Translations, Rotations, Reflections, and Glide

Translations

According to the book "A Problem Solving Approach to Mathematics for Elementary School Teachers," "any motion that preserves length or distance is an isometry (which is derived from
Greek meaning 'equal measure). A translation is an isometry and is a one-to-one correspondence between a plane and itself. Any function from a plane to itself that is one-to-one correspondence is a transformation of the plane. So a translation is also a transformation. Because transformations are functions."

In other words a translation its like a child sliding down a slide. A translation moves every point on a plane towards a specific distance, direction along a straight line.

 

 

 

 

Rotations:

 

 According to the book "A Problem Solving Approach to Mathematics for Elementary School Teachers," a rotation is another type of isometry. The textbook defines rotation as "a transformation of the plane determined by holding one point---the center---- fixed and rotating the plane about this point by a certain amount in a certain direction (a certain amount of degrees either clockwise or counterclockwise)."

 

 

 

 

Reflections

 

Reflection is another isometry.  According to the book "A Problem Solving Approach to Mathematics for Elementary School Teachers," a reflection in a line (l) " is a transformation of a plane that pairs each point P of the plane with a point P' in such a way tht the (l) is the perpendicular bisector of PP', as long as P is not (l). If P is on (l) then P=P'."

 

Think of a reflection as what you would see he figure like if you were to put it in front of a mirror. The points on the plane have to remain the same but have to change signs in order to have a reflection. Below are some examples of reflections on a plane.

                

 

 

 

Glide Reflections:

Glide reflection is another type of isometry. The definition of Glide Reflection is a transformation consisting of  a translation followed by a reflection in a line parallel to the slide arrow.

 

 

 

Symmetries

Line Symmetry

According to the book "A Problem Solving Approach to Mathematics for Elementary School Teachers," mathematically a geometric figure has a line of symmetry (l) " if its own image under a refection in (l).
According to the website "Math is Fun" "line of symmetry is another name for reflection symmetry. One half is the reflection of the other half.
The "Line of Symmetry"  is the imaginary line where you could fold the image and have both halves match exactly."
In other words Line of symmetry is when a figure, shape object or anything that when cut in fact shows the same exact thing in both sides! For example:

All things that have symmetry can also have more than just one line of symmetry. For example,
 
How many lines of symmetry does the letter A have?


Only 1 line of symmetry because only the third A with the horizontal line represents the reflection of the other side.

But unlike a other figures have more than one line of symmetry. For example:


in fact there is one shape that its symmetry is infinite, can you guess which one??

If you guessed a circle then your right! A circle has infinite symmetry.


Rotational Symmetry

 According to the book "A Problem Solving Approach to Mathematics for Elementary School Teachers," a figure has rotational symmetry " when the traced figure can be rotated less that 360 degrees about some point so that it matches the original figure." Remember its important to note that the condition  less than 360 degrees is clear and necessary because if we were to turn a figure 360 degrees it will coincide with itself.


 

 The figures above all classify to rotational symmetry!!


Point Symmetry
 According to the book "A Problem Solving Approach to Mathematics for Elementary School Teachers," "any figure that has rotational symmetry is said to have point symmetry about the turn center. The examples below are examples of figures with point symmetry:




Fact: if a figure has point symmetry it has rotational symmetry.
 

Volume

"Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance  or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic meter." (1) def. Wikipedia.com

Today I will show you how to find the volume of prisms.

According to the book "A Problem Solving Approach to Mathematics for Elementary School Teachers," the volume of a right rectangular prism "can be found by determining how many cubes are needed to build it as a solid. To find the volume count how many cubes cover the base and then how many layers of these cubes are used to fill the prism."

Lets take a look at the figure bellow:

 

 



* there are 8 x 4 cubes that cover the base, so a total of 32 cubes. There are 5 layers.

So, the volume of that rectangular prism is 32 x 5 = 160 cubic units

 

You may be wondering, how do I know how many cubes cover a rectangular prism, if only the figure on the right was showed?

 

Well, I will now provide you with the formula the of the volume of a rectangular prism. 

 

Formula: V=lwh  (length width height)

 

 

Volume of Cylinder:

Formula: pi r^2 h

 

 

 

 

Volume of Pyramids and Cones :

 

  

 

I hope you are understanding so far! make sure to follow along with the formulas and you will have no problem. Check out the video below for any doubts.

 

 

 

 

Volume of a Sphere:

 

 

                                              Formula : V = 4/3 π r³

 

 

Surface Areas

Hello everyone! Today I will be talking to you about surface areas. For those of you who don't know, surface area is a measurement used on a daily basis. For example when painting your room or home, when changing floors or adding carpets at home, it pretty much goes from measuring the walls at home to driveways, highways etc. Today you are going to learn how to find the surface area from many shapes.

Before I start in case you hadn't read the previous blogs I've posted I will recommend you to do so, in order for you to clearly understand what I am going to be talking about today.

A) Surface area of a right prism
   To find the surface area of a right prism you have to find the net of the shape. In other words you have to flatten a shape into a 2-dimensional shape.

- Net for a cube = 6 x Edges^2
-Sum of lateral faces is the sum of the surface area
-Find the lateral area by adding the lateral faces, and last
-to find the surface area you must add the lateral area and the area of the bases together.

 

 

 

 

 

Surface Area of a Cylinder

 

 

 

 

 

The formula to find the surface area of a cylinder is

  2 pi r ² + 2 pi r

 

http://www.math.com/tables/geometry/surfareas.htm    <----- more info here

 

 

 

 

Surface Area of a Pyramid



According to the textbook "A Problem Solving Approach to Mathematics," the surface area of a pyramid "is the sum of the lateral surface area of the pyramid and the area of the base."

 

 

 

Formula:  n (1/2 bl) + B     {where b: base,  B: area of base, n: # of faces}

 

 

 

 

 

 

Surface Area of a Cone

 

 

 

Formula: pi r^2 + pi rl

 

 

 

 

 

Surface Area of a Sphere

 

 

 

Formula : 4 pi r^2 

 

The Pythagorean Theorem

The Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who discovered the Pythagorean Theorem and its proof.m. There is evidence that Babylonian mathematicians understood the formula, but there lack of evidence that they used it at the times.

The Pythagorean theorem is often used to find the measures of the sides of a right triangle. It works the following way. Picture a right triangle, the side that is opposite from the right angle is know as the hypotenuse. The two sides left are known as the legs.

According to the book "A Problem Solving Approach to Mathematics for Elementary School Teachers," "interpreted in terms of area, the Pythagorean theorem states that the area of a square with the hypotenuse of a right triangle as a side is equal to the sum of the areas of the squares with the legs as sides."

The formula for the Pythagorean theorem as stated above is

Lets perform the following. Let  a=24 b=18, we must find the hypotenuse or c^2.

So, 18^2 + 24^2 = c^2
     324 + 576 = c^2
        900 = c^2
       √900 = 30

c=30


Pick's theorem

Today I have a little trick theorem for you guys. Have you ever bumped into a "weird" shape and have gotten stuck trying to find the area of the shape? Well if you have I will show you how to get rid of these problem. Pick's theorem gives an a very accurate approximation of the area of a bizarre shape.

Formula:
             A = B/2 + I - 1            (B: border Points, I: interior points)

Note: This theorem can only be performed if the shape is shown on either a geoboard or a dotted paper because is the only way to find the interior angles.

Areas of Polygons and Circles

Today I am going to be talking about what the area of a polygon is and how to find it. I need you to play close attention to the definitions and formulas that I will be providing you with! Never forget that MATH IS FUN!

"The area of a Polygon is the amount of space or region occupied by that polygon."

There are 2 methods to find areas other than the formulas and they can both be found on a geoboard or a dot paper. I will be showing you a video with one of the methods which I consider to be the clearest one which will be perfect for your future classroom.

 

when finding the area or perimeter or any type of calculation it is important not to forget the square units and how to convert them. Remember this sentence in order to remember how to convert in metric:  King      Henry     Died        Ugh     Drinking   Chocolate    Milk
             K:kilo   H:hecto-   D:deka-  U:unit  D:deci-     C:centi-       M:milli

 

Unit Name	Unit Symbol	Metric Equivalent	Imperial Equivalent
Metric...
square centimetre	cm²	1 cm² = 100 mm²	1 cm² = 0.1550 in²
square metre	m²	1 m² = 10,000 cm²	1 m² = 1.1960 yd²
hectare	ha	1 ha = 10,000 m²	1 ha = 2.4711 acres
square kilometre	km²	1 km² = 100 ha	1 km² = 0.3861 mile²
Imperial...	Unit Symbol	Imperial Equivalent	Metric Equivalent
square inch	in²		1 in² = 6.4516 cm²
square foot	ft²	1 ft² = 144 in²	1 ft² = 0.0929 m²
square yard	yd²	1 yd² = 9 ft²	1 yd² = 0.8361 m²
acre	acre	1 acre = 4840 yd²	1 acre = 4046.9 m²
square mile	mile²	1 mile² = 640 acres	1 mile² = 2.59 km²

 

Now I will provide you with the formulas needed to find the area of polygons!

Area of a square = s²

Area of a rectangle = lw    (length x width)    

Area of a triangle = bh   (1/2 x base x height) 

Area of a parallelogram = bh  (base x height) 

Area of a trapezoid = () or  1/2 x h   (b1: base 1 b2: base 2)

 

 

Lets find the area of the shape above:

1. First lets start with the rectangle laid horizontally. We know that the formula to find the area of a rectangle is A: lw  (length x width)

 

2. The length of a rectangle is the longest line, in this case the length of the rectangle is 6.5m. The width is the shortest line so the width of the first rectangle is 4m.

 

3) Now lets put it into play!   A: 6.5m x 4m = 26m^2

 

4) Lets do the same for the rectangle laid vertically!  A: 9m x 3.5m = 31.5m^2

 

5) GREAT JOB!! don't forget about changing the metric units to meter squared!

 

Area of a circle

 

area = π * (radius)²
π = 3.14

 

A: πr²

 

This is all for The Area of Polygons and Cirlces! I hope you enjoyed and I hope you are able to clear your doubts! If you need more information go to www.mathisfun.com and try how to find the areas of different shapes!



Biography

Hello everybody, my name is Laura Jaramillo and I am from Colombia. I have been living in the U.S for about 6 years but I try to visit my country every summer. I lived in Miami Beach Florida for the past 5 years but moved to Arizona last summer. I will continue my education in Washington DC for the upcoming fall semester but I am glad I got to experienced living in the valley of the sun. I currently play tennis in the school team and I love it. On my free time I try to read, workout or perform outdoor activities.

I changed my major from Communications to Elementary Education at the beginning of my sophomore year. I love children and I always thought of teaching as a possible profession so I made it happen. I have a passion for children I love to be surrounded by them and I love to watch them learn. I am totally fascinated by their hunger for learning new things which I believe is one of the main reasons I want to be a teacher. i want to be the guide to success for my future students and I will make them want to learn.

“I am not a teacher, but an awakener.”
― Robert Frost

“[Kids] don't remember what you try to teach them. They remember what you are.”
― Jim Henson

This is the love of my life :)