Irrational (and Other!) Numbers on the Number Line

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SWBAT find the approximate locations for irrational numbers - and numbers in various forms - on the number line.

Big Idea

In addition to continuing with their work, students learn one of my favorite closing structures: appreciations!

Pre-Bell: Think about what you're going to do today.

Here is today's agenda: U1L15 Agenda.  It follows a simple arc: we're going to circle up to set the stage for our work, then you're going to spend the period working on whatever you need to do, then we'll circle up at the end.  I think this is my favorite kind of agenda, and I would like to do it more often than I do.

As students enter today, I provide them with the "pre-bell" prompt: "Assess your progress on the Number Line."  I say, "Think about what you've done so far on the Number Line Project.  Think about what you're going to do today."  During the first two classes this week, students got started on the Number Line Project.  This is only our third day working on the project, and already, the class is widely differentiated, as different students experience different successes and struggles.

So I simply tell them to get ready for whatever it is they have to do today.  

Circle Up: Three Questions

5 minutes

When the bell rings, I ask everyone to circle up.  It's the first time we've done this since the first day of school.  Most students are game, but there are always a few who don't like this structure.  To these students, I say, "Join us.  This is important."  I also make a big deal of all students standing in the circle.  No sitting down, leaning on a desk, or "double-parking" outside the rest of the circle.  The purpose of this structure is to help transfer power and authority from the teacher to the students.  Even though it feels a little contradictory, I use my authority to insist that students take this responsibility seriously.

I point to today's agenda (Agenda), which includes three questions: 

  1. What have you learned so far?
  2. What question do you have?
  3. What do you want to accomplish today?

I explain that everyone has to choose one of these questions and answer it.  It doesn't matter which one.  I give everyone a few moments to think, then we go around the circle and everyone has a chance to share.  I ask for a volunteer to start, and if anyone makes it clear that they'd prefer not to speak, I give them that option by saying, "You can always pass if you don't want to share."  

When it comes to my turn to speak, I say "I want to know what 1/3 of 20 is," referring to a common hangup that students have on Part 1b of the project.  When we get all the way around the circle, I check the time and tell students how long they have to get work done today.  I say that with 5 minutes left in class, we'll circle up again, and that if anyone is ready for Part 1c, it's available. Then, we break.

Introduce Part 1c in this lesson

5 minutes

Note About Pacing

As I mentioned in the previous section, today is the third day my class will spend on the Number Line Project.  At the beginning of today's class, my students are at various levels of progress through the project.  My most attentive students and those who can get homework done outside of class will be done with Parts 1a and 1b entering class today.  Other students might be finishing up Part 1a or working through Part 1b.

To simplify these lesson plans, I'm sharing these parts of the project at the rate of one per day. Even though some of my students won't get to this part of the project until later in the week, you can get a general idea for pacing here, and adapt as necessary as you implement all or part of this project.

The Number Line Project, Part 1c: Build a Number Line

When students show me that they've completed work on Part 1b, they're ready for the next part. Part 1c is even more student-directed than Part 1b, and I verbally offer three quick instructions. First, I give students a sheet of legal-sized paper and a rule, then I share the first instruction: draw a 12-inch number line that goes from -6 to 6, and to label all the integers.  That's all I have to say for them to get started.  When they've successfully done that (and some need help figuring out that each integer should be 1 inch from its neighbors), I tell them to mark - but not label - every quarter-inch on the their line.  Some students need help here, and every time they do, I'm glad to be spending this time now.  A ruler is a tool that everyone knows, but many of my kids don't know the details of how to use one strategically (MP5).  As the year continues, I'd like my kids to use more advanced tools than that, so it's good that we start here.

Once that's done, I simply give them this handout: NLP Part 1c, and say that their task is to plot all of these numbers on the number line.  "Make sure you write each number as it is written here," I explain.  "So for 'root 4' don't write the letter 'c' or the number '2'.  Find this value on the line, and write it as you see it here."

As students dig into this activity, it again opens up all sorts of conversations.  Student questions will lead to conversations about square roots, approximation, rounding, negative numbers, and more.  When we have conversations about irrational numbers, I'll point to the learning target and say that some of these numbers can be placed exactly, while other times we can use approximations.  This is a key distinction, and I'm really glad that the learning target makes it, because part of attending to precision (MP6) is knowing when to pay close attention to it, and when to settle for an approximation.   As such, this activity sets the stage for similar conversations we'll have throughout the year.

Work Time: The Best Kind

33 minutes

Work time - or may I call it "workshop" time - is my favorite way to spend my lessons.  When everyone is working on what they need, and learning whatever they can where they're at, so many amazing lessons can happen.  This half-hour is too open-ended for me to describe everything, so here I'll share one strategy and a photo.

Strategy: Playing Telephone

I love to build kids up as the experts.  So when I see that one student has figured out how to solve a particular problem, I might refer their classmates to them when those classmates are struggling. This strategy also works when I'm introducing something new like Part 1c to one or two students at a time.  As more students finish Part 1b and are ready to move on, I like to start handing off the ruler and paper and then pointing kids in the direction of a classmate who is already started.  This way, the knowledge spreads while I continue to help the students who have fallen behind, and kids are more often inclined to be more helpful than me, by sharing tips and hints for this part of the project.

Photo: Student Work

The image I've shared here is a little piece of student work that shows why I do this project.  This student has some misunderstandings of fractions.  This is why we must engage in this work! Surfacing stuff like this: U1L15 Misconception Fractions on Number Line, at this time in the year, is so important if every student is going to learn something.  What would you say to this student?

Closing Circle: Appreciations

5 minutes

With five minutes left in class, I ask for everyone to circle up again.  I introduce another one of my favorite structures: appreciations.  Instead of going around the circle in order, now everyone can speak whenever they'd like.  The prompt is simply to call out someone you appreciate, and say a little bit about why you're appreciating them.  Maybe it's because they helped you with something today - then share what it is.  Maybe they were just nice to you.  These appreciations may or may not be related to the math work, that's ok.  

As with the opener, I wish I could write about great things I've ever experienced while circling kids up. All I can say is you have to try it!  It feels awkward and campy until you start talking, then it's just the best!  Also, this is one of those structures that might feel like it falls a little flat the first time you try it, but the next time, you'll see that kids really did appreciate it!  It can take time for 14-year-olds to admit that they like to do this, but by the end of the year they and you will be glad that you did!