Making sure your students' work is neat and orderly will help prevent them from losing 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 also will 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 two are there? (4)
How do I express that using an exponent?
Students should say to write it as "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 .
- 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 .
Next, find the prime factorization for 48 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 = 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 only be done one way, not counting the order of the factors
Illustrate what is meant by that by showing them 12 = or 12 = .
- 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 they did, have them 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 , 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 them to write the prime factorization for each one using factor trees: 24, 56, 63, and 46.
- Say: Now that you have a good idea of what a number line with negative numbers looks like, each of you will make a number line using the strip of paper I will pass out to you.
Tell them to write neatly so they can read their number lines and to be careful to place the numbers appropriately on their number line, so that it looks like the one up front.