Number Tricks, Patterns, and How to Succeed in This Class

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SWBAT understand the mastery-based grading system that is employed in this class, hear more about my expectations, and try their first number trick.

Big Idea

Everything is formative! Long before I collect my first student work, I give myself all sorts of opportunities to learn about my students and their habits of scholarship.

Opener: Who's Who at Your New Table?

5 minutes

Today's opener is on the first slide of today's Prezi

This is the same as yesterday's opener, with a catch: everyone has to sit at a different table, with completely different people.  No repeats are allowed.  It's actually the beginning of an interesting optimization problem, but I don't force that at all.  I simply ask students to find their index card from yesterday, and on the other side, write the name of their new table (Greek letter stickies are still placed at each table) along with the full names of everyone in their groups.

There are two things I want students to know by experience here:

  1. Seating in this class will be fluid.  You'll be expected to change seats frequently and work with a variety of people.  You won't have a seating arrangement, but you shouldn't get too comfortable in any one spot, either.
  2. If I give you an index card and say not to lose it, I mean it.  There are a few kids who can't find their index card today.  I simply ask them what happened to it, and I express the fact that they're going to have to do better.  When everyone is done, I collect the cards.  For me, this is really an exercise in learning names, but for kids it sets a gentle but clear tone of accountability.

Syllabus: In Greater Depth

15 minutes

Yesterday, didn't get to talk about the whole syllabus on the first day.  That's ok, I ask students to find the syllabus, and I run through it now.  I've prepared slides 2 through 7 of today's Prezi with different views of the syllabus.  I use the projector at the front of the room to display the syllabus, and I tell students to read along with me.

The Front

The front of the syllabus is full of information about this class.  I try to keep it brief, and hit the things that really matter. They are:

  • Homework - every night, and you must be willing to work hard
  • Grading Policy - based on mastery of your learning targets (we'll spend a lot of time on this)
  • Binder - you need one, right away
  • Textbook - you'll get one to take home
  • Seating Policy & Group Work - collaboration is the name of the game
  • Class Web Site - you'll be able to see agendas and find documents you need

The key here is that I expect the syllabus to be a living document.  So I'll read through this quickly and give students a chance to ask questions, but I also understand that for any of this to stick, they'll have to experience it, and we'll have to revisit it.  I give students the chance to ask questions, and most often there are series of questions that I answer with simple "Yes": "Do we really need a binder?"  "Will we have homework every night?"  "Even Fridays?"  I look forward to all the deep conversations we'll eventually have this year, and again, I know that in order to get there, we'll just have to get started.

The Back / Grading

I ask students to turn to the other side of the syllabus, which is much more important than the overview on the front.  Grading is based on mastery.  I help students fill in the blanks.  When I pull up the 6th slide of the Prezi, I write directly on it to give students pointers on how to "get the grade they want" in this class.  It looks like this: U1L2 Syllabus Notes.

Most kids won't have a firm grasp on this grading system or our SLTs until they have a few grades.  In a few weeks, after they've been assessed a few times and have some numbers to look at, they'll be able to make more sense of how this works.  

How to read an SLT

We'll become very familiar with these over time, but to start, I say that today we're going to focus on the first two Mathematical Practices:

  • MP1: I can make sense of problems and persevere in solving them.
  • MP2: I can reason abstractly and quantitatively.

With the syllabus still projected at the front of the room, I ask for a volunteer to read MP1.  I ask if there are any important words here.  The idea of "making sense of problems" is not foreign to students, but a few of them might be unclear on the word "persevere."  If no one can tell me what it means, I ask for it's opposite.  Usually at this point, someone says "quit."  I say that's right, that quitting is the opposite of perseverance.  To persevere just means that you don't give up, and you always try hard to figure things out.  I refer back to the idea of "working hard," which was my first pointer for how to succeed in the class.  It's all about growth mindset, baby.

I repeat the process with MP2.  This time, there are those two big words: abstractly and quantitatively.  "Now we're getting into it," I say.  "To begin this class, we're going to study these two words."

A Number Trick

15 minutes

"In order to show you what it means to 'reason abstractly,' I'd like to show you a number trick," I say.  But before I get started, I show everyone a quote.

What's the Best Way to Learn?

The 9th slide of today's Prezi is a quote from George Polya:

The best way to learn anything is to discover it by yourself.

I put this quote on the board and I ask for a volunteer to read it allowed.  Then I ask if everyone agrees with it or not.  After giving students a moment or two to volunteer their responses, I ask everyone to give me a thumbs up if they agree, a thumbs down if they don't, or a sideways thumb if they're somewhere in between.  This might be favorite formative assessment I know, and it's one we'll use a lot this year.  

Whatever I see in these classroom full of thumbs (it's usually a pretty even mix), I use my most serious voice to say that I really believe this, and that in this class, students will be asked to figure stuff out all the time: "When I show you this number trick, I'd like you to try to figure out how it works."

A Number Trick

On slides 10 through 17 of the Prezi are the steps of a number trick.  This is the first Number Trick that Harold Jacobs shares in Elementary Algebra, which is one of the best Algebra textbooks I've ever seen.  (It is one of two texts from which I draw heavily for inspiration in this course.)

I ask students to write their numbers and each of the steps in their notes.  I pause on each step for long enough for them to write down each step, but I don't wait for anyone who might be distracted.  Instead, I circulate a little between each step to make sure everyone is keeping up.  This being the first math we've done in the class, I can look over the shoulders of each kid, and get a quick glimpse of their confidence levels, note-taking habits, and arithmetic skills.

After showing them each step individually, slide #17 shows all steps at once.  As students finish working through the steps, I tell them to compare their answers with a neighbor.  One by one, they should get 5, no matter what number they start with.

Ideally, someone will ask if they had to pick a number between 1 and 50.  This happens about half the time, and when it does, I suggest that anyone who wants to tries the trick with either a bigger number, a negative number, or even a fraction or decimal.  Again, I can quickly learn which of my students are (at this point in their schooling) inclined to try a challenge just to satisfy their curiosity. 

At the end of the trick, some students are really impressed.  Others are not - either because it's too easy to really surprise them, or because it's over their heads that this is of any interest.  Again, I'm learning who's who (at least for now -- these dispositions may change this year!) in my class today.

While I take mental notes on my new students, I say "pretty cool right?" then, "But what's really interesting about this trick is trying to figure out how it works.  Does anyone have any ideas?"  At this point I give a few minutes for students to share their ideas with the class.  Sometimes, a student will want to come write some thoughts on the board, other times, students will want to talk to each other.  This is good time to introduce the Think/Pair/Share discussion strategy, but I usually let the conversation run free, again, just to learn about my students.

After a few minutes of these sorts of conversations, I say that tomorrow we're going to explore how the trick works.  "For now," I say, "I'd like to show you how to homework in this class."

How to Do Homework / Tonight's HW

7 minutes

The last slides of today's Prezi are about tonight's homework.  Tonight's homework is short.  My goal is to introduce procedures more than it is to get kids doing super-rigorous mathematics.

The first instruction is to write the heading on a piece of paper.  With the slide projected on the board, I sketch a piece of paper and show what the heading should look like.  I explain that the title of the assignment goes in the upper right-hand corner of the page, so it's easy to find in a binder.  I've told students that I'm going to help them make their binders useful, and this is my first tip.  I hold up a binder and show how easy it is to flip through the top corners of the pages. 

Beneath the heading, I instruct students to write the assignment instructions.  

Then, I show them the three patterns with which they'll work tonight:(U1 L2 Homework 2 of 2).  It's just three problems, and there are entry points for everyone.  I'm hoping that all students will be able to find the 5th term in each pattern, and that I'll be able to learn a bit more about their abstraction skills and work habits when I see how they go about finding (or floundering) as they look for the 100th term in each pattern.