Rectangular Solids

Introduction of Rectangular Solids: Rectangular solid is same as rectangular prism and it is a three dimensional figure. It is a rectangular prism in which it has six rectangular faces, twelve edges and eight vertices. Rectangular solid includes volume and surface area formula. The rectangular solid has three dimensions they are length width and height.

The Lesson:
We ask in practical terms two questions about 3 dimensional objects. First, how much paint does it take to paint the object (we would have to know the surface area). Second, how much water can the object hold (we would have to know the volume). We show how to calculate surface area and volume.

  A rectangular solid is a 3 dimensional object with six sides, all of which are rectangles. We first examine a cube, in which all six sides are squares. In the diagram below, a square of side 2 inches is used to form a cube with six square sides. The square clearly has an area of 4 square inches. Therefore the cube has a surface area of 48 square inches because its sides are composed of six of these squares. The cube is composed of 8 smaller cubes with a side of 1 inch. The volume of this cube is 8 cubic inches.

We note that a square has its name because its area is the square of the length of a side. The area 4 =2² . The cube has its name because the volume of a cube of the length of a side. The volume is 8 =2³. We can generalize this result for cubes by saying that the volume of a cube of side a is a³. Since the area of one side is the length of the side squared, the entire surface area of the cube is 6 x 4 = 24. This can also be generalized for any cube of side a. The surface area is 6 x a²= 6a².

In the diagram above, we show a rectangular solid at right with dimensions 5 x 2 x 3 inches. These are the measures of the length l, the width w and the height h. The area of the Front/Back rectangles is 15 square inches. The area of the Sides is 6 square inches, and the area of the Top/Bottom rectangles is 10 square inches. Adding these we get the surface area which is 62 square inches. The volume is found by multiplying the lengths of the sides as we did with the cube. The volume is 30 cubic inches.
Hope you like the above introduction of Rectangular Solids and diagram.So, kindly post your comments.

Transformations

Definition of Transformations
Transformations are a change of its point of an object on a plane. In mathematics, Transformations are of any function that replacing a set A on to a different set or on to itself. On the other hand, most often the set A has some additional algebraic or geometric structure and the term "transformation" determines a function from A to itself which protects this structure


Reflections
A reflection is like placing a mirror on the page. When describing a reflection, you need to state the line which the shape has been reflected in. The distance of each point of a shape from the line of reflection will be the same as the distance of the reflected point from the line.
For example, below is a triangle that has been reflected in the line y = x (the length of the pink lines should be the same on each side of the line y=x):

 



Rotations

"Rotation" means turning around a center:The distance from the center to any point on the shape stays the same.Every point makes a circle around the center
When describing a rotation, the centre and angle of rotation are given. If you wish to use tracing paper to help with rotations: draw the shape you wish to rotate onto the tracing paper and put this over shape. Push the end of your pencil down onto the tracing paper, where the centre of rotation is and turn the tracing paper through the appropriate angle (if you are not told whether the angle of rotation is clockwise or anticlockwise, it would usually be anticlockwise). The resultant position of the shape on the tracing paper is where the shape is rotated to.

 

Enlargements

Enlargements have a centre of enlargement and a scale factor.
1) Draw a line from the centre of enlargement to each vertex ('corner') of the shape you wish to enlarge. Measure the lengths of each of these lines.
2) If the scale factor is 2, draw a line from the centre of enlargement, through each vertex, which is twice as long as the length you measured. If the scale factor is 3, draw lines which are three times as long. If the scale factor is 1/2, draw lines which are 1/2 as long, etc.
 
The centre of enlargement is marked. Enlarge the triangle by a scale factor of 2.

Translating Word Problems

Word Problems

The hardest thing about doing word problems is taking the English words and translating them into mathematics. Usually, once you get the math equation, you're fine; the actual math involved is often fairly simple. But figuring out the actual equation can seem nearly impossible. What follows is a list of hints and helps. Be advised, however: To really learn "how to do" word problems, you will need to practice, practice, practice.

The first step to effectively translating and solving word problems is to read the problem entirely. Don't start trying to solve anything when you've only read half a sentence. Try first to get a feel for the whole problem; try first to see what information you have, and what you still need.

The second step is to work in an organized manner. Figure out what you need but don't have, and name things. Pick variables to stand for the unknowns, clearly labeling these variables with what they stand for. Draw and label pictures neatly. Explain your reasoning as you go along. And make sure you know just exactly what the problem is actually asking for. You need to do this for two reasons:

1. Working clearly will help you think clearly, and
2. figuring out what you need will help you translate your final answer back into English.

Regarding (2) above: It can be really frustrating (and embarrassing) to spend fifteen minutes solving a word problem on a test, only to realize at the end that you no longer have any idea what "x" stands for, so you have to do the whole problem over again. 

• Addition- increased by
  more than
  combined, together
  total of
  sum
  added to

• Subtraction- decreased by
  minus, less
  difference between/of
  less than, fewer than

• Multiplication- of
  times, multiplied by
  product of
  increased/decreased by a
  factor of (this type can
  involve both addition or
  subtraction and
    multiplication!)

• Division- per, a
  out of
  ratio of, quotient of
  percent (divide by 100)

• Equals- is, are, was, were, will be
  gives, yields
  sold for

Note that "per" means "divided by", as in "I drove 90 miles on three gallons of gas, so I got 30 miles per gallon". Also, "a" sometimes means "divided by", as in "When I tanked up, I paid $12.36 for three gallons, so the gas was $4.12 a gallon".

Warning: The "less than" construction is backwards in the English from what it is in the math. If you need to translate "1.5 less than x", the temptation is to write "1.5 – x". Do not do this! You can see how this is wrong by using this construction in a "real world" situation: Consider the statement, "He makes $1.50 an hour less than me." You do not figure his wage by subtracting your wage from $1.50. Instead, you subtract $1.50 from your wage. So remember; the "less than" construction is backwards.

Also note that order is important in the "quotient/ratio of" and "difference between/of" constructions. If a problems says "the ratio of x and y", it means "x divided by y", not "y divided by x". If the problem says "the difference of x and y", it means "x – y", not "y – x".