MVP Lesson 1.6 Debrief & Final Plans

10.2 MVP 1.6 Day 4

I wrote about my plans and first day for MVP Lesson 1.6 here and here. After two more days of seeing kids working on this problem, I have some thoughts.

  1. It really was unnecessary to leave the statement of the context as “15 pounds in the machine, 180 candies per pound”. We spent a ton of time talking about why that means there are 2,700 pieces of candy in the machine to begin with. Time that I think would have been better spent looking at the mathematics of this decreasing sequence.
  2. We’re moving REALLY slowly, but the kids are actually OWNING everything we’re doing. I think you could describe previous years of my classroom as “constructivism/problem-based-learning lite”. Like, I wanted kids to construct their own meaning for everything, but there was always outside pressure to hurry up through the curriculum, so I had to find a balancing act. This is the first time I feel I can actually let EVERYTHING (except for conventions of notation) come from the kids. I will admit, however, that even I am surprised at how much longer this deep learning takes.
  3. Most of my kids have figured out an explicit equation and/or a verbal recursive equation for this situation. However, they are hardcore grasping at straws when it comes to using function notation for the recursive equation. We had a PD day on Friday so they’ve had a long weekend to forget everything. My plan for Monday is to have them work through a review of stuff they’ve already figured out, and then hopefully have a class discussion in which they figure out how to write the recursive equation using function notation.

Tuesday is going to be a bit of a chaotic day. I’m going to have kids working on posters to summarize their learning. I’m also going to be handing back tons of papers and having them organize their binders. I haven’t quite figured out the logistics for how to make sure everyone is using this time productively.

On Wednesday – Friday we’re going to do MVP Lesson 1.7. I’ve been working on converting it into a Desmos activity. Stay tuned for a finalized version!

MVP Lesson 1.6 Day 1 Debrief

9.27 MVP 1.6 Day 2

I started MVP Lesson 1.6 today. I began with a notice/wonder of the context:

Screen Shot 2017-09-26 at 5.12.00 PM.png

A key question I had going in was whether to make sure kids had thought about the total amount of candy in the machine (15 lbs * 180 candies/lbs) BEFORE releasing them to group work. I decided to opt not to do that, because we’ve had some long lesson launches recently and I wanted to get kids working in their groups as quickly as I reasonably could.

Three of my four classes had a student notice that there are 2,700 candies total, so in those classes we talked about why that was true. One class didn’t have anyone notice or wonder about the total amount of candy, so they went into their groups without having discussed that. This period made a lot of tables showing # of candies as a function of pounds.

Even the other classes that had discussed the total candy in the machine during the lesson launch struggled to note during group work that the question was asking them to represent the number of candies IN the machine (not the amount of candy coming out). I saw a lot of tables showing the sequence 7, 14, 21, …

In fact, in all of my classes, I only had one group find 2,700–7=2,693. When I asked them why they had subtracted and what the 2,693 represented, they freaked out and erased it and decided to multiply instead of subtracting. I saw a lot of work that seemed like kids were just doing mathematical operations without slowing down to think about WHY they were choosing WHICH operation and what the numbers MEANT.

The focus of this lesson is comparing increasing and decreasing arithmetic sequences. Thus, tomorrow’s Warm Up is designed to prime them to make a decreasing arithmetic sequence.



The Week Ahead: Master Designer & MVP 1.6

Monday: Master Designer (an activity to promote group work)

9.25 Master Designer Group Work Norms

Some of my classes have been working great together at their #VNPS, and others less so. There’s one class in particular that has a lot of built-up animosity among students dating back many years (to middle and even elementary school).

In an effort to promote positive group interactions, I will be doing Master Designer on Monday. This activity is from Designing Groupwork: Strategies for the Heterogeneous Classroom.

Basically, each kid gets a collection of cut-out shapes (9.25 Master Designer Shapes). The “Master Designer” arranges the shapes into whatever sort of beautiful/creative design they want. Each kid, including the Master Designer, has their binder set up on their desk creating a “privacy cubicle” so that no one else can see their shapes. The Master Designer has to leave their shapes on the table and communicate through words and/or gestures how the shapes are arranged so that the other group members get their shapes to match the Master Designer’s design.

One student sits out each round and checks off when they observe positive group behaviors, such as giving clear explanations or when someone other than the Master Designer helps explain the arrangement to another group member. When someone thinks their design matches the Master Designer’s, they can ask the observer whether they are finished, but the observer can only answer with a simple “yes” or “no”. If they designs don’t match 100%, the observer can NOT say what needs to change – rather, the student who thought they were done needs to ask clarifying questions of the Master Designer to figure it out.

After each round, you rotate who is the Master Designer and who is the observer. For the first round, you can have a particularly loud and distractible student (or a shy and introverted student) be the Master Designer for each group. This forces that student to actively participate because the entire group relies on them!

The Warm Up for this lesson has two purposes:

  1. Have the students suggest the positive behaviors that the Master Designer activity reinforces. It’s important to have leading questions ready to make sure the target behaviors are identified “organically” from the discussion.
  2. Demonstrate that you can describe characteristics of shapes without fancy vocabulary. For example, if you don’t know the word “square”, you can still describe a square. You can also gesture. I’ve done this activity successfully with newcomer ELLs, and this point is important to make in the Warm Up.

Depending on how much time it takes to go over the directions for the activity, this might be the whole period, or we might have a bit of time to kill at the end. If there’s some extra time, we’ll work on practicing function notation for recursive functions: 9.25 Recursive Functions.

Tuesday and Wednesday (and maybe Thursday): MVP Lesson 1.6.

9.26 MVP 1.6 Day 1

Screen Shot 2017-09-24 at 12.14.07 PM

We’ll start with a #noticewonder of this scenario. A key thing students need to do early on in this lesson is to realize that the total amount of candy is (15 lbs)(180 candies/lb) = 2700 pieces of candy. I’m hoping that this comes out of the notice/wonder conversation, but if it doesn’t, I think I’m still going to release kids to groupwork. We spent so much time talking about the context of lesson 1.5 as a class that they really didn’t start working on the problems until day 2. I don’t want this to happen with 1.6, particularly because even if they start off thinking that the machine begins with 180 pieces of candy, they’ll still be able to grapple with the mathematics behind a decreasing arithmetic sequence.

I want the debrief to this task to bring out a lot of important mathematics:

  • recognizing that an arithmetic sequence can decrease
  • understanding that multiplying neg*pos is repeated subtraction the way pos*pos is repeated addition
  • comparing the graphs of increasing and decreasing sequences
  • writing explicit and recursive equations using function notation

As such, I expect this lesson to go at least into Wednesday, possibly Thursday with groups making summarizing posters.

If everything happens to be amazing on Tuesday and Wednesday and I feel like they really don’t need another day on this lesson, I’m planning to use Thursday to review multiplying negatives by introducing the students to Desmos with this Desmos activity.

Building this familiarity with Desmos will be particularly helpful since I’d like students to use Desmos when working on lesson 1.7 the following week. In fact, I’m even considering turning lesson 1.7 into a Desmos activity…

Friday is a Professional Development day, so the kiddos get a long weekend.

Happy teaching, everyone!

MVP Lesson 1.5 Debrief

9.21 MVP 1.5 Day 1

I wrote about my plans for lesson 1.5 here. Honestly, this was kind of a frustrating lesson. I was worried about my ELL students understanding the chain letter context in the original, so I rewrote the problem to be contextualized in twitter. Some of my colleagues made a version grounded in instagram, which probably resonated with the students more, but my twitter version had the retweet symbol integrated into the text, which I thought would be helpful for my newcomers.

Literally every class struggled to understand that the context involved multiplying at each successive step. I hadn’t had a chance to make a tree diagram ahead of time, and I think that might have contributed to the confusion. I did get a few good student-generated ones in my last class of the day, so of course none of the other periods had the benefit of seeing them until day 2.

The kids weren’t that invested in the context, and confusion about the context seemed to be distracting from the core mathematics of geometric sequences, so I didn’t really push for a deep and thorough lesson discussion at the end. We’ll see plenty more exponential functions again.

One key take-away I did have, however, was that students tend to scale their graphs by evenly spacing the numbers in their table. This means they essentially create a log scale for the y-axis and make their exponential functions look linear. I’m undecided if I want to address this by letting the mistake occur and discussing as a class, or if I want to give explicit directions on scaling axes so that they’ll see the shape of exponential functions. I feel like the ideal is letting the mistake occur and discussing it, but sometimes there are so many things that need discussing that that may not be the best use of time. TBD….

MVP 1.4 Wrap-Up: Function Notation

9.20 Function NotationIMG_2323

These are the notes that I used to introduce function notation after our wrap-up discussion on lesson 1.4. As you can see, I only got through the first example, but I think that’s fine.

Almost all of my students had never before seen the formal mathematical concept of “function”, nor had they seen function notation, so this lesson was really just about planting some seeds in their brain that still need a lot of time to mature.

When writing the function recursively, I said f(d) is the “current” and wondered aloud what the “next” would be. I wish I could remember all of the ideas they shot at me, and I wish I had the time to analyze the thinking behind those ideas, because they were all over the map!

I hope some students try to write their recursive equation using function notation when we do lesson 1.5. It’ll be interesting to see if the knowledge transfers to a geometric sequence.

MVP Lesson 1.4 Day 1 Debrief

1.4 Scott’s Workout Day 2 JH

Today, I implemented the lesson that I blogged about here. (It’s MVP Lesson 1.4.) Each class was fairly different, so I’m going to have to break it down period by period.

Period 2 (sheltered ELD): 

Some good tables and graphs. A couple of groups said the equation is 2n. One group said 2n+1, but their reasoning for the +1 was that it’s because it’s day 1. One group seemed to have a solid understanding of the equation, although I didn’t get to talk to them so I’m not sure if it’s the whole group or one kid in the group.

This was a day where I could really tell these lessons were designed for block periods, rather than the 59 min I have. Around the point that most groups started to lose focus or feel like they had done everything they could, we had about 7 minutes left in class. If it were a longer period, it would have been a perfect time to transition to a whole-class discussion.

I’m still trying to figure out how to break these lessons across a couple of days without having kids lose interest or momentum.

I ended up giving each group an 11″x17″ mini-poster and telling them to write down everything they had done on their #VNPS. This will hopefully help us launch our discussion tomorrow, and it gave me a less chaotic way to review each group’s work rather than worrying about figuring out what they did and also whether everyone was on task and helping each other.

It did, however, make the end of class a bit frantic, as kids really didn’t have a lot of time to transfer their work onto their mini-posters.

Periods 3 and 6:

Both of my gen-ed classes had about 10-15 minutes for discussion at the end. I asked them at the outset to focus on where we see that +2 in all of the various representations. They were super clear on it in the table, and after looking back over their notes, identified this as an arithmetic sequence.

They were super NOT clear on where this common difference appears on the graph. I think the concept of slope is fairly new and unfamiliar for most of them, so I’m not going to focus on it here since I know it will come up in much more depth later.

They were all pretty solid on the recursive equation, which they phrased as “Next=Current+2, Initial=3”.

In period 3, no one really attempted the explicit function, however two groups did work for #1 (how many push-ups on day 10) that could lead to an explicit equation. One group said 10*2+1=21. Another said 2*10=20+3=23. They also labeled the 2 as “growth” and the 3 as “initial”.

At the end of class, I pointed out that the group that got 23 has reasoning that seems to match the recursive equation, but that that somehow led them to the wrong answer (since we know 21 is right from the table). Hmm…. I’m hoping that comparing these will lead to a discussion on an explicit equation.

In period 6, I noted that one group said 2n+1 and another said 2n+3. We were pretty much out of time at that point, so I left that as something to be resolved tomorrow.

For both period 3 and period 6, I’m planning to start tomorrow by discussing these unresolved issues. After that, I would like to introduce formal function notation, starting with the explicit function, and then doing the recursive function. However, if they seem too squirmy to sit for notes after opening with a class discussion, I’ll have them make posters showing this pattern in multiple representations and we’ll save function notation for Wednesday.

Period 5 (sheltered ELD):

This class is super challenging. It’s huge (35 students!), it has some personalities that require constant supervision, and it’s all newcomers. On top of that, there is a HUGE discrepancy in background knowledge – more so than in any other class. There are some kids who literally wrote the explicit equation perfectly the moment they saw the bar graph, and others who struggle with the table.

I’m trying to instill a culture of collaboration, and I’m trying to combat the many status issues that this set-up engenders, but it’s been a challenge.

Like period 2, this class didn’t have time for any discussion, and they also had a bit less time (and were a bit more chaotic) than I felt would be reasonable for mini-posters, so I had them summarize their group’s work on the back of their notice wonder sheets at the end of class. I’m going to give them 10 minutes to make posters (maybe mini-posters?) tomorrow, and then do the discussion.

MVP Lessons 1.4 & 1.5

1.4 Scott’s Workout Day 1 JH

I’ve caught up to the present in writing these blog posts about using the MVP Curriculum, so I’m currently writing on Sept 16 about my plans for the upcoming week.

I’m excited about lesson 1.4, because I think it will be fairly accessible for most students, yet there’s a lot of rich discussion that can come out of it. The MVP teacher’s notes say to focus on the common difference between consecutive terms, and where we see that common difference in each representation (table, graph, explicit function, recursive function).

In my sheltered classes, I’m going to ask a student to show us what push-ups are, to make sure we’re all on the same page about the context here. I can already think of a few hyperactive kids who I’m sure will be happy to volunteer!

After reading through the scenario as a class, I want to give some time for students to notice and wonder. We’ll do this individually first, then in pairs, and then as a class.

This will serve as the launch for the lesson, and then I’ll have students work in groups at Vertical Non-Permanent Surfaces (#VNPS).

Where we go from there will be entirely dependent on what the students produce. The MVP teacher’s notes recommend sequencing student responses during the whole-class discussion as follows: table, graph, recursive equation, explicit equation – emphasizing with each where we can see the common difference.

I haven’t formally discussed function notation with my classes yet, so I’m planning to introduce that (for both explicit and recursive functions) at the end of this lesson.

This lesson is starting on a Monday, and depending on how things go, the final wrap-up with function notation may take place on either Tuesday or Wednesday. If we’re able to finish up this lesson by the end of Tuesday, then I’m going to use Wednesday’s short period to do another Illustrative Mathematics lesson to review operations with signed numbers – something I’ve seen my kids struggling with a LOT.

Either way, I plan to start lesson 1.5 on Thursday.

1.5 Don’t Break the Chain Day 1 JH

Lesson 1.5 has kind of terrified me as a teacher of sheltered classes for English learners. This is the original problem context:

1.5 Context.png

There’s so much text! And it’s so culturally dependent! After talking this over with my Math 1 co-lead as well as the ELL department chair, I have decided to rework the context to make it easier for my English learners to access.

I would love feedback on what I have made! I initially wanted to do something more tangible and concrete, but I also didn’t want the common ratio to be 2, since that’s what the kids saw with the last geometric sequence in lesson 1.3.

I don’t think a lot of kids know about chain emails, but they are on social media. I originally wanted to make the context based on Instagram, but to be honest I’m not on Instagram, and I was worried that I’d write it in a way that doesn’t make sense based on how the app works. I decided to use twitter because I think most kids are familiar with it and I understand how twitter works.

That being said, if you can think of a different, non-internet-based context for a geometric sequence, please let me know! I’m definitely not wedded to my version here.

This is what I came up with, keeping the numbers the same as in the original MVP lesson:

1.5 Twitter Context.png

Again, I’m planning to launch with #noticewonder (individually, in pairs, then as a class). I think I’m going to make a tree diagram to help explain the context in my sheltered classes. My goal will be to NOT use it, or maybe if a kid draws something similar to have them explain their tree diagram and how it represents the problem.

I don’t want to prescribe student thinking on this problem, but I also want something in my back pocket in case my kiddos are totally confused by this context.

Once they (hopefully) understand the context, it will be off to work in groups at #VNPS!

MVP Lesson 1.3

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:
Most of my kids saw the pattern today as three separate groups of dots that were growing (rather than as a triangle that’s doubling). I didn’t bother focusing on the different ways to conceptualize this because that seemed less important than the notion of doubling.
In pretty much every class I had some kids who saw the pattern with addition and some with multiplication. I sequenced addition first, then multiplication, and had them notice that with multiplication the number stays the same each time (*2).
This is the moment where my sheltered classes are going to diverge a bit. By the end of class in my sheltered classes, we had seen the pattern with addition and multiplication, not necessarily understood both well, and still had lots of questions about what happens at t=5min.
In gen ed, we ended class with a clear understanding that the number of dots is doubling, along with what the picture looks like at 5 min.
(Well, that was 6th. In 3rd they understood the doubling and a kid with the correct diagram explained how he got it, and then the kids all asked me to tell them whether or not he was right and when I put it back on them they got mad. So I left that one as some unfinished learning and made the point that the person who does the talking does the thinking, and the person who does the thinking does the learning, so they need to step up to the plate.)
I feel pretty good about sticking to my Thursday plan in gen ed. I’ve added a couple of questions to the WU to review exponents, then we’ll discuss the pattern and start working in groups.
In my sheltered classes I’m going to have them draw the picture at 5 min after the WU and then discuss as a class what that looks like before releasing them to groups. Unless tomorrow goes exceptionally well, I doubt that we’ll be finished with all the big synthesizing ideas by the end of Friday. I’ve requested to have my sheltered classes do MAP at the end of next week so it won’t be too awkward if they’re finishing 1.3 next Monday.
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:
Happy Friday!

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”.
Then on Wednesday I’m going to use an Illustrative Mathematics lesson to review adding and subtracting on the number line. This is the one I’m currently planning on:
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.


Period 2

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.

Period 3

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

  1. 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.

Period 5

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.

Period 6

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.

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 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

MVP Lesson 1.2

This is the first post in what I’m hoping will become a fairly regular series reflecting on the Mathematics Vision Project (MVP) Math 1 curriculum in my class this year. This first post is going to be a bit on the long side because it involves several days worth of reflections that I emailed out to my Math 1 team before I had decided to start blogging my thoughts.

For the most part, I’m going to just copy-paste the emails that I sent my colleagues, rather than trying to polish them up too much, because one of my colleagues described my emails as me opening up my brain for everyone to see how I was thinking about this pedagogy, and that’s really what I’d like these posts to be.

I will, however, add small bits of context and commentary where necessary.

This first email is from a Wednesday, which means our periods are 45 minutes long. We had planned as a team to wrap-up/synthesize lesson 1.2 by the end of that Friday. (Wednesdays are our only short days; all other days have 59 minute periods.)

First email, sent midday on the first day of Lesson 1.2

A few quick notes from our release time conversation today:
1) MVP is SUPER inconsistent about using words to describe recursive equations verbally. We have decided on consistently using the following:
2) The questions in lesson 1.3 mention both “recursive formula” and “explicit formula”, so those vocab words NEED to be introduced in Friday’s synthesizing discussion. Kids don’t necessarily need to be comfortable with them, but they need to have them in their notes for reference when they’re working on 1.3 next week. (Also, MVP seems to bounce around with “formula” vs. “equation” vs. “function” somewhat haphazardly).
3) I wanted to share out how far my morning classes got: Period 2 just had time to work on t=100 in their groups. Period 3 had time to do t=100 and then would have had time to discuss it but some kids were disrespectful of another student’s mistake so I had to pause for some culture resetting.

Also, having kids work on whiteboards/windows (#VNPS) was AMAZING! It took a while to set-up and explain directions, but the kids were super engaged and collaborative. A major downside is that I felt like I really struggled to monitor what was happening in each group, but that may get easier as I get more comfortable with the structure.
Second email, later that same day:
An update on my afternoon classes: period 5 struggled to understand what was going on; I think I may not have scaffolded the language barriers well enough (also that class is huge and has some super high-maintenance kids).
Period 6 had some really interesting work. We ended the day with all groups having an answer for t=100. The answers were 400, 401, and 404. If you’re

wondering where 404 came from, they counted by 4s (basically a table with only y-values) until they got to 101 at 25 minutes, then they multiplied by 4. I’m hoping for a good discussion comparing that to 400.
I’ve modified a few of my documents to reflect where the classes are going to be tomorrow, and I’ve attached them here: Growing Dots Group Challenge.
I’ll send out the GN for Friday’s lesson tomorrow.
The next email, after a full 59 minute period of class on Thursday:
Attached are the GN I have made to synthesize learning at the end of lesson 1.2. I’ve attached both the word doc and an example of how I’m envisioning using them.
To be honest, even though I know it’s not ideal to break this lesson up over the long weekend, given how today went, I think I may be doing these notes primarily on Tuesday. I really want to make sure that EVERYTHING comes from the kids (with the exception of conventions of notation). Hence “Kid 1’s Name” in my sample notes; that’ll be a shout-out to a kid who saw the pattern in a particular way.
(I actually did a better job of organizing things in class than in the sample notes I originally sent in my email, so I’m going to upload an image of that when I’m at school next week.)
Then, later that night: 
I know I said earlier that I might do the synthesizing discussion next week, but after my afternoon classes, I’ve realized that my kids have a LOT of confusion about variables and what we mean by an equation for the number of dots at t minutes. I don’t know that further struggle will be productive, so I’m going to do the notes tomorrow and hopefully give them time in class to start their HW.
My goal is still to have as much of the notes conversation be student-driven as possible.
I’ve made a new HW for tomorrow because I don’t think they’re going to have the bandwidth for me to review function notation or exponents after doing the GN.