3-1

Addition Properties

Properties – something you cannot disprove – always true. *You must memorize

these properties!

1) Commutative property of addition – changing the order of addends will not

change the sum

Ex) 4+5 = 9 5+4 = 9

Addends Sum Addends Sum

Ex) -4+3[(3+22)3+b-2] = 3[(3+22)3+b-2] +-4 – Recognize this as a properties problem.

See that addends are the same on both sides. *(Think about the messy part of the

problem as one term – say… negative four plus the “mess” equals the “mess” plus

negative four)

2) Associative property of addition – regrouping the addends will not change the

sum.

Ex) (3+4)+7 = 3 +(4+7) – We did not change the order of addends (that would make

it the commutative property)

3) Additive identity property – any number (addend) plus zero is that number. The

identity stays the same.

Ex) -4+[(3+2

2)

3+b

-2]

3+0 =

-4+[(3+2

2)

3+b

-2]

3 (A “mess” plus zero equals the “mess” - its

identity didn’t change)

4) Additive inverse (opposite) property – The sum of a number and its inverse

equals zero. (Don’t confuse with additive identity property.)

Ex) 7+-7=0 Zero will always be the answer to an additive inverse property problem.

Ex) (a2+b-c) +

-(a

2+b-c)=0 this would be read “the quantity of a

2+b-c plus the opposite

of the quantity of a2+b-c equals zero.”

3-2

Multiplication Properties

1) Commutative property of multiplication – changing the order of the factors will

not change the product.

Ex) ab=ba Two variables next to each other means multiplication (ab means the

same as a•b)

Ex) 4(3)=3(4)

2) Associative property of multiplication - changing the grouping (regrouping) of the

factors will not change the product.

Ex) (5•4)•6=5(4•6)

3) Multiplicative identity property – any number multiplied by the factor 1 is the

original number. It keeps its identity.

Ex) 6•1=6

Ex) 4[(2+b2)3+7] •1=4[(2+b2)3+7]

4) Multiplicative inverse property – any number multiplied by its multiplicative

inverse is one (1). This property involves multiplying a number by its reciprocal.

Ex) 4• 1

4

=1 1

4

4

1

= “sneaky name for one”

5) Multiplicative property of zero – any factor multiplied by zero is zero.

Ex) 7•0=0

Ex) b(10-164-5)•0 =0

6) Distributive property – a number next to a parenthesis means multiply. Distribute

multiplication over parenthesis.

Ex) 3(4+-2) = (3•4)+(3•-2) Ex) 4(7-6) = (4•7)-(4•6)

Ex) (2•-5)-(2•3) = 2(

-5-3)

4

4

3-3

Quadrilaterals

A quadrilateral is a closed plane, two dimensional figure with four sides that are line

segments. There are no curves.

≅ means congruent

means congruent (the dashes on the sides of the figure)

Quadrilateral Flowchart

*All Quadrilaterals’ diagonals bisect*

Trapezoid

Quadrilateral with:

exactly one pair of

opposite, parallel sides.

(Isosceles trapezoid – 1 set of opposite

parallel sides. The other set of opposite

sides are congruent lines.)

Rectangle

Parallelogram

with:

• Four right angles

• Diagonals are congruent

Square

Parallelogram with:

•

•

•

3-4

Quadrilateral Flowchart

Quadrilateral

(4 sided polygon )

*All Quadrilaterals’ diagonals bisect*

1 set of opposite

parallel sides. The other set of opposite

Parallelogram

Quadrilateral

with:

• 2 sets of opposite parallel sides

• Opposite angles are congruent

Diagonals are congruent

Rhombus

Parallelogram

with:

• Four congruent sides

• Diagonals are perpendicular

Square

Parallelogram with:

• Four congruent angles

• Four congruent sides

• Diagonals are congruent and

perpendicular

2 sets of opposite parallel sides

Opposite angles are congruent

Four congruent sides

Diagonals are perpendicular

3-5

21

30 15

5 10

7

A C

B E

D F

Similar Figures

Shapes that look exactly the same but can be different sizes are similar.

Similar symbol ~

Two figures are considered similar if:

a) Corresponding angles (∠ ‘s) are congruent (≅) and

b) Corresponding sides are proportional

To check if corresponding sides are proportional, first use proportional reasoning. Each side has

to increase or decrease by the same number of times. If you can’t see proportional reasoning, set

up a proportion and use labels.

Proportion

15 (big triangle) = 21 (big triangle)

5 (small triangle = 7 (small triangle)

Common mistake – to check for corresponding, do not add/subtract. (They must increase or

decrease by the same number of times, not the same number.)

SOL test example:

If the two triangles below are similar, which of the following must be true?

A DE

AC=

DE

AB

B DE

AB=

DF

BC

C DF

AB=

DE

AC

D DEsmall

ABbig=

DFsmall

ACbig

(use labels to check)

CA

B E

DF

∠A corresponds to ∠D

∠BCA corresponds to ∠EFD

������ corresponds to ������

∠A≅∠D, ∠B≅∠E, ∠C≅∠F

X3 X3

D is true because AB corresponds to DE and AC

corresponds to DF so they must be proportional by

definition of similarity.

If you don’t see the proportional

reasoning, cross multiply to see if

the answers are equal. If they are

not, then the figures aren’t similar

105 15 = 21 105

5 7

105=105 so they are similar

3-6

20

3

X

5.7 X

4.3

10.8

Similar Figures/Missing Sides

Two figures have to be similar before we can find the missing side. If they are asking

us to find the missing side, they first tell us the two figures are similar.

Ex)

12 (big triangle) = 20 (big triangle)

3 (small triangle = X (small triangle)

12X=60

12

12X=

12

60

X=5

Ex)

10.8 (big) = 4.3 (big)

5.7 (small) = X (small

10.8X=24.52

8.10

8.10 X=

8.10

51.24

X≈2.27

X4 X4

12

3-7

5 cm

7 cm

8 in

4 in

Area

The word “cover” is a good synonym for area. Area is 2 dimensional (2D) – squared, power of 2,

covering with2 dimensional (flat) squares.

Area of a parallelogram

Formula: area = bh (base times height)

Steps:

1) Always write the formula first - area=bh

2) Label height and base of the parallelogram – base is 6”, height is 6”

3) Substitute dimensions (numbers) into the formula – area = 6•5

4) Label correctly – area = 30 in.2

The height must be perpendicular to the base in a parallelogram

area=bh

area = 7•4

area = 28 cm2

Common mistake would be to use 5 cm as the height instead of 4cm.

Area of a triangle

Formula: area = 2

1 bh (base times height)

Use the same steps as for parallelogram.

area= 2

1 bh

area = 2

)4(8

area = 2

32

area = 16 in2

Volume is 3 dimensional (3D) – cubed, power of 3, filling with cubes

Perimeter is 1 dimensional (1D) – go around with line segments

5 in

6 in

4 cm

3-8

7 in

6 i

n

5 i

n

6 i

n

5 i

n

5 in

Surface Area (SA) of Rectangular Prisms

Each face of a rectangular prism (3D) is a 2D parallelogram. When finding the surface area of a

rectangular prism you are covering each face with 2D squares.

When you unfold a 3 dimensional figure it is called a net.

Formula:

SA(surface area) = 2 lw (length times width)+2lh (length times height)+2wh (width times height)

Steps:

1) Always write the formula first – SA=2lw+2lh+2wh

2) Label* length, width, height of the rectangular prism

3) Substitute dimensions (numbers) into the formula

4) Label correctly

*When you label, it does not matter which edge is length, width or height as long as you are

consistent when substituting within the formula.

Ex)

SA=2lw+2lh+2wh

SA=2(7)(6)+2(7)(5)+2(6)(5)

SA=84+70+60

SA=214 in2

Ex) To find the amount of glass needed for an aquarium with no top, you would alter the formula

as follows:

SA = lw+2lh=2wh because you don’t need to add the surface area for the top.

SA=(7)(6)+2(7)(5)+2(6)(5)

SA=42+70+60

SA=172 in2

7 in length

6 in width

5 in height

Using the

Net add:

35 in2

42 in2

30 in2

35 in2

42 in2

30 in2

= 214 in2

7 in length

6 in width

5 in height

35 in2

35 in2

42 in2

42 in2

30 in2

30 in2

3-9

Surface Area (SA) of Rectangular Prisms

Formula Explanation

SA=2lw+2lh+2wh

Part 1 Part 2 Part 3

SA= 2lw + 2lh + 2wh

Bottom and top Front and back sides

Part 1 – top and bottom faces

Part 2 – front and back faces

Part 3 – side faces

To get the area for the bottom face

and the area for the top face,

multiply the length by the width.

lw length

width

height 1

2

To get the area for the front face

and the area for the back face,

multiply the length by the height.

lh

To get the area for the side faces,

multiply the width by the height.

wh

length

width

height

1

2

length

width

height

1 2

3-10

Volume of Rectangular Prisms

Volume means fill. You are filling with 3 dimensional cubes. Even liquid is measured in 3D cubes.

Formula: v=lwh (volume = length times width times height)

volume = length times width times height

Steps:

1) Always write the formula first – v=lwh

2) Label length, width, height of the rectangular prism

3) Substitute dimensions (numbers) into the formula

4) Label correctly

Ex)

v=lwh

v=7(6)(5)

v=210 in3

Scale Factor: What you are multiplying by.

When you change one attribute by any scale factor, the volume will change by the

same scale factor. Using the above example, if the width is doubled to 12 the volume

is doubled (2•210 in3=420 in

3) as shown below.

v=lwh

v=7(12)(5)

v=420 in3

7 in length

6 in width

5 in height

• Tells how many cubes cover the bottom

face (1 layer)

• Tells how many layers

3-11

Area and Circumference of Circles

Circle terminology:

Diameter – distance across the center of a circle

Circumference – distance around the circle (perimeter)

Radius – distance from the center to any point on the circle - 2

1the diameter

Formulas:

Circumference - c=2πr (2 times pi times radius) or c=πd (pi times diameter)

Area – a=πr2 (pi times radius squared)

π≈3.14 (Circumference/diameter = π)

Ex)

c=2πr

c=2(3.14)(7)

c≈43.96 cm

Ex)

a=πr2

a=3.14(52)

a≈78.5 in2

7 cm

10 in

3-12

Surface Area and Volume of Cylinders

Surface Area

Formula: SA =2πrh+ 2πr2 (2 times pi times radius times height plus 2 times pi times

radius squared.

Formula explanation: Part 1 Part 2 Part 3

SA= 2πr X h + 2πr2

The circumference of a

circle which when

unfolded becomes the

base of a rectangle.

Multiply by height to

get the area of the

rectangle.

Add the area of the

two circles.

It is easier to visualize with a net of a cylinder:

2 circles and a rectangle.

Ex)

SA=2πrh+ 2πr2

SA=2(3.14)(3)(7)+2(3.24)(32)

SA=2(3.14)(3)(7)+2(3.24)(9)

SA=131.88+56.52

SA≈188.4 in2

Volume

Formula: v=πr2h

Ex)

v=πr2h

v=(3.14)(32)(7)

v=(3.14)(9)(7)

v≈197.82 in3

3-13

Coordinate Plane Horizontal line

Vertical Line

Point of origin is where the horizontal and vertical number lines meet at 0,0. This forms four

quadrants.

The horizontal number line is the x-axis (move left and right)

Vertical number line is the y-axis (move up and down)

Coordinate plane extends forever in all directions.

Quadrants extend forever but have 2 boundaries – the x and y- axis.

Whenever we draw a point on the coordinate plane it is called plotting a point.

The coordinates of the point are a pairs of numbers with an x value and a y value (x,y). The x-axis

is always first.

Quadrant I – both coordinates are positive (+,+)

Quadrant II - x is negative, y is positive (-,+)

Quadrant III - both coordinates are negative (-,-)

Quadrant IV –x is positive, y is negative (+,-)

Ex) On this coordinate plane what are the coordinates for points A, G, F, K?

A (-4,3)

G (5,4)

F (1,0)

K (-2,-3)

A

G

K

F

3-14

Transformations (Movements)

Movement of a figure or a point on a coordinate plane is a transformation.

There are four types of transformations:

1. Translations – slide

2. Reflections – flipping over the x or y-axis

3. Rotations – turn

4. Dilations – enlarging/shrinking (similar figures on a coordinate plane)

Translations 1-3 change the location. Transformation 4 (dilations) change the size. Of

these, rotations and dilations are the hardest to understand.

Pre-image – original point or figure (given figure) on a coordinate plane

Image – new point or figure after the transformation (movement)

Points and figures (after transformation) are marked with the prime symbol. (′)

Translation – do one point at a time.

= Arrow notation

Ex) Translate ABCD 6 units to the right and 4 units down. The translation rule in arrow notation form for this example is:

A(-5,5) (-5+6, 5-4) You can find A’ by just doing the addition or subtraction of the x and y.

A’(1,1) because -5+6=1 and 5-4=1

Coordinates:

Pre-image (original)

A(-5,5)

B(-3,5)

C(-3,1)

D(-5,1)

Image (new)

A’(1,1)

B’(3,1)

C’(3,-3)

D’(1,-3)

3-15

Transformations (Movements) cont’d. Rotation

90° clockwise – to the right

90° counter clockwise – to the left

180° can be in either direction.

The point of rotation can be any point on the figure or the point of origin (0,0)

Rotation steps:

SOL step one: using graph paper draw x, y axis then redraw the shape on the graph paper

(include original coordinates)

Step two: list the original coordinates

Step three: predict what quadrant your image will be in

Step four: turn the paper in the opposite direction of the rotation

Step five: re-plot the original points (looking at the new x,y axis)

ABCD 90° clockwise. List Coordinates: Pre-image (original) Ex) Rotate

A(-5,5), B(-3,5), C(-3,1), D(-5,1).

Turn the paper in

the opposite

direction of the

rotation. In this

example turn the

paper counter

clockwise since the

rotation is

clockwise.

Re -plot the

original points

(looking at the

new x,y axis).

Turn the

paper back.

List the image

(new)

coordinates.

A’(5,5)

B’(5,3)

C’(1,3)

D’(1,5)

3-16

Transformations (Movements) cont’d.

Reflections

You will be told to flip over either the x or y axis. Do one point at a time. Whatever

axis you are reflecting over, the value for that axis in the coordinate will stay the

same. The other value will be the inverse.

Reflection steps:

SOL step one: using graph paper draw x, y axis then redraw the shape on the graph

paper (include original coordinates)

Step two: list the original coordinates

Step three: predict what quadrant your image will be in

Step four: (one point at a time) Whatever axis you are reflecting over, count how

many spaces you are away from that axis. Plot the same number of spaces on the

other side of the axis. Points need to be lined up.

Step five: list the new coordinates.

You can check the reflection by folding the graph paper.

Ex) Reflect ABC over the y axis

Pre-image

coordinates

Image

coordinates

A(-3,0) A’(3,0 )

B(-2,-2) B’(2,-2))

C(-1,2) C’(1,2)

3-17

Transformations (Movements) cont’d. Dilations

Dilations are similar figures on a coordinate plane. Dilations can enlarge or shrink the

pre-image.

Scale factor tells how much to shrink or enlarge the pre-image. Dilations are all

multiplications even if by a fraction.

SOL NOTE: tests usually include the scale factors 4

1 , 2

1 , 2, 3, 4 for dilations because

they fit on a coordinate plane easily.

Dilation steps:

SOL step one: using graph paper draw x, y axis then redraw the shape on the graph

paper (include original coordinates)

Step two: list the original coordinates

Step three: look at the scale factor and determine if the image will be bigger or

smaller than the pre-image.

Step four: multiply the scale factor by the x and y coordinates for each point

Step five: plot and list the new coordinates

Ex) Dilate ABC by a scale factor of 2

Pre-image coordinates Multiply by scale factor Image coordinates

A(-3,0) -3x2, 0x2 A’(-6,0)

B(-2,-2) -2x2, -2x2 B’(-4,-4)

C(-1,2) -1x2, 2x2 C’(-2,4)