MVP 4.1 Debrief and Next Steps

Last week, I wrote about my plans to teach lesson 4.1 in the MVP curriculum introducing solving equations.

There were definitely both highs and lows.

First, the highs:

The 3 reads strategy went super well! My gen-ed kids were SO not into the reading out loud in unison thing, so they did their reads in partners. I found it was better to tell them to read in the story in partners for two minutes and THEN give them the prompt to answer, rather than giving them the prompt at the same time as telling them to read.

The sheltered ELL students, however, LOVED reading in unison, and gave themselves a round of applause after finishing each read-through. Most importantly, they made sense of the context in a genuine and authentic way.

 

This next one might count as both a high and a low. The following statement uses a form of the verb “to leave” both to mean “go away” and to mean “stay”:

Screen Shot 2018-03-04 at 7.20.15 PM
 The students at the front table separated into three equal-sized groups and then two groups left, leaving only one-third of the students at the table. 


I’m not joking when I say we spent a solid 15-20 minutes discussing this in the sheltered classes. That being said, they did understand it, and felt quite proud of themselves for getting the nuance. I apologized for English being such a tricky language, and one of my students muttered to himself, “Change the English.” If only I could…

Now, the lows:

The teacher notes for this lesson begin with this sentence: “In this task students will develop insights into how to extend the process of solving equations—which they have previously examined for one- or two-step equations—so that the process works with multistep equations.”

But here’s the thing, most of my students have not previously examined the process of solving one- or two-step equations.

I knew this going into the lesson, but I also wondered just how much of an impediment it would be. The lesson is designed around building an intuitive and conceptual foundation for solving equations, so maybe we could skip over one- and two-step and just hit the ground running with these multi-step equations.

The work that I saw last week has convinced me that that is not the case. My students REALLY struggled with the notion of “undoing” operations and working backwards to find a solution.

Side note: the first order that the Post-It Notes are presented in has a fundamental flaw – you can arrive at the correct answer with incorrect reasoning!

Screen Shot 2018-03-04 at 7.29.53 PM
·      Some students are sitting at the front table. (I got distracted by an incident in the back of the lunchroom, and forgot to record how many students.)
·      Each of the students at the front table has been joined by a friend, doubling the number of students at the table. 

·      Four more students have just taken seats with the students at the front table.
·      The students at the front table separated into three equal-sized groups and then two groups left, leaving only one-third of the students at the table. 

·      As the lunch period ends, there are 12 students seated at the front table.

Specifically, the process should be:

12 * 3 = 36

36–4 = 32

32/2 = 16

But students can completely neglect inverse operations and do the following:

12/3 = 4

4+4 = 8

8*2 = 16

So, in the future, I plan to change the order of the Post-It notes so that that doesn’t happen!

Not only did my students struggle with conceptualizing the need for inverse operations (and I think the multi-step nature of the problem contributed to this), they also seemed to have no prior intuition for the notion of keeping an equation balanced.

Only a couple of my classes got to the point where they were trying to solve the equations at the end of the lesson, and I mostly saw guess-and-check and some working backwards with informal notation.

Future lessons in this module need students to be able to solve equations with the traditional, formalized notation, but I’m still working on building students’ intuition and conceptual understanding.

As a result, I’m going to take a quick break from MVP and do a couple of lessons from the Illustrative Mathematics 7th grade curriculum.

Specifically, lessons 6.7 and 6.8, which use the notion of balanced hangers to develop both the intuition and the notation for solving equations.

I think that next year, I would probably want to do these lessons before MVP 4.1, rather than after it.

 

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MVP 4.1: Introducing Equations

I have been excited and terrified by lesson 4.1 since I first worked through it in PD. I’m excited because it does a phenomenal job of teaching Common Core standard A.REI.1: “Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.”

Basically, students will figure out the solution (x=15), then work backwards to construct an equation Screen Shot 2018-02-25 at 12.01.34 PM based on which operations make sense in the context of the story.

I love this approach because it focuses students’ attention on the fact that every step in a solution process is an equivalent equation. This is such a deeper and more nuanced understanding than how students usually view solving equations, namely as a procedure of steps that yields a magical answer at the end.

That is why I am excited by this lesson. I’m terrified because in order to build the equations from context, this lesson has a LOT of language demands, and unlike other lessons where some of the language might be superfluous to the mathematics, almost every piece of language in this problem is essential to building the mathematical understanding.

Oh yeah, and I got an email from one of the counselors last week that on Monday I’ll have two brand new students in my ELL class who don’t speak a word of English. Welcome to my life! (I’m just hoping they’re in different sections so my classes stay below 36.)

To see what I mean about the language demands of this lesson, check out the story context:

Screen Shot 2018-02-25 at 12.13.08 PM.png

So, how to break this down? I’m going to use the 3 reads strategy. I’ve never used this strategy before, so I can’t speak very authoritatively on it, but here’s the basic idea:

Read 1: I read the story out loud to students. They listen, then write down everything they can remember. I’m going to post a visual with some of the key vocabulary:

Screen Shot 2018-02-25 at 12.16.54 PM.png

Students will then answer the following prompt:

Read #1: What is this story about?

Write down everything you remember from the story.

Read 2: I’m going to give students the story context on a half-sheet to share with their partners. We’ll do a choral read of the story, then students will answer this prompt:

Read #2: What are the quantities in the story?

What numbers are in the story?

What do those numbers represent?

Read 3: One more choral read (maybe a partner read if they hated the choral read) with the following prompt:

Read #3: What mathematical questions can we ask about the story?

Write down math questions that you can ask about the story.

After each read, I’m going to do a think-pair-share: individual time to write, partner time to discuss, whole-class share-out.

Realistically, this is most likely going to take the full period. Students will keep the notes and questions they’ve created to use as a reference on day 2.

 

Day 2: Working Backwards

After reviewing our notes from the 3 reads strategy, I’m going to display the following paired down diagram of the story:

Screen Shot 2018-02-25 at 12.24.29 PM.png

I’m then going to have students try to work backwards to determine the number of trays in each carton at the beginning:

Screen Shot 2018-02-25 at 12.25.44 PM

Students will do this in groups, and I plan to select a student/group who showed each operation carefully to present their work. We’ll discuss why each operation was necessary, and write down the actions that led to using that operation.

The MVP teacher notes suggest introducing equations to represent each stage of this undoing process:

Screen Shot 2018-02-25 at 12.29.36 PM.png

We’ll do this in guided notes for students to keep for reference. The notes will have the paired down story diagram at the top.

We’ll then pretend that we don’t know that the answer is 15, and we’ll try to solve the equation Screen Shot 2018-02-25 at 12.01.34 PM by undoing the operations and connecting each step back to the original story context.

Days 3–5: Building Equations from Post-It Notes

The rest of the week will be spent on the Post-It Notes story context in this lesson. On the first day, which according to this current plan is a Wednesday and has a shortened period, we’ll just decode each Post-It Note and make sure we understand the operation involved. I’ll leave the class hanging with the question of whether or not the order matters:

Screen Shot 2018-02-25 at 12.34.41 PM.png

Then, on Thursday we’ll do question 2:

Screen Shot 2018-02-25 at 12.36.19 PM

And maybe start question 3? I’m hoping that by the end of Friday most students will have a solid understanding of question 3 and be ready for a class discussion about it. That’s the goal anyway…

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These are the lesson materials I have so far:

2.26 MVP 4.1 Day 1

2.27 MVP 4.1 Day 2

2.27 MVP 4.1 Sequence of Events Diagram

2.28 MVP 4.1 Day 3 Post-it Notes

Alcohol and Asymptotes: A Real-World Context for Rational Functions

This post is in response to the following tweet:

When I was a fellow with Math for America DC, my amazing mentor, Rosalie Dance, observed a lesson I did on transformations of y=1/and said I should bring in some real-world examples to explain asymptotes. To which I responded, “What?? There are no real-world examples of asymptotes!”

It turns out there are. And not only that, Rosalie had coauthored an incredible lesson on them.

Maybe I’ll give a more detailed breakdown of how I teach this lesson at some point in the future, but my single biggest piece of advice is this:

Do all of the problems ahead of time, at least two or three timesSeriously. The strength of this lesson is that it has students interpret the meaning of asymptotes in context. This is subtle and nuanced, and you need to be completely familiar with what quality answers look like.

Let me know if you have any questions about how to implement this!

Here’s the file: Asymptotes and Alcohol

Beginning Module 3 (MVP Math 1)

Last week, I did lesson 3.1 of the Mathematics Vision Project (MVP) Math 1 Module 3. This week, I’m doing lesson 3.2 and I am both excited and apprehensive. In this post, I will explain my excitement and apprehension, as well as give a summary of how I’m planning to structure my lessons this week.

I’m excited because I think that the contexts used in this lesson provide a low entry-point, so I anticipate most students being able to feel like they immediately understand (at least part of) what’s happening.

The context is that some kids are rafting down a river. One kid measures the depth of the water:

Screen Shot 2018-01-28 at 3.43.35 PM

Another kid measures the total distance they’ve traveled: Screen Shot 2018-01-28 at 3.43.48 PM.png

I’m apprehensive because there is a LOT of vocabulary used to describe the features of these functions that my students are either only somewhat comfortable with, or straight-up have never even been taught. Part of this is because of instructional decisions that I made earlier in the year.

I found module 2 to be incredibly challenging to teach, because it assumed that students had mastered middle school content standards that frankly many of my students have never been exposed to, much less mastered. I found that a lot of the linear contexts were not well suited to developing proportional reasoning for students who have never been asked to reason proportionally before.

Critiquing module 2 really needs to be a blog post in and of itself, but suffice it to say that I never really solidified “domain” or “intercept” or some other vocabulary terms that are necessary for lesson 3.2 and are supposed to be mastered by the end of module 2.

When planning for 3.2, I had about 30 seconds in which I considered front-loading vocabulary so that I could ask students the content questions and preserve the wording of the questions used in the original MVP materials.

But then I came to my senses and remembered that vocabulary words are meaningless if students don’t have a concept they’re already thinking about that they’re ready to attach the vocabulary word to.

So, rather than telling students to “interpret/describe the key features (increasing, decreasing, domain, range, maximum, minimum, intercepts) of the relationship”, I’m asking the following questions:

  1. When is the depth increasing?
  2. When is the depth decreasing?
  3. How long was Alonzo measuring the depth for? When did he start measuring? When did he stop measuring?
  4. What is the range of river depths that Alonzo measured?
  5. What is the minimum depth of the river?
  6. What is the maximum depth of the river?

By focusing on the word “when” in questions 1 and 2, I’m hoping to draw students’ attention to the x-axis when describing intervals of increase and decrease (they also saw this in a Warm Up last week).

I’m hoping that students’ answers to questions 3 and 4 will help lead to a more formalized definition of domain and range.

I’m planning to spend about a week on this lesson. Here’s the preliminary breakdown:

Monday

Introduce the context, notice/wonder about the table, begin working on the questions in random groups at #VNPS.

I highly doubt that we’ll have time to debrief on this day, but I’m hoping that kids will be able to make decent progress. Thus, I’m planning to have them copy their work from the whiteboards onto paper at the end of class so that they have a record of it the next day.

Tuesday

Ideally, if students were mostly able to finish answering the questions on Monday, I’ll give them some time at the beginning of the period to make posters displaying their work. The reason I’d like the work on posters rather than whiteboards is so that they’ll have a (large) record saved for later in the week when we’re discussing the analysis questions involving both the table and the graph contexts.

Once their posters are made, we’ll discuss the answers as a class. This is where I’m hoping to motivate the words domain and range (and maybe interval??).

A note about whole-class debriefs: depending on the class, these have sometimes been incredibly painful, with me feeling like I’m pulling teeth trying to get kids to listen to and engage with each other. I think they might focus a little more if they have something to DO when other groups are presenting their ideas, so I’m going to have students fill out the following note organizer:

Screen Shot 2018-01-28 at 4.21.35 PM.png

I’ve never used this structure before, so we’ll see how it goes!

Wednesday

I’m currently planning for us to spend some time noticing and wondering about the graph, and then have kids start working on the interpretation questions at #VNPS. Alternatively, depending on how Tuesday’s debrief goes, I might want to spend some time on Wednesday solidifying vocabulary words.

Thursday

At this point, I feel like it’s hard to predict because it’s so dependent on how Tuesday and Wednesday shake-out, but I’m anticipating some combination of debriefing the graph questions and/or solidifying vocabulary.

Friday

Most likely this will be spent working on the follow-up analysis questions at the end of the lesson. By this point, I should be able to use the vocabulary and wording in the original MVP materials.

ALSO, I’m finally going to take a few minutes of class time to do something that I’ve been meaning to do forever: The Mathematicians Project. This is an awesome idea that the inimitable Annie Perkins had to help students see themselves as mathematicians by learning about mathematicians other than old white dudes.

I wanted to do the Mathematicians Project earlier this year, but felt a little too overwhelmed with being at a new school and implementing a new curriculum. But then I’ve watched in horror as we deal with the consequences of having blatant racists run the country, and I decided that it really was time to start the Mathematicians Project (among many other things!).

Since I’m starting the project at the beginning of Black History Month, I’m going to begin by focusing on Black mathematicians, and this Friday we’ll talk about Katherine Johnson (played by Taraji P. Henson in Hidden Figures).

I plan to continue the project for the rest of the year, and definitely want to highlight Latinx (particularly Central American), Arab (particularly Yemeni), Chinese, and Filipino mathematicians, because those are the primary demographics of my students.

Here’s my blurb on Katherine Johnson:

Word Doc: The Mathematician’s Project

PDF: The Mathematician’s Project

And here are the materials I have so far for lesson 3.2:

Word Doc: 1.29 MVP 3.2 Floating Down the River

PDF: 1.29 MVP 3.2 Floating Down the River

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.