The calendar worked out such that I was scheduled to introduce lesson 1.3 on a Wednesday, which meant only 45 minutes of instruction.
Here are my thoughts before the lesson:
I think that 45 minutes is an awkwardly short amount of time to launch and really dig into 1.3. As a result, I’m just going to discuss the first two questions on Wednesday (How do you see the pattern? What’s the next figure?), and then have Thursday be the main productive struggle time for the harder questions (recursive and explicit equations).
This is my debrief from that first day:
After lesson 1.3, day 2:
Not gonna lie, my kids were super chatty and off-task today, and a significant number were just in a bad mood in general. I say this as a preface to saying that I’m not really where I want to be with this lesson, and I’m having to adjust accordingly.
Also, what happened and what kids understood was wildly different in each class, so I’m going to have to break this down period by period rather than giving a broad overview.
Period 2: Super chatty and unfocused during the WU (also I spent some extra time teaching/practicing English words and pronunciation for exponents), so we only had about 10 min in groups. Most groups had good discussions, but not a lot of crystallization and clarity on the big questions of the day.
Period 3: Pretty stuck on the explicit equation. One group wrote the recursive as previous*2, and when I said there should be an equals sign in front, someone else suggested Current=previous*2. Another group said current*2 but we didn’t get a chance to discuss what should go in front.
There was a lot of awkward silence as I asked questions and no one said anything. I think my main focus for this class is to get them to listen to each other and to realize that I’m not going to swoop in and tell them the answer (or tell them whether a student’s answer is correct).
Period 5: Two groups got 3*2^t. Or rather, two kids got that; I’m not really sure how well they were able to explain it to their group members. No one had a recursive equation.
Period 6: Everyone was super confused by the explicit equation. However, a lot of groups had decent recursive equations, and we had a pretty good discussion at the end of class comparing the various recursive equations and collectively deciding that both Current=previous*2 and Next=current*2 are correct, and that you have to specify Initial=3.
So….where to go from here? Each class’s needs are pretty different, but I’m hoping that by having a warm up tomorrow that models what explicit and recursive equations look like, the kids will have some basic sort of structure to be aiming for. (I think that was a shortcoming of my lesson today – I had kids look back at their previous sequence notes for the words “recursive” and “explicit”, and that was enough for some kids to know what today’s questions were asking but definitely not for all).
I’m really hoping to end Friday with some solid class discussions. I doubt that I will have time for the GN that I made. I’m sort of leaning towards Monday being a day to have kids make posters that compare arithmetic and geometric sequences, and then starting 1.4 on Tuesday.
I’ve put my 1.3 Day 3 materials in the dropbox. I’m having a crisis of conscience about homework, so the homework that I’m giving over the weekend is not math but rather a reflection on how students interact with homework. But I haven’t had dinner yet and that’s a conversation for another day….
After lesson 1.3, day 3, which you will see was a rather eventful day for me:
I think most of you already know about the excitement I had today, but for those who don’t: one of my little darlings in 2nd set off a stink bomb that made the room barely inhabitable. As a result, both my 2nd and my 3rd had to be relocated.
So here’s where I am:
Period 2: ….had quite a bit of disruption at the beginning of class. However, once we moved to another room they were actually quite lovely and had some great discussions in their groups. This is the class that was already behind, so our end-of-class discussion was just about how we can represent the doubling of the dots using either addition or multiplication. I honestly need to think over the weekend about how I want to proceed with this group.
Period 3: This class is always a bit of a struggle, and the lack of structure from being relocated to another room definitely did not help. Some of the groups made tables where they drew arrows to show the multiplication by 2 between rows. I asked them how we might represent this repeated multiplication and a few thought of exponents, but they’re still not super clear on how and why their equations might be correct.
This is also the class that REFUSES to listen to each other or be the ones evaluating whether proposed answers are correct. They’re waiting for me to swoop in and be the evaluator. As much as I know that it would be easier to do so and we would “cover” more material, I am hyper-vigilant to what sorts of habits we’re developing as a class, and I do not want us to get in the routine of me saying which answers are right. That’s their job.
But I also am recognizing that they need some more structure and support to feel comfortable taking that intellectual risk.
Here’s my plan: I’m going to make cards with all of the equations that different groups came up with. On Monday, they’re going to work in groups at their tables to sort the equations and determine which ones they think are correct. They’re also going to have to draw pictures or give verbal explanations to explain WHY their equations are correct and why each operation is what it is.
These are the equations they created that I’m going to make into cards: Current=previous*2, Next=current*2, 2^t, (2^t)*3, (t^1)*3.
Period 5: A couple of kids got 3*2^t, and we sort of talked about it at the end of class, but the discussion was rushed and I know a lot of kids aren’t comfortable with it. Not a lot of recursive equations, and I’m wondering if I need to do more to review/teach the English vocab words “next”, “current”, and “previous”. With this class, I’m sort of debating between doing Guided Notes vs. some sort of card sort similar to what I’m doing with period 3.
Period 6: This is the class that had a good discussion on recursive equations yesterday. Today I really pushed them to focus on the explicit.
They mostly struggled, and I think a lot of kids are not really clear on the concept of writing an equation with a variable that can then be substituted for any value. For example, one group had a table showing the repeated multiplication by 2, and as I drew their attention to that they wrote the equation (2^5)*t*3 because their table went to 4 min so they were thinking about the next step. I even asked them which numbers were important to the pattern and why, going through 3, 2, and 5 individually. They could explain why 3 and 2 mattered and how 5 was only important for 5 min, then they rewrote their equation. I don’t remember what it was, but it still had a 5 (!).
By the end of class, the three equation possibilities that had surfaced were y=3^x, 2^t, and t^2.
I’m wondering if a card sort would help them as well. On the one hand it’s a nice structure that will give them multiple things to look at and discuss. On the other hand, I’m not sure it addresses the underlying issue that students have of simply being confused by the notion of a variable and its purpose in an equation. I need to think about this some more.
In terms of a bigger picture for next week, I don’t really want to launch something that I can’t finish before MAP on Thurs/Fri.
So, on Tuesday I think we will do a synthesizing discussion and/or Guided Notes. Depending on how Monday goes, this might just be focused on 1.3, or, ideally, it’ll be a comparison of arithmetic and geometric sequences.
To be honest, I think it will be nice to have a little extra time to wrap up this lesson, particularly because it will allow me to introduce function notation for recursive equations (as opposed to expressing them verbally). I fear that some of my students think explicit means “has t” and recursive means “has words”.
Later this weekend I’ll send out the sort cards that I’m making for Monday, and I’ll let you all know when I finalize my plans. I haven’t looked carefully enough at the IM materials for Wednesday to know the extent to which I might tweak them, but I will send anything I create.
That last email was after a long Friday! The next email was sent two days later, on Sunday night:
Here are my finalized plans for tomorrow, and an overview for the coming week.
My WU for the sheltered classes reviews the vocab “current” “next” and “previous”. These English words have been barriers to students’ ability to create recursive equations.
Period 2 honestly still needs independent work time in groups, so that’s what I’m giving them.
I’m doing a card sort with students’ equations from last week. I’m going to ask them the following three questions in their groups:
1.Sort the cards into two piles:
recursive equations and explicit equations
- Decide which equations accurately describe the pattern of growing dots.
3.Make your thinking visible:
Draw pictures and write words to explain why your chosen equations accurately represent the pattern of dots.
I’m hoping that after this sorting activity we’ll be able to have a productive class discussion with STUDENT ideas and reasoning taking center stage.
I’m going to have them make posters showing their thinking and multiple representations. I’m hoping that the conversations students have making these posters will diffuse more of the understanding around the explicit equation, and that the WU might help support the recursive equation.
A card sort focused on the explicit equation, possibly followed by making posters.
On Wednesday I’m going to use an Illustrative Mathematics lesson to discuss representing addition of signed numbers on the number line. https://im.openupresources.org/7/teachers/5/2.html
I’ve mostly formatted the lesson for myself, but still need to finish the HW. This lesson starts with a “Which One Doesn’t Belong?” as the Warm Up. Tuesday’s lesson will be entirely dependent on what happens tomorrow (probably some combo of discussion/notes/posters), but I’m going to introduce the WODB structure in Tuesday’s WU. This is a great structure to promote discourse with a low entry-point. See here http://wodb.ca/ for tons of examples, and check out the hashtag #WODB.
I don’t know how helpful my card sorts will be for other people since they’re very specific to the equations that my students came up with.
1.3 Growing Growing Dots Day 4 JH