The Correct Way to Use the Area Model
Is there a WRONG way to use the area model? I think so, but you may disagree with me. I don’t know where this started, but I am very tired of seeing area models being modeled incorrectly. It is so prevalent that I even saw one of the keynote speakers at an NCTM Regional Conference use the model incorrectly in her presentation. So somewhere, some curriculum (maybe???) has shown teachers a way to use the area model that turns the area model into just a PROCEDURE instead of a MODEL of the problem.
Compare these two versions of the area model:
How teachers are doing it: every model is ‘sliced’ down the middle, making four equally sized areas.
How it should be done: ‘slice’ the sides proportionally, which is hard because I know even my example here isn’t perfect, but I try to get as close as I can without getting a ruler out. However, the only time you should slice a side in half is if you are decomposing that side in half (14 becomes 7+7).
What’s the difference????? Proportionality of your ‘slices.’ The power of the area model is that it gives us another opportunity to talk about how numbers relate to each other. If I cut the 14 into a 10 and a 4, the part that gets sliced into the ‘4’ should be a little less than half the size of the 10. Plus, we really do want the Representation of the Area Model to connect to the Concrete Area Model we can build with base 10 blocks. Models should be a MODEL of what actually happens. When you have a 10×10 area that is larger than a 10×4 area. (FYI, the images below were created using The Math Learning Center’s wonderful Number Pieces app.)
Am I just getting upset about something that I shouldn’t??? I really emphasize with teachers that their area models need to be proportional in order to help the kids make connections. Otherwise, kids just think it is magic and the area model does not become a model…it’s just another procedure. What are your thoughts?
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I pretty much agree with you, especially during the ages that kids are still shaky on how to get the area of a rectangle, but remember they’ll use the area model later for situations where you can’t know how to make it proportional (e.g. modeling (x + 2)(3x-1) before you know what value x has) and/or the “area” doesn’t really make sense as an area (e.g. -1 times 2). Even before that, kids modeling (say) 5034 x 31 would have a tough time making a proportionally accurate drawing! So making it more abstract and less quantitative isn’t the end of the world to me, although I think you’re right that we should try to make it reasonably accurate when we can.
Thanks for the insight, Julie. I agree that later the area model becomes more of an abstract model when we start using it for binomials. But just like all procedures/algorithms we have, we first want to start with developing the understanding before we move to the abstract procedures so that they understand what is going on in the procedure.
I really dig this post Christina and you’re not being picky at all!
Not paying attention to how we partition the area model suggests that structure isn’t important. Structure is crucial in building a conceptual understanding of multiplication AND DIVISION! If the area model is used beyond a set of procedures, then students can see the area as both product and dividend. Shameless plug here but I think you won’t mind …I just posted this week about the importance of students working with area models for both division and multiplication (http://gfletchy.com/2015/03/03/undoing-concrete-models/). If students are asked to reason with models for both operations, the area model becomes way more than a procedure. In which case…structure is everything!
Graham,
I’m totally using your division models with the tiles and base 10 blocks. My pre-service teachers have been working on area model with multiplication and division and they are connecting the multiplication, but they aren’t liking the area model for division. I’m going to try your video with them and the first problem you showed in your post. Thanks for sharing! Love your stuff.
Thanks Christina…All of us are smarter than one of us. Glad I could help out and pay it backwards:-)
Graham, see my reply below. And, btw, I will always consider you famous because I love your video on Memorization vs. Memory. One of the best out there, IMO.
Ditto that! Here is that talk in case anyone missed it: https://www.youtube.com/watch?v=1I0Hi-wPd_E
I agree with everyone on this blog. I remember back in the late 90s when I presented middle school professional development and tried to use alge-blocks with teacher! It was such a foreign model to them- that tray and then add in positive and negative quadrants- whew! If I would have started with the area model for whole numbers, they may have gotten it! I think I will try to determine who is still teaching middle school from then and reach out to them again, because I still , as a coach, do not see any concept development with middle school aged children.
Doesn’t Graham have to stop using the 3-D cubes and start using base 10 flats if we’re picky? (JK, Graham…nice post…although I’m dying to give you linear/edge pieces to use in the photo to show dimensions!) 🙂
And, on being picky with the array…I have mixed feelings. We model a lot with concrete manipulatives before moving to base ten grid paper, which really keeps the model proportional. Then we move to sketches. My sketches aren’t as bad as the equal portions shown in this post, but I’m not that accurate with sketching either; I guess I figured it didn’t matter that much once kids thoroughly got the hang of what the model represented…esp. since, as Julie mentioned, it will eventually come to represent x anyway. Maybe I should try harder to be accurate!
🙂 true point about the base 10 blocks. I think you do have a good point that IF we spend time developing the distributive property, then when we move to the open area models (sketches) we can be more abstract. Problem is, I know a lot of textbooks don’t give students enough development time with the concept before moving to the abstract and so then the sketches seem unrelated to the problem.
Great post! This is something I’ve been thinking about as well. But maybe it’s OK to use the four equal sized boxes not as an area model, but just as a way to better organize the distributed multiplication. The “bow-tie” can be confusing and also misses the place value.
As for the proportional area model, I’ve been playing around with something like cutting out the 4 different sized rectangles (drawn to scale in cm) with the products written inside, give the kids the pieces, and see if they can reassemble it and come up with the factors. Then have kids make their own and mix up like three or four sets to get put back together like a puzzle.
Oh, I am totally stealing that activity, Joe. Thank you.
I have been talking about how important proportionality is when we introduce this pictorial model, I have emailed my teachers this link and pray they read your post and the comments left here, thank you for sharing I love your blog.
Love this post! I enjoy teaching the area models with base ten blocks. We use it for decimal multiplication in fifth grade and using the correct sizings really helps to emphasize the influence of place value on the position of the decimal.