Download Prime Factorization

Teacher’s Name: Mary Anne Bertola
Subject: Mathematics
Grade: 6
Date: October 9, 2008
Topic: Prime Factorization
Essential Questions/Big Ideas:
 How are numbers related to one another?
 How can we determine if a number is prime or composite?
Learning Objectives:
VA SOL – Number and Number Sense
6.3
The student will
a) find common multiples and factors, including least common multiple and
greatest common factor; and
b) identify and describe prime and composite numbers; and
c) identify and describe the characteristics of even and odd integers.
NCTM Standards – Number and Operations Standard for Grades 6-8
In grades 6-8 all students should –
Use factors, multiples, prime factorization, and relatively prime numbers to solve
problems.
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Students will be able to define factor, prime number, composite number, and prime
factorization. (knowledge)
Given a number, students will be able to generate a prime factorization for that number.
(application)
Students will be able to compare prime and composite numbers. (analysis)
Students will understand the relationship between divisibility rules and prime
factorization. (evaluation)
Student and Teacher Activities with Estimated Time Blocks:
Time
Warm-up/Homework Check –
15 minutes
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Teacher
(what the teacher will do)
Display warm-up at front
of class for students to see
While students are
completing warm-up
questions, determine
attendance by checking
homework for completion
Lead class discussion of
warm-up answers
Once warm-up is
completed, recite answers
to homework problems
Lead class discussion of
homework answers and
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Students
(what the students will do)
Settle down at beginning
of class and get out
homework from previous
night
Copy down tonight’s
homework from front of
class
Answer warm-up
questions
Correct warm-up answers
and ask questions if
appropriate
Correct homework
answers and ask questions
Lecture: Prime Factorization –
15 minutes
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Gizmo: Finding Factors with
Area Models – 15 minutes
(student centered instruction)
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Closure Activity – 5 minutes
(formative assessment)
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answer student questions
Supply definitions for
factor, prime number,
composite number, and
prime factorization
Provide examples for
finding the factors of a
number, determining if the
number is prime or
composite, and finding the
prime factorization of the
number
Establish how divisibility
rules are related to prime
factorization
Help clarify student
understanding
Display Finding Factors
with Area Models Gizmo
on projector
Review with students how
the prime factorization is
on the left and the area
model is on the right
Select various numbers to
show how prime
factorization can be
modeled through area of
rectangles
Select individual students
to choose a number and
change the rectangles in
the area model section
(interactive learning)
Hand out assessment
questions after the activity
Collect assessment
questions
Display exit card questions
on the board for students
to answer
Hand out homework
worksheet
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
if appropriate
Copy notes regarding the
definitions for factor,
prime number, composite
number, and prime
factorization (including
examples)
Participate in discussion of
how divisibility rules are
related to prime
factorization
Ask questions if
appropriate
Participate in gizmo
Help select various
numbers to show how
prime factorization can be
modeled through area of
rectangles
Complete assessment
questions worksheet with a
partner after activity is
completed
Hand in assessment
questions
Answer exit card questions
prior to leaving class
Write down any questions
regarding the lesson on the
exit card to be discussed
tomorrow
Put away homework
worksheet and complete
for tomorrow’s lesson
Materials Needed for Lesson:
Gizmo: Finding Factors with Area Models
 Projector
 Computer with internet access
Resources:
(2008). Finding factors with area models. Retrieved October 11, 2008, from Explore Learning
Website:
http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=2
18
Bailey, R. et. al. ( 2005). Mathematics applications and concepts: Course 1. New York:
McGraw-Hill Glencoe.
Ferrini-Mundy, J. et. al. (2000). Principles and standards for school mathematics. Reston, VA:
The National Council of Teachers of Mathematics, Inc.
Schroder, K. et. al. (2001). Mathematics standards of learning for Virginia public schools.
Richmond, VA: Commonwealth of Virginia, Board of Education.
Notes on Prime Factorization
I. Warm-Up
1. Tell whether each number is divisible by 2, 3, 4, 5, 6, 9, or 10. Then classify each
number as even or odd. Remember to explain your reasoning.
a. 60
2: Yes; the ones digit, 0, is divisible by 2
3: Yes; the sum of the digits, 6, is divisible by 3
4: Yes; the number formed by the last two digits, 60, is divisible by 4
(trivial)
5: Yes; the ones digit is 0
6: Yes; the number is divisible by both 2 and 3
9: No; the sum of the digits, 6, is not divisible by 9
10: Yes; the ones digit is 0
The number is even.
b. 17, 256
2: Yes; the ones digit, 6, is divisible by 2
3: Yes; the sum of the digits, 21, is divisible by 3
4: Yes; the number formed by the last two digits, 56, is divisible by 4
5: No; the ones digit is not 0 or 5
6: Yes; the number is divisible by both 2 and 3
9: No; the sum of the digits, 21, is not divisible by 9
10: No; the ones digit is not 0
The number is even.
II. Definitions
 When two or more numbers are multiplied, each number is called a factor of the product.
 A whole number that has exactly two unique factors, 1 and the number itself, is a prime
number.
 A number greater than 1 with more than two factors is a composite number.
 Every composite number can be expressed as a product of prime numbers. This is called
a prime factorization of the number. A factor tree can be used to find the prime
factorization of a number.
 We can find factors of a number by applying divisibility rules to that number.
 Is it possible to have numbers which are neither prime nor composite?
 Concept Summary:
Number
Prime
Composite
Neither prime nor
composite
Definition
A whole number that has
exactly two factors, 1 and
the number itself.
A number greater than 1
with more than two factors.
1 has only one factor.
0 has an infinite number of
factors.
Examples
11, 13, 23
6, 10, 18
0, 1
III. Examples
 1 × 7 = 7 The factors of 7 are 1 and 7.
 1 × 6 = 6 and 2 × 3 = 6 The factors of 6 are 1 and 6, and 2 and 3.
 Tell whether each number is prime, composite, or neither.
o 28  The factors of 28 are 1 and 28, 2 and 14, and 4 and 7. Since 28 has more
than two factors, it is a composite number.
o 11  The factors of 11 are 1 and 11. Since there are exactly two factors, 1 and
the number itself, 11 is a prime number.
 Find the prime factorization of 54.
54
Write the number that is being factored at the top.
54
2 × 27 Choose any pair of whole number factors of 54
3 × 18
2 × 3 × 9 Continue to factor any number that is not prime
3×2×9
2×3×3×3
3×2×3×3
Note: Except for the order, the prime factors are the same. The prime factorization of 54
is 2 × 3 × 3 × 3 or 2 × 33 .
IV. Exit Card
1. The state of Kentucky has 120 counties. Write 120 as a product of primes.
Answer: 120 = 2 × 2 × 2 × 3 × 5 or 23 × 3 × 5
2. Give an example of a number that is not composite.
Answer: Based on student input. Example: any prime number
3. Write down any questions you have regarding today’s lesson.
Name: _________________________
Assessment Questions: Finding Factors with Area Models Gizmo
1. Which of the following is not a prime number?
a. 13
b. 15
c. 17
d. 19
2. The number 105 is:
a. a prime number.
b. a composite number.
c. a prime factorization.
d. None of the above.
3. Which of the following is not a prime factorization?
a. 2 • 2 • 5 • 7
b. 22 • 5 • 7
c. 2 • 2 • 7 • 5
d. 2 • 2 • 5 • 8
4. What is the prime factorization of 36?
Name: _________________________
Homework Worksheet: Prime Factorization
1. Tell whether each number is prime, composite, or neither. Explain your reasoning.
a. 56
b. 114
c. 291
2. Find the prime factorization of each number.
a. 102
b. 55
c. 126
3.
Animal
Cheetah
Antelope
Lion
Coyote
Hyena
Speed (mph)
70
60
50
43
40
Animal
Rabbit
Giraffe
Grizzly Bear
Elephant
Squirrel
a. Which speed(s) are prime numbers?
Speed (mph)
35
32
30
25
12
b. Which speed(s) have a prime factorization whose factors are all equal?
c. Which speeds have a prime factorization of exactly three factors?
4. To find the volume of a box, you can multiply its height, length, and width. The measure of
the volume of a box is 357. Find its possible dimensions.
l
h
w