Fractions. If just thinking about them makes you shudder, you’re not alone. For kids who struggle with math, fractions are often the first concept that truly baffles them. And for many students, struggling with fractions is only the beginning of their math struggles. A deep understanding of fractions lays the groundwork for much of the math that comes later, so if students don’t understand them it’s often hard to recover and find success later on.
But why? Why are fractions so difficult for students to understand? And why does a lack of understanding them lead to so many other math difficulties?
It has a lot to do with how our number system is built and how fractions change our understanding of that number system. Have you ever noticed how fractions with a denominator (the bottom number) of 10 weren’t as difficult to understand? Fractions like 1/10 and 3/10 probably made more sense to you; they were even easier to convert into decimals. If you recall, 1/10=0.1, 2/10=0.2, 3/10=0.3 and so on. Pretty straightforward, but this is no mere coincidence! Our entire number system is built on our basic understanding of the number 10. In fact, our number system is literally called a “Base 10” number system! Ten is the first double digit number and it’s also the first time we encounter regrouping - you’ll notice that numbers get a new, special name for each additional group of 10 (twenty, thirty, forty, etc). Any whole number (1,2,3,4,5,6,etc) or number with a “10” in it probably generally makes pretty good sense to kids because they’re all “Base 10” numbers and they work well with each other.
The biggest reason fractions are so difficult is because each fraction with a different denominator is in an entirely different number system! In a fraction, the denominator tells you what base you’re in. For example, 1/2 is “Base 2”, 2/11 is “Base 11”, and 14/25 is “Base 25.” And numbers in different bases don’t work very well with each other! But even these number are related to our basic “Base 10” numbers. Think about the most common fraction: 1/2. Just about everyone knows that 1/2=0.5. But have you ever wondered why 1/2=0.5? Did you notice that 2 (the denominator of the fraction) times 5 (in the decimal) equals 10? Definitely not a coincidence there either! Throughout our number system, you’ll find many hidden 1s, 10s, 100s, and 1000s if you know where to look. These types of numbers (a 1 followed by only zeros) are so special, they even have their own name, “Powers of 10” (there’s that 10 again!). People often prefer decimals to fractions because decimals are already in one of only a few different bases, all of which are “Powers of 10” (tenths, hundreths, thousandths, etc).
But let’s get back to how fractions with different denominators, or bases, are incompatible with each other. One common way in which we see this is when we try to add and subtract them. You may not recall what common denominators are, but maybe you feel the sense of dread as you try to remember the pain they once caused! Common denominators are a way of putting numbers in the same base so that they can be combined. It essentially comes down to giving the fractions a new “name” (denominator, base, and name are all interchangeable here).
Here’s an example that might make more sense: If I have 3 apples and you have 2 apples, we could combine those to say that we have 5 apples, right? If you have 4 bananas and I have 5 bananas, we could combine those, too, and say we have 9 bananas. This works because the things we’re combining have the same name, be it apples or bananas. But what if I have 3 apples and you have 4 bananas? Can we combine those and say we have 7 banapples?! Of course not! In this case - when the name of the fruits, bananas and apples, are not the same - we must change them to a name that they have in common when we combine them. This is just like fractions with different denominators! Before we can combine our apples and bananas, we must change their base, or “re-name” these items with a name they have in common. Rather than saying we have 7 banapples, we could say we have 7 pieces of fruit because apples and bananas are both types of fruit. By renaming apples and bananas, we can now combine them in a way that makes sense to everyone. This is how we can add 1/3 and 1/4 even though they’re different bases. We must first give them a new base that they have in common, in this case 12, and then we can add them once they have the same name.
At Mathnasium, our instructors are specially trained in a variety of instructional strategies - including changing the “name” of a fraction - that will help students to better understand concepts they struggle with! No matter the topic, we know the math and we have the tools to break it down and explain it in a way that makes sense to students.
Give us a call today to see how we can help make math make sense - (805) 525-MATH (6284)!
Mathnasium meets your child where they are and helps them with the customized program they need, for any level of mathematics.