39 Addition and Subtraction of Fractions

You will need: Centimeter Strips (Material Cards 16A-16L), Fraction Tiles (Material Card 25)

Exploration Activity

Set-Up: Take out your fraction tiles and line up [latex]\frac{5}{12}[/latex] and [latex]\frac{1}{3}[/latex] in one long line. Put them directly next to each other so you have one single strip of length [latex]\frac{5}{12}+\frac{1}{3}[/latex].

1. Underneath your set-up, lay out the full row of fourths. (All four pieces of size [latex]\frac{1}{4}[/latex].) Line them up so your first [latex]\frac{1}{4}[/latex] piece starts at the same place as your first [latex]\frac{1}{12}[/latex] piece.How many [latex]\frac{1}{4}[/latex] pieces does it take to exactly match [latex]\frac{5}{12}+\frac{1}{3}[/latex]?
   

2. Remove the fourth pieces. Now underneath your set-up, lay out the full eighths strip. Again, line it up so your first [latex]\frac{1}{8}[/latex] piece starts at the same place as your first [latex]\frac{1}{12}[/latex] piece.How many [latex]\frac{1}{8}[/latex] pieces does it take to exactly match [latex]\frac{5}{12}+\frac{1}{3}[/latex]?
   

3. Consider your answers to parts (a) and (b). How do the two fractions compare? Explain why this makes sense.

4. Remove the eighth pieces. Now, line up a single [latex]\frac{1}{3}[/latex] piece with twelfths underneath. How many [latex]\frac{1}{12}[/latex] pieces does it take to equal a single [latex]\frac{1}{3}[/latex] piece?
   

5. Based on your answer to (d), what would you expect [latex]\frac{5}{12}+\frac{1}{3}[/latex] to be in terms of twelfth pieces?
   

6. Compare your answer from (e) to your answer from (a). Why does this make sense?

Addition of Fractions with the Same Denominator

As we do with other types of numbers, we can add or subtract fractions. When two fractions have a common denominator, addition and subtraction are straightforward. You are just counting how many of that type of piece you have. Let’s add [latex]\frac{3}{10}[/latex] and [latex]\frac{5}{10}[/latex] using fraction tiles.

Many elementary classrooms have a supply of fraction tiles/bars, but these are limited because they do not have many different denominators. Typically they only have denominators of 1, 2, 3, 4, 5, 6, 8, 10, and 12.

Another way to visualize addition of fractions is with our C-strips.

Example 2 shows us that [latex]\frac{7}{11}[/latex] [latex]+[/latex] [latex]\frac{3}{11}[/latex] = [latex]\frac{10}{11}[/latex].

The advantage to fraction tiles is that they are automatically made to have the same unit size. With C-strips, you have to determine which piece will represent a unit, but this makes the C-strips more flexible, meaning you can model a wider variety of problems with C-strips than you can with fraction tiles.

The same denominator indicates the same unit, so when we add two fractions with a common denominator, we just keep the denominator and add the numerators to get the new numerator. This is because we are just counting how many pieces of that size we have. In general, we can write the rule as below:

[latex]\displaystyle\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}[/latex].

Addition of Fractions with Different Denominators

When two fractions have different denominators, we cannot add them directly. Let’s consider [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex]. These pieces are different sizes. It is like trying to add 1 teaspoon and 1 tablespoon. We need to convert them to the same type of item before we can add them together. In other words, we need to use equivalent fractions! But, which denominator should we use?

You may already know which denominator to use, but let’s suppose you are a student that is unsure. We can use fraction bars to check which denominators will give us equivalent fractions for both [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex].

If we line up fourth pieces with our [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex] piece, we see that [latex]\frac{2}{4}[/latex] lines up perfectly with [latex]\frac{1}{2}[/latex], but no amount of fourth pieces will line up with the [latex]\frac{1}{3}[/latex] piece. Continuing through with various denominators, we see that sixth and twelfth pieces will line up with both [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex]. We can use either 6 or 12 as our common denominator, so let us use 6.

According to our fraction bars, [latex]\frac{1}{3}=\frac{2}{6}[/latex] and [latex]\frac{1}{2}=\frac{3}{6}[/latex]. Thus, [latex]\frac{1}{3}+\frac{1}{2}[/latex] is the same as [latex]\frac{2}{6}+\frac{3}{6}[/latex].

Once we have a common fraction type, we count how many pieces we have.

Answer: [latex]\displaystyle\frac{5}{6}[/latex]

Perhaps your school does not have access to fraction tiles/bars. We can also find a common denominator using C-strips. Again, it comes down to multiples.

Reconsider [latex]\frac{1}{3}+\frac{1}{2}[/latex]. Which [latex]C[/latex]-strip should we use to represent one unit? Notice that [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex] have two different units! If we are going to add fractions, they need to come from the same unit.

We need to represent [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex] using the same unit. This requires a unit that is divisible by both [latex]3[/latex] and [latex]2[/latex]. In other words, the unit should be one of the common multiples of [latex]3[/latex] and [latex]2[/latex]. For convenience, we usually choose the least common multiple ([latex]LCM[/latex]) of the two denominators, which is [latex]6[/latex]. Let the C-strip [latex]D[/latex] which has length six represent one unit. Then [latex]\frac{1}{3}[/latex] can be represented by [latex]R[/latex] (length 2) and [latex]\frac{1}{2}[/latex] (length 3) can be represented by [latex]L[/latex].

Therefore, the sum of [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex] would be represented by a C-strip with length [latex]5[/latex], that is, [latex]Y[/latex], which represents [latex]\frac{5}{6}[/latex] when [latex]D[/latex] is a unit.

In summary, what we’ve done with the C-strips can be written as

[latex]\displaystyle \frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}[/latex].

Our fraction tiles and C-strips are limited by the sizes we have. You can combine C-strips to get larger size unit pieces, but there is also another model we can use to visualize this process: the dot model.

In Example 3, we used the dot model to visualize the following math procedure:

[latex]\displaystyle \frac{2}{5}+\frac{3}{7}=\frac{14}{35}+\frac{15}{35}=\frac{29}{35}[/latex].

Subtraction of fractions is similar to addition in that it all comes down to needing a common unit.

Subtraction of Fractions with the Same Denominator

When dealing with fractions that have the same denominator, you can either use the missing-addend visualization or take-away visualization of subtraction.

Both of these models show that [latex]\frac{7}{10}-\frac{3}{10}=\frac{4}{10}[/latex].

Let’s consider a problem with our C-strips. As with addition, we need to identify which strip will represent a unit. Once we have all of our relevant pieces, we can use the missing addend model.

Example 5, shows that [latex]\frac{7}{11}-\frac{3}{11}=\frac{4}{11}[/latex]. What we see is that if fractions have the same denominator, they come from the same unit, so we can keep the denominator and just subtract the numerators. In general,

[latex]\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}[/latex].

Subtraction of Fractions with Different Denominators

At this point you may have guessed, if two fractions have different denominators we need to convert them to same unit/piece before we can subtract them. Our missing-addend model with fraction tiles is actually the only one that does not require we find a common denominator.

If we want to find [latex]\frac{11}{12}-\frac{1}{6}[/latex], we can line up the fraction tiles with [latex]\frac{1}{6}[/latex] underneath the eleven pieces of size [latex]\frac{1}{12}[/latex].

We need to know what can fill the gap. One tile that works are one-fourth pieces.

Thus, [latex]\frac{11}{12}-\frac{1}{6}=\frac{3}{4}[/latex].

Note that this is not the only tile we could have used. Both [latex]\frac{6}{8}[/latex] and [latex]\frac{9}{12}[/latex] would have worked since they are equivalent to [latex]\frac{3}{4}[/latex].

Fraction tiles are the easiest to use because they have a set unit size and we work within that unit size, but this makes them inflexible. If we want to solve [latex]\frac{2}{3}-\frac{2}{5}[/latex], no combination of fraction tiles will work. We can however use our C-strips.

Example 6 shows us [latex]\frac{2}{3}-\frac{2}{5}=\frac{10}{15}-\frac{6}{15}=\frac{4}{15}[/latex].

Let’s see an example of subtraction with our dot model. Here we will use the take-away model of subtraction.

In Example 7, we used the dot model to visualize: [latex]\frac{3}{7}-\frac{2}{5}=\frac{15}{35}-\frac{14}{35}=\frac{1}{35}[/latex].

Algorithm for Addition and Subtraction of Fractions – Common Denominator

This algorithm was covered in the Pre Unit Review Add and Subtract Fractions. This is one of the most forgotten algorithms, so now that you can visualize adding and subtracting fractions, let’s revisit this topic.

Mathematically, what was done above is to write the two fractions into their equivalent forms with a common denominator, then add or subtract the numerators.  This is because, in order to add or subtract fractions that do not have a common denominator, we must first rewrite each fraction as an equivalent fraction so that both fractions have a common denominator. As numbers in the denominator get larger, it is not feasible to draw out dots or use C-strips. Thus, it is important for children to understand the physical models deeply enough to move beyond them to an algorithm with numbers.

The easiest way to find a common denominator is to convert each fraction to an equivalent fraction using the other fraction’s denominator. Consider the fractions [latex]\frac{4}{9}[/latex] and [latex]\frac{5}{12}[/latex]. If we want a common denominator we can convert each fraction as follows:

[latex]\displaystyle \frac{4}{9}=\frac{4\times 12}{9\times 12}=\frac{48}{108}[/latex]

[latex]\displaystyle \frac{5}{12}=\frac{5\times 9}{12\times 9}=\frac{45}{108}[/latex]

Now we have to fractions of the same denominator and can add, subtract, compare, as desired. For example, [latex]\frac{4}{9}+\frac{5}{12}[/latex] is

[latex]\displaystyle \frac{4}{9}+\frac{5}{12}=\frac{48}{108}+\frac{45}{108}=\frac{93}{108}[/latex].

Unless the context makes sense otherwise, we do not like to leave our answer as an unsimplified fraction. Simplifying [latex]\frac{93}{108}[/latex] gives

[latex]\displaystyle \frac{93}{108}=\frac{31\times 3}{36\times 3}=\frac{31}{36}[/latex].

If finding the least common multiple of the denominators is too time consuming, you can always fall back on the “multiply by the other denominator” technique.

Estimation of Addition and Subtraction

It is important to practice number sense when it comes to fractions. If you enter the numbers incorrectly into your calculator and get a bad answer, you’ll be able to catch it. Not only is it a good way to check your answer, but it will make you more comfortable with fractions whenever they come up in life!

When we solved this problem earlier our answer was [latex]\frac{31}{36}[/latex]. This is close to our estimate of 1.

Applications of Addition and Subtraction of Fractions

Adding and subtracting fractions is part of everyday life. The key with word problems is to understand what is happening and use a model to mimic the action. Since we are dealing with something real, it is easier to imagine what is going on.