Semester I: Study Guide 8th Grade Mathematics

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)

Irrational Numbers Anytime you see Imperfect square root  answer gives a decimal Non-repeating, non-terminating decimals  0.6758321…

Not a Real Number

Terminating Decimal  Fraction

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

Repeating Decimal  Fraction *Base 10* 0.9 0.90 One place value Two place values ÷ 9 ÷ 99

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?

Location of Irrational Numbers

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

Irrational Numbers on a Number line

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

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

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

Cube a Number

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

Finding the Cube Root

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

Laws of Exponents: Zero Exponent

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

Laws of Exponents…

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

Laws of Exponents…

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

Laws of Exponents…

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

Keep the base, multiply the exponents!

Laws of Exponents… Negative Exponent

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

Laws of Exponents… Negative Fraction

Fraction with Negative Exponent  Just Flip It!

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

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”)

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”)

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!”

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!”

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!

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!

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?

Scientific Notation vs. Standard Notation

Scientific Notation: Positive Exponent

Scientific Notation: Positive Exponent It is positive because you will see NO Decimals 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 END of the number

Scientific Notation: Negative Exponent

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

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

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

Scientific Notation: Multiplication

Scientific Notation: Division

Working with the Slope of a Line

Four Lines of a Slope

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

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

Finding the Slope of a Graph

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

Finding the Slope Using a Chart/Table

𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑜𝑓 𝑦 𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑜𝑓 𝑥 m =

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

Standard Form  Slope-Intercept

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?

Graphing Using the Slope Intercept

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.

Using the Point-Slope Form  Slope Intercept

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

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

Proportional Relationship: Graph

a

Proportional Relationship: Chart

𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑜𝑓 𝑦 𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑜𝑓 𝑥 a =