Fractions and Decimals on the Number Line

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SWBAT locate rational numbers on the number line, and get comfortable expressing these numbers as both fractions and decimals.

Big Idea

Counting is easy, right? We hope to find that counting by fractions is as well!

Opener: Simple Expressions on a Number Line

10 minutes

Here is today's opener: Opener Sept17.pdf.  As usual, I give students a few minutes to try this on their own.  I circulate and encourage them to get started, and I answer questions without giving away answers.  After a few minutes of that - generally enough time for me to take two laps around the room and record attendance, I get started by guiding students through the problem.

This is a dual-purpose opener.  The first is directly related to our work on the Number Line Project, which began yesterday.  I want students to be able to create a number line, with attention to scale, and to accurately place numbers on it.  About this I am very direct with my kids.  

The second, sneakier purpose of this opener is that it gives me a chance to see how my students understand simple algebraic expressions.  I don't make a big deal out of this at all.  I simply pay attention to which of my students are familiar with the content, and which are foundering.  

There are different ways to approach this problem.  A common method applied by my students is to draw a line, place the first value on it, which is A=2, then B=3, and to continue to build the line as they go.  Some can use this method effectively; I demonstrate a more organized approach, in order to give a preview of an upcoming part of the project.  

I say that, first of all, before I can draw a number line, I need to know what numbers will be on it.  I ask for help evaluating each of these expressions.  That word, "evaluating", is a word students have seen in their textbook homework, so I use it here to make it feel natural.  Only after we're successful finding the value of each expression can we consider the number line we need.  I ask, "What are the lowest and highest numbers we'll need on this number line?"  Students now see pretty quickly that the line will go from -1 to 6.  I scale as carefully as I can, noting for students that I'm sketching on a whiteboard without the help of a ruler.  After I sketch my line, I say, "I know this is probably not perfect, but it's good enough, right?"  Then I place each point and expression on the line.  It looks like this: U1L14 Opener.jpg

This is a preview of what students will do on Part 1c of the Number Line Project.  They will be asked to make a number line that goes from -6 to 6, and then to place a variety of numbers on the line.  For more on that, see tomorrow's lesson: Irrational Numbers on the Number Line.

The Number Line Project, Part 1b: From -1 to 1

30 minutes

Here's a picture of today's agenda: U1L14 Agenda.  It's of a kind that will grow more common as the year progresses - giving students the choice of what to do, based on what they've done so far.  I try to give assignments that differentiate for students naturally, and in this project, each part builds upon the previous part.  As students finish Part 1a, then 1b is a sensible next step.  But if they need help on Part 1a, then this project meets them there.

About Part 1b

On Part 1b of the Number Line Project, students will create a number line on graph paper, then plot a set of unit fractions and their opposites in between -1 and 1.  This is a great task that gets students started, then gives a teacher many entry points for teachable moments.  While Part 1a was highly structured and algorithmic, this new task is a little more open-ended.  When students are done with Part 1a, they show me, I take a glance at it - especially anything with fractions - then I sign off on their homework sheet and give them Part 1b.  I hand them a new sheet of graph paper, and the instructions (NLP Part 1b.pdf), which are on a half-sheet of letter paper.  I simply tell them to read the instructions carefully, and to let me know if they have questions.

Of course, many students will have questions, but I want to encourage their independence.  Some will have questions about the heading - even that's worthwhile at this time of year.  Some will have questions about what to do after they place -1 and 1, and I'll say start at 0.  Where should it go?  Some of my conversations with students are about how to count on graph paper: do you count the lines or the spaces in between?  Other conversations delve a little deeper, like which fraction is the most difficult to place, and why?  All conversations are precisely what students need to learn a bit today.  

There are 20 spaces between 0 and 1, so any unit fraction with a factor of 20 in the denominator won't be so bad.  That only leaves 1/3 as a challenge.  Some students are quick to place 1/2, but then stuck.  There are all sorts of great opportunities to talk about fractions and what they mean.  

Work Time & Finishing Part 1a

In a perfect world, all of my students would be done with Part 1a and ready to move on to Part 1b.  In reality, about half are done, and ready for the next part.  That's fine.  Those with incomplete work on 1a have this class period to continue to grapple with whatever they find challenging.  I've already described this part of the project, so here I'll share an example of student work: U1L14 the sooner I see these errors.jpg.  This student was a little behind when I saw this, but this is exactly why we do this work in the first place!  The sooner I see how this student thinks about repeatedly adding a decimal value, the sooner I can work with him to correct the error.

Next Steps and Recording Homework

3 minutes

With a few minutes left in class, I put this week's homework sheet back up on the screen.  (It was originally distributed yesterday.)  I ask, "What's your homework tonight?"  Students who are keyed into the kind of work we're doing shout what they're going to do.  Others need a little encouragement to record their next steps.  I say that I want everyone to work on something tonight.  I say, "You don't have to do this whole project at once.  But if you can spend 20 to 30 minutes each night, you'll be in great shape by the end.  Just spend a little while finishing whatever you're working on."

I try to see that everyone has written something for tonight's homework before the bell rings.