Showing posts with label Algebra. Show all posts
Visual and Geometric Proof of Algebraic Identities
There are a lot of algebraic identities. One of the most commonly used identity is the identity
(a+b)^2 = a^2 + 2ab + b^2.
Did you ever wondered where did it came from?
There are various ways on how to prove this algebraic identity. One of which is using the visual and geometric way. It used basic concepts such as areas of squares and rectangles to derive the identity. In the following video, the prerequisite concepts are also included to support the derivation process. The visual proof or geometric representation is shown in detailed and in step-by-step manner. This is for you to easily understand the concept. You may watch the following video:
The same algebraic identity can also be proven using basic algebraic processes. Some of the prerequisite concepts included are multiplication law of indices and distributive property of multiplication. These prerequisites help in the derivation process of the algebraic identity (a+b)^2 = a^2 + 2ab + b^2. You may watch the complete details in this video:
Further, the algebraic identity can also be used as a guide in expanding the square of any binomials. An acronym S-2P-S is introduced in the following video for you to easily remember the process of expanding the square of binomials the fastest way. Here is the complete discussion of the acronym with various examples:
Hope you will learn from these videos about algebraic identities.
Your comments and suggestions are welcome here. Write them in the comment box below. Thank you and God bless!
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PYTHAGOREAN THEOREM EXPLORATION 2 (CUT-OUTS)
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| Pythagorean Theorem Exploration 2 Cutouts |
You may download this template for personal use and for your math class activity. The length of the sides on the second page measures 6.5 inches. This is also the measure of the length of the hypotenuse of the right triangle on the first page.
Your comments and suggestions are welcome here. Write them in the comment box below.
Don't forget to like and share.... :)
Thank you and God bless!
Don't forget to like and share.... :)
Thank you and God bless!
PYTHAGOREAN THEOREM (Exploration 2)
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| The Pythagorean Theorem |
Start with a cutout of a right triangle.
Cut three more cutouts.
Let us name the sides of the right triangle with c as the hypotenuse.
Now, let us form a square using the four (4) triangles. This time, use the hypotenuse as the sides of the square.
Let us take a look at the different parts of the square we formed. If we separate the whole square, we can get its area in terms of c.
The middle part is also a square with sides equal to the difference of sides b and a. The area can be obtained as
On the other hand, the area of the four triangles outlining the sides of the whole square is as follows.
Combining their areas, we obtain
Thus, c^2 = a^2 + b^2.
Your questions, comments and suggestions are welcome here. Kindly write them in the comment box below.
1 = 2?
Maybe you are wondering why or how can 1 be equal to 2. Let us look at the following proof:
First, let us have two real numbers a and b where
Using the addition property of equality, let us add b on both sides.
If we simplify both sides by combining like terms, it will become
Now let us subtract 2a on both sides
Combining like terms, we will arrive at
The right side has a common factor, which is 2. Using distributive property, it can be rewritten as
If you notice, both sides has a common factor, which is b - a. To simplify the equation, let us divide both sides by the common factor.
The process will arrive at
Are you convinced? No no no...
Seems like the proof is valid but there is something wrong with it. Look over the proof once again. Can you identify which of the process is not valid?
There is nothing wrong with the given. Real numbers can be equal. There is nothing wrong also with adding b and subtracting 2a on both sides. Likewise, there is nothing wrong with combining like terms on both sides. Then, where is the mistake?
There is nothing wrong with dividing both sides by any number but in this case it becomes invalid. The reason is that b - a = 0 since a = b. Subtracting equal numbers will yield 0. Since b - a = 0, then the result will be UNDEFINED. We cannot also cancel out b - a on both sides because of that.
Therefore, 1 is not equal to 2.
Here is a copy of the proof in pdf form. You may download and print it for educational purposes. You may share it to your friends. Your comments and suggestions are also welcome here.
DAMATH BOARDS
Here are the variations of the DAMATH board available for download. You may choose which one is applicable for your class/school.
If you have any suggestions regarding the form or appearance of the board, please let me know. Just write it in the comment box below. Thank you!
GRAPHING PAPER (SIZE A4) FOR DOWNLOAD
.jpg "graph paper")
This is the second version of the downloadable graphing paper that can be used for math and science classes. The size of this grid paper is a4 (8.27in. x 11.69 in or 210mm x 297 mm). You may directly print it also.
You are also free to write your comments and suggestions about this below (using the comment box). Thank you!
GRAPHING PAPER (LETTER SIZE) FOR DOWNLOAD
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This is a graphing paper in letter size (8.5 in. x 11 in.). You can download this pdf file using the download arrow below. You can directly print it also.
I hope this may help you in any way. If there are any suggestions or comments about this graphing paper, please inform me through the comment box below. Thank you!
