Lesson: Prime Factorization Developing the Concept

Now that students can find the prime factorization for numbers that are familiar products, it is time for them to use their rules for divisibility to find the prime factorization of unfamiliar numbers.

• Write the number 91 on the board.
• Say: Yesterday, we wrote some numbers in their prime factorization form.
• Ask: Who can write 91 as a product of prime numbers?
Many students might say it can't be done, because they will recognize that 2, 3, 4, 5, 9, and 10 don't divide it. They may not see that 7 divides it, which it does. If they don't recognize that 7 divides 91, show them that 7 does divide it. The prime factorization of 91 is 7 x 13.
• Write the number 240 on the board.
• Ask: Who can tell me two numbers whose product is 240?
Students will probably say 10 and 24. If not, ask them to use their rules for divisibility to see if they can find two numbers. Create a factor tree for 240 like the one below.
• Ask: How many factors of 2 are there in the prime factorization of 240? (four)
Who can tell me how to write the prime factorization of 240? (24 x 3 x 5)
• Say: It was better to start this process with two factors like 10 and 24 or 20 and 12 than to take two factors like 2 and 120 or 3 and 80, because each of the previous numbers can be broken down and the end result will probably take fewer steps.
• Say: Since the prime factorization of this number is 24 x 3 x 5, the only prime numbers that divide this number are 2, 3, and 5. Prime numbers like 7 and 11 will not divide the number, because they do not appear in the prime factorization of the number.
• Write the number 180 on the board.
• Ask: What two numbers might we start with to find the prime factorization of 180?
(If students say 10 and 18, commend them on their answer.)
What other numbers could we use?
If no one says 9 and 20, mention it as another possibility. Have half the students use 10 and 18 and the other half use 9 and 20. Have two students put the two factor trees on the board for the class to see.
• Ask: If the prime factorization of a number is 2² x 5 x 7, what can you tell me about the number?
They should say that the number is even and it ends in zero, since both 2 and 5 divide the number. They may also tell you other things, such as it is a composite number, it is greater than 100, that three is not a factor of the number, and so on. Questions like the one above get at a depth of understanding about the prime factorization of a number.
• Ask: If the prime factorization of a number is 3³ x 11, what can you tell me about this number?
Students should answer that the sum of its digits is a multiple of nine and the number is an odd number. They might also tell you that it is a composite number, five is not a factor of the number, and so on.

Give students the following numbers and ask them to find the prime factorization: 231, 117, and 175. Also give students the following prime factorizations of numbers and ask them to write down at least two things they know about the numbers represented: 3² x 5², 2³ x 3 x 13, and 2² x 3 x5.

Wrap-Up and Assessment Hints
Finding the prime factorization of numbers will strengthen your students' basic facts and understanding of multiplication. Students who do not know the basic multiplication facts well may struggle with this, because they do not recognize products such as 24 or 63 readily. Turning the problem around and giving students the prime factorization of a number and asking them what they know about the number without multiplying it out is a good way to assess their understanding of the divisibility rules, the concept of factor, and multiplication in general.