Students will work on the questions below based on what they learned from playing the Factor Game.
After Cam plays the Factor Game, he says, "Bigger numbers must have more factors than smaller numbers, so I always try to circle small numbers on my turn. That way my opponent won't get very may points."
Do you think Cam's strategy is a good one? Why or why not?
Some students may think the strategy is good, because they don't have a full understanding of prime numbers. Other students may have noticed that some of the larger numbers didn't have a lot of factors, which will lead to a discussion of prime and composite numbers.
I will review the definition of prime and composite numbers. Students may already know these terms, but I will suggest they take notes on the definitions and characteristics.
prime number - a number that only has two factors, 1 and itself
composite number - a number that has more than two factors
For future assistance with identifying prime numbers, I will share some important facts.
- The numbers 0 and 1 are neither prime nor composite because....
- 1 has only 1 factor - itself
- 0 has an endless number of factors.
- 2 is the only even prime number
For method 1, I share with students a story from my childhood. "When I was a little girl I really looked up to my older brother. He was six years older and I wanted to do everything he did. Occasionally he let me hang out with him and his friends. One day they went exploring in the woods and they let me tag alone. Well, they decided to climb a tree. I was scared because the tree was pretty high with lots of branches and leaves. I didn't want them to make fun of me so I decided to try it. I shocked them and myself and climbed up several branches. I believed I gained a little respect from the boys that day."
For method 2, I ask students to share their favorite type of cake. Answers vary from vanilla, red velvet, chocolate, carrot, ... After several students have shared, I ask them to imagine creating a birthday cake with several of their favorite layers.
I will explain to students that there's often more than one way to do math. I will show them 2 different methods of finding the prime factorization of a number and explain that they can choose the method that they prefer.
Method 1: The Factor Tree Method
Many students may already be familiar with this method, but I will work through an example. Prime Factorization - Method 1 I use the analogy of climbing a tree. If a number is composite there are more branches to climb. If a number is prime, they've reached a leaf and it needs to be circled. A common mistake made by students is to circle a number that is composite and not factor it completely.
Method 2: The Division (Birthday Cake) Method
This method is usually new to students. I explain that it is time to create their birthday cake. What's most important to remember for this method is that the divisor, the number outside the cake must be a prime number. I will work through an example with students. Prime Factorization - Method 2 I will explain that if the quotient is a composite number then they need to add another layer to their cake. If the quotient is prime, then they are ready for the candle (the number 1). A common mistake made by students is to use a composite number as the number outside the cake.
I will give students a few problems to work through. I will suggest to students that they may want to try both methods, before they make a choice of their preferred method. As students, work I will circulate to make sure that students are using the methods correctly. Common mistakes are dividing incorrectly and with method 2, using a composite number as the divisor. I will also remind students that finding factors is easier when they use the divisibility rules.
Using either method, find the prime factorization of:
I will encourage students to discuss and check their answers with their group. (MP3)
I will discuss with students any questions or comments they may have. Students may share that their answers were different from someone at their group and that they realized that they made one of the common mistakes.
This is an opportunity to clarify any misconceptions that students might have and for students to share their understanding with their classmates.