Sometimes they don't realize that you can add them, but often times they realize they are like terms but end up messing with the exponents. To me, it's really just a matter of how they are looking at it. My best attempt at getting my students to see this problem the way that I see it is by telling them that if everything in a term is the same except for the coefficient(1) is the same you can think of it as a banana. Therefore, the problem above is simply:

Now, they see the answer is 4, but don't totally get it right away, so the banana explanation is quickly followed by:

"So, when you add a banana and 3 bananas, does the fact that you're adding bananas change?"

"No!"

"So the answer is 4 bananas, and what did we say was a banana?", etc

During that exchange the lightbulb goes off and they get it.

Just like the rainbow rule, I'll go through this with a class the first time time it comes up naturally in some problem, and then refresh it as it comes up throughout the year. This method seems to reinforce that it's a tool, and not just "this is how you do this kind of problem". If I have time, I might go off on a tangent about how it works with anything even if they haven't seen it before and go through some quick examples with trig, logs, integrals or crazy fractions and roots. Also like rainbow rule, it helps kids put a name to something they struggle with and attaches an intuitive process to it.

(1)Okay, I usually say "number in front" instead of coefficient at this juncture, but we're trying to make it simple right? Please let me keep my math teacher card.

I use apples instead of bananas (really!) but otherwise, sounds like we're on the same wavelength.

ReplyDeleteApparently @samjshah does this with apples and "blahs". Which makes me start singing I like to blah blah blah apples and bananas....

ReplyDeleteAnd please, delete this comment once Sam's happy.

I can definitely use a simplified version of this in my 8th grade math class right now. They're struggling with like terms - completely baffling, since they had it cold at the beginning of the year.

ReplyDeleteThanks for the idea!

I use the apples too, maybe from seeing the others mentioned? My students will mess with the exponents too. I get specific with my fruits by saying "If we have green apples and red apples in a basket and I tell you to find the number of green apples, do you count all the apples?" That always helps them until they forget two days later. Hopefully it sticks with a few of them though.

ReplyDeleteI like this one better than the rainbows. This one is about why (to me) and the rainbow isn't. This one says, focus on the fact that you have 1 of these monsters and 3 more of the

ReplyDeletesame kindof monsters, so there's 4 of these monsters. (I like monsters, because it looks like a scary monster to the students.)The rainbow seems to me to just be a memory trick. Could be useful, but I'm guessing there's a way to work with those problems that helps students see why they work the way they do.

I used a similar discussion with my students. It's amazing how something as simple as counting and single digit arithmetic can be complicated by what amounts to little more than a label.

ReplyDeleteThe only difficulty I have when doing this (I use apples as well) is extending it to multiplication, because when you multiply, 1 apple times 1 apple is an apple squared.

ReplyDeleteI use this too, but I don't always use bananas. I usually start with apples or bananas the first few times and then I let the students choose... we end up combining like terms by using pizza, elephants, or whatever pops in their head!

ReplyDeleteIt's cool that a lot of us are on the same wavelength here. This was something that always just made sense to me in math class, and my first year or two I couldn't understand why my kids just didn't get it. Through some trial and error I found that naming it "The Banana Rule" and referring to it vs. just telling them it's like adding fruit seems to help it stay memorable for longer.

ReplyDeleteFinally, after going through this a couple times with my class this year we started referring to adding unlike terms as "Making fruit salad" which is pretty fun.

Thanks for the comments!

Would the same concept apply to any variable in algebra? I'm wondering if some students with language issues might be confused by x's, y's, z's, and n's, but more comfortable with fruit or some other symbol in their equations.

ReplyDeleteJust a thought.

Amazing how we all think alike. Mine are usually animals... 2m^2 + m^2 = 2 square monkeys + another square monkey. I'm also in danger of losing my math teacher card, but we gotta do what works!

ReplyDeleteWow. I taught this the same way, almost verbatim (apples and bananas). Maybe we're related.

ReplyDeleteI'm two years two late to this conversation, but I also use this same basic idea with combining like terms. No one else has pointed out a possible "why" in the comments to explain the amount of unintended group-think on this one. So I'll assume the reason why I teach it this way applies to everyone ... we all grew up hearing the saying "comparing apples to oranges" which obviously puts us in a fruit mindset when combining like and unlike terms

ReplyDeleteLOL...I use cats and garbage trucks. They just don't go together....

ReplyDeleteAfter I give them the order of operations rationale, I tell them to call each variable by a noun - but try to make them really different. If it is a constant, we call it dollars. So, 3a + 5b + 4 would be apples, bloodhound and dollars. If they try to given me 8ab, I tell them to picture what a cross between a bloodhound and apple would look like. If they are categorizing them in the room, they would be in 3 different piles

ReplyDeleteYep, we math teachers often think alike... but I've always used cows!

ReplyDeleteI use Algebra Fruit for combining terms because there are no banapples so 5a +3b does not equal 8banapples! Of course, there are pluots and such like but ...

ReplyDelete