Concept #1: Establishing the Measurement Principle and Placing Positive Whole Numbers on the Number Line
The Measurement Principle
In order to remedy the siloed nature my students’ view of a number, and to help them develop a unified sense of number, I plan first to give numbers meaning by exploring where numbers come from. Without exploring this very basic principle, there is no foundation to build on. Before getting to the number line in greater detail, my students will participate in a process of measuring everyday objects, from first principle, without using a standard tool. Each group of students will be asked to measure anything in their book bags. They will be asked to determine how big is their object with no other context besides making meaning for themselves. Throughout this process, I expect, and will encourage, my students to try to acquire some unit of measure to compare their object to. Students will innately try to discover that their object is 4 fingers, or 3 pen caps in length. During this process, students will not be able to exactly align their “unit” into multiples. The object will be a multiple plus some leftover length. The answers that I will get from the class will be vary from exact to in between unit lengths.
Next, the students will be asked to team up with another group and start to combine their objects. This will allow students to compare how they measured their individual object and come to a conclusion about which unit measurement is most efficient. This should lead them to conclude that we need some sort of standardized unit. The development of the unit is of the upmost importance because this will come into play when whole and rational numbers are developed later in this unit.
Students will see that numbers are generated from standardized measurement. First, the unit must be introduced, whether the length is a bar or a fixed counting manipulative, it does not matter. Below, I have defined the unit arbitrarily in Figure 1. This is compared to other bars that are multiples of that unit length. Students will then repeat the above exercise but using the standard unit.
Figure 1
An arbitrary object will be chosen. It can be measured in terms of the unit by putting unit lengths together and lining them up against the object. There are no gaps in between unit lengths. We now can say that this object has a length of 4 units with no leftover lengths.
Figure 2
Next, we can see that the length of the object is defined as the number of unit lengths that fit into it, without overlap or gaps. Later students will see that is exactly the principle for placing numbers on the number line. The broken line below in Figure 3 has measure not exactly 4 units, but a little less. We know that in terms of the unit measure that this second object is between 3 and 4 units in length.
Figure 3
This leads to error in measurement and may create greater error when measuring more than one object together. This is not an ideal procedure when measuring; we will need take parts of this unit measure to more accurately measure objects.
Placing Positive Whole Numbers on the Number Line
Once students have grasped the measurement principle, they can apply this understanding to the number line. They can use it as a viewpoint from which to see numbers in terms of distance and length. By having students measure objects using standardized units, I have laid the foundation for the number line. The importance of the unit is that it can be used to develop the rest of the whole numbers to the right of zero. We must do the following:
i) Choose an origin. “0”
ii) Choose a unit interval to represent the distance from “0 to 1.”
Now that we have a fixed standard model, we can start to place each whole number as a multiple of that distance. In other words, the location of whole numbers on the number line depends on what is chosen as the 1. Every whole number following 1 is a multiple of the unit distance i.e. the distance from 0 to 1.
Note, this implicitly gives us an orientation. Going to the right of the origin will give birth to positive numbers. Later we will see inversely, that laying off multiples to the left of the origin will give negative us whole numbers. Keeping in mind the measurement principle:
iii) The distance from 0 to n is n times the unit distance.
Along with orientation, this will give us 3,4,5...and -3,-4,-5... as well. This will allow my students to be making the connection between counting number and measurement number.
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