*Understanding proper fractions: the zero-one relationship that a student must know before introducing the algorithm related to fractions.*

One of our students, a 6^{th} grader, said lightly that it doesn’t matter if your math is bad in elementary and middle school, as long as it’s good in high-school. Well .. really? Foundational gaps do not happen overnight so it is very unlikely you can close the gaps really quick. There are no silver bullets for curing long-term problems with math, especially for middle and high school students.

Just after our last blog “FLUENCY IN FRACTIONS IS NECESSARY FOR SUCCESS IN ALGEBRA .. AND BEYOND” where it says that the typical mistake students make when ordering fractions is to put proper fractions before 0, it happened again. Just like most kids who did our assessment, a highschooler did the same mistake; a sign that they missed to understand this concept back in upper elementary. *It’s not uncommon to see students who are able to do fractions operations but actually they missed the basic understanding of the value of proper fractions. *

Let’s talk about the zero-one relationship then. Understanding this concept is part of an important math learning process for the primary grades, and also the foundation of a focused remedial program for middle school and high school.

**The Zero-One Relationship**

The *zero-one *relationship defines all other relationships in mathematics. When the distance from 0 to 1 is known, all other relationships are locked-in. When you know how big 1 is, then the size of 2, 3, and even the fractions 1/2, 3/4, and so on, are automatically determined and easily demonstrated.

The number 1, in addition to being part of the sequence “0, 1, 2...,” also represents a *whole*, one whole thing. This notion of 1 representing the whole is the basis on which Proportional Thinking, including the important skills of computing ratio and percent, is built.

The numbers between 0 and 1 (the proper fractions) have some often-overlooked characteristics. Here’s a good example:

**Multiplication – Proper Fractions**

You know that when we multiply by a number bigger than 1, the answer is larger than the number being multiplied. For instance:

2 x 6 = 12, where the answer (12) is larger than the number being multiplied (6).

(This question says, “How much is 6, two times?”)

But when we multiply by a number smaller than 1, the answer is smaller than the number being multiplied. For example:

1/2 x 6 = 3, where the answer (3) is smaller than the number being multiplied (6).

(This question says, “How much is half of 6?” Obviously, half of any number is always smaller than the number itself.)

So, multiplying does not always make things bigger. It depends on what the multiplier is.

**Division – Proper Fractions**

Similarly, dividing does not always make things smaller.

When we divide by a number bigger than 1, the answer is smaller than the number being divided. For instance:

12 : 2 = 6, where the answer (6) is smaller than the number being divided (12).

(This question says, “**How many 2s are there in 12**?”)

But *be aware*: when we divide by a number smaller than 1, the answer is bigger than the number being divided. Notice that:

6 : 1/2 = 12, where the answer (12) is larger than the number being divided (6).

(This question says, “**How many halves are there in 6 wholes**?” When you think about it, there must be more *halves *than *wholes *in any number.)

*Please pay close attention, the language of “how many of these are in this?” is so important to build students’ understanding before introducing the algorithm related to fractions.*

The nature and behavior of the numbers between 0 and 1 should be studied as a continuing strand throughout the math learning.