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Math 8 Cheat Sheets Knack Academics Nicholas Cragg Knack Publishing www.knackacademics.com [email protected] 604.505.2867 M8 Txt Page 1
Math 8 Cheat Sheets
Knack Academics
Nicholas CraggKnack Publishing
604.505.2867
M8 Txt Page 1
Check Answer
Expressions
BEDMAS FractionsNumber Line
Equations Graphing TOV
휋2⎯⎯(푥 + 2)±0123456789
+−×÷(3) √9⎯⎯ =≠
Geometry
Probability
2푥 + 3 = 5
푃(퐻) =12⎯⎯
푥 y 1 4 2 5
12⎯⎯+
13⎯⎯
푥 + 푥 = 2푥
0.50 = 50%
Conversions
Exponents
×100푐푚1푚⎯⎯⎯⎯⎯⎯
Hyp
휃
Opp
푦 = 푚푥 + 푏
푥 × 푥 = 푥
√12⎯⎯⎯ = 2√3⎯⎯
푠푖푛휃 =표ℎ⎯⎯
Distribution/FOIL
(3 + 2) × 6
Trig
Decimals/% Pythagoras
푎 + 푏 = 푐
Numbers
√2⎯⎯ , 0. 6,−1,0,1 …
Polynomials
5푥 − 3푥 = 2푥
Inequalities
푥 ≥ 2(푥 + 2)(푥 + 3)
푥 + 5푥 + 6
Factoring
Circles Geometry
Radicals
LCD
M8910 - 0.0 - Math in Pictures
M8 Txt Page 2
Check Answer
PolynomialsTransform
Quadratics
Reciprocals
Sequences
Radicals
Inequalities
AbsolutesRationals
Exponentials
Combinatorics
FunctionsLogarithms
Trig
푥________ , ________ , ________
√푥 + 1⎯⎯⎯⎯⎯ = 5
푥 ≥ 21푥⎯⎯
|푥|
푦 = 2푓(푥 + 2)푥 …
푠푒푐휃 =1
푐표푠휃⎯⎯⎯⎯ 2
log 푥1
푥 + 2⎯⎯⎯⎯⎯
푛푃푟
푔 푓(푥)
퐴푇푆
퐶
Trigonometry
1푥⎯⎯+
12⎯⎯=
14⎯⎯
Radicals
√푥⎯⎯
푅퐴퐷!
Rationals
5! 퐿푖푚 푦 =푑푦푑푥⎯⎯⎯
푑푎푑푡⎯⎯⎯ 푓(푥)푑푥
Limits
푥 → 푎
Derivatives
Optimization(퐶푃)
(퐼푃)
Rates Integrals
Intersections
M11/12/Calc - 0.0 - Math in Pictures
M8 Txt Page 3
Page Layout
Title/Date/InitialsClean Notes/One Sided1 question type per pageNew page if ⎯ downNew stuff at back1a) off to left
Calendars/Todos
Binder ReminderPhone AppPaper Calendar viewPaper Checkboxes
Todo
Organization
Binder Nightly. 1 binder!At home binder #2Syllabus /RemindersDuotangsTwo handed flipping
Inventory
Tool Box*Calendar/BinderCalculator (Charged)Pencil CasePen/RedWhite EraserRuler/ProtractorUSB/StickiesBlack MarkerWhiteout
Daily
Recite/Think what you have learned5 minutes of Math on paperTell someone what you learnedApply learning to daily lifeStudy with classmates
Homework
Make Notes on NotesBinder open to notesDo homeworkDo parts you can/Leave spaceRedo homeworkExtra questionsMake Table of ContentsMake ChecklistsDon’t erase!
Misc
Sum up info/highlightSimilar ExampleI don't knowShow more workWrite down moreDo a Methods SheetDo a Formula SheetDo a Remember SheetCircle/Double AnswerAnswer Question!Arbitrary #sPoint to where you lookDon’t pick up Calc!Use Graph PaperGuess* and Check (Logical)Don’t GuessTest TheoriesPractice CalculatorCheck your EquipmentPhotocopy hand-in's
Formulas
= same spotNo half-fractionsWrite down the formulaLike Terms!
Tests
Ask Teacher!OutlineFormat/LayoutFormulasCalculator?Graph/Spare PaperStudy/w/ClassmatesCheat sheetsCalm/BreatheWhat's NextWrite down Q's after
Extra Marks
RetestsCorrectionsBonusMake-up Assignments
Office Hours
Teacher helpDo parts of questionsExtra Questions
Life
Its on you!Balance!Water/FoodExercise/RestGet MotivatedStay on Top of ThingsFind something to doSet Goals/Get a MentorHave Idols/Make PlansFollow ThroughConfidenceSet an examplePlanting seeds
Trouble Shooting
Copy down Question CorrectlyCheck with your CalculatorCheck each OperationFriends/Teachers!/Family
Word Problems
Let StatementsEquationsSubstitution*Solve
Get Ahead
WebsiteMeet TeacherRead Textbook/Photo Q'sPrint Material
Logic!
General
Go Slower/Be CarefulDraw/Label a diagramWrite/Circle GivensDown the PageSubstitute with BracketsEquations side by sideWork Off to RightEnglish SentencesCheck AnswerDo Reverse/Try AgainMake up questionsCalculator/PencilHighlighterGraph paperMake Review SheetsMake up QuestionsTeacher Office HoursApply to LifeUse Your Hands/Arms
Ideals
Do Both Ways!AmbitionSimplifyPassionPoetryPracticePhilosophyDetermination
Questions:What do you know/how to do? List it!What could you do? Do it!
Table of Contents
Front CoverMath 8910 in PicturesMath 1112Calc in PicturesTable of Contents/BootcampMath 8MethodsRemember 3) Pythagoras 5/7) Geometry 6) Fractions 8) BEDMAS 9) Graphing 10) Equations 11) ProbabilityMath 9MethodsRemember 1) Symmetry 2) Numbers 3) Exponents 5) Polynomials 6) Graphing 9) Inequalities 10) CirclesMath 10MethodsRemember 1) Conversions 2) Area/Volume 3) Trigonometry 4) Exponents/Radicals 5) Polynomials6/7/8/9) Graphing End Page
M8 - 0.0 - Student Bootcamp
M8 Txt Page 4
M8 Txt Page 5
Scale/Similar TrianglesMultiply/divide by the scale factor
푆푐푎푙푒 퐹푎푐푡표푟 =퐿푎푟푔푒푟 #푆푚푎푙푙푒푟 #⎯⎯⎯⎯⎯⎯⎯⎯⎯
Substitution:푎 = 2 3푎 =? 3(2) = 6
Probability/Odds in Favour : Odds Against
푃푟표푏푎푏푖푙푖푡푦 =# 표푓 푓푎푣표푟푎푏푙푒 표푢푡푐표푚푒푠
푡표푡푎푙 표푢푡푐표푚푒푠⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
Methods: ×/Table/Tree (× Branches + Leaves)
Percentages/Fractions/Decimals:
푃푎푟푡푇표푡푎푙⎯⎯⎯⎯⎯= 퐷푒푐푖푚푎푙
퐷푒푐푖푚푎푙 × 100 = % (푑푒푐푖푚푎푙 푡푤표 푡표 푟푖푔ℎ푡)퐷푒푐푖푚푎푙 = % ÷ 100 (푑푒푐푖푚푎푙 푡푤표 푡표 푙푒푓푡)
Equal Fractions/RatiosDo to Top/Do to BottomDo to Left/Do to RightDirection!Multiply Tops/Multiply BottomsFlip and MultiplyLCD: Lowest Common DenominatorMixed/Improper
Geometry:Pythagorean RelationshipPerimeter, Area, Volume Net Surface Area: Lay Shape Flat Formulas
BEDMASIntegers (side by side and × & ÷)Same Plus: Two signs side by side that are the SAME equal a POSITIVE number.
−(−4) = +4 − 4 × −4 = +16
Different Minus: Two signs side by side that are DIFFERENT equal a NEGATIVE number. +(−4) = −4 + 4 × −4 = −16 Adding Subtracting: Number-line
Expressions/Equations: Numbers then LettersGolden Rule: Do to the Left/Do to the RightOpposite Operation to Both Sides (SAMDEB)Adding/Subtracting Both SidesMultiplying/Dividing Both SidesDistribution/Combine Like TermsSquare Root/Square Both SidesExpressions/Equations 1/2 Variable/sMultiply by the LCDDo to one Do to all (× & ÷)Add/Subtract Multiply/Divide by "푥"Rearrange Order of Terms/Mirror EquationIsolate a Variable
2(푥 + 4)2푥 + 8
Put 2 in for "푎" with brackets!
푃푎푟푡 = 퐷ecimal × 푇표푡푎푙
1%
Square Roots√4⎯⎯ = √2 × 2⎯⎯⎯⎯⎯ = 2
√36⎯⎯⎯ √49⎯⎯⎯
6 7
√ ⎯⎯⎯
6.238
Estimating Square/s/Roots
6 7 8
*−2 − 2 = −4 ≠ 4
퐼푆푂퐹⎯⎯⎯=
%100⎯⎯⎯
푀푢푙푡푖푝푙푖푒푟 = 1 ± 푟
Substitution
퐼푛푖푡푖푎푙 × 푀푢푙푡푖푝푙푖푒푟 = 퐹푖푛푎푙
Bottom Top
Words ProbsLet StatementsEquation/s(Arbitrary#'s)IsolateSubstituteSolve (Algebra)SubstituteSolveAnswer!Check Answer!(Eliminate)
Multiply by the decimal, Then Add
Multiply by the % you want to be!
⬚
⬚⎯⎯=
⬚
⬚⎯⎯
푂푣푒푟 = 푂푣푒푟
%퐶ℎ푎푛푔푒 =퐹푖푛푎푙 − 퐼푛푖푡푖푎푙
퐼푛푖푡푖푎푙⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
푎푏⎯⎯= 푐
푎 = 푐푏
Long Division
Graphing: Table of Values TOV푥 − 푖푛푡푒푟푐푒푝푡: 푦 = 0,푝푢푡 0 푖푛 푓표푟 푦 푎푛푑 푠표푙푣푒. (푥, 0)푦 − 푖푛푡푒푟푐푒푝푡: 푥 = 0,푝푢푡 0 푖푛 푓표푟 푥 푎푛푑 푠표푙푣푒. (0,푦)
#: # #: #
Geometry퐴 = 푙푤
퐴 =푏ℎ2⎯⎯⎯
퐴 = 휋푟퐶 = 2휋푟푎 + 푏 = 푐푎푏⎯⎯=
푐푑⎯⎯
푉 = 퐴 × ℎ,
푉 =13⎯⎯퐴 × ℎ
푉 =43⎯⎯휋푟
푆퐴 = 4휋푟
(퐵푎푠푒 푚푢푠푡 푏푒 푠푎푚푒 푎푠 푇표푝)
LCM 8 푎푛푑 12 = 24
GCF 8 푎푛푑 12 = 4
푆퐴∗ = 휋푟 + 휋푟푠(푆푝ℎ푒푟푒)
(푃푟푖푠푚푠)
(푃푦푟푎푚푖푑 푎푛푑 퐶표푛푒∗)
PythagorasSimilarShapes
M8 - 0.0 - Methods
M8 Txt Page 6
푎 + 1 = 푎 + 1
Can't add or subtract unlike terms.Can only add or subtract like terms.
2 × 푥 = 2(푥) = 2푥 = 푥22 × 3 = 2(3) = 6
Algebra
Every Number is Over "1."
−23⎯⎯=
−23⎯⎯⎯=
2−3⎯⎯⎯≠
−2−3⎯⎯⎯
−2−3⎯⎯⎯= +
23⎯⎯
Square Roots (Radicals)3 × 3 = 9 √9⎯⎯ = 3
12푥⎯⎯⎯≠
12⎯⎯푥
The area of a square with side lengths 3 is 9. The square root of 9 is 3.
푥2⎯⎯=
1푥2⎯⎯⎯=
12⎯⎯푥 =
12⎯⎯× 푥
√−9⎯⎯⎯ = 푢푛푑푒푓푖푛푒푑Can't square/even root a negative
Factor trees: You will always end up with the same prime factors
Probability :퐴푛푑 = 푀푢푙푡푖푝푙푦 푂푅 = 퐴푑푑
Exponent : Do not multiply the base by the exponent!
Pythagoras: Right Angle Triangles 90"c" is always the hypotenuse, the longest sideBigger square − Smaller square = Other smaller square.
−푥 = −1푥1 × 푥 = 1푥 = 푥 = 푥 푥 = 1푥
Same Plus+ × + = +− × − = ++ ÷ + = +− ÷ − = +
Different Minus+ × − = −− × + = −+ ÷ − = −− ÷ + = −
푥 + 푥 = 2푥푥 × 푥 = 푥
Geometry: (Units)Adjacent sides of net areas are the same length.The width of rectangle in the cylinders net area is the circumference of the circle
푉 = (푎푟푒푎 표푓 푏푎푠푒) × ℎ푒푖푔ℎ푡*Base must be identical to Top
15% 푓푒푤푒푟 푑표푒푠푛 푡 푚푒푎푛 0.15푥퐼푡 푚푒푎푛푠 85% = (1 − 0.15)푥 = 0.85푥
13⎯⎯+
22⎯⎯≠
ퟏ + ퟐퟑ + ퟐ⎯⎯⎯⎯⎯ 푥 =
1푥1⎯⎯⎯
3742.347 ThousandthsHundredths
Tenths
Ones
Tens
Hundreds
Thousands
3742.347 = 3000 + 700 + 40 + 2 + 0.3 + 0.04 + 0.007
3 = 32 = 15 = 5푥
Coefficients/Fractions/Exponents
4 =41⎯⎯
80⎯⎯≠ 0 Undefined
08⎯⎯= 0 Calculator!
63⎯⎯= 2
62⎯⎯= 3
Switch 2and 3
Graphing: TOV! Watch out for increments
RatiosPart-to-PartPart-to-Total
Goes into2 : Even 0,2,4,6,83 : Digits add to multiples of 35 : Ends in 5 or 07,11,13…: Long Division
10
8 2Break up a Line
# ≠ # 푁표 푠표푙푢푡푖표푛# = # 퐼푛푓푖푛푖푡푒 푆표푙푢푡푖표푛푠
√−8⎯⎯⎯= −2
푠푝푒푒푑 =푑푖푠푡푎푛푐푒푡푖푚푒
⎯⎯⎯⎯⎯⎯⎯⎯ 푠 =푑푡⎯⎯ 푠 =
500푘푚ℎ푟
⎯⎯⎯⎯⎯⎯⎯
Unit Rate: =#1⎯⎯푟푎푡푒 =
푞푢푎푛푡푖푡푦푡푖푚푒
⎯⎯⎯⎯⎯⎯⎯⎯
Place Value
Odds - Part:Part % - Part:Total
Scientific Notation: 6.02 × 10 = 6020
Brackets around the Top, Brackets around the Bottom, Divide on your Calculator(Or do Top then Bottom)
Calculator: (2 + 3)(4 + 6)⎯⎯⎯⎯⎯⎯⎯=
510⎯⎯⎯=
12⎯⎯
"푇ℎ푒 푠푢푚 표푓 2 + 3 "All" over the sum of 4 + 6"
M8 - 0.0 - Remember
M8 Txt Page 7
퐴 = 퐿푤
(퐶표푛푒 푎푛푑 푃푦푟푎푚푖푑)(푃푟푖푠푚: 퐵푎푠푒 푚푢푠푡 푏푒 푠푎푚푒 푎푠 푡표푝)
푎푏⎯⎯=
푐푑⎯⎯
퐴 =푏ℎ2⎯⎯⎯
퐴 = 휋푟퐶 = 2휋푟
푉 = 퐴 × ℎ 푉 =
13⎯⎯퐴 × ℎ
푎 + 푏 = 푐
푆퐴 = 4휋푟
퐿
푤
Square & Rectangle
P = 2퐿 + 2푤
ℎ
푏
ac
P = 푎 + 푏 + 푐
Triangle푟
Circle
Scalene Isosceles Equilateral Right Angle Similar
Cube &Rectangular
푟
ℎ
Triangular
푏
ℎ퐻
푠h
SpherePyramid
b
s s rh
r
Cone
푆퐴 = 휋푟 + 휋푟푠Surface Area: Lay it Flat!
Tick Identical Lines/Angles!
Above Prisms/Pyramids!* Combinations/Cuts!*
푢푛푖푡푠
푢푛푖푡푠
푢푛푖푡푠퐶 = 휋푑
Shade the Base!
푟 =푑2⎯⎯
PrismsCylinder
푉 =43⎯⎯휋푟
Ellipse(Stretched Circle)Quadrilaterals (4)Trapezoid (2 Parallel)
퐴 =(푎 + 푏)ℎ
2⎯⎯⎯⎯⎯⎯⎯⎯
Parallelogram 퐴 = 푏ℎ(2x2Parallel)Rhombus(Square Parallelogram)Kite/DiamondAgonsPent 5, Hex 6Hept 7, Oct 8Non 9, Dec 10Dodec 12Regular* (Same)Semi/Hemi (Half)
Base
2휋푟
Pythag! 퐴 = 휋푟푠
s
s
s
Sphere
M8 - 5.0 - Geometry Review
M8 Txt Page 8
VolumeSurface Area
Cube
Rectangular Prism
Shape
푆퐴 = 푠 × 6s
푙푤
ℎ 푆퐴 = 2(푙푤 + 푙ℎ + 푤ℎ) 푉 = 푙푤ℎ
푉 = 푠
푉 = 휋푟 ℎ
푆퐴 = 푠푢푚 푠푖푑푒 푓푎푐푒푠
푆퐴 = 푏ℎ + 2푠퐻 + 푏퐻ℎ
푆퐴 = 휋푟 + 휋푟푠
푉 =13⎯⎯× 푙 × 푤 × ℎ
푉 =13⎯⎯× 푙 × 푤 × ℎ
h
ℎ푟
푉 =푏ℎ2⎯⎯⎯× 퐻
퐻
Cylinder
푆퐴 = 2푏푠 + 푏
푠
푉 =43⎯⎯휋푟
Sphere
Cone
Square-Based Pyramid
b
Triangular Prism
s
s
r
푆퐴 = 2휋푟 + 2휋푟ℎ
h
General Right Pyramid
푏
푉 =13⎯⎯× (휋푟 ) × ℎ
r
푉 = 퐴푟푒푎 × ℎ푒푖푔ℎ푡
푉 =13⎯⎯퐴푟푒푎 × ℎ푒푖푔ℎ푡
푆퐴 = 4휋푟
Square
퐴 = 푠푃 = 4푠
Area & PerimeterRectangle
퐴 = 푙 × 푤푃 = 2푙 + 2푤
Triangles
퐴 =푏ℎ2⎯⎯⎯
푝 = 푎 + 푏 + 푐
퐴 = 휋푟퐶 = 2휋푟
b (flat)
h r
Scalene, Isosceles, Equilateral, Right, Similar푎 + 푏 = 푐
푎푏⎯⎯=
푐푑⎯⎯60 90180
M8 - 5.0 - SA/V Table Review
M8 Txt Page 9
Multiplying Fractions
Dividing Fractions
Adding Fractions
Subtracting Fractions
Mixed numbers to Improper Fractions
12⎯⎯×
13⎯⎯=
1 × 12 × 3⎯⎯⎯⎯⎯=
16⎯⎯
Multiply topsMultiply bottoms
14⎯⎯÷
23⎯⎯=
14⎯⎯×
32⎯⎯=
38⎯⎯ Flip second fraction and multiply
23⎯⎯+
14⎯⎯=
2 × 43 × 4⎯⎯⎯⎯⎯+
1 × 34 × 3⎯⎯⎯⎯⎯=
812⎯⎯⎯+
312⎯⎯⎯=
1112⎯⎯⎯
퐿퐶퐷 = 12Multiply top and bottom of first fraction by 4Multiply top and bottom of second fraction by 3Add numerators
13⎯⎯−
16⎯⎯=
1 × 23 × 2⎯⎯⎯⎯⎯−
16⎯⎯=
26⎯⎯−
16⎯⎯=
16⎯⎯
퐿퐶퐷 = 6Multiply top and bottom of first fraction by 2Subtract numerators
235⎯⎯ =
푏표푡푡표푚× 푙푒푓푡 + 푡표푝푏표푡푡표푚
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
=5 × 2 + 3
5⎯⎯⎯⎯⎯⎯⎯⎯ =
135⎯⎯⎯
Improper Fractions to Mixed numbers
134⎯⎯⎯= 3
14⎯⎯
Bottom goes into top 3 times (this number goes in front of the fraction)Remainder goes in the numerator
Multiply the bottom by the left and add the top (this number goes in the numerator of the improper fraction)
Simplification24⎯⎯=
2 ÷ 24 ÷ 2⎯⎯⎯⎯⎯=
12⎯⎯ Divide the top and bottom
Expansion
12⎯⎯=
1 × 22 × 2⎯⎯⎯⎯⎯=
24⎯⎯ Multiply the top and bottom
12⎯⎯+
13⎯⎯=
3 × 13 × 2⎯⎯⎯⎯⎯+
1 × 23 × 2⎯⎯⎯⎯⎯=⬚
6⎯⎯+
⬚
6⎯⎯=
36⎯⎯+
26⎯⎯=
56⎯⎯퐿퐶퐷 = 6
Multiply Both Sides by the LCD
14⎯⎯×
23⎯⎯=
14⎯⎯×
23⎯⎯=
16⎯⎯
2
1
Cross-simplification
푥2⎯⎯+
14⎯⎯=
34⎯⎯
푥2⎯⎯+
14⎯⎯=
34⎯⎯ × 4
2푥 + 1 = 3
Multiply and Top & Bottomby the LCD
3 + 2푥⎯⎯
1 − 2푥⎯⎯
⎯⎯⎯⎯⎯
3 + 2푥⎯⎯ × 푥
1− 2푥⎯⎯ × 푥
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
3푥 + 2푥 − 2⎯⎯⎯⎯⎯⎯
Or Add Fractions, Top & Bottom,Flip and Multiply
M8 - 6.0 - Fraction Operations Review
M8 Txt Page 10
푎푏⎯⎯+
푐푏⎯⎯=
푎 + 푐푏
⎯⎯⎯⎯⎯
(푥 − 2)(푥 + 1)푥 − 푥 − 2
푥(푥 + 1)푥 + 푥
푎푏⎯⎯×
푐푑⎯⎯=
푎푐푏푑⎯⎯⎯
푎푏⎯⎯÷
푐푑⎯⎯=
푎푏⎯⎯×
푑푐⎯⎯=
푎푑푏푐⎯⎯⎯
(푥 + 1)(푥 − 1)푥 − 1
푎푏푐⎯⎯⎯⎯⎯ =
푎 ÷푏푐⎯⎯=
푎 ×푐푏⎯⎯=
푎푐푏⎯⎯⎯
푎푏⎯⎯
푐⎯⎯⎯ =
푎푏⎯⎯÷ 푐 =푎푏⎯⎯×
1푐⎯⎯=
푎푏푐⎯⎯⎯
푎푏⎯⎯푐푑⎯⎯⎯⎯⎯ =
푎푏⎯⎯÷
푐푑⎯⎯=
푎푏⎯⎯×
푑푐⎯⎯=
푎푑푏푐⎯⎯⎯
푎 ÷푏푐⎯⎯=
푎 ×푐푏⎯⎯=
푎푐푏⎯⎯⎯
푎푏⎯⎯÷ 푐 =푎푏⎯⎯×
1푐⎯⎯=
푎푏푐⎯⎯⎯
푎 ×푏푐⎯⎯=
푎푏푐⎯⎯⎯
푎푏⎯⎯× 푐 =
푎푐푏⎯⎯⎯
푥 × 푥 = 푥 푥푥⎯⎯⎯= 푥
푥푥⎯⎯= 1푥
푥⎯⎯⎯= 푥
푥푥⎯⎯⎯= 푥푥 × 푥 = 푥
푎 + 푏 + 푐 = 푎 + 푐 + 푏 푎푏 = 푏푎
1푎⎯⎯× 푎 = 1
푎푏⎯⎯+
푐푏푑⎯⎯⎯=
푎푑 + 푐푏푑
⎯⎯⎯⎯⎯⎯푎푏⎯⎯+
푐푑⎯⎯=
푎푑 + 푐푏푏푑
⎯⎯⎯⎯⎯⎯⎯푎푏⎯⎯⎯+
푐푏⎯⎯=
푎 + 푐푏푏
⎯⎯⎯⎯⎯⎯푎푏⎯⎯+
푐푏 + 1⎯⎯⎯⎯⎯=
푎(푏 + 1) + 푐푏푏(푏 + 1)⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
푎 + 푏푐
⎯⎯⎯⎯⎯=푎푐⎯⎯+
푏푐⎯⎯
푥 × 푥 = 푥푥푥⎯⎯⎯= 푥
(푥 ) = 푥 × 푥 =1푥⎯⎯⎯ 푥⎯⎯ = √푥⎯⎯⎯
푥 + 푎푥
⎯⎯⎯⎯⎯ ≠ 1 + 푎
(2푥 + 1)(푥 + 1)2푥 + 3푥 + 1
2 − 1푥
⎯⎯⎯⎯⎯⎯⎯=1
2⎯⎯⎯− 1
푥⎯⎯⎯⎯⎯⎯≠
−12 푥⎯⎯⎯⎯⎯
푎푏⎯⎯=
푐푑⎯⎯
푑푎푏⎯⎯⎯= 푐 푑푎
푐⎯⎯⎯= 푏 푑 =
푐푏푎⎯⎯⎯푎 =
푐푏푑⎯⎯⎯
푎푏⎯⎯=
푐푑⎯⎯
푑푎 = 푐푏
CrossMultiply
MirrorRearrange
Distribution
Fractions
Exponents
푎푏 + 푐⎯⎯⎯⎯⎯≠
푎푏⎯⎯+
푎푐⎯⎯
Multiply
Divide
Add/Subtract
Common MistakesSeparate푥 + 푎푥
⎯⎯⎯⎯⎯=푥푥⎯⎯+
푎푥⎯⎯
CommonMistakes
1푥⎯⎯⎯= 푥
1푥⎯⎯+ 1
2 + 3푥⎯⎯
⎯⎯⎯⎯⎯=1 + 푥2푥 + 3⎯⎯⎯⎯⎯⎯
Multiply Top/BottomBy LCD
푥 − 푥 − 2(푥 − 2)(푥 + 1)푥 + 푥
푥(푥 + 1)
푥 − 1(푥 + 1)(푥 − 1)
FOIL Conjugates: FL
2푥 + 3푥 + 1(2푥 + 1)(푥 + 1)
Factoring
푥2⎯⎯=
12⎯⎯
푥2⎯⎯=
12⎯⎯
푥 = 1
푥4⎯⎯=
14⎯⎯+
24⎯⎯
MultiplyBoth SidesBy LCD
푅푒푐푖푝푟표푐푎푙: 푎푏⎯⎯=
푏푎⎯⎯
푎 + 푐 "All" over b
M8 - 6.0 - Fractions/LCD/Exponents/Distribution Theory
M8 Txt Page 11
푚 = −퐴퐵⎯⎯
풙 풚 −2 −1 0 1 2
풙 풚0
0
Increments of x by denominator of slope away from zero. Or y-coefficient.
푦 =23⎯⎯푥 + 1
풙 풚 −3 0 3
푦 = 푓(푥)
푦 − 푦 = 푚(푥 − 푥 )y coordinate x coordinateslope
y-interceptslope
퐴푥 + 퐵푦 = 퐶
Slope- Intercept
푦 = 푚푥 + 푏
Slope-Point
퐺푒푛푒푟푎푙/푆푡푎푛푑푎푟푑
퐴푥 + 퐵푦 + 퐶 = 0
푁표 퐹푟푎푐푡푖표푛푠푋 푡푒푟푚 푝표푠푖푡푖푣푒+푥,푦, # = 0∗
푦 − 푖푛푡 = ±퐶퐵⎯⎯
(0, 푏)
푆푙표푝푒 = 푚 = 푟푖푠푒푟푢푛⎯⎯⎯⎯= 푦 − 푦
푥 − 푥⎯⎯⎯⎯⎯⎯⎯(푥 ,푦 ) (5,4)
(2,4)(푥, 푦)
(푥 ,푦 ) (−2,3)
Step 1: Find Intercepts푦 − 푖푛푡: 푥 = 0 (0,푦), 푝푢푡 푧푒푟표 푖푛 푓표푟 푥 푎푛푑 푠표푙푣푒푥 − 푖푛푡: 푦 = 0 (푥, 0), 푝푢푡 푧푒푟표 푖푛 푓표푟 푦 푎푛푑 푠표푙푣푒
Graph Steps1) Plot y-int2) Use SlopeFind Equation1) Find y-int2) Find Slope
+1푦 =
푚 = 푚푚 = −
1푚⎯⎯
ParallelSame slope Perpendicular 90
Negative Reciprocal Slope
푦 = −퐴퐵⎯⎯푥 ±
퐶퐵⎯⎯
(0,3)
(0,1)
(2,0) (4,0)푥 (퐼푛푝푢푡)
푦 (푂푢푡푝푢푡)
푦 푑푒푝푒푛푑푠 표푛 푥!
101
102
50 55
(2,3)(푥, 푦)
(−4,−2)
SkipIncrements
Graph Steps1) Plot Point2) Use SlopeFind Equation1) Find Point2) Find Slope
푥
푦
푥
푦푚 = 0
푚 = 푢푛푑
(1,2) (3,2)푦 = 2 (1,4)
(1,2)
푥 = 1
Draw a Graph and Count!
푦
푥
+− 푚 == 푚
푥 y 0 푏 1 3 2 5
∆푥∆푦⎯⎯⎯ 퐶표푛푠푖푠푡푒푛푡∗
푦 = 푏; 푥 = 0!
푦 =∆푦∆푥⎯⎯⎯푥 + 푏
∆푥∆푥
∆푦∆푦
푦 = 푚푥 + 푏
Up toRight
Down toRight
(푥 ,푦 ) (3,5)
M8 - 9.0 - Graphing Review
M8 Txt Page 12
1. 푎 + 1 = 3
푎 + 1 = 3 −1 − 1
푎 + 1 = 3 −1 − 1
푎 = 2
Subtract 1 from both sides+1 − 1 = 0. Cross it off.
2. 푎 − 1 = 2
푎 − 1 = 2 +1 + 1
푎 − 1 = 2 +1 + 1
푎 = 3
Add 1 to both sides
−1 + 1 = 0. Cross it off.
3. 2푎 = 4
2푎2⎯⎯⎯=
42⎯⎯
2푎2⎯⎯⎯=
42⎯⎯
푎 = 2
Divide both sides by the number in front of the letter.
⎯= 1. Cross it off.
4. 푎2⎯⎯= 3
푎2⎯⎯=2 × 3 × 2
푎2⎯⎯=2 × 3 × 2
푎 = 6
Multiply both sides by the number below the letter.
⎯= 1. Cross it off.
5. 2푎⎯⎯= 1
2푎⎯⎯=
푎 × 1 × 푎
2푎⎯⎯=
푎 × 1 × 푎
2 = 푎
Multiply both sides by 푎.
⎯= 1. Cross it off.
푥 = 25
푥⎯⎯⎯ = √25⎯⎯⎯
푥 = ±5
Square/Rootbothsides.
6.
7. 푥 + 4 = 2푥−푥 − 푥 4 = 푥
Subtract x from both sides
I'm not afraid of adding/subtracting x from both sides.
Work on the complicated side!
4(푥 − 3) = 8
4푥 − 12 = 8 …
Multiply the outside by all of the inside.
3푥 + 3푥 = 4 + 8 6푥 = 12
6푥6⎯⎯⎯=
126⎯⎯⎯
푥 = 2
Combine like Terms
2푥 − 3 + 3푥 = 5푥 − 3
푥2⎯⎯+
14⎯⎯=
34⎯⎯
푥2⎯⎯+
14⎯⎯=
34⎯⎯ × 4
2푥 + 1 = 3 …
Multiply both sides by the LCD (and simplify at the same time.)
Bring it up and Multiply. Bring it down. So Easy. (Over = Over!) (1…!)
√푥⎯⎯ = 5
푥 = 25(√푥⎯⎯ ) = (5)
(푥 + 2) = (푥 + 2)(푥 + 2)
8.
9.
10.
11.
푎 + 1 = 푎 + 1
Divide by −1 −푥 = 5 푥 = −5
Check Answer!(2) + 1 = 3 Substitution
Bring it over!Change the Sign!
Cross Multiply
12.
푥 + 2 푎푙푙 푠푞푢푎푟푒푑
푎 + 1 = 3 − 1 푎 = 2
푎 − 1 = 2 + 1 푎 = 3
푎2⎯⎯= 3 × 2푎 = 6
2푎 =42⎯⎯
푎 = 2
M8 - 10.0 - Solving Equations (OPPOSITE OPERATION BOTH SIDES)
M8 Txt Page 13
21 different outcomes if same colored dice.
36 different outcomes if different colored dice.
(2,1) = (1,2)
Sum of two dice
Product of two dice
RED
DIE
BLACK DIE
BLACK DIEBLACK
DIE
Rolling Two Dice
Hearts ♥ Diamonds ♦ Spades ♠ Clubs ♣
Ace ♥ Ace ♦ Ace ♠ Ace ♣
2 ♥ 2 ♦ 2 ♠ 2 ♣
3 ♥ 3 ♦ 3 ♠ 3 ♣
4 ♥ 4 ♦ 4 ♠ 4 ♣
5 ♥ 5 ♦ 5 ♠ 5 ♣
6 ♥ 6 ♦ 6 ♠ 6 ♣
7 ♥ 7 ♦ 7 ♠ 7 ♣
8 ♥ 8 ♦ 8 ♠ 8 ♣
9 ♥ 9 ♦ 9 ♠ 9 ♣
10 ♥ 10 ♦ 10 ♠ 10 ♣
Jack ♥ Jack ♦ Jack ♠ Jack ♣
Queen ♥ Queen ♦ Queen ♠ Queen ♣
King♥ King ♦ King ♠ King ♣
Pick a Card (4 Suits/13 Cards per Suit/52 Cards) What is the probability of drawing an Ace?
Now we remove an the Ace of Spades (Ace ♠) from the deck. What is the new probability of getting an Ace?
푃(퐴|퐴♠) =3 푎푐푒푠
51 푡표푡푎푙 푐푎푟푑푠⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯=
351⎯⎯⎯
퐴푐푒 ♥ + 퐴푐푒 ♦ + 퐴푐푒 ♣All the cards except Ace ♠ (51 cards)⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
푃(퐴) =4 푎푐푒푠
52 푡표푡푎푙 푐푎푟푑푠⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯=
113⎯⎯⎯
퐴푐푒 ♣ + 퐴푐푒 ♥ + 퐴푐푒 ♦ + 퐴푐푒 ♣52 Cards
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
Dependent
M8 - 11.0 - Dice/Cards Tables
M8 Txt Page 14
The End
M8 Txt Page 15
