Multiples, Common Multiples and Lowest Common Multiples

Resourses:

http://www.mathgoodies.com/Lessons/vol3/lcm.html

http://webmath.com/intlcm.html

http://www.mathsteacher.com.au/year8/ch01_arithmetic/03_mult/mult.htm

Multiples
The multiples of a number are its products with the natural numbers 1, 2, 3, 4, 5, ....
Example 1
1 x 8 = 8
2 x 8 =16
3 x 8 =24
4 x 8 =32
5 x 8 =40
So, the multiples of 8 are 8, 16, 24, 32, 40 and so on.
Note:
The multiples of a number are obtained by multiplying the number by each of the natural numbers.

Example 2
Write down the first five multiples of 9.
Solution:
1 x 9 = 9
2 x 9 =18
3 x 9 =27
4 x 9 =36
5 x 9 =45
The multiples of 9 are obtained by multiplying 9 with the natural numbers 1, 2, 3, 4, 5 …
So, the first five multiples of 9 are 9, 18, 27, 36 and 45.

Common Multiples
Common multiples are multiples that are common to two or more numbers.

Example 3
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, …
Multiples of 3 are 3, 6, 9, 12, 15, 18, …
So, common multiples of 2 and 3 are 6, 12, 18, …
Example 4
Find the common multiples of 3 and 4.
Solution:
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, …
So, the common multiples of 3 and 4 are 12, 24, 36, …

Lowest Common Multiple
The lowest common multiple (LCM) of two or more numbers is the smallest common multiple.
Example 5
Multiples of 8 are 8, 16, 24, 32, …
Multiples of 6 are 6, 12, 18, 24, …
The LCM of 6 and 8 is 24
.
Example 6
Find the lowest common multiple of 2 and 5.
List the multiples of 5 and stop when you find a multiple of 2.
Multiples of 5 are 5, 10, …
Multiples of 2 are 2, 4, 6, 8, 10, …
The LCM of 2 and 5 is 10

NUMBER PATTERNS AND SEQUENCES
Real-Life Examples
Try to discover geometric patterns in nature (shells, flowers, animals, pine cones, rocks); in architecture (archways, doorways, stairways, floor tiles, windows); in clothing and home fashion (designer labels, t-shirt logos, neckties, quilts, wallpaper, and floor tile); and in technology (computer-generated graphics and logos).

Create, Extend and Explain Number Patterns
Have fun with your family by making up number patterns. Encourage other family members to discover each pattern, predict the next three numbers in the pattern, and explain how the pattern works.
5, 10, 15, 20, 25, 30 … (35, 40, 45)
Add 5 to the previous number.
Count by five’s.

2, 4, 6, 8, 10, 12 … (14, 16, 18)
Add 2 to the previous number.
List the even numbers.
Count by 2.

1, 3, 5, 7, 9, 11 … (13, 15, 17)
Add 2 to the previous number.
List the odd numbers.

1, 2, 4, 8, 16 … (32, 64, 128)
Double the previous number.
Multiply the previous number by 2.
Add 1 to the first number, then 2 to the next number, then 4, then 8, etc.

1, 4, 9, 16, 25, 36, 49 … (64, 81, 100)
Multiply each number by itself (1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16).
Find the perfect squares of the counting numbers.
Add 3 to the first number, then 5 to the next number, then 7, then 9, then 11, etc.

Write Words on Calculators?

This is a fun activity to do when you are bored in your Math class!

Calculator Words. Type 200 Words Using A Simple Calculator - These bloopers are hilarious
CALCULATOR WORDS
Use a calculator to find these words by doing the following calculations and then turning your calculator upside down.
a. 3357 -2223 to get a place you wouldn’t go to. (………………..…….)
b. 300 000 + 18 830 to get a girl’s name. (…………………..….)
c. 85 423 + 294 496 to get something you might do if you’re happy or embarrassed about something. (…………………..….)
d. 6411 − 897 to get a noise you wouldn’t want to hear while bushwalking. (………..…………….)
e. 63 552 ÷ 64 to get something you eat. (…………………..….)
f. 203 × 15 to get something you wear. (……………..……….)
g. 52 043 ÷ 71 to get an animal. (………………..…….)
h. 52 360 ÷ 17 to get a musical instrument. (…..………………….)
i. 4417 x 8 to get some animals. (………………..…….)
j. 923 x 5 to get something you need your breath to do. (……..……………….)
k. 23 x 19 + 7301 to get something you often hear at school. (……..……………….)
l. 888 888 ÷ 2 – 65 638 to get something you shouldn’t do when you eat. (……..……..………..)
m. 4536 ÷ 81 + 261 to get something you should never do. (……………….……..)
n. 237 023 x 2 ÷ 421 x 2 + 4853 to get something you find in the garden. (……….……………..)
o. 6716 ÷ 73 x 125 136 ÷ 16 – 642 187 to get something you may find on the beach. (………………….…..)
p. 7 + 700 + 7000 + 10 000 + 300 000 + 5 000 000 to get something that you shouldn’t eat too much of. (……..……………....)

The following videos show more examples of the application of PEMDAS

COMBINED OPERATIONS
If the expression consists of parenthesis, exponents, +, –, × and ÷, then the operations MUST be performed in the following order.
Always work on the calculations within parenthesis first if any.
Next, calculate the exponents.
Then, carry out multiplication or division, working from left to right.
Lastly, do addition or subtraction, working from left to right.

The order to perform combined operations is called the PEMDAS rule.
Note: A common mnemonic for PEMDAS is Please Excuse My Dear Aunt Sally.
Example:
Evaluate 10 ÷ 2 + 12 ÷ 2 × 3
Using the PEMDAS rule, we need to evaluate the division and multiplication before subtraction and addition. It is recommended that you put in parenthesis to remind yourself the order of operation.
Solution:
10 ÷ 2 + 12 ÷ 2 × 3
= ( 10 ÷ 2) + (12 ÷ 2 × 3)
= 5 + 18
= 23

HAPPY NEW YEAR.

WELCOME BACK TO SCHOOL.

........................................................................

WHOLE NUMBERS
Divisibility Tests

Divisibility by 2
A whole number is divisible by 2 if the digit in its units position is even, (either 0, 2, 4, 6, or 8).
Examples:
The number 84 is divisible by 2 since the digit in the units position is 4, which is even.The number 333336 is divisible by 2 since the digit in the units position is 6, which is even.The number 1297000 is divisible by 2 since the digit in the units position is 0, which is even.

Divisibility by 3
A whole number is divisible by 3 if the sum of all its digits is divisible by 3.
Examples:
The number 177 is divisible by three, since the sum of its digits is 15, which is divisible by 3.The number 8882151 is divisible by three, since the sum of its digits is 33, which is divisible by 3.The number 162345 is divisible by three, since the sum of its digits is 21, which is divisible by 3.
If a number is not divisible by 3, the remainder when it is divided by 3 is the same as the remainder when the sum of its digits is divided by 3.
Examples:
The number 3248 is not divisible by 3, since the sum of its digits is 17, which is not divisible by 3. When 3248 is divided by 3, the remainder is 2, since when 17, the sum of its digits, is divided by three, the remainder is 2.
The number 172345 is not divisible by 3, since the sum of its digits is 22, which is not divisible by 3. When 172345 is divided by 3, the remainder is 1, since when 22, the sum of its digits, is divided by three, the remainder is 1.

Divisibility by 4
A whole number is divisible by 4 if the number formed by the last two digits is divisible by 4.
Examples:
The number 3124 is divisible by 4 since the number formed by its last two digits, 24, is divisible by 4.The number 1333336 is divisible by 4 since the number formed by its last two digits, 36, is divisible by 4.The number 1297000 is divisible by 4 since the number formed by its last two digits, 0, is divisible by 4.
If a number is not divisible by 4, the remainder when the number is divided by 4 is the same as the remainder when the last two digits are divided by 4.
Example:
The number 172345 is not divisible by 4, since the number formed by its last two digits, 45, is not divisible by 4. When 172345 is divided by 4, the remainder is 1, since when 45 is divided by 4, the remainder is 1.

Divisibility by 5
A whole number is divisible by 5 if the digit in its units position is 0 or 5.
Examples:
The number 95 is divisible by 5 since the last digit is 5.The number 343370 is divisible by 5 since the last digit is 0. The number 129700195 is divisible by 5 since the last digit is 5.
If a number is not divisible by 5, the remainder when it is divided by 5 is the same as the remainder when the last digit is divided by 5.
Example:
The number 145632 is not divisible by 5, since the last digit is 2. When 145632 is divided by 5, the remainder is 2, since 2 divided by 5 is 0 with a remainder of 2.
The number 7332899 is not divisible by 5, since the last digit is 9. When 7332899 is divided by 5, the remainder is 4, since 9 divided by 5 is 1 with a remainder of 4.

Divisibility by 6
A number is divisible by 6 if it is divisible by 2 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 6: its units digit is even and the sum of its digits is divisible by 3.
Examples:
The number 714558 is divisible by 6, since its units digit is even, and the sum of its digits is 30, which is divisible by 3. The number 297663 is not divisible by 6, since its units digit is not even.The number 367942 is not divisible by 6, since it is not divisible by 3. The sum of its digits is 31, which is not divisible by 3, so the number 367942 is not divisible by 3.

Divisibility by 8
A whole number is divisible by 8 if the number formed by the last three digits is divisible by 8.
Examples:
The number 88863024 is divisible by 8 since the number formed by its last three digits, 24, is divisible by 8.The number 17723000 is divisible by 8 since the number formed by its last three digits, 0, is divisible by 8.The number 339122483984 is divisible by 8 since the number formed by its last three digits, 984, is divisible by 8.
If a number is not divisible by 8, the remainder when the number is divided by 8 is the same as the remainder when the last three digits are divided by 8.
Example:
The number 172045 is not divisible by 8, since the number formed by its last three digits, 45, is not divisible by 8. When 172345 is divided by 8, the remainder is 5, since when 45 is divided by 8, the remainder is 5.

Divisibility by 9
A whole number is divisible by 9 if the sum of all its digits is divisible by 9.
Examples:
The number 1737 is divisible by nine, since the sum of its digits is 18, which is divisible by 9.The number 8882451 is divisible by nine, since the sum of its digits is 36, which is divisible by 9.The number 762345 is divisible by nine, since the sum of its digits is 27, which is divisible by 9.
If a number is not divisible by 9, the remainder when it is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.
Examples:
The number 3248 is not divisible by 9, since the sum of its digits is 17, which is not divisible by 9. When 3248 is divided by 9, the remainder is 8, since when 17, the sum of its digits, is divided by 9, the remainder is 8.
The number 172345 is not divisible by 9, since the sum of its digits is 22, which is not divisible by 9. When 172345 is divided by 9, the remainder is 4, since when 22, the sum of its digits, is divided by 9, the remainder is 4.

Divisibility by 10
A whole number is divisible by 10 if the digit in its units position is 0.
Examples:
The number 1229570 is divisible by 10 since the last digit is 0.The number 676767000 is divisible by 10 since the last digit is 0.The number 129700190 is divisible by 10 since the last digit is 0.
If a number is not divisible by 10, the remainder when it is divided by 10 is the same as the units digit.
Examples:
The number 145632 is not divisible by 10, since the last digit is 2. When 145632 is divided by 10, the remainder is 2, since the units digit is 2.The number 7332899 is not divisible by 10, since the last digit is 9. When 7332899 is divided by 10, the remainder is 4, since the units digit is 9.

Divisibility by 11
Starting with the units digit, add every other digit and remember this number. Form a new number by adding the digits that remain. If the difference between these two numbers is divisible by 11, then the original number is divisible by 11.
Examples:
Is the number 824472 divisible by 11? Starting with the units digit, add every other number:2 + 4 + 2 = 8. Then add the remaining numbers: 7 + 4 + 8 = 19. Since the difference between these two sums is 11, which is divisible by 11, 824472 is divisible by 11.
Is the number 49137 divisible by 11? Starting with the units digit, add every other number:7 + 1 + 4 = 12. Then add the remaining numbers: 3 + 9 = 12. Since the difference between these two sums is 0, which is divisible by 11, 49137 is divisible by 11.
Is the number 16370706 divisible by 11? Starting with the units digit, add every other number:6 + 7 + 7 + 6 = 26. Then add the remaining numbers: 0 + 0 + 3 + 1=4. Since the difference between these two sums is 22, which is divisible by 11, 16370706 is divisible by 11.

Divisibility by 12
A number is divisible by 12 if it is divisible by 4 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 12: its last two digits are divisible by 4 and the sum of its digits is divisible by 3.
Examples:
The number 724560 is divisible by 12, since the number formed by its last two digits, 60, is divisible by 4, and the sum of its digits is 30, which is divisible by 3.The number 36297414 is not divisible by 12, since the number formed by its last two digits, 14, is not divisible by 4.The number 367744 is not divisible by 12, since it is not divisible by 3. The sum of its digits is 29, which is not divisible by 3, so the number 367942 is not divisible by 3.

Divisibility by 15
A number is divisible by 15 if it is divisible by 3 and divisible by 5. We can use each of the divisibility tests to check if a number is divisible by 15: its units digit is 0 or 5, and the sum of its digits is divisible by 3.
Example:
The number 7145580 is divisible by 15, since its units digit is even, and the sum of its digits is 30, which is divisible by 3.

Divisibility by 16
A whole number is divisible by 16 if the number formed by the last four digits is divisible by 16.
Examples:
The number 898630032 is divisible by 16 since the number formed by its last four digits, 32, is divisible by 16.The number 1772300000 is divisible by 16 since the number formed by its last four digits, 0, is divisible by 16.The number 339122481296 is divisible by 16 since the number formed by its last four digits, 1296, is divisible by 16.
If a number is not divisible by 16, the remainder when the number is divided by 16 is the same as the remainder when the last four digits are divided by 16.
Example:
The number 172411045 is not divisible by 16, since the number formed by its last four digits, 1045, is not divisible by 16. When 172411045 is divided by 16, the remainder is 5, since when 1045 is divided by 16, the remainder is 5.

Divisibility by 18
A number is divisible by 18 if it is divisible by 2 and divisible by 9. We can use each of the divisibility tests to check if a number is divisible by 18: its units digit is even and the sum of its digits is divisible by 9.
Examples:
The number 7145586 is divisible by 18, since its units digit is even, and the sum of its digits is 36, which is divisible by 9. The number 2976633 is not divisible by 18, since its units digit is not even.The number 367942 is not divisible by 18, since it is not divisible by 9. The sum of its digits is 31, which is not divisible by 9, so the number 367942 is not divisible by 9.

Divisibility by 20
A number is divisible by 20 if its units digit is 0, and its tens digit is even. In other words, the last two digits form one of the numbers 0, 20, 40, 60, or 80.
Examples:
The number 3351002760 is divisible by 20, since the number formed by its last two digits is 60.The number 802199730000 is divisible by 20, since the number formed by its last two digits is 0.

Divisibility by 22
A number is divisible by 22 if it is divisible by the numbers 2 and 11. We can use each of the divisibility tests to check if a number is divisible by 22: its units digit is even, and the difference between the sums of every other digit is divisible by 11.
Example:
Is the number 117524 divisible by 22? The units digit is even, so it is divisible by 2. The two sums of every other digit are 4 + 5 + 1 = 10 and 2 + 7 + 1 = 10, which have a difference of 0. Since 0 is divisible by 11, 117524 is divisible by 11. Thus, 117524 is divisible by 22, since it is divisible by both 2 and 11.

Divisibility by 25
A number is divisible by 25 if the number formed by the last two digits is any of 0, 25, 50, or 75 (the number formed by its last two digits is divisible by 25).
Examples:
The number 73224050 is divisible by 25, since its last two digits form the number 50.The number 1008922200 is divisible by 25, since its last two digits form the number 0.

Who Owns The fish?
Albert Einstein [allegedly] wrote this riddle early on in his career. He said that 98% of the world's population would not be able to solve it (there are no tricks, just pure logic). The question is: Who owns the fish? Here's the riddle:
In a street, there are five houses, painted five different colors. In each house live a person of different nationality. The five homeowners each drink a different kind of beverage, smoke a different brand of cigarette and keep a different pet.
1. The Brit [Englishman] lives in the red house.
2. The Swede has a dog.
3. The Dane drinks tea.
4. The green house is on the left of the white house.
5. The owner of the green house drinks coffee.
6. The person who smokes Pall Mall has birds.
7. In the yellow house, they smoke Dunhill.
8. The man living in the middle house drinks milk.
9. The Norwegian lives in the first house.
10. The man who smokes Blend lives next to the house with cats.
11. The horseman lives next to the man who smokes Dunhill.
12. The man who smokes Blue Master drinks beer.
13. The German smokes Prince.
14. The Norwegian lives next door to the blue house.
15. The man who smokes Blend has a neighbor who drinks water.

AFTER END OF YEAR EXAM

Online Crossword Puzzles

Please visit:
http://www.eduplace.com/kids/hmsc/content/crossword/


Contents:

Continuity of Life

Classifying Organisms

Cell Structure and Function

Reproduction and Heredity

Change Over Time


The Changing Environment

Cycles in the Biosphere

Earth’s Ecosystems

Populations


The Dynamic Earth

The Rock Cycle The Dynamic Earth

Earth’s Energy Resources


Earth in the Universe

Global Weather Systems

Earth, Moon, and Sun

The Solar System and Beyond


Matter and Its Properties

Composition of Matter

Physical and Chemical Changes


Energy, Forces, and Motion

Energy Light and Its Properties

Electricity and Magnetism Motion, Gravity, and Work

Using Games in Education
Games provide an excellent environment to explore ideas of computational thinking. The fact that many games are available both in a non-computerized form and in a computerized form helps to create this excellent learning environment. A modern education prepares students to be productive and responsible adult citizens in a world in which mind/brain and computer working
together is a common approach to solving problems and accomplishing tasks.

Puzzles
A puzzle is a type of game. To better under the purpose, think about some popular puzzles such as crossword puzzles, jigsaw puzzles, and logic puzzles (often called brain teasers). In every case, the puzzle-solver’s goal is to solve a particular mentally challenging problem or accomplish a particular mentally challenging task.

Many people are hooked on certain types of puzzles. For example, some people routinely start the day by spending time on the crossword puzzle in their morning newspaper. In some sense, they have a type of addiction to crossword puzzles. The fun is in meeting the challenge of the puzzle—making some or a lot of progress in completing the puzzle.

Crossword puzzles draw upon one’s general knowledge, recall of words defined or suggested by short definitions or pieces of information, and spelling ability. Through study and practice, a person learns some useful strategies and can make considerable gains in crossword puzzlesolving expertise. Doing a crossword puzzle is like doing a certain type of brain exercise. In recent years, research has provided evidence that such brain exercises help stave of the dementia and Alzheimer’s disease that are so common in old people.

From an educational point of view, it is clear that solving crossword puzzles helps to maintain and improve one’s vocabulary, spelling skills, and knowledge of many miscellaneous tidbits of information. Solving crossword puzzles tends to contribute to one’s self esteem. For many people, their expertise in solving crossword puzzles plays a role in their social interaction with other people.

Want to read more about Using Game In Education, visit:
http://darkwing.uoregon.edu/~moursund/Books/Games/Games.pdf

Why 09/09/09 Is So Special?

Numerologists aren't the only ones excited about Wednesday's date.

Not only does the date look good in marketing promotions, but it also represents the last set of repeating, single-digit dates that we'll see for almost a century (until January 1, 2101), or a millennium (mark your calendars for January 1, 3001), depending on how you want to count it.
Though technically there's nothing special about the symmetrical date, some concerned with the history and meaning of numbers ascribe powerful significance to 09/09/09. For cultures in which the number nine is lucky, Sept. 9 is anticipated - while others might see the date as an ominous warning.

Math magic

Modern numerologists - who operate outside the realm of real science - believe that mystical significance or vibrations can be assigned to each numeral one through nine, and different combinations of the digits produce tangible results in life depending on their application.
As the final numeral, the number nine holds special rank. It is associated with forgiveness, compassion and success on the positive side as well as arrogance and self-righteousness on the negative, according to numerologists.

Though usually discredited as bogus, numerologists do have a famous predecessor to look to. Pythagoras, the Greek mathematician and father of the famous theorem, is also credited with popularizing numerology in ancient times.

"Pythagoras most of all seems to have honored and advanced the study concerned with numbers, having taken it away from the use of merchants and likening all things to numbers," wrote Aristoxenus, an ancient Greek historian, in the 4th century B.C.As part of his obsession with numbers both mathematically and divine, and like many mathematicians before and since, Pythagoras noted that nine in particular had many unique properties. Any grade-schooler could tell you, for example, that the sum of the two-digits resulting from nine multiplied by any other single-digit number will equal nine. So 9x3=27, and 2+7=9.

Multiply nine by any two, three or four-digit number and the sums of those will also break down to nine. For example: 9x62 = 558; 5+5+8=18; 1+8=9.
Sept. 9 also happens to be the 252nd day of the year (2 + 5 +2)...

Loving 9

Both China and Japan have strong feelings about the number nine. Those feelings just happen to be on opposite ends of the spectrum. The Chinese pulled out all the stops to celebrate their lucky number eight during last year's Summer Olympics, ringing the games in at 8 p.m. on 08/08/08. What many might not realize is that nine comes in second on their list of auspicious digits and is associated with long life, due to how similar its pronunciation is to the local word for long-lasting (eight sounds like wealth).

Historically, ancient Chinese emperors associated themselves closely with the number nine, which appeared prominently in architecture and royal dress, often in the form of nine fearsome dragons. The imperial dynasties were so convinced of the power of the number nine that the palace complex at Beijing's Forbidden City is rumored to have been built with 9,999 rooms.

Japanese emperors would have never worn a robe with nine dragons, however.
In Japanese, the word for nine is a homophone for the word for suffering, so the number is considered highly unlucky - second only to four, which sounds like death.

Many Japanese will go so far as to avoid room numbers including nine at hotels or hospitals, if the building planners haven't already eliminated them altogether.

Original Story: Why 09/09/09 Is So SpecialLiveScience.com

Menjelang Sambutan Ulangtahun Kemerdekaan

Effective Study Skills
Students often fail to secure good grades despite studying for hours. This is because they have never studied effectively. Unless you know how to make your study hours effective, you cannot get the expected results. Here we will give you some glimpses of effective study skills.

• A study is called effective if you can remember and recall the major segments of the syllabus that you have read. Do not try to cram chapters that you fail to understand. You will never be able to remember them. The easiest thing to do is to seek the assistance of your classmates or the teachers to understand the confusing portions.
• 
  You have to decide what is most important to you. Make your schedule to address those important needs. You should never compromise with your priorities. Never try to finish too many things at a time. Study all the subjects on a regular basis while allotting some time for revisions. Keep yourself motivated. It is better to know the theme of the chapter that you are planning to study. Reading it line by line will then help you understand it better.
• 
  Effective study skills also stress on reading methods. You will see that there are some words that have fonts in bold or italics; you should pay special attention to these. You can also try to find whether the current text refers to any other works of the author that you have read earlier.
• 
  Effective study skills include listening to the explanations given by teachers. You should read and understand the notes that you have taken in the classroom. Following these aforesaid effective study skills will save you disappointments and help you get the desired results.
  
  

Solid Geometry

Solid geometry is concerned with three-dimensional shapes. Some examples of three-dimensional shapes are cubes, rectangular solids, prisms, cylinders, spheres, cones and pyramids.

Cubes
A cube is a three-dimensional figure with six matching square sides.

The figure above shows a cube.

The dotted lines indicate edges hidden from your view.

If s is the length of one of its sides,

the Volume of the cube = s x s x s.

Since the cube has six square-shape sides,

the Surface area of a cube = 6 x s x s

Cuboid

In a cuboid, the length, width and height may be of different lengths.

The volume of the above cuboid would be the product of the length, width and height that is
Volume of rectangular solid = lwh
Surface area of rectangular solid = 2(lw + wh + lh)

Prisms
A prism is a solid that has two congruent parallel bases that are polygons. The polygons form the bases of the prism and the length of the edge joining the two bases is called the height.

A rectangular solid is a prism with a rectangle-shaped base.
The volume of a prism is given by the product of the area of its base and its height.

Cylinders
A cylinder is a solid with two congruent circles joined by a curved surface.
In the above figure, the radius of the circular base is r and the height is h.

The volume of the cylinder is the area of the base × height.

Spheres
A sphere is a solid with all its points the same distance from the center.

Cones

A circular cone has a circular base, which is connected by a curved surface to its vertex. A cone is called a right circular cone, if the line from the vertex of the cone to the center of its base is perpendicular to the base.

Pyramids
A pyramid is a solid with a polygon base and connected by triangular faces to its vertex. A pyramid is a regular pyramid if its base is a regular polygon and the triangular faces are all congruent isosceles triangles.

Symmetry / Reflection

• A plane figure is symmetrical about a line if it is divided into two identical (coincident) parts by that line. The line is called its line (or axis) of symmetry.
• A plane figure is symmetrical about a point if every line segment joining two points of the figure and passing through the point is bisected at that point. The point is called its point (or center) of symmetry.
• A plane figure has a rotational symmetry if on rotation through some angle ( 180°) about a point it looks the same as it did in its starting position.
• If A° ( 180°) is the smallest angle through which a figure can be rotated and still looks the same, then it has a rotational symmetry of order 360/A.
• The reflection (or image) of a point P in a line AB is a point P' such that AB is the perpendicular bisector of the line segment PP'.
  To find the reflection (or image) of a point P in a line ABFrom P, draw PM perpendicular to AB and produce PM to P' such that MP'= MP, then P' is the reflection (or image) of P in the line AB.
• The reflection of the point P (x, y) in the x-axis is the point P'(x, -y).
• The reflection of the point P (x, y) in the y-axis is the point P'(-x, y).
• If a point P (x, y) is rotated through 180° (clockwise or anti-clockwise) about the origin to the point P', then co-ordinates of P' are (-x,-y).
• If a point P (x, y) is rotated through 90° clockwise about the origin to the point P', then co-ordinates of P' are (y,-x).
• If a point P (x, y) is rotated through 90° anti-clockwise about the origin to the point P', then co-ordinates of P' are (-y,x).

RIDDLE

Brain Teaser