Our 3rd grade team has been working on area the past few days. This team of teachers is awesome to work with because (A.) they put up with me and (B.) their willingness to try “new ideas” and push themselves as educators. Some lessons have gone really well and others totally blew up which is fine because learning is taking place…on our end!
Conceptually speaking, area is one of the most difficult concepts taught in 3rd grade and unfortunately the formula (l x w) is much easier to teach than allowing students the opportunity to struggle and make sense of the math. Here is a typical area question that 3rd grade students encounter in textbooks and on standardized tests:
Because when 3rd grade students look at a picture frame this is what they wonder! Really?!?!
I have an extremely difficult time with this problem for two reasons:
- students don’t care about the area of a picture frame
- many students immediately jump to the formula with limited understanding of why it works.
This problem couldn’t be any further from the real world and nonetheless problematic then memorizing multiplication facts. One of the teachers asked me what an authentic assessment task would look like for area. Below are the fruits of our collaborative effort!
We started by asking the students what type of flooring they prefer the most (carpet, tile, laminate, hardwood).
T: Imagine you covered all the floors in your house with the flooring you wanted!
S1: I’d put carpet all over the house to keep my feet warm but it would cost a lot of money?
T: Oh yeah, how much would it cost?
Students wrote down an estimated cost to re-floor their house. It was quite interesting to see as some students wrote down $30. Students were quick to intercept this thinking by checking the reasonableness of each others’ estimates.
S1: We don’t know how much flooring we need! I can’t figure the price out because I don’t know how big my house is!
Me: Yes we do! Tell me your address. (I typed it in and the sketch below appeared on the projector) Did you want to carpet the bonus room as well? (I asked sarcastically)
Floor plan of the student’s house taken from our county’s public database.
S1: Hey… how did you do that? That looks like my house! That is my house!
The class erupts in laughter and awe!
Sidenote: Every county in the United States has public records database with parcel maps and building sketches. All you do is find the website, type in the street address and you’ll get a floor plan.
The student got up and showed everyone where the different rooms in his house were located. At that point every student wanted to see their own house and the trap was set. Before we started the task, the teachers and I went and searched for every student’s address and printed their floor plan in advance. The fact that each student had their house, gave the task purpose and made it meaningful. We had 2 students that lived in a trailer, which unfortunately doesn’t show up on parcel maps. We had them figure the area of the teacher’s house, which to them was even more special than doing their own home (it’s a K-5 thing because students think that we live at school).
Then a student said it “Hey Mr. Fletcher! Multiplying length x width won’t work because my house is not a square and there are parts missing!
Me: Oh man I didn’t recognize that!!!!! What can we do? (this question was addressed to the class) After about 5 minutes of brain storming…
S3: We could cut our houses into smaller squares and add them up!
And they were off! Unfortunately we didn’t get to finish today but we will tomorrow and we will give the price of flooring to students. It was great to see students reason and struggle with this task. Certain students did not partition the floor plan the most efficient way which lead to confusion down the road but that’s okay. Some students eventually realized that they did not need to floor the garage or the outside patio which was great because it all came back to making sense of the problem.
From the outside this task might appear to be “too difficult” for 3rd grade students but it comes back to making the Math Accessible. Here are 2 of many ways we made the math accessible to students:
Use of tiles to construct the partitioned area.
Use of graph paper to represent their house. The blue part is 2 x 1/2 sq unit to be addressed tomorrow
By using the tiles, graph paper, Base-Ten Grid Paper (for larger floor plans) we made the math accessible to students while engaging them in the Standards for Mathematical Practice…AND THAT’S WHAT IT’S ALL ABOUT!!!’
Not quite a finished product!
As students were working I remembered a little puzzle from Math Pickle. It would be a great ticket out the door after we wrap up tomorrow. Holy cow, I just realized that when we’re done with this task, we would have addressed area without every looking at a picture frame!!!!!!!
Thanks Joe!!! Every time you post I’m immediately able to take something back to the classroom because you continue to address key issues and how to conceptually work through them. I have added you to my blog as well.
(all readers except Joe)….subscribe here for some wicked-awesome K-5 insight: http://exit10a.blogspot.com/
Back to A&P-I think that teaching the two in isolation is a great idea. Although my intuition tells me to intertwine as many topics as possible, this might be one of those instances to go against the norm.
I was talking to a teacher today and it’s really funny that you brought up the grid paper because there were some students that counted the corner square when counting the perimeter (squares around the figure).
It is a huge misconception that was still there this year…but definitely minimized. Loving the idea of twist ties and I’ll definitely be pulling that out at the first opportunity I get.
Cheers and keep sharing!!!!
Graham,
Great work here. I’m going to put a link your blog on mine, which I should have done a long time ago! And my assignment for later is to look at the public records data base.
The struggle with area and perimeter continues. A former supervisor of mine who is now in academia suggests separating the two topics and teaching them months apart rather than teaching them together. Also one thing that has always bothered me about questions like the photo frame: in the picture the side lengths are not really 10 and 8 inches. I wonder what a kid who has no real concept of scale (or has a teacher who does not try to address the issue) thinks about that. I also think that sometimes using grid paper works against us when kids start counting squares instead of sides of squares to find perimeter.
One cool thing to try with perimeter is to construct a rectangle with anglegs (or even with straws and twist ties) measure each segment and add, then unsnap it at one vertex and lay all the sides out in one long segment and measure. If kids can visualize perimeter in this way, as the measure of all the sides laid end to end, it may help.