Do you see this common mistake?
What is the fraction indicated by the red marker?
Why do students make this error? Some students see a number line divided into 8 equal parts, then started the tick marks from zero until they reach the red marker (5). Hence they deduce that the fraction is 5/8.
The ability to place and identify fractions on the number line is an important step in the teaching of fractions. In the 3rd grade Common Core Standards (CCSS.Math.3.NF.2):
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
However, students who are new to fractions and who see fractions as sub-division of physical objects (pizza, pie, etc), or shapes (rectangles, circles, etc), may find it hard to visualize fractions as being part of the number line.
There are two related challenges when it comes to fractions on the number line:
- Place given fractions on the number line.
- Name values when given a fractional position on the number line.
Some students struggle with both and in general have difficulty reconciling the fact that fractions is just another value on the number line.
The cause of students’ inability to see the fractional values on the number line can be traced to their difficulty linking sub-dividing along the number line to the sub-dividing of pictorial or physical objects.
Before this, students are generally used to seeing fractions as a part of a shape or physical object, thus sub-dividing is very concrete and intuitive. When placed on the number line, however, students failed to see the relationship between the number 1 on the number line and the one whole pizza that they used to slice in their mind.
One way we can ease students into seeing fractions as values on the number line is to show sub-divided shapes along with the numbers on the number line, e.g.
However, while this is ok for proper fractions less than 1, some students might have problem with fractions greater than one. For instance, using the same example of 5/4, students might count the number of shaded quadrants (5) and use that as the numerator and count the total number of equal parts (8) and use that as the denominator, hence deduce the fraction is 5/8.
Another way to present the number line is to use only the sub-divided parts, i.e. do not include the rest of the whole.
Notice the quadrants are still arrange in their usual orientation so that students can intuitively see the whole when all four are present. For the 5/4 example, it will simply be 5 fourths of a circle. This also makes it easy for students since it is in line with the way we name fractions. In this case, 5/4 is called “five-fourths”, and intuitively students would know that it refers to five units of fourths. Besides shapes, we can also use familiar physical objects, e.g. pizza
Placing fractions and seeing them along the number line lets the student appreciate that fractions are just another subset of the numerical system and gives them the ability to relate fractions to the already familiar set of whole numbers. It is an important step in building up the student’s intuition of the “value” of a fraction, and to have the intuition to know that something is wrong when 5/8 appears in between the whole numbers 1 and 2, as in the opening example.
This article is part of a series of blog posts on Fractions:
- >> Read the next post on Fractions: Teaching Equivalent Fractions
- << Read the previous post on Fractions: Understanding the Concept of Same Whole in Fractions
- Or start from the beginning: Understanding Fractions as Equal Parts
More Fraction Resources
For more fraction resources, refer to our main fractions page.