Thursday, 3 January 2013

Frugal Fractions

Ice cube trays, floor tiles, egg cartons, Lego bricks, wall tiles, chocolate bars, pizzas, notes in an octave, an analogue clock face, flower petals, graph paper, oranges, a flat of canning jars, a football field; what do all of these things have in common?

Perhaps my title gave it away, but each of these things (and many more) are common examples of fractions in everyday life. Each of these involves a whole divided into parts--a whole dozen eggs, a whole tray of ice cubes, a whole orange, etc.

If you have already visited my math page, you may sensed where I am going with this post.

It is my belief that the purchase of expensive, specialized math manipulatives is unnecessary and does little to accomplish one of the goals some profess to have by using them, that of relating mathematical concepts with the "real world".

There are exceptions, of course. I have loved using pattern blocks with my students and kids for many years, and would not hesitate to recommend them. Their uses are many, both in terms of math and art. And were I to be in a hurry, I might consider purchasing or borrowing a class set of base-ten manipulatives, although the students would miss out on some learning involved in measuring and cutting their own sets.

But for fractions, I believe we can do much better using everyday and homemade items.

Chances are, if you were to read the above list, you would see you have at least one of the concrete examples available to you in your home right now. What makes a good example of an everyday fraction set? These are my criteria:

1. It must have equal parts that form a whole that can be somehow identified (floor tiles of a whole room, or ones in the area marked off with masking tape, for example).

2. It must be present, measurable and real in order to be considered concrete (time is not so concrete, but a calendar or analogue clock face can be).

Some examples will be easier to use than others. If you choose to use an egg carton, hard-boiled eggs, marbles or other place holders might be well advised, or you can cut up the carton into egg sections and use a second one to hold the cut up pieces.

Using food containers alone you could probably find examples of halves, thirds, quarters, fifths, sixths, eighths (hotdog buns), tenths, twelfths (hotdogs), 20ths, 24ths, etc. Some quantities seem to be more common than others, which is another cause for classroom discussion.

There will likely become a time not far into the discussion of fractions where some smaller and more manageable examples will become more practical.

I often have my students use construction paper or coloured copy paper (which has the bonus of being recyclable), to create their own fraction sets.

Along with providing an inexpensive solution, the actual making of the manipulatives helps to reinforce the concepts. The language alone contributes to making mathematical connections as you must divide the paper into equal sections. Other related skills include measurement, reviewing the concept of equivalency, fine motor skills of marking and cutting, division, area, perimeter, and can spin off in other directions as well--even into Pythagoras' famous theorem if you start working on right-angled triangles as your base shape.

So, how do you start?

First, you need to start with a base shape. For this example, I'll use a rectangle.

Next, you need to decide on the shape's dimensions. I'll start with a 10 cm by 24 cm shape, mainly because these dimensions are convenient for easy division.

To reduce the amount of work involved, have students cut four of these in one go. Give each student four sheets of different coloured paper (using a sheet of 1 cm graph paper as one of these sheets can help provide a stronger visual representation). Show them how to use paper clips to hold the pieces together in place as they cut. Once they have measured and cut their pieces, they should have four rectangles of the same size but of different colours.

Now they choose one colour (usually centimeter graph paper, but you can vary this if desired) to keep whole as a base, and the other three to divide into equal parts. Remind students to stop after dividing each colour to label the fraction on each piece; for example, if you divided the yellow sheet into halves, label each half as 1/2 before moving to the next sheet in order to avoid confusion.* Using a paper clip to hold all of the same sized pieces together also helps keep things organized. Provide a large envelope for each student or pair to store their set.

If you used a 10 cm by 24 cm guide, you might suggest they divide one into halves, another into thirds and another into quarters. Or you can use higher denominators as you wish. If all the students are working with the same dimensions, you could have them work with a partner and provide each pair with 8 different colours so that they can work with additional fractions, such as fifths (a bit challenging to figure out, but definitely possible!), sixths, sevenths (very challenging as these dimensions do not divide evenly by 7) eighths, etc.

For halves and quarters, you can have students fold the paper to divide it, then cut or use the "lick and tear" method to separate sections. You can then challenge them to find ways to fold the paper equally into thirds, fifths and other fractions not divisible by 2, or simply ask them to measure and cut, depending on your time and intended focus.

Students can now use their labeled fraction kits to compare different fractions, add and subtract fractions, etc. Which is larger, 3/5 or 2/3? You will know they understand the concept when they can explain it both visually and by making the operational connection to division, ie. 3/4 = three divided by four = 0.75 = 75%. Showing the relationship between halves, quarters and eighths and/or thirds and sixths is a good way to start a discussion on finding the lowest common denominator, or as I like to introduce it, the simplest way to show that amount (but it is important that they also learn the proper terminology!). In fact, it is a good idea to review the terminology often. Remembering that multiplication can also mean "groups of" or just "of" can help make the concepts more accessible. Likewise, a reminder that the "denominator" is "down" and shows how many parts you need to make a whole, and that the "numerator" is the "number of parts of the denominator that are actually there" can go a long way for many students. Others will find it easier to directly treat all fractions as division questions, so providing parallel terminology will help those students make that connection. For example, in 1/2, the 2 is the divisor and the 1 is the dividend; the quotient would be 0.5, and the remainder would be 0.

You can repeat this exercise with circles (have students draw a circle using a compass), a square, or even a right-angled triangle. When you use a circle, some of the students may wish to start exploring circular geometry, which can be a good way to double-check answers.

A related discussion to explore might be how to represent a fraction with the denominator of 0. What might dividing by 0 really mean, and how mathematicians work with / around such problems.

For classroom use, it may be useful to make a teacher copy of each type of set from coloured transparencies.

For homeschooling, try ordering a pizza and ask for it to remain uncut (or make your own pizza). Use this to reinforce size comparisons in fractions, ie., assuming you like pizza, would you rather have 2/5 or 1/3 of one?

Or try making some lasagne or magic bars (shown in the photo at the top).
Beware though, you may be in for some very detailed measuring!

*You can also show students how 1/2 is the same as "1 of 2"; etc. to show the relationship in terms of language already encountered in arithmetic, since "of" is a multiplication word. This helps show that fractions are actually little division problems too, and also helps reinforce the connection between multiplication and division.