June 2012 - Page 4 of 8 - Math and Multimedia

Visualizing the Difference of Two Squares

June 18, 2012 GB High School Algebra, High School Mathematics

Geometric objects are powerful representations that can be used to visualize algebraic properties of mathematical objects. Proofs without words are examples of such visual representations. In this post, we relate the difference of two squares to the areas of squares and rectangles.

The difference of two squares states that for all numbers $a$ and $b$ , $a^2 - b^2 = (a + b)(a-b)$ . The visual representation below, however, only covers the condition that $a^2 - b^2 \geq 0$ (Why?).

To proceed with the visual proof, we create a square with side length $a$ as shown in (1). Then, we cut a square with side length $b$ from its corner as shown in (2). Since the area of the larger square is $a^2$ and the area of the smaller square is $b^2$ , the area of the remaining figure is $a^2 - b^2$ . » Read more

3 comments difference of two squares, proofs without words, special products and factoring, visual representations

Math Teachers at Play Carnival 51 at Math Mama Writes

June 17, 2012 GB Blog Carnival

Math Mama Writes has a staggering collection of 51 articles from the blogosphere for the 51st edition of the Math Teachers at Play Blog Carnival. I particularly liked the opening trivia about 51.

I’m a pentagonal number,

I have just two factors,

and if you put me in base 2, 4 or 16,

I’m a palindrome.

I wonder:

Is there anyone else like me in the number universe?

On the other hand, the Math and Multimedia Carnival is currently dormant and will resume next month. It will be hosted by Mathematics, Learning and Web 2.0. You may now submit your entries at the submission form.

Leave a comment blog carnival, math carnival, math teachers at play, mtap carnival

Triangular Numbers and the Sum of the First n Positive Integers

June 16, 2012 GB High School Mathematics, High School Number Theory

The numbers 1, 3, 6, 10, 15, … are called triangular numbers because they could be arranged in the form of triangles. Triangular numbers is one of the polygonal numbers — numbers that can represented by dots to form regular polygons.

Finding the nth triangular number is quite easy. All we have to do is form a rectangle using the “dot representation” of two triangular numbers. For example, we want to find the fourth triangular number, we create dots representing two triangular numbers, and then use them to form a rectangle. The area of the formed rectangle is $4(4+1)$ . » Read more

2 comments polygonal numbers, rectangular numbers, sum of the first n positive integers, triangular numbers

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