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6th Grade

Factors and Multiple

2015-10-20

www.njctl.org

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Factors and MultiplesClick on the topic to go to that section

· Glossary & Standards

· Greatest Common Factor· Least Common Multiple· GCF and LCM Word Problems

· Divisibility Rules for 3 & 9· Even and Odd Numbers

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Even and Odd Numbers

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Warm-Up ExerciseThink about the following questions and write your answers in your notes.

1) What is an even number?

2) List some examples of even numbers.

3) What is an odd number?

4) List some examples of odd numbers.

Derived from

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What happens when

we add two even numbers? Will we

always get an even number?

What do you think?

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Drag the paw prints into the box to model 6 + 8

+

Circle pairs of paw prints to determine if any of the paw prints are left over.

Will the sum be even or odd every time two even numbers are added together? Why or why not?

Adding Even Numbers

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Drag the paw prints into the box to model 9 + 5

+

Circle pairs of paw prints to determine if any of the paw prints are left over.

Will the sum be even or odd every time two odd numbers are added together? Why or why not?

Adding Odd Numbers

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Drag the paw prints into the box to model 7 + 8

+

Circle pairs of paw prints to determine if any of the paw prints are left over.

Will the sum be even or odd every time an odd and even number are added together? Why or why not?

If the first addend was even and the second was odd, then would your answer change? Why or why not?

Adding Odd and Even Numbers

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1 The product of two even numbers is even.

TrueFalse

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Explain your answer.

2 The product of two odd numbers is

A oddB even

Multiplication is repeated addition. If you add an odd number over and over, then the sum will switch between even and odd. Since you are adding the number an odd number of times, your product will be odd.

Click to Reveal

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3 The product of 13 x 8 is

A oddB even

Explain your answer.

13 x 8 is equivalent to saying 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13.Since you are adding it an even number of times, the product will be even.

Click to Reveal

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4 The sum of 32,877 + 14,521 is

A oddB even

Explain your answer.

If you model the numbers using dots and circle all the pairs, the single dots leftover from each number will create a pair and none will be leftover making the sum an even number.

Click to Reveal

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5 The product of 12 x 9 is

A oddB even

Explain your answer.

12 x 9 is equivalent to 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12.No matter how many times you add 12, since it is even the sum will always be even.

Click to Reveal

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6 The sum of 8,972 + 1,999 is

A oddB even

Explain your answer.

If you model the problem using dots and circle all the pairs, then there will be one dot leftover since one of the addends is odd.

Click to Reveal

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7 The sum of 9 + 10 + 11 + 12 + 13 is

A oddB even

Explain your answer.

The first two addends will result in an odd number. By adding another odd number, the sum is even. Adding an even number will result in an even number. Since the last addend is odd, the final answer will be odd.

Click to Reveal

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8 The product of 250 x 19 is

A oddB even

Explain your answer.

The product of an odd and even number will always result in an even number. Click to Reveal

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9 The product of 15 x 0 is

A oddB even

Explain your answer.

0 is an even number and the product of any even number and odd number is always even.Click to Reveal

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Divisibility Rules for 3 and 9

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Below is a list of numbers. Drag each number in the circle(s) that is a factor of the number. You may place some numbers in more than one circle.

2

8

4

10

5

24 36 80 115 214 360 975 4,678 29,785 414,940

Derived from

Let's review!

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2: If and only if its last digit is 0, 2, 4, 6, or 8.

4: If and only if its last two digits are a number divisible by 4.

5: If and only if its last digit is 0 or 5.

8: If and only if its last three digits are a number divisible by 8.

10: If and only if its last digit is 0.

Divisibility Rules

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Divisibility Rule for 3What factor do the numbers 12, 15, 27, and 66 have in common?

They are all divisible by 3.

Now, take each of those numbers and calculate the sum of its digits.

12 1 + 2 = 3

15 ________

27 ________ 66 ________

What do all these sums have in common?

They are all divisible by 3!Click

Click

A number is divisible by 3if the sum of the number's digits is divisible by 3.Click

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Divisibility Rule for 9What factor do the numbers 18, 27, 45, and 99 have in common?

They are all divisible by 9.

Now, take each of those numbers and calculate the sum of its digits.

18 1 + 8 = 9

27 ________

45 ________ 99 ________

What do all these sums have in common?

They are all divisible by 9!Click

Click

A number is divisible by 9if the sum of the number's digits is divisible by 9.Click

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Check if the numbers in the chart are divisible by 3 or 9.Put a check mark in the box in the correct column.

Divisible by 3 Divisible by 9

228

531

735

1,476

Try these!

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10 468 is divisible by: (choose all that apply)

A 2B 3C 4D 5E 8F 9G 10

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11 Is any number divisible by 9 also divisible by 3? Explain.

YesNo

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12 Is 135 divisible by 3?

YesNo

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13 Any number divisible by 3 is also divisible by 9.

TrueFalse

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14 The number 129 is divisible by 9.

TrueFalse

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15 Is 24,981 divisible by 3?

If it is, type the quotient. If it is not, type 00.

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1. Make a table listing all the possible first moves, proper factors, your score and your partner's score. Here's an example:

2. What number is the best first move? Why?

3. Choosing what number as your first move would make you lose your next turn? Why?

4. What is the worst first move other than the number you chose in Question 3?

First Move Proper Factors My Score Partner's

Score1 None Lose a Turn 02 1 2 13 1 3 14 1, 2 4 3

more questions

Discussion Questions

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5. On your table, circle all the first moves that only allow your partner to score one point. These numbers have a special name. What are these numbers called?

Are all these numbers good first moves? Explain.

6. On your table, draw a triangle around all the first moves that allow your partner to score more than one point. These numbers also have a special name. What are these numbers called?

Are these numbers good first moves? Explain.

Discussion Questions Continued

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ActivityParty Favors!You are planning a party and want to give your guests party favors. You have 24 chocolate bars and 36 lollipops.

Discussion QuestionsWhat is the greatest number of party favors you can make if each bag must have exactly the same number of chocolate bars and exactly the same number of lollipops? You do not want any candy left over. Explain.

Could you make a different number of party favors so that the candy is shared equally? If so, describe each possibility.

Which possibility allows you to invite the greatest number of guests? Why?

Uh-oh! Your little brother ate 6 of your lollipops. Now what is the greatest number of party favors you can make so that the candy is shared equally?

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We can use prime factorization

Greatest Common Factor

to find the greatest common factor (GCF).

1. Factor the given numbers into primes.

2. Circle the factors that are common.

3. Multiply the common factors together to find the greatest common factor.

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16 Is 54 divisible by 3 and 9?

YesNo

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17 Is 15,516 divisible by 9?

If it is, type the quotient. If it is not, type 00.

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18 Which of the following numbers is divisible by 3, 4 and 5?

A 45B 54C 60D 80

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19 The number 126 is divisible by: (choose all that apply)

A 2B 3C 4D 5E 8F 9G 10

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20 The number 120 is divisible by: (choose all that apply)

A 2B 3C 4D 5E 8F 9G 10

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Greatest CommonFactor

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The Greatest Common Factor is 2 x 2 = 4

Use prime factorization to find the greatest common factor of 12 and 16.

12 16

3 4 4 4

3 2 2 2 2 2 2

12 = 2 x 2 x 3 16 = 2 x 2 x 2 x 2

Prime Factorization

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2

2

2

16

8

4

22

1

3

1

6

3

2

2

12

12 = 2 x 2 x 3 16 = 2 x 2 x 2 x 2

The Greatest Common Factor is 2 x 2 = 4

Use prime factorization to find the greatest common factor of 12 and 16.

Another way to find Prime Factorization...

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Use prime factorization to find the greatest common factor of 60 and 72.

60 72

6 10 6 12

2 3 2 5 2 3 3 4

2 3 2 5 2 3 3 2 2

60 = 2 x 2 x 3 x 5 72 = 2 x 2 x 2 x 3 x 3

GCF is 2 x 2 x 3 = 12

Example

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2

2

3

60

30

15

55

1

2 72

2

2

36

18

93

1

33

Use prime factorization to find the greatest common factor of 60 and 72.

60 = 2 x 2 x 3 x 5

GCF is 2 x 2 x 3 = 12

72 = 2 x 2 x 2 x 3 x 3

Example

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Use prime factorization to find the greatest common factor of 36 and 90.

36 90

6 6 9 10

2 3 2 3 3 3 2 5

36 = 2 x 2 x 3 x 3 90 = 2 x 3 x 3 x 5

GCF is 2 x 3 x 3 = 18

Example

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2

2

3

36

18

9

33

1

2

3

3

90

45

15

55

1

36 = 2 x 2 x 3 x 3 90 = 2 x 3 x 3 x 5

GCF is 2 x 3 x 3 = 18

Use prime factorization to find the greatest common factor of 36 and 90.

Example

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21 Find the GCF of 18 and 44.

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22 Find the GCF of 28 and 70.

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23 Find the GCF of 55 and 110.

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24 Find the GCF of 52 and 78.

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25 Find the GCF of 72 and 75.

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26 What is the greatest common factor of 16 and 48.

Enter your answer in the box.

From PARCC EOY sample test non-calculator #13

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Review of factors,

Interactive Website

Play the Factor Game a few times with a partner. Be sure to take turns going first. Find moves that will help you score more points than your partner. Be sure to write down strategies or patterns you use or find.

Answer the Discussion Questions.

prime numbers and

composite numbers.

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Player 1 chose 24 to earn 24 points.

Player 2 finds 1, 2, 3, 4, 6, 8, 12 and earns 36 points.

Player 2 chose 28 to earn 28 points.

Player 1 finds 7 and 14 are the only available factors and earns 21 points.

(Rows and Columns can be adjusted prior to starting the game)

Game

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Relatively PrimeTwo or more numbers are relatively prime if their greatest common factor is 1.

Example:15 and 32 are relatively prime because their GCF is 1.

Name two numbers that are relatively prime.

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27 Seven and 35 are not relatively prime.

True

False

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28 Identify at least two numbers that are relatively prime to 9.

A 16B 15C 28D 36

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29 Name a number that is relatively prime to 20.

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30 Name a number that is relatively prime to 5 and 18.

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31 Choose two numbers that are relatively prime.

A 7

B 14

C 15

D 49

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Least CommonMultiple

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Text-to-World Connection

1. Use what you know about factor pairs to evaluate George Banks' mathematical thinking. Is his thinking accurate? What mathematical relationship is he missing?

2. How many hot dogs came in a pack? Buns?

3. How many "superfluous" buns did George Banks remove from each package? How many packages did he do this to?

4. How many buns did he want to buy? Was his thinking correct? Did he end up with 24 hot dog buns?

5. Was there a more logical way for him to do this? What was he missing?

6. What is the significance of the number 24?

(Click for Link to Video Clip)

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A multiple of a whole number is the product of the number and any nonzero whole number.

A multiple that is shared by two or more numbers is a common multiple.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ...

Multiples of 14: 14, 28, 42, 56, 70, 84,...

Least Common Multiple

The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 6 and 14 is 42.

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There are 2 ways to find the LCM:

1. List the multiples of each number until you find the first one they have in common.

2. Write the prime factorization of each number. Multiply all factors together. Use common factors only once (in other words, use the highest exponent for a repeated factor).

Least Common Multiple

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EXAMPLE: 6 and 8

Multiples of 6: 6, 12, 18, 24, 30Multiples of 8: 8, 16, 24

LCM = 24

Prime Factorization: 6 8

2 3 2 4

2 2 2

2 3 2 3 LCM: 23 3 = 8 3 = 24

Least Common MultipleSlide 66 / 113

Find the least common multiple of 18 and 24.

Multiples of 18: 18, 36, 54, 72, ...

Multiples of 24: 24, 48, 72, ...

LCM: 72

Prime Factorization: 18 24

2 9 6 4

2 3 3 3 2 2 2

2 32 23 3 LCM: 23 32 = 8 9 = 72

Example

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32 Find the least common multiple of 10 and 14.

A 2

B 20

C 70

D 140

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33 Find the least common multiple of 6 and 14.

A 10

B 30

C 42

D 150

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34 Find the least common multiple of 9 and 15.

A 3

B 45

C 60

D 135

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35 Find the least common multiple of 6 and 9.

A 3

B 12

C 18

D 36

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36 Find the least common multiple of 16 and 20.

A 80

B 100

C 240

D 320

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37 Find the LCM of 12 and 20.

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38 Find the LCM of 24 and 60.

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39 Find the LCM of 15 and 18.

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40 Find the LCM of 24 and 32.

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41 Find the LCM of 15 and 35.

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42 Find the LCM of 20 and 75.

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Uses a venn diagram to find the GCF and LCM for extra practice.

Interactive Website

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GCF and LCM Word Problems

Return toTable ofContents

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How can you tell is a word problem requires you to use Greatest Common Factor or Least Common Multiple to solve?

Question

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GCF Problems

Do we have to split things into smaller sections?

Are we trying to figure out how many people we can invite?

Are we trying to arrange something into rows or groups?

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LCM Problems

Do we have an event that is or will be repeating over and over?

Will we have to purchase or get multiple items in order to have enough?

Are we trying to figure out when something will happen again at the same time?

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Samantha has two pieces of cloth. One piece is 72 inches wide and the other piece is 90 inches wide. She wants to cut both pieces into strips of equal width that are as wide as possible. How wide should she cut the strips?

What is the question: How wide should she cut the strips?

Important information: One cloth is 72 inches wide. The other is 90 inches wide.

Is this a GCF or LCM problem?

Does she need smaller or larger pieces?This is a GCF problem because we are cutting or "dividing" the pieces of cloth into smaller pieces (factor) of 72 and 90.

click

Example

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90 inches

Use the greatest common factor to determine the greatest width possible.The greatest common factor represents the greatest width possible not the number of pieces, because all the pieces need to be of equal length. This is called making a Bar Model.

72 inches

18 inches

Bar Modeling

click

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Ben exercises every 12 days and Isabel every 8 days. Ben and Isabel both exercised today. How many days will it be until they exercise together again?

What is the question: How many days until they exercise together again?

Important information: Ben exercises every 12 days Isabel exercises every 8 days

Is this a GCF or LCM problem?

Are they repeating the event over and over or splitting up the days?

This is a LCM problem because they are repeating the event to find out when they will exercise together again.

click

ExampleSlide 86 / 113

Ben exercises in:

Isabel exercises in:

Bar ModelingUse the least common multiple to determine the least amount of days possible.

The least common multiple represents the number of days not how many times they will exercise.

12 Days

8 Days

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43 Mrs. Evans has 90 crayons and 15 pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets an equal number of crayons and an equal number of paper?

A GCF Problem

B LCM Problem

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44 Mrs. Evans has 90 crayons and 15 pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets an equal number of crayons and an equal number of paper?

A 3

B 5

C 15

D 90

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45 How many crayons and pieces of paper does each student receive if there are 15 students in the class?

A 30 crayons and 10 pieces of paper

B 12 crayons and pieces of paper

C 18 crayons and 6 pieces of paper

D 6 crayons and 1 piece of paper

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46 Rosa is making a game board that is 16 inches by 24 inches. She wants to use square tiles. What is the largest tile she can use?

A GCF Problem

B LCM Problem

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47 Rosa is making a game board that is 16 inches by 24 inches. She wants to use square tiles. What is the largest tile she can use?

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48 How many tiles will she need?

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49 Y100 gave away a $100 bill for every 12th caller. Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $100 bill and a concert ticket?

A GCF Problem

B LCM Problem

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50 Y100 gave away a $100 bill for every 12th caller. Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $100 bill and a concert ticket?

A 36

B 3

C 108

D 6

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51 There are two ferris wheels at the state fair. The children's ferris wheel takes 8 minutes to rotate fully. The bigger ferris wheel takes 12 minutes to rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris wheel. If they both start at the bottom, how many minutes will it take for both of them to meet at the bottom at the same time?

A GCF Problem

B LCM Problem

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52 There are two ferris wheels at the state fair. The children's ferris wheel takes 8 minutes to rotate fully. The bigger ferris wheel takes 12 minutes to rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris wheel. If they both start at the bottom, how many minutes will it take for both of them to meet at the bottom at the same time?

A 2

B 4

C 24

D 96

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53How many rotations will each ferris wheel complete before they meet at the bottom at the same time? (Input the answer for the small ferris wheel.)

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54Sean has 8-inch pieces of toy train track and Ruth has 18-inch pieces of train track. How many of each piece would each child need to build tracks that are equal in length?

A GCF Problem

B LCM Problem

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55What is the length of the track that each child will build?

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56 I am planting 50 apple trees and 30 peach trees. I want the same number and type of trees per row. What is the maximum number of trees I can plant per row?

A GCF Problem

B LCM Problem

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Glossary & Standards

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Standards for Mathematical Practice

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

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Back to

Instruction

Bar Model

Part

Whole

One part

Whole # of partsx

A diagram that uses bars to show the relationship between

two or more numbers.

Whole

Part Part

Part + Part = WholeWhole - Part = Part

Larger Amount

Smaller Amount

Difference

Large - Difference = SmallLarge - Small = Difference

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Composite NumberA number that has

more than two factors.

121 x 12

2 x 6

3 x 46 factors

3 x 5 = 15Any number with factors other than one and itself is

composite.

131 x 13

Only 2 factors.

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Instruction

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Exponent

32Base

Exponent

"3 to the second power"

32= x 33

3 = x x 33 3332

x 2333

x 33

A small, raised number that shows how many times the

base is used as a factor.

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FactorA whole number that can divide into another

number with no remainder.A whole number that multiplies with

another number to make a third number.

15 3 5

3 is a factor of 15

3 x 5 = 15

3 and 5 are factors of 15

1635 .1R

3 is not a factor of 16

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Instruction

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Greatest Common Factor (GCF)The largest number that will divide two or more numbers

without a remainder.

12: 1, 2, 3, 4, 6, 1216: 1, 2, 4, 8, 16 Common Factors

are 1, 2, 4

GCF is 4

12 = 2 x 2 x 3

16 = 2 x 2 x 2 x 2

GCF = 2 x 2 GCF is 4

Using Prime Factorization 1 and 2 are

common factors, but not the greatest

common factor.

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Instruction

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Least Common Multiple (LCM)The smallest number that two or more numbers share as a

multiple.

9 = 3 x 3

15 = 3 x 5LCM = 3 x 3 x 5

LCM is 45

Using Prime Factorization9: 9, 18, 27, 36, 45

15: 15, 30, 45

LCM is 45

2: 2, 4, 6, 84: 4, 8

4 is the LCM, not 8

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Instruction

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MultipleThe product of two whole numbers is a multiple of each of those numbers.

3 x 5 = 1515 is a

multiple of 3.

2 x 6 = 12

Factors Product / Multiple

4 x 5 = 205 and 4 are

factors of 20, not multiples.

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Instruction

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Prime Factorization

A number written as the product of all its prime factors.

18 = 2 x 3 x 3

18 = 2 x 32

18 = 1 x 2 x 3 x 3Only prime numbers are included in prime

factorizations.

There is only one for any number.

or

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Prime NumberA positive integer that is

greater than 1 and has exactly two factors, one and itself.

1One is not a prime

number, because it has only one factor.

2Two is the only

even prime number.

2, 3, 5, 7, 11, 13, 17, 19,

23, 29

Prime #s to 30

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Proper FactorAll of the factors of a number

other than one and itself.

6: 1, 2, 3, 6Proper Factors:

2 and 3

9: 1, 3, 9

Proper Factor: 3

7: 1, 7The number 7

does not have any proper factors.

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Relatively PrimeTwo numbers who only

have 1 as a common factor.

8: 1, 2, 4, 815: 1, 3, 5Only Common

Factor is 1

All prime numbers are

relatively prime to

every other number.

9: 1, 3, 915: 1, 3, 5, 15

Common Factors:

1 and 3

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