Even and Odd Numbers GCF and LCM Word Problems Glossary & - - PDF document

SLIDE 1

Slide 1 / 113 Slide 2 / 113

6th Grade

Factors and Multiple

2015-10-20 www.njctl.org

Slide 3 / 113 Factors and Multiples

Click 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

Slide 4 / 113

Even and Odd Numbers

Return to Table of Contents

Slide 5 / 113 Warm-Up Exercise

Think 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

Slide 6 / 113

What happens when we add two even numbers? Will we always get an even number?

What do you think?

SLIDE 2

Slide 7 / 113

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 Slide 8 / 113

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 Slide 9 / 113

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 Slide 10 / 113

1 The product of two even numbers is even. True False

Slide 11 / 113

Explain your answer.

2 The product of two odd numbers is A

• dd

B even

Multiplication is repeated addition. If you add an odd number

• ver and over, then the sum will switch between even and
• dd. Since you are adding the number an odd number of times,

your product will be odd. Click to Reveal

Slide 12 / 113

3 The product of 13 x 8 is A

• dd

B 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

SLIDE 3

Slide 13 / 113

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

• dd

B 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

Slide 14 / 113

5 The product of 12 x 9 is A

• dd

B 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

Slide 15 / 113

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

• dd

B 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

Slide 16 / 113

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

• dd

B 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

Slide 17 / 113

8 The product of 250 x 19 is A

• dd

B even

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

Slide 18 / 113

9 The product of 15 x 0 is A

• dd

B even

Explain your answer. 0 is an even number and the product of any even number and

• dd number is always even.

Click to Reveal

SLIDE 4

Slide 19 / 113

Divisibility Rules for 3 and 9

Return to Table of Contents

Slide 20 / 113

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! Slide 21 / 113

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 Slide 22 / 113 Divisibility Rule for 3

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

Click

Slide 23 / 113 Divisibility Rule for 9

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

Click

Slide 24 / 113

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!

SLIDE 5

Slide 25 / 113

10 468 is divisible by: (choose all that apply) A 2 B 3 C 4 D 5 E 8 F 9 G 10

Slide 26 / 113

11 Is any number divisible by 9 also divisible by 3? Explain. Yes No

Slide 27 / 113

12 Is 135 divisible by 3? Yes No

Slide 28 / 113

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

Slide 29 / 113

14 The number 129 is divisible by 9. True False

Slide 30 / 113

15 Is 24,981 divisible by 3? If it is, type the quotient. If it is not, type 00.

SLIDE 6

Slide 31 / 113

• 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 Score 1 None Lose a Turn 2 1 2 1 3 1 3 1 4 1, 2 4 3

more questions

Discussion Questions Slide 32 / 113

• 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 Slide 33 / 113 Activity

Party Favors! You are planning a party and want to give your guests party

• favors. You have 24 chocolate bars and 36 lollipops.

Discussion Questions What 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?

Slide 34 / 113

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.

Slide 35 / 113

16 Is 54 divisible by 3 and 9? Yes No

Slide 36 / 113

17 Is 15,516 divisible by 9? If it is, type the quotient. If it is not, type 00.

SLIDE 7

Slide 37 / 113

18 Which of the following numbers is divisible by 3, 4 and 5? A 45 B 54 C 60 D 80

Slide 38 / 113

19 The number 126 is divisible by: (choose all that apply) A 2 B 3 C 4 D 5 E 8 F 9 G 10

Slide 39 / 113

20 The number 120 is divisible by: (choose all that apply) A 2 B 3 C 4 D 5 E 8 F 9 G 10

Slide 40 / 113

Greatest Common Factor

Return to Table of Contents

Slide 41 / 113

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 Slide 42 / 113

2 2 2 1 6 8 4 2 2 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...

SLIDE 8

Slide 43 / 113

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 Slide 44 / 113

2 2 3 6 3 15 5 5 1 2 7 2 2 2 3 6 1 8 9 3 1 3 3 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 Slide 45 / 113

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 Slide 46 / 113

2 2 3 3 6 1 8 9 3 3 1 2 3 3 9 4 5 1 5 5 5 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 Slide 47 / 113

21 Find the GCF of 18 and 44.

Slide 48 / 113

22 Find the GCF of 28 and 70.

SLIDE 9

Slide 49 / 113

23 Find the GCF of 55 and 110.

Slide 50 / 113

24 Find the GCF of 52 and 78.

Slide 51 / 113

25 Find the GCF of 72 and 75.

Slide 52 / 113

26 What is the greatest common factor

• f 16 and 48.

Enter your answer in the box.

From PARCC EOY sample test non-calculator #13

Slide 53 / 113

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.

Slide 54 / 113

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

SLIDE 10

Slide 55 / 113 Relatively Prime

Two 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.

Slide 56 / 113

27 Seven and 35 are not relatively prime. True False

Slide 57 / 113

28 Identify at least two numbers that are relatively prime to 9. A 16 B 15 C 28 D 36

Slide 58 / 113

29 Name a number that is relatively prime to 20.

Slide 59 / 113

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

Slide 60 / 113

31 Choose two numbers that are relatively prime. A 7 B 14 C 15 D 49

SLIDE 11

Slide 61 / 113

Least Common Multiple

Return to Table of Contents

Slide 62 / 113 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)

Slide 63 / 113

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.

Slide 64 / 113

There are 2 ways to find the LCM:

• 1. List the multiples of each number until you find the first
• ne they have in common.
• 2. Write the prime factorization of each number. Multiply

all factors together. Use common factors only once (in

• ther words, use the highest exponent for a repeated

factor).

Least Common Multiple Slide 65 / 113

EXAMPLE: 6 and 8 Multiples of 6: 6, 12, 18, 24, 30 Multiples 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 Multiple Slide 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 3

2

23 3 LCM: 23 3

2 = 8 9 = 72

Example

SLIDE 12

Slide 67 / 113

32 Find the least common multiple

• f 10 and 14.

A 2 B 20 C 70 D 140

Slide 68 / 113

33 Find the least common multiple

• f 6 and 14.

A 10 B 30 C 42 D 150

Slide 69 / 113

34 Find the least common multiple

• f 9 and 15.

A 3 B 45 C 60 D 135

Slide 70 / 113

35 Find the least common multiple

• f 6 and 9.

A 3 B 12 C 18 D 36

Slide 71 / 113

36 Find the least common multiple

• f 16 and 20.

A 80 B 100 C 240 D 320

Slide 72 / 113

37 Find the LCM of 12 and 20.

SLIDE 13

Slide 73 / 113

38 Find the LCM of 24 and 60.

Slide 74 / 113

39 Find the LCM of 15 and 18.

Slide 75 / 113

40 Find the LCM of 24 and 32.

Slide 76 / 113

41 Find the LCM of 15 and 35.

Slide 77 / 113

42 Find the LCM of 20 and 75.

Slide 78 / 113

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

Interactive Website

SLIDE 14

Slide 79 / 113

GCF and LCM Word Problems

Return to Table of Contents

Slide 80 / 113

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

Question Slide 81 / 113 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

• r groups?

Slide 82 / 113 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?

Slide 83 / 113

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 Slide 84 / 113

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

SLIDE 15

Slide 85 / 113

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

Example Slide 86 / 113

Ben exercises in: Isabel exercises in:

Bar Modeling

Use 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

Slide 87 / 113

43 Mrs. Evans has 90 crayons and 15 pieces of paper to give to her students. What is the largest number

• f 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

Slide 88 / 113

44 Mrs. Evans has 90 crayons and 15 pieces of paper to give to her students. What is the largest number

• f 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

Slide 89 / 113

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

Slide 90 / 113

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

SLIDE 16

Slide 91 / 113

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?

Slide 92 / 113

48 How many tiles will she need?

Slide 93 / 113

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

Slide 94 / 113

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

Slide 95 / 113

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

Slide 96 / 113

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

SLIDE 17

Slide 97 / 113

53 How 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.)

Slide 98 / 113

54 Sean 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

Slide 99 / 113

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

Slide 100 / 113

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

Slide 101 / 113

Glossary & Standards

Return to Table of Contents

Slide 102 / 113

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

• thers.

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.

SLIDE 18

Slide 103 / 113

Back to Instruction

Bar Model

Part Whole One part Whole # of parts

x

A diagram that uses bars to show the relationship between two or more numbers.

Whole

Part Part

Part + Part = Whole Whole - Part = Part Larger Amount

Smaller Amount

Difference

Large - Difference = Small Large - Small = Difference

Slide 104 / 113

Composite Number

A number that has more than two factors. 12

1 x 12 2 x 6 3 x 4 6 factors

3 x 5 = 15

Any number with factors other than

• ne and itself is

composite.

13

1 x 13

Only 2 factors.

Back to Instruction

Slide 105 / 113

Exponent

3

2

Base

Exponent "3 to the second power"

3

2= x

3 3

3 = x x 3 3 3

3

3

2 x

2 3 3

3 x

3 3

A small, raised number that shows how many times the base is used as a factor.

Back to Instruction

Slide 106 / 113

Factor

A 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

16 3 5 .1

R 3 is not a factor of 16

Back to Instruction

Slide 107 / 113

Greatest Common Factor (GCF)

The largest number that will divide two or more numbers without a remainder.

12: 1, 2, 3, 4, 6, 12 16: 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.

Back to Instruction

Slide 108 / 113

Least Common Multiple (LCM)

The smallest number that two

• r more numbers share as a

multiple.

9 = 3 x 3 15 = 3 x 5 LCM = 3 x 3 x 5 LCM is 45

Using Prime Factorization

9: 9, 18, 27, 36, 45 15: 15, 30, 45

LCM is 45

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

4 is the LCM, not 8

Back to Instruction

SLIDE 19

Slide 109 / 113

Multiple

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

3 x 5 = 15

15 is a multiple of 3.

2 x 6 = 12

Factors Product /

Multiple

4 x 5 = 20

5 and 4 are factors of 20, not multiples.

Back to Instruction

Slide 110 / 113

Prime Factorization

A number written as the product

• f all its prime factors.

18 = 2 x 3 x 3

18 = 2 x 3

2 18 = 1 x 2 x 3 x 3

Only prime numbers are included in prime factorizations.

There is

• nly one

for any number.

• r

Back to Instruction

Slide 111 / 113

Prime Number

A positive integer that is greater than 1 and has exactly two factors, one and itself.

1

One is not a prime number, because it has

• nly one factor.

2

Two is the only even prime number.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Prime #s to 30

Back to Instruction

Slide 112 / 113

Proper Factor

All of the factors of a number

• ther than one and itself.

6: 1, 2, 3, 6

Proper Factors: 2 and 3

9: 1, 3, 9

Proper Factor: 3

7: 1, 7

The number 7 does not have any proper factors.

Back to Instruction

Slide 113 / 113

Relatively Prime

Two numbers who only have 1 as a common factor.

8: 1, 2, 4, 8 15: 1, 3, 5

Only Common Factor is 1 All prime numbers are relatively prime to every other number.

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

Common Factors: 1 and 3

Back to Instruction