My 7th graders just finished state testing yesterday and I wanted to help set the stage for what I'm going to expect from them next year as 8th graders. The 7th grade curriculum is not much different than grades 2-6; it's very skill based. My kids are very good duplicators, but I want them to become good applicators and eventually creators. I explicitly said that to them today. I want them to focus on representing their ideas as many ways as they can.
Anyway, the theory goes that you don’t understand a mathematical concept until you understand it in TWO modalities. I do very well with visual knowledge, so my notes of understanding are full of color and pictures and mindmaps and arrows linking concepts, and I highlight the holy hell out of math books. However, I don’t believe I KNOW a concept until I can explain it verbally, because I can barely understand anything if someone just talks it at me.
First swipe is through my best modality, second swipe is through my worst modality. The whole “learning style” thing may be overstated, but it remains true that getting students to understand things in a variety of modalities seems like the way to go.
So today the plan was to:
- Have a little chat re: expectations
- Work through some pattern recognition via sequences
- See if I can get them to describe the patterns at least two ways
- Do an activity with algebra tiles
Chat went well. And I showed this slide:Note: I'm pretty sure you get sequences 1 and 2, but just in case, sequence 3 has to do with words. The first term has one one (11), the second term as two ones (21) and so on. So n5 = 111221.
Kids are really good at finding the next numbers but pretty lazy about writing a good description of the rule. The had to add, "for example" to many of their written rules. The challenge was to have them write a rule that was short but stood alone. The got it.
Challenge: Can we write a rule in Mathanese?
I explained subscripts and how they can be used to name different values for the same variable. For example: n5 represents the 5th term in the sequence. It didn't take long for them to recognize that:
n5 = n4 + n3
"So, if n5 = n4 + n3, what does nk equal?"
"nk = nj + ni."
This led to a great discussion on how the alphabet doesn't nec-ess-a-ri-ly have one to one correspondence with the whole numbers. Good talk nonetheless.
We ultimately settle on nk = nk-1 + nk-2 and for the most part the class was good with that.
Sequence three provoked something I hadn't considered before. Luke asked if the fifth term could be 1231 or even 3112. He explained:
"Each of the previous terms are describing what comes before it, but does it have to be as you read it left to right? Could it just be describing the amount of times the digit shows up in the term."
"So if I would have given you the fifth term, would your rule work?"
Then Joe comes out of nowhere.
"Are there more possible right answers?"
This is just too good to be true. We now have two questions we need answered:
I let them choose which question interested them the most and split the class into two groups to investigate. Group one came awful close to saying that nk = nk-1 + k - 1. They're not quite there so I'll let that one marinate for the night. Group two came to the conclusion that there were no other possible 5th terms. They were a bit disappointed because they thought that they got the wrong answer.
"No, no, no. You had a question. You investigated it. And you decided that the answer was no. Can you explain why the answer is no?"
"Then that's a good day's work."
Gonna have to finish the algebra tiles tomorrow.
Maybe you do this already with sequences, but what I tell the students that helps them "see the light" on writing formulas (recursive or not) is that they should NEVER do any math once they see the pattern. So, your first sequence would end up being 2, 2+1, 2+1+2, 2+1+2+3, ... Once you break it down into the pattern, then it's much easier to connect the subscript to the current term, I think.
I also like the "Give two possible answers for the next number in the sequence given the pattern for the first four" problem. I end up telling them that if you are only given a finite number of terms in a sequence, you can NEVER definitively say what the next term will be. Even 1, 2, 3, 4, ... could be a sequence that continues to just repeat 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, ... and you just happened to stop at the worst time possible.
To be honest, I've never really gotten into the notation in sequences with my 7th graders. Most of what happend in this lesson just kinda happened. I saw where they were taking it, I liked it and we went with it.
I like what you describe. Re-writing the terms using what they all have in common will make defining a variable much easier. Thanks.
I've been reading your blog for a while, but this is my first post. Just wanted to let you know that I'm enjoying and learning.
@Calc Dave: I think of "never do any math" as don't simplify. "Simplifying" is actually one of my pet peeves and I try my best to never use the term (although I sometimes fail because it is quite ingrained). Instead I talk about different forms and the form for the context that is most informative. So in your case, 2+1+2+3 is much more informative than 8.
@David: You mentioned that your students were looking for a different pattern that worked for your 3rd sequence. It's ugly, but if you consider 1 to be the 0th term, (1/3)(590x^3-1770x^2+1210x+3) works. I'd give you the "unsimplified" expression, but that would take away all the fun of how I figured this out. :)
Teacher or Student? Thanks for stopping by; I'm glad you're finding this blog helpful.
I'll check out your rule...thanks.
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