Basic Number Properties

There are four basic properties of numbers: commutative, associative, distributive, and identity. You should be familiar with each of these. It is especially important to understand these properties once you reach advanced math such as algebra and calculus.

Commutative Property

a. Addition. When two numbers are added, the sum is the same regardless of the order in which the numbers are added.

3 + 5 = 8 or 5 + 3 = 8

b. Multiplication. When two numbers are multiplied together, the product is the same regardless of the order in which the numbers are multiplied.

3 x 5 = 15 or 5 x 3 = 15

Associative Property

a. Addition. When three or more numbers are added, the sum is the same regardless of the way in which the numbers are grouped.

6 + (4 + 3) = 13 or (6 + 4) + 3 = 13

b. Multiplication. When three or more numbers are multiplied, the product is the same regardless of the way in which the numbers are grouped.

6 x (4 x 3) = 72 or (6 x 4) x 3 = 72

Distributive Property

The sum of two numbers times a third number is equal to the sum of each addend times the third number.

5 x (7 + 2) = 45 or 5 x 7 + 5 x 2 = 45

Identity Property

a. Addition. The sum of any number and zero is that number.

12 + 0 = 12

b. Multiplication, The product of any number and one is that number.

18 x 1 = 18

Knowing these properties of numbers will improve your understanding and mastery of math.

The RQWQCQ Strategy for Solving Math Word Problems

RQWQCQ is a good strategy to use when solving math word problems. Each of the letters in RQWQCQ stands for a step in the strategy.

Read
Read the entire problem to learn what it is about. You may find it helpful to read the problem out loud, form a picture of the problem in your mind, or draw a picture of the problem.

Question
Find the question to be answered in the problem. Often the question is directly stated. When it is not stated, you will have to identify the question to be answered.

Write
Write the facts you need to answer the question. It is helpful to cross out any facts presented in the problem that are not needed to answer the question. Sometimes, all of facts presented in the problem are needed to answer the question.

Question
Ask yourself "What computations must I do to answer the question?"

Compute
Set up the problem on paper and do the computations. Check your computations for accuracy and make any needed corrections. Once you have done this, circle your answer.

Question
Look at your answer and ask yourself: "Is my answer possible?" You may find that your answer is not possible because it does not fit with the facts presented in the problem. When this happens, go back through the steps of RQWQCQ until you arrive at an answer that is possible.

Use RQWQCQ to help you correctly solve math word problems.

Temperature Scales

Temperature is the level of heat in a gas, liquid, or solid. Three scales are commonly used for measuring temperature. The Celsius and Fahrenheit scales are the most common. The Kelvin scale is primarily used in scientific experiments.

Celsius Scale

The Celsius scale was invented in 1742 by the Swedish astronomer, Anders Celsius. This scale divides the range of temperature between the freezing and boiling temperatures of water into 100 equal parts. You will sometimes find this scale identified as the centigrade scale. Temperatures on the Celsius scale are known as degree Celsius (ºC).

Fahrenheit Scale

The Fahrenheit scale was established by the German-Dutch physicist, Gabriel Daniel Fahrenheit, in 1724. While many countries now use the Celsius scale, the Fahrenheit scale is widely used in the United States. It divides the difference between the melting and boiling points of water into 180 equal intervals. Temperatures on the Fahrenheit scale are known as degree Fahrenheit (ºF).

Kelvin Scale

The Kelvin scale is named after William Thompson Kelvin, a British physicist who devised it in 1848. It extends the Celsius scale down to absolute zero, a hypothetical temperature characterized by a complete absence of heat energy. Temperatures on this scale are called Kelvins (K).

Converting Temperatures

It is sometimes necessary to convert temperature from one scale to another. Here is how to do this.

1. To convert from ºC to ºF use the formula: ºF = ºC x 1.8 + 32.
2. To convert from ºF to ºC use the formula: ºC = (ºF-32) ÷ 1.8.
3. To convert from K to ºC use the formula: ºC = K – 273.15
4. To convert from ºC to K use the formula: K = ºC + 273.15.
5. To convert from ºF to K use the formula: K = 5/9 (ºF – 32) + 273.15.
6. To convert from K to ºF use the formula: ºF = 1.8(K – 273.15) + 32.

Comparing Temperatures

Here are some common comparisons between temperatures on the Celsius and Fahrenheit scales.

TEMPERATURE	ºC	ºF
Boiling point of water	100	212
Freezing point of water	0	32
Average human body temperature	37	98.6
Comfortable room temperature	20 to 25	68 to 77

You probably refer to temperature every day. Be sure about the scale you are using.

Roman Numerals

You can guess from their name that Roman Numerals originated in ancient Rome. They were created as a simple means of counting in which certain letters are given values as numerals (a numeral is a written symbol referring to a number). The original system of Roman Numerals was modified during the Middle Ages and is the system we still use today.

Seven letters form the basis of Roman Numerals. Each letter stands for a number as shown here.

ROMAN NUMERALS

I = 1	V = 5
X = 10	L = 50
C = 100	D = 500
M = 1,000

Reading and Writing Roman Numerals

Numbers are represented by combining the letters shown above. There are several rules to follow.

1. If one or more letters are placed after another letter of greater value, add that amount.

   VI = 6 (5 + 1 = 6)
   XXVII = 27 (10 + 10 + 5 + 1 + 1 = 27)
   MDC = 1,600 (1,000 + 500 + 100 = 1,600)

2. A letter cannot be repeated more than three times.

   30 = XXX (10 + 10 + 10 = 30)
   40 = XL (50 - 10 = 40) You cannot write 40 as XXXX.

3. If a letter is placed before another letter of greater value, subtract that amount.

   IX = 9 (10 - 1 = 9)
   XL = 40 (50 - 10 = 40)
   CML = 950 (900 + 50 = 950)

4. You can only subtract powers of 10 (I, X, C).

   95 = XCV (100 - 10 + 5 = 95)
   You cannot write 95 as VC because V is not a power of 10.

5. You cannot subtract more than one number from another number.

   18 = XVIII (10 + 5 + 1 + 1 + 1 = 18)
   You cannot write 18 as IIXX.

By the way, there is no zero in the system of Roman Numerals.

This is all pretty complicated. Why should you even bother learning about Roman Numerals?

You will find that Roman Numerals are used all around you. Here are just some of the ways they are used today:

• They are used in outlines.
• They are used on the faces of clocks and watches.
• They are used to number pages that come before the main pages of a book.
• They are used to identify kings and queens (Henry VIII of England).
• They are used to identify sporting events (The New York Giants won Super Bowl XLII).

Roman Numerals were created long ago. They are still with us today.

Measurement Units

A measurement system is a set of units which can be used to specify anything which can be measured. There are various measurement systems used across the world. The system used in the United States is the U.S. customary system. Here are the common units used in this system along with examples to give you a frame of reference.

Length

Inch (in): The distance between the knuckles on your index finger is approximately one inch.

Foot (ft): One foot equals 12 inches. An official professional football is about one foot long.

Yard (yd): One yard equals three feet. A baseball bat is about one yard long.

Mile (mi): One mile equals 5,280 feet. A mile is approximately the distance a championship distance runner can run in just under four minutes.

Weight

Ounce (oz): A slice of bread usually weighs a little less than one ounce.

Pound (lb): One pound equals 16 ounces. A loaf of white bread usually weighs a little more than one pound.

Ton (T): A ton is 2,000 pounds. The famous Liberty Bell in Philadelphia weighs about one ton.

Capacity

Cup (c): A standard baby bottle holds about one cup of juice.

Pint (pt): One pint equals two cups. A pint of ice cream is just about right for four people to share.

Quart (qt): One quart equals two pints. Motor oil typically comes in a quart-sized container.

Gallon (gal): One gallon equals four quarts. A large container of milk contains one gallon.

Knowing these measurement units will help you in school and in everyday life.

Tips and tricks

How to Take Notes

Notes are a valuable entity to classroom learning. They accompany your textbook knowledge and complement the teaching. Classroom learning comes from a combination of textbook information and outside information from the teacher. It is necessary to take notes in class because not all of the information you will learn will come from textbooks. Consequently, in high school, college, and graduate school classes, you will "take notes" or jot down information for yourself to study at a later date.

In today's day of changing and transforming technology, people are taking notes in various methods. Many people still take notes in the old fashioned method of shorthand (pen on paper). Others record lectures and listen to them at later dates, jotting down the information they believe to be important. And, now students are bringing their laptops to class and typing directly into their computers.

• Shorthand

Shorthand note taking is still the most popular and foolproof method of taking notes. Students bring pens, pencils, and paper (notebook ruled paper) to class and listen to the teacher lecture. They write down important information in specific codes that they understand. Handwriting becomes illegible to everyone but the student. Eventually, students re-write and copy their handwritten notes into legible copies.

• Recording

While discouraged, this method of note taking is prevalent in schools. Students sit in class to listen and soak in as much information as possible. They bring in a mini recording device and tape the lecture (as discouraged as this practice may be). Later, they listen to the lecture and slowly write down the important facts.

• PowerPoint Presentations

Lucky for students, some professors now utilize the PowerPoint functions on their computers to give lectures. Notes are given directly to the students, as the PowerPoint presentation serves as an outline for the lecture. If your teacher gives you a printed copy of the presentation (an outline for the lecture), use it as your baseline for note taking. Write little facts given in lecture directly on the outline provided.

• Laptops

While laptops in class are not especially prominent in high school, they are growing in college, and are ubiquitous in graduate school coursework. Students bring laptops directly to class and take notes on their computers. The benefits of note taking on computers is that many people type faster than they can hand write, and consequently write down more notes (and take in more information). Another benefit of this type of note taking is that the notes will be legible later on when you study your notes.

If you are unsure as to what information to write down (in hand or on a computer), then simply ask your teacher. Generally, the teacher will tell you if information is important for you to write down. If the teacher does not tell you, then you should not write down everything said in class. It will be virtually impossible to write down everything said. Therefore, you must write down just the bare minimum. You will know what is important to write down because it will be written on the board, on a transparency, or the teacher will tell you. It is far better to sit in class and understand what is being discussed, then try to write down every word from the outside. You may look at your notes and understand nothing.

Tips and tricks

How to Take Notes

Notes are a valuable entity to classroom learning. They accompany your textbook knowledge and complement the teaching. Classroom learning comes from a combination of textbook information and outside information from the teacher. It is necessary to take notes in class because not all of the information you will learn will come from textbooks. Consequently, in high school, college, and graduate school classes, you will "take notes" or jot down information for yourself to study at a later date.

In today's day of changing and transforming technology, people are taking notes in various methods. Many people still take notes in the old fashioned method of shorthand (pen on paper). Others record lectures and listen to them at later dates, jotting down the information they believe to be important. And, now students are bringing their laptops to class and typing directly into their computers.

• Shorthand

Shorthand note taking is still the most popular and foolproof method of taking notes. Students bring pens, pencils, and paper (notebook ruled paper) to class and listen to the teacher lecture. They write down important information in specific codes that they understand. Handwriting becomes illegible to everyone but the student. Eventually, students re-write and copy their handwritten notes into legible copies.

• Recording

While discouraged, this method of note taking is prevalent in schools. Students sit in class to listen and soak in as much information as possible. They bring in a mini recording device and tape the lecture (as discouraged as this practice may be). Later, they listen to the lecture and slowly write down the important facts.

• PowerPoint Presentations

Lucky for students, some professors now utilize the PowerPoint functions on their computers to give lectures. Notes are given directly to the students, as the PowerPoint presentation serves as an outline for the lecture. If your teacher gives you a printed copy of the presentation (an outline for the lecture), use it as your baseline for note taking. Write little facts given in lecture directly on the outline provided.

• Laptops

While laptops in class are not especially prominent in high school, they are growing in college, and are ubiquitous in graduate school coursework. Students bring laptops directly to class and take notes on their computers. The benefits of note taking on computers is that many people type faster than they can hand write, and consequently write down more notes (and take in more information). Another benefit of this type of note taking is that the notes will be legible later on when you study your notes.

If you are unsure as to what information to write down (in hand or on a computer), then simply ask your teacher. Generally, the teacher will tell you if information is important for you to write down. If the teacher does not tell you, then you should not write down everything said in class. It will be virtually impossible to write down everything said. Therefore, you must write down just the bare minimum. You will know what is important to write down because it will be written on the board, on a transparency, or the teacher will tell you. It is far better to sit in class and understand what is being discussed, then try to write down every word from the outside. You may look at your notes and understand nothing.