Algebra 1 Systems of Inequalities - Dan Wekselgreene's Ohio Jones & the Templo de los Dulces treasure map

Some of Dan Wekselgreene's early puzzles, lessons, and projects are truly love poems for Algebra 1 students. And I have loved his Ohio Jones and the Templo de los Dulces systems of inequalities puzzle since the first time I read about it, did it, and used it.

So there was never any question that I would use it with my Algebra 1 students this year. The only question was, how would I make it accessible to my blind student?

Susan Osterhaus of the Texas School for the Blind and Visually Impaired has been generous beyond words with her ideas for teaching math to blind students. Her web site is filled with ideas, best practices, and links to resources for making mathematics accessible to blind students. I cannot recommend it strongly enough.

Here is how I adapted this activity:

FRONT PAGE OF DAN'S WORKSHEET

I typed out each of the three clues as a quote by a statue — i.e., Statue 1 says...

Then I plugged each quote into an online Braille Translator (I like http://brailletranslator.org) and downloaded the Braille text file as an image. I copied and pasted each image file onto an Omni Graffle document (though you could also use Word or Pages) next to the regular text quote. That way the student and the paraprofessional aide could easily collaborate and share information.

This took three pages, but it worked.

I traced the basic map at the bottom of the page but without all the grid lines. This became the "map" for this part of the puzzle. 

Then I copied these pages onto capsule paper and ran them through the PIAF (Pictures In A Flash) machine to create a tactile worksheet with Braille and a raised map. The PIAF machine (affectionately known around the math office as "the toaster") takes the capsule paper with all its delicious black carbon-heavy areas and raises them to create a tactile graphic that can be read by a Braille-literate blind person.

After solving the system and figuring out the target region, my student used Wikki Stix on the map to make a graph.

BACK PAGE — THE MAP

I traced the "big picture" outline of the map to remove as much visual noise and clutter as possible from the main image.  I added Braille labels to indicate the start and the hint at the end of the map:

I scanned this file, printed it on capsule paper, and ran it through the PIAF machine. Again, the student worked on Braille graph paper, then transferred her results to her tactile treasure map using Wikki Stix.

For each "sector" of the map, my student used Braille graph paper and Wikki Stix while her classmates used pencil and the grid on the worksheet.

It was such a joy to see her as just another team member at her table, doing mathematics and solving a puzzle. It was even more exciting to see how her table mates appreciated her mathematical skills.

All in all, a successful experience in creating an inclusive classroom!

My reduced version of the Teacher Packet (including the worksheet and instructions) plus the Braille-ready package are all on the Math Teacher's Wiki.

Wikki Stix are available in a big box on Amazon or any kids' art supply store.

Algebra 1 Inequalities – A minor 'How People Learn' unit

Here's a perfectly imperfect model unit of how I use the How People Learn stages and cycles in a typical unit in my Algebra 1 class. I'm documenting this for myself, so anybody else who finds this useful is just icing on the cake! :)

All of the files I use are in this downloadable zip file on the Math Teacher Wiki:

     Algebra 1 Inequalities unit

Here's a rough overview of how this works.

ESSENTIAL QUESTION:
How are algebraic inequalities related to our basic number sense concepts of "more than" and "less than," and how can we use this understanding to build a more generalizeable algebraic understanding?

STAGE 1 - hands-on introductory task

(1) Deleted scene (readers' theater activity):  groups read the deleted scene about how to do Talking Points, which also contains a review of number sense concepts of more than and less than.

(2) Talking Points - set #1 — more than and less than: what does your group think?
Students follow the protocol they have just learned and do set #1 of Talking Points that active prior knowledge about which is more and which is less, given two sets of quantities.

STAGE 2 - initial provision of an expert model

Each day is different, but usually I spend about 10-15 minutes working with the whole class to "do some notes" (combination of mini-lecture and note-taking and modeling). This is often the last thing we do in the class period.

STAGE 3 - deliberate practice with metacognitive self-monitoring

Practice happens at both a macro- and a micro- level. On the micro- level, each day's homework (which is completely distinct from the day's classwork) gives a chance to practice and review. Then the first activity of the next day's class is comparing answers in your table groups and answering all questions that groups or students are able to answer for each other or for themselves. I take only Burning Questions (like only taking group questions during Complex Instruction problem-solving tasks).

At the macro-level, we are building toward two big days of Speed Dating, which is differentiated deliberate practice with metacognitive self-monitoring and peer tutoring or reteaching as needed.

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We cycle back through all of these for several days, as you can see, with a new set of Talking Points each day that students have to work through and puzzle over. Each day's Talking Points build in a new piece of knowledge that is in students' Zone of Proximal Development so that they can encounter it, wrestle with it, and formalize their understanding of it. Then they get the nightly chance to practice some more.

This is a highly Vygotskian model of learning.

Eventually we need to introduce a new concept into our exploration of greater than and less than. I call this concept the concept of "betweenness." We do this through (5) another deleted scene.

Once again, we are investigating numbers and quantities, but we are extending our investigation to more abstract conceptions of quantities. We go back to Talking Points. We do some problem-solving. We struggle together. We organize our learning.

STAGE 4 - transfer task

I don't have a particularly great transfer task yet for this unit. That's why it's such a good one to use as an introduction or reintroduction to my Talking Points norms and practices.

If you have a super-terrific transfer task for Algebra 1-level linear inequalities (not yet at systems), I'm all ears. Please let us know about it in the comments section.

Let me know what you think if you try any of this!
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A Postscript —
Today was the kind of day that makes all this work worth it. To review for tomorrow's unit test, we did group whiteboarding of some pretty hard problems. And even though many kids were still stumped by some of the harder problems, they felt excited. They were understanding it.

And doing it.

Doing math.

Now that is the kind of thing that makes it worth waking up at 5 in the morning five days a week for ten months of the year straight at near-poverty wages. :)

Betweenness and non-betweenness: absolute value inequalities and Patrick Callahan

I felt a little nervous about having Patrick Callahan come to observe my classroom yesterday, but in the end, it was fun. I had asked one of our security guards, to bring him down to my room when he arrived at our school. He walked in as he always does, all mathematical open-mindedness and pedagogical curiosity.

And we got started.

I felt anxious about having him observe my conceptual lessons about betweenness and non-betweenness. I have never seen anything even close to how I understand and talk about absolute value and inequalities. I talk about boundary points and betweenness and I have students hold up their fists and point their thumbs to show me their understanding. “Is this a situation of betweenness — or NON-betweenness?” I demonstrate with my own fists, swinging my thumbs inward or outward. “Your fists are the boundary points and your thumbs are how you shade your graph on the number line. So is this a situation of betweenness... or NON-betweenness?”

If it is a situation of "betweenness," then students point their thumbs inward towards each other, touching the tips together. If it is NON-betweenness, then they point their thumbs outward in either direction, like a group of indecisive hitchhikers. And once we have done this analysis, then we can do whatever calculations we may need to find our boundary points.

So much of advanced algebra and precalculus depends on having this kind of deep conceptual understanding and thinking. Am I looking for quantities that are GREATER than...? or LESS than? Is this quantity going to be positive? or negative?

For me, the whole thing is intimately hooked together with the real number line. And with number sense. 

When we started last week, we began with an inquiry into “more than” and “less than” and widened our thinking outward from there.We connected more than and less than to number line thinking. I always emphasize Number-Line-Order and Number-Line-Thinking in my Algebra 1 classes. If they think about the number line, then they can anchor their thoughts in their bodies. LHS (or Left-Hand Side) and RHS (Right-Hand-Side) are fundamental ways of thinking in algebra. These ideas are eternal and unchanging. The number line is the foundation of everything. It gives you the “true north” of the real number system.

So we always ground our thinking in our bodies. I ask, “Left Hand Side or Right Hand Side?” “Is this a situation of betweenness or non-betweenness?” “OK, now that we know that, now what?”

I also anchor this unit in what they know about logical reasoning. They have an intuitive sense of how many possible cases a situation may present. I've been a huge Yogi Berra philosophy fan all my life, so I believe that when you come to a fork in the road, you should take it. When you come to a fork in the road, you can go left or you can go right. Or you can stay right where you are. Three possible cases. Over and over I ask them, “What’s going on here? How do you know?”

Absolute value inequalities are either situations of betweenness or situations of non-betweenness. Figure that out and then everything else will run smoothly. Then all you have to do is to use what you already know.

Once students have gotten that figured out, it’s just one more small step to combining their new knowledge with their existing knowledge. Follow the order of operations and common sense. Plus everything you know about the real number line and multiple representations. Then things can naturally unfold the right way.

But I always come back to number sense to what we know about the real number line. Numbers are the ground, the foundation.

So when Patrick walked in yesterday — this world-class mathematician and math education expert — what he encountered was my bootcamp in algebraic thinking. “Hold up your fists! Is this a situtation of betweenness or non-betweenness?”  "How do you know?" And then my waiting until everybody’s thumbs are pointing in the same direction.

It is Logic 101 and numbers and anchoring our thoughts about numbers in our bodies. Like the ancient Greeks and Babylonians and Egyptians before us.

Our next step is to solidify our thinking through what How People Learn calls “deliberate practice with metacognitive awareness.” We are going to do two days of Speed Dating. Now I have to make up Speed Dating cards and a test to use on Thursday. 

And then to document my thinking.

When the class ended, Patrick came up to my tech podium and was excited. He grabbled a whiteboard marker and started sketching and pouring out ideas.

For me, that was the best possible review I could have gotten on this lesson. A five-unicorn review. A direct hit. :)

NEW STRATEGY: having students introduce themselves to Talking Points by way of a deleted scene

I wrote another one of my "deleted scenes" from various Hollywood movies as a way to get students to introduce themselves to Talking Points.

I think this is my best idea yet.

Here is a link to the deleted scene, on the Math Teacher Wiki:

                 Intro to Talking Points and inequalities - Harry Potter.pdf

Students do so much better a job of monitoring their own use of the structure. I think this makes them "own it" more.

After only two weeks, they now groan dramatically at my corny situations. But they secretly (and not-so-secretly) love it. They dive right in, choosing roles and reading dramatically.

The whole thing takes about 7 minutes.

Let me know what you think!

The concept of betweenness

I am coming to believe that, much like the concept of substitution, developing a deep understanding of the idea of betweenness is a huge part of the psychological and conceptual work of Algebra 1.

I have been dissatisfied for years now with the fact that we tell students about the geometric interpretation of absolute value, but we don't really get them to live it. And yet that idea of the "distance from zero" on the real number line is not something we give students time to really marinate in.

And you know what? That feels dumb to me.

So this week and part of the next, my students and I are really wallowing in that idea.

I've taken a page from my studies as a young piano student. In the study of the piano, there are certain studies of technique that really force you to slow down and take apart the finger movements. There are specific figures that you have to practice over and over and over so that they become part of your finger memory. How they feel in your fingers is how you come to relate to them.

This is not just about developing automaticity, although that is a side benefit. This is about learning to feel these foundational figures in your bones. In your body. They become so fundamental that as you learn and grow as a musician, you come to feel them when you see them coming up in a new score you are studying.

The technique does not replace musicianship. The technique supports the musicianship.

I've been noticing lately how my own experience of absolute value is about noticing boundary points at the periphery of my mathematical perception. I see them out of the corner of my mathematical mind's eye. And how an inequality is said to relate to them defines how I relate to those boundary points.

So I am taking the risk of sharing this mathematical experience with my students.

As with young piano students, we take this slowly. One figure at a time. Right now we are only dealing with the case of an absolute value being less than a nonnegative quantity. We are dealing with situations of betweenness, where an inequality presents us with a figural situation that is going to wind up with a quantity being between two boundary points.

That is all. And that is enough.

I see the effort in their faces and in their fingers as they rewrite, revise, calculate, solve, and sketch graphs. I see them noticing and wondering whether they need to use a closed dot or and open dot.

And I hear them developing the confidence that comes from experience in developing a relationship with these quantities.

They are not following rules. They are listening to their own deeper wisdom. Everybody knows something about the situation of being "between" other things. Betweenness is one of the most elemental human ideas.

They are making friends with mathematics.

Seating charts for equity

I can't imagine traveling to new lands and not wanting to try their cuisine. But there really are people who bring their own food with them. One of the best things about traveling in my opinion is being educated  in the sense of the Latin root word — being led out of my own ignorance.

The same is true for me about attending a large, great school. It always has been. From the moment I arrive in a great new school, I feel excited and open to meeting and learning with all different kinds of people from different cultures and backgrounds. I want to expand my own limited world view.

But it seems inevitable that, without outside intervention, I often end up knowing and hanging out with the other Buddhists and Jews in any room. Cultural affinity is a force that possesses a tractor beam all its own. Fortunately, I am not the first to have noticed noticed this.

Our amazing counseling department and our Peer Resources program noticed this phenomenon too, and when they did their most recent student survey of our very large, urban, diverse student body, they put in some questions about this in their student well-being section. And the results were very moving to me.

Students overwhelmingly reported that when they first arrived at our school, they felt enormous pressure to connect with their cultural affinity groups. And for this reason, they reported, they deeply appreciate seating charts in classes that take this pressure away. This practice overwhelmingly helped them to feel that they fit in here and that those who are different from them in some ways are more like them in other ways than they are inclined to believe. It also created a zone of psychological and emotional safety to explore social connections with others not as "Others" but as fellow explorers in a safe space.

These findings touched my heart. Our kids' deeper wisdom never fail to blow me away.

So I sit here on the Sunday before the first day of Spring term making up seating charts, making sure that everybody arrives in my classes in the same boat as everybody else, and with the same opportunity to experience connection with others in as safe a space as I can create.

I will also pre-make Seating Charts #2, #3, and #4 so that it's convenient for me to change the seating without having to think. Sometimes "don't think" is the best rule.

I don't have any scintillating conclusions to draw here. I just wanted to document for myself what I am doing and why so that when I forget, I can more easily remember.

Happy New Year! Ewok Logic Puzzle

Here's your new emergency back-up sub plan. You're welcome. ;)