Desmos plus INBs — Conic Sections Edition

One of the things I have always been frustrated with is the crappy way example graphs look in student notebooks.

Well, no more.

For my conic sections notes sessions in Precalculus, I'm using Desmos-created graphs with all their equational and slider glory.

Here's how I ensure kids have readable, meaningful examples in their INBs:

I've created some modified graphs of the Desmos parabola graphs — one with a vertical axis and one with a horizontal axis.

I take a screen shot of the equation drawer PLUS the graph and paste it into an Omni Graffle document. For those of you playing our home game on the Mac, that's:

• Press Cmd-Shift-4 to enter screen grab mode
• Select the region of the Desmos window that you want to use as your graphic (this pastes it directly onto the Mac OS X clipboard)
• Paste into a blank Omni Graffle document (from omnigroup.com )
• Resize to fit your needs, then
• Select, copy, and paste as many times as you need to create the master for your tiny handout

I arrange them 3-UP on the photocopy master so the tiny handout will fit onto a standards notebook/INB page.

Here are the files for my photocopy masters:

• Parabola geometric def example VERTICAL AXIS.pdf
• Parabola geometric def example HORIZONTAL AXIS.pdf

Chop, glue, annotate.

I recognize this is totally old school, but everything old is new again.

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.

New strategy for introducing INBs: complex instruction approach

After months of not feeling like my best teacher self in the classroom, I got fed up and spent all weekend tearing stuff down and rebuilding from the ground up.

INBs are something I know well — something that work for students. So I decided to take what I had available and, as Sam would say, turn what I DON'T know into what I DO know. Love those Calculus mottos.

So I rebuilt my version of the exponential functions unit in terms of INBs. But that meant, I would have to introduce INBs.

As one girl said, "New marking period, new me!" The kids just went with it and really took to it.

Here is what I did.

ON EACH GROUP TABLE: I placed a sample INB that began with a single-sheet Table of Contents (p. 1), an Exponential Functions pocket page (p. 3), and had pages numbered through page 7. There were TOC sheets and glue sticks on the table.

SMART BOARD: on the projector, I put a countdown timer (set for 15 minutes) and an agenda slide that said,

• New seats!
• Choose a notebook! Good colors still available!
• Make your notebook look like the sample notebook on your table 

As soon as the bell rang, I hit Start on the timer, which counted down like a bomb in a James Bond movie.

Alfred Hitchcock once said, if you want to create suspense, place a ticking time bomb under a card table at which four people are playing bridge. This seemed like good advice for introducing INBs to my students.

I think because it was a familiar, group work task approach to an unfamiliar problem, all the kids simply went went with it. "How did you make the pocket? Do you fold it this way? Where does the table of contents go? What does 'TOC' mean? What goes on page 5?" And so on and so on.

I circulated, taking attendance and making notes about participation. When students would ask me a question about how to do something, I would ask them first, "Is this a group question?" If not, they knew what was going to happen. If it was, I was happy to help them get unstuck.

Then came the acid test: the actual note-taking.

I was concerned, but they were riveted. They felt a lot more ownership over their own learning process.

There are still plenty of groupworthy tasks coming up, but at least now they have a container for their notes and reflection process.

I'm going to do a "Five Things" reflection (trace your hand on a RHS page and write down five important things from the day's lesson or group work) and notes for a "Four Summary Statements" poster, but I finally feel like I have a framework to help kids organize their learning.

I've even created a web site with links to photos of my master INB in case they miss class and need to copy the notes. Here's a link to the Box.com photo files, along with a picture of page 5:

We only got through half as much as I wanted us to get through, but they were amazed at how many notes we had in such a small and convenient space.

It feels good to be back!

Substitution with stars

This one is for Max, who asked about it on Twitter, and for Ashli, who interviewed me for her Infinite Tangents podcasts.

As Ashli and I were talking about some of the struggles we see as young adolescents make the transition from concrete thinking to abstraction, I mentioned substitution.

For many learners, there comes a point in their journey when abstraction shows up as a very polite ladder to be scaled. But for others (and I count myself among this number), abstraction showed up as the edge of a cliff looking out over a giant canyon chasm. A chasm without a bridge.

This chasm appears whenever students need to apply the substitution property of equality — namely, the principle that if one algebraic expression is equivalent to another, then that equivalence will be durable enough to withstand the seismic shift that might occur if one were asked to make it in order to solve a system of equations.

Here is how I have tinkered with the concept and procedures.

Most kids understand the idea that a dollar is worth one hundred cents and that one hundred cents is equivalent to the value of one dollar. I would characterize this as a robust conceptual understanding of the ideas of substitution and of equivalence.

One dime is equivalent to ten cents. Seventy-five pennies are equivalent to three quarters. You get the idea.

We play a game. "I have in my hand a dollar bill. Here are the rules. When George's face is up, it's worth one dollar. When George is face down, it's worth one hundred cents. Now, here's my question."

I pause.

"Do you care which side is facing up when I hand it to you?"

No one has yet told me they care.

"OK. So now, let's say that I take this little green paper star I have here on the document camera. Everybody take a little paper star in whatever color you like."

Autonomy and choice are important. I have a student pass around a bowl of brightly colored little paper stars I made using a Martha Stewart shape punch I got at Michael's.

Everybody chooses a star and wonders what kind of crazy thing I am going to have them do next.

We consider a system of equations which I have them write down in their INB (on a right-hand-side page):

We use some noticing and wondering on this little gem, and eventually we identify that y is, in fact, equivalent to 11x – 16.

On one side of our little paper star, we write "y" while on the other side, we write "11x-16":

I think this becomes a tangible metaphor for the process we are considering. The important thing seems to be, we are all taking a step out over the edge of the cliff together.

We flip our little stars over on our desks several times. This seems to give everybody a chance to get comfortable with things. One side up displays "y." The other side up displays "11x–16." Over and over and over. The more students handle their tools, the more comfortable they get with the concepts and ideas they represent.

Then we rewrite equation #1 on our INB page a little bigger and with a properly labeled blank where the "y" lived just a few short moments ago:

"Hey, look!" somebody usually says. "It looks like a Mad Lib!"

Exactly. It looks like a Mad Lib. Gauss probably starts spinning in his grave.

"Can we play Mad Libs?" "I love Mad Libs!" "We did Mad Libs in fifth grade!" "We have a lot of Mad Libs at my house!" "I'll bring in my Mad Libs books!" "No, mine!"

It usually takes a few minutes to calm the people down. This is middle school.

I now ask students to place their star y-side-up in the blank staring back at us.

Then it's time to ask everybody to buckle up. "Are you ready?"

When everybody can assure me that they are ready, we flip the star. Flip it! For good measure, we tape it down with Scotch tape. Very satisfying.

A little distributive property action, a little combining of like terms, and our usual fancy footwork to finish solving for x.

Some students stick with substitution stars for every single problem they encounter for a week. Maybe two. I let them use the stars for as long as they want. I consider them a form of algebraic training wheels, like all good manipulatives. But eventually, everybody gets comfortable making the shift to abstraction and the Ziploc bag of little stars goes back into my rolling backpack for another year.

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I'd like to thank the Academy and Martha Stewart for my fabulous star puncher, without which, this idea would never have arisen.

I wore out my first star puncher, so I've added a link above for my new paper punch that works much better for making substitution stars. Only eight bucks at Amazon. What's not to like? :)

#made4math | Words into Math - Taming Troublesome Phrases with an interactive foldable translator

It's been busy here in the Intergalactic Cheesemonkeysf R&D Laboratories
(see trusty assistant hard at work, right). Ever since Twitter Math Camp 12, I've been working on implementing all the lessons and activities I learned about in person from my fabulous math teacher tweeps!

I'm using the Interactive Notebook structure that Megan Golding-Hayes showed us, and I'm also incorporating a lot of Julie Reulbach's foldables. The most helpful insight (out of many) I received from Julie was the idea of using a foldable as a way of getting kids to SLOW DOWN and trust the steps of the process as they're working on word problems. So I've made a nifty little foldable like hers that will go into an INB pocket the first week and will be usable on all quizzes and tests.

One of the reasons I like having students develop tools they can use on tests is that many of the discouraged math learners just don't trust their own learning. They have a habit of "collapsing" when they encounter a first speed bump. So from the perspective of encouraging students' courage in problem-solving, it is good to allow them to have tools they can use, even if the tools are sometimes nothing more than a security blanket — a talisman or a good-luck charm they can touch as a tangible reminder of their own courage and resourcefulness. So a four-step problem-solving foldable serves double duty: it acts both as a checklist (as in Atul Gawande's New Yorker piece and book) and as a reminder to have courage and perseverance in working through problems.

However many students have a habit of either not using the tools or finding the tools too complicated or frustrating. Nowhere has this been more evident than when I've given them approved lists of words and phrases they should stop, consider, and look up if need be. The charts and lists seem to turn into giant floating word clouds that signify nothing. So I wanted to come up with a slightly more interactive than usual foldable that students could use as a way of isolating and decoding some of the most troublesome words and phrases they get hung up on. Not only does it slow them down, it gives them a focal task that redirects an anxious mind.

After a lot of research on both blogs and on Pinterest ("PINTEREST!" #drinkinggame), I came up with the idea of a folded sleeve with a sliding chart insert, containing the phrases that often confuse kids or cause them to second-guess their translations from words into math. Here's what the finished product looks like:

Here is a close-up:

I used OmniGraffle to make the sleeve template and I used Pages, Preview, and Adobe Acrobat to make the insert. I'm linking to the Troublesome Phrase Translator sleeve, a generic sleeve you can customize for your own fiendish purposes, and a PDF of my exact insert (Troublesome Phrase Translator INSERT). 

If you want to make your own inserts, you'll need to set up your own table (Word, Pages, Excel, etc) making sure that your row height is exactly 1/4 inch. Your LHS cells should be 1 9/16" wide and your RHS cells should be 1/2 inch wide. You can have about 19 or 20 rows, depending on what you put in them.

Sometimes a little magical thinking is just the thing to displace a discouraged learner's anxiety (or freaked-out-ness) for that extra second it might take to recommit to the process of solving a problem. If that helps me hang onto just one extra student a day, it's a win. But usually I find that a tool like this will encourage multiple students to encourage each other's confidence as well, which is an even bigger win in my book!