Illustrating Division of Fractions

  Things have changed since I first started teaching.  One of the big changes is the desire to have students understand the concepts behind the math.  When I was in school, you didn't learn about concepts, you learned to follow a prescribed process. 

You didn't vary from that process.  It was accepted you didn't really need to know what you were doing, you just needed to know enough to follow the steps to obtain the correct answer.

Over the years, I've discovered my students often did not understand or know the steps needed to solve a problem.

For instance, why do we change a division problem with fractions into a multiplication with its reciprocal?  After quite a lot of looking I finally found an explanation which makes sense.  When you divide one fraction by another such as 3/4 / 2/3  you multiply the top and bottom by 3/2 to get one in the denominator.

In all the years I've taught, this is the first time I've seen an explanation of why you multiply by the reciprocal.  When I got my teaching credentials, we used the "Flip the right one, not the wrong one."  Even at the local community college, we were encouraged to use that but never this other explanation.  I wish I'd known it then.
 
The explanation is so clear and easy to see but its not as clear when you try to draw a picture to show the same thing.  I honestly tried to figure it out myself.

 I drew the first block divided into quarters.  I divided the second block into thirds.  I had to divide each quarter into 3 more so the whole block consisted of 12 segments.  I then divided the 12 segments by 3 so I knew each 3rd equaled four segments from the first block.

So 3/4 equals 9 segments while 2/3 equals 8 segments.  3/4 / 2/3 is 9/8 or 1 1/8.  I am not going to tell you how many times I watched the Learn Zillions video before I finally got the hang of it.  I think I watched it close to 15 times. 

Yes I'm stretching my mind learning this but its cool.  We all need to learn something new every day.  Let me know what you think.  I look forward to hearing from you.

Dividing Fractions

  The other day I was looking for new ways to present the concept of dividing fractions so students understand why they end up multiplying by the reciprocal.  I know as a teacher I have trouble teaching this because it is one of those processes we teach as the way its done.  I found a way but it is not as clear as this method.

I will start out by thinking Fawn Nguyen and her blog Finding Ways for this method.  It is cool and makes a lot of sense.  Look at the division problem 3/4 divided by 2/3.

Fawn uses rectangles as a way of creating the base for the problem so for 3/4 she recommends a 3 by 4 rectangle and the same 3 by 4 rectangle for 2/3.  3/4th of the squares are colored in to represent 3/4th of 1 while 2/3rds of the other rectangle is colored in to represent 2/3rds of 1.

3/4 and 2/3 are colored in.

 Now you count the number of shaded in squares for 2/3 and you discover 8 squares are colored in.  This is the number of squares that make 1 so you count the number of squares in the 3/4ths and find there are 9 squares colored in so you know you have 8 squares or 1 plus 1/8th left over or 1 1/8

 If you do the math you find it is correct because 3/4/2/3 is 3/4 * 3/2 or 9/8 = 1 1/8

This is one of the cooler ways I've seen to illustrate the concept for division.  Most of the sites simply tell people to invert and multiply but they do not provide a conceptual drawing to help explain.  I like this method and I'm going to play around with it to see if it works in a certain form.  I'll report back in a while on it.  Enjoy playing with this idea.

Fraction division by Braining camp

This is the other free app offered by Braining Camp.  It has the lesson, questions, manipulative and challenge or assessment just like the area app.  The one thing this app offers that the other one does not is the use of badges.  The student can earn up to thee badges for each section.  Although most high school students should be solid in their division of fractions, many of my students are not and I see this being used as either a scaffolding step or for remediation. 

I have noticed that many of my students have learned the algorithms to divide fractions but they do knot always understand the concept so this app helps students with understanding why they are dividing fractions the way they do.  I plan to recommend this app to the 5th and 6 the grade teachers for use in their classrooms.