The Divisibility RulesA number is divisible by

2 if it is even

3 if the sum of its digits are divisible by 3

4 if the number formed by its last two digits is divi sible by 4

5 if the units digit is 0 or 5

6 if it is even and the sum of its digits is divisibl e by 3(if it is divisible by 2 and 3, it is divisible by 6)

9 if the sum of its digits is divisible by 9

10 if the units digit is 0

12 if it is divisible by 3 and 4

25 if the last two digits are 00, 25, 50, or 75

Prime Numbers Factor TreesPrime numbers are natural (whole) numbers that are only divisible by themselves and 1.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, . . .

Use factor trees to show the prime factorization of composite numbers.

Composite numbers are the result of the multiplication of prime numbers.

0 and 1 are neither prime nor composite.

Example of a factor tree:

42

2 21

3 7

The prime factorization of 42:

(((( )))) (((( )))) (((( ))))42 2 3 7====

Lowest Common Multiple (LCM)

Greatest Common Factor (GCF)

To find the LCM of two numbers we write the prime factorization of each number and then use the highest power from each factorization with the factors that are common to both numbers.

To find the GCF of two numbers we write the prime factorization of each number and then use the lowest power from each factorization with the factors that are common to both numbers.

(((( ))))LCM 36,24 ====

(((( )))) (((( ))))2 236 2 3==== (((( )))) (((( ))))3

24 2 3====

comparecompare

(((( ))))23 (((( ))))3

2

(((( )))) (((( ))))232 3 72==== (((( ))))GCF 100,120 ====

(((( )))) (((( ))))2 2100 2 5==== (((( )))) (((( )))) (((( ))))3

120 2 3 5====(((( ))))22 (((( ))))5

(((( )))) (((( ))))22 5 20====

comparecompare

Order of Operations The IntegersBEMDAS or BEDMAS

BracketsExponentsMultiplyDivideAddSubtract

(((( ))))23 2 5 2 3 4+ × − − ×+ × − − ×+ × − − ×+ × − − ×(((( ))))2 42 2 33 5+ × −+ × −+ × −+ × − ××××−−−−(((( ))))23 2 5 22 1+ × − −+ × − −+ × − −+ × − −(((( ))))23 2 5 2 12+ × −+ × −+ × −+ × − −−−−

23 2 5 10+ × −+ × −+ × −+ × − −−−−2 2 5 103 + × − −+ × − −+ × − −+ × − −

2 5 109 + × − −+ × − −+ × − −+ × − −2 59 10++++ ×××× − −− −− −− −

9 010 1+ − −+ − −+ − −+ − −9 010 1++++ − −− −− −− −9 110 0++++ ++++9 110 0++++ ++++

19 10++++19 10++++

29

Positive and negative whole numbers that include zero.

. . . 3, 2, 1,0,1,2,3, . . .− − −− − −− − −− − −

Operations Fractions(((( )))) (((( ))))2 3 6====

(((( )))) (((( ))))2 3 6− = −− = −− = −− = −

(((( )))) (((( ))))2 3 6− = −− = −− = −− = −

(((( )))) (((( ))))2 3 6− − =− − =− − =− − =

8 42

==== 8 42

= −= −= −= −−−−−8 4

2−−−− = −= −= −= − 8 4

2−−−− ====−−−−

8 2 10+ =+ =+ =+ = 8 2 6− + = −− + = −− + = −− + = −

8 2 6− =− =− =− = 8 2 10− − = −− − = −− − = −− − = −

8 2− + =− + =− + =− + = 8 2− − =− − =− − =− − =8 2 6− =− =− =− = 8 2 10+ =+ =+ =+ =

8 2+ − =+ − =+ − =+ − =8 2 6− =− =− =− =

13

==== 23

====

2 5 7 7, , , and 3 6 12 9

Comparing FractionsFirst, we need to have a common denominator for all the fractions we want to compare.

33

====

(((( ))))LCM 3,6,9,12 ==== 36

Rename with common denominators:

7 4 289 4 36

×××× ====××××5 6 306 6 36

×××× ====××××7 3 21

12 3 36×××× ====××××

In increasing order:

2 12 243 12 36

×××× ====××××7

1256

79

23

712

23

79

56

Adding Fractions Subtracting FractionsGet a common denominator for the fractions being added and then add the numerators ( do not add the denominators ).

3 58 12

++++

924

1024

++++

9 1024++++

1924

Get a common denominator for the fractions being subtracted then subtract the numerators ( do not subtract the denominators ).

52 2160 60

−−−−

52 2160−−−−

3160

13 715 20

−−−−

38

512

++++

33

××××××××

22

××××××××

13 715

30

44 32

−−−−× ×× ×× ×× ×× ×× ×× ×× ×

1315

720

−−−−−−−−

Powers of Numbers23 ==== (((( )))) (((( ))))3 3 ==== 9

23− =− =− =− = (((( )))) (((( ))))3 3− =− =− =− = 9−−−−

(((( ))))23− =− =− =− = (((( )))) (((( ))))3 3− − =− − =− − =− − = 9

(((( )))) (((( )))) (((( )))) (((( ))))3 3 3 3 ==== 4313 ==== 315 ==== 503 ==== 105 ==== 1

Square Roots

3 cm

(((( ))))2area 3 cm====

29 cm====

length of sides area====29 cm====

3 cm====

0 0==== 1 1==== 4 2==== 9 3====

16 4==== 25 5==== 36 6====

49 7==== 64 8==== 81 9====

100 10==== 121 11==== 144 12====

33

÷÷÷÷÷÷÷÷

Multiplying Fractions Dividing FractionsMultiply the top by the top and the bottom by the bottom. Reduce the final result if possible.

3 105 21

××××

3 105 21

××××××××30

105

301

5505

÷÷÷÷÷÷÷÷

621

====

Flip the second fraction, then multiply the top by the top and the bottom by the bottom. Reduce the final result if possible.

3 64 11

÷÷÷÷

34

116

××××

(((( )))) (((( ))))(((( )))) (((( ))))3 114 6

3324

118

====

3324

====

33

÷÷÷÷÷÷÷÷

621

27

====

Mixed Fractions and Improper Fractions

Mixed Fraction: 2 34

Improper Fraction:

Whole Number

Proper Fraction

175

Numerator is larger than the denominator

Converting between mixed and improper fraction:

2 34

==== (((( )))) (((( ))))2 44

3++++==== 8 3

4++++ ==== 11

4mixed

fractionimproper fraction

175

15 25

==== ++++ 15 25 5

= += += += + 235

= += += += + 235

====

mixed fraction

improper fraction

33

÷÷÷÷÷÷÷÷

Adding & Subtracting Mixed FractionsConvert the mixed fractions into improper fractions. Get a common denominator and add the numerators.

Multiplying & Dividing Mixed FractionsConvert the mixed fractions into improper fractions. Perform the multiplication or division the same way as you would with proper fractions.

3 12 3 ?4 6

+ =+ =+ =+ =

(((( )))) (((( )))) (((( )))) (((( ))))2 4 3 3 6 14 6

+ ++ ++ ++ +++++

8 3 18 14 6+ ++ ++ ++ +++++

11 194 6

++++

11 194 6

3 23 2

++++× ×× ×× ×× ×× ×× ×× ×× ×

33 3812 12

++++ 7112

====

2 11 4 ?3 2

× =× =× =× =

(((( )))) (((( )))) (((( )))) (((( ))))1 3 2 4 2 13 2

+ ++ ++ ++ +××××

3 2 8 13 2+ ++ ++ ++ +××××

5 93 2

××××

(((( )))) (((( ))))(((( )))) (((( ))))5 93 2

456

==== 152

====

Decimals1 2 3 4 . 5 6 7

Converting Decimals to Fractions

thousandshundreds

tensones

tenthshundredths

thousandths

Rounding DecimalsRound to the nearest tenth:

1 7. 3 5 2

tenths

1 7. 3 5 2

1 7. 4

We will round the 3 up to a 4.

Hundredths is greater than or equal to 5.

0.3

tenths

03

1==== 0.67

hundredths

067

10====

Special Fractions and Repeating Decimals13

==== 0.333... ==== 0.3

23

==== 0.666... ==== 0.6

19

==== 0.111... ==== 0.1

29

==== 0.222... ==== 0.2

49

==== 0.444... ==== 0.4

59

==== 0.555... ==== 0.5

Converting Special Fractions to Decimals34

==== 34

2525

×××××××× ==== 75

100==== 0.75

25

==== 2 225

×××××××× ==== 4

10==== 0.4

58

==== 58

125125

×××××××× ==== 625

1000==== 0.625

Note that if we can make the denominator a power of 10, it is easy to convert the fraction into a decimal.

Decimals, Fractions and Percents

25%

percent

25100

====

fraction

25100

====

0.25====

decimal

Note that 2510

2520 5

÷÷÷÷÷÷÷÷ ==== 1

4

Arrange the List in Increasing Order

29%, , and 0.1225

9% ==== 910

==== 0.09

225

==== 8100

====2 4425

×××××××× ==== 0.08

0.08 0.09 0.12

2 , 9%, 0.1225

Percents and Fractions in Word Problems

20% of 45

cross-multiply

20100 45

x====

(((( )))) (((( )))) (((( )))) (((( ))))20 45 100 x====

(((( )))) (((( )))) (((( ))))900 100 x====

900100

x====

9x ====

2020%100

====

20 $45100 1

(((( )))) (((( ))))20 45100

900100

9

multiplyor

Percents and Fractions in Word Problems

A shirt is regularly sold for $24. This week it is on sale for 1/3 off. How much is the shirt this week?

1 $243 1

====

$243

Multiply to get the discount

$8====

Sale Price $24 $8= −= −= −= − $16====

The Metric System Triangles

Km Hm Dam m dm cm mm÷10

×10

÷10÷10÷10÷10÷10

×10 ×10 ×10 ×10 ×10

15.3748 Km m? ====

15.3748 Km m15 374.8====

85 cm m? ====

08 .5 cm m85 ====

height

base

height

base

height

basea

b

90 180ba + + ° =+ + ° =+ + ° =+ + ° =∠∠∠∠ ∠∠∠∠ °°°°180 90ba ++++ ∠∠∠∠ ° −° −° −° −∠∠∠∠ = °= °= °= °

90ba ∠∠∠∠∠∠∠∠ + = °+ = °+ = °+ = °

b

ca

180ca b ∠∠∠∠∠∠∠∠ + =+ =+ =+ =∠∠∠∠+ °+ °+ °+ °

Types of Triangles

equilateral

scalene

right triangleisosceles

isosceles right triangle

Perimeter and Area of Polygons

d a b

c

c a

b

a a

a

b

a

b c

d

a

a

b

Scalene triangle

P ba c= + += + += + += + +(((( )))) (((( ))))

2A

d c====

Isosceles triangle

P a a b= + += + += + += + +

(((( )))) (((( ))))2

Ac b

====

2P ba= += += += +

Equilateral triangle

P a a a= + += + += + += + +

(((( )))) (((( ))))2

Ab a

====

3P a====

Quadrilateral

b da cP = + += + += + += + + ++++

Square

a a a aP = + + += + + += + + += + + +

(((( )))) (((( ))))A a a====4P a====

Rectangleb ba aP = + + += + + += + + += + + +

(((( )))) (((( ))))A ba====2 2P a b= += += += +

b

ac

d

b

a

a c

d

b

h

Parallelogram

Rhombusa a a aP = + + += + + += + + += + + +4P a====

Trapezoid

b ba aP = + + += + + += + + += + + +

(((( )))) (((( ))))A cb====2 2P a b= += += += +

(((( )))) (((( ))))2

Ad b

====

b da cP = + += + += + += + + ++++(((( ))))

2A

b hd+ ×+ ×+ ×+ ×====

Perimeter and Area of Polygons

P an= ×= ×= ×= ×

Perimeter of Regular Polygons

a

a

a

perimeterlength of the sidenumber of sides

P

na

============3P a= ×= ×= ×= ×

4P a= ×= ×= ×= ×

5P a= ×= ×= ×= ×

Note that if a polygon is a regular polygon, all of its sides are of equal length and all of its interior angles are equal.

Classifying Quadrilaterals

Quadrilateral:

Any polygon that has 4 sides. Square, rectangle, parallelogram, rhombus, right trapezoid and isosceles trapezoid.

isoscelestrapezoid

righttrapezoid

Parallelogram:

Any quadrilateral with two pairs of opposing parallel sides. (Square, rectangle, parallelogram and rhombus.)

Parallel Lines and Equal Anglesvertically opposite angles

corresponding angles

alternate interior angles

alternate exterior angles

a b∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠

a b∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠

a b∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠

a b∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠

ba

b

a

b

a

b

a

Similar FiguresTwo polygons are similar when both shapes have the same interior angles and have proportional matching sides .

A

B

C

5

7

860°

82° 38°D

E

F

10

14

16

60°

82° 38°

ABC DEF∆ ∆∆ ∆∆ ∆∆ ∆∼

A D∠ = ∠∠ = ∠∠ = ∠∠ = ∠B E∠ = ∠∠ = ∠∠ = ∠∠ = ∠C F∠ = ∠∠ = ∠∠ = ∠∠ = ∠

5 110 2

mABmDE

= == == == = 8 116 2

mBCmEF

= == == == = 7 114 2

mACmDF

= == == == =

SimilarityIf rectangle ABCD is similar to rectangle EFGH, what is the length of side FG?

A

B C

D

2 cm

4 cm

E

F G

H

10 cm

x

210

4x

====

(((( )))) (((( )))) (((( )))) (((( ))))2 10 4x ====

2 40x ====402

x ====

20x ====20 cmmFG ====

Parallel Lines and Similar Triangles

A

C

E

B D

A

C

E

B D

CACE BCD∆ ∆∆ ∆∆ ∆∆ ∆∼

A B∠ = ∠∠ = ∠∠ = ∠∠ = ∠ C C∠ = ∠∠ = ∠∠ = ∠∠ = ∠ E D∠ = ∠∠ = ∠∠ = ∠∠ = ∠

Calculating the MeanMean = Average

2, 3, 5, 10, 13, 18

2 3 5 10 13Avera6

8ge 1+ + + + ++ + + + ++ + + + ++ + + + +====

Average651====

Average 8.5====

Calculate the mean of the following six data points.

Circle GraphMake a circle graph using the results in the bar graph.

Determine the number of degrees for each color.

Red Green

12 603

3x==== °°°° 12 60

53

x==== °°°° 12 604

3x==== °°°°

(((( )))) (((( ))))3 360 12x====1080 12x====1080

12x==== 90= °= °= °= °

(((( )))) (((( ))))5 360 12x====1800 12x====1800

12x==== 150= °= °= °= °

(((( )))) (((( ))))4 360 12x====1440 12x====1440

12x==== 120= °= °= °= °

Blue

Bar Graph

6543210

# of

stu

dent

s

Favorite Colorblue red green

green

blue 3 students5 students4 students

total 12 students

red====

Circle Graph

Blue 90= °= °= °= °Red

Green 120= °= °= °= °

150= °= °= °= °

90°°°°

90150240

°°°°+ °+ °+ °+ °

°°°°

Trace a circle. Mark the divisions on the perimeter.

Draw in the lines.Label the sections and give it a title.

Our Favorite Colors

Geometric Transformations

Translation RotationReflection

tA

B

C

A'

B'

C'

4

2

A

B

C

A'

B'

C'

A

B

C

90R °°°°

A'

B'

C'

Finding the Rule of an Equation for a Table of Values

x 1 2 3 4

y 8 11 14 17

y mx b= += += += ++1 +1 +1

+3 +3 +3

31

y x b= += += += +

3y x b= += += += +

3 5y x= += += += +

3

1

is the rulex 1 2 3 4

y 8 11 14 17

x

y

-1

-3

0

5

Change in y

Change in x cc

hha

annge

ge i

nin

yx

What is ywhen x = 0?