1. Semester I: Study Guide
8th Grade Mathematics

3. Rational Numbers Whole Numbers: 0,1,2,3,4,5,6…
Natural Numbers: 1, 2, 3, 4, 5, 6…
Integers: positive and negative numbers
Perfect Square Roots: a radical that gives a whole number
Fractions: ALL fractions are rational
Decimals: terminating (0.45)
repeating (0.45)

5. Irrational Numbers Anytime you see
Imperfect square root  answer gives a decimal
Non-repeating, non-terminating decimals  …

6. Not a Real Number

8. Terminating Decimal  Fraction

9. Terminating Decimal  Fraction
Steps to find:
Place the numerator OVER the base ten (# of place value in the numerator = the number of zeros in the denominator)
Find the factors of the numerator
Find the factors of the denominator
Find the GCF of the two
Divide to simplify the fraction

11. Repeating Decimal  Fraction
*Base 10*
One place value Two place values
÷ ÷ 99

15. When to Divide or Simplify Fractions?
Big Number in Numerator
Small Number in Denominator
𝟑𝟔 𝟒 = 9
Small Number in Numerator
Big Number in Denominator
𝟒 𝟏𝟐 = 𝟏 𝟑 GCF = 4
GCF?

16. Location of Irrational Numbers

17. Between which two numbers does the number fall
Between which two numbers does the number fall? 𝟑𝟖 = fall between 𝟑𝟔 & 𝟒𝟗 𝟑𝟖 falls between 6 &7

18. Irrational Numbers on a Number line

19. Irrational Numbers on a Number Line
Draw the number line and number it. Remember 10 clicks = 1 whole number
ABOVE the numbers, write their square roots
Determine the closest click to where your square goes
Label it

21. Squaring a Number
Draw a box and divide by the number of digits in the number
Draw diagonals to cut each square into TWO right angles
Write the first digits on the top, then the second same digits on the side
Multiply
Add the diagonals

23. Square Root  what number times itself
Use the ± in your answer

24. Cube a Number

25. How many times the base it multiplied to itself?
= 6 23

26. Finding the Cube Root

27. Never use the ± symbol for the cube root UNLESS the – is OUTSIDE the radical when you don’t know the answer, use the calculator and MULTIPLY  number to itself 3 times find a starting point and keep going until you find the answer

28. Laws of Exponents: Zero Exponent

29. Any number/variable raised to the zero power = 1
When raised to zero power, just scratch that term out!

30. Laws of Exponents…

31. Laws of Exponents: Multiplication
(5m3n4)(6m2n3)
(5*6) (m3+2)(n4+3)
30m5n7
Keep the base; ADD the exponents

32. Laws of Exponents…

33. Keep the base, subtract the exponents!
Draw your columns down to divide the terms
Divide/Simplify the coefficients
Keep the base
Subtract the exponents
Pay attention to your answer….make sure there is NO negative exponent

34. Laws of Exponents…

35. Keep the base, multiply the exponents!
When you only have ONE set of ( )

36. Keep the base, multiply the exponents!

37. Laws of Exponents… Negative Exponent

38. 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 (𝒕𝒐𝒑) 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 𝒏𝒆𝒈𝒂𝒕𝒊𝒗𝒆 (𝒃𝒐𝒕𝒕𝒐𝒎,𝒘𝒊𝒕𝒉𝒐𝒖𝒕 𝒕𝒉𝒆 𝒏𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝒔𝒊𝒈𝒏)

39. Laws of Exponents… Negative Fraction

40. Fraction with Negative Exponent  Just Flip It!

41. Solving Equations… Always “I.I.S” an equation I (isolate) I (inverse)
S (solve)

42. One-Step Equations: Addition
Steps to find the solution
Draw a line down the = sign
Isolate the variable: Put a box around the variable
Inverse the number OUTSIDE the box
Solve it (Laws of Integers  “give me the finger”)

43. One-Step Equation: Subtraction
Steps to find the solution
Draw a line down the = sign
Isolate the variable: Put a box around the variable
Inverse the number OUTSIDE the box
Solve it (Laws of Integers  “give me the finger”)

44. One-Step Equation: Multiplication
Steps to find the solution:
Draw a line down the = sign
Isolate the variable: Put a box around the variable
Inverse the number OUTSIDE the box
“anytime there is a number, next to a variable, next to an equal sign, you always DIVIDE!”

45. One-Step Equation: Division
Steps to find the solution:
Draw a line down the = sign
Isolate the variable: Put a box around the variable
Inverse the number OUTSIDE the box
“Circle it. Move it over. Multiply!”

46. Solving Two-Step Equations
Steps to find the solution:
Box in the coefficient and variable
Perform the inverse operation to the number outside the box
Solve it
Perform the 2nd inverse operation (multiply? Divide?)
Solve it!

47. Solving Multi-Step Equations…
Draw the line down the = sign
Left Column
Right Column
Do you need to apply the laws of integers? –(- ) - (+ )
neg* neg -> positive neg * positive -> negative
Do you need to do the distributive property?
Do you need to combine like terms?
Did you box in the variable?
Did you perform the inverse operation? Do you need to do it twice?
Is the variable by itself?
Is the number by itself? (yes -> then solve the equation)
Do you need to get your Beyoncé’ on???
Inverse ALL variables to the LEFT!

48. Solving Equations: Word Problems
Problem Solving 101: BUCK the Problem
Did you box in the question? What will your variable be = what is the math unknown in the question?
Did you go sentence by sentence and circle the “math stuff”?
Did you scratch out EVERYTHING that is NOT circled?
Did you T.T.P? (Translate. Put it on Top. Put it Together!)
Did you SOLVE the equation?

49. Scientific Notation vs. Standard Notation

50. Scientific Notation: Positive Exponent

51. Scientific Notation: Positive Exponent
It is positive because you will see NO Decimals
Step 1: circle the FIRST number (between )
Step 2: Place a decimal BEHIND the circle
Step 3: Count the number of jumps to get to the END of the number

52. Scientific Notation: Negative Exponent

53. Scientific Notation: Negative Exponent
It is negative because the decimal comes BEFORE the zeroes.
Step 1: circle the FIRST number (between 1- 10)
Step 2: Place a decimal BEHIND the circle
Step 3: Count the number of jumps to get to the beginning of the Zeroes

54. Standard Notation: Positive Exponent
Move to the right the number of times given by the exponent!

55. Standard Notation: Negative Exponent!
Move to the left the number of times given by the exponent!

56. Scientific Notation: Multiplication

57. Scientific Notation: Division

58. Working with the Slope of a Line

59. Four Lines of a Slope

60. No Slope vs. Undefined Slope
m= 𝟎 𝒏𝒖𝒎𝒃𝒆𝒓
m = 𝒏𝒖𝒎𝒃𝒆𝒓 𝟎

61. Finding the Slope of Two Points
Label x1, y1, x2, y2
Write the formula and place ( ) around each variable
Take your eraser and erase y1 and substitute the value. Continue with the rest of the variables
Solve it

62. Finding the Slope of a Graph

63. Finding the Slope using a Graph
m = "𝑠𝑖𝑑𝑒" "𝑏𝑎𝑠𝑒"
Write the word “side” and “base” on the graph
Find 2 points and draw a RIGHT triangle
Count the moves for the “side” and for the “base”
Substitute the value
Divide or simplify

64. Finding the Slope Using a Chart/Table

65. 𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑜𝑓 𝑦 𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑜𝑓 𝑥
m =

67. Slope intercept  y = mx + b
Write y = mx + b ABOVE the equation
Circle the m and the number and sign below it
Circle the +b and the number and sign below it
Write m = and b = and substitute the values in the circles
m = slope
b = y-intercept (or where the line crosses the y-axis

68. Standard Form  Slope-Intercept

69. Convert from standard to slope intercept: Checklist
Did you draw a line down the = sign?
Did you circle x and its coefficient?
Did you draw an arrow to inverse the circle behind the = sign?
Did you divide all 3 terms by the coefficient with y?

70. Graphing Using the Slope Intercept

71. Steps for Graphing using the slope-intercept
Place the y-intercept (b) on the graph on the y-axis
From that point on the y-axis, do the rise, then the run (the slope)
Draw a line through those two points.

72. Using the Point-Slope Form  Slope Intercept

73. Using the Point Slope Formula: When Slope is Given
On the side of the problem, write m = x = y =
Write the formula
Substitute the values in the formula
Perform the distributive property for “x”
Box in “y” and inverse the number outside the box to the number to the right

75. Proportional Relationship EXISTS if…
The line goes through the point of origin (0,0)
Constant proportionality  the same rate of the proportion
Proportional Relationship

76. Proportional Relationship: Graph

77. a

78. Proportional Relationship: Chart

79. 𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑜𝑓 𝑦 𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑜𝑓 𝑥
a =