Download Growing a Factor Tree

Growing a Factor Tree
An Introduction to Prime
Factorization by Mrs. Gress
Objectives
• This project was designed for Mrs. Gress’
fifth grade students.
• There are two objectives for this lesson:
– Objective #1: After using the stand-alone
instructional resource, you will be able to
define the term prime factorization and
identify examples of prime factorization.
– Objective #2: After using the stand-alone
instructional resource, you will be able to
solve prime factorization problems using a
factor tree.
Related Math GLCEs
• N.MR.05.01 Understand the meaning of
division of whole numbers with and
without remainders; relate division to
fractions and to repeated subtraction.
• N.MR.05.07 Find the prime factorization
of numbers from 2 through 50, express in
exponential notation, e.g., 24 = 2 3 x 31,
and understand that every whole number
greater than 1 is either prime or can be
expressed as a product of primes.
Introduction
Today you will be
learning how to grow a
factor tree! Your job is
to navigate through this
presentation on your
own. By doing so you
will learn what the term
prime factorization
means and how to find
the prime factorization
of a number by making
a factor tree.
A Brief Review
• Before you begin growing your
factor tree we must review some key
terms!
• What do you know about each of
these terms?
– Product
– Factor
– Prime Number
– Composite Number
What does “product” mean?
• A product is the answer to a
multiplication problem.
7  3  21
Product
What are the products?
6  8  48
5 * 60  300
If you said 48,
300, 36 and 160
then you’re
right!
36  12  3
If you need
more practice
click the button
below.
20  8  160
Click here for the answers.
More
Practice
More practice with products
9  9  81
60  6  360
If you said 81,
360, 18 and 56
then you’re right!
18  3  6
56  7  8
Click here for the answers.
Great Job!
If you need more
help please see Mrs.
Gress.
What does “factor” mean?
• Factors are the numbers that are
multiplied together to get a
product.
7  3  21
7 and 3 are factors of the number 21.
What are the factors?
6  8  48
5 * 60  300
The factors are 6 and 8
The factors are 5 and 60
36  12  3
The factors are 12 and 3
20  8  160
The factors are 20 and 8
Click here for the answers.
What are prime numbers?
• A prime number is a number that has
exactly two factors, itself and 1.
An example of a prime number is 7
7 is a prime number because the only
numbers that will divide into it evenly
are 1 and 7.
Examples of Prime #s
2, 3, 5, 7, 11, 13, 17, 19
You should memorize the first
five prime numbers!
Special Note: The number 1
is not a prime number!
What are composite numbers?
• A composite number is a number that
has more than two factors.
An example of a composite number is 8
8 is a composite number because it
has more than two factors. Its factors
are 1, 2, 4, 8. *Hint: Remember you
can make a factor rainbow!
Examples of Composite #s
4, 6, 8, 9, 10, 12, 14, 15
Special Note: Every whole
number from 2 on is either
prime or composite.
The Lonely Number
One is not a prime
because it does
not have exactly
two different
factors!
1
One is not a
composite
because it does
not have more
than two factors!
One is a special case! It is neither
prime nor composite
Back to prime number practice
Practice with Primes
Click on the prime number in
the list below.
9
Nice try but not quite! 9 is composite
because it has three factors, 1, 3, and 9.
3
That’s right! 3 is prime because it has
exactly two factors, 1 and 3.
10
Nice try but not quite! 10 is composite
because it has four factors, 1, 2, 5, and 10.
More Practice with Primes
Click on the prime number in
the list below.
1
15
5
Nice try but not quite! 1 is neither prime
nor composite. Click here to read more
about this special number.
Nice try but not quite! 15 is composite
because it has four factors, 1, 3, 5, and 15.
That’s right! 5 is prime because it has
exactly two factors, 1 and 5.
Practice with Composites
Click on the composite
number in the list below.
2
Nice try, but not quite! 2 is a prime
because it has exactly two factors, 1 and 2.
7
Nice try, but not quite! 7 is prime because
it has exactly two factors, 1 and 7.
12
That’s right! 12 is composite because it
has six factors, 1, 2, 3, 4, 6, and 12.
More Practice with Composites
Click on the composite
number in the list below.
6
That’s right! 6 is a composite because it
has four factors, 1, 2, 3, and 6.
23
Nice try, but not quite! 23 is prime because
it has exactly two factors, 1 and 23.
3
Nice try, but not quite! 3 is prime because
it has exactly two factors, 1 and 3.
What is Prime Factorization?
Look at the
equations on the
right. What
pattern do you
notice? Write
down what you
notice in your
math notebook.
Click here for a hint?
2  2  2  2  3  48
2  2  3  5  60
7  7  49
3  3  5  45
2  2  3  3  5  180
Look at the factors? What kind
of numbers are they?
What does “factor” mean?
• Factors are the numbers that are
multiplied together to get a
product.
7  3  21
7 and 3 are factors of the number 21.
Back to Prime
Factorization
A Definition
• Hopefully you noticed that each of
the factors in the equations are
prime numbers.
• Prime Factorization is a way to write
a composite number as a product of
prime factors.
3  3  5  45
3  5  45
2
Growing a Factor Tree
• To find the prime factorization of a
number you can grow a factor tree!
• Let’s get started.
Getting Started
You
might
Can
you
think
Let’s that
grow180
notice
of one factor
a tree
of in
has
a
ZERO
pair for 180?
the
itsfactors
ONES
This
should
be
of 180.
PLACE
which
two
numbers
means
it is a
that
multiply
multiple
of 10.
together
So…to
give you 180.
180
10
10 x •= 180
10 x 18 = 180
Click here to continue
Click here to continue
18
What’s Next?
Now you
have to
find
factor
pairs for
10 and
18.
180
10
18
We “grow” this “tree”
downwards since that
is how we write in
English (and we can’t
be sure how big it will
be - we could run out
of paper if we grew it
upwards).
Keep Growing Down
180
First find two
factors of 10.
Next find
factors for 18.
Click here to continue
Click here to continue
10
2 x 5 = 10
2
18
5
6
3
6 x 3 = 18
Are We Done Yet???
Since 2, 3, and 5
are prime
numbers they do
not grow “new
branches” they
just grow down
alone.
Click here to continue
180
10
18
2
5
2
5 2
6
3
Since 6 is NOT
a prime
number - it is a
COMPOSITE
NUMBER - it
still has
factors. Since
it is an EVEN
NUMBER we
see that:
6=2x3
3 3
Are We Done Yet???
180
Now that the
bottom row of
our tree is made
up of all prime
factors we have
found the prime
factorization.
There is only
one more step!
10
18
2
5
2
5 2
6
3
3 3
The Final Step
180
10
18
2
5
2
5 2
6
All we have left to do is
write the equation. Make
sure you write the
numbers in order from
least to greatest!
3
3 3
2  2  3  3  5  180
2  3  5  180
2
2
So….
The prime factorization of the
composite number 180 is…
2  2  3  3  5  180
or
2  3  5  180
2
2
Now You’re Ready…
to grow your own factor tree!
You will need to draw your factor tree in
your math notebook.
Find the prime factorization of 63
Remember, first
you find two
factors of 63.
63
9
7
These two factors
multiply together to give
you the product 63.
Now you try! First check to see if either
number is prime. If a number is prime
bring it straight down. (Remember 1 is not
a prime number so it should not be part of
your prime factorization!)
Find the prime factorization of 63
Your factor tree
should now look
like this!
63
9
7
7
Next, look at the other number and find
two factors that multiply together to give
you that number. Extend your tree!
Find the prime factorization of 63
Your factor tree
should now look
like this!
3
63
9
7
3
7
Now, if all of your numbers are prime
then you are done. If not, keep extending
your tree.
Find the prime factorization of 63
63
Since 3, 3, and 7
are all prime
numbers you are
done!
All you need to do is
write it as an
equation.
9
3
7
3
7
3  3  7  63 or 3  7  63
2
It’s Quiz Time
• Let’s see what you know!
• You will need a blank piece of paper.
Please make sure you put your
name, number, and the date on the
top.
• You will need to draw a factor tree
for each of the problems. This will
be turned in to Mrs. Gress when you
are done.
Question #1
What is the prime factorization of 27?
(Remember to draw your factor tree on your paper before you
try to answer the question!)
A. 3 9
B. 3  3  3
C. 3 9 1
That’s Correct!
27
9
3
3
3
3
Since 3 is a prime number the prime
factorization of 27 is 3 x 3 x 3.
Not Quite! Try again!
9 is a composite number so 3 x 9 can’t
be the prime factorization of 27
because all the numbers are not prime.
Not Quite! Try again!
9 is a composite number and 1 is neither
prime nor composite, so 3 x 9 x 1 can’t
be the prime factorization of 27 because
all the numbers are not prime.
Question #2
What is the prime factorization of 40?
(Remember to draw your factor tree on your paper before you
try to answer the question!)
A. 2  2  2  5
B. 2  2  5
C. 4  2  5
That’s Correct!
40
8
4
2
5
2
5
2 2
5
Since 2 and 5 are both prime
numbers the prime factorization of
40 is 2 x 2 x 2 x 5.
Not Quite! Try again!
2 x 2 x 5 equals 20, not 40 so it can’t be
the prime factorization of 40.
Not Quite! Try again!
4 is a composite number, so 4 x 2 x 5
can’t be the prime factorization of 40
because all the numbers are not prime.
Question #3
What is the prime factorization of 100?
(Remember to draw your factor tree on your paper before you
try to answer the question!)
A. 2  2  5  5
B. 4  5  5
C. 1 2  2  5  5
That’s Correct!
100
10 10
2
52
5
Since 2 and 5 are both prime
numbers the prime factorization of
100 is 2 x 2 x 5 x 5.
Not Quite! Try again!
4 is a composite number, so 4 x 5 x 5
can’t be the prime factorization of 100
because all the numbers are not prime.
Not Quite! Try again!
4 is a composite number, so 4 x 2 x 5
can’t be the prime factorization of 40
because all the numbers must be prime.
Question #4
What is the prime factorization of 36?
(Remember to draw your factor tree on your paper before you
try to answer the question!)
A. 6  2  3
2  2  3 3
C. 6 6
B.
That’s Correct!
36
6
2
6
32
3
Since 2 and 3 are both prime
numbers the prime factorization of
36 is 2 x 2 x 3 x 3.
Not Quite! Try again!
6 is a composite number, so 6 x 2 x 3
can’t be the prime factorization of 36
because all the numbers are not prime.
Not Quite! Try again!
6 is a composite number, so 6 x 6
can’t be the prime factorization of
36 because all the numbers are not
prime.
Question #5
What is the prime factorization of 14?
(Remember to draw your factor tree on your paper before you
try to answer the question!)
A. 11 2  7
2  3 4
C. 2 7
B.
That’s Correct!
14
2
7
Since 2 and 7 are both prime
numbers the prime factorization of
14 is 2 x 7.
Not Quite! Try again!
1 is neither prime nor composite, so
1 x 1 x 2 x 7 can’t be the prime
factorization of 14 because all the
numbers are not prime.
Not Quite! Try again!
2 x 3 x 4 equals 24, not 12 so it can’t
be the prime factorization of 12.
Question #6
What is the prime factorization of 110?
(Remember to draw your factor tree on your paper before you
try to answer the question!)
2  2  5 11
B. 2  5 11
C. 1011
A.
That’s Correct!
110
10
2
5
11
11
Since 2, 5, and 11 are all prime
numbers the prime factorization of
110 is 2 x 5 x 11.
Not Quite! Try again!
2 x 2 x 5 x 11 equals 220, not 110
so it can’t be the prime
factorization of 110.
Not Quite! Try again!
10 is a composite number, so 10 x 11
can’t be the prime factorization of
110 because all the numbers are not
prime.
Question #7
What is the prime factorization of 63?
(Remember to draw your factor tree on your paper before you
try to answer the question!)
3 3 7
B. 9 7
C. 7 7
A.
That’s Correct!
63
9
3
7
3
7
Since 3 and 7 are both prime
numbers the prime factorization of
63 is 3 x 3 x 7.
Not Quite! Try again!
9 is a composite number, so 9 x 7
can’t be the prime factorization of
63 because all the numbers are not
prime.
Not Quite! Try again!
7 x 7 equals 49, not 63 so it can’t be
the prime factorization of 63.
Question #8
What is the prime factorization of 250?
(Remember to draw your factor tree on your paper before you
try to answer the question!)
2  2  5 5
B. 10  5  5
C. 2  5  5  5
A.
That’s Correct!
250
10
2
25
5 5
5
Since 2 and 5 are both prime
numbers the prime factorization of
250 is 2 x 5 x 5 x 5.
Not Quite! Try again!
10 is a composite number, so
10 x 5 x 5 can’t be the prime
factorization of 150 because all
the numbers are not prime.
Not Quite! Try again!
2 x 2 x 5 x 5 equals 100, not 250 so it
can’t be the prime factorization of
100.
Congratulations!
• You have done a great job growing
your factor trees!
• Please turn in the eight factor trees
from the quiz to Mrs. Gress.
and
Don’t forget to keep practicing!