I remember the very first time I stood up in front of a room of teenagers and asked them to do something. I nervously gave them a scattered lecture on the intricacies of y = mx + b. As I was talking they were writing down the things I was saying, and whatever I put on the whiteboard they also put into their notebook. During the lecture I even asked the students some questions, and a few of them even raised their hands and offer up answers. Next I told them to get out their workbooks and there was a huge rustling of paper as they actually did it. I told them to go to section 5-4 and do problems #1 – 20 or something like that, don’t remember the exact numbers. Either way, in unison the class asked me “which page number is that?”, I mean it was probably only two students but it felt like they were all asking. I learned students prefer page numbers to section numbers.
At that point it was about answering individual questions. So I basically just floated up and down the rows, or at least it seemed like I floated because I don’t remember hearing my footsteps. Or maybe I just ignored them because the sounds in the room were really beautiful – I was hearing words I wasn’t used to hearing teenagers say, like “slope” and “intercept”. And I was hearing words more familiar to me like “yesterday” and “that’s cool”. The students all knew each other because it was the middle of the year. I was just there for one day as a requirement before beginning a student teaching assignment. I was a guest in their house.
At the end of the period they all turned in their papers to me – full of calculations and circled answers. And their names were all at the top right even though I never asked them to do that. Then a bell sounded and they all packed up and left. I looked at their papers, more specifically their names, and thought about how cool it would be if I actually knew who they were. If I was actually their teacher.
I was amazed at the whole experience. And I’m not saying it was the ideal class, nor am I advocating for any particular teaching strategy – I’m just saying I was amazed.
“What’s your name, what’s your monomial?”
This is a great activity that was inspired by Matt Vaudrey’s Equation Speed Dating. In this lesson each student gets to create their own monomial – which I constrained to having to be even and with a variable. Then they break up their paper into three columns: Partner / Our Binomial / Our Rectangle. The students pick a partner and join each others monomials together to create “Our Binomial”. Then they factor their binomial and represent it as a rectangle by labeling it’s dimensions and indicating the area. I circulate the room and once it appears every group is finished, I have everyone get up and find a new partner. I’m demanding here that all students get up out of their seats and move somewhere new.
After a couple rounds I started having them draw their monomial and their partners monomial as separate rectangles, and then draw them together.
I have been focusing on a geometric approach to factoring, so the rectangle column was a great addition to previous times when I have done this activity but only asked for the solution.
The column “Our Binomial” does a nice job reinforcing that a binomial is the combination of two monomials.
Don’t require them to say “what’s your monomial?”, “do you agree that our binomials is…. “, but inspire them to say it by modeling it. A lot of my students were saying it because I was giving them messages that anytime they get the chance to say “monomial” or “binomial” they need to take it.
Tell the students not to move onto a new partner when they are finished. They need to wait until you tell them to switch partners.
Remind them that you are really counting on the partners to catch any errors! Because you can’t do the problems on the board since every pair is working a different problem. “And yes, you are the partner I am counting on for someone else.”
“What’s your name, what’s your monomial?” No that’s not a pickup line for Speed Dating… or is it?
Here is a released question from Smarter Balanced (I even answered it!!!):
Ok I lied. That was an edited version of a Smarter Balanced question – here’s the original:
Now all of a sudden my answer doesn’t seem sufficient anymore :( Here’s my best guess at a popular student answer:
This word “explain” is keeping me up at night lately. In this problem I’m not sure adding the word explain to the end gains us enough to warrant it. To achieve Common Core we can’t just throw the word “explain” after every problem we did last year and call it a day. By the way I’m not saying that’s what the Smarter Balanced Consortium did on this particular problem. But this use of the word “explain” does bring two things to mind:
1. It’s hard to explain your mathematical reasoning without access to drawing diagrams.
2. If we ask students to explain something – it should be something worth explaining.
With respect to #1 – my focus this year has been on explanations through multiple representations. Basically I have students make connections between diagrams, tables, graphs, mathematical symbols, and written descriptions. I feel underwhelmed asking students to explain with just a typed explanation. I want explanations to look like this:
In the student work above – image if it was only the conclusion. Look at how much would be lost.
There are certainly better answers to the rectangle problem from Smarter Balanced than I offered up here. I actually really like the problem itself, I just do not think having them explain it gains us much versus just solving it.
It’s hard to explain the word explain. It’s a word that only makes sense to me until I try to explain it.
I will categorize this post as “sometimes you just need a worksheet”. #SYJNAW for my twitter peeps.
I have always kind of disliked teaching the circles unit in geometry because of all the different rules – tangent / secant angles, chord-chord sides, chord-chord angles, blah blah. This year I put together a learning segment on circles that involved satellites in geostationary orbit. It was based on my experiences working at Lockheed Martin and my engineering background. I will write about it when I have time. But for now I will just attach a couple worksheets I made of problems that I put on a homework, or threw in a test. I figured I would just share these, because you know… some times you just need a worksheet.
These problems themselves involve tangents, central angles, and trig functions. The actual learning unit is very similar, but requires the students to contextualize and decontextualize. So without further comment – here’s some of the practice problems I used:
The Goods: (sorry I only have pdf’s, I create things with Adobe Illustrator)
I created this visual pattern as a followup to the I Rule! exercise from MVP. It is intended to be more difficult than I Rule!. When I gave this to my students, I included a linear T-Block just like MVP does for I Rule!
I asked them two questions:
1. How many squares are in the 10th sequence
2. How many squares are in the nth sequence
Only a couple of my students actually got to the right answer, but the effort was tremendous. I had students coming to me during lunch and saying they had asked all their friends and they couldn’t figure it out. Students were telling me they worked with their parents and couldn’t get it. I had a student (who failed first semester mind you) tell me that her and her two math tutors stayed 45 minutes after their session working on it and couldn’t figure it out. She had two pages of work. I have a couple students who get 100% on everything they touch, and they didn’t figure it out. So yay me! I challenged them :)
Here’s the I Rule! pattern:
Check out many more visual patterns at visualpatterns.org – a site created and curated by my conference buddy Fawn Nguyen (@fawnpnguyen)
Here is the worksheet I used, not sure if I will include the linear T next year.