The first thing the student notices in his first algebra class is that numbers are mixed with letters. Previous mathematics classes were all about numbers. Consider the English language problem: "What number is doubled, then increased by three to equal 11?" In this problem, we are looking for something that can be manipulated to produce 11. The algebraic way to say this is: 2X + 3 = 11. When you write it this way, it is easy to see by trial and error that X = 4. Tip No. 1: know what the variable is. In this problem, X is a number we are looking for.
Exponents show another use of letters in algebra -- a way to make general statements that are true for any numbers. A^3 = A X A X A. This is true no matter what A is, so 5^3 = 5 X 5 X 5 and 100^3 = 100 X 100 X 100. Rules for exponents are also introduced using letters. A^n X A^m = A^(n + m). These patterns cover an infinite number of cases, such as 5^2 X 5^3 = 5^5 and 7^1 X 7^2 = 7^3. Tip No. 2: In algebra, letters are often used to make general statements about concepts.
Polynomials can be used to solve difficult problems. For example, suppose a farmer has 100 feet of fence and wants to build a pig pen that has a barn for one side and a fence for the other three sides. What shape would enclose the maximum area? Express the area in terms of fence length perpendicular to the barn: A = F(100 - 2F) = -2F^2 + 100 F. Graphing this, we get a parabola whose high point is at F = 25. The pen with the maximum area is 25 X 50 feet. Tip No. 3: knowing about polynomials can help you solve real-world problems.
Cartesian coordinates are a way to assign an address to every point in space and to describe the relationship between things as pictures. This can make a lot of problems easier. For example, turning the problem about fencing an area into a mathematical relationship between fence length and area enclosed and then graphing the relationship gives us a picture of the relationship. We can look at this picture and tell immediately what shape gives the maximum enclosed area. Cartesian coordinates are one of the most important inventions in algebra. Tip No. 4: Graphing function using the Cartesian coordinates allows you to see important characteristics of a function.