Math Basics is the first section that you reach if you use the "next"
button. We recommend that you work your way through this section first. In Math
Basics, you can jump to two different starting points based on your grade level -
6th grade and younger, and 7th through 12th grade. In Math
Basics, you can review topics you may have come across in school, and can
use the pages as a detailed reference with animated examples. Math Basics
contains information you need to learn the great tricks that
are presented in the Mental Math classroom. The Math Basics home page
provides a hyperlinked list of all topics covered in the Math Basics
classroom which you can use to travel directly to a specific
topic. However, we recommend that you travel through our entire site
first by only using the NEXT buttons and exploring the
other links embedded into each page.
The Memorization classroom contains very important lists
and tables. Memorizing the data in
this classroom will help you in solving advanced problems quickly
without having to look for a specific problem or even use a
calculator. You will be amazed! Trust us on this one.
This classroom contains information tables that we, ourselves,
memorized back in the 6th grade, and we still remember
them today! The contents include formulas, fractions to decimal
conversions, a list of prime numbers, squares of whole numbers, and
much more. Take time to familiarize yourself with the information
presented in this classroom. It will save you countless hours in your math
future.
You will probably find that the Mental Math classroom can be
a lot of fun. It presents many
simple, yet helpful "tricks" to aid you in your quest for math
brilliance. Some tricks may take a half hour to master, but then,
there are some that are very easy to pick up. An important key to learning
these is to practice, practice, and practice! Over time,
the tricks can easily become second nature, and your calculators just might
start collecting a little dust! Have a look through this section; we
bet you will find something interesting.
The Homework classroom provides you with a range of materials that you
can take with you even when you are off the
Internet. Here, you will find a quick link to all the materials
that can be downloaded from our site. A very handy feature incorporated
into Simply Number Sense located here is the option to print a paper
copy of our complete site quickly and easily, without having to print each
page individually. You do have the option to print out only certain portions
of our site, also. Also included in this classroom is a link to all
tests and quizzes that you can download to test your skills and
knowledge. The tests that are printable as they are
available in PDF format.
Quiz Central is exactly as it sounds. It is a growing collection of
quizzes that will test your speed and accuracy using the mental
math tricks that you have learned. You can
print out available Adobe® Acrobat® PDF quizzes that you are
free to distribute to your friends and/or teachers!
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In order to serve you best, we have provided two different starting points at the bottom of this page. The first one is for students who either need a quick, basic review of math topics, or are younger students. The second one is for students in 7th grade or above. If you do decide to click on "7th grade or above", you will skip past the following topics:
Topics accessible after all those mentioned, or by clicking on "7th Grade or Above," include:
Now that you have had a rundown of what is available, click on either of
the two starting points below, or use the Math Basics Directory above
to begin your learning adventure! Pay attention, gain knowledge, and have fun!
The first two basic operations in mathematics are addition and subtraction. The chalkboard below illustrates the simple combining or "putting together" of two groups of circles. This demonstrates the operation addition performed on two whole numbers, two and three (known as addends,) which result in a sum (total) of five circles. Subtraction is the inverse or opposite of addition, so when three circles are subtracted, or taken away, from the group of five, the result is two remaining circles. Likewise,when two circles are subtracted from the group of five, there will be three circles left. A subtraction problem is always written in the form minuend - subtrahend = difference.
The other two basic operations are multiplication and division. These two operations also act as inverses of each other. Moreover, multiplication and division can be thought of as successive addition and successive subtraction, respectively. The numbers that are multiplied together are known as factors, while the answer is the product. Although the two numbers multiplied together are often called factors, they are also known as the multiplicand and multiplier, with the multiplicand being the number to the left of the multiplication sign. In division, the number you are dividing is known as the dividend, the number you are dividing by is called the divisor, and the answer is the quotient.
On the chalkboard below, you can observe the multiplication and then, the division of another group of circles. Try to visualize the successive addition and subtraction inherent to these operations, as well as how each operation acts as an opposite to the other.
These four operations, addition, subtraction, multiplication, and division,
in conjunction with numbers and the number
line, act as the foundation for almost all math.
A review of the number line is essential to our lesson in mathematics. Understanding the basic concept and utilizing a picture of the number line (as seen below) can, many times, be most helpful in solving seemingly complex problems.
The number line shown above is also known as the real number line. The line represents all real numbers. Real numbers include all positive and negative numbers, and zero. All whole numbers as well as fractions (or decimals) are included.
Use of the number line is most helpful when adding and subtracting negative numbers. In the example below, we can use the number line to solve the problem 4+(-3)=?. In order to solve this, let's start and place a marker on the four (4) on the number line. The plus sign indicates that the direction we will move our marker is to the right. However, we now must change the direction because we encounter a negative sign with the three (3). So, let's move our marker exactly three places to the left. Our marker ends up at positive one (1). Thus, 4+(-3)=1.
A quick way to solve a problem such as the previous one is to rewrite the problem as 4 + -3=?. Now, just memorize the conversions below, and you can quickly simplify your problem.
According to the conversion table, 4 + -3 can now be simplified to 4-3 (since a positive and a negative make a negative), which of course is equal to one (1). You wouldn't believe how useful these combination conversions are, so, you best familiarize yourself with them now!
Another use for the number line is when one is dealing with
inequalities [>, <, and the equal sign (=) with a slash (/) through.] Remember that the values
of numbers get bigger as you move towards the right
on the number line. It follows then, that the values of the numbers get smaller as you
move towards the left on the number line. Thus, for example, if you want to see if
five (5) is greater than or less than
negative eight (-8), you'll
see that your markers will indicate that -8 is smaller, since it's more towards the left side
of the number line.
The set of whole numbers (i.e. -2,-1,0,1,2,...) can be broken down into two general subsets; even numbers and odd numbers. This concept is a very simple one, yet it is very important. Later in the section Mental Math, we will show you tricks to solving certain types of math problems that depend on even or odd numbers.
In the animated example presented below, a group of twelve sticks is separated into two piles. Each group contains an equal number of sticks. This division of the sticks can be represented mathematically by the equation 12÷2 = 6. The twelve sticks were divided into two groups, and each resulting pile had six sticks. Based on the ability to give each pile an equal number of sticks, the group of twelve is considered to be even.
In the next example, a group of nine sticks is separated into two piles. This time two groups of an equal number of sticks is impossible. The closest we can get is either one group of five sticks and one group of four, or two groups of four with one stick left over. Due to the fact that we can not split the nine sticks into two equal groups without the existence of a remainder, we designate the group of nine as odd.
As you may have noticed, the test to determine whether a number is even or odd is simply to divide the number by two. If there is no remainder in the answer, the number is even; otherwise, the number is odd.
A shortcut in determining whether a number is even or odd,
is to look at the right-most digit (the one's place), and if that digit is
zero, two, four, six, or eight, the number is even. If that digit is any of
the remaining (one, three, five, seven, and nine), then the number is odd.
The chalkboard below shows another illustration of the concept of even and odd numbers.
A solid knowledge of the concepts of digits and digit places is necessary to understand how our numbering system works. As children, we were taught to start counting at one, as in, "One circle, two circles, three circles." Thus, the set of counting numbers starts with one and includes all whole numbers greater than one.
The whole numbers one through nine all use only one digit. Whole numbers greater than nine begin to use multiple digits, as well as multiple digit places. As numbers continue to use mutiple digits, new digit places are introduced. Starting from the decimal point and going left, the first digit place is called the ones place. After that, the next nine digit places (again, going to the left) are the tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, and billions, consecutively.
Beyond the set of whole numbers, there are numbers that are less than one, but, greater than zero; they are known as fractional parts. The name lends itself well, as these numbers are parts, or fractions, of one whole. For instance, say you have a half of a pizza. Intuitively, one understands that this is less than a whole pizza. In this case, the pizza represents one (1). Thus, you have more than nothing (0), but less than the whole (1). A half of a pizza can be represented in two ways. The first is as 1/2. This symbolizes the relationship that the part has to the whole. The whole pizza consists of two parts (or halves) but you have only one; one of two parts (1/2). 1/2 can also be written as .5 (because 1 ÷ 2 = .5, but more on that later.)
Starting from the decimal point and going to the right, the digit places are utilized for precision purposes. Especially in higher math and science, numbers are often written with, at least, three or four decimal places. All fractional digit place names end in "ths." The first six digit places to the right of the decimal are tenths, hundredths, thousandths, ten thousandths, hundred thousandths, and millionths. You should recognize these names as being the same ones as those to the left of the decimal place except that they all end in ths.
The number represented below is an example of, probably, one of the
most complex you will come
across in mathematics. This number, when correctly written out, is very long:
Five billion, three hundred forty-six million, two hundred seventy-one thousand,
three hundred twenty-four and nine thousand, seven hundred sixty-eight ten
thousandths. [Note that only the last digit place name will carry the ths.
The numbers to the right of the decimal will be written out and read as if they were whole numbers,
except that the name of the final digit place will be added. In this case,
ten thousandths.]
To review, take a look at the chalkboard below. The digit places for most of the digits in the number on the chalkboard above have been labeled with their names.
Any number less than one (1), but greater than zero (0), is a fraction. Understanding this concept is very important. A fraction is a part of a larger whole, where the whole is represented by the number one. One way that might help you understand this is by thinking of the number zero as representing emptiness or nothing, while the number one represents completeness.
Most often, you will be asked to work with numbers that are written in decimal or fractional form (i.e. .75, 3/4, etc.). If the number happens to be written as a percent, it is a good idea to convert the percent to another form (the form that most of the other numbers in the problem appear as) before starting the problem. By first doing this, you can often avoid a silly and very common mistake. Also, you need to recognize that fractions can be read in different ways. For example, 3/4 (which is equal to .75), can be read as "three-fourths", or "seventy-five hundredths."
Fractions can always be converted from one form to either of the two remaining forms. When performing the basic operations with fractions, it is often helpful to change all the numbers to the same form. The decimal form and the fractional form are the easiest forms to compute with. Percents are usually converted to a different form before operations are performed.
The chalkboards below demonstrate the conversions between
fractional forms. The basic rules for converting fractions
between forms are outlined below the chalkboard.
Follow along with each chalkboard while noting the
step-by-step instructions for converting.
To convert fraction to percent and percent to fraction, first, you must convert the number to its decimal form. Please follow the instructions above for converting fractions and percents to decimals.
When multiplying numbers in decimal form, the lining up of the decimal point is not necessary. Just multiply the two numbers as you would if they were whole numbers. Afterwards, count every digit to the right of either numbers' decimal place. Obtain the sum of these two numbers and then move the decimal place in your answer left that number of digits. Follow along with the example below.
Dividing two numbers in decimal form is essentially the same as whole numbers except, like multiplying decimals, you must also transform the answer by moving the decimal place after you finish the division. Perform the division while ignoring the decimal points, and then for every digit to the right of the decimal place in the divisor, move the decimal place in your answer one digit to the right. You can follow along with the example below.
Operating on fractions can be very difficult for students who have a hard time remembering the different rules. However, patience and practice is the key to mastering this math topic. To make life easier, always reduce the fractions as much as possible before operating on the fractions. If a problem is simplified beforehand, the operations themselves become simpler, and you will not have to worry about having to reduce your final answer. Your answer will already be in its simplest form. You may want to take a quick look at our GCFs and LCMs section for information on greatest common factors and least common multiples before jumping into operating on fractions.
Multiplying and dividing fractions are much simpler than
adding and subtracting. Once again, remember to reduce both
fractions first before doing the operations.
Mixed numbers and improper fractions are very closely related as they are two forms in which numbers can be written. For example, the number 6.75 can either be represented as both a mixed number or as an improper fraction. As it is now, 6.75 is written in decimal notation. As a mixed number, 6.75 is equal to 6 3/4. As an improper fraction, 6.75 is equal to 27/4. We explain why this is below. Often times, it is necessary to convert between these three numerical representations.
As discussed on the last page, be sure to always try to reduce and simplify before transforming the number.
Now, if you want to further transform the mixed number to an improper fraction, follow these next steps. First, realize that your answer will consist of a numerator and a denominator. To obtain the numerator of the answer, simply multiply the denominator by the whole number, and then add that to the numerator. This shortcut can be visualized as a clockwise computation (shown below). Now, the denominator of the improper fraction is simply the denominator of the fractional part of the mixed number. If you are a bit confused, follow along with the chalkboard below and observe the conversion from decimal form to a mixed number, and finally to an improper fraction.
Most math teachers require that their students write answers to math problems in simplest form. This serves two purposes. One, teachers can easily check for correct answers. There may be an infinite number of equivalent answers to a particular problem, so, having all students write their answers in one form serves to rid the teachers of the painstaking task of having to examine thirty different answers. Second, oftentimes, by simplifying an answer, the answer will make "sense." This serves the purpose of giving students a boost of confidence in the answer that they have come up with.
Simplifying or reducing fractions is a technique that comes in handy quite often. In order to reduce fractions, one must find a common divisor for the numerator and denominator of the fraction. If a common divisor is found, both parts of the fraction (the numerator and the denominator) are each divided by the divisor. This process, finding a common divisor and dividing both parts of the fraction by it, is repeated until no common divisors except for the number one (1) can be found. At this point, the fraction is considered reduced (simplified) as far as possible. To see how to reduce the fraction 18/24, follow along with the chalkboard below.
Although the example shown is for proper fractions (fractions less than one), the same method works for improper fractions (where the numerator is larger than the denominator, and therefore, the value of the fraction is more than one.)
The two numbers without common divisors are said to be
relatively prime
to each other. This is a bit
of an advanced concept at this point. For more information on
relatively prime numbers, as well as a useful factoring technique, you
can jump ahead and visit our page on
Prime Numbers.
We have added a section on currency and sales taxes & gratuities since these are a couple of the most common areas of everyday life where we are required to use our math skills.
The United States' (US) system of currency is based on the dollar. The symbol representing the dollar is this: "$". For instance, one dollar can be represented as $1 or $1.00.
Currency in the US comes in two forms - coins, and bills. Coins most often represent fractions of a dollar, while bills represent either one dollar or multiple dollars. The penny is the smallest denomination (value) of U.S. currency. It is worth one one-hundredth of a dollar and is expressed in written form as $0.01 or 1¢. The symbol "¢" is known as the cent. The penny is worth 1 cent, and 100 cents = 1 dollar. The next coin, the nickel, is worth five pennies or $0.05. A dime is worth ten pennies or $0.10, and the quarter is worth 25 pennies or $0.25. Although these are the most common types of coins used, half-dollars ($0.50) and dollars can also be found in the form of a coin; however, they are not widely used.
The common forms of bills come in the following denominations: the dollar-bill, the five-dollar bill, the ten-dollar bill, and the twenty-dollar bill. Also available from your local bank are bills in denominations of two, fifty, one hundred, and even one thousand dollars!
You can add, subtract, multiply, and divide currency in the very same manner that you would operate on decimals. You should also be able to recognize the most common currency conversions such as, five nickles = one quarter, four quarters = one dollar bill, and five one dollar bills = five dollar bill, and so on.
Taxes seem to be a necessary evil inherent to our society. When one goes to the store to purchase an item, the store is often required to charge extra for the sale of each item. This charge is known as a sales tax. Taxes usually are expressed as a percentage of the amount charged for the item.
To determine the amount of sales tax on an item: first, take the percentage of tax and change it into decimal form. If you remember, this is accomplished by dividing the percentage by 100 and removing the percent sign. An easy way to divide by 100 is to simply move the decimal point two places to the left. This decimal number is then multiplied by the price of the item. The calculated amount is the sales tax. In order to obtain the the total cost of the item, add the price of the item to the sales tax amount that you have calculated. Remember to always round off currency to the nearest cent.
Often times at restaurants, you will be served your food by either a waiter or a waitress. Luckily, your server will include the amount of tax into your bill. However, it is courtesy to give your server what's known as a gratuity, or more commonly, a tip. A common tip is about 15% of the food and beverage amount. Note that we said about.
No one expects you to carry around a calculator to dinner, so, when you try to compute the tip, you would normally estimate. When trying to estimate 15%, it's easier if you first estimate 10% of the bill. For example, let's say the bill is $23.74. We would first find what 10% of $23.74 is. This is easy, as taking 10% of a number is the same as muliplying the number by 0.1, which is the equivalent of moving the decimal point one place to the left. Thus, 10% of $23.74 is $2.374. Since we are estimating, let's go ahead and round this to $2.40. Now, since this is 10% of the bill (and we still need another 5%, let's now take half of the $2.40, which is $1.20. We now know that 10% = $2.40 and 5% = $1.20. 10%+5% = 15%, and $2.40+$1.20 = $3.60. Thus, about 15% of the bill is equal to $3.60! Just add this to the bill and you can leave the table. If you couldn't follow along just now, check out the same example on the chalkboard below. [The (~) symbol means "estimate."]
Whole numbers greater than one (1) come in two flavors, composite and prime. The set of composite numbers are all those which have more than two, positive, whole number factors (besides one and the number itself.) For example, the number four is composite because it also has two as a factor besides one and four. In fact, two (2) is the only even number which is not composite.
Those numbers which only have two factors are considered prime. Prime numbers have only two factors: the number one (1) and itself. Thus, numbers can not be prime and composite at the same time. Numbers less than two are excluded from these definitions. The first twenty-five prime numbers are on the chalkboard below. If you can find a number on there with more than two, positive, whole number factors, let us know!
As we have discussed before, simplifying a problem before you start to solve it is very helpful. A technique known as prime factoring is available. For instance, the number 96 can be broken down into the multiplication problem 2x2x2x2x2x3. This can also be written as 25x3. By changing 96 into this form, it is easier to find what are known as the least common multiple (LCM) or the greatest common factor (GCF) of two numbers. The LCM and GCF are discussed in a later section accessible directly from here. Prime factorization can be used in all sorts of problems, and it is also especially useful in helping to reduce fractions (which is really finding GCFs). An example of how to create what is known as a prime factor tree is shown on the screen below. Note that this is only one shape the factor tree can take; however, the resulting prime numbers will always be the same.
One last thought on prime numbers. Any two numbers which
do not have common prime factors have a relationship in
which they are said to be relatively prime to each
other. One example of this would be any two prime numbers
(as they would each only have one and themselves as
factors,) or two composite numbers such as 6 and 35. 6
has the distinct factors of 2 and 3, while 35 has the
distinct factors of 5 and 7.
Our numbering system consists of digits known as Arabic numerals (1, 2, 3...) However, another common numbering system used around the world utilizes Roman numerals. These numerals are most often used to either denote the copyright date shown at the end of a television show or on the inside of a book, or as an alternative to numbering entries in an outline.
Roman numerals are simply letters and are usually expressed in upper case. Here are the most commonly used Roman numerals and their equivalent, or values, in Arabic numbers:
There are some rules that must be observed when writing Roman
numerals.
At first these rules may seem a bit hard to grasp,
however, over time they become all too natural.
The GCF of a group of numbers is defined as the largest integer
that all the numbers are divisible by. For example, the GCF of
9 and 12 is 3. By first prime factoring the numbers, we
can find easily see that three is the GCF. The chalkboard below
shows a bit more complex problem, finding the GCF of three
numbers, 12, 18, and 54.
The LCM is the smallest integer multiple of a group of
numbers. To find the LCM of a group numbers, you should
start listing multiples of each number horizontally. The
first number you come across that appears on each row is
the LCM. You can see an example of this on the chalkboard
below.
A quick tip for finding the LCM of two numbers. Let's take
12 and 18. First find the GCF of the two numbers. In this
case, the GCF is 6. In order to find the LCM, divide one of
the number by the GCF (12÷6=2) and
multiply that answer by the other
number. 2x18=36, and 36 is the LCM!
The second type of inverse in basic mathematics pertains to addition. Known appropriately as an additive inverse, the sum of the inverse and the number is always zero (0). The additive inverse of a number is always the number with the sign changed. Therefore, the additive inverse of 6 is -6. Additive inverses are also known as opposites.
Looking forward into higher mathematics, you will be asked not only to work with inverses of numbers but rather of functions, so it is pertinant that you understand the basic concept of an inverse presented here.
In contrast to inverses, the concept of an identity is
not as quickly understood. However, after a little thought,
the concept seems too intuitive. Identities are
"ineffectual transformations;" esentially a transformation that
does nothing. The identity for addition is zero. This is
because for a number, N, N+0=N. The result is
that which you started with. For multiplication, the identity
is one because any number multiplied by one is itself. For
a number, N, Nx1 = N.
The definition of mean is simply the average. So, in order to find the mean, add up all the data in the set and then divide by the number of data elements. Do not assume you did something wrong if you get a decimal. Decimals almost never indicate that you have done something incorrectly. The animation underneath the next paragraph includes an example of finding the mean of a set.
The median of a set of numbers is the middle
number of the set after the numbers have been arranged in
ascending order. You may be asking why this is useful. Well,
statisticians use the median as well as
other statistical data in order to graphically display the center
of a set. This definition is really too watered-down, but the
question is really beyond the scope of this site once
again. However, the important thing is that you be able to find
the median if asked. It is considered common knowledge, and you
may see a question about it on your college entrance exams such
as the SAT or ACT. There is one little catch to finding the
median of the set. If the set of numbers has an even number of
elements in the set, then the median is actually found by taking
the mean of the two middle numbers. Now aren't you glad we just
explained what a mean is?! The animation below gives an example
of finding the mean and median of a set.
Moving on, the mode of a set of numbers is the number that appears most often in the set. Why one would like to know this varies but we can easily make a reasonable guess. Simply scan the dat for the number that appears most often and make sure no others have the same number of elements. If there is more than one number that have the most number of elements, combine these numbers into one set, and you have your mode. The animation at the bottom of the page includes an example of finding the mode of a set.
The range of a set of numbers is the simplest of all to find
since it has no little catches. The range is simply the
diference of the largest number in the set minus the smallest
number. The animation below includes an example of finding
the range of a set.
Subsets have an interesting property. The number of subsets a
particular set has can be determined easily by the formula
2n, where n is the number of elements in the
set. Each set consists of what are known as proper and
improper subsets. For each set, there is always
exactly one improper subset (the null set). All other
subsets are proper. Thus, the formula to find all proper
subsets of a set is 2n-1.
Note that even though it is customary to write sets in
ascending order when possible, an unordered set is
not incorrectly written.
When given two or more sets, the union of
the sets is a collection of all the elements that are in at
least one of the sets. When a union is taken of two sets, it is
assured that all elements in the union are different. Each element
in the union can not be repeated in the unified set even if it
appears in more than one of the original sets. An example of
taking the union of three sets is shown on the chalkboard.
As shown above, the union of two sets is symbolized by a 'U'. Note that you may also see union in written context symbolized by the word "or" or by the addition symbol '+'.
The other basic set operation is the intersection. The intersection
is found by taking all elements within the sets that appear
in all the sets. Simply put, if the element does not
exist in every set, then the element can not be a part of the
intersection. Thus, the resulting set must not have a greater
number of elements than the original set with the fewest
number of elements. An example of taking the interesection of two
sets is shown below.
As with unions, intersections can be represented in several ways. First, as shown above, an intersection is most often represented by an upside-down "U". The concept is also inherent to the word "and" as well as the mathematical multiplication symbols "x" or "*"
You will find another use of unions and intersections when studying
statistics. These operations are often applied to categorical data
in order to help a statistician anylize data. These are
especially important when anylizing different samples within a
particular population. Also, in computer science and engineering,
the concept is very important when dealing with logic
or electricity circuits.
Sometimes, there just are not enough facts to make a good decision. Maybe there are just too many possibilities to predict what to do or what is going to happen. If we are able to find the probability that something is going to happen because of our decision, we can make a better decision.
Luckily for us, we are not going to be forced to make any life-changing decisions. The first question that is posed to us is, "What is the probability of rolling a '6' on a normal, unweighted die?" Well, this seems fairly easy. There are six sides and only one of them contains the desired outcome. Our chances of getting a '6' are 1 possibility out of 6, or 1/6. In fact, the probability of getting any of the numbers 1-6 on the die is 1/6.
Well that was easy, right? Now, we are asked, "What are
the odds of rolling a '6' on a normal, unweighted
die?" Wait just a second there. Is that not the same
question, just phrased differently? Actually, it is not the
same question at all. They may seem the same, but the
formulas for figuring probability and odds are
different. The chalkboard shows the differences in the two
formulas.
Now, do you have answer to the odds question? Yes, the answer is 1:5; one favorable outcome for every five unfavorable outcomes. Also, odds are often expressed with a colon ':' between the two values.
We are now asked to find the probability and odds that
a sum of seven is rolled on two dice. Now we
have to consider all the combinations of the two dice. We
no longer are dealing with six possibilities, rather it is
now thirty-six (6x6)! Using a simple table of all
possibilities (as shown below), we see that six times a
sum of seven is produced. Thus, the probability
of rolling seven is 6/36=1/6, and the odds
of rolling a seven is 6:30=1:5. The table below represents
the rolling of two dice, A and B. The bold values next to
each letter correspond to the value rolled on that die.
| + | A1 | A2 | A3 | A4 | A5 | A6 |
| B1 | 2 | 3 | 4 | 5 | 6 | 7 |
| B2 | 3 | 4 | 5 | 6 | 7 | 8 |
| B3 | 4 | 5 | 6 | 7 | 8 | 9 |
| B4 | 5 | 6 | 7 | 8 | 9 | 10 |
| B5 | 6 | 7 | 8 | 9 | 10 | 11 |
| B6 | 7 | 8 | 9 | 10 | 11 | 12 |
One last topic associated with probability:
independence. This concept is important as
probabilities change if events (such as the
rolling of a single die) are not independent. In our
example of rolling two dice, we assumed that the rolling
of one die did not affect the rolling of the other
die. Had they affected each other, the probability
of getting a '6' on the second die would not have
necessarily been 1/6. For now, just try to understand
what independence is and why it is important.
It is best we work with an example. Let's take the even
numbers starting at two and increasing (2,4,6,8,10,...) This
is an example of an arithmetic sequence because each
successive term is produced by adding the same number (-2) to
the previous term. An example of a geometric sequence is
2,4,8,16,32. Instead of adding two to the previous term,
we multiply the previous term by two. You can
now take a look at the chalkboard below and follow
along with the examples just given.
Another topic very closely related to sequences are
series. Series can be created simply by
replacing the commas with addition signs. Sequences and series
are fundamental to certain higher math, and you will most
likely study them in depth in your Algbera II and
Calculus classes in high school.
Also up until this point, if you wanted to multiply the same number, N, by itself, you would write N*N. If you needed to multiply N by itself five times, youwould have to write out N*N*N*N*N. However, mathematicians have made things much easier by creating what are called exponents.
N*N is equivalent to writing N2 with an
exponent. In this example, the number two is the
exponent. Exponents are also known as
powers. N2 is read "N to the second
power", or more commonly, "N squared." The most commonly
seen powers in lower level mathematics are squares and
cubes (denoted with an exponent of 3). The process of
multiplying a number though the use of exponents is often
called "raising an expression to a power." Raising to a
power of two is called "squaring an expression." Examples
of how to read exponents are written on the chalkboard
below.
Now that you understand raising numbers to powers, you
need to know how to undo this process. The inverse process
of raising an expression to a power is to "take a root of an
expression". For example, the inverse to squaring would be to
take the square root. Taking the square root of a number is
most often difficult without a calculator. Unless the number
is a perfect square (an integer squared,) then a calculator
is necessary to get an accurate estimation of the true
root. A root is symbolized by a radical
(shown below) in conjunction noting the degree of the root
(equivalent to the power). Writing the root as an exponent
only requires that the reciprocal of the root be used as the
exponent. Note that a radical sign without indication of a
root always represents a square root. Examples of roots are
shown on the chalkboard below.
An equation is defined as two expressions separated by equals sign (=) indicating that the two expressions are equivalent. When discussing expressions and equations, note that one simplifies expressions while one solves equations. Also, if the two expressions are indeed not equal, then one of the three inequality signs, greater than (>), less than (<), or not equal ('=' with a slash through) should be used in place of the '='.
Now, we can explain how to graph an equation. Shown
below this paragraph is the XY-plane. Also known as
the Cartesian plane, a two dimensional graph, or
simply as a graph, the
XY-plane is fundamental to the studies of high school
algebra, trigonometry, and calculus. The XY-plane can be thought
of as an infinitely thin piece of paper that extends in the directions
of its axes infinitely (a rather large sheet of paper!)
The graph above is adequate for representing equations with no more
than two variables. This is why the graph is two
dimensional. Identification of points (places) on the XY-plane
are named according to the Cartesian coordinate system. The
graph below is a representation of the equation y=2x+1. Using the
coordinate system, ordered pairs (x,y) that make the
equation true are plotted on the graph. Notice that points are
written with the x-coordinate (value) 1st and y-coordinate
second, and that the graph
of the equation (shown below) is a line. This is because the
equation has no variable which has an exponent other than one. An
equation that represents a line is known as a linear
equation. If a third coordinate was introduced by an equation,
it would be the z-coordinate, and th ordered triple would be (x,y,z).
In the graph above, the two variables are represented by their own axis. Each axis is a number line that extends infinitely in both directions. On the XY-plane, it is customary to draw the x-axis horizontally and the y-axis vertically. The point at which the axes cross is known as the origin. The x and y values at this point form the ordered pair (0,0).
There are two points specifically marked on the graph; (0,1) and (2,5). In order to graph a line, only two points must be plotted in order to draw the entire line. The point (0,1) is significant because it is the y-intercept. An intercept is a point at which a graph (the drawing of the equation) crosses an axis. The other point, (2,5), is an arbitrary point. However, we made sure that both points satisfied the original equation, y=2x+1. If you want to check that these two points work, simply substitute the points back into the equation and check to see that both sides of the '=' are equivalent.
In case you do not understand how to plot a point on the graph,
continue to read here. Otherwise, feel free to move on to the next
section. In order to find the point (2,5) on the graph,
we placed our finger at where the axes cross each other,
which again represents the coordinate (0,0). From
there, we must travel two units to the right (because positive
numbers are on the right side of the number line) while then traveling
five units up (positives increase up from the origin on the y-axis)
in order to reach the desired point. This process is quite
intuitive and should become second nature very quickly.
As you know, if you are given any two points on a line, you can quickly draw a stright line thorough the two points and graph the line. The distance between these two points is unimportant, but what do you do if you want to come up with the equation which represents the line? First, you need to know the three most common equation forms for a line and their properties. These are listed in the table below.
Before moving on, let us discuss slope just a bit. The slope of a line as you already know is the change in y divided by the change in x. This is also known as rise over run. Given two points, (x1,y1) and (x2,y2), you can use the formula associated with rise/run: (y2-y1)/(x2-x1). If you like, you can also memorize the short formula for slope when a linear equation is written in standard form, m=-B/A. However, this is not necessary since you can easily transform the general standard form equation into y=mx+b form. Go ahead and try this on paper, and you will see how we got m=-B/A. It isn't magic!
Now, we can discuss point-slope form. From above, you know that given both a point as well as the slope of the line, you can create an equation that represents the line. It is important that you can distinguish between (x1,y1), and (x,y). (x1,y1) represents a single point that one already knows lies on the graph. (x,y) is the representation for all points on the graph. Thus, if you know what the coordinates of one point, plug those coordinates into (x1 and y1).
The screen below shows how to draw a graph given a point and the slope, as well as when given the slope and y-intercept.
When two lines converge at a point,
an angle is formed. Angles are measured in the unit,
degrees. A full circle (as shown below), has an angle
measure of 360°. Take a look at the screen below to
see what an angle is.
The example below this paragraph shows how angles are formed
by lines that cross each other. By observing each example
on the chalkboard, try and guess how we might distinguish
between the types of angles. A discussion of these
angles are provided below the chalkboard.
On the chalkboard above, example #1 shows four angles that were created by the converging lines. Each angle is denoted by an arc. Angles lettered a and c are known as acute because the measure of the angles are less than 90°. Angles b and d are obtuse as their angle measurements are greater than 90°. In example #2, all four angles are of the same measure. We are assured of this is as each angle has been marked with the right angle marker. Instead of an arc, all four angles are marked with a symbol that forms a small box in the angle. Every right angle is exactly 90°.
Pairs of angles that add to 90° are called complementary
angles. The relationship between angles that have a sum of 180°
is known as supplementary. A single line segment is
also known as a straight angle as it measures
exactly 180°.
The XY-Plane and Graphing Equations were your introduction to the two-dimensional (2D) world. This world is very interesting and worthy of analysis. The subject of this page is specifically various 2D shapes and their properties.
Two-dimensional figures lie on one plane and are flat. All the figures that we will handle here are regular and closed. Each side of each figure is a line segment, (a line that does not stretch infinitely but rather has definite (finite) endpoints. The perimeter of a two-dimensional object is defined as the distance around the shape (figure). The measurement for the inside of the shape is known as the area of the figure.
The fewest number of sides a shape can have is three. Every shape has the same number of internal angles as it does sides, and the same applies to each shape's external angles. The total number of degrees that a shape has internally can be found by the formula 180°*(n-2), where n is the number of sides the shape has. The total number of degrees that a shape has externally is always exactly 360°. Because the number of external degrees is always constant, references to "degrees" refer to inside the shape from here on out.
All three sided figures are known as triangles. Every
triangle's internal angles total 180°. The formula for finding the
area of a triangle is to multiply the triangle's base and
height together and divide by two (A=b*h/2). This is the
same as multiplying the base and height together then multiplying
the product by 1/2. We will not go any
further into detail about triangles as they are the focus
of our next page. Triangles come in all
forms and sizes, and the chalkboard below shows only a few of the
infinite possibilities.
Another type of two-dimensional figure is a quadrilateral, or four-sided figure. The measure of the degrees in all quadrilaterals is 360°. Also, the area of most quadrilaterals can be found by the formula A=b*h. The most commonly known type of quadrilateral is the square. A square has four sides of equal length and four right internal angles. It is sometimes referred to as the "perfect parallelogram." The next chalkboard on this page contains drawings of the different types of quadrilaterals discussed in the next few paragraphs.
A rectangle is another type of quadrilateral with four right angles. However, the width and height of rectangles are not necessarily equal. By this definition, all squares are rectangles, but not vice-versa. This is an example of where the converse of a statement is false. The converse statement here is, "all rectangles are squares." This is not true because squares must have four sides of equal length.
There is another subcategory of quadrilaterals known as parallelograms. Both squares and rectangles do fit into this category. A parallelogram is defined as a quadrilateral with two pairs of parallel sides. Another parallelogram besides the rectangle and square that fits the definition of a parallelogram is the rhombus. A rhombus is virtually a rectangle without necessarily having right angles.
There are two other types of quadrilaterals we would like to show you at this time. The first is a trapezoid. A trapezoid is a quadrilateral with exactly one pair of parallel sides. It is not possible for the parallel sides to be of equal lengh. If they were, the shape would result in a rectangle with two sets of parallel sides which would deny the shape of its status as a trapezoid. A trapezoid's area can not be calculated by the formula A=b*h. Instead, use the formula A=(b1+b2)*h/2.
The last type of quadrilateral that we need to introduce is the kite. The shape of a kite is commonly referred to as a diamond. A kite has two sets of sides equal in length, but has no sides that are parallel. Equal sides are adjacent to each other. Because there is no "base" of a kite, a different formula is needed to find its area. The formula A=M*m/2 can be used where M and m each represent the distances between opposite vertices on the kite. M represents the major axis and m represents the minor axis.
Take a good amount of time remembering which general shapes are
associated with which names. It is important that you can identify
both the similarities as well as the differences between the
different types of parallelograms.
I believed we fibbed when we said that all shapes had sides
consisting of line segments. Well, on the chalkboard below
are two shapes. The one on the left is known as a
circle, and the one on the right is appropriately
referred to as a semi-circle. The line drawn though the
center of the circle and connecting opposite sides of the
circle is known as the diameter. Half of the
diameter is defined as the circle's radius.
The perimeter of a circle has a special name -
circumference. An interesting property of circles is
the relationship between the circumference and diameter of
all circles. A direct variation exists between the two, and
a special symbol called pi is used to identify this. Pi
represents an irrational number (a number that can not be
expressed in fractional form) and is identified by a special
symbol that is shown on the chalkboard below. In order to calculate
the circuference or area of a circle, you must use pi (approximately
equal to 3.14159 or 22/7) and either the raidus or diameter must be
known. The formulas are also shown on the chalkboard below. Note
the symbol for pi (
Finally, we present you with a table that you can use for regular
shapes. The table is in descending order based on the number of
sides the shape has. Then name of the shape and the sum of
each shape's angles are listed. On the chalkboard below the
table, some of the shapes are drawn for you to visualize and remember.
| # of Sides | Shape Names | Sum Interior Angles |
| 3 | Triangle | 180° |
| 4 | Quadrilateral | 360° |
| 5 | Pentagon | 540° |
| 6 | Hexagon | 720° |
| 7 | Septagon | 900° |
| 8 | Octagon | 1080° |
| 9 | Nonagon | 1260° |
| 10 | Decagon | 1440° |
| 12 | Dodecagon | 1800° |
Notice the relative angle measurements of each riangle. The first triangle above has angle measurements that all less than 90°. For this reason we would call the triangle an acute triangle. If a triangle has a single right angle we appropriately call it a right triangle. Triangle number two fits this definition. When neither of these definitions fit (when one angle is greater than 90°), we call it an obtuse triangle.
We have already identified the three triangles based upon the
measure of their angles. Now we can do the same with their side
measurement. The first triangle has all three sides of equal
length. We know this is true because of the 'tick marks' on
each side. Since each side has exactly one tick mark, we
infer that all sides are of equal length. We can now call this
triangle an acute-equilateral triangle. Notice how we
combine the two properties of the triangle. This gives persons
who may not be able to see a triangle a better mental picture
about what you are describing. The second
triangle has only two sides that are equal in length so we would
say it is an isosceles triangle. Combining its
description based on its angles we would now describe it as a
right-isosceles triangle. The third and final triangle
has no sides that are equal in measure. Since the triangle
has one angle greater than 90° we call it an
obtuse-scalene triangle. Refer back to the triangles
above and make sure you can identify each by both angle measure
as well as side length. Also, try identifying each of the
triangles on the chalkboard below.
As you already know, the area of any triangle can be found by the formula A=b*h/2. However, what consitutes the base of a traingle, and what about the height? Well, all triangles have three distinct bases as well as three distinct heights. However, each base corresponds to a single height. You can not mix and match for the purposes of finding the area of the triangle.
Each side of the triangle constitues one of the bases. For each
base, the height is defined as the distance between the highest
point on the triangle and the base along a line perpendicular
to the base. To perependicular lines form right angles at the
point where they cross each other. If the perpendicular line
does not touch the base itself, then the base is extended
for the purposes of measuring the height. Examples of finding
heights are shown on the chalkboard.
Three-dimensional (3D) objects are often referred to as solids. These solids occupy three planes (space) as opposed to one for 2D shapes. The surface of a solid can be calculated as an area. However, the inside of a solid is not referred to as area, but rather it is known as the volume. Formulas for finding the volume of a 3D solid are often either V=Bh, where 'V' is volume, 'B' is the area of the base, and 'h' is the height of the solid, or the formula is derived from this general formula.
Probably the most basic of all 3D solids is the cube. A cube is a six sided object with all edges of equal measure. The volume of a cube can easily be found by cubing an edge! Now you can understand why raising a number to the power of three is referred to as cubing. Another property that all solids have in addition to volume is surface area. The surface area is the sum of the areas of all exposed surfaces of an object. A cube has six sides, and its surface area can be calculated by adding the areas of each side together. Because each side of a cube is the same size square, simply use the formula S=6*edge2. Common examples of cubes include dice, boxes, as well as the basic shape of a televeision.
Another common 3D figure is the sphere. The volume of the sphere is very different as it is found by using the formula V=4/3*pi*r3. If you were to slice a sphere, the resulting cross section would be a circle. This is the reason there is a "pi" in the formula for the volume. Examples of spheres in everday life include baseballs, basketballs, and globes. The chalkboard below has drawings of both a cube and a sphere.
The next type of solid can be thought of as a bunch of circles of the same size stacked directly on top of each other. This type of solid is known as a cylinder. In order to find the volume of a cylinder, multiply the area of the circle by the height of the solid. Common examples of cylinders include dog food cans, rolled up pieces of paper, and lamp posts.
Cones and square pyramids, two other types of solids, can be thought of as fractional parts of a cylinder and cube, respectively. The formulas for finding their volumes are simply those of the cylinder and cube, but then divided by three.
The last type of solid we want to discuss are prisms. Cubes and cylinders are prisms with special types of bases (a square and circle). Prisms can be visualized as any type of shape repeatedly stacked on top of itself.
Finally, we present you with a table that you can use for regular
solids. The table is organized with solids having similar
properties next to each other. The name of each solid, as well
as formulas for finding certain properties of each solid are
listed next to the name of each solid.
| Solid Names | Volume Formula | Surface Area Formulas |
| Sphere | 4*pi*r3/3 | 4*pi*radius2 |
| Square Pyramid | edge2*height/3 | base+4*(area of one triangle) |
| Triangular Pyramid | base*height/3 | base+3*(area of one trianlge) |
| Rectangular Prism | length*width*height | 2*(length*width+length*height+width*height) |
| Cube | edge3 | 6*edge2 |
| Triangular Prism | base*height | 2*base+(3*area of one rectangle) |
| Cylinder | pi*radius2*height | 2*pi*radius*(radius+height) |
| Cone | pi*radius2*height/3 | pi*radius*(radius+slant) |
The ten digits we use range from zero through nine. Let's now consider another base, base two. In base two, the only digits available are zero and one. This means that the first few numbers in base two are not 0,1,2,3,4... Remember, 2,3, and 4 do not exist in base two. So, what are the first few numbers in base two? They are 0,1,10,11,100. Thus, 4 in base 10 represents 100 in base 2. This equality is written as 410=1002. WOW, that does seem like an awful large number, but it is correct.
When naming digit places in base ten, you start with the one's place, the ten's place, the hundred's place, the thousands place, and so on. This would be the same as the 100 place, 101 place, 102 place, 103 place, 104 place, and so on. This is also the same for other bases. For example, in base 2, the first digit place would be 20, then 21, 22, 23, and so on. If you understand this concept, converting back and forth between bases will become fairly simple in short time.
Base 10 is also known as decimal. Base 2 is known as binary, and is commonly used for computer programming. The two values (0 and 1) represent "switches" that can either be on or off. As you discovered, base two numbers use many digit places in order to represent some very small numbers in base 10. For efficiency, programmers now use base 16 or hexadecimal. In hexadecimal 16 digits exist; 0-9 and A-F. 1510=F16. By using hexadecimal, larger numbers can be stored using less digits. For example, one billion in base ten contains ten digits. However, using hexadecimal, the same number can be represented using only eight digit places.
It is easier to convert a number from some base to base ten than vice-versa, so that is what we will start with. When converting from any base to base 10, you must first label each place value for each digit in the number you are trying to convert. Let's take 1012 as our example. We first label the place value of the rightmost one as 20 or as one. We then label the zero as 21 or as two. Finally, we label the leftmost digit's place as 22 or as four. Now, we can multiply the place value (our labels) for each digit by its corresponding digit. By adding these values together, the resulting sum is our base two number transformed into base 10. This process would look like this: (4*1)+(2*0)+(1*1)=5. We can confidently say that 1012=510!
Now we get to take 5 in base 10 and convert it back to base two. In order to do this, we must first find the greatest power of two that is not greater than five. This would be 22=4. 23 would be too large as it equals eight. So we divide five by four and get one with a remainder of one. The whole number part we use as the digit for the 22. For now, we know our answer is 1??. We then take the remainder from the division problem and divide by the next smaller power of 2, 22. 1/(22) is equal to 0 with a remainder of one. We use the 0 for the 22's place. Our answer is now 10?. We once again take the remainder of one and divide by the next smaller power of 2, 21. This time we get one with a remainder of zero for our answer. We use the one as the digit for the 21's place, and since we have a remainder of zero, we are all done! Our answer is 510=1012.
A final note; if you are trying to convert a number from one base
to another and neither base is base 10, you should first convert the
number to base ten. If you know of an easier method,
let us know! We'de love
to learn something new.
The "Review Now" button will take you to the Math Basics Review to view our PowerPoint® presentation. Don't worry, no extra plug-ins are required! This presentation will allow you to quickly review the main points of every Math Basics page without having to search through all the text and graphics. You get exactly what you need, simply the basic fundamentals of math.
The next button will whisk you away to our Memorization
classroom. There, you will atempt to memorize information that can
only make working math problems quicker and easier than before. Just
remember to go at your own pace and you will be fine. We'll still be
here tomorrow so take your time and keep coming back for more!
One final thought: all your hard work will pay off in the end!
We suggest that when trying to memorize these squares, memorize them in order, and say them out loud in this manner: "Fifteen squared is two two five. Fifteen squared is two two five. Sixteen squared is two five six. Sixteen squared is two five six." Notice that we repeat each square twice, and that we break each number down into its digits. All we can say is that, to this day, our mind recognizes that "thirty-six squared is one two nine six!" By the way, we memorized these squares back in 1991 and 1992, and all thirty-six squares are on the tip of our tongues at this very moment!
| 1 2=1 | 132=169 | 252=625 |
| 2 2=4 | 142=196 | 262=676 |
| 3 2=9 | 152=225 | 272=729 |
| 4 2=16 | 162=256 | 282=784 |
| 5 2=25 | 172=289 | 292=841 |
| 6 2=36 | 182=324 | 302=900 |
| 7 2=49 | 192=361 | 312=961 |
| 8 2=64 | 202=400 | 322=1024 |
| 9 2=81 | 212=441 | 332=1089 |
| 102=100 | 222=484 | 342=1156 |
| 112=121 | 232=529 | 352=1225 |
| 122=144 | 242=576 | 362=1296 |
We don't necessarily recommend that you memorize more squares than these, but, if you really are ambitious, the table of squares continues past 100 here.
| 13=1 | 8 3=512 | 153=3375 |
| 23=8 | 9 3=729 | 163=4096 |
| 33=27 | 103=1000 | 173=4913 |
| 43=64 | 113=1331 | 183=5832 |
| 53=125 | 123=1728 | 193=6859 |
| 63=216 | 133=2197 | 203=8000 |
| 73=343 | 143=2744 | 213=9261 |
E-mail us and let us know if you've got all 21 memorized!
| 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 |
| 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 |
The first ten factorials are provided in the table below. We are sure you will come across factorials in at least one of your math classes. Having these memorized has saved us a lot of time when doing homework simply because we didn't have to go to our calculator only to repeatedly enter 7*6*5*4*3*... Feel free to let us know when you have all these factorials memorized!
| 0! = 1 |
| 1! = 1 |
| 2! = 2 |
| 3! = 6 |
| 4! = 24 |
| 5! = 120 |
| 6! = 720 |
| 7! = 5040 |
| 8! = 40320 |
| 9! = 362880 |
| 10! = 3628800 |
| Length Equivalents |
| 12 inches = 1 foot |
| 3 feet = 1 yard |
| 36 inches = 1 yard |
| 1760 yards = 1 mile |
| 5280 feet = 1 mile |
| 16 1/2 feet = 1 rod |
| 5 1/2 yards = 1 rod |
This seems like a good time to give you an introduction to the metric system of measurement. This system is less complicated than the more commonly used system in the United States. Most countries use the metric system because its measurements use the base 10 counting system, and therefore, it is simpler for conversion purposes. There are some prefixes that you should memorize in order to identify the power of 10 when working with the metric system.
| Metric Prefixes | |
| kilo- (k) | 1000 |
| hecto- (h) | 100 |
| deka- (dk or da) | 10 |
| deci- (d) | .1 |
| centi- (c) | .01 |
| milli- (m) | .001 |
The following table gives three common conversions from the system used to represent lengths in the United States to the Metric System.
| Metric Conversions |
| 39.37 inches (in) = 1 meter (m) |
| 1 inch (in) = 2.54 centimeters (cm) |
| 1 mile (mi) = 1.6 kilometers (km) |
You probably already have most of the following time conversions memorized, because we use them every day. You know the saying, "Time is money, money is time." Well then, it does seem that, if you don't have these conversions committed to memory, you should take this opportunity to do so now :)!
| Time Equivalents |
| 60 seconds = 1 minute |
| 60 minutes = 1 hour |
| 24 hours = 1 day |
| 7 days = 1 week |
| 28-31 days = 1 month |
| 12 months = 1 year |
| 52 weeks = 1 year |
| 365 days = 1 year |
| 366 days = 1 leap year |
| 10 years = 1 decade |
| 100 years = 1 century |
| 1000 years = 1 millenium |
We do realize that you just can not remember every conversion there is. We also realize that, understandably, people prefer shortcuts and the easy way out. Below is our conversion calculator that you may use if you have a javascript enabled browser. If you don't have this type of browser, we recommend that you go to www.microsoft.com and download the most current version of Internet Explorer 4+, or to another site offering a free javascript compatible browser.
| Type the number you want to convert: Then, click the buttons for the desired conversion: | ||||||||
| From: | CM | In. | Feet | Yards | Meters | Rods | KM | Mi. |
| To: | CM | In. | Feet | Yards | Meters | Rods | KM | Mi. |
| | ||||||||
| Capacity Equivalents |
| 2 tablespoons = 1 fluid ounce |
| 8 fluid ounces = 1 cup |
| 2 cups = 1 pint |
| 2 pints = 1 quart |
| 4 quarts = 1 gallon |
| 2 gallons = 1 peck |
| 4 pecks = 1 bushel |
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| 250 milliliters = 1 metric cup |
Once again, we realize that you may prefer to use a "conversion calculator" when having to convert lots of measurements. You may use ours, provided below, if you have a javascript-enabled browser. If you don't have this type of browser, we recommend that you go to www.microsoft.com and download the most current version of Internet Explorer 4+, or to another site offering a free javascript compatible browser.
| Type the number you want to convert:
Then, click the buttons for the desired conversion: | ||||||||
| From: | tbsps. |
fl.oz. |
Cups |
Pints |
Quarts |
Gallons |
Pecks |
Bushels |
| To: | tbsps. |
fl.oz. |
Cups |
Pints |
Quarts |
Gallons |
Pecks |
Bushels |
| | ||||||||
All of these weight conversions are very commonly used. Note that "ounces" is different from "fluid ounces" (above), and that a ton differs in meaning from a metric ton.
| Weight Equivalents |
| 16 ounces = 1 pound |
| 2000 pounds = 1 ton |
| 2.2 pounds = l kilogram |
| 1000 kilograms = 1 metric ton |
We have decided to add these three numbering conversions to this page. The first conversion is by far the most commonly used. The other two do come up every once and a while, so it is helpful to, at least, be familiar with the vocabulary.
| Numbering |
| 12 items = 1 dozen |
| 144 items = 1 gross |
| 24 items = 1 quire |
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Clicking on the "Next" button will transport you to our
Mental Math home page. The Mental Math tutorial will introduce
you to many simple math tricks that we have learned over
time. We expect them to make your math experience much less dependent
on a calculator.
Of course, it will take time to
master these techniques. Practice, practice, and more practice
will enable you to master each
and every trick. Come visit us daily to brush up on each trick.
You'll soon discover that is well worth your time and effort!
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To multiply by any multiple of 10, move the decimal point to the right for as many zeros that are in the multiple of 10. ex: 3.5 x 10^4 = 35000 To divide by a multiple of 10, move the decimal point to the left for as many zeros that are in the multiple of 10. ex: 934 / 100 = 9.34 |
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Congratulations to you again! You have reached the end of the Mental Math section and hopefully you have learned a few useful tricks if not all of them. Many of these tricks that we have shown you can become very useful when you forget your calculator at home. Therefore, you should make sure that you come back and visit this classroom often to refresh your memory of these tricks.
By
clicking on the "Next" button, you will be taken to our Homework
classroom... effectively the classroom you can take with you! Here you
can print out our site easily or even download tests that have been
created especially for practicing mental math! We call them our NUMBER
SENSE tests!!!
We have provided an easy to use "printer friendly" section that can be used to print portions of our site without wasting paper. You can even print the entire Simply Number Sense site in one print job! Click here to go to this Printer Friendly section.
Also, you once again have the option of downloading mental math tests in PDF format. These can also be printed out on plain paper, and you can distribute these to your friends! Click here to jump to Quiz Central!
| Simply Number Sense Classrooms |
| Homeroom |
| Math Basics |
| Memorization |
| Mental Math |
| Homework |
| All Five |
If you would like to print out the entire Simply Number Sense site,
click HERE. Naturally, if you
do choose this option, please expect an extensive wait time.
