On adding fractions

The first thing I do when students arrive in my math class at VISST is make sure we’re all on the same page about fractions, which are a common stumbling block for math learners. While we can all agree that 2+3=5, it’s not true that  $$$\frac23+\frac32=\frac55$$$. In fact, we can’t add the two fractions until we transform them using something called a common denominator. Why did the universe have to make math so difficult?! In this post I’ll dive into the way I teach adding fractions at VISST.

An important part of understanding fractions, in my view, is understanding the major intuitive distinction between the numerator and the denominator. When my students see a fraction like $$$\frac23$$$, I want them to think of the “3” as the unit size, and the “2” as how many of these units we have. I like to use analogies, so I start by asking them, “If you travel 2 km and then you travel another 400 m, how far have you travelled? In other words, what is 2 km + 400 m?” Early high school students generally (correctly) answer 2.4 km or 2400 m without hesitation. “Great,” I respond. “How did you figure that out?” We explore how they put the two quantities into the same units first; that is, they rephrased the question as either 2000 m + 400 m = 2400 m or 2 km + 0.4 km = 2.4 km. “It turns out,” I might say, “that when you have a fraction, the denominator is kind of like the units, that is like the km or m.”

Let’s take the addition problem $$$\frac16+\frac23$$$ and write it out this way: 1 sixth + 2 thirds. Here, I am deliberately using letters instead of numbers for the denominators, to highlight the analogy with our distance addition, 2 km + 400 m. As with the distances, we can’t directly add 1 sixth + 2 thirds because the two terms have different units. But, we can do this:

1 sixth + 2 thirds = 1 sixth + 4 sixths = 5 sixths, more conventionally written as $$$\frac16+\frac23=\frac56$$$.

Now, we sometimes get into a slightly more inconvenient situation with fractions. Let’s return to our original problem of $$$\frac23+\frac32$$$. We can write it out as 2 thirds + 3 halves and recognize that we need to do a unit conversion. However, it’s not very convenient to convert between thirds and halves. With km and m, we have that 1 km = 1000 m, and with thirds and sixths we have that 1 third = 2 sixths. But with halves and thirds, well, 1 half equals how many thirds, exactly?$$$†$$$ It’s in these types of situations, where the two units in question are not nice multiples of each other,  where the common denominator comes in. To keep our numbers nice and clean, our best bet is to convert both units to a new unit. It would be unnecessary to convert both our km and m to cm, but here we have to bite the bullet and do it. Choosing the best common denominator can be a separate lesson (it turns out to be the least common multiple of the two denominators), but for starters we’ll just use the rule of multiplying the two denominators: when the denominators are 2 and 3, we’ll convert to the unit of sixths because $$$2\times 3=6$$$. Here we go:

2 thirds + 3 halves = 4 sixths + 9 sixths = 13 sixths or $$$\frac{13}{6}$$$. Hooray!

Now, students are very capable of memorizing a set of steps for performing mathematical operations, so the triumph here (if indeed one exists) is not that we can now add fractions. Rather, it’s that students now have an intuitive understanding of why we add fractions the way we do, and will never again be tempted by the allure of $$$\frac23+\frac32=\frac55$$$. Indeed, we wouldn’t add 2 km + 400 m = 402 kmm!

† In fact one half equals 1.5 thirds, so you could say 2 thirds + 3 halves = 2 thirds + 4.5 thirds = 6.5 thirds or $$$\frac{6.5}{3}$$$. This is not technically incorrect but opens a can of worms that we don’t want to open at this point in the lesson.

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