Algebra An Introduction

The History of Algebra The history of algebra began In Ancient Egypt and Babylon where basic equations were first solved. Our modern view of algebra first arose in the 9 th Century B.C.E in Baghdad, Persia, in a book by a mathematician named al-Khwarizmi Al-jabru is the Arabic word for restoration or completion and is the root of our word algebra. Algebra made its way into Europe in the 12 th Century through a Latin translation of al-Khwarizmi’s text and was first used by Leonardo Fibonacci. A French mathematician named René Descartes was the first to use x, y and z to represent unknown values in his treatise La Géométrie, in 1637.

Arithmetic vs Algebra Algebra uses all the same operations as other areas of maths In algebra we have a new part to our sum called the unknown. This unknown is typically written as a letter.

Problem John recently started working at the local dog grooming parlor. He gets €5 an hour and €2 for every dog that he washes. 1. How much money will John earn after washing the following amounts of dogs? 5 Dogs: _______ 10 Dogs: _________ 14 Dogs: ________ 25 Dogs: _______ 2. How much money does John earn after 100 dogs? _________ 3. Explain how you found your answer for the amount for 100 dogs:

Problem There is a pattern to how John is earning his money so we could represent what John is doing using algebra. Dog s ExpressionSavings (1) (2) (3) (4) (5)15

Key Words Below is what we call an expression. It’s a formula we use to find an unknown value. Each part of the expression is called a term. Constant (Value doesn’t change) Coefficient (The number in front of the variable) Variable (Value changes)

Writing Expressions All of the expressions below mean ‘2 times d’ 2 X d =2d 2d(This is the most common way to represent ‘2 times d’) 2  d 2(d) (2)(d) (2)d

Simplifying Expressions Each part of our expression or sum is considered a different term, or a separate part. When we want to simplify an expression we look to see if any terms have something in common. In the expression below there are 5 terms 3 of these terms are different Note: We can only add or subtract terms that are the same. Next we group them together and finish our sum

Simplifying Expressions Remember when simplifying expressions we look to see if any of the terms have something in common. Think about the following expression There are 3 terms 2ab and 3ba are considered the same, even though the order has changed. This is an example of the commutative property

Evaluating an Expression When we evaluate an expression we find its value. Our answer should always be a number We use a method called substitution. We swap the letter with a number and work out the sum. For example; Evaluate the following sums if b=3

Simplifying Expressions with Brackets Two things to remember Brackets in maths can mean multiplication The distributive rule To evaluate an expression like we must use the distributive method Why? We can’t add or subtract terms that are different. Distributive Method

Simplifying Expressions with Brackets No matter how many brackets are in an expression or how many terms in each bracket we always use the distributive method. Use the distributive rule to remove the brackets Remember the integer rules Group the terms and finish

Simplifying Expressions with Indices, Powers or Exponents When we covered natural numbers we saw that if a number was multiplied by itself we could write it in exponent form For example The same rules apply when multiplying letters We can’t go any further as x 2 and x are different terms As there is an x 2 term this is called a quadratic expression It is called quadratic as the Latin word quadratum means square

Simplifying Expressions with Indices, Powers or Exponents More examples We must multiply the coefficients and the letters There are two x terms being multiplied so the new term is x 2 There are two x terms being multiplied one is x 2 and the other is x so the new term is x 3 The sum is

Expanding Two Brackets I can find the area of this rectangle in two ways. I could add the individual lengths and widths and multiply Or I could find the area of the 4 smaller rectangles and add them together Either way I get an answer of 60 2

Expanding Two Brackets 4 3 This time we would have to find the area of the smaller rectangles and add them to find the total area. If we do this we get

Expanding Two Brackets We could think of the previous rectangle as The general method is to split the first bracket and multiply both terms by the second bracket. This is what we did with the rectangle The second bracket repeats

Expanding Two Brackets

Writing Expressions One of the most useful areas of algebra is using it to solve a problem. This requires you to be able to translate a sentence or paragraph into an algebraic expression. + – x÷ Plus, sum, increased by Minus, difference, less than Times, product, equal groups of Divided by, quotient

Words to expressions John types 62 words per minute. Write an expression to represent this fact Use m to represent words Per minute means multiplication 62m Robert is 4 years older than Emily. Write Robert’s age as an expression of Emily’s Use a to represent the age. Older than means more so use addition 4 + a