Y1
Slope =
change
y = mx + b
Run
X2 – X1
time
· All of the above are ways to look at rates. The second equation is the easiest way to calculate a rate, especially from looking at a graph. Rates will often be written using the word “per” followed by a unit of time, such as cases per year, grams per minute, or miles per hour. The word “per” means to divide, so miles per gallon is actually the number of miles driven by one gallon.
· Rates are calculating how much an amount changes in a given amount of time.
Scientific Notation
Science often deals with large numbers. The number of hydrogen atoms in a liter of water, for example, is almost 70,000,000,000,000,000,000,000,000. On the other hand, the width of our galaxy is 9,315,000,000,000,000 km. To write out such huge numbers every time you used them would be a lot of trouble. If you were performing a series of calculations, working with long numbers could be time-consuming and confusing.
· Breaking Down Large Numbers
To deal with this problem, scientists often write numbers in scientific notation. In scientific notation, all numbers are expressed as two numbers multiplied together. The first is a number between one and ten, and the second is a power of ten. For example, 20 is written as 2 x 101, 200 is 2 x 102 (2 x 100), and 2,000 is 2 x 103 (2 x 1,000). This method makes it possible to write very large numbers in a simple way.
Thousand = 103 = 1,000 (notice how many zeros there are)
Million = 106 = 1,000,000 (300 million people in the US)
Billion = 109 = 1,000,000,000 (people on Earth = 7 billion)
Trillion = 1012 = 1,000,000,000,000 (National Debt)
· Using Decimals in Scientific Notation
In many cases, decimals must be used to keep the proper number of significant figures. For example, the radius of Earth is 6,370,000 m. To keep three significant figures and have the first number in scientific notation be between one and ten, this number must be expressed as 6.37 x 106 m. Note that the number of the exponent, in this case 6, is the number of places the decimal point was moved to the left.
· Writing Very Small Numbers
Very small numbers can also be expressed in scientific notation. To express a number less than one, negative exponents are used. For example, the lifetime of a tiny particle called a pion is just 0.000 000 026 of a second, or 2.6 x 10-8 s. In this case, the exponent is the number of places the decimal point was moved to the right.
· When using very large numbers, scientific notation is often easiest to manipulate. For example, the US population is 300 million people or 3 x 108.
· When adding or subtracting, exponents must be the same. Add the numbers in front of the ten and keep that exponent the same.
· When multiplying or dividing, multiply or divide the number in front of the ten and add the exponents if multiplying or subtract the exponents if dividing.
Online resource: http://www.chem.tamu.edu/class/fyp/mathrev/mr-scnot.html
Dimensional Analysis / Unit Conversion
Dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions and units of measure and tracking these dimensions as calculations or comparisons are performed. Converting from one dimensional unit to another is often somewhat complex. Dimensional analysis is a widely used technique for performing such conversions using the rules of algebra.
You should be able to convert any unit into any other unit accurately if given the conversion factor. Online resources are available:
http://www.chemprofessor.com/dimension_text.htm
http://www.chem.tamu.edu/class/fyp/mathrev/mr-da.html
Prefixes
A metric prefix or SI prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. Each prefix has a unique symbol that is added to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand; one kilogram is equal to one thousand grams. The prefix centi-, likewise, may be added to meter to indicate division by one hundred; one centimeter is equal to one hundredth of a meter.
Metric Conversion
Easy way to remember the order:
King
Henry
Died
By
Drinking
Chocolate
Milk
Kilo
Hecto
Deka
Base Unit
Deci
Centi
Milli
1,000
100
10
1
0.1
0.01
0.001
Online resource: http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT_RESOURCE/U06_L2_T1_text_final.html
Long Division and Multiplication
You should be able to do basic calculations by hand, including values with decimals and scientific notation. Many students struggle in this area because CALCULATORS ARE NOT ALLOWED ON THE AP EXAM. Make sure you are comfortable with these calculations.
Online resources:
http://www.mathsisfun.com/dividing-decimals.html
http://www.tutors4you.com/tutorialondecimals.htm
The APES exam has a significant amount of math and does not allow the use of calculators! Most students find that with a little practice, the math is not difficult, but as many of us have not had practice with setting up and solving problems without a calculator in a long time, in the beginning it can be daunting.
The following is meant to help you review basic math and science numeracy skills.
Percentage
17% = = 0.17
· Remember that “percent” literally means divided by 100
· Percentage is a measure of the part of the whole, or part divided by whole
For Example: 15 million is what percentage of the US population?
= 0.05 = 5%
For Example: What is 20% of this $15 bill so that I can give a good tip?
$15 x 0.20 = $15 x = $3
Rates
Rise
=
Y2 – Y1
Slope =
change
y = mx + b
Run
X2 – X1
time
· All of the above are ways to look at rates. The second equation is the easiest way to calculate a rate, especially from looking at a graph. Rates will often be written using the word “per” followed by a unit of time, such as cases per year, grams per minute, or miles per hour. The word “per” means to divide, so miles per gallon is actually the number of miles driven by one gallon.
· Rates are calculating how much an amount changes in a given amount of time.
Scientific Notation
Science often deals with large numbers. The number of hydrogen atoms in a liter of water, for example, is almost 70,000,000,000,000,000,000,000,000. On the other hand, the width of our galaxy is 9,315,000,000,000,000 km. To write out such huge numbers every time you used them would be a lot of trouble. If you were performing a series of calculations, working with long numbers could be time-consuming and confusing.
· Breaking Down Large Numbers
To deal with this problem, scientists often write numbers in scientific notation. In scientific notation, all numbers are expressed as two numbers multiplied together. The first is a number between one and ten, and the second is a power of ten. For example, 20 is written as 2 x 101, 200 is 2 x 102 (2 x 100), and 2,000 is 2 x 103 (2 x 1,000). This method makes it possible to write very large numbers in a simple way.
Thousand = 103 = 1,000 (notice how many zeros there are)
Million = 106 = 1,000,000 (300 million people in the US)
Billion = 109 = 1,000,000,000 (people on Earth = 7 billion)
Trillion = 1012 = 1,000,000,000,000 (National Debt)
· Using Decimals in Scientific Notation
In many cases, decimals must be used to keep the proper number of significant figures. For example, the radius of Earth is 6,370,000 m. To keep three significant figures and have the first number in scientific notation be between one and ten, this number must be expressed as 6.37 x 106 m. Note that the number of the exponent, in this case 6, is the number of places the decimal point was moved to the left.
· Writing Very Small Numbers
Very small numbers can also be expressed in scientific notation. To express a number less than one, negative exponents are used. For example, the lifetime of a tiny particle called a pion is just 0.000 000 026 of a second, or 2.6 x 10-8 s. In this case, the exponent is the number of places the decimal point was moved to the right.
· When using very large numbers, scientific notation is often easiest to manipulate. For example, the US population is 300 million people or 3 x 108.
· When adding or subtracting, exponents must be the same. Add the numbers in front of the ten and keep that exponent the same.
· When multiplying or dividing, multiply or divide the number in front of the ten and add the exponents if multiplying or subtract the exponents if dividing.
Online resource: http://www.chem.tamu.edu/class/fyp/mathrev/mr-scnot.html
Dimensional Analysis / Unit Conversion
Dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions and units of measure and tracking these dimensions as calculations or comparisons are performed. Converting from one dimensional unit to another is often somewhat complex. Dimensional analysis is a widely used technique for performing such conversions using the rules of algebra.
You should be able to convert any unit into any other unit accurately if given the conversion factor. Online resources are available:
http://www.chemprofessor.com/dimension_text.htm
http://www.chem.tamu.edu/class/fyp/mathrev/mr-da.html
Prefixes
A metric prefix or SI prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. Each prefix has a unique symbol that is added to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand; one kilogram is equal to one thousand grams. The prefix centi-, likewise, may be added to meter to indicate division by one hundred; one centimeter is equal to one hundredth of a meter.
Metric Conversion
Easy way to remember the order:
King
Henry
Died
By
Drinking
Chocolate
Milk
Kilo
Hecto
Deka
Base Unit
Deci
Centi
Milli
1,000
100
10
1
0.1
0.01
0.001
Online resource: http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT_RESOURCE/U06_L2_T1_text_final.html
Long Division and Multiplication
You should be able to do basic calculations by hand, including values with decimals and scientific notation. Many students struggle in this area because CALCULATORS ARE NOT ALLOWED ON THE AP EXAM. Make sure you are comfortable with these calculations.
Online resources:
http://www.mathsisfun.com/dividing-decimals.html
http://www.tutors4you.com/tutorialondecimals.htm
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