INTRODUCTION 

          One of the major difficulties young people often have in elementary mathematics lies in the area of fractions.  Even many adults look back on this part of their schooling with apprehension and concern. 

          This is in large part due to the lack of accurate, powerful, and easy to use models to illustrate the underlying concepts.  Lacking such unifying representations the learner will fall back on poorly understood and developed procedures.  Adding insult to injury these rule-based procedures are often misapplied with erratic results.  An example familiar to nearly every teacher will bear this out:

          In this case the student has confused two fraction “rules”, the addition rule and the multiplication rule.  As a result the procedure to multiply numerators and denominators together to get the answer, which is appropriate when multiplying, is misapplied to addition.  Lacking a model or referent this result is accepted without question. 

I remember one case where I asked the student if their answer (which had been derived using this particular “method”) made sense to them.  The response was immediate, and discouraging. 

“No, Dr. Connell.  But it makes as much sense as anything else I have done in math today.”

This situation clearly cannot be allowed to continue. 

To combat this situation we will be developing a powerful and flexible set of representations that may be used to model fractions and to generate the operations of addition, subtraction, multiplication, and division.  These models will require practice on the part of the teacher and I encourage you to work with them yourself before attempting to use them in your instruction.  Most of us were taught using the procedures, or rules, and will tend to revert back to their use when we are under pressure.  We want to become comfortable enough with these models that we never feel the pressure to revert back to the manner in which we were taught.

SHARING 

Sharing provides a powerful set of experiences to use in building meaning for the denominator of a fraction[1].  Nearly every child has had the opportunity to share with another – whether it is toys, time, money, or any of a broad variety of materials.  This provides an important background within which we can begin to develop fractions understandings. 

For example, most children can tell you that it would not be fair to share twelve pennies with three people like this:

Person 1 à 1 Penny

Person 2 à 1 Penny

Person 3 à 10 Pennies

To share, at least to share fairly, would require that each person in the sharing operation receive the same amount.  We will need to emphasize to the children that in our sharing activities in mathematics that we will always want to share fairly. 

The fair sharing version of our earlier example becomes:

Person 1 à 4 Penny

Person 2 à 4 Penny

Person 3 à 4 Penny

We will build on this notion of Fair Sharing as we develop our fraction models.  In order for this to take place we will begin by introducing some vocabulary, in this case the use of the words Share With for the fraction bar.

Thus  is read as “Share With” two,  would be read as “Share with” five, and so on.

As you can see there are many different types of objects that this action of “Share With” can be applied to.  I recommend that students be given the opportunity to “Share” both discrete objects (such as counters, pennies, buttons, pennies, etc.) and continuous objects (clock faces, meter sticks, graph paper chunks, etc.). 

For example, our earlier penny example will often lead to the children’s adoption of a “dealing out” strategy where they will physically “deal out” the twelve pennies into three groups until each group has the same amount. 

Such experiences and understandings are essential for a full understanding of mathematical sharing (and hence of the denominator and of division itself) to become established.

For the purpose of our eventual fraction development the object which we eventually want our children to be performing the “Share With” action upon is that of a unit square.  We will refer to this as a “Cake” due to the familiarity of the children with cakes, and several important actions (such as cutting!), which may be performed upon cakes.  As we will later see this cake model will have many advantages over the more traditional “Pie” models that are often used.

When we couple our earlier “Share With” understandings (i.e., to share an object is to divide it into a specified number of equal partitions) with the unit square based “Cake” we have both the appropriate action and object upon which to build fractions.

Let us see what happens when we perform the action of sharing with three upon the cake.  We are, of course, asking the child to divide the cake[2] sketch into three equal parts. 

We will begin by representing the problem:

Perform the following sharing on the cake:

This activity will need to be repeated until the students are able to quickly and accurately “Share” any instruction with a cake quickly and accurately. 

There are a few understandings that must take place at this time.  First among these is that each cake – whether drawn on the board or sketched on the students’ paper – represents the same cake and is exactly the same size and shape.  The students will need to play a bit of “Let’s pretend” at this point.  As an aside, I have found that the more time the teacher spends on trying to make everything look perfect the less the students understand the underlying concepts we are trying to develop. 

The second understanding is that although each share may not look perfect they really are all the same size.  We just can’t draw perfection!  These two “let’s pretend” items ensure that the students are performing equal partitioning upon equal items, essential if our later explanations are to be effective in the long term for them. 

These requirements do not appear to be overtaxing to the students and quickly become a part of the background knowledge brought to bear as they work with these actions and objects.  When we consider the intricacies of most fantasy worlds the children are familiar with – Pokémon, for example – you can easily see that this is easily accomplished.

SHARING and TAKING 

The activities outlined in the preceding section were designed to develop a robust understanding of the “Share With”, or Denominator, portion of the fraction symbols.  The step from this beginning to a full blown understanding of the fraction symbol as a series of well understood actions upon objects with known properties – the cake – is very small. 

All that is needed is to develop the conceptual background for the “Take” portion, or Numerator, of the fraction symbol.  This can be done now quite easily.  When we see the symbol ½, for example, we read it as “Share with two and take one share”. 

Thus instead of an abstracted symbol a fraction becomes the instruction for performing well-understood actions upon our cake (unit square) object.  In actual use the symbol ½should tell us to draw a cake, share it with two and then “Take” one of these two shares.  This can be sketched out as follows:

We record the action of “Taking” the shares by filling in the corresponding shares.  As we will show in the next section the suggested method to use when representing multiple fractions is to always show the first one as being “Shared” horizontally with the shares “Taken” from top to bottom.  The second fraction will be “Shared” vertically with shares “Taken” from left to right[3]. 

Although most of our work will involve a maximum of two fractions, subsequent fractions would be represented in an alternating fashion – horizontal with shares taken from top to bottom, and vertical with shares taken from left to right.

At this point the students will now have all of the understandings necessary to represent any fraction using this model.  They will also have enough background to identify the appropriate fraction symbol (ex., ½ ) when presented with it’s sketch.  These two abilities alone will put them in a stronger conceptual position then where we find most of our students today.

SHARING and TAKING

Practice Activity

INSTRUCTIONS

Instructions:      Flip nine pennies and record the number of heads and tails.  Use the larger number as your “Share With” and the smaller number as your “Take” number.  Then perform the operation on a cake and record your work on a recording sheet.

Example:           I have four heads and five tails.   My share and take instruction is to share with five and take four shares.  It looks like this à

NAME   _______________

DATE    _______________

SHARING and TAKING

Practice Activity 

Record of Action Sheet

Now that we can represent any fraction it is time to begin developing some further actions – in this case the standard operations – which we may perform upon this more elaborated sketch object. 

In the development of any number type I always use a series of questions, the response to which leads the students to explore and develop the meanings behind comparisons, subtraction, and addition. 

These questions are as follows:

Ø      What does it look like?

o       This question requires the student to represent the number being developed.

Ø      Which is larger?

o       This question is asked after the student has represented two numbers and requires the student to compare the numbers represented.  For the case of fractions this will require the development of some important understandings and extensions of the cake model.

Ø      By how much?

o       This question refers to the Which is larger? question and requires the student to exactly quantify the size of the differences.

Ø      How many do we have in all?

o       This question is asked after the student has represented and compared the two numbers.  It requires the student to add the numbers represented. 

The manner of addressing the second of these questions for the case of fractions will be the topic of this section.  As mentioned, the first step in performing a comparison is to identify and represent the two fractions to be compared. 

Let us imagine that we have performed the Sharing and Taking activity.  We can use this recording sheet as the source for our pairs of fractions for comparison.  In the course of filling in this sheet the students have already performed most of the representations necessary to address each of the questions described above. 

We will begin by altering the recording sheet slightly to better contrast the pairs of fractions that we wish to compare and to make recording of newly needed information easier.  The modified recording form would look like this:

 For illustrative purposes let us imagine that we are going to compare:

We will begin by representing these two fractions on the recording form using the suggestions outlined in the Sharing and Taking Section.  Following this initial representation our form will look like this:

To make these sketches in a form that we might compare the fractions represented it is necessary to use an old “Baker’s” trick.  The nice thing about cakes is that they can be cut in two directions, horizontally and vertically.  All we need to do to make pieces of the same size in our example is to perform the cut on each cake as has been performed on the other.  This actually sounds more difficult than it is. 

Let’s see what we would actually do in our example.  We would cut the first cake (which has been cut in half horizontally) into fourths vertically.  We would then cut the second cake (which has been cut into fourths vertically) in half horizontally.  When we have finished with this our recording form will look like this:

We have just performed via actions on our cake model the equivalent of finding a common denominator.  In common sense terms, all we need to do next is to count the number of equal sized pieces (reflecting the new numerators) and decide which instruction gives us the most cake.  The final sketch showing these steps would be:

This example shows how easy it is, in a very common sense and step-by-step fashion, to develop comparison of fractions – assuming, of course, that we really know and can represent what it is that we are comparing!

It should also serve to show that it is possible to perform sophisticated operations with fractions without the prior necessity of developing a formal vocabulary.  This is an instance were the vocabulary is best developed via the actions themselves.  The terms denominator and numerator now have a rich context within which to have meaning thanks to the students work with sharing and taking.

SUBTRACTION

 At this point we have successfully represented and compared our two fractions.  Our next question to address is that of  “By how much?”  in reference to the difference in sizes.  Since we have already found out which fraction is larger we do not have to worry about ending up in negative number terrain and can address the differences in size. 

This question is now easily addressed; let’s look at the ending sketch from our comparison activity:

 We can see that the ¾ is larger then ½.  We can further see that the size of this difference is 2 pieces of the newly double cut cake.  Now all we need to do is to decide how to record this.  We will start by noting that each cake is now shared with a total 8 shares (following our double cut).  So, using our convention of using the fraction symbol to record “Share with” and “Take” we can show this difference as follows:

 Of course if we happen to notice that share with eight and take two is the same as share with four and take one so much the better!

ADDITION

 As you can no doubt tell by now, the question of  “How many do we have in all?” is easily answered at this point. 

Once again we will begin by looking at the ending sketch from our comparison activity:

 We can see that these two instructions give us a total of 10 pieces of the newly double cut cake.  We might even notice that we have 1 full cake and 2 pieces left over with which to begin filling another cake. 

Now, as before, all we need to do is to decide how to record this.  We already know that each cake is shared with a total 8 shares (following our double cut).  So, using our convention of using the fraction symbol to record “Share with” and “Take” we can show this addition as follows:

MULTIPLICATION

 If multiplication of whole numbers has been taught using an area model[4] then multiplication of fractions will be a piece of cake!  (Joke intended).  If not, we will need to revisit the area model for multiplication. 

Assuming the worst, we will need to rebuild what multiplication looks like from an area perspective.  When we represent a multiplication problem using the area model there are a series of related fact families that are simultaneously presented to the student. 

Let’s look at the example of 2 x 3 = 6. The sketch of this would be a rectangle 2 units in one direction and 3 in the other giving rise to an area of 6 square units.  This would look like the following: 

 As an aside this same sketch can be used to illustrate each of the following related facts:

2 x 3 = 6

3 x 2 = 6

6/3 = 2

6/2 = 3

 This clearly shows how this single sketch can be used to represent the relationships between and among four related facts (fact families).   I hope that it might encourage you to use this model.  The links between division, fractions, and multiplication that can be encouraged via this model is, in itself, worthwhile to develop and encourage.

With this as background let’s take another look at multiplication of fractions.  The sketch of 2 x 3 showed a rectangle  2 units in one direction and 3 in the other.  In a like fashion the sketch of ½  x ¾  should show us a rectangle ½ in one direction and ¾ in the other.

This is easily understood and quite easily drawn given the background we have developed thus far.  We will use a single cake as our base and then look at the pieces that are described.  Looking at the ½ first we see:

We will now draw the ¾ portion resulting in:

 As we can see from the picture there are 3 small pieces which are found in the rectangle that is  ½ in one direction and ¾ in the other.  We can also see that there are now a total of 8 pieces making up the full cake.  So the total area we are describing in this multiplication problem is

To make the link to the standard multiplication of fractions procedure crystal clear we can look back to the picture.  The number of small pieces we have, i.e. the numerator of our answer can be described by the rectangle that is 1 x 3.  The total size of the cake, i.e. the denominator of our answer, is described by the rectangle that is 2 x 4.  This is identical to the procedure of multiplying numerator x numerator and denominator x denominator. 

DIVISION 

For those of use who learned to divide fractions by following the old poem, “Ours is not to question why, just invert then multiply” it may come as a bit of a shock to see just how easy division is.  Assuming, of course, that you can represent what it is that you are dividing!

Let’s go back to our initial comparison example:

 When we are asking the question “What is ½ divided by ¾?” what we are really asking is “How many times will the ¾ fit into the ½?”

Let’s take a common sense approach as we attempt to answer this.  The ¾ cake gives us a total of 6 small pieces, the ½ gives us 4 small pieces.  I can see by looking at the picture that all 6 of those pieces will not fit.  In fact, only 4 of the 6 will fit.  In “Share With” and “Take” terms that is:

It is just that easy!

For those unbelievers, let’s compare this result with our standard procedure of invert and multiply.

Just to further illustrate this method, we’ll ask the other division question which might be of interest.

The remaining question that might be asked is“What is ¾ divided by½?”   As you should be able to tell by this time, what we are really asking is “How many times will the ½ fit into the ¾?”

Remembering what we know, we see that  ¾ cake gives us a total of 6 small pieces, and ½ gives us 4 small pieces.  I can see by looking at the picture that all 4 of those pieces will fit into the 6.  In fact, not only will all 4 of them fit 1 full time, but another 2 of the 4 pieces will fit into the 6 as well!  In “Share With” and “Take” terms that is:

And just to be completely compulsive, let’s check this result against the standard procedure:

I would suggest that for the case of division of fractions this method not only makes sense, but also is actually easier to both teach and to understand.

CONCLUDING THOUGHTS

 We must ensure that our students have a firm grasp of all of the essential concepts underlying fractions. 

These should include: 

Ø      How do we represent a fraction?

Ø      What does this representation mean?

Ø      How can we compare fractions in a meaningful fashion?

Ø      What does this comparison show us?

Ø      How can we model and understand the basic operations that may be performed upon fractions?

o       Addition

o       Subtraction

o       Multiplication

o       Division

Ø      How do we link between our sketches and the procedures that we derive from these sketches?

 I hope that these materials will show you that these are not impossible tasks.  In fact, once we are clear how to represent what is meant by a fraction everything else falls into place.  This is as it should be, and serves to illustrate the importance of understanding the concepts.  Once a concept is truly understood the procedures will fall into place quite easily.

In closing, however, let me stress the importance to you of practicing with these models.  I will never forget a teacher-candidate in one of my early mathematics methods courses at my first professorship.  We had struggled together long and hard to develop some of the understandings for her that are reflected in these pages.

Finally she was able to use the models, but I noticed that she always checked her work using the procedures.  I joked with her about this, as she had been the first to admit that these rules were meaningless to her.  I will never forget her face as she said,” Dr. Connell, I would like to believe these models but my rules just won’t let me!”

We need to practice with these new methods until not only will our rules let us believe them, but that they will become our default manner of representing and solving fractional problems.  As you explore these models further you will no doubt find that the cross-multiply procedures are just waiting for you to develop and use – once you understand the model itself. 

Michael L. Connell, Ph.D.

344 Farrish Hall

University of Houston

Houston, TX  77204-5872

713-743-8677

MKahnl@aol.com