A c o m m o n f a c t o r i s a n u m b e r t h a t i s

a f a c t o r o f t w o o r m o r e n u m b e r s .

T h e g r e a te s t o f t h e c o m m o n

f a c t o r s i s c a l l e d t h e g r e a te s t

c o m m o n f a c t o r ( G C F ) .

Greatest Common Factor

Ladder method:

F i n d t h e G C F o f 1 2 & 1 8

1 . L i s t y o u r n u m b e r s a n d c r e a te f i r s t s te p o n t h e l a d d e r .

2 . U s e a p r i m e n u m b e r t h a t w i l l d i v i d e b o t h n u m b e r s , w r i te t h e n u m b e r o n t h e l e f t .

3 . U n d e r t h e f i r s t s te p , w r i te h o w m a n y t i m e s t h e f a c t o r g o e s i n t o t h e n u m b e r s .

4 . R e p e a t s te p s 2 & 3 u n t i l t h e n u m b e r s c a n ’ t b e b r o k e n d o w n a n y m o r e .

5 . To f i n d t h e G C F , m u l t i p l yt h e p r i m e f a c t o r s .

12 182

6 93

2 3

Find the GCF of 1 6 & 2 4 using the ladder method

Find the GCF of 1 8 & 3 0using the ladder method

Remember, that a prime number is a whole number that has exactly two

factors, 1 & itself.

1

G C F : 2 x 3 = 6

16 242

8 122

4 62

2 3G C F = 8

18 302

9 153

3 5

G C F = 6

Least Common Multiple

Multiples:

A multiple of a number is the

product of a number and any whole

number.

Multiples of 3: ___, ___, ___, ___

A common multiple, is a multiple

that two numbers have in

common. The smallest of the

common multiples is the Least

Common Multiple (LCM)

To find the Least Common Multiple:

Ladder method:1 . L i s t y o u r n u m b e r s a n d

c r e a te f i r s t s te p o n t h e

l a d d e r .

2 . U s e a p r i m e n u m b e r t h a t

w i l l d i v i d e b o t h n u m b e r s ,

w r i te t h e n u m b e r o n t h e

l e f t .

3 . U n d e r t h e f i r s t s te p , w r i te

h o w m a n y t i m e s t h e

f a c t o r g o e s i n t o t h e

n u m b e r s .

4 . R e p e a t s te p s 2 & 3 u n t i l

t h e n u m b e r s c a n ’ t b e

b r o k e n d o w n a n y m o r e .

5 . To f i n d t h e L C M , m u l t i p l y a l l

o f t h e n u m b e r s o u t s i d e o f

t h e s te p s .

Find the LCM of 8 & 4

8 42

4 22

2 1

Example

Find the LCM of 3, 5, 7:

2

LCM: 2 x 2 x 2 x 1 = 8

3 51

3 5

7

7

LCM: 1 x 3 x 5 x 7 = 105

Practice: Finding GCF & LCMFind the LCM of 8 & 12

24

Find the GCF of 90 & 75

15

Fourteen boys and 21 girls will be equally divided into groups. Find the greatest

number of groups that can be created if no one is left out.

7

Find the LCM of 9& 7

63

3Answer Key

RATIOSA r a t i o i s a c o m p a r i s o n o f

t w o q u a nt i t i e s b y d i v i s i o n .

Ratios are often expressed in simplest form.

W r i te t h e r a t i o i n s i m p l e s t f o r m t h a t c o m p a r e s t h e n u m b e r o f b a s ke t b a l l s to t h e n u m b e r o f s o c c e r b a l l s .

Practice:

W r i te t h e r a t i o i n s i m p l e s t f o r m t h a t c o m p a r e s t h e n u m b e r o f s ta r s to t h e

n u m b e r o f f l o w e r s .

Several students named their favorite ice cream flavor. a. Write the ratio in simplest form that compares the

number of students who chose chocolate to the total number of students. 7:18

b. Write the ratio in simplest form that compares the number of students who chose vanilla to the number of students who chose cookies n’ cream.

1:3

F a v o r i t e f l a v o r o f i c e c r e a m

Flavor # of responses

Chocolate 7

Vanilla 2

Cookies n’ Cream

6

Mint 2

Strawberry 1

Answer Key6

3: 1

rates• A rate is a ratio comparing two quantities of different kinds of

units.

Samantha picked 45 oranges in 5 minutes

• A unit rate has a denominator of 1 unit when the rate is written as

a fraction.

• To write a rate as a unit rate, divide the numerator and the

denominator by the denominator.

Whatever you do to the top,You do to the bottom!

“Per ______” means “for every 1 unit.” Wherever you see “per ____” that’s the unit that will be your denominator.

practiceAna downloaded 35 songs in 5 minutes. How many songs

did she download per minute?

Martha is baking several loaves of bread to sell in her

bakery. She used 9 cups of water & 12 cups of flour. How

much water was used per cup of flour?

Answer Key 9

7 songs

1 minute

0.75 cups of water

1 cup of flour

Ratio Tables• A table with columns filled with pairs of numbers

that have the same ratio is called a ratio table.

• Equivalent ratios express the same relationship

between quantities

• Multiplying or dividing two related quantities by

the same number is called scaling.

Soda 1 2 3

Juice 3 6 9

Are the ratios in the table equivalent?

Cans of corn are on sale at 10 for $4. Find the cost of 15 cans.

Cans of corn 10 5 15

Cost in $ 4 2 6

1 2

Yes, all ratios will be 1/3 when simplified

Ratio Tables:Practice

To make yellow icing, you mix 6 drops of yellow food coloring with 1 cup of white icing. How much yellow food coloring should you mix with 6 cups if white icing to get the same shade?

To make cranberry jam you need 12 cups of sugar for every 16 cups of cranberries. Find the amount of sugar needed for 4 cups of cranberries

Joe mows lawns during his summer vacation to earn money. He took 14 hours last week to mow 8 lawns. At this rate, how many lawns could he mow in 49 hours?

Drops of yellow 6 30

Cups of icing 1 5

Sugar (c) 12 6 3

Cranberries (c) 16 8 4

Number of Hours 14 7 49

Number of Lawns 8 4 28

1 3

Graph Ratio TablesCoordinate Plane

1 6

• T h e c o o r d i n a te p l a n e i s

f o r m e d w h e n t w o

p e r p e n d i c u l a r n u m b e r l i n e s

i nte r s e c t a t t h e i r z e r o

p o i nt s .

• T h i s p o i nt i s c a l l e d t h e

o r i g i n .

• T h e h o r i z o nta l n u m b e r l i n e

i s c a l l e d t h e x - a x i s .

• T h e v e r t i c a l n u m b e r l i n e i s

c a l l e d t h e y - a x i s .

Label the Coordinate Plane:

Graphing Ordered Pairs

• A n o r d e r e d p a i r s u c h a s ( 2 , 3 ) i s a p a i r o f n u m b e r s

u s e d to l o c a te a p o i nt o n t h e c o o r d i n a te p l a n e .

• T h e f i r s t n u m b e r i n a n o r d e r e d p a i r i s t h e

x - c o o r d i n a te , a n d t h e s e c o n d n u m b e r i s t h e

y - c o o r d i n a te .

(4 ,6)

or i gi n

Y - a x i s

x - a x i s

x - c o o r d i n a te y - c o o r d i n a te

1 7

You can express information in a table as a set of ordered pairs. To see patterns, __________ the ordered

pairs on the coordinate plane.

T h e ta b l e s h o w s t h e

c o s t i n d o l l a r s to

c r e a te C D s o f

d i g i ta l p h oto s a t a

p h oto s h o p .

C o st to C rea te C d s

N u m b e r o f C d ’ s

x

C o st i n d o l l a r s

y1 3

2 6

3 9

O r d e re d p a i r s : ( 1 , 3 ) , ( 2 , 6 ) , ( 3 , 9 )

1 . G r a p h t h e o r d e r e d p a i r s :

2 . D e s c r i b e t h e

p a t te r n i n t h e g r a p h

1 2

1

2

3

3

4

4

5

5

6

6

7

7

8

8

9 10

9

10

P o i nt s a p p e a r i n a

s t r a i g ht l i n e , E a c h

p o i nt i s o n e u n i t

to t h e r i g ht a n d

t h r e e u n i t s u p .

C o s t i n c r e a s e s b y

$ 3 f o r e v e r y C D .

T h e r a t i o o f p a g e s to p h oto s i s 1 : 4

w h i l e t h e r a t i o f o r I v a n a i s 1 : 6 . O n t h e

g r a p h , b ot h s e t s o f p o i nt s a r e i n a

s t r a i g ht l i n e b u t I v a n a ’ s l i n e i s

s te e p e r

Practice: Graph Ratio Tables

1 8

• T w o f r i e n d s a r e m a k i n g s c r a p b o o k s . A n d r e a p l a c e s 4 p h o t o s o n e a c h p a g e o f h e r s c r a p b o o k . I v a n a p l a c e s 6 p h o t o s o n e a c h p a g e o f h e r s c r a p b o o k .

• M a k e a t a b l e f o r e a c h s c r a p b o o k t h a t s h o w s t h e t o t a l n u m b e r s o f p h o t o s p l a c e d , I f e a c h b o o k h a s 1 , 2 , 3 , o r 4 p a g e s . L i s t t h e i n f o r m a t i o n a s o r d e r e d p a i r s ( p a g e s , p h o t o s )

Andrea’s Scrapbook

Pages, x Photos, y Ordered

Pairs

1 4 (1,4)

2 8 (2,8)

3 12 (3,12)

4 16 (4,16)

Ivana’s Scrapbook

Pages, x Photos, y Ordered

Pairs

1 6 (1,6)

2 12 (2,12)

3 18 (3,18)

4 24 (4,24)

G r a p h t h e o r d e r e d p a i r s f o r e a c h f r i e n d o n t h e s a m e

c o o r d i n a t e p l a n e .

1 2

3

6

3

9

4

1 2

5

1 5

6

1 8

7

2 1242730

# of pages

# o

f P

hoto

s

H o w d o e s t h e r a t i o o f e a c h p a g e t o p h o t o s

c o m p a r e f o r e a c h p e r s o n ? H o w i s t h i s s h o w n o n t h e

g r a p h ?

Equivalent Ratios

2 1

There are different ways to determine if two ratios or rates are equivalent.

1.Use unit rates20 miles in 5 hours; 45 miles in 9 hours

20 miles5 hours

4 miles1 hour

= 45 miles9 hours

5 miles1 hour

=

Since the rates do/don’t have the same unit rate they are/are not equivalent

3 t-shirts for $21; 5 t-shirts for $35

$213 t-shirts

$71 t-shirt

= $355 t-shirts

=

Since the rates do/don’t have the same unit rate they are/are not equvalent.

2. Use Equivalent FractionsIf a unit rate is not easily found

Whatever you do to the numerator, you must do to the denominator.

3 free throws made out of 7 attempts; 9 free throws made out of 14 attempts

3 free throws7 attempts

9 free throws14 attempts

=

What you did to thenumerator is not what you did to the denominator so the rates are not equivalent.

Are these rates equivalent? Explain.

$71 t-shirt

x3

x2

Equivalent Ratios:Practice

22

Determine if each pair of rates is equivalent using the unit rate method:36 t-shirts in 3 boxes;60 t-shirts in 6 boxes.

Not equivalent

Club A raised $168 by washing 42 cars. Club B raised $152 by washing 38 cars.

equivalent

Determine if the pair of rates is equivalent using the equivalent fractions method:Selena is comparing the cost of two packages of DVD’s. A package of 6 DVD’s cost $90 and a package of 3 DVD’s cost $45. Are the rates equivalent?

equivalent

Ratio & Rate ProblemsPart-to-Part Part-to-Whole

provide the relationship between two distinct groups

provide the relationship between a particular group and

the whole population.

Label the following ratios as part-to-part of part-to-whole:

2 red flowers out of 10 flowers

Part to whole

4 boys to 6 girls

Part to Part

Identify the part and the whole of the following ratio:

2 red flowers out of 10 flowers

Part: 2 red flowers

Whole: 10 flowers

5% out of 100%

Part: 5%

Whole: 100%

2 5

You can use equations with ________________ ratios to solve ratio and rate problems.

Rockway Middle School has 300 students out. 1 out of 3 students in Ms. Martinez’s class are left handed. Use this ratio to predict how many students in the entire middle school are left-handed.

Write the ratio as a fractionPart=numerator; Whole=denominator

Highlight KEY

words

13

100300x100

x100 100 students @ Rockway Middle School are left handed

26

The Garcia’s drove 120 miles on 6 gallons of gas. At this rate, how many miles can they drive on 4 gallons of gas?

If you can’t get from one number to another, use a “MIDDLE MAN”

Middle Man: number(s) that help you get from one number to another

Milesgal

Milesgal

Middle Man

PracticeThe ratio of the number of text messages sent by Sandra to the number of text messages sent by Samantha is 3 to 4. Sandra sent 24 text messages, how many did Samantha send?

*Highlight key words*

There are 810 calories in 3 slices of pizza. How many caloriesare there in 7 slices of pizza?

1206

4023

3x2

x2 804

34

1824x6

x6

8103

18907x7

x7 567021 3

3

1890 calories in 7 slices of pizza

Samantha sent 18 text messages

Convert Measurement UnitsCustomary conversions

Type of Measure Larger Unit à Smaller Unit

Length 1 foot (ft)1 yard (yd)I mile (mi)

===

12 inches (in.)3 feet

5,280 feet

Weight 1 pound (ln)1 ton (T)

==

16 ounces (oz)2,000 pounds (lb)

Capacity 1 cup ( c )1 pint (pt)

1 quart (qt)1 gallon (gal)

====

8 fluid ounces (oz)2 cups

2 pints4 quarts

• Each relationship in the table can be written as a unit ratio.• A unit ratio is just like a unit rate, where the denominator is a 1.

q Step one: Identify the units being compared & write the relationship as a unit ratio.q Units: feet & inches

q Write the relationship as a unit ratio. (larger unit is always the denominator)q Unit ratio: 12 in

1 ftq Write the given information as a fraction

q In this case, write 20 feet as a fraction:

q Divide out common units:

q Multiply 20 x 12 in = 240 inches

20 ft1

20 ft1

12 in1 ftx

Convert 20 feet to inchesConvert Larger units to smaller units:

Convert 15 quarts to gallons

q Step one: Identify the units being compared & write the relationship as a unit ratio.q Units: quarts & gallons

q Write the relationship as a ratio. (smaller unit is always the denominator)q ratio:

q Write the given information as a fractionq In this case, write 15 quarts as a fraction:

q Divide out common units:

q Multiply/Simplify

Convert smaller units to larger units:

1 gal4 qt

15 qt1

15 qt1

1 gal4 qtx

15 gal4

= 3.75 gallons