Problem Set: Number Lines

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SWBAT draw number lines with attention to scale.

Big Idea

A number line gives us a way to visualize order of operations, bridging the gap between abstract and quantitative reasoning . As we begin, students pay close attention to scale.

Opener: Order of Operations

3 minutes

Today's Order of Ops opener consists of two arithmetic expressions that students are asked to evaluate.  The first one is from last night's homework, and includes a fraction in front of some parentheses; kids always have questions about this one.  The second example is heavy on grouping symbols, and I like to give the class a chance to try to make sense of it.  Today's exit slip includes an expression similar to the second example.

As students give these a try, I walk around to glance at the progress students have made on last night's homework.  If they have questions about the homework (whose content was the same as this opener), I can spend a few minutes addressing these after going over the opener. 

I say that I want students to think about keeping great notes.  After giving them a few minutes to grapple, I show them how to simplify the expression one step at a time; this comprises the day's notes.  I'm touching on order of operations here, because it's important for students to grasp it, but I'm not dwelling on it for long.  If we spend a little time each day, and we begin to use these skills in context, I hope to see that students are confident in what they know and can do.

Number Line Problems

35 minutes

Today, students begin the first problem set of the year.  It is due at the start of class tomorrow, so whatever is not done today in class serves as tonight’s homework.  The problem set is called Number Line Problems.

When it comes time to graph functions and to create statistical representations later in this class, an initial roadblock that many students face is their inability to properly construct and scale their axes.  Rather than knowing this is coming and then bemoaning it, the CCS gives us a chance to preempt this by including the standards 8.NS.2 and N-Q.1, which I combine into the learning target

SLT 1.2: I can create a number line with attention to scale, and I can locate the exact and approximate locations of all sorts of numbers on a number line.

I find that taking the time to really think about number lines now - both with this problem set, and with a project that begins next week - pays great dividends later in the class.

What is a Problem Set?

To get started, I distribute the handout, and tell students that they’re going to learn about another new class structure now.  First, I give the class some notes about problem sets.  I post the document on the board, I fill in the blanks, and ask students to do the same on their handouts.  Here are the notes: Notes on Problem Sets.  With all of that in mind, I explain what it means to complete a problem set. 

Everyone gets a sheet of graph paper.  I make a really big deal of writing the heading correctly.  I tell students that by starting with attention to details like this, they set themselves up to learn more and be better organized later.

Next we read the learning targets.  Mathematical Practice #1 will be on every problem set.  Practice #2 we’ve seen before, and number lines will help us make some connections between quantitative and abstract reasoning.  Then we read SLT 1.2.  I ask for a volunteer to read the learning target aloud, then I invite students to shout out the key words in the SLT.  I really want to draw their attention to the word “scale”.  I show them a sketch of what I call a "screwed up ruler," which has the 1 and 2 inch marks very close together, and the 3 inch mark way beyond the 2.  I ask, "What’s wrong with this?"  We consider the idea that a well-defined scale gives equal space to equivalent values on a number line.

How Work Time Works

As everyone gets started on the work, I say that I want students to collaborate on this assignment.  "If you have a question," I say, "try talking to each other before you talk to me."  I emphasize conversation, and I hope that the structure of this problem set allows it to occur naturally.  I emphasize hard work.  When I see students who are off task or taking their time to get started, I provide little leeway.  I counter this by demonstrating endless patience for mathematical questions of any sort.

What I Glean and What Students Learn / Common Conversations

Moving forward, I'll use this problem set to identify what my students know and to plan my next instructional steps with each student.  Although a lot this work will happen after I collect the problem sets tomorrow, it begins as soon as kids get to work.

The first problem on the set says to draw a number line that goes from 0 to 10, and it students have any reaction to this, it's just to say, "Oh this is easy!"  Then they get to the second problem, and say, "Wait you want me to count up to 500?"  This is where I say, "Well, there aren't 500 squares on your graph paper.  And if I ask you to count to 500, would you really want to count by 1's?"  I start counting, with a little wink.  For some students, that's enough to get them going, but if I see they're hesitating, I continue, "What would you like to count by?"  Most say 100's, others say 10's or 50's.  Whatever the case, I tell them to give it a try.  This conversation continues on each problem.

The placement of zero is another big deal, and in every class, there are always students who place zero closer to 1 than 1 is to 2 (or in the case of the second problem, 100 closer to zero than to 200).  This is exactly why it's worth our time to complete this set.  I move from table to table making sure that they scale is impeccable on the first four number lines that all students draw.  I often use the line, "I know this is going to sound really picky, but it's going to be very important later..." when I open a conversation about the location of 0.

As the problem set continues into problems that span both the negative and positive ends of the number line, the question of 0's location can puzzle even more students, and again, that's exactly why we're doing this work.

We also have the opportunity to begin to clear up some misunderstandings about the relationship between the values of numbers in decimal form.  Which is greater, for example: 0.5 or 0.25?  And once we get that figured out, how do -0.5 and -0.25 look on the number line?  Each problem here gives me a chance to figure out what my students know, and for them to shore up some knowledge that I took for granted earlier in my career.  Even for my brightest students, this assignment is an introduction to the kind of attention to detail that I'll look for as the year continues.

So with all that in mind, here are snapshots of some student work.  What "sort-of picky" advice would you give to these kids?

Exit Slip: Order of Operations

5 minutes

With five minutes left in class, I distribute today's exit slip.  I took a quick glance to see that students are doing homework, but I usually don't collect and grade textbook homework.  Instead, I give this quick slip to get a look at what my students know and can with the order of operations.

I say, "Show me perfect steps, because I'm not asking for 20 problems right now.  I'm giving you 5 minutes to give me perfect work on these two problems."  I'll use this exit slip to continue learning what level of knowledge my new students bring to class.

The document is made to print six to a page, which serves two purposes: it saves paper, and it makes for a small stack that I can stick in my pocket and flip through whenever I have a free moment.