Arithmetic Sequences, Day 2

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SWBAT see structure in arithmetic sequences as they prepare to study linear functions.

Big Idea

I know that there's an intimate relationship between arithmetic sequences and linear functions, but kids shouldn't have to take my word for that if they can see it for themselves!

Opener: Picking Up Where We Left Off

5 minutes

This is a continuation of the next lesson about lines and points on the plane.  It's also possible that we'll spend just a little time finishing up yesterday's work before I give students some introductory notes about finding the slope of a line through two points.  My decisions on pacing are adapted to each moment, and so each class will get as far as I can help them go. 

Usually, today's class will begin with most students working on problem #5 on the second page of yesterday's notes, so that's what I project on the board as they enter the room.  I instruct everyone to find the work from yesterday, and pick up where they left off.  I say that students may work in groups or independently to get this done.

As students get to work, I return exit slips from yesterday and use these to frame today's work.  I've written brief feedback without grades on each student's exit slip.  To kids who submitted perfect work, I tell them that we're going to continue this work of writing rules for arithmetic sequences today.  For anyone whose work needed revision, I spend a moment explaining what I saw on their exit slip.  In about five minutes, I can check in personally with every student in the class.

Review Notes: How to Write a Rule

5 minutes

By the time I'm done returning exit slips, students have had a few minutes to recall yesterday's work, to get started, and to start asking questions.  Usually, enough questions are about writing function rules for arithmetic sequences that I start with that.  

We review the shortcut that I described yesterday: find the common difference,  then figure out what would come before the first term, and use these to construct the rule.  Yesterday, some students understood this, and today it clicks for another subset of kids. 

Notes and Work Cycle, Plus Additional Problems

23 minutes

Work and Notes

The tasks for #5 and #6 generate plenty of conversations.  I move around the room, listening in on conversations, turning questions back on students, and leading when I have to.  If many students have the same questions at the same time, I might give a mini-lecture about it.  Sometimes this lesson results in students busily working on their own, and sometimes there's more class discussion.  I'm ready for either, and it really depends on the class.

As students finish on #5, I check their work and tell them to get going on #6.  Problem #6 is where today's real work happens.  This is a great task because the instructions are simple, and the complexity grows gradually.  This work is integral to what we'll be doing in the next few days.  My role is to figure out how to lay the strongest foundations possible, and that's what I'm thinking about in my conversations with kids.

Review and Scaffolding

With everyone working at a different pace, I have to be flexible about the way I help students.  A lot of students breeze through 6a-c, but then get stuck on Exercise 6d.  I want to help them, but I don't want to give anything away.  I want to give everyone a chance to get stuck on this problem, and then to work to figure it out.  So, if that happens, I write a few scaffolding problems on the board.  These give students a chance to think about what's happening here.  I have conversations with small groups of kids who are at the same place, and then I ask, "What if there are three or four spaces between the 10 and the 22?  Then what is the common difference?  Try to fill in the blanks."  When they get to the example of four spaces in between 10 and 22, the common difference is no longer an integer.  This allows students to chew on the concept of how we might develop an algorithm for finding the common difference (rather than falling back on guess and check), which in turn sets the stage for calculating slope.  Here are a few more examples of problems I use to guide today's conversations.

Extension Problems

If students finish #6 quickly, then I show them this extension problem.  I ask "which one is more difficult than the others, and why?"  It's a chance for students to explore the factors of 360, and I'm always pleasantly surprised by how compelling students find this task to be.

Further Extension: To Linear Functions

If it fits into today's lesson, I show students how to make a table from an arithmetic sequence.  For example, we can think of that sequence in Exercise 6d as having two points - or table rows: (1, 3) and (4,45).  Then we can consider the rows in between as (2, __) and (3, __), and we can move toward defining slope.

Along these same lines, some students will finish the extension problem.  For them, I want to continue to push them toward thinking about linear functions.  I might give them the 5th and 10th terms in a sequence, and say, "what if these terms are 6 and 8?"  This leads students to thinking about common differences that are fractions and again, pushes them toward slope, which is the focus of tomorrow's lesson.

Group Challenge: Fill in the Blanks

10 minutes

With about 10 minutes left in class, I want to check in with how students are doing and I want them to continue talking to each other about these tasks, because talking about what they understand so far helps everyone make sense of these important concepts.

This Closing Challenge is easy to understand and a bit of a challenge to complete, which makes it high-engagement, and students are excited to try to work through it together.  Of course, for anyone who really gets the algorithm, it's not too hard.  The ideal scenario is that the algorithm will become clear to kids as they work on this problem, which will set them up perfectly for the discussion of slope on the coordinate plane that will be our focus for the rest of the week.