Reflections On Dividing Fractions

 I find it difficult to teach students to divide fractions using diagrams.  I realized that if I had difficulty picturing the process, it means I don't fully understand the concept.  I can do the process swiftly and come up with the answer.  That is because I learned the process only.

When I was growing up, they didn't worry about students being able to visualize concepts.  They were concerned with the process.  They wanted to know that you could follow the algorithm correctly and they used lots of short cuts and sayings to help you remember the steps

Over the years, there has been a move to providing visualization for the concept as it is being taught.  After repeatedly looking at pictures on line, I think I finally figured out how to read the pictures showing dividing fractions. I also finally connected the idea of part to whole when dealing with dividing fractions.  

For instance, I have 3/4th of a pizza I want to divide into 2 pieces.  So the whole instead of four pieces making up the whole, I have 8 pieces of pizza if I had a whole pizza but only 6 pieces are relevant.  Now we want 3 parts of the 6 so we each part of the pizza is 3/8th.  On the other hand, if I have the same 3/4th of a pizza and I divide it into quarters, I am asking myself, how many fourths are there, so Im looking at the parts which means there are 3 one quarter sections.

So if I have 3/4 divided by 1/8, it means I start with 3/4th of a pizza and I want 8 pieces instead of four which means each piece is divided in half and my pizza now has 6 total pieces which is what I needed to find.  On the other hand if I have two pizzas and I divide them into quarters, I end up with 8 pieces because each pizza is divided into four and four times two is eight.

As far as improper fractions go, I figured out that one needs to convert the improper fraction into a mixed number.  The student would then draw the mixed number using circles or rectangles with the fractional part in it's own circle or square.  After it's all set up, then apply the division.  It's important to see it all in context and as a whole.  This makes it so much easier to "see" what is happening.

Honestly every time I've tried to figure it out with pictures, I'd stumble around until it made sense and then promptly forgot how it worked.  I think that is because I never took time to verbalize what was happening.  The verbalization allowed me to focus on the concept while moving it from short term to long term memory.  

The next thing I want to figure out is how to apply visualization to algebraic fractions.  I'm not sure how to draw 1/(x-1) or 1/(x-1)/1/(x+2).  My next step is to see if I can figure out how to express algebraic fractions while showing addition, subtraction, multiplication, and division of those same algebraic fractions.

Let me know what you think, I'd love to hear.  Have a great day.

Dividing Fractions in Real Life

While writing yesterday's entry, I realized I could not name any believable situations when a person would need to divide fractions.

I started with Dr Math and his examples included one my students would roll their eyes on.  It was about having 3/4 of a pizza left and wanting to know if four people could share it so each person got 2/5th of a pizza.

There is always someone in the class who would say, yes they could all share but not necessarily what they want since its in fourths, not fifths.  Another might say, who cares, they'll eat what they eat.

Another example which is more realistic states you have so much land.  You want to subdivide the land into smaller plots but the county requires that the septic system needs x amount of land.  How many plots can you sell.  This is better but its hard for my students because the houses either use a suction system or a honey bucket ( a bucket for collecting human waste)  They have no idea what a septic system is.

Another site provided some great real life examples such as you buy 7.5 yards of material to make up several pot holders.  Each pot holder requires 3/4 foot, how many pot holders can you make.  Or so many feet of rope that must be divided into lengths of 1/2 foot, how many strands will you have. Or the same using ribbon.  These are actually one's I've done myself when making things for the house.

Dividing fractions occurs in cooking when you want reduce a recipe.  I have tons of recipes which make enough for 6 and there is only me.  This means I either eat or freeze the rest of the meal or I cut the recipe so it makes enough for 2 or 3 people.  My students have such large families, they don't usually try to reduce a recipe.  Cooking around here is just throw stuff in till you have the right amount of food. 

The reality is we can come up with real life situations for dividing fractions but most of those situations are not ones people will do such as figuring out exactly how much detergent is used in each load if you have 50 oz of detergent.  I just scoop and dump.  I don't sit there and calculate the number of ounces used each time.  Besides, most instructions require you to use more than is actually needed.

A few years ago, I taught a remedial math class where I had students draw pictures visualizing dividing fractions.  I am the first to admit that it was hard because I'd never done it.  Fortunately, I had a book which showed a few problems so I could figure it out but it was not easy.

I'd love to hear some real examples which are used by people so I can share them with my students.
Thanks for reading, have a good day.