The first week of school I tried out Factor Craze on my students.
It’s great timing for the beginning of school as we are reviewing prime factorization, exponential notation, and exponents in Pre-Algebra. We had just finished working on the Sieve of Eratosthenes the day before, and this proved very helpful for the problem.
I gave zero directions, and most students started out the same way, listing the numbers that go with each rule. I let them work and work.
During this work I had some questions and heard many great comments.
- Does one count as a factor?
- There has to be an easier way to do this.
- Guys, let’s look for a pattern.
- There’s got to be a pattern here!
There were having an impossible time finding any pattern (well, any pattern that actually worked). They were very creative however, and came up with many patterns that worked for some numbers.
Finally, I couldn’t take it anymore and asked the class, “What have we been working on the past couple of days?” “Prime factorization.” they answered. I gave them the hint, “Well, maybe you should look at the prime factors of all of those numbers you have listed. They looked at me doubtfully, but were happy to have any hint at this point in time so they went to it.
Not long after, students actually started screaming, “I found it! I see a pattern!” Well, they thought they had found part of a pattern, but weren’t quite there. I told them how excited I was that they had found a pattern! Then I asked them to test the next number and see if their pattern worked! “Oh no! The three’s rule doesn’t work for 4! Maybe it’s all ODD numbers!” After they tested AGAIN, “Oh no, it doesn’t work for 9!” It took each group a while, but eventually they all discovered the rule for exactly three factors. I encouraged them to write down their rule in words. Then, I asked them if they could make it more, efficient or algebraic. They were all able to make “a prime number squared” into p^2.
Once this happened, they were inspired! Most of them quickly discovered that this worked for each question. And then I said, “That is sooo awesome! Hey! Look at your 4’s, does your rule work for all numbers listed? What about 6, 10, 14, 15,… ?” I was very excited when someone asked me if there could be more than one rule that worked, because I really didn’t want to give them another hint.
At the end of the class, all students had discovered at least one rule. I had them all take notes of what they had done on the giant whiteboards into their graph books. For homework, I told them that I wanted them to continue to work on the problem, and see if they could discover any more rules. Then, I told them that they could work together tonight, with anyone they wanted in both 7th grade classes. At that point, students started making plans to chat on Skype and Google + that night. I’m not going to lie, that was a pretty exciting moment for me.
The next day they all came in excited to tell me the new rules they had found. I made a chart and posted it up on the board, and let all of the students come and write their rules.
The students had discovered all of the rules, although they did need just a small bit of questioning to help realize that they couldn’t write prime times a different prime as prime squared. Luckily, they remembered their subscripts. So, we used p sub 1 and p sub 2.