Solari Garren
History and Technology
History
The development of Algebra can be seen through Egyptian Algebra,
Babylonian Algebra, Greek Geometric Algebra, Diophantine Algebra, Hindu Algebra,
Arabic Algebra, European Algebra and Modern Algebra. Algebra has been developed
over the past 4000 years. Modern or abstract algebra is only about 200 years old
(The History of Algebra).
Algebra was first seen in Egypt. Written around 1650 B.C. the Rhind papyrus
is where most of the available information on Egyptian algebra comes from. At this
point they could solve problems equivalent to linear equations in one variable. They
did not yet use symbols. Their problems were all solved verbally. Babylonian
Algebra (1800-1600 B.C.) was a little more advanced than the Egyptians. They had a
better number system and were solving quadratic equations with a general formula
however; they only recognized one solution, the positive. The Babylonians were also
solving simple systems of equations, two equations with two variables. They were,
like the Egyptians, solving these rhetorically. For the Greeks they focused on algebra
geometrically. They dealt and solved problems that could be addressed and proven
geometrically. They were able to do about the same as the Babylonians. The form of
geometric algebra made progress difficult (The History of Algebra).
Diophantus, a Greek mathematician from around the third century had many
contributions to mathematics. He wrote three books, Arithmetica, On Polygonal
Numbers and Porisms (Allen). He made progress solving what are now called
Diophantine equations. These equations involved sums, powers and products of
integers. The only solutions were integers (Diophantine Equations). Diophantus was
the first to introduce symbolism to algebra (Allen).
Hindu mathematics dates from around 800 B.C. It was influenced by
astronomy and astrology. About 600 A.D. the Hindu’s developed a base-10 number
system and began to treat zero as a number. They also began to use negative
numbers. Other cultures did not have physical representations of negative numbers
so they were reluctant to consider them numbers at all. The Hindus also started to
use symbolism. They solved quadratic equations recognizing both negative and
irrational solutions and that quadratics have two roots (The History of Algebra).
Much progress was made through Arabic algebra during the seventh and
eight centuries. They picked up where the Hindu’s were expanding on their number
system. The Hindu-Arabic numeration system is what is used today. They did take a
step backwards however and rejected negative numbers. The name Algebra came
from the Arabs. It was adapted from an Arabic word that was in the title of a book
on the subject written about 830. The Arabs also made strides in solving cubic
equations (The History of Algebra).
In Europe after 1500 advances were still being made. Irrationals and
negatives were being used but complex numbers had not yet been thought of.
Algebra was still mostly rhetoric. A Frenchman, Viète, used letters to represent
numbers in the sixteenth century. In 1637 Descartes published La Géométrie. This
linked algebra and geometry and made full use of symbolic algebra (The History of
Algebra).
In the nineteenth century abstract algebra first appeared. Vectors, matrices
and transformations were introduced. Later groups, fields and rings would be seen
as well (The History of Algebra). Algebra has been around a long time but much of
what it looks like today is relatively recent. The symbolism and abstract algebra has
all happened within the last few hundred years.
Technology
There are several different types of technology I want to use when teaching
this unit on solving linear equations in one variable. These include SMARTboard,
Algebra Tiles, Algebra Balance Scales and Hands on Equations. These different
technologies include both virtual manipulatives and hangs on, physical
manipulatives, which help students to understand solving equations.
The Algebra Balance Scales and the Hands on Equations are similar concepts.
Both these manipulatives have a “balance” to help students to see and understand
that equations must stay balanced. What you do to one side, you must do to the
other. The Algebra Balance Scales are a virtual manipulative. An equation can be
typed in then you drag the appropriate tiles to each side of the scale so it is
balanced. The scales will be crooked, or unbalance, if the equation is not balanced.
Hands on Equations is a similar concept except it is a physical manipulative.
Students can actually work with it. Hand on Equations however, is not an actual
scale. It is a bar to represent a balance.
Another technology that I would use to teach this unit are Algebra Tiles.
These are a hands on or virtual manipulative using squares and rectangles to
represent variables and numbers. These can help students see what they are doing
when they solve an equation. Algebra tiles can be used on a SMARTboard by the
teacher and/or each student can have their own paper set.
A SMARTboard is the main type of technology I want to use when teaching
my lesson. With SMARTboard board I can easily do virtual manipulatives with the
students and algebra tiles as a class. The SMARTboard is a great interactive tool to
have in a classroom. It is very versatile. There are many tools available with the
SMARTboard software and available for download as well. There are different
things like jeopardy available that would be great for review also.
These are all different technologies I want to use to help the students to
understand how to solve linear equations in one variable and why these methods
word. If they understand the basics and how and why it works it will not be such a
great leap to apply what they know to real world problems.