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Slide 1 / 113 Slide 2 / 113
6th Grade
Factors and Multiple
2015-10-20
www.njctl.org
Slide 3 / 113
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
Slide 4 / 113
Even and Odd Numbers
Return toTable ofContents
Slide 5 / 113
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
Slide 6 / 113
What happens when
we add two even numbers? Will we
always get an even number?
What do you think?
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.
TrueFalse
Slide 11 / 113
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
Slide 12 / 113
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
Slide 13 / 113
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
Slide 14 / 113
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
Slide 15 / 113
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
Slide 16 / 113
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
Slide 17 / 113
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
Slide 18 / 113
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
Slide 19 / 113
Divisibility Rules for 3 and 9
Return toTable ofContents
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 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
Slide 23 / 113
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
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 25 / 113
10 468 is divisible by: (choose all that apply)
A 2B 3C 4D 5E 8F 9G 10
Slide 26 / 113
11 Is any number divisible by 9 also divisible by 3? Explain.
YesNo
Slide 27 / 113
12 Is 135 divisible by 3?
YesNo
Slide 28 / 113
13 Any number divisible by 3 is also divisible by 9.
TrueFalse
Slide 29 / 113
14 The number 129 is divisible by 9.
TrueFalse
Slide 30 / 113
15 Is 24,981 divisible by 3?
If it is, type the quotient. If it is not, type 00.
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
Score1 None Lose a Turn 02 1 2 13 1 3 14 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
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.
Slide 35 / 113
16 Is 54 divisible by 3 and 9?
YesNo
Slide 36 / 113
17 Is 15,516 divisible by 9?
If it is, type the quotient. If it is not, type 00.
Slide 37 / 113
18 Which of the following numbers is divisible by 3, 4 and 5?
A 45B 54C 60D 80
Slide 38 / 113
19 The number 126 is divisible by: (choose all that apply)
A 2B 3C 4D 5E 8F 9G 10
Slide 39 / 113
20 The number 120 is divisible by: (choose all that apply)
A 2B 3C 4D 5E 8F 9G 10
Slide 40 / 113
Greatest CommonFactor
Return toTable ofContents
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
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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...
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
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
Slide 47 / 113
21 Find the GCF of 18 and 44.
Slide 48 / 113
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.
Slide 52 / 113
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
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 55 / 113
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.
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 16B 15C 28D 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 61 / 113
Least CommonMultiple
Return toTable ofContents
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)
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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.
Slide 64 / 113
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
Slide 65 / 113
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
Slide 67 / 113
32 Find the least common multiple of 10 and 14.
A 2
B 20
C 70
D 140
Slide 68 / 113
33 Find the least common multiple of 6 and 14.
A 10
B 30
C 42
D 150
Slide 69 / 113
34 Find the least common multiple of 9 and 15.
A 3
B 45
C 60
D 135
Slide 70 / 113
35 Find the least common multiple of 6 and 9.
A 3
B 12
C 18
D 36
Slide 71 / 113
36 Find the least common multiple of 16 and 20.
A 80
B 100
C 240
D 320
Slide 72 / 113
37 Find the LCM of 12 and 20.
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.
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41 Find the LCM of 15 and 35.
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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 79 / 113
GCF and LCM Word Problems
Return toTable ofContents
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 or 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 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
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
Slide 87 / 113
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
Slide 88 / 113
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
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 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 97 / 113
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.)
Slide 98 / 113
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
Slide 99 / 113
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
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 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.
Slide 103 / 113
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
Slide 104 / 113
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.
Back to
Instruction
Slide 105 / 113
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.
Back to
Instruction
Slide 106 / 113
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
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, 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.
Back to
Instruction
Slide 108 / 113
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
Back to
Instruction
Slide 109 / 113
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.
Back to
Instruction
Slide 110 / 113
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
Back to
Instruction
Slide 111 / 113
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
Back to
Instruction
Slide 112 / 113
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.
Back to
Instruction
Slide 113 / 113
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
Back to
Instruction