Saturday, 30 April 2011
square and square roots
First learn about Squares, then Square Roots are easy.
How to Square A Number
To square a number, just multiply it by itself ...
Example: What is 3 squared?
| 3 Squared | = |  | = 3 × 3 = 9 |
"Squared" is often written as a little 2 like this:
This says "4 Squared equals 16"
(the little 2 says the number appears twice in multiplying)
Squares From 12 to 62
| 1 Squared | = | 12 | = | 1 × 1 | = | 1 |
| 2 Squared | = | 22 | = | 2 × 2 | = | 4 |
| 3 Squared | = | 32 | = | 3 × 3 | = | 9 |
| 4 Squared | = | 42 | = | 4 × 4 | = | 16 |
| 5 Squared | = | 52 | = | 5 × 5 | = | 25 |
| 6 Squared | = | 62 | = | 6 × 6 | = | 36 |
| You can also find the squares on theMultiplication Table: |  |
Negative Numbers
You can also square negative numbers.
Example: What happens when you square -5 ?
Answer:
(-5) × (-5) = 25
(because a negative times a negative gives a positive)
That was interesting!
When you square a negative number you get a positive result.
Just the same as if you had squared a positive number:
We look at that in more detail in Squares and Square Roots in Algebra.
Square Roots
A square root goes the other way:
3 squared is 9, so a square root of 9 is 3
A square root of a number is ...
... a value that can be multiplied by itself to give the original number.
A square root of 9 is ...
... 3, because when 3 is multiplied by itself you get 9.
It is like asking:
What can I multiply by itself to get this?
 | To help you remember think of the root of a tree: "I know the tree, but what is the root that produced it?" In this case the tree is "9", and the root is "3". |
Here are some more squares and square roots:
 | ||
4 | 16 | |
5 | 25 | |
6 | 36 | |
Decimal Numbers
You can also square decimal numbers.
Try the sliders below. Note: the numbers here are only shown to 2 decimal places.
Using the sliders (remembering it is only accurate to 2 decimal places):
- What is the square root of 8?
- What is the square root of 9?
- What is the square root of 10?
- What is 1 squared?
- What is 1.1 squared?
- What is 2.6 squared?
The Square Root Symbol
 | This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards. It is called the radical, and always makes math look important! |
You can use it like this:
you would say "square root of 9 equals 3"
Example: What is √25?
Well, we just happen to know that 25 = 5 × 5, so if you multiply 5 by itself (5 × 5) you will get 25.
So the answer is:
√25 = 5
Example: What is √36 ?
Answer: 6 × 6 = 36, so √36 = 6
Perfect Squares
The perfect squares are the squares of the whole numbers:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | etc | |
| Perfect Squares: | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 | ... |
Try to remember at least the first 10 of those.
Calculating Square Roots
It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.
Example: what is √10?
Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.
- Let's try 3.5: 3.5 × 3.5 = 12.25
- Let's try 3.2: 3.2 × 3.2 = 10.24
- Let's try 3.1: 3.1 × 3.1 = 9.61
- ...
Getting closer to 10, but it will take a long time to get a good answer!
 | At this point, I get out my calculator and it says: 3.1622776601683793319988935444327 But the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation ! |
Note: numbers like that are called Irrational Numbers, if you want to know more.
The Easiest Way to Calculate a Square Root
 | Use your calculator's square root button! |
And also use your common sense to make sure you have the right answer.
A Fun Way to Calculate a Square Root
There is a fun method for calculating a square root that gets more and more accurate each time around:
| a) start with a guess (let's guess 4 is the square root of 10) | |
 | b) divide by the guess (10/4 = 2.5) c) add that to the guess (4 + 2.5 = 6.5) d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25) e) now, set that as the new guess, and start at b) again |
- Our first attempt got us from 4 to 3.25
- Going again (b to e) gets us: 3.163
- Going again (b to e) gets us: 3.1623
And so, after 3 times around the answer is 3.1623, which is pretty good, because:
3.1623 x 3.1623 = 10.00014
Now ... why don't you try calculating the square root of 2 this way?
How to Guess
What if you have to guess the square root for a difficult number such as "82,163" ... ?
In that case I would think to myself "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 would be a good start.
Physical Education In Secondary Schools
My philosophy of physical education in the secondary setting deals with
students becoming physically fit. Also that students learn how to communicate
with others, especially in a team setting. My philosophy also holds in it
different teaching methods. Methods that physical education teachers can use to
get the most out of their students. With this philosophy of physical education I
plan to use it toward becoming a good teacher.
The major reason for having physical education in secondary schools is
to lead the students toward a better lifestyles and to be physically fit. There
was an article published In 1992 by, The National Association for Sport and
Physical Education (NASPE), stating what "The Physically Fit Person" should
consist of. This article states five factors of a physical fit person. One
factor is that a physically fit person "has learned skills necessary to perform
a variety of physical activities". The second factor, "is the person physically
fit". The third factor is "does this person participate regularly in physical
activity". The fourth is a physically fit person "knows the implications of and
the benefits from involvement in physical activity". The last factor is that a
physically fit person "values physical activity and its contributions to a
healthful lifestyle". The plan as physical educators is to have each child be a
physically fit person, by their exit of high school. This will take some work
but I believe it could be done.
Another very important issue to be learned in secondary physical
education is teamwork. This is a very important aspect in everyday life. This
helps people think of others as much, or more, than themselves. Today some work
places are going to the team oriented style of work. The worker needs to be
able to express ideas amongst his/her teammates and come with solutions together.
A good activity that could be introduced in secondary physical education
programs is a...
students becoming physically fit. Also that students learn how to communicate
with others, especially in a team setting. My philosophy also holds in it
different teaching methods. Methods that physical education teachers can use to
get the most out of their students. With this philosophy of physical education I
plan to use it toward becoming a good teacher.
The major reason for having physical education in secondary schools is
to lead the students toward a better lifestyles and to be physically fit. There
was an article published In 1992 by, The National Association for Sport and
Physical Education (NASPE), stating what "The Physically Fit Person" should
consist of. This article states five factors of a physical fit person. One
factor is that a physically fit person "has learned skills necessary to perform
a variety of physical activities". The second factor, "is the person physically
fit". The third factor is "does this person participate regularly in physical
activity". The fourth is a physically fit person "knows the implications of and
the benefits from involvement in physical activity". The last factor is that a
physically fit person "values physical activity and its contributions to a
healthful lifestyle". The plan as physical educators is to have each child be a
physically fit person, by their exit of high school. This will take some work
but I believe it could be done.
Another very important issue to be learned in secondary physical
education is teamwork. This is a very important aspect in everyday life. This
helps people think of others as much, or more, than themselves. Today some work
places are going to the team oriented style of work. The worker needs to be
able to express ideas amongst his/her teammates and come with solutions together.
A good activity that could be introduced in secondary physical education
programs is a...
ibuku.
Ibu
Setiap deras degupan jantungku
Engkau mampu merasanya
Ketika aku:
Sedih
Gembira
Gelisah
Naluri ibumu
Bagaikan sebati dalam diriku
Ketika aku sedih
Kaulah ubat kepada duka laraku
Menceriakan dan menghakis
Perasaan yang melunturkan
Semangatku
Ketika aku gembira
Engkau juga turut berkongsi
Kegembiraan yang menyinari saat
Aku hidup di muka bumi ini
Sebagai manusia
Pabila gelisah menyelubungi jiwaku
Engkau masih bersamaku
Menolong menyelesaikan
Masalah yang tak sudah-sudah mengusutkan
Perjalanan hidupku
Ibu
Selagi hayat dikandung badan
Aku akan sentiasa mendampingimu
Membelai dan menjagamu hingga ke hari tua
Ibu
Aku tahu kasih sayang yang kucurahkan kepadamu
Tak setanding kasih sayang ibu kepada anaknya
Namun, terimalah keikhklasanku
Untuk menyayangi
Insan yang mulia ini
Setiap deras degupan jantungku
Engkau mampu merasanya
Ketika aku:
Sedih
Gembira
Gelisah
Naluri ibumu
Bagaikan sebati dalam diriku
Ketika aku sedih
Kaulah ubat kepada duka laraku
Menceriakan dan menghakis
Perasaan yang melunturkan
Semangatku
Ketika aku gembira
Engkau juga turut berkongsi
Kegembiraan yang menyinari saat
Aku hidup di muka bumi ini
Sebagai manusia
Pabila gelisah menyelubungi jiwaku
Engkau masih bersamaku
Menolong menyelesaikan
Masalah yang tak sudah-sudah mengusutkan
Perjalanan hidupku
Ibu
Selagi hayat dikandung badan
Aku akan sentiasa mendampingimu
Membelai dan menjagamu hingga ke hari tua
Ibu
Aku tahu kasih sayang yang kucurahkan kepadamu
Tak setanding kasih sayang ibu kepada anaknya
Namun, terimalah keikhklasanku
Untuk menyayangi
Insan yang mulia ini
Descriptive Essays
The thick, burnt scent of roasted coffee tickled the tip of my nose just seconds before the old, faithful alarm blared a distorted top-forty through its tiny top speaker. Wiping away the grit of last night's sleep, the starch white sunlight blinded me momentarily as I slung my arm like an elephant trunk along the top of the alarm, searching for the snooze button. While stretching hands and feet to the four posts of my bed, my eyes opened after several watery blinks. I crawled out of the comforter, edging awkwardly like a butterfly from a cocoon, swinging my legs over the side of the bed. The dusty pebbles on the chilled, wood floor sent ripples spiraling from my ankles to the nape of my neck when my feet hit the floor. Grabbing the apricot, terri-cloth robe, recently bathed in fabric softener and October wind, I knotted it tightly at my waist like a prestigious coat of armor and headed downstairs to battle the morning.
When I was little, I loved visiting my relatives. Although I was a country girl, I didn't have any brothers or sisters that were close to my age, nor did my parents enjoy some of the outdoor events that my extended family did. But my aunts and uncles in Manchester, Tennessee, were always doing something fun, especially my Uncle George. Uncle George loved to laugh and he loved to fish and he loved me.
Uncle George and I would head out before it got light. The trek to the pond was always exciting. The earth smelled fresh and new, promising warmth, and as the birds awoke, they'd tentatively practice the prologues to their songs. We'd walk past the apple trees, and I could smell the sharpness of the rotten fruit that had dropped to the ground. Occasionally, I'd slip on a peel, so I learned to be careful not to run too quickly. We'd walk past the water troughs where the tadpoles were busy wiggling their way to froghood and pick up...
When I was little, I loved visiting my relatives. Although I was a country girl, I didn't have any brothers or sisters that were close to my age, nor did my parents enjoy some of the outdoor events that my extended family did. But my aunts and uncles in Manchester, Tennessee, were always doing something fun, especially my Uncle George. Uncle George loved to laugh and he loved to fish and he loved me.
Uncle George and I would head out before it got light. The trek to the pond was always exciting. The earth smelled fresh and new, promising warmth, and as the birds awoke, they'd tentatively practice the prologues to their songs. We'd walk past the apple trees, and I could smell the sharpness of the rotten fruit that had dropped to the ground. Occasionally, I'd slip on a peel, so I learned to be careful not to run too quickly. We'd walk past the water troughs where the tadpoles were busy wiggling their way to froghood and pick up...
Ambitious..
If I can become anything else in my next life, I would like to become a soccer player. I would like to be a soccer player because I love to play soccer, the salary is good, and also because I can become a famous person. These are the reasons why I would like to be a human in my next life. I would like to be a soccer player.
I would like to be a soccer player because I love to play soccer. I want to have a job that I can enjoy, and feel good about. Playing soccer makes me feel good, and I enjoy it. That is why I would like to become a soccer player in my next life.
Money is important in life. If I become a soccer player, I can enjoy my life and also earn a lot of money. Soccer players have good salaries. That is why I would like to become a soccer player.
Being happy and wealthy is also important, but if I can be popular and famous, it would make my life good. I would like to be a soccer player because it is a popular job. Soccer players are famous, and people like them.
In my next life, I would like to be a human, so that I can become a soccer player. Soccer is what I love, and in life, people need to enjoy their jobs. Also, we need money, and love. If I become a soccer player, I can earn a lot of money, and become famous and popular. That is why I would like to be a soccer player.
I would like to be a soccer player because I love to play soccer. I want to have a job that I can enjoy, and feel good about. Playing soccer makes me feel good, and I enjoy it. That is why I would like to become a soccer player in my next life.
Money is important in life. If I become a soccer player, I can enjoy my life and also earn a lot of money. Soccer players have good salaries. That is why I would like to become a soccer player.
Being happy and wealthy is also important, but if I can be popular and famous, it would make my life good. I would like to be a soccer player because it is a popular job. Soccer players are famous, and people like them.
In my next life, I would like to be a human, so that I can become a soccer player. Soccer is what I love, and in life, people need to enjoy their jobs. Also, we need money, and love. If I become a soccer player, I can earn a lot of money, and become famous and popular. That is why I would like to be a soccer player.
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