1. Confidential1 Our Lesson: Review of Integers

2. Confidential2 opposites Integers are all of the positive whole numbers, their opposites and zero Example …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

3. Confidential3 We use VARIABLES when we don't know the number. For example2 yards = x feet We can also write VARIABLE EXPRESSIONS such as s + 9 8 n 16 x h VARIABLES AND ABSOLUTE VALUES

4. Confidential4 The VARIABLES in an expression can be replaced with any number. 3 + x If I substitute a 5 for the x...... I have 3 + 5 or 8 This is how we Evaluate (or find the value of) the Expression Therefore we Evaluate Expressions by finding the value of the Expression when we replace the Variable with a Number.

5. Confidential5 The ABSOLUTE VALUE of a number is its distance from zero on the number line. The Absolute Value of 4 is written as follows: 4

6. Confidential6 Distance is Positive. Therefore, since Absolute Value is the distance from zero, Absolute Value is always Positive. 9 = 9 -11 = 11

7. Confidential7 1 = 1 -47 = 47 11 = 11 -5 = 5 Let’s Find

8. Confidential8 We will explore 3 different METHODS METHOD 1 - ZERO PAIRS + The red chips are NEGATIVE. The grey chips are POSITIVE - - - - + + ADDING POSITIVE AND NEGATIVE INTEGERS

9. Confidential9 - + 1 red chip1 grey chip + = 0 We call 1 red chip and 1 grey chip a ZERO PAIR Zero Pair

10. Confidential10 METHOD 2 - Number Line We can use a number line to find the sum of 2 integers. 10234567 89 10 -2-3-4-5-6-7-8-9-10 WE ALWAYS START AT ZERO!

11. Confidential11 Now, let's add (-2) + (-8) 1. Start at zero and move to the left (in a negative direction) 2 spaces. 2. Next, move further to the left 8 spaces. 3. Your answer is -10. (-2) + (-8) = -10 102 34567 89 10 -2 -3-4-5-6-7-8-9 -10

12. Confidential12 Method 3 - Rules for adding integers Adding the The sum of 2 positive numbers is POSITIVE. Same sign The sum of 2 negative numbers is NEGATIVE. Example: 4 + 7 = 11 (-3) + (-6) = (-9) 11 + 8 = (-9) + (-1) = 19 -10

13. Confidential13 Adding Different Signs - 1. Find the absolute value of the 2 numbers. 2. Subtract the smaller number from the larger number. 3. Take the sign of the larger number for your answer. Example:(-2) + 9 = 1.)2 9 Absolute Values 2.)9 - 2 Step Subtract Step3.) +7 Sign of the larger number

14. Confidential14 Let us try a few problems Remember your Addition Rules! 1.)(-7) + (-11) = (-18) 2.)22 + 7 = 29 3.)3 + (-10) = (-7) 4.)(-8) + 20 = 12 5.)(-15) + (-5) = (-20)

15. Confidential15 Two methods for subtracting integers. METHOD 1 - Zero Pairs The red chips are NEGATIVE. The grey chips are POSITIVE. + + + - -- - - Subtracting Positive and Negative Integers

16. Confidential16 For example, let's subtract 5 - (-3) = We begin with 5 POSITIVE (grey) chips. + +++ +

17. Confidential17 5 - (-3) = We begin with 5 POSITIVE (grey) chips. + + + + Now, we need to take away (or Subtract) 3 NEGATIVE (red) chips but we don't have any!! So we must ADD 3 Zero Pairs............... - - - + + + + +

18. Confidential18 + + + + + -- - ++ + Now, we can take away (or Subtract) the 3 NEGATIVE (red) chips. And we are left with 8 POSITIVE (grey) chips. Therefore, 5 - (-3) = 8 5 - (-3) =

19. Confidential19 METHOD 2 - Rule for Subtracting Integers When we subtract an integer, we add its opposite. For example, (-10) - 7 = (-10) + (-7) = (-17) we add its opposite

20. Confidential20 Of course, if we are going to ADD its opposite, we must remember the rules for ADDING INTEGERS. Adding Different Signs - 1. Find the absolute value of the 2 numbers. 2. Subtract the smaller number from the larger number. 3. Take the sign of the larger number for your answer. Adding the Same Sign - The sum of 2 positive numbers is POSITIVE. The sum of 2 negative numbers is NEGATIVE.

21. Confidential21 You try a few. (-4) - (-4) =(-2) - 4 =12 - (-8) = (-4) + 4 = 0(-2) + (-4) =(-6)12 + 8 = 20

22. Confidential22

23. Confidential23 Game time Click here to play a gameplay

24. Confidential24 We are very familiar with Addition and it is easy for us to add 4 + 4 + 4 = 12 We can also think of this as three groups of 4 OR 3 x 4 = 12 Multiplying and Dividing Integers

25. Confidential25 So, let's look at (-4) + (-4) + (-4) = -12 We know from our Addition Rules that when we add NEGATIVE numbers together, our sum is NEGATIVE. We can also look at this as three groups of (-4) OR 3 x (-4) = -12

26. Confidential26 On a number line, this might look like...... 3 x (-4) = (-12) 40 -4 -8 -12 1 23

27. Confidential27 Rules for Multiplying Integers The product of two integers with the SAME sign is always POSITIVE. The product of two integers with DIFFERENT signs is always NEGATIVE. 6 x 7 = +42 (-5) x (-4) = +20 8 x (-4) = -32 (-9) x 3 = -27

28. Confidential28 Let's try a few! Remember: SAME SIGNS Positive DIFFERENT SIGNS Negative 1.)-6 x 44 = (-264) 2.)100 x -5 = (-500) 3.)(-8) x (-9) = 72

29. Confidential29 Let's look at that relationship with POSITIVE and NEGATIVE numbers! DIVISIONMULTIPLICATION (-3) X 5 = -15 Remember that a Negative times a Positive number equals a Negative number. (-15) ÷5 = (-3) And, in division, a Negative divided by a Positive number ALSO equals a Negative number.

30. Confidential30 RULES FOR DIVIDING INTEGERS The quotient of two integers with the same sign is always POSITIVE. The quotient of two integers with different signs is always NEGATIVE. 244 = 6 (-12)(-3) = 4 32(-8) = -4 (-27)3 = -9 Remember that the QUOTIENT is the answer to a division problem

31. Confidential31 This is very similar to our MULTIPLICATION rule. Signs are the same result is POSITIVE Signs are different result is NEGATIVE

32. Confidential32 1.) 27 (-3) = -9 2.) 24 6 = 4 3.) -35 7 = -5 Let's try a few!

33. Confidential33 Order of Operations It tells us which operation to perform first so that everyone gets the same final answer!

34. Confidential34 Order of Operations 1. Parentheses 2. Exponents 3. Multiplication Division 4. Addition Subtraction Simplifying the expressions inside grouping symbols examples: (3+5) or (4*6) Find the value of all powers examples: 2 3 or 4 2 Perform multiplication or division in the order in which it occurs when reading the expression from left to right. Perform addition or subtraction in the order in which it occurs when reading the expression from left to right. P E M D A S

35. Confidential35 We can remember the Order of Operations as PEMDAS P E M D A S arentheses xponents ultiplication ivision ddition ubtraction "Please Excuse My Dear Aunt Sally"

36. Confidential36 P E M D A S arentheses xponents ultiplication ivision ddition ubtraction Or "Purple Eggplants Make Delicious Afternoon Snacks!"

37. Confidential37 P E M D A S arentheses xponents ultiplication ivision ddition ubtraction whichever comes first "Purple Eggplants Make Delicious Afternoon Snacks!" 10 - 2 + 8 = Read the expression from left to right. 8 + 8 = We perform Addition and Subtraction in the order in which they occur. So do the Subtraction first! Then do the Addition. 16 Numerical Expression

38. Confidential38 Sometimes we have operations "nested" within the parentheses which must be completed first. Example: 60 ÷ (12 + 2 3 ) x 9 = 60 ÷ (12 + 8) x 9 = 60 ÷ 20 x 9 = 3 x 9 = 27 We want to do the Parentheses but first we must take care of the Exponents WITHIN the Parentheses. Then we can solve the Parentheses. Division comes before Multiplication when reading the expression from left to right. Finally we can perform the Multiplication.

39. Confidential39 Practice Problems (26 + 5) x 2 - 15 = 47 5 2 + 18 ÷ 2 = 34 5 + (2 - 3) = 4 28 - (2 2 + 14) + 6 = 16

40. Confidential40 Algebraic Expressions consist of Variables, Numbers, and Operations. Example: 4x + 8 Variables are symbols (usually letters) that represent Numbers. Example: x or y or a or b etc. Evaluating Expressions

41. Confidential41 Algebraic Expressions consist of Numbers, Operations, and Variables. 3 + n is an "ALGEBRAIC EXPRESSION" Numbers Operations Variables

42. Confidential42 b 10 x less than 3 3 - x Word phrases can be translated into VARIABLE EXPRESSIONS. Word Phrase Variable Expression 2 more than n 2 + n k times 8 k x 8 a number b divided by 10

43. Confidential43 The Variables in an Expression can be replaced with Numbers. Example: In the Expression 4b - 2, we can replace b with a Number. We can Evaluate Expressions by finding the value of the Expression when we replace the Variable with a Number. We can write multiplication as 3 x a OR 3a OR 3 a.

44. Confidential44 Let's Evaluate the Algebraic Expression 16 + b if b = 25 We replace b with the number 25 16 + b = 16 + 5 = 21 The Value of the Algebraic Expression when b = 25 is 21

45. Confidential45 Let's Evaluate the Algebraic Expression x - y if x = 22 and y = 7 In other words, in the Expression x - y, let's replace x with the number 64 and replace y with the number 27. x - y = 22 -7 = 15

46. Confidential46 Your Turn! Evaluate each expression if a = -5 and b =3 1.)a + 4 = -1 2.)a - b = -8 3.)a x b = -15 4.) 4a – 3b = -29

47. Confidential47 Complete the Table Algebraic Expressions Name the Variables Name the Numbers Name the Operations a and b x, y,z 12, 5 2, 12, -5 +, x, ^x, - and + 12a + 5b 2 2x – 12y + (-5z)

48. Confidential48 1) A hot iron piece was at 800 degrees. It was left for cooling every two minutes the change of temperature was -20 degrees. What will be the temperature of this iron piece after 15 minutes 650 degrees

49. Confidential49 Ben and Frank had a cycle race. The race was conducted in six sections. In the first section Frank gained 10 seconds. After that he gained 20 seconds then lost one minute, gained 15 seconds lost 27 seconds and finally gained 41 seconds over his friend Ben. Who lost the race? Frank Lost the race by one second

50. Confidential50 The sum of two numbers is 1 and their product is -30. What are the two numbers? -5 and 6

51. Confidential51 You had a great lesson today! Be sure to practice what you learned!