Concrete way to teach addition and subtraction of fractions

Another way to represent addition and subtraction of two fractions is by subdividing rectangles both horizontally and vertically.

Say you want to show how to add $frac{1}{3}$ and $frac{2}{5}$. Draw three equal sized rectangles. Subdivide the first in threes with horizontal lines. Subdivide the second rectangle in fifths with vertical lines. The students probably understand these representations of the fractions already.

The third rectangle is used to represent the sum. Subdivide that rectangle both horizontally and vertically to get fifteens. Now you must ensure that the students understand that this is actually fifteenths, and that they can see how many fifteenths are covered by $frac{1}{3}$ and $frac{2}{5}$. The first two rectangles should make this obvious to most students.

This representation of the sum of fractions naturally shows how adding thirds and fifths leads to fifteenths. That is, it clearly shows how a common denominator solves the problem. It also shows why the common denominator is found by multiplication.

The representation is not perfect, though. It does not naturally explain the least common denominator, it does not extend to three or more summands without adding dimensions, and it is awkward with improper fractions.

Suggestion: Do not bother having students do calculations with two fractions until they can do a comparable operation with a fraction and a whole number.

Example: Before teaching, say, $frac{3}{2}-frac{4}{3}=__$, ensure they can answer something like $4 - frac{1}{5}=__$. Make sure all students can draw that on a number line and with area diagrams. This is how you ensure that their knowledge of fractions will integrate with their knowledge of whole numbers.

Once students are ready to add and subtract fractions that already have common denominators, ensure they can do so on a number line and with area diagrams and with fraction tiles.

Then, when it comes time to do $frac{3}{2}+frac{4}{3}=__$, you can:

1. Put the halves tiles and the thirds tiles together
2. Pretend to be horribly confused by how much that is
3. Estimate it to be a little under 3 wholes
4. Demonstrate how sixths tiles come to the rescue. :)

I like the cutting paper approach the best because it is cheap and easy, and it forces students to confront basic fractions by folding/cutting them out. But here’s another thought...

For you to demonstrate: have several clear-plastic, cylindrical cups, each marked off in different fractions of a whole (maybe start with 1/2’s, 1/3’s, 1/4’s, 1/6’s, 1/8’s, 1/10’s and 1/12’s). Have another larger cylinder marked off in whole numbers of the smaller cups.

Write the problem $frac{3}{2} + frac{4}{3}$ and have students make a guess/estimate, and then use the 1/2 and 1/3 cups to fill up with colored liquid or rice (which is why you will demonstrate this). Add the liquid to the large cylinder to see that “yeah, it’s between 2 and 3”, as the students should have guessed. Then work backward, pouring away whole numbers worth of smaller cups until you have less than a whole cup left In the large cylinder. Now try the smaller cups one-by-one until you find one in which the remaining liquid fills up to a marker line. Finally, declare the answer as the mixed number $2 frac{5}{6}$.

I might do this with two nice examples (where I have the appropriately-marked cups) and then try the first one out again, but pour each of the original fractions into the 1/6’s cup before adding them together in the large cylinder. Lots of things you could do with these.

Use a piece of paper as your whole. To teach $3/2 + 4/3$ do the following.

1. Give each child/group of children 6 pieces of paper.
2. One piece of paper should be left as a whole - the students can write 1 whole on the paper.
3. Have the students fold 2 of the pieces in half lengthwise, label each half as $1/2$, and cut out the halves.
4. Have the students fold 2 pieces of papers in thirds widthwise, label each third as $1/3$ and cut out the thirds.
5. Have the students take $3/2$ and $4/3$ and try to add them. You will have 2 wholes and $1/2$ and $1/3$ Discuss what to do with the leftover pieces...
6. Next take the last piece of paper and fold in sixths by folding in half lengthwise and in thirds widthwise. Label each piece $1/6$ and cut out the pieces.
7. Put the some of the sixths from part 6 on the leftover pieces in part 5 to show that $1/2=3/6$ and $1/3 = 2/6$. Together they make $5/6$

Hope this works for you.

Use a language approach.

This is not meant to be a snide remark; I came across a particular TED talk a while ago that illustrated how math is just another language and we must first learn the proper syntax.

Of course, we know that adding fractions require the use of common denominators, so I would first introduce the topic of equivalent fractions. If the concept of common denominators is new to the student, perhaps using something concrete like denominations of coins can help bridge that gap.