8th Grade Semester Objectives

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Rules of Signed Numbers: Addition and Subtraction

Negative + Negative = Bigger Negative

Positive + Positive = Bigger PositiveUnlike Signs: Take the absolute difference (Ignore signs) and then use the sign of the larger number

Multiplication and Division: LIKE signs = POSITIVE UNLIKE signs = Negative

Negative X Negative = Positive Negative X Positive = NegativePositive X Positive = Positive Positive x Negative = Negative

Review of Fractions and DecimalsFractions must always be reduced into “Simplest Terms”Do not leave Improper Fractions: where the Numerator is greater than the Denominator.Terminating Decimal Numbers may be expressed as a fraction.

.8 = 810

=45when reduced .375 =

3751000

=1540

whenreducedby 25 thenthen 38when reducedby 5.

Repeating decimals may also be expressed as a Fraction: .3 = 39=1

3 .46 = 4699 .327 =

327999

Some numbers like “Pi” or the √2 are Non-repeating and Non-terminating and are called Irrational Numbers.

8th Grade Semester Objectives

FACTORS and PRIME Numbers:All numbers are made up of other numbers! Numbers that are multiplied together to form another number are called FACTORS:Factors are usually considered to be WHOLE NUMBERS (Integers) and most often the positive values:The Factors of 12: (1, 12); (2, 6); (3, 4)

But since Like Signs: Negative x Negative = Positive: (-1, -12) (-2, -6) (-3, -4) could also be factors. Numbers that only have the factor of ONE (1) and the number are called PRIME Numbers.Numbers that are made up of other factors are known as Composite Numbers.

PRIME Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 59, 83, 93, 972 is the only “EVEN” Prime Number.1 is not considered either a Prime or a Composite number.

An “Odd” number ending in ‘5’ cannot be Prime since “5” would be a factor of it.Not all “ODD” numbers are Prime! Some are divisible by “3” (9, 21, 27)

Addition and Subtraction of Fractions: Fractions must have a Common Denominator.

Find the Least Common Multiple and use the Identify Fractions of: nn .

Given a problem of: ab+ cd , you can always get a Common Denominator but it might not be the LEAST COMMON DENOMINATOR.

( dd x ab )+( cd x b

b )= (ad+bc )

(bd) Add Numerators but just keep one denominator.

Rules for LCM: 1) See if the smaller Denominator goes into the Larger. If it does. The larger number is the LCM. Since 6 goes into

12, then 12 is the smallest or LCD. 56+

712=

22 x 5

6 +712=

(5∗2)+712 =

(10 )+(7)12 =

1712

=1 512

2) If both the Denominators are PRIME, Multiply the two numbers.3) If one of the Denominators is 2 and the other is Odd, Multiply two denominators.4) Look for the Least Common Multiple. (since 3 is prime, and 4 is not a multiple of 3, Multiply numbers together)

14+ 2

3=¿

(ad+bc )(bd)

=(1∗3 )+(2∗4)

(3∗4) =

3+812

=1112

16+ 3

8=¿ You could us (6 x 8) as the common denominator but the numbers get large and need to be reduced

at the end.

8th Grade Semester Objectives

24 is a Multiple of 6 and 8: so ( 44 x 1

6 )+( 38 x 3

3 )=(ad+bc)LCM =

(1∗4 )+(3∗3)24 =

4+924

=1324

8th Grade Semester Objectives

Multiplication of Fractions:There is no need to get a common denominator.

You can get the product of the Numerators and Denominators and then REDUCE ab x cd= acbd

Or reduce first. Depending on the numbers you can decide which is easier.

406x 2

15=80

90=8

9 406

x 215

by5 : 86x 2

3∧thenby 2: 8

3x 1

3=8

9Reducedat the Beginning

1118

x 266

Reduce Fi rst by 11:= 118

x 26∧thenby 2:=1

9x 1

6= 1

54

If you don’t reduce you will get 221,188

whichthen needs¿bereduced !

8th Grade Semester Objectives

SQUARES and SQUARE ROOTSSquare of a Number means: an number: N times itself or N x N or expressed as N2

A Cube of a Number means: a Number times itself 3 times or: N x N x N or expressed as N3

PERFECT SQUARES are the inverse of Squares: the Radical symbol √❑ is the Square Root:

A number raised to the ½ or (.5) power is also a Square Root: 4912 = 7.

Prefect Squares are: 4, 9, 16, 25, 36, 49, 64, 81 and 100 (Integers >=2 have a perfect square.)While it is true the there is also a negative (-) root, it usually is not used: √16 = ±4 since - 4 x – 4 = 16 Like signs = +

Approximation of Square Roots: Mixed Number ModelIf you had to approximate the square root of 34 it would be between√25>√34<√36. It is between 5 and 6.

So an easy approximation as a Mixed number is: 5 911

.

The Numberatior isthe difference ¿ the target (34) ∧thelow (25 )∨34−25=9The Denominator is the Sum of the Low and High Perfect Squares: 5 + 6 = 11

Perfect Square Factorization Model: Mathematically, this would be a more correct and precise value!This method is very similar to the method used to find Least Common Denominators that uses the PRIME numbers to find the factors of a number. In this Model, you use Perfect Squares as your factor rather than a Prime. If you do not see perfect squares, you can just factor it.

Simplify: √144 √72 √4500

√6 x√24 √4 x √18 ¿√45 x100

√6 x√4 √9 x√2 = √9 x5 x100 so :√6x √6 x√4 so: √4 x √9 x√2 = 3 √5 x10

6 x 2 2 x 3 √2 = 30 √5

12 6 √2

8th Grade Semester Objectives

DIVISIBILITY RULES:When doing long division or reduction of fractions or finding (FACTORS) of numbers: (You need to know your Times Tables)There are some rules to help you finding the factors of a number:A number is divisible by:

2 if it is even (0, 2, 3, 6, 8)

3 if the sum of the digits is divisible by 3: 147: 1 + 4 + 7 = 12 Yes! 438: 4 + 3 + 8 = 15 Yes

4 if the last two digits in the number are divisible by 4.

5 if the number ends in a Zero (0) or Five (5)

6 If the number is divisible by 2 and by 3 then it would also be divisible by 6.

7 (There is process for ‘7’ which is interesting but not required).

8 If the last 3 digits are divisible by 8 then the number is divisible by 8.

9 If the sum of the digits is divisible by 9 the number is divisible by 9. 216: 2 + 1 + 6 = 9 Yes! 81: 8 + 1 = 9

Language of Math (Coded Words making Expressions and Solving “Word Problems”ADDITION:

a) The SUM of a number and 2b) 3 more than a numberc) 7 plus a numberd) 16 added to a numbere) A number increased by 9f) The sum of TWO numbers

SUBTRACTION:a) 4 less than a numberb) 10 Minus a numberc) A number decreased by 6d) A number subtracted from 12e) The difference between two numbers

MULTIPLICATION:a) 14 times a numberb) A number multiplied by 8

c) (OF) 34 of a number

d) Triple (three times) a numbere) The product of two numbers

DIVISION:a) The QUOTIENT of 6 an and a numberb) The QUOTIENT of a number and 6c) A number Divided by 15d) Half of a number

8th Grade Semester Objectives

DISTRIBUTIVE PROPERTY: You might recall a problem like: 7(2 + 4) would become: 7(6) which equals 42. BUT, 7(2 + 4) can also be distributed as: 7(2) + 7(4) or 14 + 28 = 42 (same result but achieved in a different way).

4x + 28 can also be “simplified” by factoring out something in common: 4(x + 7).As you get into harder math, some problems cannot be solved as easily and another approach is required:3(x + 4) distributed is 3(x) + 3(4) or 3x + 12. -3( 4 – z) = (-3 * 4) + (-3 * -z) = -12 + 3z

In general terms: a(b + c) = ab + ac BE CAREFUL OF THE RULES OF SIGNED NUMBERS!

MULTIPLY or DIVIDELike Signs: POSITIVE Unlike Signs: NEGATIVE+Positive x +Positive = +Positive +Positive x –Negative = -Negative-Negative x –Negative = +Positive -Negative x +Positive = -NegativeAlso recall there is really no such this as SUBTRACTION, But Addition of the

OPPOSITE!ADDING or SUBTRACTINGLIKE SIGNS: +Positive + +Positive = +Bigger Positive

-Negative + -Negative = -Bigger NegativeUNLIKE SIGNS: Take the “Absolute Difference” between the two values and use the

sign of the Larger value:4 – 7 becomes: 4 + -7 unlike sings Difference is: 7 – 4 =-3

7 is a (-) so -3 or -3 a) 4(3 + 2) 4(5) = 20 or 4(3 + 2) = 4(3) + 4(2) - 12 + 8 = 20b) 4(3 – 2) 4(3 + -2) 4(1) = 4

Yes! (3-2) = 1 but when problems get more complex you will see why this is important.4(3) + 4(-2) 12 + -8 = 4

c) 6 – 4(3 – 2) 6 – 4(1) 6 – 4 = 2If you distribute and don’t change the signs you get: 6 – 4(3) - 4(-2) 6 – 12 - - 8 = 2If you change the Subtraction to ADDITION OF THE Opposite, it becomes a little clearer:

6 – 4(3 – 2) 6 + -4(3 + -2) 6 + -4(3) + -4(-2) 6 + -12 + +8 -6 + 8 = 2d) 6x – 4(-x -2) 6x + -4(-x + -2) 6x + -4(-x) + -4(-2) 6x + 4x + 8 10x + 8e) 3(x + 5) + 2(2x – 2) 3(x) + 3(5) + 2(2x) +2(-2) 3x + 15 + 4x – 4 7x +11

Changing signs really helps with complex problems and when VARIABLES are involved!

8th Grade Semester Objectives

EXPRESSIONS and EQUATIONS: (1 and 2 Steps)Vocabulary: Expression: Equation Variable

Constant Coefficient Combine Like Terms Inverse OperationExamples:

Expressions: 3x + 2 + 4y + 5x 2z – 5 + 15z + 3x 20 – 4(x -2 + y) 3( 2 + x) * 4( 7 + x)

1 Step Equations: 3 + N = 18 n – 5 = 20 Q = 20 = = -30n9 = 5

3 + n + 7 = 14 6n = 38 4n + 2n = 242 Step Equations: 3n + 2 = 10 5n + 1 – 3n + 4 = 252 Step Move 7n – 6 = 3n + 22 4n + 3 – 2n = 20 - 4 + 1Multi-step 2(N + 3) = 26 4(3 – x) = 20 + 2x 6(3 – y) = 20 + 2(4 – 2y)

Don’t Call Me After Midnight:Distribute Combine Move by (Adding or Subtracting) Multiply or Divide

GENERIC EQUATIONS Formulas

PURCHASE PROBLEMS: Total = Item1 + item2 +…..Itemn

Sometimes the values needed are a PRODUCT of a Quantity X Price3 Hamburgers x $2 + 4 Drinks x($1) *3Fries x (Price) = 18

SPENDING PROBLEMS: BEGIN – (PQ) = ENDBeginning amount P = Price or Cost of each itemQ = Quantity of itemsEnd = Amount of money leftSPENT = Begin – End

Price Per Item (per means /)SPENT

QUANTITY SPENT = (Begin – End) so (Begin−End)QUANTITY

8th Grade Semester Objectives

Scientific Notation:Scientific Notation is way to express very large or small numbers using a Factor of 10 and Exponents.

The FORM is ONE digit to the left of the Decimal and up to 3 digits to the right of the decimal. N.ddd x 10X Large Numbers have a Positive Exponent and Small Numbers have a Negative Exponent.

(Remember a negative exponent means Reciprocal so :( 10−2= 1102 ).

The Standard Form of a Number is the “Normal” way of looking at the number: 5,239,223 5.239 x 106 1,328,005,203,223 5.328 x 1012

.6319831203 6.31 x 10-1 .0000000000215974 2.159 x 10-11

Working with Exponents: Vocabulary for: Squares, Square Roots and Exponents1) Base2) Exponent3) Power4) Radical5) Radicand6) Root7) N2

8) N3

9) Index10) Cubed11) Squared12) Prime Factorization13) Square Factorization14) Order of Operations15) Laws of Exponents16) Negation17) Inverse Operation18) Integer19) Whole Number20) Rational Number21) Terminating Decimal22) Standard form

23) Exponential form24) Repeating Decimal25) N0 = 126) N1 = N27) N

12=N .5=√N28) N13= 3√N

29) N−x= 1N x

30)1

N− x=N x

31) ( ab )−n

=( ba )n

8th Grade Semester Objectives

RULES of EXPONENTSN0 = 1

N1 = N

N2 = N x N (or N Squared or N to the 2nd Power)

N3= N x N x N (or N Cubed or N to the 3rd Power)

√N = Square Root of N. What number (q) times itself gives N: q x q = N or √N∨ 2√N3√N = Cubed Root of N. What number (q) times itself three times gives N: q x q x q = N

N12=N .5=√N N

13= 3√N N

1T =T√N

N-1 = 1N 10-1 =

110 = .1

N-2 = 1N 2 10-2 =

1100 = .01

( ab )n

= an

bn Example: ( 49 )

2

=1681

=.19753….∨( 49 )

2

=(.4)2=¿ .19753….

( ab )−n

= bn

an (Make a Reciprocal of the fraction)

( 49 )

−2

= 1

( 49 )

2=11÷ 16

81so : 1

1x 81

16=2.25∨(4

9 )−2

=¿ ( 9

4 )2

=8116

=2.25

Multiplying with Exponents:BASES MUST BE EQUAL: basea xbaseb 33 x34=(3 x3 x 3 ) (3 x 3x 3 x3 )

RULE: base(a+b) 3(3 +4) = 37

It still works with “-“negative Exponents: 56 x5−4=5 (6+−4 )=52

56 x 154 =

(5 x5 x5 x5 x 5 )1

X 1(5 x5 x5 x5)

=5 x51

=52

Dividing with Exponents:

BASES MUST BE EQUAL: basea÷baseb 26÷24=26

24 =2x 2x 2x 2x 2

2x 2x 2x 2=2 x2

1=4

RULE: base(a-b) 2(6-4) = 22 = 4It still works with “-“negative Exponents: 56÷5−4=5(6−−4 )=510

56÷ 154 =

56

1× 54

1=(5 x5 x 5x 5x 5 ) X (5 x5 x5 x 5)

Bases to a Power: Multiply Exponents:(na )b = nab

8th Grade Semester Objectives

( 42 )3 = ( 42 ) (42 ) (42 )∨( 46 ) Compound Factors to a Power:(mna )b = mbnab

( 4 x2)3 = ( 4 x2) ( 4 x2 ) ( 4 x2 )∨43 x6

GraphingOn a “plane”, graphing is done with Ordered Pairs on a Cartesian coordinate system: X axis HorizontalY axis VerticalOrigin Where x and y axis cross: (0, 0)Point Coordinate pair

Ordered pair(x, y)

X abscissae domainY ordinate rangeQuadrants: I (+, +)

II (-, +) III (-, -) IV (+, -)

Given the Equation of a Line in Standard form of Ax + By = C, you can prove if a point satisfies the equation by substituting into the equation.

Is (0, 4) on the line: 3x + y = 4: 3(0) + 4 = 4 : 4 = 4 YES

For Linear Equations in Standard Form, you can solve for the X and Y values by substituting in a ZERO for x to solve for y and 0 for y to solve for x.

2x – 7y = 14: When x = 0, y=-2: When y = 0, x = 7

The SLOPE of a line is defined as slope=riserun

=Change∈ yChange∈x

= ∆ y∆ x

Slope (m) = ∆ y∆x

=y2− y1

x2−x1 It does not matter what points you substitute in:

(6, 8) and (-2, -4): y2− y1

x2−x1=−4−8

−2−6=−12

−8 OR y2− y1

x2−x1=8−−4

6−−2=+12

+8

8th Grade Semester Objectives

The Slope Intercept Formula is y = mx + b. m is the slope and b is the Y intercept.

If you have two ordered pairs, you can calculate the slope. If you only have 1 ordered pair and also know the slope, you can use the Point-Slope Equation of a

line: y – y1 = m(x – x1) where x1 and y1 are the known point.

When looking at ordered pairs, you can try to judge a Line of Best Fit. This is open to some interpretation and judgement as to what points make the best choice for drawing a line to represent all the data. Sometimes you need to throw data out.

Parallel lines are lines that have equal slopes.

Perpendicular Lines have Inverse Reciprocal slopes.

RadicalsMany times in math you will need to “SOLVE” problems that involve “RADICALS” (Square Root Sign). The trouble with most Radicals is they ARE NOT Perfect squares and thus the answers can only be approximated using a calculator. BUT, you can simplify them using various rules!Rules to Remember:

a) Never leave aRadical∈theDen ominator !

b) √N√N

=1 (IDENTITY RADICAL) is used to “remove the radical”

c) Look for FACTORS and Perfect Factors. Place Perfect Squares first!d) Products can be Distributed: (sometimes it’s easier to reduce and

sometimes it is easier to multiply out)e) √N x √N=Nf) √n2 = ng) √n4 = n2

√ 185

√ 185

x√ 55

= √18 x √5√5 x √5 = √9√2√5

5=3√10

5

8th Grade Semester Objectives

DISTANCE AND RATE problems fall into some general formats:

The basic formula is DISTANCE = RATE x TIME. d=rt ; r=dt; t=d

r

RATE is assumed to be an AVERAGE rate (Speed) since there may be starting and stopping as well as changes during the course of the trip. The second thing to be careful about is the Units of Time. Sometimes you will get an answer like 3.65 Hours which in some cases needs to be converted into real time of Hours and Minutes.

Up-Stream and Down-Stream or Head-Wind and Tail-Wind problems.

In these problems, the rate of the FLOW is either added or subtracted from the average rate of movement.

In Up-stream problems you are going AGAINST the flow so you true or net rate is: RATE – flow.If you had a “RATE” of zero, the river would push you backward or in the opposite (negative) direction. In order to go forward, you have to move faster than the current of the wind or river.

In Down-Stream problems, you are going WITH the flow so your true or net rate is: RATE + flow.If you had a RATE of zero, you would still be moved in a positive direction.Three other “classic” word problem situations are:

1) Leave from the same place. Go in the same direction. When will they be a certain distance apart?

f = Fast s=Slowdf = rf x tf--------------------------------------------ds =rs x ts---------------------/ distance apart {DA}/

df – ds = DA or df – DA = ds or ds + DA = df (or DA = t(rf –rs)

2) Leave from the same place and go in the opposite directions, When will they be a certain distance apart? df = rf x tf-------------------------------------------- ds =rs x ts

<---------------------- {………………………………………………………………………………………..}

df + ds = N_distance (or N_distance = t(rf +rs)

3) Leave from different places heading in the opposite directions; When will they pass each other) or crash? df = rf x tf-------------------------------------------- | d s =rs x ts

<----------------------df + ds = N_distance (or N_distance = t(rf +rs)