The Number Line in the Common Core

Using rods and ten blocks to recognize lengths

In order to build some background understanding for students, I plan to begin with having students work with Cuisenaire rods, base ten blocks, and other tools for connecting length with counting. In the initial experiences my students will not assign numbers to the rods but I will develop the idea that putting rods together is a version of addition. The act of placing the rods end to end without gaps will be reinforced, I will explicitly teach this process to my students so that they will be able to use the manipulatives in a mathematical way that will build their conceptual understanding that we are creating measurement sticks. This activity will allow my students to visualize varying lengths and begin to join lengths and compare lengths without assigning values.  As students work with the manipulatives, I will encourage them to describe their trains in terms of how many units they have assembled. “My train is 5 units in length”, will enable them to make the connection between length and number. Once students become comfortable with this connection between number and length, the concepts of joining trains end-to-end as addition will give them visual and concrete meaning. Using base ten pieces provides for an excellent transition from single-digit addition to multiple-digit numbers.

In our seminar we distinguished five Stages of Addition.

Stage 1 Putting bars together;

Stage 2 Measuring against a unit length;

Stage 3 Addition on the number line (number ray);

Stage 4 Vector addition of signed numbers on a number line;

Stage 5 Addition as a translation of the number line. 

My unit will focus on Stages 1, 2 and 3.

Stage 1: Putting lengths/ bars together

Length measurement has its own notion of addition and we shall begin by putting lengths together without worrying about the number, allowing students to compare lengths and make observations about this.

a. putting two lengths together (combining- analog of addition)

b.  comparing lengths (analog of subtraction)

As students engage with the manipulatives, they are building understanding of addition properties, and also that numbers can also describe length measurement.

Stage 2: Measuring against a unit length, connect numbers to counting via measurement

At this stage, a unit is designated and then addition and subtraction can be assigned numbers by measuring in terms of the unit. The operations of addition and subtraction of lengths can be checked to agree with the already learned ideas in terms of cardinality of sets.

= 1 unit

A visual representation of 5+3=8

There are five units that are placed in a train or end to end. Then we place the three units on the end of the train to create a length of 8 units. Five units added to three units will create a length of 8 units.

This can be refined to fractions by creating fractional units. Using Cuisenaire rods, I will establish one rod as the whole, and students will then investigate which rod they will need 2 of to cover the whole unit, I will explain that since we need two units to cover the whole, this new rod is ½ the size of the whole unit. We will repeat the process with the unit fractions 1/3 and ¼.  At this point, I will emphasize the relationship between one whole unit and two equal units of ½ to cover the whole unit.  This can be extended to then include equivalent fractions for later lessons, however I will not be addressing equivalency in this unit.

Geometrically the process is the same, as the bars don’t care how long they are.

Stage 3 Addition on the number line (number ray)

The rods we are using do not have an intrinsic value.  As we place them together and place them on a number line, value is assigned. I will be using number lines that correspond with the Cuisenaire rods. This will allow the bars to form a standard unit for the students to compare their trains to the numbers on the number line. In this stage the number line will create a standard model and an origin will be chosen. As the origin is determined, the rods then become a measurement from the origin. The number line then will give a standard exemplar for each length. So for the expression 5+3, take a train of 5 units then place the next train of 3 where the train of 2 ends. You have now created an addition model combining 5 to 3 end to end. The unit will measure 5 units from the origin. This models the principle of slide, as you place one train onto the end of an existing train along the number line.