How Can an Abstraction Show Me How Things Work?

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SWBAT represent a number trick abstractly, by using both symbolic and algebraic representations.

Big Idea

Abstraction is a such an interesting word, because everyone does it, yet few know what the word means.

Opener: Homework Check & Syllabus Check

10 minutes

As today's class begins, I've posted the solutions to last night's homework on the board, and I give the class a chance to check their answers.  Students will have questions about this assignments, so after they look at the answers, I'll give them a chance to ask.

It's the third day of class, and I'm still establishing structures and routines.  Today's opener is another structure that students can expect regularly.  It also sets the tone for the purpose of homework.  Many students will be surprised and little confused that I'm not collecting this homework.  Even though I told them about this policy yesterday, it's still a shock to some.

I walk around with a clipboard, just making sure I know names, and seeing my first glimpse of who does the homework, how much they do, and whether or not they have any strategies for coming up with the 10th and 100th term.  I want to slow down and have all sorts of conversations right now, but I force myself to move quickly.

When I finish my lap around the room, I'll take a few questions.  I particularly want to address why the 100th term of the first sequence is 693, not 700, so I direct the conversation toward that.  For the rest, my next steps depend on how well kids seem to be getting this.  Whatever they've done, I don't want to spend too long on this, because there will be time for that over the next few class periods.

Our Two Example Number Tricks

10 minutes

For today and the next few days of class, we'll be moving back and forth between number patterns and number tricks.  After starting by checking homework and briefly addressing number patterns, it's back to the number tricks.

I've prepared two number tricks in this document: Two Example Number Tricks.  First is the one we tried yesterday, next is the new one that we'll try today.  I start by putting the first one on the board, and I simply ask, "What does this trick do?"  Everyone remembers that no matter what you start with, this trick will always end with 5.

I say that next I'm going to show everyone another number trick.  "It's the same deal as yesterday," I say.  "You can start with any number, but don't tell me what it is.  Then, follow each of the following steps."  I scroll down to the second page of Two Example Number Tricks, and I give everyone a few minutes to give it a try.

As students finish up, they compare notes with their partners.  After yesterday's trick, many of them are surprised to find that they don't all have the same numbers.  Some students check their work to see if they did anything wrong.  As answers are confirmed, there's this great moment of, "Ok, so what's the trick?"  

I start asking everyone for their "ending numbers", and to their amazement, I am able to guess everyone's starting number.  Now they all try to figure out what I'm doing, and soon they do: this trick always adds 9 to the starting number.  There are options here for how you might go about revealing this to kids; I find that given a few minutes, someone will figure it out, then if necessary, I'll help them check to see if they're conjecture is right.

Either way, kids have now seen a trick similar to yesterday's with a slightly different sort of result.  Now the real trick will be to see if we can figure out how these work.

Using Abstractions

10 minutes

Today's guiding question, "How can an abstraction show me how things work?" is written atop today's agenda.  I make a reference to it, then say, "Now that you've seen two number tricks, I'd like to show how you they work.  To do that, I'm going to show you some other ways to think about these tricks.  I'm going to show you some abstractions."

Review the Learning Target: MP2

Mathematical Practice #2 is also written on today's agenda.  I point to it.  I ask the class if any of them have ever "reasoned abstractly".  When they reply with a chorus of no's and I don't know's, I say that indeed they have.  I tell everyone to imagine a slice of pizza.  I mime what it's like to hold a slice of pizza, and I share that I always fold my pizza in half.  I ask the class if they can imagine what it's like to hold a slice of pizza.  I ask them if they can imagine what it would smell like if I had a bunch of pizzas in the room.  I ask them to think about how the pizza would taste.  "Are you all with me?"  I ask.  Of course they are.  "Now you're thinking abstractly," I say.  "If you can think about something that's not here, or if I can say the word pizza and you know what I'm talking about without actually seeing actual pizza, then you can reason abstractly.  As humans we all have this ability to reason abstractly."   

How Can an Abstraction Show Me How Things Work?

I go on to explain that another way to think about "reasoning abstractly" is just to think about finding different ways to say or represent the same thing.  This may be heady stuff for 9th graders, but this word is important, and I want to get them thinking about it.  

The thesis of today's lesson is that another thing that happens when we reason abstractly is that we might come up with other insights about how things work.  I want to show the kids that by reasoning abstractly we may be able to learn a little more about these number tricks.

Making a Chart and Using Abstractions

On the screen, I return to the first number trick, and I draw a chart around it, like this: NTP Setting Up for Abstractions.  As I do so, I say, "Charts and tables are such great tools.  Whenever I'm trying to figure something out, I find that these are always helpful."  I take a quick trip around the room to see if everyone has this set up in their notebooks.  A few students don't realize that I expect them to have this in their notes, so I give them a gentle nudge saying that this is important and they should make sure to copy it down.

Then I say that one way to reason abstractly is translate from words into symbols.  I ask the class how we might use a symbol to represent "any number," and borrowing from Harold Jacobs, I use an empty square as my symbol for that unknown, "because I could write any number in this box."  To represent known quantities, I use dots.  So for the second step, I draw the original number from step 1, along with seven dots that represent the 7 I just added to my original number.  The rest of the steps proceed as seen here: NTP Symbols and Algebra.

Some kids love the symbols, others are feeling helplessly confused by this representation.  I'm watching to see who is who: as always, learning about my kids.  After we finish the symbols column, it's time for algebra, and what's awesome here is how quickly it usually goes.  Suddenly, after seeing these brand new symbols, the ideas of algebraic representation feel easier to a lot of students.  When we're done filling in that third column, I ask the class which they prefer, the symbols or the algebra.  The zeal with which my students respond is always disproportionate to the real-life significance of this task.  I love it.

If there is time, I set students to the task of doing the same thing with the second number trick.  If there's not, I save that for tomorrow's opener.

Homework: The Number Trick Project, Part 1

13 minutes

I distribute Part 1 of the first project of the year, which is tonight's homework.  At this point, the idea of "homework" is still more significant to most of my students than the idea of "projects".  That's ok, this year will change that.

I give everyone a brief tour of the assignment.  I point out that the heading on handouts I create will be the same as the heading that I taught the class to write on yesterday's homework assignment.  Beneath the heading, I show students that there are three Student Learning Targets.  There are two Mathematical Practices, and one content target.  "As we work through this project, you'll understand more and more about how SLT's work," I say.  "For now, I want you to know that when I grade this project, I'll be looking for evidence that you can do these three things."

Then I describe the three tasks.  The first two are very similar to what we've just done with the example number tricks.  For the third one, I provide symbols and students have to write the words.  I write the symbols by hand before making copies, as shown here.  If students need clarification on what the third task is asking, I point to the middle column of the example that's still on the board.  "You're taking these symbols," I move my hand down the second column before pointing back to the first column, "and changing them into words like these."

We usually have a few minutes to get going on this.  I give students some time to get started and I encourage them to discuss the work with their group mates.  I also have a few minutes right here to help students with any practical concerns - this being the third day of school - they have as they get to know the building, the bell schedule, etc.