This year's enlightenment came unexpectedly when I placed a seemingly easy warm-up problem on the board yesterday. We've pretty much discussed similarity in as much detail as possible, so my students' brains are fairly swollen from the amount of knowledge they've gained up until this point (more on that in a second). The problem looked like this:
"My classroom is about 30ft long and takes up about 700 sq ft. If I want to draw it to scale so that it has a length of 5in, how much space on my paper do I need to reserve for my classroom?"
No problem, right? Calculate your scale factor, square it, use that to calculate your new, smaller area. Done and done. I figured we'd be able to answer this question plus the three others I had on the board within 10 minutes tops. Right? Sounds reasonable? I mean, it is just a simple problem, nothing new. We've been talking about these relationships for the last two or three days. Welp, as it turns out, I'm bad at predicting these kind of things.
Here is brief visual as to what ensued (if only my whiteboard could've captured everything that was said. Unfortunately I had to erase some awesome ideas because I was running out of room):
(I apologize for my poor panoramic cut, slice, paste job)
This single problem generated more discussion than anything we've discussed all semester. My whole 10 minute idea went out the window, we spend ONE HOUR talking about this problem. Now, I'm not upset by this. I believe there are many teachers out there who at this point in the blog would begin writing about how annoyed they were that they had to reteach topics and what not. I had quite the opposite experience; this was one of the best hours of my teaching career! I had 10 different students show 10 different ways on how to solve this thing. It was ridonkulous (that's right, I went there).
Typically with warm up problems I #hashtag students to put their work up and explain to the class what they did and then we discuss as a class. For this problem, I had heard a good amount of students talking about it, so I decided that we would talk about it together rather than have someone put their solution up. I asked for some suggestions as to what to do and I had a student offer to calculate the width of my actual classroom, calculate the scale factor, find the similar width, and use the dimensions to calculate area of the scale drawing. No problem. Another student converted feet to inches first, calculated scale factor, squared it and used that to calculate new area. Awesome. Another student set up a proportion by first converting everything to inches, then calculating the width of the classroom and setting up 360/279.96 = 5/x. Cool. Another said (and this is where it started to get interesting) to set up a proportion like this: 700/30=x/5. This was a nice way to discuss how because we have area, the proportion should be set up as 700/30^2 = x/5^2 to keep the relationships the same. Great, some learning is going on. Yet another student said, "Mr. Brandt, I did differently. I think I did it wrong, but I still got the same answer." She set up 30ft/5in to get a scale factor of 6 and went from there. Another student set up 360in/5in to get a scale factor of 72 and continued on. Both worked. There were more solutions offered and all of those processes worked. Everything kept working out! I looked at those last two options and said, "Now wait a minute. We've got two different scale factors for the same problem, but they're both giving us the same answer." Discussion ensued as to why this works. I had a former math teacher from my IU observing yesterday and he started in on the discussion. My students still had some misunderstanding (and to be quite honest, at this point I had seen so many different solutions that my brain was slowly disintegrating) so they went and got another math teacher from across the hall and asked him his opinion. At one point, there were 27 students and three math teachers all working on this problem trying to come up with explanations as to why all of these different solutions worked. It was awesome!! I still had some skeptical students that weren't convinced all this was mathematically valid, so they had me come up with an alternate example so they could test each one out. I put another problem up and everyone got to work. I was pumped! This was education, this was school.
Now, I didn't go into as much detail as I should've for the sake of space and time (and your attention span) but these students really were working for an hour on this one problem. The entire time the entire class was discussing it, either with me or their neighbors. They wanted to bounce their ideas off of someone else to see if they were making sense. When I saw this whole process start to happen, I got nervous that I was going to lose some students. I wasn't sure if I should continue with all of these explanations or just say "Here's one solution, let's move on." But, instead, I put some faith in my students and let it ride. Best decision ever. We were discussing how to set up proportions and make sure that everything was in the right position, how to identify what proportion to set up, how to identify which relationship we've calculated, and the big question: do units matter when calculating similar figures (outside of ft, ft^2, ft^3)? It turns out, they don't (which was news to me). All throughout high school, I was programmed to always make sure I had consistent units, so I convert out of habit. My students didn't and they got the correct answer. Sa-weet!
There are some teachers that I can think of that would've taken this opportunity to say "Woah, woah, woah. We don't have time for this. We've got an answer and we've got other things to cover. Let's move on." And as I said earlier, this thought crossed my mind out of fear of what would happen, but not because of the curriculum. My students were learning; wait, I mean they WANTED TO LEARN. They were begging for this understanding. I believe had I said "Too bad kids" they would've revolted, they were that into it. Who am I to complain when my students want to learn geometry? I believe if I've inspired that kind of passion and desire for learning in my classroom than I've done my job. This is the end goal of education. My students got creative, solved the problem in many different ways, collaborated, discussed, debated, asked 'experts,' rinse, and repeat.
This kind of problem solving and critical thinking needs to exist more in more classes. As I write this, I got an email from a student asking me how she can learn how to think because she can't do anything outside of the examples she sees in class. Students need to be taught how to be creative and apply their knowledge to all kinds of situations. The question of 'why' needs to be answered by teachers, and the relationships and connections to other topics need to be emphasized so students can generate that genuine curiosity and amazement. I know that when I was in high school, my performance skyrocketed when I started noticing how everything was related. The best teachers that I had included this information in their teaching and for the ones that didn't I asked for it. I see too many teachers who are very by-the-book and if its not in the book than it can't be discussed. Books no longer run the curriculum. Teachers need to teach the students, not what's on a pre-made worksheet.
As my students walked out of my class, there was a mix of emotions: excited and overwhelmed, but in a good way. Minds exploded in my classroom yesterday, and I feel bad for the janitor. I'm hoping my room has since been sanitized so that I can do it all over again.