3.1: Factoring Composite Numbers
This activity is intended to be used with Algebra I, Chapter 2, Lesson 1.
ID: 10891
Time required: 15 minutes
Activity Overview
In this activity, students will work with composite numbers to find their prime factorizations in exponent form. They will create factor trees and use division as a means to finding prime factors. An extension involving finding a common denominator for two fractions is given at the end of the activity.
Topic: From Arithmetic to Algebra
 Prime factorization
 Factor tree and prime factors
 Prime factorization in exponent form
Teacher Preparation and Notes
 If an interactive white board is available, create a number sort with a Venn diagram. For example, intersecting circles for Primes and Multiples of 8. The students would be given a list of a few numbers to place into the proper part of the diagram. In this case, there would be no intersection. For another example, use Primes and Numbers less than \begin{align*}40\end{align*}. There would be quite a few possibilities for numbers contained in the intersection. (credit: Teaching Reading in Mathematics, M. Barton and C. Heiderma, 2000)
Associated Materials
 Student Worksheet: Factoring Composite Numbers, http://www.ck12.org/flexr/chapter/9612
Problem 1 – A Frayer Square for prime
Discuss students’ Frayer Squares as a class, recording their responses on the board or an overhead transparency. Use this activity to assess students’ knowledge of the concepts prime and composite.
A prime number can be divided evenly only by itself and 1. A composite number is a nonzero number that is not prime.
Problem 2 – Exploring a factor tree for a composite number
Examine the factor tree of \begin{align*}24\end{align*} with students and assess their prior knowledge of factorization and factor trees.
The number at the top of the tree is the number being factored. Moving up the factor tree represents multiplication and moving down it represents division. The circled numbers, or leaves of the tree, are prime. They make up the prime factorization.
Discussion Questions:
 What if the factor tree starts out with two different factors than the ones shown? Will you still get the same answer? Challenge students to change the factor tree using two different starting factors.
 Does it matter in what order I list the final prime factors? Why might a teacher prefer that the numbers be listed in ascending order?
 Why is the exponent form used for the prime factorization?
Problem 3 – Exploring division as a means to finding prime factors
Demonstrate how to find the prime factorization of \begin{align*}30\end{align*} while students follow along on their calculators. The factor trees on their worksheets help students track their progress.
Students will follow this example to complete the prime factorization and factor tree for the number \begin{align*}36\end{align*}.
Things students can do after working through this problem:
 How are prime factors used to create GCF and LCM? Demonstrate with an example.
 Explain how Eratosthenes’ Sieve works to a younger student, to a peer, or to an adult.
Problem 4  Factoring on your own
Students can demonstrate the use of prime factors in selecting a common denominator for fraction addition or subtraction problems.
They can use either method from the activity, factor tree or division, to find the prime factorization of each denominator.
Students can use the common denominator to simplify the expressions.
\begin{align*}& \frac{a}{126}+\frac{a}{84} && \frac{5x}{78}\frac{x}{66} && \frac{n}{30}\frac{n}{63}\\ & \frac{a}{2 \cdot 3 \cdot 3 \cdot 7}+\frac{a}{2 \cdot 2 \cdot 3 \cdot 7} && \frac{5x}{2 \cdot 3 \cdot 13}\frac{x}{2 \cdot 3 \cdot 11} && \frac{n}{2 \cdot 3 \cdot 5}+\frac{n}{3 \cdot 3 \cdot 7}\\ & \left(\frac{2}{2}\right)\frac{a}{2 \cdot 3 \cdot 3 \cdot 7}+\left(\frac{3}{3}\right)\frac{a}{2 \cdot 2\cdot 3 \cdot 7} && \left(\frac{11}{11}\right)\frac{5x}{2 \cdot 3 \cdot 13}\left(\frac{13}{13}\right)\frac{x}{2 \cdot 3 \cdot 11} && \left(\frac{3 \cdot 7}{3 \cdot 7}\right)\frac{n}{2 \cdot 3 \cdot 5}+\left(\frac{2 \cdot 5}{2 \cdot 5}\right)\frac{n}{3 \cdot 3 \cdot 7}\\ & \frac{2a}{2 \cdot 2 \cdot 3 \cdot 3 \cdot 7}+\frac{3a}{2 \cdot 2 \cdot 3 \cdot 3 \cdot 7} && \frac{55x}{2 \cdot 3 \cdot 11 \cdot 13}\frac{13x}{2 \cdot 3 \cdot 11 \cdot 13} && \frac{21n}{2 \cdot 2 \cdot 3 \cdot 3 \cdot 7}+\frac{10n}{2 \cdot 2 \cdot 3 \cdot 3 \cdot 7}\\ & \frac{5a}{2 \cdot 2 \cdot 3 \cdot 3 \cdot 7}=\frac{5a}{252} && \frac{42x}{2 \cdot 2 \cdot 11 \cdot 13}=\frac{42x}{858}=\frac{7x}{143} && \frac{31n}{2 \cdot 3 \cdot 3 \cdot 5 \cdot 7}=\frac{31n}{630}\end{align*}
Solutions
Problem 1
Definition:  Fun facts: 



Examples  Nonexamples 


Problem 2
 The exponent is three because there are three \begin{align*}2s\end{align*}.
 The number \begin{align*}24\end{align*} is not a prime number because it has factors other than \begin{align*}1\end{align*} and itself.
Problem 3
 There are no exponents in this example because there is only one of each factor.
 \begin{align*}36 = 2^2 \cdot 3^2\end{align*}; factor tree will vary
Problem 4
Factor trees will vary.
 \begin{align*}27 = 3^3\end{align*}
 \begin{align*}56 = 2^3 \cdot 7\end{align*}
 \begin{align*}72 = 2^3 \cdot 3^2\end{align*}