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 – 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 an algebraic expression – To find the value of an algebraic expression by substituting numbers for variables. m + 8m = = 10 r – 3r = 55 – 3 = 2

Simplify – Combine like terms and complete all operations m 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 Expressions for 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 + 8 n + 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 n2n - 3 y 5 * * * *

Exponents Exponents influence only that which they touch directly. For example: 40 - d 2 + cd * 3 (for c= 2 and d= 5) 40 - (5 * 5) + (2 * 5) * (10 * 3) Now you try one: 40 + (-d) 2 + cd *3

Check your answer: 40 + (-5 * -5) + (5 * 2) * (10 * 3) *Note: In this case you multiply (-5) * (-5) because of the parentheses.

Try one more: 40 - d 2 + cd * 3 (for c = 2 and d= -5)

Check your answer: 40 - (-5)(-5) + (-5) (2) * (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: * 4 = 11 * 4 = 44 Or does it equal * 4 = = 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 Powers 3) Complete Multiplication and Division from Left to Right 4) Complete Addition and Subtraction from Left to Right

Back to Our Example For example: * 4 = 11 * 4 = 44 Or does it equal * 4 = = 20 Using order of operations, we do the multiplication first. So what’s our answer? 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) (-3.9) 7 + (-4) [ ]

Try It! [ ] + [ ] [ ] -5 -3/4 1/2 -47/8 3/4 0

Check your answer: -4 + (-5) 7/8 + (-3/4) 3/4 + 1/2 0 + (-1) [ ] -91/8 1 1/4

Scalar Multiplication (Matrices) [ ] (-0.1)(-47) (-0.1)(13) (-0.1)(-7.9) (-0.1)(0.2) (-0.1)(-64) (-0.1)(0) [ ]

Check your answer: [ ]

Try It! [ ] 3/5 * ____ 3/5* _____ -25 3/ /9 -15

Check your answer: 3/5 * -25 = -15 3/5 * 35= 21 3/5 * 10/9= 2/3 3/5 * -15= -9 [ ] /3 -9

For more practice: Go to pages 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 greater than zero

Definition Negative number – a less than zero

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

Definition 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

Negative Numbers Are Used to Show Debt Let’s say your parents bought a car but had 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? = = -14

Solve the Problems = (+3) + (+4) = = =

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? = = 4 Larger abs. value Answer = - 4

Solve These Problems = = (+3) + (-4) = = = = -2 5 – 3 = – 9 = 0 9 – 5 = 4 7 – 6 = 1 4 – 3 = 1 7 – 4 = 3

One Way to Add Integers Is With a Number Line When the number is positive, count to the right. When the number is negative, count to the left. +-

Adding on a Number Line =-2

Adding Integers Is With a Number Line =+2

Adding Integers Is With a Number Line =-4

Integer Subtraction Rule Subtracting a negative number is the same as adding its opposite. Change the signs and add. 2 – (-7) is the same as 2 + (+7) = 9!

Integer Subtraction Rule Subtracting a negative number is the same as adding its opposite. Change the signs and add. 2 – (-7) is the same as 2 + (+7) = 9!

Here are some more examples. 12 – (-8) 12 + (+8) = – (-11) -3 + (+11) = 8

Check Your Answers 1. 8 – (-12) = = – (-30) = = – 17 – (-3) = = –52 – 5 = (-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) -45 2) ) +36 3) ) -5 4) ) -40 5) ) -30 6) ) +55 7) ) -40 8) 0 18) -28 9) ) ) ) +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) -2 2) -2 12) +2 3) +5 13) -5 4) -8 14) -2 5) -2 15) -5 6) -3 16) +2 7) +2 17) -4 8) 0 18) -2 9) +3 19) -2 10) -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 = 3x 3 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) 5/16 d) -4.5(b- 3)

Check your answers : a) 4y +6 b) 1.5q + 8 c) 2 1/2 + 15/16 r d) -4.5b

by D. Fisher

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

3 + 7 = Commutative Property of Addition

8 + 0 = 8 Identity Property of Addition.

6 4 = 4 6 Commutative Property of Multiplication

2(5) = 5(2) Commutative Property of Multiplication

3(2 + 5) = Distributive Property

6(78) = (67)8 Associative Property of Multiplication

6(3 – 2n) = 18 – 12n Distributive Property

2x + 3 = 3 + 2x Commutative Property of Addition

ab = ba Commutative Property of Multiplication

a + 0 = a Identity Property of Addition

a(bc) = (ab)c Associative Property of Multiplication

a1 = a Identity Property of Multiplication

a +b = b + a Commutative Property of Addition

a(b + c) = ab + ac Distributive Property

a + (b + c) = (a +b) + c Associative Property of Addition

The Coordinate Plane By: Christine Berg Edited By:VTHamilton

Definition The plane formed when 2 perpendicular number lines intersect at their zero points 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

I II III IV

Ordered Pair A pair of numbers (x, y) assigned to a point on the coordinate plane

Ordered Pair (x, y) X-coordinate Y-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