Open Number Lines - Just the Basics

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SWBAT model and solve multiplication problems with a factor of 1, 2, 5 or 10 on an open number line.

Big Idea

Open number lines are a model for mathematical thinking that can be used in multiplication and division, as well as addition and subtraction.


6 minutes

Ask students to think about experiences they have had with number lines in the past.  After a "30 seconds, think" (during which they are silent - especially for all the students in our interactive classrooms who prefer quiet or silence), have them share w/a partner.  Monitor peer conversations for a minute.  Ask them to share out.  Record 4 or 5 responses in a visible place (whiteboard, type up and project, and so on).  Now ask them how they think a number line might be used in multiplication.  30 seconds, think.  Do not have them share out at this time.  It's important that students know that their thinking doesn't have to be externally validated to be worthwhile!  

Remind students that a line goes on forever.  A line segment is a piece of a line with two endpoints. Today we will be creating a line segment, starting at zero and ending up with our product, but we will draw it on a line.  The number lines in open number lines aren't pre-labeled or segmented.  This allows for flexible use and thinking.

Direct Instruction

6 minutes

I can't assume that my students come to third grade with experience in using an open number line. I know it is a critical "mental model" fundamental to mathematical thinking, so initially rather than having my students use the open number line, I model a few examples that demonstrate how to use it.

Show Me Video 1

 In today's lesson, I show students three different examples, briefly, and then we move to the interactive modeling where they solve multiplication problems on an open number line along with me.

Show Me Video 2 

The 3 examples covered here took less than 6 minutes of instructional time, and then I moved immediately to the interactive modeling part of the lesson.  As I said, you may want to make you own examples, but it's important to do a short demonstration before the students are asked to engage in the process.

Interactive Modeling

9 minutes

This part of the lesson is easily done on scrap paper or whiteboards.  I have included the paper version because there are times - and students - who do much better with a pre-formatted structure.

In my own classroom, I gave this paper to about seven children and had the rest do it on their whiteboards.  I gave it to students w/poor motors skills, students who needed a greater degree of accountability than an erasable whiteboard, and two students who are such perfectionists that they would have taken all day trying to make the number line look perfect w/out even beginning to solve the equation.

The equations are simple x1, x2, x5 and x10 problems so that you can be certain the the students understand the process before moving on to work more independently. Number patterns of 1s, 2s, 5s, and 10s have been practiced by students since kindergarten (e.g., skip counting). Now, we can use this pattern knowledge to develop multiplicative thinking (3.OA.9).

Peer Practice

26 minutes

I did give the students a worksheet for this part of the assignment because for me, with this particular class, many of them need to know they will be held accountable with a permanent record in order to live up to their full potential.  

Again, this could easily be done on whiteboards or scrap paper, with the problems projected on a screen, available on a class website, available in a document emailed to students, or written on the whiteboard/chalkboard!


5 minutes

Throughout this series of four exercises using open number lines and multiplication equations, I'm asking students to identify patterns they see.  My emphasis is on patterns with twos, fives, and tens, as those are at the core of students' ability to decompose, but whenever possible I also nudge students into thinking about other patterns:

  • What do they notice about problems in which one of the factors is even?
  • What do they notice about problems with two odd numbered factors?
  • What do they notice about problems in which both factors are even?
  • What patterns do they see with twos (looks like counting by twos to ten "0" and then repeats).
  • What patterns do they see w/fives (digit in ones place is always a zero or 5) and so on.