Factors and Multiples - WordPress.com · Adult: 2,100 30 𝑖 or 70 ... In a recent year, Joey...

L E S S O N 1 - 1 P A R T 1

Factors and Multiples

Vocabulary

Greatest Common Factor (GCF) – the greatest number that is a factor of two or more numbers

In other words, ask “what is the highest value these numbers have in common?”

Example 1

1. There are one-slice servings of three types of cake on a table. Each row has an equal number of servings and only one type of cake. What is the

number of servings in each row?

To solve this problem, use common

factors.

Factors of 10: 1,2,5,10

Factors of 15: 1, 3, 5, 15

Factors of 20: 1, 2, 4, 5, 10, 20

The GCF of 10, 15, and 20 is 5. So, the greatest number of pieces of cake that can be placed in each row is 5.

greatest

Got It?

a. Lana earned $49 on Friday, $42 on Saturday, and $21 on Sunday selling bracelets. She sold each bracelet for the same amount. What is the most she could have charged for each bracelet?

Example 2

Find the GCF of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

GCF: 6.

Got It?

Find the GCF for each problem.

b. 12, 66

c. 18, 30, 24

d. 20, 32, 36

A GCF of 1?

Can a GCF ever be 1?

Let’s take 13 and 17…

13 = 13 x 1 and 17 = 17 x 1

They only have 1 in common…

L E S S O N 1 - 1 P A R T 2

Factors and Multiples

Vocabulary

Least Common Multiple (LCM) – the least number that is a multiple of two or more whole numbers

Example 3

Find the LCM of 2 and 3

Method 1 – Use a number line

Method 2 – Use an organized list.

Multiples of 2: 2, 4, 6, 8, 10, 12

Multiples of 3: 3, 6, 9, 12, 15

LCM: 6

Example 4

Find the LCM of 14 and 21.

Method 3 –Write the prime factorization of each number.

14 21

7 x 2 7 x 3

Multiply using each common prime factor only once.

So, the LCM is 7 x 2 x 3 or 42.

Got It? 3 & 4

Find the Least Common Multiple for each problem.

e. 2, 6

f. 4, 5, 10

g. 3, 5, 7

Example 5

Ernesto has painting class every 2 weeks. Kamala has a pottery class every 5 weeks. Ernesto and Kamala met at the art building for class this week. How many weeks will it be until they see each other again?

2: 2, 4, 6, 8, 10, 12, 14,…

5: 5, 10, 15, 20, 25, 30,…

LCM: 10

So, Ernesto and Kamala will see each other again in 10 weeks.

In Conclusion:

What operation do you think of when finding the GCF?

What operation do you think of when finding the LCM?

L E S S O N 1 - 2

Ratios

Vocabulary

ratio - a comparison of two quantities (amounts) by division.

Example: A ratio of 2 red paper clips to 6 blue paper

clips can be written in 3 ways.

2 to 6 2:6 26

Just like fractions, ratios are often expressed in the simplest form.

Example 1

Write the ratio in simplest form that compares the number of red paper clips to the number of blue paper clips. Then explain its meaning. Write the ratio as a fraction. Then simplify.

26

= 13

The ratio is 1/3, or 1 to 3, or 1:3. “Explain its meaning” = For every 1 red paper clip there are 3 blue paper clips.

Got It? 1

Write the ratio in simplest form that compares the number of suns to the number of moons. Then explain its meaning.

Example 2

Several students named their favorite flavor of gum. Write the ratio that compares the number who chose fruit to the total number of students.

321

= 17

The ratio is 17

, 1 to 7, or 1:7.

So, 1 out of every 7 students preferred fruit-flavored gum.

Fruit: 3 Total: 9 + 8 + 3 + 1 = 21

Example 3

Monday’s yogurt sales are recorded in the table. Write the ratio that compares the sales of strawberry yogurt to the total sales. Then explain it’s meaning.

Strawberry: 8

Total: 3 + 6 +7 + 8 = 24

What is the ratio? 8

24 =

1

3

So, ____ out of every ____ yogurt cups sold were strawberry.

Got It? 2 & 3

A pet store sold the animals listed in the table in one week. Write the ratio of cats to pets sold that week. Then explain its meaning.

Example 4

Katy wants to divide her 30 flowers into two groups, so that the ratio is 2 to 3.

Step 1: Use a diagram to show a ratio of 2 to 3.

Step 2: There are 5 equal sections. So, each section represents 30 ÷ 5 or 6 flowers.

There are 12 flowers in one group and 18 in the other.

6 6

6 6 6

Got It? 4

Divide 28 cans of soda into two groups so the ratio is 3 to 4.

1.

2. 28 ÷ how many boxes = _______

3. Ratio = blue: red or ______:______

Pictures

Draw at least two pictures of 1:4.

1. Start with 1:4…that’s your first picture.

Pictures

Draw at least two pictures of 1:4.

2. Copy 1:4…and if you add 1 triangle, you add 4 more clouds.

L E S S O N 1 - 3

Rates

Vocabulary

rate - ratio comparing two quantities (amounts) of different kinds of units.

unit rate – a rate with a denominator of 1

Example: Desiree typed a text message with 15 characters in 5 seconds. Ratio Rate Unit Rate

15:5 15 characters

5 seconds

3 characters1 second

Vocabulary Continued

unit price - the cost per unit (1 item).

Example:

Rate Unit price

$36

4 𝑡𝑖𝑐𝑘𝑒𝑡𝑠

$9

1 𝑡𝑖𝑐𝑘𝑒𝑡

What is the similar between a “unit rate” and a “unit price”?

Things to consider:

1. Rate is a fraction.

2. 𝐹𝑖𝑟𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 (𝑤𝑖𝑡ℎ 𝑙𝑎𝑏𝑒𝑙)

𝑆𝑒𝑐𝑜𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 (𝑤𝑖𝑡ℎ 𝑙𝑎𝑏𝑒𝑙)

EXCEPTIONS:

**Time ALWAYS goes on the bottom**

AND

**Money AWAYS goes on the top**

Example 1

Samantha picked 45 oranges in 5 minutes. Write this rate as a unit rate.

Write the rate as a fraction. 45 𝑜𝑟𝑎𝑛𝑔𝑒𝑠

5 𝑚𝑖𝑛𝑢𝑡𝑒𝑠

Divide the numerator and denominator by the denominator.

45 𝑜𝑟𝑎𝑛𝑔𝑒𝑠 ÷ 5 = 9 𝑜𝑟𝑎𝑛𝑔𝑒𝑠

5 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 ÷ 5 = 1 𝑚𝑖𝑛𝑢𝑡𝑒

So, the unit rate is 9 𝑜𝑟𝑎𝑛𝑔𝑒𝑠

1 𝑚𝑖𝑛𝑢𝑡𝑒 or 9 oranges per minute.

Example 2

The Australian dragonfly can travel 18 miles in 30 minutes. How far can the dragonfly travel in 1 minute?

Write the rate as a fraction. 18 𝑚𝑖𝑙𝑒𝑠

30 𝑚𝑖𝑛𝑢𝑡𝑒𝑠

Since you cannot divide the numerator by the denominator, simplify the fraction.

Example 2 Continued

Simplify the fraction. 18 𝑚𝑖𝑙𝑒𝑠 ÷ 6 = 3 𝑚𝑖𝑙𝑒𝑠

30 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 ÷ 6 = 5 𝑚𝑖𝑛𝑢𝑡𝑒𝑠

It can be written as 3 𝑚𝑖𝑙𝑒𝑠

5 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 or as a unit rate of

3

5

mile to 1 minute.

Answer: 3

5 mile per minute.

Got It? 1 & 2

a. Ana downloaded 35 songs in 5 minutes. How many songs did she download per minute? b. Jonathan is baking several loaves of bread to sell in his bakery. He used 9 cups of water and 12 cups of whole wheat flour. How much water was used per cup of flour?

Example 3

An adult’s heart beats about 2,100 times every 30 minutes. A baby’s heart beats about 2,600 times every 20 minutes. How many more beats does a baby’s heart beat in 60 minutes than an adult’s heart?

Step 1: Find the unit rates.

Adult: 2,100 𝑏𝑒𝑎𝑡𝑠

30 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 or

70 𝑏𝑒𝑎𝑡𝑠

1 𝑚𝑖𝑛𝑢𝑡𝑒

Baby: 2,600 𝑏𝑒𝑎𝑡𝑠

20 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 or

130 𝑏𝑒𝑎𝑡𝑠

1 𝑚𝑖𝑛𝑢𝑡𝑒

Example 3 Continued

Step 2: Using the unit rates for each, determine the number of beats in 60 minutes.

Adult: 70 X 60 = 4,200 beats

Baby: 130 X 60 = 7,800 beats

Step 3: Find the difference.

7,800 – 4,200 = 3,600

Answer: 3,600 more times

Got It? 3

c. A hummingbird’s heart rate while resting is about 7,500 beats every 30 minutes. How many more beats does a hummingbird’s heart beat in 60 minutes than a human baby’s heart?

Example 4 – Unit Price

Four potted plants cost $88. What is the price per plant?

Write the rate as a fraction. $88

4 𝑝𝑙𝑎𝑛𝑡𝑠

Simplify. $22

1 𝑝𝑙𝑎𝑛𝑡

So, the price per potted plant is $22.00.

LET’S GO OVER #10…

Homework Question

L E S S O N 1 - 4

Ratio Tables

Vocabulary

ratio table – a table where the columns are filled with pairs of numbers that have the same ratio.

equivalent ratios - express the same relationship between quantities.

scaling - multiplying or dividing two related quantities by the same number

Example 1

To make yellow frosting, you mix 6 drops of yellow food coloring with 1 cup of white frosting. How much yellow food coloring should you mix with 5 cups of white icing to get the same shade?

Use a ratio table.

Think: What did I multiply 1

by to get to 5? 5

Multiply 6 by 5 to fill in the table

Answer: 30 drops

Example 2

In a recent year, Joey Chestnut won a hot dog eating competition by eating nearly 66 hot dogs in 12 minutes. If he ate at a constant rate, determine how many hot dogs he ate every 2 minutes.

Divide each quantity by one or more common factors until you reach a quantity of 2 minutes.

Answer: about 11 hot dogs

Got It? 1& 2

a. A patient receives 1 liter of IV fluids every 8 hours. At that rate, find how many hours it will take to receive 4 liters of IV fluids.

b. 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.

Example 3

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

There is no whole number by which you can multiply 10 to get 15. So, scale back to 5 and then scale forward to 15.

First divide by 2, then multiply by 3.

In other words, scale back, then scale forward.

So, 15 cans of corn would cost $6.

Example 4

Joe mows lawns during the 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?

Is there a whole number by which you can multiply 14 to get to 49?

NO

So, scale back to 7,

then scale forward to 49.

Answer: 28 lawns

Got It? 3 & 4

A child’s height measures 105 centimeters. Estimate her height in inches.

Example 5

On her vacation, Leya exchanged $50 American and received $60 Canadian. Use a ratio table to find how many Canadian dollars she would receive for $20 American.

Set up a ratio table. Use scaling to find the desired quantity.

Answer:$24 Canadian

Got It? 5

Lamika buys 12 packs of juice boxes that are on sale and pays a total of $48. Use a ratio table to determine how much Lamika will pay to buy 8 more packs of juice.

Explain #6 on the hw

L E S S O N 1 - 5

Graph Ratio Tables

Vocabulary

Vocabulary

coordinate plane – 2 number lines crossing

origin – point (0,0)

x-axis – the horizontal number line

y-axis – the vertical number line

ordered pair - a pair of numbers used to locate a point on the coordinate plane

x-coordinate – the 1st # in an ordered pair

y-coordinate – the 2nd # in an ordered pair

graph – to place a dot on a coordinate plane at an ordered pair

Example 1 - Table

The table shows the cost in dollars to create CDs of digital photos at a photo shop. The table also shows this information as ordered pairs (number of CDs, cost in dollars).

Graph the ordered pairs.

Example 1- Graph

Start at the origin. Use the x-coordinate and move along the x-axis. Then use the y-coordinate and move along the y-axis.

Draw a dot at each point.

Example 2 - Notes

Describe the pattern in the graph.

1. Do the points make a line or

curve?

The points appear in a line.

2. How much does it go up/down

each time?

The cost increases by $3 for every CD.

Got It?

The table shows Gloria’s earnings for 1,2, and 3 hours. The table also lists this information as ordered pairs (hours, earnings).

a. Graph the ordered pairs.

b. Describe the pattern in the graph.

Got It?

1. Do the points make a line or curve? 2. How much does it go up/down each time?

Example 3

Two friends are making scrapbooks. Renee places 4 photos on each page of her scrapbook. Gina places 6 photos on each page of her scrapbook.

Make a table for each scrapbook that shows the total number of photos placed, if each book as 1, 2, 3, or 4 pages. List the information as ordered pairs (pages, photos).

Example 4 - Notes

Graph the ordered pairs for each friend on the same coordinate plane.

1. Which one is places more photos per page?

Gina 2. Which graph is steeper?

Gina

Need More Practice? (optional)

Two friends are each saving money in their bank accounts. Marcus saves $10 each week while David saves $15 each week.

a. Make a table for each friend that shows the total amount saved for 1, 2, 3, or 4 weeks. List the information as ordered pairs (weeks, total $ saved).

b. Graph the ordered pairs for each friend on the same coordinate plane.

c. How do the ratios of Marcus’s savings and David’s savings compare? How is this shown on the graph?

Need More Practice? (optional)

L E S S O N 1 - 6

Equivalent ratios

Example 1

Is this pair of rates equivalent? Explain by showing work.

20 miles in 5 hours; 45 miles in 9 hours.

1. Write each rate as a fraction.

𝟐𝟎 𝒎𝒊𝒍𝒆𝒔

𝟓 𝒉𝒐𝒖𝒓𝒔

𝟒𝟓 𝒎𝒊𝒍𝒆𝒔

𝟗 𝒉𝒐𝒖𝒓𝒔

2. Then find the unit rates.

𝟐𝟎 𝒎𝒊𝒍𝒆𝒔

𝟓 𝒉𝒐𝒖𝒓𝒔 =

𝟒 𝒎𝒊𝒍𝒆𝒔

𝟏 𝒉𝒐𝒖𝒓

𝟒𝟓 𝒎𝒊𝒍𝒆𝒔

𝟗 𝒉𝒐𝒖𝒓𝒔 =

𝟓 𝒎𝒊𝒍𝒆𝒔

𝟏 𝒉𝒐𝒖𝒓

3. Explain.

Since the rates do not have the same unit rate, they are not equivalent.

Example 2

Is this pair of rates equivalent? Explain by showing work.

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

1. Write each rate as a fraction.

$𝟐𝟏

𝟑 𝒕−𝒔𝒉𝒊𝒓𝒕𝒔

$𝟑𝟓

𝟓 𝒕−𝒔𝒉𝒊𝒓𝒕𝒔

2. Find its unit rate.

$𝟐𝟏

𝟑 𝒕−𝒔𝒉𝒊𝒓𝒕𝒔=

$𝟕

𝟏 𝒕−𝒔𝒉𝒊𝒓𝒕

$𝟑𝟓

𝟓 𝒕−𝒔𝒉𝒊𝒓𝒕𝒔 =

$𝟕

𝟏 𝒕−𝒔𝒉𝒊𝒓𝒕

3. Explain.

Since they have the same unit rate, they are equivalent.

Got It?

Determine if each pair of rates is equivalent. Explain your reasoning.

a. 36 T-shirts in 3 boxes; 60 T-shirts in 6 boxes

b. 42 flowers in 7 vases; 54 flowers in 9 vases

Example 3

Felisa read the first 60 pages of a book in 3 days. She read the last 90 pages in 6 days. Are these reading rates equivalent? Explain by showing work.

Find the unit rates:

𝟔𝟎 𝒑𝒂𝒈𝒆𝒔

𝟑 𝒅𝒂𝒚𝒔 =

𝟐𝟎 𝒑𝒂𝒈𝒆𝒔

𝟏 𝒅𝒂𝒚

𝟗𝟎 𝒑𝒂𝒈𝒆𝒔

𝟔 𝒅𝒂𝒚𝒔 =

𝟏𝟓 𝒑𝒂𝒈𝒆𝒔

𝟏 𝒅𝒂𝒚

Since the rates do not have the same unit rate, they are not equivalent.

Got It?

Marcia made 10 bracelets for 5 friends. Jen made 12 bracelets for 4 friends. Are these rates equivalent? Explain your reasoning.

Example 4

If you can’t find unit rate, use equivalent fractions.

Determine if the pair of ratios is equivalent. Explain by showing work.

3 baskets made out of 7 attempts; 9 baskets made out of 14 attempts

1. Write each ratio as a fraction.

𝟑 𝒇𝒓𝒆𝒆 𝒕𝒉𝒓𝒐𝒘𝒔

𝟕 𝒂𝒕𝒕𝒆𝒎𝒑𝒕𝒔

𝟗 𝒇𝒓𝒆𝒆 𝒕𝒉𝒓𝒐𝒘𝒔

𝟏𝟒 𝒂𝒕𝒕𝒆𝒎𝒑𝒕𝒔

2. Make the denominator the same.

𝟑 𝒇𝒓𝒆𝒆 𝒕𝒉𝒓𝒐𝒘𝒔

𝟕 𝒂𝒕𝒕𝒆𝒎𝒑𝒕𝒔 =

𝟔 𝒇𝒓𝒆𝒆 𝒕𝒉𝒓𝒐𝒘𝒔

𝟏𝟒 𝒂𝒕𝒕𝒆𝒎𝒑𝒕𝒔

𝟗 𝒇𝒓𝒆𝒆 𝒕𝒉𝒓𝒐𝒘𝒔

𝟏𝟒 𝒂𝒕𝒕𝒆𝒎𝒑𝒕𝒔

3. Explain.

Since the fractions are not equivalent, the ratios are not equivalent.

Got It?

Mrs. Jeffries has 12 girls out of 16 students on the Student Council. The Earth Day Committee has 4 girls out of 8 students. Are the ratios equivalent? Explain by showing work.

L E S S O N 1 - 7

Ratio and Rate Problems

Example 1

Heritage Middle School has 150 students. Two out of three students in Mrs. Mason’s class gel their hair. Use this ratio to predict how many students in the entire middle school use gel in their hair.

Method 1 – Use equivalent fractions.

2

3 =

?

150

𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑤ℎ𝑜 𝑝𝑟𝑒𝑓𝑒𝑟 𝑔𝑒𝑙

𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠

Example 1- Continued

Method 2 – Use a bar diagram.

How many students will fill in each section, if there are 150 total students?

Answer: 100 students

Got It?

In a survey, four out of five people preferred creamy over chunky peanut butter. There are 120 people shopping at the grocery store. Use the survey to predict how many people in the store would prefer creamy peanut butter.

Example 2

The ratio of the number of text messages sent by Lucas to the number of text messages sent by his sister is 3 to 4. Lucas sent 18 text messages. How many text messages did his sister send?

Method 1 – Use equivalent fractions.

3

4 =

18

? (Make sure that the numerators of

both fractions are for Lucas)

Answer: 24 text messages

Example 2 - Continued

Method 2 – Use a bar diagram.

How many text messages are in each section? How many total did Lucas’s sister send?

Answer: 24 messages

Got It?

A survey found that 12 out of every 15 people in the U.S. prefer eating at a restaurant over cooking at home. If 400 people selected eating at a restaurant on the survey, how many people took the survey?

Example 3

The Millers drove 105 miles on 4 gallons of gas. At this rate, how many miles can they drive on 6 gallons of gas?

Got It?

There are 810 Calories in 3 scoops of vanilla ice cream. How many Calories are there in 7 scoops of ice cream?

Example 4

Jeremy drove his motorcycle 120 miles in 3 hours. At this rate, how many miles can he drive in 5 hours? At what rate did he drive his motorcycle?

𝟏𝟐𝟎 𝒎𝒊𝒍𝒆𝒔

𝟑 𝒉𝒐𝒖𝒓𝒔 =

? 𝒎𝒊𝒍𝒆𝒔

𝟏 𝒉𝒐𝒖𝒓 Divide both by 3 to find the

unit rate.

𝟒𝟎 𝒎𝒊𝒍𝒆𝒔

𝟏 𝒉𝒐𝒖𝒓 x 5 hours = 200 miles

So, Jeremy can drive 200 miles in 5 hours driving at a rate of 40 miles per hour.

Got It?

While resting, a human takes in about 5 liters of air in 30 seconds. At this rate, how many liters of air does he take in during 150 seconds?