algebraic expressions. education's purpose is to replace an empty mind with an open one....

Algebraic Expressions

Education's purpose is to replace an empty mind with an open one.

Malcolm Forbes

Expressions

Math expressions represent a convenient way to translate verbal expressions

What is the area of a rectangle?

Length times WidthIf the length is 3 meters and the width is 2 meters, what is the area?

A = L x WA = 3 x 2 = 6 m2

A, L and W are the variables. It is any letter that represents an unknown number.

An algebraic expression is a quantity that contains numbers and variables.

x + y , 3a2 a , 3x + 2y z

A term is a number, a variable, or a product of numbers and variables.

Terms

Algebraic Expression

Components of an Algebraic Expression

Constant term: fancy name for a number

Variable term: terms with lettersExample: 3xy – 4z + 17

Variable expression with 3 terms: 3xy, -4z, 172 variable terms and 1 constant term

Variable Terms

Consist of two partsThe variable(letter) part The number part

Example:2xy has a coefficient of 2-6j has a coefficient of –6W has a coefficient of 1

In expressions, there are many different ways to write multiplication.

1) ab 2) a • b 3) a(b) or (a)b 4) (a)(b) 5) a x b

We are not going to use the multiplication symbol any more. Why?

Division, on the other hand, is written as:

1)

2) x ÷ 3

x

3

Example of evaluating an expression.

Evaluate 3xy – 2x + 7y when x = 2 and y = 3

3(2)(3) – 2(2) + 7(3)18 – 4 + 2114 + 2135The value of the expression is 35.

Word Phrases as Algebraic Expressions

Addition

(+)

Subtraction

(–)

Multiplication

(·)

Division

()sum

plus

added to

more than

increased by

total

difference

minus

subtract

less than

decreased by

less

product

times

multiply

of

double/triple

quotient

divide

shared equally

among

ratio of


Write as an algebraic expression. Use x to represent “a number.”

In words:

5 decreased by a number

5

Translate: 5decreased by a number

– x


Write as an algebraic expression. Use x to represent “a number.”

The quotient of a number and 12

In words: a number

Translate: x

and 12

12 or 12x

The quotient of

Write an algebraic expression for the following

m increased by 5

7 times the product of x and t.

m + 5

7(xt) or 7xt

Write an algebraic expression for the following

11 less than 4 times a number.

two more than 6 times a number.

4n - 11

6n + 2

Which of the following expressions represents: 7 times a number decreased by 13

1. 7x + 132. 7x - 133. 13 - 7x4. 13 + 7x

Which one of the following expressions represents:28 less than three times a number

1. 28 - 3x2. 3x - 283. 28 + 3x4. 3x + 28

Which of the following verbal expressions represents:

2x + 91. 9 increased by twice a

number

2. a number increased by nine

3. twice a number decreased by 9

4. 9 less than twice a number

Which of the following verbal expressions represents:

x2 + 2x1. the sum of a number squared

and twice a number

2. the sum of a number and twice the number

3. twice a number less than the number squared

4. the sum of a number and twice the number squared

Which of the following expressions represents: four less than the cube of a number

1. 4 – x3

2. 4 – 3x3. 3x – 44. x3 – 4

Terms

Like termsTerms with the same variable part

Same means same letter(s) and power(s)

2x, -5x¾x2, 7x2

31xy, 4xy

Terms

We simplify variable expressions by combining like terms.To combine like terms, work with the

coefficients of the like terms

Combining Like Terms

t + t + t+ t + t

There are five variables which are like terms therefore we simply add them like we would if they were numbers.

t + t + t+ t + t = 5t


4t + 3t + t = (4+3+1)t = 8t

2t2 + 8 – 5t2

Rearrange the variables so that all like terms are side by side.2t2 – 5t2 + 8 (notice that the sign in front of the number came with the number)

2t2 – 5t2 + 8 = -3t2 + 8


2t + 3t2 – 2t –t2

3t2 – t2 + 2t – 2t (collecting like terms)2t2 + 02t2

Which figure below models the simplification of - 4x - 5 + 7x + 7 using these tiles?

1.

2.

3.

4.


3x + 5 – 9x =

-5 +3b – 7 – 5b =

– 6x + 5

-5 – 7 = -12

3b – 5b = -2b

– 2b - 12

Simplifying Algebraic Expressions

Rewrite using as few symbols as possible Use the distributive property if necessary

to remove parentheses. Combine like terms More often than not will have numbers and

letters in the final answer.

Distributive Property

a(b c) a b a c

a(b c) a b a c or


Objective - To use the distributive property to simplify numerical and variable expressions.

3(4 5) 3(9) 27Order of Operations


3(4) 3(5)12 15

27It works!

Why use the distributive property?3(x 2) 3(x) 3(2) 3x 6

Simplify using the distributive property.1)

2)

3)

4)

5)

6)

5(x 3)5 3

5x 15

6(y 7)6 7

6y 42

3(m 8)3 8

3m 24

4(3 y)4 3 4 y

12 4y

10(x 7)10 x 10 7

10x 70

4(k 2)4 k 4 2

4k 8

5 x

6 y

3 m

Geometric Model for Distributive Property

Two ways to find the area of the rectangle.

4

5 2

As a whole As two parts

A w l

4 5 2

Geometric Model for Distributive Property

Two ways to find the area of the rectangle.

4

5 2

As a whole As two parts

A w l

4 5 2 4 5 4 2

4 5 4 2

same

4 5 2 4 5 4 2

Scientific Notation

A short-hand way of writinglarge numbers without writing all of the zeros.

The Distance From the Sun to the Earth

93,000,000

Step 1

Move decimal left Leave only one number in front of

decimal

Step 2

Write number without zeros

Step 3 Count how many places you moved decimal

If moved to left, make it positive If moved to right, make it negative

Make that your power of ten

93,000,000 --- Standard Form

9.3 x 107 --- Scientific Notation

Practice Problem

1) 98,500,000 = 9.85 x 10?

2) 64,100,000,000 = 6.41 x 10?

3) 279,000,000 = 2.79 x 10?

4) 4,200,000 = 4.2 x 10?

5) 0.000013 = 1.3 x 10?

Write in scientific notation. Decide the power of ten.

9.85 x 107

6.41 x 1010

2.79 x 108

4.2 x 106

1.3 x 10-5

Complete Practice Problems

1) 50,0002) 7,200,0003) 802,000,000,0004) 0.000000000631

Write in scientific notation.

1) 5 x 104 2) 7.2 x 106 3) 8.02 x 1011 4) 6.31 x 10-10

Write in Standard Form

6.27 x 106

9.01 x 104

3.95 x 10-3

6,270,000 90,100 0.00395

Positive exponent → move to rightNegative exponent → move to left

algebraic expressions. education's purpose is to replace an empty mind with an open one....

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