How to teach sum of fractions to students?

Question

I think almost every middle school student in my country has learned sum of two fractions in this non reflexive way (I'm included when I was kid), doing the following steps:

1. They calculate the lcm.
2. They draw a long bar above the lcm.
3. They divide the lcm by the denominator of the first fraction and multiply the result by the numerator. The result of this operation they put in the numerator above the long bar.
4. They do the same with the second fraction.
5. They sum the two numbers above the long line

So how is the better way to teach sum of fractions to students? I want they know what they are doing.

Answer 1

First, I would have them really understand equivalent fractions. There are a lot of ways to write the number represented by the fraction $\frac23$. We can call it $\frac23,\frac46,\frac{20}{30},\frac{-2}{-3},$ etc., etc. Similarly, there are many ways to write the number represented by the fraction $\frac45$: as $\frac8{10},\frac{20}{25},$ etc.

Once that's clear, they should practice adding "like fractions", i.e., fractions with the same denominator. We can add $2$ sevenths and $3$ sevenths to obtain $5$ sevenths for the same reason we can add $2$ dogs and $3$ dogs to obtain $5$ dogs: they are things of the same kind. Calling them "like fractions" also has the advantage of preparing them for adding "like terms" when they get to algebra.

Finally, in order to add fractions, we must first write them as "like fractions", which requires finding a "common denominator". Whether it is the least common denominator, or some other common denominator, is not very important at this point. The can practice figuring out what a common denominator might be for various pairs of denominators, and this practice can be done independently of any adding or subtracting.

Now they are ready to "add unlike fractions". First, find a common denominator, then write each fraction as an equivalent fraction with that denominator, and then finally add the resulting like fractions.

Putting it all together for this example, we start out by noting that we have unlike fractions: thirds and fifths. A common denominator for $3$ and $5$ could be $15$ (that's the first one most students will name). Thus we write:

$$\frac{2}{3}=\frac{?}{15} \,\,\,\,\,\,,\,\,\,\, \frac45=\frac{?}{15}$$

Using what we know about equivalent fractions, we decide to multiply the first numerator by $5$, and the second one by $3$:

$$\frac{2}{3}=\frac{10}{15} \,\,\,\,\,\,,\,\,\,\, \frac45=\frac{12}{15}$$

Now, having obtained like fractions, add them:

$$\frac{10}{15}+\frac{12}{15}=\frac{22}{15}$$

As a final step, if the resulting fraction is not in lowest terms, we can reduce it. In this case, no reduction is necessary.

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I like this method because it can be taught as distinct skills, each of which is independently meaningful:

• Recognize like fractions vs. unlike fractions
• Identify a common denominator for two different denominators
• Write fractions as equivalent fractions with a given denominator
• Add and subtract like fractions
• Determine if a fraction is in lowest terms and reduce it if necessary

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If I were to teach any mechanical algorithm for adding fractions, I'd probably go straight to the "bowtie" method, also called the "butterfly" method. This is illustrated in many videos and webpages; here is one example.

Answer 2

When teaching, I always try to relate the topic back to something the students understand and experience with. I most often default to money but depending on the grade level that may or may not work. When it comes to fractions, I first like to talk about the terminology to help the students know what a fraction really means. I note that the word "denominator" has the same root as "denomination", as in the denomination of a dollar bill.

For fractions, the numerator tells you how many things you have but the denominator tells you how big each one is. Once students get a core understanding of this, you can then start to talk about how you compare two different fractions, both with matching and non-matching denominators.

I also like to use pizza. So 2/3 of a pizza is a pizza sliced into 3 equal pieces and you have two pieces left. Similarly for 4/5 of a pizza. How much total pizza does that make? Help them realize that the 6 slices are different sizes so they cannot be compared directly. But, would it be possible to cut them into thinner slices so that they match? This leads to the idea of the least common denominator. Then cut the pieces and count them up!

Answer 3

I can't say I have much pedagogical advice, but I think it is worth thinking about fractions from a much older point of view, before "fractions" and "division" were unified.

The purpose of the vinculum was grouping, and it was one of the competitors to parentheses. By putting a number under the vinculum, you could specify the denomination against which the top number was reckoned. Using the terminology in Mike Pierce's answer, it gave the units. It's not the fractions per se that are being added, but rather numbers are added together in a way that respects their denominations.

Some strategies I might consider:

• Write the denominator smaller than the numerator, strengthening the idea that it is a different kind of number. Maybe also temporarily write the plus sign at the level of the numerators.
• Write fractions like $1/3$ instead of vertically. Or temporarily use a new notation like $1/3^{\text{rd}}$, $1\text{ third}$, $1\text{ of }3$, $\frac{1}{\text{thirds}}$ (which should be understood as a vinculum indicating grouping, not division. Example: $\frac{1}{\text{thirds}}+\frac{1}{\text{fifths}}=\frac{8}{\text{fifteenths}}$).
• Use units everywhere, though it can be awkward to get right! Define an eighths-pizza to a pizza with eight slices, etc. Then $\frac{1\text{ slice}}{\text{eighths-pizza}}$ and $\frac{1\text{ slice}}{\text{tenths-pizza}}$ is in total $\frac{9\text{ slices}}{\text{fortieths-pizza}}$. It's worth realizing that it is not actually an equality, though, because the slices will probably be cut differently, so it should be pointed out that we only care about how much pizza we have.

These strategies are based on a pedagogical theory I once heard that is OK to slightly misrepresent material if it will help students reach a point where they may properly master it. The misrepresentation here is changing the commonly used notation, which I think is forgivable, since the notation is not the concept.

My hope would be that a notation that doesn't suggest a fake addition rule would cause students to think a little harder, ideally with the effect that they invent common denominators for themselves. Changing denominations would be be the lesson before this one.

Answer 4

A fundamental thing that students need understand to really know what they're doing when they add fractions is why we have to find the LCM of the denominators, or why we need to make the fractions have "like denominators." I would set up the problem up like this:

Say you've got to add the fractions $\frac{2}{3}$ and $\frac{4}{5}$. I would immediately rewrite this problem, and say that we need to add $2 \text{ thirds}$ and $4 \text{ fifths}$. Writing out the denominators as units should make it much clearer to students why we cannot add them immediately (then say something about comparing apples and oranges if you really need to). The idea is that we need to convert thirds and fifths into a common unit so that we can compare the two quantities and add them together.

I'm not sure the best way to approach teaching how to find this common unit though. I definitely would not use the term LCM. Maybe just tell the kids after they are comfortable adding fractions, "this number is commonly called the LCM." But I wouldn't even focus on the "least" aspect of finding a common multiple: it's way more important to just find a common unit for the two fractions. But it shouldn't be hard to show them how to convert units in general. For example, you can show that them that $1 \text{ third}$ is the same as $2\text{ sixths}$ by just drawing a pie chart subdivided into thirds, shading a single slice, and then subdividing each slice once more.

Doing this conversion for a bit, and looking back to the original problem, I would hope that students would see that we can talk about both fractions in term of $\text{fifteenths}$. We can convert $2\text{thirds}$ to $10\text{ fifteenths}$, and we can convert $4\text{ fifths}$ into $12\text{ fifteenths}$. Then comparing apples to apples, we have $22\text{ fifteenths}$, which we may write as $\frac{22}{15}$ if we'd like.

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Reading back, this answer is poorly written. But I trust that my meaning is clear. I'll try to write it better later.

Answer 5

I know I'm late to the discussion; but like to share a thought that I think complements the answers from above:

The way G Tony Jacobs's answer suggests to progress seems to be common ground. In order to explain the „why“ – as opposed to the „how“ - for adding fractions, this thread only mentions the pizza model.

Now, to me, if using a model (like the pizza-model) in this context, there are two questions I usually ask:

1. How does the model explain what the numbers are?

A good model for numbers explains the numbers as objects. For fractions, thats not an easy task! But I guess this is where G Tony Jacobs's answer starts, when aiming at students understanding like and unlike fractions.

2. Does the model explain all elementary operations (addition, subtraction, multiplication, division).

In a recent article, J. Dixon and J. Tobias (2013) do a nice job explaining how to use the pizza-model for all operations. But getting to division, all over the sudden the pizza gets a different shape: it’s not round but rectangular! But then, it could have been rectangular to begin with...

Now the pizza model explains fractions as „part of a whole“. But there are other aspects of fractions, that need to be considered: Fractions in measurement situations, fractions from sharing, fractions as rates used for comparison, fractions as intensive quantities (like density)...

So, creating links between procedures and conceptual understanding, I use more then the pizza model: you can use paper strips and actually fold fractions (see e.g. Paper Folding to Model Addition of Fractions... on YouTube), use the number line, and so on.

In my experience, talking about different models and what addition in these models look like, discussing the transition from one model to another, provides a good background in order to follow the procedures with more understanding.

Answer 6

I am going to try to add to the excellent answers here with a point I like to make and use for many of the techniques that we apply. This point is to emphasize the properties of real numbers and how they appear in our notation and techniques. This conversation happens with the students early in my college algebra class as the reasoning appears at many points later in the class.

All of the great techniques we use break down to an application of the properties of real numbers applied to our problem at hand. I try to emphasize this with students as often as possible so that they have a justification for parts of the process. Also as we develop techniques it can help organize thought.

The overall concept I want to teach is when we have something we don't have a technique for, one of our jobs is to make it look like something we already have a technique for. But we are restricted in that we can not change the value of the quantity.

I use common denominators as an early example of this idea, when we are going over properties of real numbers to emphasize the power of multiplying by one. Also this emphasizes what the properties of real numbers mean when they are used. The discussion is not meant to be the technique the students use to do this but it is meant to be a justification for the techniques they have been using. A paraphrase the discussion below and skip some points focusing on the things I want to highlight in the discussion. It should be noted the notation I am using in latex is not how exactly how it comes out on the board but is close.

In the case of fractions we are applying distribution and identity multiplication. So the first understanding the student needs to internalize is that for fractions to add or subtract they need a common denominator.

An Example of the reasoning:

$\frac{3}{8}+\frac{7}{8}$

We can write these as factors:

$\frac{1}{8}\cdot3+\frac{1}{8}\cdot7$

Here we want to note that this is the RHS of the distribution property.

$a(b+c)=ab+ac$

So we can go from the RHS to LHS. With what we were given we get.

$\frac{1}{8}\left(3+7\right)$

This completes:

$\frac{1}{8}\cdot10=\frac{10}{8}=\frac{5}{2}$

So we extend the idea

$\frac{3}{8}+\frac{5}{6}$

We again make factors

$\frac{1}{8}\cdot 3 + \frac{1}{6}\cdot 5$

Looking at what we did earlier we want to make this look like the problem we solved before. This requires the fractions to be equal. We say it would be nice if the the fractions were the same. If multiply 8 by something are there any numbers divisible by 6.

At this point we discuss the techniques for finding the least common multiple. That is not my focus here and this discussion with the class is longer than these few sentences. But we find 24 would be a nice denominator.

Here we have to dig into another property of real numbers and that is the multiplicative identity:

$a \cdot 1 = a$

We want to change the $\frac{1}{8}$ so that we have our nice denominator of 24. But we can not do it in a way that changes the value of the quantity. So I have to multiply my fraction by the number $1$.

$\frac{1}{8}\cdot \frac{?}{3}$ What number replaces the question mark so that we are multiplying this by the number 1.

We complete this argument and arrange our original problem like this:

$\left(\frac{1}{8}\cdot\frac{3}{3}\right)\cdot3+\left(\frac{1}{6}\cdot\frac{4}{4}\right)\cdot5$

We pause and talk about our goal was to have the fraction common to both terms. We wanted $\frac{1}{24}$ so we organize a little bit more. Here the discussion talks about how fraction multiplication works and justifying the next rearrangement.

$\left(\frac{1}{8}\cdot\frac{1}{3}\right)\left(\cdot3\cdot3\right)+\left(\frac{1}{6}\cdot\frac{1}{4}\right)\left(\cdot4\cdot5\right)$

Finishing:

$\frac{1}{24}\cdot9+\frac{1}{24}\cdot20$

$\frac{1}{24}\left(9+20\right)=\frac{1}{24}\cdot29=\frac{29}{24}$

I want to note the point isn't to give a technique for the students to use, but to develop the reasoning for why the technique is going to work. It sets the stage for later techniques where you need to find other forms of the number 1 or you need to add by zero (complete the square).

I remember in graduate electrodynamics facing the dreaded Jackson textbook and this horrible problem that none of us were getting after hours of work and pages of attempts. The instructor in one line simplified the problem by multiplying a unit vector divided by a unit vector, the vector multiplication transformed the entire problem to something familiar and the person in front of me exclaimed in agitation we just multiplied by 1. Setting the stage for this type of thinking can extend the concepts of arithmetic into the concepts of mathematics and pattern manipulation and problem solving. Somewhere in the early 80s when I was in 5th or 6th grade I remember having exercises for a few years where you could only solve the problems by using the properties of real numbers and you had to state what you did on each step. I think the idea was develop this idea of building the techniques up. Though it was lost to me at the time I do think it affected how I thought of the process.

Answer 7

I would suggest to give names to operations and objects. If you want to add two fractions, first find their common denominator. This can be the lcm but sometimes other options are more suiting (for prime denominators it's just the product, no need to involve the lcm, sometimes a power of ten as common denominator is better than the lcm, ... - let them be flexible here).

Then you need to expand each of the fraction to the common denominator by multiplying the nominator and the denominator by the suiting expansion factor.

Then you need to combine the fraction to one by taking the common denominator as denominator and the sum of the nominators as nominator, so you write a big fraction line.

Basically, those are the same steps as mentions in some of the other answers, but I would like to emphasize the importance of naming things to give them meaning and give the algorithm more flexibility and structure - you then will have distinct steps which make sense by their own, not only as part of some algorithm.

Answer 8

I'm not a teacher. I don't know how most students really learn. This is just an idea and it should be taken with a grain of salt. It's the job of teachers to not rely on this answer, and do their own research based on this answer.

I think it will work so much better for the teacher to give the students a worked example in class. They should pick two fractions one of which has 4 as a denominator and the other of which has 6 as a denominator both of which are in lowest terms. Next, they should convert both fractions into another form with 12 as the denominator. For example, they could be shown that $\frac{3}{4} + \frac{5}{6}$ can be reexpressed as $\frac{9}{12} + \frac{10}{12}$. Then they could teach them that the way to do it is to find the lcm of the denominators of each of the fractions and then convert each fraction to the form whose denominator is the lcm of the original denominators. They should word it as the lcm of the original denominators. If they call it the lowest common denominator, some students might have forgotten what "lowest common denominator" means and be confused. It's so straight forward to just tell them to convert each fraction to the form where the denominator is the lcm of the original denominators. I think students will be very capable of teaching themselves how to do that. On the other hand, I think that if the teacher replaces the instructions to do it with the instructions on how to do it, some students will see it as a big long list of instructions on how to calculate a sum and not understand what they're saying.

By picking that problem specifically or something like it, both fractions will be in lowest terms. Not only that but also neither denominator is a multiple of the other denominator nor is the lcm of them their product. That way, the students will learn how to do it when neither denominator is a multiple of the other nor is the lcm of them their product and the original fractions are in lowest terms.