Content Background
The Measurement Principle and Placing Whole Numbers on the Number Line
The number line is the primary model used in this unit to increase students’ number sense. The number line model assists students’ conceptualization of numbers and the relationships between them. The conceptualization comes from the student ability to visualize the numbers, the position of said numbers, and the operations done to them. This model will develop gradually so students can develop this understanding at any level.
In order to re-teach whole number placement onto the number line in a way that students may conceptualize the process, they must first look at numbers as a measurement of length. The linear number line model assists students in understanding numbers by way of the measurement principle. According to the measurement principle, in order to determine a number, an origin has to be determined in addition to a unit interval before you can label points by numbers.
The Measurement Principle: The number labeling a point tells how far the point is from the origin/endpoint, as a multiple of the unit distance.
The endpoint itself is labeled 0, since it is at 0 distance from itself. The unit interval [0, 1] has unit length, and 1 is at distance 1 from 0. Then 2 is twice as far from 0 as 1 is, so the interval [0, 2] is composed of two intervals of unit length, namely [0, 1] and [1, 2] and similarly with larger whole numbers. That is to say, the label of a point is determined by the ratio of the unit length to the distance from the point to the origin.
Traditionally in the math classroom, students have made connections between numbers and the number line by means of counting up from left to right without a conceptual understanding of the number placement. Instead I will address counting to students by discussing the origin and units of measurement being used to deepen understanding. I will initially address the geometry of the line, discussing the distance and origin. Next, we will discuss the orientation or direction of the line. This is done by determining the “ends” of the lines in regard to the direction of positive or negative movement. Finally, numbers can be placed on the number line accordingly.
Fractions as Ratios/Placement of Fractions
When looking at the conceptualization of fractions as ratios, students will use the number line as a model to strengthen their understanding. Students must look at fractions as numbers. The students will use the number line to see fractions as numbers and begin by again looking at the origin, 0 and looking at the interval from 0 to 1, where 1 represents the unit whole. Teachers will instruct students on how to partition and place fractions on to the number line. They partition, place, count, and compare fractions. The conceptualization comes into place as students connect the measurement principle to the distances between fractions.
“The number 1/2 goes in the middle of the unit interval, so that [0, 1/2] and [1/2, 1] have the same length, which is 1/2. The label 1/2 goes with the middle point, because then the point 1 is two times as far from 0 as 1/2 is. Similarly, the points 1/3 and 2/3 are the points that partition the interval [0, 1] into three equal subintervals, so that each interval has 1/3 the length of [0, 1]. The label 1/3 goes with the interval that starts at 0, since the point 1 will then be 3 times as far from 0 as is 1/3. The label 2/3 goes on the other 1/3-division point, since there are two intervals of length 1/3 between it and 0, or in other words, it is 2 times as far from 0 as 1/3 is. Similar reasoning serves to locate fractions with larger denominators on the number line” 1
I will work carefully with my students to make sure they know how to place fractions on the number line. This is not understood by many students and is an important misconception to address since students are often only taught to work with fractions in isolation. Students need to understand that placing fractions involves the same process as placing whole numbers. If students understand how a fraction n/d on a number line with a fixed denominator will create a variety of points with equal intervals the length 1/d. This visual will help generate a deeper more conceptualized understanding of fractions on the number line as they will see it’s similarity to the placement on whole numbers on the number line. Once they reach this point of conceptualization, it will be easier for students to be able to compare, contrast, and perform operations with fractions.
I am eager as a teacher to revise my student’s conceptual understanding of fractions to measurement because the previous memorization and visual aids have not been successful in the long term. The linear model is more effective than the frequently used pie model because of the support of unitization. When my students can visualize the unit fraction as the basic structure for all numbers and fractions on the number line, I believe that they will gain conceptual understanding of the fractions. This model also lends itself to instruction that can interchange representations from concrete to pictorial to abstract, which differs from previous student learning.
Adding on Number Line as Combining
With the number line model, the students are at the point where they are able to accurately envision numbers on the line as distances from the origin. It is now appropriate to use representations of addition on the number line for computation. To begin conceptualization, students will use bar models on the number line to represent numbers (or lengths) and then begin the process of putting the lengths together. The concept being taught to students is the idea of seeing addition as the combining of lengths. It is important next to make the connection to the measurement principle so students understand the connection to measurement by addressing the numbers and the units. This is visualized by discussing the unit length, its connection to the given number or length, and the combination of two or more lengths, followed by measurement to reach your answer or total number of units. This will continue to be refined when discussing fractions by using the fractional unit. It is important with fractional addition that students understand that geometrically the process is the same and even when a unit is defined the bar lengths may differ in length.
Here we have divided the unit interval into intervals of length 1/24. This lets us see that each 1/3 of the interval is 8/24, and each 1/8 of the interval is 3/24. This means that 3/8 + 1/3 = (3x3)/24 + 8/24 = 9/24 + 8/24 = 17/24. The number line does the computation for us!
Next students will use the number line as the arena for solving addition problems with attention brought to orientation and direction when working with signed numbers. I am in hopes that using the number line for addition will continue to support students as they develop deeper number sense and number relationships. Previous student learning has relied on simple models and memorization that often resulted in a low level of conceptual understanding and computational facility needed in higher grades. The number line is a simple tool that if used after students understand the measurement principle, will assist in creating a conceptual and procedural understanding due to the strong grasp of position and measurement in addition.
Subtracting on Number Line as Comparing
Subtracting on the number line will be done similarly to addition, but instead of combining lengths, students will compare the lengths. When introducing subtraction, students must concretely see the bars but instead of putting the bars together they must line the bars up in order to compare them. Students will continue to bear in mind the measurement principle when comparing the two bars distances from the origin. It is important to show students how to use the number line to measure the differences in the unit lengths. This can be looked at by the subtraction of the units or finding the “missing part” of the two lengths. Once this is understood, I will continue to practice subtraction with students but with the introduction of vector addition of signed numbers on the number line. This is the point where addition and subtraction start to become unified.
Students have traditionally been taught to see subtraction as taking one number away from another, as opposed to comparing two numbers. This basic understanding does not lead to a conceptualization of subtraction and leads to misconceptions when students are working with fractions and signed numbers. The number line model allows students to continue to see how numbers and operations are all connected to measurement. By transitioning their thinking from separation to comparison, the students will have an easier time understanding subtraction as a translation on the number line, which will ultimately have a greater impact on their ability to make connections from arithmetic to Algebra.
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