I didn’t have the greatest teaching day today, so I thought I’d try to remind myself that I CAN do this teaching thing by describing an activity from a few days ago that was successful.
Factor Craze, which I think I found via Fawn Nguyen, is one of NCTM’s monthly “Problems to Ponder”, and it asks:
Which numbers have exactly three factors?
Which numbers have exactly four factors?
Which numbers have exactly five factors?
This problem was a great introduction to factoring with my FST (2nd half of an extended Algebra 2) kids. They saw factoring last year, but this year I wanted them to really understand how they were coming up with the equivalent expression instead of following a list of steps from the teacher. So I used Factor Craze to spark some conversations about factors.
I have my students seated in groups, but I had them think on their own for a minute before working with their group. I actually started with the question Which numbers have exactly two factors?, which may seem rather elementary for high school juniors and seniors, but as I suspected, many had very little knowledge or experience with the concept of prime numbers.
Most groups started by writing down examples of numbers that had the required number of factors, but I prompted them with, “What’s a way to describe ALL numbers that have exactly ___ factors?”. All groups eventually came up with prime for exactly two.
When they got to exactly three, most groups found out that 4 and 9 worked. I asked them if there was anything special about numbers 4 and 9. “Oh, oh! They’re perfect squares! Perfect squares have exactly three factors!”
So I respond with, “Do all perfect squares? What about 16 and 25?”
“16 doesn’t work. Oh. But 25 does!”
So I say, “Nice. So some perfect squares but not all perfect squares. What type of perfect squares work?”
And so on. Most groups figured out that squares of prime numbers have exactly three factors. Only one group in each class was able to delve into exactly four factors before we ran out of time.
I really liked how this problem posed a challenge for every student. For some, just remembering what prime numbers are like was a challenge. For others, it was recognizing a theme for exactly four factors. Either way, all students were developing an understanding of factors.
We later moved on to greatest common factors and factoring expressions, and I think laying the ground work with Factor Craze made a difference.