tools of algebra : variables and expressions; exponents and pemdas; working with integers; applying...

Tools of Algebra:

Variables and Expressions; Exponents and PEMDAS;

Working with Integers; Applying the Distributive Property; and Identifying

Properties of Real Numbers Compiled and adapted by

Lauren McCluskey

Credits• “Algebra I” by Monica and Bob

Yuskaitis• “Interesting Integers” by Monica and

Bob Yuskaitis• “Multiplying Integers”• “Dividing Integers” • “Order of Operations” • “Properties” by D. Fisher• “Coordinate Plane” by Christine Berg• Prentice Hall Algebra I

Algebra I

By Monica Yuskaitis

Variable

• Variable – A variable is a letter or symbol that represents a number (unknown quantity).

• 8 + n = 12

Expression

• Algebraic expression – a group of numbers, symbols, and variables that express an operation or a series of operations.

• m + 8• r – 3

Evaluate

• Evaluate an algebraic expression – To find the value of an algebraic expression by substituting numbers for variables.

• m + 8 m = 2 2 + 8 = 10• r – 3 r = 5 5 – 3 = 2

Simplify

Simplify – Combine like terms and complete all operations

• m + 8 + m 2 m + 8• (2m x 2) + 8n 4m + 8n

Words That Lead to Addition

• Sum

• More than

• Increased

• Plus

• Altogether

Words That Lead to Subtraction

• Decreased

• Less

• Difference

• Minus

• How many more

Write Algebraic Expressionsfor These Word Phrases

• Ten more than a number

• A number decrease by 5

• 6 less than a number

• A number increased by 8

• The sum of a number & 9

• 4 more than a number

n + 10

w - 5

x - 6

n + 8n + 9

y + 4

Types of Equations:

• Equations may be:

• ‘True’ (when the expressions on both sides of the equal sign are equivalent)

• ‘False’ (when the expressions on both sides of the equal sign are not equivalent)

• ‘Open sentences’ (when they contain one or more variables.

Complete This Table

n 2n - 3 y5 *2 - 3

10 *2 - 3

21 *2 - 3

32 *2 - 3

173961

7

ExponentsExponents influence only that which they touch directly. For example:

40 - d2 + cd * 3

(for c= 2 and d= 5)

40 - (5 * 5) + (2 * 5) * 3

40 - 25 + (10 * 3)

40 - 25 + 30

70 - 25

50

Now you try one: 40 + (-d)2 + cd *3

Check your answer:

40 + (-5 * -5) + (5 * 2) * 340 + 25 + (10 * 3)40 + 25 + 30 95

*Note: In this case you multiply (-5) * (-5) because of the parentheses.

Try one more:

40 - d2 + cd * 3

(for c = 2 and d= -5)

Check your answer:

40 - (-5)(-5) + (-5) (2) * 3

40 - (25) + [(-10) * 3]

40 + (-25) + (-30)

40 + (-55)

-15

Exploring Real Numbers:

• Natural Numbers may also be known as counting numbers {1, 2, 3…}

• Whole Numbers include zero {0, 1, 2, …}

• Integers include negative numbers

{…-2, -1, 0, 1, 2, …}

Exploring Real Numbers

• Rational Numbers can be written as either terminating or repeating decimals

• Irrational Numbers do not terminate or repeat when written as decimals

• Real Numbers include all rational and irrational numbers

Order of Operations

A standard way to simplify mathematical

expressions and equations.

Purpose• Avoids Confusion• Gives Consistency

For example:8 + 3 * 4 = 11 * 4 = 44

Or does it equal8 + 3 * 4 = 8 + 12 = 20

Order of operations are a set of rules that mathematicians have agreed to follow to avoid mass CONFUSION when simplifying mathematical expressions or equations. Without these simple, but important rules, learning mathematics would

be maddening.

The Rules

1) Simplify within Grouping Symbols

( ), { }, [ ], | |2) Simplify Exponents

Raise to Powers3) Complete Multiplication and

Division from Left to Right4) Complete Addition and

Subtraction from Left to Right

Back to Our Example

For example:

8 + 3 * 4 = 11 * 4 = 44

Or does it equal

8 + 3 * 4 = 8 + 12 = 20Using order of operations, we do the multiplication first. So what’s ouranswer? 20

How can we remember it?

• Parenthesis - Please

• Exponents - Excuse

• Multiplication - My

• Division - Dear

• Addition - Aunt

• Subtraction - Sally

OR: ‘PEMDAS’

Adding / Subtracting Real Numbers

• Inverse property:

“For every real number n, there is an additive inverse -n such that

n + (-n) = 0.

We use the inverse property to solve equations.

from Prentice hall Algebra I

Matrices

by Lauren McCluskey

Adding Matrices

[ ] [ ] -5 + (-3) 2.7 + (-3.9) 7 + (-4) -3 + 2

[ ]

-5 2.77 -3

-3 -3.9-4 2

-8 -1.23 -1

Try It!

[ ] + [ ]

[ ]

-5 -3/4

1/2 -1-4 7/8

3/4 0

Check your answer:

-4 + (-5) 7/8 + (-3/4)3/4 + 1/2 0 + (-1)

[ ]-9 1/8

1 1/4 -1

Scalar Multiplication (Matrices)

[ ] (-0.1)(-47) (-0.1)(13) (-0.1)(-7.9)

(-0.1)(0.2) (-0.1)(-64) (-0.1)(0)

[ ]

-47-0.1 13 -7.90.2 -64 0

Check your answer:

[ ]4.7 -1.3 0.79-0.02 6.4 0

Try It!

[ ]

3/5 * ____ 3/5* _____

3/5 * ____ 3/5* _____

-253/5 3510/9 -15

Check your answer:

3/5 * -25 = -15 3/5 * 35= 21

3/5 * 10/9= 2/3 3/5 * -15= -9

[ ]

-15 212/3 -9

For more practice:

• Go to pages 27-30 for +;

or pages 43 and 45 for * (scalar).

Interesting Integers!

What You Will Learn• Some definitions related to

integers.

• Rules for adding and subtracting integers.

• A method for proving that a rule is true.

Definition

• Positive number – a number greater than zero.

0 1 2 3 4 5 6

Definition

• Negative number – a number less than zero.

0 1 2 3 4 5 6-1-2-3-4-5-6

Definition

• Opposite Numbers – numbers that are the same distance from zero in the opposite direction

0 1 2 3 4 5 6-1-2-3-4-5-6

Definition

• Integers – Integers are all the whole numbers and all of their opposites on the negative number line including zero.

7 opposite -7

Definition• Absolute Value – The size of a

number with or without the negative sign.

The absolute value of 9 or of –9 is 9.

Negative Numbers Are Used to Measure Temperature

Negative Numbers Are Used to Measure Under Sea Level

0102030

-10-20-30-40-50

Negative Numbers Are Used to Show Debt

Let’s say your parents bought a car buthad to get a loan from the bank for $5,000.When counting all their money they add in -$5.000 to show they still owe the bank.

Integer Addition Rules• Rule #1 – If the signs are the same,

pretend the signs aren’t there. Add the numbers and then put the sign of the addends in front of your answer.

OR:

• Think Teams: Which team won? How much did they win by?

9 + 5 = 14 -9 + -5 = -14

Solve the Problems• -3 + -5 =

• (+3) + (+4) =

• -6 + -7 =

• -9 + -9 =

-8

-18

-13

7

Integer Addition Rules• Rule #2 – If the signs are different pretend the

signs aren’t there. Subtract the smaller from the larger one and put the sign of the one with the larger absolute value in front of your answer.

OR• Think Teams: Which team won? How much

did they win by?

-9 + +5 =9 - 5 = 4

Larger abs. value

Answer = - 4

Solve These Problems

• 3 + -5 =• -4 + 7 =• (+3) + (-4) =• -6 + 7 = • 5 + -9 =• -9 + 9 =

-25 – 3 = 2

0 -4

1-1

3

9 – 9 = 0

9 – 5 = 4

7 – 6 = 14 – 3 = 1

7 – 4 = 3

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

When the number is positive, countto the right.

When the number is negative, countto the left.

+-

Adding on a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+3 + -5 = -2

Adding Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+6 + -4 = +2

Adding Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+3 + -7 = -4

Integer Subtraction RuleSubtracting a negative number is the same as adding its opposite. Change the signs and add.

2 – (-7) is the same as

2 + (+7)

2 + 7 = 9!

Here are some more examples.

12 – (-8)

12 + (+8)

12 + 8 = 20

-3 – (-11)

-3 + (+11)

-3 + 11 = 8

Check Your Answers

1. 8 – (-12) = 8 + 12 = 20

2. 22 – (-30) = 22 + 30 = 52

3. – 17 – (-3) = -17 + 3 = -14

4. –52 – 5 = -52 + (-5) = -57

Multiplying / Dividing Real Numbers

Multiplying and dividing positive and negative numbers is easy when you remember the rules:

positive * positive = positive [+*+ = +] negative * negative = negative [- * - = -]positive * negative = negative [+ * - = - ]

MULTIPLYING INTEGERS

Problem 1

(-3)( 5)=

Problem 2

(5)(-3) =

Problem 3

(-2)(-10) =

Problem 4

(-3)(8) =

Problem 5

(-6)(8) =

Problem 6

(3)(-9)=

Problem 7

(-7)(-3) =

Problem 8

(-9)(0) =

Problem 9

(-9)(-7) =

Problem 10

(16)(-10) =

Problem 11

(9)(-5) =

Problem 12

(-4)(-9) =

Problem 13

(5)(-1) =

Problem 14

(10)(-4) =

Problem 15

(15)(-2) =

Problem 16

(-5)(-11) =

Problem 17

(-10)(4) =

Problem 18

(-4)(7) =

Problem 19

(-12)(5) =

Problem 20

(-8)(-4) =

Check your answers:1)-15 11) -452) -15 12) +363) +20 13) -5 4) -24 14) -405) -48 15) -30 6) -27 16) +557) +21 17) -408) 0 18) -289) +63 19) -6010) -160 20) +32

DIVIDING INTEGERS

1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE

2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE

Remember:

Problem 1

(-6)÷( 3)=

Problem 2

(6) ÷(-3) =

Problem 3

(-10) ÷ (-2) =

Problem 4

(-16) ÷ (2) =

Problem 5

(-12) ÷ (6) =

Problem 6

(9) ÷ (-3)=

Problem 7

(-8) ÷ (-4) =

Problem 8

(-9) ÷ (0) =

Problem 9

(-21) ÷ (-7) =

Problem 10

(16) ÷ (-8) =

Problem 11

(10) ÷ (-5) =

Problem 12

(-12) ÷ (-6) =

Problem 13

(5) ÷ (-1) =

Problem 14

(10) ÷ (-5) =

Problem 15

(15) ÷ (-3) =

Problem 16

(-22) ÷ (-11) =

Problem 17

(-16) ÷ (4) =

Problem 18

(-14) ÷ (7) =

Problem 19

(-12) ÷ (6) =

Problem 20

(-8) ÷ (-4) =

Check your answers: 1)-2 11) -22) -2 12) +23) +5 13) -54) -8 14) -25) -2 15) -56) -3 16) +27) +2 17) -48) 0 18) -29) +3 19) -210) -2 20) +2

The Distributive Property

•You can use the Distributive Property

to multiply a sum or difference

by a number.

•You can also use the Distributive Property

to simplify algebraic expressions by

removing the parentheses.

A good way to remember how to apply the Distributive Property is to visualize 2 rectangles:

3*1= 3

3 * x = 3x3

1x

So 3( x + 1) =(3 * x) + (3* 1)=3x + 3

Try It!

a) 2/3(6y + 9)b) 0.25(6q + 32)c) (8- 3r) 5/16d) -4.5(b- 3)

Check your answers:

a) 4y +6

b) 1.5q + 8

c) 2 1/2 + 15/16 r

d) -4.5b + 13.5

by D. Fisher

(2 + 1) + 4 = 2 + (1 + 4)Associative Associative Property of Property of AdditionAddition

3 + 7 = 7 + 3

Commutative Commutative Property of Property of AdditionAddition

8 + 0 = 8

Identity Identity Property of Property of AdditionAddition

..

6 • 4 = 4 • 6

Commutative Commutative Property of Property of

MultiplicationMultiplication

2(5) = 5(2)



3(2 + 5) = 3•2 + 3•5Distributive Distributive

PropertyProperty

6(7•8) = (6•7)8

Associative Associative Property of Property of


6(3 – 2n) = 18 – 12n

Distributive Distributive PropertyProperty

2x + 3 = 3 + 2x


ab = ba



a + 0 = a

Identity Identity Property of Property of AdditionAddition

a(bc) = (ab)c

Associative Associative Property of Property of


a•1 = a

Identity Identity Property of Property of


a +b = b + a


a(b + c) = ab + ac

Distributive Distributive PropertyProperty

a + (b + c) = (a +b) + c

Associative Associative Property of Property of AdditionAddition

The Coordinate

PlaneBy: Christine BergEdited By:VTHamilton

DefinitionThe plane formed when 2

perpendicular number lines intersect at their

zero points

Coordinate Plane

Coordinate Plane

The perpendicular number lines form a

grid on the plane

X-axis

The horizontal number line

•Positive to the right

•Negative to the left

Y-axis

The vertical

number line

•Positive upward

•Negative downward

Origin

Where the x and y axes intersect at their zero

points

Quadrants

The x and y axes divide the coordinate plane into 4 parts called

quadrants

III

III IV

Ordered Pair

A pair of numbers (x , y) assigned to a point

on the coordinate plane

Ordered Pair

(x , y)

X-coordinateX-coordinate

Y-coordinateY-coordinate

Plotting a Point

Step 1:

Begin at the Origin

Plotting a Point

Step 2:

Locate x on the x-axis

Plotting a Point

Step 3:

Move up or down to the value of y

Plotting a Point

Step 4:

Draw a dot and label the point

tools of algebra : variables and expressions; exponents and pemdas; working with integers; applying...

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