Download CMP3_G6_PT_AAG_3-2

At a Glance | Problem 3.2
3.2
Pacing 1 Day
Finding the Longest Factor String
Focus Question How many unique prime factorizations of a number are there?
Launch
Review prime factorization as you introduce Problem 3.2. You can preface
the discussion by showing the launch video.
Suggested Questions
• Yesterday we found the longest string of factors for 840. How could
we communicate our method to someone else?
• Write the following without exponents: 33 × 72 × 11. What is
the number?
Key Vocabulary
• exponent
• factorization
• prime
factorization
• 81 can be written as 9 × 9, 92, 3 × 3 × 3 × 3, or 34. Which factorization
is the prime factorization?
• Write the prime factorization of 18 using exponents.
Tell students that they will continue to find the prime factorization of a number and
use it to answer questions about the number.
Explore
Look for interesting strategies to share when you summarize.
Suggested Questions
• Did everyone in your group get the same prime factorization for 36?
• Are all the numbers in your string prime?
• Did you keep track of all the prime factors as you found them?
• Did you multiply your string of numbers to check that the product was 36?
Summarize
Call on students to share different strategies you observed during the Explore.
Suggested Questions
• How can we rewrite the prime factorization of 36 using exponents?
• Do we all need to use the same strategy to find the prime factorization of
a number?
• What will the prime factorization of a multiple of 36 have in common with the
prime factorization of 36?
• How can you tell whether a number is prime or composite by looking at its
prime factorization? Give an example.
• How can you tell whether a number is even or odd by looking at its prime
factorization? Give an example.
Prime Time
1
Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.
Investigation 3
At a Glance | Problem 3.2
A C E
B. 1. a. 10 = 2 × 5
b. 100 = 2 × 2 × 5 × 5 = 22 × 52
c. 1,000 = 2 × 2 × 2 × 5 × 5 × 5 = 23 × 53
d. 10,000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5
= 24 × 54
Assignment Guide for Problem 3.2
Applications: 5–20 | Connections: 31–36
Extensions: 51–53
2. 270,000 is the same as 27 × 10,000.
Therefore, 270,000 = 3 × 3 × 3 × 104.
By using the observation from
Question B, part (1), we can write
the prime factorization of 270,000 as
3×3×3×2×2×2×2×5×5×5×5
or 33 × 24 × 54.
Answers to Problem 3.2
A. 1. 36 = 2 × 2 × 3 × 3; 36 = 22 × 32
2. A student might choose the factor pair
(4, 9), write out the factorization of 36 as
36 = 2 × 2 × 3 × 3, and circle 2 × 2 and
3 × 3.
C. 1. 24 × 32 × 5 = 720
2. Yes; 22 × 3 = 12 is a factor of 720,
because 12 × 60 = 720, or because the
factorization 22 × 3 is included in the
factorization 24 × 32 × 5.
36 = 4 × 9
Make sure students see that the prime
factorization of 36 is composed of the
prime factorization of 4 and the prime
factorization of 9.
3. Yes; 25 × 32 × 5 = 1,440 is twice 720 (i.e.,
2 × 24 × 32 × 5).
D. Since 6 = 2 × 3 and 12 = 2 × 2 × 3,
2 × 52 × 6 × 11 × 12 becomes
24 × 32 × 52 × 11.
Since 25 = 5 × 5 and 44 = 2 × 2 × 11,
3 × 11 × 25 × 44 becomes 22 × 3 × 52 × 112.
Since 122 = (2 × 2 × 3)2 and 55 = 5 × 11,
5 × 122 × 55 becomes 24 × 32 × 52 × 11.
Since 10 = 2 × 5, 22 = 2 × 11, and
33 = 3 × 11, 5 × 10 × 22 × 33 becomes
22 × 3 × 52 × 112.
Therefore, 3 × 11 × 25 × 44 and
5 × 10 × 22 × 33 are equivalent, and
2 × 52 × 6 × 11 × 12 and 5 × 122 × 55 are
equivalent. Note: Look for other ways that
students might answer this question.
3. To find the first number in a factor pair,
first choose some of the prime numbers
in the factorization. Multiply these primes
together. To find the second factor in the
pair, multiply the remaining prime factors.
For example, a student might circle
2 × 3 × 3 and 2 to find the factor pair
(18, 2).
4. (2, 18):
(3, 12):
(4, 9):
(6, 6):
36 = 2 × 2 × 3 × 3
36 = 2 × (2 × 3 × 3)
36 = 2 × 18
36 = 2 × 2 × 3 × 3
36 = (2 × 2 × 3) × 3
36 = 12 × 3
36 = 2 × 2 × 3 × 3
36 = (2 × 2) × (3 × 3)
36 = 4 × 9
36 = 2 × 2 × 3 × 3
36 = 2 × 3 × 2 × 3
36 = (2 × 3) × (2 × 3)
36 = 6 × 6
E. There is only one (unique) set of prime
factors for any number. For example,
the prime factors of 330 are 2, 3, 5 and
11, and the prime factorization of 330 is
2 × 3 × 5 × 11. There is no other possible
set of prime numbers that can be multiplied
to make 330. The prime numbers cannot
be broken down further into a product of
primes. 3 can be written as 1 × 3, but 1 is not
a prime.
5. One possible multiple of 36 is 72.
The prime factorization of 72 contains the
prime factorization of 36.
Prime Time
2
Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved.
Investigation 3