Lesson 2: Math Concepts for Physical Science

Introduction
The recipe for this cake required knowledge of measurement and the use of fractions. You are planning a party, and a friend asks you how how far away your house is from hers. How would you explain the distance to your home? You could describe the number of miles it is, tell her how many minutes it takes to get there, or even use the number of blocks. No matter how you explain it, you have to use numbers. Scientists rely on numbers and math to help them describe their ideas and observations. Math is critical to science, from the study of gravity to chemical equations to climate change. In this lesson, you will review basic math concepts that will be helpful to you in this course. You will also learn how to express very large and very small numbers using scientific notation. Use the Practice Problems after each section to test your understanding of each concept.
What Math Concepts Are Used in Physical Science?
In science, numbers are used for comparisons and measurements. Below, you can review some of the different ways to compare numbers, such as mean and decimals. You will also have a chance to review how to do some basic calculations involving numbers with decimals, fractions, and/or percentages as well as review basic algebra skills.
Mean Mean, sometimes called average, gives the "middle of the road" for a set of data. To find the mean, take the sum of all the data in a set and divide it by the total number of items of data. Steps for Finding the Mean       1. Find the sum of all the numbers in the set.      2. Count how many numbers are in the set. 3. Divide the sum of all the numbers by how many numbers there are in the set. For example: Find the mean of the set. 2, 2, 3, 0, 4, 0, 6, 5, 17, 10, 9, 2, 1, 9 Find the sum of all the numbers in the set.   2 + 2 + 3 + 0 + 4 + 0 + 6 + 5 + 17 + 10 + 9 + 2 + 1 + 9 = 70                               Count how many numbers are in the set.   2, 2, 3, 0, 4, 0, 6, 5, 17, 10, 9, 2, 1, 9 →14 numbers          Divide the sum of all the numbers by how many numbers there are in the set to find the mean, which is 5.   70 ÷ 14 = 5
Decimals Greater Than and Less Than The smaller side of the sign points to the smaller number 4 < 9 3 > 2 4 is less than 9 3 is greater than 2 Numbers with decimals are used often in science. They allow for very precise values when collecting or comparing data and taking measurements. It is important to understand the value of decimal places and to be able to compare decimal numbers. For example, which of the following numbers is greater? 1.45         1.513 Compare decimal numbers by comparing one digit at a time, from left to right. Make sure to compare digits in the same place values. Using a place value chart is helpful when doing this. Look at the place value charts below. In the ones place, the numbers are equal.   hundreds tens ones . tenths hundredths thousandths                      100 10 1 .1 .01 .001 1.45 → 1 . 4 5 1.513 → 1 . 5 1 3     In the tenths place, 1.513 has the greater digit.   hundreds tens ones . tenths hundredths thousandths                      100 10 1 .1 .01 .001 1.45 → 1 . 4 5 1.513 → 1 . 5 1 3 You do not have to compare any more digits. You now know that 1.513 is the greater number. 1.45 < 1.513
Reviewing Decimal Computations In this course it is important to be able to add, subtract, multiply, and divide numbers with decimals. Use the table to review the steps and solve the problems below. Adding and Subtracting Decimals 1. Line up the decimal points in the numbers. 2. If the numbers end in different place values, add zeros to make them end in the same place value. 3. Add or subtract the numbers. Place the decimal point in the answer directly beneath where it is in the problem. For example: Solve. 4.75 + 0.38 Write the numbers with the decimal points lined up. Use a place value chart if it helps. 10 1 .1 .01 .001 4 . 7 5   0 . 3 8               Complete the addition. Carry numbers as when adding whole numbers. Pull the decimal point straight down into the answer. 1 1 4 . 7 5   + 0 . 3 8 5 . 1 3 Multiplying decimals is a lot like multiplying whole numbers. There are just a few extra steps at the end. Multiplying Decimals 1. Line up the numbers to the right. Do not line up the decimal points. 2. Multiply as with whole numbers. 3. Count the total number of digits to the right of the decimal point in both numbers you multiplied. In your answer, the product, count that number of places from the right and place the decimal. Notice that the first step of multiplying is not the same as for adding and subtracting. With multiplying, only think about the decimal point after completing the multiplication. For example: Solve. 3.14 × 1.8 Write the numbers with their digits lined up on the right. Pay no attention to the decimal points yet.     3.14 ×   1.8                                        Multiply the numbers. Remember the problem is not done until the decimal point is placed.      1  3    3.14 ×   1.8   2512   3140    5.652       Count the number of digits to the right of the decimal point in both the numbers you multiplied together. Then count the same number of digits from the right of the answer. This is where the decimal goes to complete the problem.    3.14 ×   1.8   5.652
Fractions Numbers may also be expressed as fractions. A fraction is a portion of a whole number. Recall that the numerator is the number on top and the denominator is the number on the bottom. Use fractions to show rates and proportions, which are different ways to compare numbers. A rate is a comparison of two values using units. For example, a rate is used when comparing the number of miles a car has driven to the number of gallons of gas it has used. A unit rate is a rate with a denominator of 1, such as a mile per gallon. Fractions are most often expressed in simplest form. When a fraction is in simplest form, the numerator and denominator have no common factor except 1. Simplest form is obtained when all of the factors shared by the numerator and denominator have been cancelled out by factoring, reducing, or simplifying. For example: Reduce the fraction below to its simplest form. 15 25 Find a factor that is common to both the numerator and the denominator. Write the numerator and denominator as products of their factors. 15 = 3 × 5 25 5 × 5                                   Cancel out the factors that are in both the numerator and the denominator. 3 ×  5  5 ×  5        3  is the simplest form of   15 . 5 25 3 5 Reviewing Fraction Computations It is important to know how to multiply and divide fractions because they are used in this course to convert between different units of measurement. Follow these steps to multiply fractions. Steps for Multiplying Fractions      1. Multiply one numerator by the other. This is the numerator of the product.        2. Multiply one denominator by the other. This is the denominator of the product.   a  ×   c  =  a × c  =  ac b d b × d bd Always multiply straight across. Never multiply the top by the bottom or the bottom by the top.  For example: Solve. 4  ×  2 5 9 Multiply the first numerator by the second numerator. 4  ×   2  =   4 × 2  =   8 5 9 5 × 9                                                   Multiply the first denominator by the second denominator. 4  ×   2  =   4 × 2  =   8 5 9 5 × 9 45       Make sure the answer is in simplest form. 8 45 Dividing fractions is almost the same as multiplying them. To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is the inverse. To find the reciprocal, flip over the fraction. The denominator becomes the numerator and the numerator becomes the denominator. For example, the reciprocal of 2/3 is 3/2. Once you find the reciprocal, rewrite the problem as multiplication. Steps for Dividing Fractions       1. Rewrite the problem as a multiplication problem by taking the reciprocal of the second fraction.         2. Multiply one numerator by the other and one denominator by the other.   a  ÷   c  =   a  ×   d  =   ad b d b c bc For example: Solve. 7 ÷ 1 15 4 Write the first fraction. 7 15             Add a multiplication sign. 7  ×   15       Write the reciprocal of the second fraction. 7  ×   4 15 1                                                Multiply. Make sure the answer is in simplest form. 7  ×   4  =  7 × 4  =   28 15 1 15 × 1 15
Practice Problems Solve each problem on your paper. Then click on the "Check Answers" button to check your answers. 1. Polly has taken 5 clarinet playing tests in band class this semester. Her scores are shown below. What is Polly's mean test score? Test # Score 1 114 2 115 3 98 4 115 5 107 2. Solve. 5 - 1.53 3. Solve. 6.41 × 1.2 4. Solve. 7  ×   3 9 5. Solve. 2  ÷  5 9 8
Percentages In physical sciences like chemistry and physics, percentages are used to describe and compare many different types of data. A percentage is a way of expressing a number as a fraction of 100. Percentages can also be expressed as small numbers less than 1, as shown below. 23%  =    23  =  0.23  100 The different ways of expressing percentages can be used to solve problems. Calculate the percentage     1. Convert the percentage into a decimal.       2. Multiply the decimal by the whole number.   How do you calculate percentage? For example: A teacher buys a 6-foot sub sandwich for a classroom celebration. He cuts it into 24 equal pieces. By the end of class, 75% of the sandwich has been eaten. How many pieces did the students eat? Convert the percentage to a decimal. 75% = 0.75                                                         Multiply the decimal by the total number of slices that make up the whole sandwich to find the number of pieces eaten.   0.75 × 24 = 18 The students ate 18 pieces. To convert a fraction to a percentage, divide the numerator by the denominator. The result is a decimal number. Convert it to a percentage by multiplying by 100. For example: A teacher buys a 6-foot sub sandwich for a classroom celebration. He cuts it into 24 equal pieces. By the end of class, 18 pieces of the sandwich have been eaten. What percentage of the sandwich did the students eat? Write the numbers out as a fraction. The students ate 18 out of the total 24 pieces.   18 24           Simplify.   18 = 3 24 4           Divide the numerator by the denominator.   3  =  0.75  4           Convert the fraction to a percent by multiplying by 100.   0.75 × 100 = 75%   The students ate 75% of the sandwich.
Algebra Many concepts in physical science are explained and solved using basic algebra. For this, an understanding of variables is necessary. This section is a review of those basic algebra skills. Tips for Solving Algebraic Equations      1. Isolate the variable by using the opposite operation.     2. Always perform the same operation on both sides of the equation.   3. Remember the order of operations. 4. When the variable is by itself, the equation is solved. For example: Solve for m. m + 5 = 2 Isolate m by subtracting 5. Remember to do the same thing to both sides.   m + 5 - 5 = 2 - 5                                                 Complete the operations. Notice that the 5 is cancelled out by subtracting 5.   m = -3
Scientific Notation 100  =   1  = 1  101  =   10  =   10  102  =   10 × 10  =   100  103  =   10 × 10 × 10  =   1,000  104  =   10 × 10 × 10 × 10  =   10,000           10-1  =   0.1  =  0.10 10-2  =   0.1 × 0.1  =  0.01 10-3  =   0.1 × 0.1 × 0.1  =  0.001 10-4  =  0.1 × 0.1 × 0.1 × 0.1  =  0.0001 Scientific notation is a shorthand method used to express very large or very small numbers. In scientific notation, the numbers are written as values between 1 and 10, and multiplied by a power of 10. Look at the table. Each power of 10 adds another zero, and moves the decimal point one place to the right. If the exponent is negative, the decimal point moves that number of times to the left. Each power of 10 adds another zero between the number and the decimal. Remember that any number to the first power is equal to itself. Any non-zero number to the zero power is equal to 1. When reading numbers in scientific notation, look at the exponent on the 10. Moving the decimal to the right or left that number of times produces the original number. 2.345 × 102     ®     234.5 4.16 × 107     ®     41,600,000 2.7 × 1011     ®     270,000,000,000 9.996 × 10-3     ® 0.00996 6.75 × 10-9     ®  0.00000000675 To convert a number into scientific notation follow the steps in the table below. Convert Very Small or Very Large Numbers to Scientific Notation     1. Move the decimal point to the right or left until there is only one non-zero digit to the left of the decimal point. If there is no decimal point, add a decimal point at the end of a number, and move it in the correct direction.       2. Count the number of times the decimal was moved from its original position. The number of places the decimal has to be moved is the value of the exponent on the 10.   How do you write 810,000,000 in scientific notation? 1. Rewrite the number by adding a decimal point on the end: 810,000,000.0 2. Move the decimal as shown to find the first number. 3. Count the number of decimal places and then write the 10 with that value as the exponent. ®  8.1 × 108
Think About It Why would it be especially useful for astronomers to use scientific notation?
For example: The planet Mercury is about 57 million km from the Sun. What is this distance in scientific notation? Write the number out. Be sure to include the decimal point.   57,000,000.0                                                   Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.   57,000,000.0 ® 5.7       Count the number of times the decimal was moved from its original position to find the exponent on the 10.     5. 7 0 0 0 0 0 0 .0   7 6 5 4 3 2 1         Write the number in scientific notation.   5.7 × 10^7