I
spend a lot of time in schools helping math teachers and students use
various
software programs. Some of these programs I especially like because
they usually reveal something interesting about the learners and
their approach to solving problems. One example is Fraction
Darts which is a microworld designed to help students with comparing
fractions. The context is an engaging darts-like game
where the object is to pop a balloon located on a
number line between 0 and 1 by entering a number in fractional form.
Where the dart lands in relation to the balloon offers a clue as to the
selection of the next "fraction to throw."
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In one of my staff development sessions I had
two veteran 6th grade teachers play a few rounds of Fraction Darts.
Given their many years of teaching 6th grade math I assumed that they
would like the program and find it easy to use. I was right about them
liking the program, but I was surprised that they found the activity
also very challenging! |
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Here's a game of Darts in
progress. Each teacher has taken a turn throwing a dart. The first
teacher tried 3/4. Noticing that was too large, the second teacher
chose 5/8. Since these teachers had a lot of years under their belts
teaching
fractions, I just assumed they would take the (obvious?) strategy of
finding a common
denominator to choose their next dart. For example if they chose 8 for a common denominator then
they would need a number between 5/8 and 6/8. With fractional notation
this is difficult to "intuit" since it is not obvious what is
in between 5/8 & 6/8. (Fraction Darts won't allow 5.5/8 which is intutive.) Of course 16 is a better choice for the
common
denominator because you get 10/16 and 12/16 then the number in between, 11/16, is easy
to determine. However these teachers took a different
route. Here's the dialogue that followed (as best as I can remember it.) I'll call the teachers Alice and John.
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Alice: So I need to throw something bigger than 5/8 but smaller
that 3/4. Hmm.. Let me try making the denominator [in 5/8] smaller. Say 5/7?
John: 5/7 made it bigger, but by too much.
Alice: I'll try 5/9.
John: That made it too small - even smaller than 5/8.
Alice: We now need something smaller than 5/7 and bigger than 5/8.
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Alice: Smaller than 5/7? Then it must also be smaller than 10/14. Right? So 10/15 should be smaller right?
John: Let's try it. (The balloon pops.) Bingo!
Alice and John continued enthusiastically playing several
rounds that deepened their understanding of how fractions work.
So what did they learn from this experience?
1. You can make a fraction smaller if you leave the numerator alone and increase the denominator. (For example, 4/6 is smaller than 4/5.)
2. You can make a fraction larger if you leave the
numerator alone and decrease the denominator. (Revisiting the previous example, 4/5 is bigger than 4/6.)
This appears at first glance to be counter intuitive. But it works because in 4/6 you are dividing your unit
into more pieces than 4/ 5 so each piece of 4/6 will be smaller than 4/5.
My teachers were both pleased as punch that they had come up with
those conclusions. In all their years of teaching fractions in conventional ways they had not thought about them like this.
As time for my inservice was coming to a close the teachers were
investigating another question: What happens to the fraction if you add 1
to (or subtract 1 from) both numerator and denominator? (It works, but can you see why?)
I asked them later if there was a more efficient way
to find a number that "lives in between" two fractions? They didn't know but they would think
about it as well. (Anyone reading this wish to offer your idea about this?)
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