Human Coordinate Plane

So, I haven’t had a lot that I wanted to blog about recently, until I decided to do something a fun my 7th graders on Friday.  I went to home depot and got a 50 foot rope and a can of black spray paint.  I cut the rope in half and spray painted every foot.  I laid one rope piece across the other, and voila! Coordinate Plane. Setup wise, if I had to do it again I would’ve gotten more rope and painted more than a foot apart to give the students some more space, but it went pretty well all the same.

When I initially took them out there, I gave them all a note card that was folded in half.  The outside had a coordinate, and the inside had instructions on changes in the x and y values (x+3 and y-7) of the point.  The changes in those x and y values actually corresponded to another person’s coordinate.  So I had each student find their coordinate and then I chose a random student to start.  I asked him, given his changes in x and y, in what direction will he be moving?  I then asked him to predict who was standing in his new spot.  After he did this, I had him count off the changes in both numbers, and take the place of the person standing there.  That person then followed the same pointing in the general direction of movement then predict who was standing in their new spot directions.  We did this until everyone had moved and the last person had taken the first person’s place.  The only thing I had trouble within this lesson was that my second class is much smaller than my first, but I used the same set of cards, so it didn’t work out quite as cleanly.  I should’ve made them their own set.

Next I wanted to graph some equations.  My students do not know slope or y-intercept or how to graph lines, they simply no how to graph points (x,y) and how to connect those dots to make shapes.  We have been working on functions and two-variable equations recently, and graphing x and y values that make those true, so I asked them to do that.  I split them into two groups, so I could have two lines, and at first I had one do x+y=5 and the other do x+y=7.  Very quickly someone came up with the fact that they were parallel.  We did some other things (graphed x-y=5 and 2x+y=5 and x+2y=5) and they really started to get that they always made straight lines.  We also had good observations about 2x+y=5 being steep and x+2y being flat.

The best part was when we got back to the classroom, in both classes, I had students say they wanted to do more things like that.  They thought it was fun and they felt like they really understood what they were doing better with coordinate planes.  I was also a little shy at first even taking them outside, because I’m new and hadn’t seen others do it very much, but I had multiple teachers tell me that they were glad I was getting the students outside and doing active things.

Have you done anything with a human coordinate plane before?  Please let me know your successes, failures, and ideas in the comments!

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Thank Yous to the MTBoS

First, I want to thank everyone for their kind words here and on Twitter about my last post.  It made me feel like I am doing at least a few things right in my classroom this year.

This week, I was working with 7th graders on scale drawings, and I spent quite a bit of classtime working on a problem where we first scaled the sides and then found the area of the room and then we found the area of the unscaled room and then scaled and we found different answers (although we eventually found that if you square the scale and then scale the unscaled area, you end up with the same answer).  I thought we had a good talk about this, but then a student asked me if their answer was right to another problem, and they had just multiplied the area and the used the scale, which we had literally just proven not to work. I was a good opportunity, though, to let her table partner explain why that didn’t work.

I also have recently realized that I have planned a lot of interesting problem based activities for my 6th and 7th graders, but ever since we started graphing in Algebra, I have found it hard to find good activities.  I have also gotten some pushback, from students and parents, when we try thinking activities that I “haven’t been teaching them anything,” which I have tried to explain to both parties, but I can’t always get them to understand that my class will be a little different than what they have had previously.  Because we were moving away from graphing for a little bit and into solving systems of equations, I thought it would be fun for them to explore finding the intersections of systems of equations.  But that ended up more like pulling teeth.  When I asked them how we might find where they had the same output and input, one student suggested making a function table and looking at the values.  We had some good thoughts about what numbers to plug in (always start with 0 and see if the outputs are getting closer or further away) but when we realized that the answer to the first pair of equations was a fraction, the class got derailed again.  We talked about graphing both lines, which would be a good suggestion if we could perfectly tell from a graph what fraction this might be, but unfortunately that wouldn’t work either.  It took quite a bit of prompting to get them to think about setting equations equal to one another from slope intercept form.  And then it became apparent how many of them we just not paying attention when I asked them to explore 5 pairs of problems for homework and I got comments that ranged from “what were the answers to the ones we worked” (I had already erased them) and “what do I do with the equations to find the answers” (which I refused to answer since we had spent a good 30-40 minutes working on just that).  I also worry that there wont’ be actual thinking about this on their part and it will come down to asking their parents, reading something in the book, or just googling it, but there’s no helping that at this point.  Oh well.  I guess we will try again on Tuesday.

6th grade on Friday was a good time, though.  We played single-step equation Bingo.  They showed some great strategies.  They still struggle with solving singles-step problems with fractions, so when we had those, many would copy them down and use time when they had an easier problem (such as 8x=8) making sure they got the correct answer or checking their answer to fraction problems.

After a tough week I was pretty bummed on Saturday morning and I was cruising Twitter looking for some guidance and and motivation.  I would like to thank @brennemania for linking to Emergent Math’s excellent Problem Based Learning Starter Kit and helping me find some of that guidance and motivation.  Reading this (as well as Dan Meyer’s Unengagables article linked to in this post) helped me realize that I wasn’t the only person struggling and that I could do good things if I kept trying.  I feel like I need to bookmark both of those and read them once every couple of weeks to keep myself motivated.  This is the biggest thing that has helped me since finding the MTBoS is support to pick me up when I am feeling down and the inspiration to try new things and learn from my mistakes.  Thank you to everyone who I have interacted with here or on Twitter for helping me stay positive and giving me feedback on my teaching!

Students Writing Their Own Problems

While I was participating in the Twitter #MSMathChat this week, we got onto the subject of having students write problems, which is a strategy I used while my students were working on adding and subtracting negative fractions and mixed numbers.  @JustinAion and @J_Lanier strongly requested that I do a blog write-up of how I structured this activity, so here it is.

I have a full class set of individual-sized whiteboard that I find incredibly useful in situations like this.  I gave every student a whiteboard, and I asked them to draw a line own the middle and label one side as addition and one side as subtraction.  I then wrote a fraction or a mixed number on the board with the instructions “Write one expression for addition and one for subtraction that contain at least one negative fraction and have an answer of the number written on the board.”  I also asked that they not write the answer, only the expression.  I then collected the whiteboards and handed them back out so every had someone else’s expressions, and I asked them to make sure that the expression was equivalent to the number on the board.  I then asked a few students to share.  Sharing was very open, because if the written problem was wrong, no one had to know who wrote it because no one had their own board back.  As they shared their equation, I asked if they thought it was a correct expression, and then had another student walk through how to get the answer.

I started with 1 2/5, and most students began writing 2 term expressions, but one student in particular wrote 1 2/5 + 1/5 + -1/5, which she thought was gaming the system, but actually gave me a great opportunity to review the inverse property of addition.  Once she had written one with 3 terms, other students felt more free to write in more than two terms, and I got some pretty long equations.  This made for some excellent expressions (I wish I had taken pictures), and forced a bunch of different students to think about complicated problems.  After the students had figured out how to “game” the system by writing simple problems, I would throw a more difficult rule onto my instructions (For subtraction, one number must be a whole number, for addition, you had to use fractions or mixed number with different denominators, etc).  As I added rules, it turned into a nice back-and-forth with them trying to fit their “easy” problems into my new rules and me trying to come up with new rules to push them,

Overall, I enjoyed this more than continually having to come up with my own problems for them to solve, and it offered them a different perspective on the operations.  They seemed to like writing their own and solving other students’ problems.  I can’t think of anything off the top of my head that I would change, outside of giving very specific instruction about them not doodling on the whiteboards.  I actually just used a similar activity today where they were writing algebraic addition and subtraction equations with a specified value for X.

“Mr. Speer, you’re the reason I like math”

This was a very welcome comment after a long week capped off by a rather scattered Friday.  Our schedule Friday was messy because of events in the middle and at the end of the day, which caused me to miss both of my 6th grade classes, which put me off from the beginning.  Throw in that one of our other teachers was out and her homeroom was being rambunctious for their sub, which meant that I had to help some with crowd control, and this comment made me feel very good at the end of the day.

I have found that I expected a little too much maturity right out of the gate, from my 6th graders particularly, so I am moving toward giving them some more structure.  This is going to include a more solid schedule for our daily problem (Mind Bender Monday, Table Question Tuesday, Numeracy Wednesday, Throwback Thursday, and Find the Flub Friday (I stole those last two from someone, but I cannot at this point remember whom… If it was you, sorry for not including credit, but I will if you leave a comment)).  I have also put a little more structure into my classroom, with my tables still forming groups of 2 or 3, but facing more in rows to keep attention at the topic at hand.  I am hoping that this will help some organization for them, and as they progress this year that putting the students in groups of 4-5 will be more of an option.

I have been having some thoughts because of a conversation with my Assistant Principal and a few tweets from Matt Vaudrey (@mrvaudrey) and Sadie Estrella (@waheabug).  My Assistant Principal is worried about our standardized test scores in computation, so she asked me if I give timed tests, which I do not, but I could see if I could find time for.  But the tweets by Matt and Sadie indicate that math should not be associated with speed, and that there is research to support this, so I feel a little stuck.  What I would like to do, particularly for my 6th graders who I have for a shortened period on Wednesdays, is to simply work with them on numeracy one day a week.  We will put away the “content” and simply work on number sense and related skills.  I would like to know if anyone has some good games to build these skills that don’t just seem like worksheets in a fake game shell.

I also struggle with this idea that math is not about speed.  I see our work with basic math concepts to be parallel with reading literacy.  I understand that reading literacy is not about speed either, except that to a degree it is.  We want to get to a point where we see a word, like “cantaloupe,” and we don’t have to sound it out, but we recognize it on sight and we also have an idea in our head what exactly that words stands for.  For my students, I would like them to see a problem like 7+8 in the same way.  So we don’t have to count up from 7 or get 5 and carry the one each time, but that they simply see that problem as equivalent to 15, 6+9, etc.  And I guess I don’t see how that isn’t about speed.  And even if I can draw a line between literacy and speed, how do I explain that to people who aren’t math teachers?  Because I feel that most of them will see that as memorization, even if I feel that there is more to it than that.