Tuesday, February 26, 2019

The Importance of the Unit, Part II: Fractions as Units

How many times have we heard, “I hate fractions!” from both children and adults? So many times, because after students spend years learning how numbers work, suddenly they are introduced to a new kind of number that breaks all the rules they learned before.

When we add or subtract fractions, we don’t add them the way we add whole numbers, but we have to remember a whole set of new rules. And when we multiply by a fraction, sometimes the product is bigger, and sometimes it’s smaller.

Dividing fractions makes no sense -- so students learn “keep-change-flip,” which if you ask students (or teachers) to explain why it works, you will get blank stares and stuttering. Not only that, but many students don’t remember which fraction to “flip,” or why to flip the divisor and not the dividend. But this mnemonic is so popular that people create videos and other funny ways to remember it and how to apply it. Still, errors abound.

And then there are “improper” fractions too -- and if you ask some students, that term should apply to all fractions, because they don’t follow proper number rules!

So as soon as decimals appear on the horizon, many students will avoid using fractions at all costs, and will run straight for the calculator and converting to decimals in high school. This makes calculating slope, ratios, and a number of other concepts more challenging, concepts that would be easier with a strong grasp of fractions.


Unlike students who learned in procedural ways, in which operations with fractions appear mysterious and illogical, when students learn them conceptually, fractions often become their favourite numbers in math. How does this work?

In early grades, students are introduced to the basic fractions they are most likely to encounter in everyday life: halves, thirds, and fourths. These are introduced in Singapore Math using manipulatives like fraction tiles and circles, and are then mirrored with pictures of these units. Students are given stories involving fractions that relate to real life as well, such as sharing pizza or other equally divisible items. The Fraction Tiles and Fraction Discs tools on Matholia can be used to model these units.

When students are introduced visually to fractions, they are then given the concept that a fraction is a unit just like any other unit. They have been introduced to counting by ones, tens, apples, and so on; counting by halves or thirds is just counting by another type of unit.

Using a number line, students can then count by units in later grades to find non-unit fractions and mixed numbers. For example: one-fourth, two-fourths, three-fourths, four-fourths (or one whole), five-fourths , etc.

Students practice converting back and forth between different forms, including equivalent fractions and mixed numbers. This makes adding and subtracting more intuitive, because the big idea is -- you have to have the same unit to add or subtract! In other words, how to you add two apples and three oranges? You can’t -- but you can convert both to fruit, and then you can add.

In the same vein, how do you add ⅓ and ⅖? You don’t -- but you find the same value in an equivalent unit, in which you can then add or subtract.

In future articles, visual ways to find equivalent fractions, and the way this approach relates to multiplication and division of fractions, will be explored.

Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.

The Matholia Team

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