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.


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