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Test Taking Tips. Read each question carefully. Read the directions for the test carefully. For Multiple Choice Tests Check each answer – if impossible or silly cross it out. Back plug (substitute) – one of them has to be the answer For factoring – Work the problem backwards

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slide4

For Multiple Choice Tests

  • Check each answer – if impossible or silly cross it out.
  • Back plug (substitute) –
  • one of them has to be the answer
  • For factoring – Work the problem backwards
  • Sketch a picture
  • Graph the points
  • Use the y= function on calculator to match graphs
slide5

Do the Easy Ones First

Then go Back and

do the Hard Ones!

slide6

Beware of the Sucker Answer

Make sure you answer

the question that is asked!

Double check the question

before you fill in the bubble!!

slide8

X 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45

4 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

6 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

7 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105

8 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112120

9 0 9 18 27 36 45 54 63 72 81 90 99 108117 126135

10 0 10 20 30 40 50 60 70 80 90 100 110 120130 140150

11 0 11 22 33 44 55 66 77 88 99 110121 132143 154 165

12 0 12 24 36 48 60 72 84 96 108120 132144 156168 180

13 0 13 26 39 52 65 78 91 104 117130 143156 169182 195

14 0 14 28 42 56 70 84 98 112 126140 154168 182196 210

15 0 15 30 45 60 75 90 105 120 135150 165180 195 210225

Factors Multiples Perfect Squares

(6 ) (4 ) = 24

slide9

C

B

A

Geometry Basics

A

Point (Name with 1 capital letter)

Line (Name with 2 capital letters,

B

)

A

Ray (Name with 2 capital letter,

)

A

B

Angle (Name with 3 letters.

Middle letter is vertex

C

)

B

A

Line Segment (Name with two letters, AB)

A

B

Plane (Name with 3 non-collinear points, ABC)

slide10

90

  • Complementary Angles
  • Right Angles
  • Symbol (┌ or ┐)
  • Perpendicular ┴
  • A corner
  • 180
  • Straight Angle (line)
  • Supplementary Angles
  • Half Circle
  • Sum of angles in a triangle

GEOMETRY MAGIC NUMBERS

Also called linear pair

  • 360
  • Circle
  • Sum of angles in a 4 sided figure (quadrilateral)
slide11

GEOMETRY MAGIC NUMBERS

  • Complementary Angles
  • Right Angles
  • Symbol (┌ or ┐)
  • Perpendicular ┴
  • A corner

90

slide12

GEOMETRY MAGIC NUMBERS

180

  • Straight Angle (line)
  • Supplementary Angles
  • Linear Pair
  • Half Circle
  • Sum of angles in a triangle
slide13

Supplementary Angles

Two Angles are Supplementary if they add up to 180 degrees.

HINT: S Straight or S Splits

Thanks to http://www.mathsisfun.com/geometry/complementary-angles.html

slide14

Vertical Angles

Angles opposite each other when two lines cross

They are called "Vertical" because they share the same Vertex (or corner point)

vertex

Vertical angles are congruent and their measures are equal:

http://www.mathwarehouse.com/geometry/angle/interactive-vertical-angles.php

slide15

Complementary Angles

Two Angles are Complementary if they add up to 90 degrees (a Right Angle).

HINT: C Corner or C looks like a corner

Thanks to http://www.mathsisfun.com/geometry/complementary-angles.html

slide16

Linear Pairs

Angles on one side of a straight line will always add to 180 degrees.

If a line is split into 2 and you know one angle you can always find the other one by subtracting from 180

25°

A° = 180 – 25°

A° = 155°

slide17

Right Angles

A right angle is equal to 90°

Notice the special symbol like a box in the angle. If you see this, it is a right angle. 90˚ is rarely written.

If you see the box in the corner, you are being told it is a right angle.

90°

90°

Notice: Two right angles make a straight line

slide18

Properties of Equality

  • Addition Property: If a=b, then a+c=b+c
  • Subtraction Property: If a=b, then a-c=b-c
  • Multiplication Property: If a=b, then ac=bc
  • Division Property: if a=b and c doesn’t equal 0, then a/c=b/c
  • Substitution Property: If a=b, you may replace a with b in any equation containing a and the resulting equation will still be true.
slide19

Properties of Equality

Reflexive Property:

For any real number a, a=a

Symmetric Property:

For all real numbers a and b, if a=b, then b=a

Transitive Property:

For all real numbers a, b, and c, if

a=b b=c a=c a=c

slide20

Conditionals & Bi-conditionals

EXAMPLES:

IFtoday is Saturday, THENwe have no school.

“IF-THEN ” statements like the ones above are called CONDITIONALS.

To make a bi-conditional, take off the IF and replace the THEN with “IF AND ONLY IF”

Today is Saturday, IF AND ONLY IF we have no school.

slide21

Conditionals

Conditional statements have two parts…

The part following the wordIF is the HYPOTHESIS

The part following the word THEN is the CONCLUSION

IFtoday is Saturday, THENwe have no school.

Hypothesis: today is Saturday

Conclusion: we have no school

slide22

Converse

The of a conditional statement is formed by exchanging the HYPTHESIS and the CONCLUSION.

CONVERSE

CONDITIONAL:IFit is snowing, THENwe will have a snow day.

CONVERSE:

IFwe will have a snow day, THEN it is snowing.

slide23

Counterexample

A Counterexample is an example that proves a statement false.

Conditional Statement: IFan animal lives in water, THENit is a fish.

* This conditional statement would be false.

You can show that the statement is false because you can give onecounterexample. *

Counterexample: Whales live in water, but whales are mammals, not fish.

slide24

If-Then Transitive Property

Given

If Athen B, and if B then C.

If sirens shriek,

then dogs

howl

If dogs

howl,

then cats

freak.

You can conclude:

If A then C.

If sirens

shriek,

then cats

freak.

slide25

Quadrilaterals

( 4 sides )

Parallelogram

Rectangle

Rhombus

Square

Trapezoid

Isosceles Trapezoid

Kite

slide26

Rectangle

All angles are congruent (90 ˚ )

Congruent Sides

Congruent Angles

Parallel Sides

Diagonals are congruent

Congruent Sides

Congruent Angles

Parallel Sides

Opposite sides

Opposite sides parallel

Parallelogram

Opposite angles

Diagonals bisect each other

Consecutive angles are supplementary

slide27

Rhombus

Congruent Sides

Congruent Angles

Parallel Sides

Congruent Sides

Congruent Angles

Parallel Sides

Diagonals are perpendicular

All sides are congruent

Diagonals bisect angles

Square

Diagonals are perpendicular and congruent

Diagonals bisect each other

All sides are congruent

All angles are congruent

Angles = 90°

slide28

Isosceles Trapezoid

Congruent Sides

Congruent Angles

Parallel Sides

Congruent Sides

Congruent Angles

Parallel Sides

Diagonals are congruent

Trapezoid

Kite

Diagonals are perpendicular

geometry in motion

Reflection

Dilation

Reflection

Geometry in Motion

Rotation

Reflection

Reflection

Transformation

slide30

A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.

Let's examine some translations related to coordinate geometry.

In the example, notice how each vertex moves the same distance in the same direction.

6 units to the right

slide31

Translations

In this next example, the "slide"  moves the figure7 units to the left and 3 units down.

There are 3 different ways to describe a translation

slide32

When you reflect a point across the y-axis, the y-coordinate remains the same,

the x-coordinate changes!

slide33

When you reflect a point across the x-axis, the x-coordinate remains the same, and the y-coordinate changes!

examples of the most common rotations
Examples of the Most Common Rotations

Counterclockwise rotation by 180° about the origin:

A is rotated to its image A'. The general rule for a rotation by 180° about the origin is

(x,y)

(-x, -y)

slide35

Examples of the Most Common Rotations

Counter clockwise rotation by 90° about the origin:

A is rotated 90° to its image A'. The general rule for a rotation by 90° about the origin is

(x,y)

(-y, x)

slide36

Dilations always involve a change in size.

Dilations

Dilations

Dilations

Dilations

Dilations

Dilations

Dilations

Dilations

Dilations

Dilations

slide37

You are probably familiar with the word "dilate" as it relates to the eye.  The pupil of the eye dilates (gets larger or smaller) depending upon the amount of light striking the eye.

slide38

Dilations - Example 1: If the scale factor is greater than 1, the image is an enlargement (bigger).

PROBLEM:  Draw the dilation image of triangle ABC with scale factor of 2.

OBSERVE: Notice how EVERY coordinate of the original triangle has been multiplied by the scale factor (2).

HINT: Dilations involve multiplication!

slide39

Dilations Example 2: If the scale factor is between 0 and 1, the image is a reduction (smaller).

PROBLEM:  Draw the dilation image of pentagon ABCDE with a scale factor of 1/3.

OBSERVE: Notice how EVERY coordinate of the original pentagon has been multiplied by the scale factor (1/3).

HINT: Multiplying by 1/3 is the same as dividing by 3!

slide40

Transversal

Angles

Parallel Lines

Exterior

1

2

4

3

Interior

6

5

8

7

slide41

Same Slope

Parallel Lines

y2 – y1

x2 – x1

Slopes are Negative Reciprocal

Flip and Change Sign

or

Perpendicular

slope

y = mx + b

lines

slide42

Slope – Intercept Form

y = mx + b

Slope- directions

Rise

Run

It’s a line address

Example 2

To Graph:

Example 1

y = -3X+0

y=-3X

Starts at 0

rise/run =3/-1

Directions areup 3, over -1

y=2X+1

Starts at 1

Rise/run = 2/1

Directions areup 2,over 1

Thanks to http://www.mathsisfun.com/equation_of_line.html

slide43

Linear Equations, Standard Form ax + by = c

  • Solving for y, It’s a football Game
  • Y VS Everybody Else
  • Follow football rules

Play Football

Lettersvs Numbers

Example: Solve for Y

2x – 7y = 12

Just 3 easy steps

1. -7y = 12 – 2x X is offside, Penalty change signs

2. -7y = (12-2x) Huddle up ( )

3. y = (12-2x) / -7Man on man defense

Now you are ready to enter it into the calculator or graph it

WATCH YOUR SIGNS!!

slide44

Find Equation of the Line:

y = mx + b

To find m – Solve the equation for y and use m

or use the

y2 – y1

x2 – x1 formula

I need slope (m) & the y-intercept (b)

To find b - Plug x, y and m into the line equation and solve for b.

MY ANSWER:

y = x +

slide45

Formulas

Line Stuff

Slope: m =

)

Midpoint: (x, y) = (

,

Distance: d =

  • Polygons:

Sum of the interior measures:

Sum of the exterior measures: 360°

Measure of the interior angle in a regular polygon:

Measure of the exterior angle in a regular polygon: 360°

slide46

Sum of the Angles of a Polygon.

Sum of Exterior Angles is 360

slide47

Floor Rugs

Area

Floor Plan

Examples of things you’d find the area of.

Tiles or floors

Acres

slide48

Perimeter – Path around the Outside

No Trespassing – Go all the way Around!

slide49

Area Formulas

h

a

b

Area of Plane Shapes

h

h

b

b

  • b2

h

r

b1

slide50

Perimeter Formulas

a

a

b

Area of Plane Shapes

c

a

h

b

b

b2

d

a

r

b1

slide51

A

b

c

B

C

a

Trigonometry for Any Triangle

Law of Sines

sin(A) = sin(B) = sin(C)

a bc

Law of Cosines

a² = b² + c² – 2bc * cos(A)

b² = a² + c² - 2ac * cos(B)

c² = a² + b² – 2ab * cos(C)

cos(A) = (a² – b² – c²)

(-2ab)

To convert from:

Degrees to radians – multiply by π 180

Radians to degrees – multiply by 180 π

slide52

3 Trig Functions:

SOH

CAH

TOA

opp

opp

adj

Θ

Sin

Cos

Tan

Θ

=

=

=

hyp

hyp

adj

Θ

slide53

Trigonometry Functions

(Be sure your calculator is in degrees)

Trigonometry is the study of how the sides and angles of a right triangle are related to each other.

Hyp is always across from right angle. Adj and Opp change depending on Θ

3 Sides:

1. Hypotenuse - Across from right angle.

2. Opposite - Across from angle Θ.

3.Adjacent– Next to angle.

Θ

hyp

opp

hyp

adj

Θ

adj

opp

slide54

To Solve:

  • Use chart to organize information
  • Set up ratio
  • Cross multiply
  • Solve for X