Download 6th Grade Even and Odd Numbers

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6th Grade
Factors and Multiple
2015-10-20
www.njctl.org
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Factors and Multiples
Click on the topic to go to that section
·
·
·
·
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Even and Odd Numbers
Divisibility Rules for 3 & 9
Greatest Common Factor
Least Common Multiple
GCF and LCM Word Problems
Glossary & Standards
Even and Odd Numbers
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Table of
Contents
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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
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What do you think?
What happens
when
we add two even
numbers? Will we
always get an even
number?
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Adding Even Numbers
Adding Odd Numbers
Drag the paw prints into the
box to model 6 + 8
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.
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?
Will the sum be even or odd every time two odd numbers
are added together? Why or why not?
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Adding Odd and Even Numbers
Drag the paw prints into the
box to model 7 + 8
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1 The product of two even numbers is even.
True
False
+
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?
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2 The product of two odd numbers is
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3 The product of 13 x 8 is
A
odd
A
odd
B
even
B
even
Explain your answer.
Multiplication is repeated addition. If you add an odd number
over and over, then the sum will switch between even and
Click to Reveal
odd. Since you are adding the number an odd number of times,
your product will be odd.
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
of times, the product will
Click number
to Reveal
be even.
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4 The sum of 32,877 + 14,521 is
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5 The product of 12 x 9 is
A
odd
A
odd
B
even
B
even
Explain your answer.
Explain your answer.
If you model the numbers using dots and circle all the pairs, the
Click
to Reveal
single dots leftover from each
number
will create a pair and none
will be leftover making the sum an even number.
12 x 9 is equivalent to 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12.
to Reveal
No matter how many timesClick
you add
12, since it is even the sum
will always be even.
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6 The sum of 8,972 + 1,999 is
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7 The sum of 9 + 10 + 11 + 12 + 13 is
A
odd
A
odd
B
even
B
even
Explain your answer.
Explain your answer.
If you model the problem using
dots
and circle all the pairs, then
Click to
Reveal
there will be one dot leftover since one of the addends is odd.
The first two addends will result in an odd number. By adding
another odd number, the sum
is to
even.
Adding an even number
Click
Reveal
will result in an even number. Since the last addend is odd, the
final answer will be odd.
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8 The product of 250 x 19 is
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9 The product of 15 x 0 is
A
odd
A
odd
B
even
B
even
Explain your answer.
The product of an odd and even number will always result in an
Click to Reveal
even number.
Explain your answer.
0 is an even number and the product of any even number and
Click to Reveal
odd number is always even.
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Let's review!
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.
24
36
Divisibility Rules
for 3 and 9
80
115
214
360
975
4,678
4
2
5
Derived from
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Divisibility Rules
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Divisibility Rule for 3
What factor do the numbers 12, 15, 27, and 66 have in common?
They are allClick
divisible by 3.
2: If and only if its last digit is 0, 2, 4, 6, or 8.
Now, take each of those numbers and calculate
the sum of its digits.
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.
12
1+2=3
8: If and only if its last three digits are a number divisible by 8.
15
________
10: If and only if its last digit is 0.
27
________
66
________
What do all these sums
have in common?
They are all Click
divisible by 3!
A number is divisible by 3
if the sum of the number'sClick
digits is divisible by 3.
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Divisibility Rule for 9
Try these!
What factor do the numbers 18, 27, 45, and 99 have in common?
They are allClick
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 Click
divisible by 9!
A number is divisible by 9
Click
if the sum of the number's digits is divisible by 9.
414,940
10
8
Return to
Table of
Contents
29,785
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
228
531
735
1,476
Divisible by 9
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10 468 is divisible by: (choose all that apply)
A
2
B
3
C
4
D
5
E
8
F
9
G
10
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11 Is any number divisible by 9 also divisible by 3?
Explain.
Yes
No
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12 Is 135 divisible by 3?
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13 Any number divisible by 3 is also divisible by 9.
Yes
True
No
False
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14 The number 129 is divisible by 9.
True
False
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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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Discussion Questions
Discussion Questions
Continued
1. Make a table listing all the possible first moves, proper factors,
your score and your partner's score. Here's an example:
First Move
1
2
3
4
Proper
Factors
None
1
1
1, 2
Partner's
Score
Lose a Turn
0
2
1
3
1
4
3
My Score
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?
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.
4. What is the worst first move other than the number you
chose in Question 3?
more questions
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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.
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Greatest Common Factor
We can use prime factorization
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.
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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16 Is 54 divisible by 3 and 9?
Yes
No
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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
45
B
54
C
60
D
80
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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
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20 The number 120 is divisible by: (choose all that apply)
A
2
B
3
C
4
Greatest Common
Factor
D
5
E
8
F
9
G
10
Return to
Table of
Contents
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Prime Factorization
Another way to find Prime
Factorization...
Use prime factorization to find the greatest common
factor of 12 and 16.
12
3
3
16
4
2
4
2
12 = 2 x 2 x 3
2
4
2
2
2
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.
2 16
2 12
2 8
2 6
2 4
3 3
2 2
1
1
12 = 2 x 2 x 3
16 = 2 x 2 x 2 x 2
The Greatest Common Factor is 2 x 2 = 4
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Example
Example
Use prime factorization to find the greatest common
factor of 60 and 72.
60
6
6
2 72
2 36
2 30
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
Use prime factorization to find the greatest common factor of
60 and 72.
2 60
72
10
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2 18
3 15
3 9
5 5
72 = 2 x 2 x 2 x 3 x 3
3 3
1
1
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
GCF is 2 x 2 x 3 = 12
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Example
Example
Use prime factorization to find the greatest common
factor of 36 and 90.
36
6
2
90
6
3
2
9
3
36 = 2 x 2 x 3 x 3
3
2
2 36
2 90
3 45
3 9
5
90 = 2 x 3 x 3 x 5
GCF is 2 x 3 x 3 = 18
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21 Find the GCF of 18 and 44.
Use prime factorization to find the greatest common factor of
36 and 90.
2 18
10
3
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3 15
3 3
5 5
1
1
90 = 2 x 3 x 3 x 5
36 = 2 x 2 x 3 x 3
GCF is 2 x 3 x 3 = 18
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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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25 Find the GCF of 72 and 75.
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24 Find the GCF of 52 and 78.
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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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Interactive Website
Review of factors,
prime numbers and
composite numbers.
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.
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Game
(Rows and Columns can be adjusted prior to starting the game)
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.
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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.
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27 Seven and 35 are not relatively prime.
True
False
Name two numbers that are relatively prime.
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28 Identify at least two numbers that are relatively
prime to 9.
A
16
B
15
C
28
D
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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Text-to-World Connection
(Click for Link to Video Clip)
1. Use what you know about factor pairs to evaluate George
Banks' mathematical thinking. Is his thinking accurate? What
mathematical relationship is he missing?
Least Common
Multiple
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?
Return to
Table of
Contents
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?
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Least Common Multiple
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,...
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.
Least Common Multiple
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).
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Least Common Multiple
EXAMPLE: 6 and 8
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Example
Find the least common multiple of 18 and 24.
Multiples of 6: 6, 12, 18, 24, 30
Multiples of 8: 8, 16, 24
Multiples of 18: 18, 36, 54, 72, ...
Multiples of 24: 24, 48, 72, ...
LCM = 24
LCM: 72
Prime Factorization:
18
24
Prime Factorization:
6
8
2 3
2
2
4
2 3 3
2 2 2
2 3
2
3
9
2 32
LCM: 2 3 = 8 3 = 24
3
6
4
3 2 2 2
23 3
LCM: 23 32 = 8 9 = 72
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32 Find the least common multiple
of 10 and 14.
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33 Find the least common multiple
of 6 and 14.
A
2
A
10
B
20
B
30
C
70
C
42
D
140
D
150
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34 Find the least common multiple
of 9 and 15.
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35 Find the least common multiple
of 6 and 9.
A
3
A
3
B
45
B
12
C
60
C
18
D
135
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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40 Find the LCM of 24 and 32.
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42 Find the LCM of 20 and 75.
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39 Find the LCM of 15 and 18.
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41 Find the LCM of 15 and 35.
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Interactive Website
Uses a venn diagram to find the GCF and LCM for extra practice.
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Question
How can you tell is a word problem requires you to use
Greatest Common Factor or Least Common Multiple to solve?
GCF and LCM Word Problems
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Table of
Contents
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GCF Problems
LCM Problems
Do we have an event that is or will be
repeating over and over?
Do we have to split things into smaller
sections?
Will we have to purchase or get multiple
items in order to have enough?
Are we trying to figure out how many people
we can invite?
Are we trying to figure out when
something will happen again at the same
time?
Are we trying to arrange something into rows
or groups?
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Bar Modeling
Example
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?
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
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.
90 inches
Is this a GCF or LCM problem?
Does she need smaller or larger pieces?
clickis a GCF problem because we are cutting or
This
"dividing" the pieces of cloth into smaller pieces
(factor) of 72 and 90.
click
18 inches
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Bar Modeling
Example
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
click
is a LCM problem because they are repeating the
event to find out when they will exercise together again.
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.
Ben exercises in:
12 Days
Isabel exercises in:
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?
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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
GCF Problem
A
3
B
LCM Problem
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?
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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
GCF Problem
A
2
B
LCM Problem
B
4
C
24
D
96
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53
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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.)
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
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55
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What 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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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.
Glossary
& Standards
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Contents
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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Composite Number
Bar Model
A diagram that uses bars to
show the relationship between
two or more numbers.
A number that has
more than two factors.
12
Whole
Whole
Part
Part
Part + Part = Whole
Whole - Part = Part
Smaller
Amount
Difference
3x4
6 factors
One part
Large - Small = Difference
Large - Difference = Small
x # of parts
Whole
1 x 13
Any number with
factors other than
one and itself is
composite.
2x6
Part
13
3 x 5 = 15
1 x 12
Larger Amount
Only 2
factors.
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Instruction
Instruction
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Factor
Exponent
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.
A small, raised number that
shows how many times the
base is used as a factor.
Exponent
3
2
2
2
Base
"3 to the
second power"
3 = 3x3 3 3x 2
33 = 3x3 x3 33 3x 3
15
3
5
3 is a factor of
15
3 x 5 = 15
5 R.1
3 16
3 and 5 are
factors of 15
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, 12
16: 1, 2, 4, 8,
16 Common Factors
are 1, 2, 4
GCF is 4
Using Prime
Factorization
12 = 2 x 2 x 3
16 = 2 x 2 x 2 x 2
GCF = 2 x 2
GCF is 4
1 and 2 are
common
factors, but not
the greatest
common
factor.
Least Common Multiple (LCM)
The smallest number that two
or more numbers share as a
multiple.
9: 9, 18, 27, 36, 45
15: 15, 30, 45
LCM is
45
Back to
Instruction
Using Prime
Factorization
9=3x3
15 = 3 x 5
LCM = 3 x 3 x 5
LCM is 45
2: 2, 4, 6, 8
4: 4, 8
4 is the
LCM, not 8
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Multiple
Prime Factorization
The product of two whole
numbers is a multiple of
each of those numbers.
A number written as the product
of all its prime factors.
3 x 5 = 15
15 is a
multiple of 3.
2 x 6 = 12
4 x 5 = 20
Factors Product /
5 and 4 are
factors of 20, not
multiples.
Multiple
18 = 2 x 3 x 3
or
18 = 2 x 3
2
There is
only one
for any
number.
18 = 1 x 2 x 3 x 3
Only prime numbers
are included in prime
factorizations.
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Instruction
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Prime Number
Proper Factor
A positive integer that is
greater than 1 and has exactly
two factors, one and itself.
Prime #s
to 30
2, 3, 5, 7, 11,
13, 17, 19,
23, 29
2
Two is the only
even prime
number.
All of the factors of a number
other than one and itself.
1
6: 1, 2, 3, 6
One is not a prime
number, because it has
only one factor.
Proper Factors:
2 and 3
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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
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9: 1, 3, 9
7: 1, 7
Proper Factor:
3
The number 7
does not have any
proper factors.
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