Introducing the Concept: Finding Prime Factors Making sure your students' work is neat and orderly will help prevent them from losing factors when constructing factor trees. 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 Take one of the pairs of factors and create a factor tree for the prime factorization of 48 where all students can see it. Ask : How many factors of two are there? If they don't, remind them that the exponent tells how many times the base is taken as a factor. 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?
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. Say : Now let's try one on your own. Find the prime factorization of 60 by creating a factor tree for Have all students independently factor As they complete their factorizations, observe what students do and take note of different approaches and visual representations. Ask for a student volunteer to factor 60 for the entire class to see.
Ask : Who factored 60 differently? Have students who factored 60 differently either by starting with different factors or by visually representing the factor tree differently show their work to the class. Ask students to describe similarities and differences in the factorizations. If no one used different factors, show the class a factorization that starts with a different set of factors for 60 and have students identify similarities and differences between your factor tree and other students'.
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. Developing the Concept: Product of Prime Numbers Now that students can find the prime factorization for numbers which are familiar products, it is time for them to use their rules for divisibility and other notions to find the prime factorization of unfamiliar numbers.
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 try to see if 7 divides it, which it does.
If they don't recognize that 7 divides 91, demonstrate it for them. Next, write the number on the board. Ask : Who can tell me two numbers whose product is ? Students are likely to say 10 and If not, ask them to use their rules for divisibility to see if they can find two numbers. Create a factor tree for like the one below. Ask : How many factors of two are there in the prime factorization of ?
If you start with 2 and , you end up with the same prime factorization in the end, but you end up with a "one-sided tree" that some students may find more difficult to work with. Have students identify ways that they prefer to factor and guide them to explain their reasoning. A prime number is a natural number, greater than 1, that can be divided by only itself and 1. Another definition: A prime number is a positive integer that has exactly two different factors: itself and 1.
Example 2. The only factors of 19 are 1 and 19, so 19 is a prime number. That is, 19 is divisible by only 1 and 19, so it is prime. Example 3. The factors of 27 are 1, 3, 9, and 27, so it is not prime. The only even prime number is 2; thereafter, any even number may be divided by 2. The numbers 0 and 1 are not prime numbers. The prime numbers less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and Composite numbers. A composite number is a natural number divisible by more than just 1 and itself.
But 6 is not a prime number, so we need to go further. Example: What is the prime factorization of ? Can we divide exactly by 2? The next prime, 5, does not work. Example: What is the prime factorization of 17? Hang on
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