I have found it very difficult to draw a pictorial model that shows why you have to turn the problem of dividing one fraction by another fraction into a multiplication problem. If I struggle with it, you know our students will struggle with it since it seems counter intuitive.
If we take the problem 1/2 divided by 1/6 we get three as an answer but most students think that the 3 represents the whole number not the parts you get. So 1/2 divided by 1/6 gives us 3 but 3 what's.
The 1/6th means you divide the whole circle into 6 pieces just like the picture below. Then you look at 1/2 of the circle so there are three parts in that half. Thus the answer is 3.
Unfortunately, when students get a whole number as an answer, they automatically assume the answer is 3 as in the whole number rather than 3 parts make up half of the whole or 1/2 of the whole that is divided into 6 is 3 pieces.
It's a bit easier when you end up with a fractional answer but its still hard for them to understand the answer refers to parts the fraction is divided into. If you divide 1/2 by 1/5, you would follow the same process of dividing the whole into 5 equal pieces and taking half of that so you get 5/2 or 2 1/2 which is 2 1/2 pieces make up the 1/2.
I realize this approach is probably going to result in people disagreeing with me but its one of the few ways I can think of to get students that 2 1/2 is not two full circles and 1/2 of it, it is two full parts and 1/2 of a part.
This may be why people have difficulty in creating a visual representation of division so the picture meets the context of the of the equation of problem. If you have a better way of explaining it, I would love to hear your suggestions. Have a great day. In the mean time, I'm going to try to figure out how to use legos to explore this topic.
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