Lesson: Prime Factorization
Introducing the Concept
By making sure your students' work is neat and orderly, they will be less likely to lose factors when constructing factor trees. Have them check their prime factorizations by multiplying the factors to see if they get the original number.
Prerequisite Skills and Concepts: Students will need to know and be able to use exponents. They will also find it helpful to know the rules of divisibility for 2, 3, 4, 5, 9, and 10.
- Write the number 48 on the board.
- Ask: Who can give me two numbers whose product is 48?
Students should identify pairs of numbers like 6 and 8, 4 and 12, or 3 and 16. Take one of the pairs of factors and create a factor tree for the prime factorization of 48 on the board or on an overhead transparency as shown below.
- Ask: How many factors of 2 are there? (four)
How do I express that by using an exponent?
Students should answer 24. If they don't, remind them that the exponent tells how many times the base is taken as a factor. Finish writing the prime factorization on the board as 24 x 3. - Say: When we write a composite number as the product of prime numbers, we have written the prime factorization for the number. In this case, the prime factorization of 48 is 24 x 3.
Next, find the prime factorization for 48 by using a different set of factors. - Ask: What do you notice about the prime factorization of 48 for this set of factors?
Students should notice that the prime factorization of 48 = 24 x 3 for both of them. - Say: There is a theorem in mathematics that says when we factor a number into a product of prime numbers, it can be done only one way, not counting the order of the factors.
Illustrate what is meant by that by showing them 12 = 2² x 3 or 12 = 3 x 2². - Say: Now let's try one on your own. Find the prime factorization of 60 by creating a factor tree for 60.
Have someone come to the board and show how to find the prime factorization of 60. - Ask: Did anyone else start with a different set of factors for 60?
If there are students who have, ask them to show their work as well. If not, show them by starting with a different set of factors for 60. - Ask: If I said the prime factorization of 36 is 2² x 9, would I be right?
The students should say no, because 9 is not a prime number. If they don't, remind them that the prime factorization of a number means all the factors must be prime, and 9 is not a prime number.Place the following composite numbers on the board and ask students to write the prime factorization for each one using factor trees: 24, 56, 63, and 46.