Attending to Precision: INBs and group work (Interactive Notebooks)

I love new beginnings, but I am only so-so with early middles. Now that kids have started their INB journey, we've arrived at that crucial moment between the beginning and the first INB check. This, as the saying goes, is where the rubber meets the road.

I find that kids never understand at this stage why I insist on being so darned nit-picky about their notebooks. Every day someone new asks me why this or that HAS to go on the right-hand side or EXACTLY on page 5.

One of the many reasons why this is important, I have learned, is that it is all about teaching strategies for attending to precision — Mathematical Practice Standard #6, which is defined this way in the standards documents:

Mathematically proficient students try to communicate precisely to others.• They try to use clear definitions in discussion with others and in their own reasoning.When making mathematical arguments about a solution, strategy, or conjecture (see MP.3), mathematically proficient elementary students learn to craft careful explanations that communicate their reasoning by referring specifically to each important mathematical element, describing the relationships among them, and connecting their words clearly to their representations. They state the meaning of the symbols they choose, including us- ing the equal sign consistently and appropriately.• They are careful about specifying units of measure,• and labeling axes to clarify the correspondence with quantities in a problem.• They calculate ac- curately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 

The problem, I find, is that this description of precision is precise only at the theoretical level. On the front lines, it's unrealistic because most kids never get to this level of precision.

And that is because their notes and their work are generally quite a mess.

A big part of teaching students to attend to precision is giving them a structure for being an impeccable warrior as a math student — that is to say, taking and keeping good notes, noticing and keeping track of your own progress as a learner, preserving your homework in a predictable place that is not, let us say, the very bottom of your backpack, crushed into a handful of loose raisins.

It means stepping up your game as a student of mathematics and presenting your work in a way that makes it possible for others to notice the care with which you are specifying units, crafting careful explanations, describing relationships, and so on. And it means presenting your work in this way ALWAYS — in all things, in all times, wherever you go.

INBs are an incredibly low-barrier-to-entry, accessible structure for teaching attention to precision. There are no students who cannot benefit from having a clear, common, and predictable structure for organizing their learning. INBs are also a great leveler. For those of us who are focused on creating equity in our classrooms, INBs offer all students a chance to prove both to themselves and others that they are indeed smart in mathematics. As I saw the other night at Back To School Night, my strongest note-keeping students are rarely the top students computationally speaking. But they are the ones who can always find what they are looking for — a major advantage on an open-notes test.

INBs are also a phenomenal formative assessment tool. Flipping through a students INB gives me an incredible snapshot of where and when they were truly attending to precision and where they were fuzzing out. Blank spaces and lack of color or highlighter on specific notes pages give me a targeted spot for further formative assessment. In my experience, it is exceedingly rare for a student who thoroughly understands a topic to write no notes or diagrams on that page. If anything, they are the ones who are most likely to appreciate the chance to consolidate their understanding.

So I am sticking with it and zooming in on some of the areas where kids' understanding fell apart last week. We'll be reviewing how to convert from percentages to decimals and how to document and analyze the iterative process of calculating compound interest because that is where my students' notes fell apart.

I'll be astonished — but will report back honestly — if these on-the-fly assessments prove to have been inaccurate.

Using Exeter problems as an intro to algebra tiles

We are an Apple 1-1 school, so I am always happy to figure out lessons my students can use their laptops for.

I also like to use manipulatives in Algebra 1. It's not easy to get all students to accept the need to use multiple representations (such as an area model), but they help enormously to extend kids' conceptual understanding of the distributive property — plus they make a return appearance a lot when we get to the Festival of Factoring in the late winter.

So the National Library of Virtual Manipulatives seemed like a natural fit. But what problems to use to introduce them?

Enter the Exeter Math 1 problem sets.

I have been using the Exeter problems with my advanced 8th grade students taking Algebra 1 almost every week during our Problem-Solving Workshops on block days. Each page is a self-contained "problem set" that builds from simplest principles and often loops back on itself later in the page. This gives students a chance to give themselves a pat on the back for having discovered and developed an intuition for activating their own prior knowledge. I then have them write up one of the problems they solved as a problem of the week to give them practice in blending symbolic and graphical representations with verbal representations (don't forget the verbal representations!). So much Common Core math in such a small span of time!

I will write more about using the Exeter problems as a resource for long-form problem-wrestling with my students, but here I just want to talk about the specifics of introducing algebra tiles.

One of the features of the Exeter problems that we do not get to take much advantage of is the way they build page over page. They will introduce part of a concept or skill in a problem on page 5, say, then introduce the next part of the concept in a problem on, say, page 8. This makes so much sense if you are teaching using one page per day and working through all of them. But for those of us who dip in and nick out once a week, that isn't really possible.

But... by the time we got to around page 21, it dawned on me that I could collect the five problems they use that introduce algebra tiles and put them on a single sheet of paper.

Then I could give that to students during Problem-Solving Workshop, along with a quick intro to the NLVM, to let them teach themselves how to use algebra tiles!

So that is what I did this week.  :)

A couple of programming notes if you want to try this yourself with your students:
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1. Start at the very beginning with NLVM and PREPARE FOR TECH HICCUPS
NLVM can be extremely persnickety. This is probably due to some perverse desire to help us cultivate CC standards of mathematical practice #1. Encourage yourself and others to persevere.

Your network may have restrictions on how students can use Java-enabled apps on school equipment. We had some hiccups getting NLVM to run on everybody's system at first. Firefox seems to be the most reliable browser for NLVM. Also, you need to have the most up-to-date version of Java on the student's system.

On our network and systems, students can only update Java by logging out and in again or by restarting their computers. No matter how many times I explained this, some kids still didn't quite figure it out. So much for being "digital natives." Plan to go around to each kid the first time to help them get their systems up and running.

Our system throws up a modal "Security Warning" dialog that forces you to check "I accept the risk" and "Run" before NLVM will load in the browser window. Again, a minor pain in the butt, but you do need to make sure that every kid gets through the security gauntlet to use the system.

Refresh the browser window if need be and be patient which Java and the applet cooperate in loading.
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2. Get everybody to the *FIRST* page of the Algebra Tiles site on NLVM
For reasons that pass my understanding, NLVM dumps you into the sixth page of the algebra tiles site (the activity panel on the right, which loads as "Multiplying Binomials - 1").

You need to have students click the leftward-ho button at the top of this right-hand panel SIX TIMES to get back to the first page, which is called "Distributive Law - 1."

This is stupid but necessary because on the first two pages of this site, you can do things you need at the beginning that will quickly drop away as students gain fluency.

For example, the Distributive Law pages are the only ones where you can easily represent both multiplication over addition AND an area addition model in the same window.

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3. Familiarize yourself with the syntax of the NLVM Algebra Tiles pages
You'll need to tinker with this a bit, to get comfortable with the syntax of the applet, but there are two essential features of the Algebra Tiles distributive property pages:

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4. Click to CREATE tiles in the workspace; drag to MOVE tiles
You can create instances of any of the area blocks that are possible by CLICKING them in the menu bar along the bottom of the workspace. When you click the "x" button, for example, NOTICE that NLVM creates a single instance of a 1-by-x rectangle in the workspace. You can create as many "instantiations" of any of these blocks as you need for any expression you want to represent.

NOTICE that you can drag these critters around in the workspace and add them up, like LEGOs. Or you can drag them into the x-axis tray or the y-axis tray to represent lengths and widths of various area blocks of multiplication.

ALSO NOTICE that you can mouse over the corner of a block in the workspace to rotate it into the position you need.

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5. How to show multiplication over addition (i.e.,  how to show  x (y+2) :
The x- and y-axis are basically x- and y-axis "trays" that students can drag tiles into. Drag a 1-by-x tile into the y-axis tray and it creates an x coefficient. Drag a 1-b-y tile and two unit blocks into the x-axis tray and they become the quantity in parenthesis over which your x coefficient will drape itself in multiplication.

NOTICE that as blocks snap into place in the second axis tray you fill, a red area outline appears in the main workspace between the x-axis tray and the y-axis tray.

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6. How students can confirm for themselves that area addition and multiplication over addition produce equivalent area values (i.e.,  how to show that  x (y+2) = xy + 2x :
In this window, students can create blocks to fill in this red outline and verify for themselves that the area they get using the distributive property is equivalent to the area they can get using the area addition postulate approach.

Have students click to create blocks and then drag them around to fill the red outlined area perfectly.
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MATERIALS

My bastardized worksheet of the five Exeter Math 1 problems that introduce algebra tiles and an area model can be found here on the Math Teacher's Wiki.

NLVM Distributive Property pages are here and 6 pages to the left.

Enjoy!