When a Fraction Means Whole: Dividing Fractions

If you have been around me long enough you have experienced one of these moments.  This particular Ah Ha moment came after teaching a lesson on dividing fractions. I felt like I had failed because I could not show how the algorithm worked. I was taught to change the division to multiplication and the divisor to it's reciprocal.
So 1/2  divided by 3/4, became 1/2 multiplied by 4/3. I had no problem showing this but really did not understand why. As a teacher, I wanted to be able to explain why we did that. I had to ask my students for a few days to investigate it further.
So talking it over with a few teacher friends, we began drawing a multitude of arrays trying to fit the pieces together and see if there was a visual pattern for the how the numbers came together. We continued to draw on the paper table cloths at the restaurant and it all clicked.

It was important to recognize that the divisor is the whole group you are trying to create.
So in 1/2 divided by 3/4. It means that I want a to see how many groups of 3/4 I can make with an amount of 1/2.
What I got out of this exploration was a clearer idea of using the language of whole number division in relation to the fractions.

*** Also, a teacher I worked with realized that we do not even need to multiple by the divisor. You can simply cross multiply and then simplify your answer.
For Example: 1/2 divided by 3/4= (4*1)/(2*3)=4/6=2/3...

A new way to teach an algorithm and a great visual to show the parts and whole.***

Fraction Fun Run!!!

For years I have been running at this local park. Over the last few days I discovered that as I was running, I stated adding, subtracting, and computing equivalent fractions in my head as a gained distance. It became a game to me and kept my mind off the fact that I was running. You see, I know that two laps around equaled a whole mile. As the first lap finished I realized I was half done. As I continued to run, I kept chunking my fractions in relation to the mile and saw that the track visually broke up nicely into fourths. My body was getting more tired coming into the third mile and I started to notice that each fourth of the lap could be broken up in half by a conveniently placed tree or trashcan. The comparisons kept evolving; from the mile,  to the lap, and my total run, I was fraction machine.  Soon I was counting down from 16ths to 8ths, 4ths and halves. People say that when you run, you go into a zone. I seem to go into an endless zone of calculations.

Right Brain Math

I learned about these methods from a math content class I am taking through UCLA. My husband and I took a look at the youtube link and sat there in awe of the methods shown for multiplication tables. We then continued to look further for patterns and literally laughed out loud because we were having so much fun relearning our tables.
There are lots of students that simply get turned off by needing to memorize their tables and then this holds them up for future content. If they can visualize patterns or better yet find enjoyment from learning something in a new way, they are better off in the long run.
I am positive that even seasoned math teachers and learners will find the Right Brain Math content to be a beautiful surprise.
Check it out and let me know what you think.

http://www.youtube.com/watch?v=yoac4xzIhSw

If I only had a Brain!

Watching the Wizard of Oz, made me think of something that each and eveyone one of us goes through. A lack of self belief. We go through life doubting ourselves and thinking that something or someone outside ourselves is what will take us to that special place, and change what we are feeling. And although it is true that something outside of us can inspire and remind us,  it is within us that all is possible and all is whole.

How does this relate to mathematics???.... An infinite number of examples could be posed, but I am refelcting on the teaching aspect of math. My job is not only to convey concepts to students and help them explore the fundamentals of Math. My most important job is the motivational part. I have to help the students believe that they can and will succeed in our class. It comes from the way we offer questions and give answers. All has to come from a place where the student understands that they are the ones figuring out the problems posed.

So students, know that you can and you will discover the way to your answers!