October 23, 2024

Lyle Lee Jenkins

How many of you remember hating fractions as a child, or even now as an adult?

Now we know there are some of you who may say they are simple, what is the issue? Adapting these methods when it comes to working with fractions will have your students learning more about fractions than you ever thought possible and having fun doing it!

The reason is, there is no “bridge” between using manipulatives to understand the concept of a fraction, and addition, subtraction, division and multiplication of fractions. What we are going to do today is teach the “bridge”. It is so important so we don’t go from what a fraction is all about to immediately adding, subtracting, multiplying and dividing without this bridge. The students will use the manipulatives to compute the fractions without needing to understand common denominators right away.

We have Fraction Slices^{TM }and we have the book that pairs with them, *How to Create Math Experts with Fraction Slices ^{TM}* both items are available in the store on our website as well as Amazon. For the book, in line with the rest of the educational books we publish, there is a QR code in the back that allows educators to print worksheets from the book on demand for their classroom. The slices are made of thick translucent plastic acrylic and each fractional measurement has its own assigned color. The slices have to be somewhat transparent so that students can easily perform operations that require slices to be placed on top of each other.

The first thing the students do is identify the colors. What color is the one whole slice, the half slices, the quarter slices, on and on. Knowing the colors and their appropriate measurement helps students identify them as they are working through problems later on.

Once students have become familiar with what each Fraction Slice^{TM} is they will move on to experimenting with greater than, less than and equal equations. Through manipulation students will learn how many thirds equal one sixth and how many fourths equal one half. Students need this step, they don’t need a formula, they need to just look at the slices and experiment with them. Students who struggle and are the most nervous when it comes to learning fractions, will start off successfully with these tools.

Then, we can move onto addition. This is CRAZY, the first addition problem the students do is with uncommon denominators, NOT common denominators.

The question is ⅓ + ⅙, what do we say to the kids? Adding means putting things together, just like you have always done. Since kindergarten they have known that addition means putting things together, just because they are in a higher grade, the concept doesn’t change.

- They will pick up the ⅓ piece and the ⅙ piece and put them together.
- Tell students to cover up the two fraction slices with as many pieces that they want or need as long as they only use one color to cover them. Since they can only use one color to cover the slices we are removing the common denominator issue from the equation for now.
- Almost always, the kids will reach over and pick up the green ½ piece and notice that it covers the slices perfectly and write ½ on their paper.
- That’s how easy it is all the way through!

One of the funny ones to watch a student work on is ⅓ + ¼, they try all sorts of things and nothing works until they finally try the one-twelfth pieces and realize that if you put 7 of the one-twelfth pieces on the ⅓ and ¼ pieces it covers them perfectly.

As we move into subtraction, it is important to note that there are two concepts of subtraction. The first type of subtraction is “take away”. You have this and you take away that, how much is left? The other concept in subtraction is comparison, how much bigger is one than the other? If you ask how much taller one person is than another person, we are not cutting off the difference, we compare it. So when we do subtraction with the Fraction Slices^{TM} we are *comparing*.

The first question is ¾ - ½:

- Students grab three of the ¼ pieces and one ½ piece.
- Then they will put the ¼ pieces together on the table and put the ½ piece on top.
- “How much bigger is ¾ than ½?”
- Students will find what piece or pieces fill that space left over using only one color (remember it can only be ONE color).
- They will discover that the ⅙ piece fills in the open space next to the ½ piece and they write ⅙ as the answer on their paper.

Usually the Fraction Slices^{TM} works well with a group of about four kids sharing one set and they are able to work through the pages feeling confident in their fraction subtraction accomplishments.

Division? Why division? Division is actually the easiest, it's easier than addition and subtraction. I will never forget, many years ago, a school district had its own exam. There were two fraction division questions on the exam for fifth graders. One question was ⅕ divided by ⅕ and the other was ½ divided by ¼. In this group of fifth graders, 90% of them missed both of these questions. They couldn’t remember what was supposed to be turned upside down or what they were supposed to do. It is such an easy question, one fifth goes into one fifth one time, one fourth goes into one half twice. That is how simple it is.

When you look at the first question in our book, ⅓ ÷ ⅙, one sixth fits into one third two times, simple as that. Here’s how we got there:

- Grab one ⅓ piece and a couple ⅙ pieces.
- Place the ⅓ piece on the table and simply see how many of the ⅙ pieces it takes to perfectly cover the ⅓ piece.

There is a concept later in the book where it gets more difficult, instead of how many times does a ¼ go into one half, you turn it around and say how many ½ pieces go into one ¼ piece? That in turn changes the question to how MUCH of the ½ piece fits into the ¼ piece? Students need this initial division of fraction success so they can carry that confidence and understanding onto the more challenging concepts. With early success students will be eager to learn how to manipulate more complex fractions with a pencil and paper, but the foundation and understanding needs to be there first.

Multiplication with whole numbers makes rectangles and squares, if you have a zig zag it's not a multiplication problem. If you take the problem 3 × 2 you make a rectangle that is three across and two down or vice versa. Once the rectangle is made you will see there are six pieces that make up the rectangle. The same concept applies to multiplying fractions.

Let’s work through ⅔ × ½:

- First grab two ½ pieces and lay them together on the table.
- Then grab three of the ⅓ pieces and lay them on top of the ½ pieces in the opposite direction creating a cross pattern.
- At this point you should have a perfect square of the two colored pieces stacked on top of each other.
- If you look at the small rectangles that were created from crossing the pieces on top of each other you will see that it has made six rectangles each equaling one sixth of the whole square.
- Now students will know that anytime you are multiplying ½ and ⅓ that you are making sixths, and they now have their denominator for their answer.
- Now how do we get the numerator? Our problem was ⅔ so remove the extra ⅓ pieces so that only two remain in the square.
- Do the same thing for the ½ piece, so there is only one ½ piece remaining.
- Now how many of those little rectangles representing ⅙ remain overlapped?
- There are two rectangles remaining, so students will write 2 in the numerator place.
- This now provides them with the answer ⅔ × ½ = 2/6.

Using this method, multiplying fractions becomes so easy and students can confidently get the right answer. We never want students, years after learning about fraction multiplication to ask, “Huh, how come when you multiply fractions you get a smaller number?” This should be understood from the beginning, that is what we are doing, we are showing what it looks like and what is actually happening when you multiply fractions.

Like most other books, *How to Create Math Experts with Fraction Slices ^{TM} *ends with putting it all together. There is a table with two fractions and students will perform all operations with those two provided fractions to fill in the table. Then there is a bonus exercise where portions of a table are filled in and students have to solve for the answer, or maybe even the fractions themselves!

Fractions do not need to be a forbidden word or a source of anxiety for educators or students. Secondary math teachers dream that one day they will receive students with a deep understanding of fractions. We can do this! There are more ways to practice fractions, similar to what we have done with Fraction Slices^{TM}, none of them are as complete for that bridge, but we do fractions with Tangrams, Geo-Boards, Pattern Blocks and Geo-Blocks. All of these tools together build an extremely strong understanding of fractions. Implementing these tools will not only boost student understanding but they will have a lot of fun along the way!