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Fractris
Posted
April 2, 2001 as a Math Forum EPoW
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Introduction: In this problem,
students fill up rows in a Tetris-like
game by making combinations of certain
fractions that add up to 1.
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Where’s the Math: This
problem deals with basic addition
of fractions. However, some of the
questions also encourage students
to investigate different ways to
get the fractions to add up to 1.
By multiplying all the fractions
by 12, all the fractions are converted
to integers, and the problem becomes
finding all the ways to get the
numbers 1-6 to add up to 12. This
involves combinatorics and number
theory.
Standards: Number
& operations
Role of Components: AgentSheets
is used to run the Fractris game.
A button panel allows the user to stop and start the game.
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Try the applet!
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Sample submitted solutions:
From: Mary, age 13
School: Taipei American School, Taipei,
Taiwan
1. If the computer sends down a 1/3 block,
how can you finish the row with the fewest
number of blocks and without using the same
size block twice?
If the computer sent down a 1/3 block, the
fastest way to fill up the row would be
to put down a 1/2 block, then a 1/6 block.
2. If the computer sent down 1/5, would
you be able to fill the row? If so, how
could you do it with the fewest blocks?
If not, explain why not and tell how close
you could get to completing the row.
If the computer sent down a 1/5 block, then
it would be impossible to fill up the row.
When you change all of the fractions to
have the number 60 as their denominator,
then no matter what combination of fractions
you put down, you won't be able to fill
up the row. However, you can come very close
to filling the row by putting down a 1/4
and a 1/2. This way, the added sum comes
to 57/60, which is the closest you can come
to filling the row.
3. What do all the fractions in the Fractris
game (1/2, 1/3, 1/4, 1/6, 1/12, 5/12) have
in common?
All of them are factors of 12.
From: Julius, age
13
School: Taipei American School, Taipei,
Taiwan
1. If the computer sends down a 1/3 block,
how can you finish the row with the fewest
number of blocks and without using the same
size block twice?
You can use a 1/2 block and a 1/6 block to
fill it up.
2. If the computer sent down 1/5, would
you be able to fill the row? If so, how could
you do it with the fewest blocks? If not,
explain why not and tell how close you could
get to completing the row.
No, you can not. The denominator in 1/5 is
not a multiple or factor of the denominator
in the other fractions. You can get as close
as 57/60 of the row completed.
3. What do all the fractions in the Fractris
game (1/2, 1/3, 1/4, 1/6, 1/12, 5/12) have
in common?
They all have denominators that are factors
of 12.
Bonus:
What are all the different combinations of
the fractions 1/2, 1/3, 1/4, 1/6. 1/12, and
5/12 that will sum to 1 without using any
fraction twice? Explain how you know that
you have found all the ways.
1/2 + 1/3 + 1/6
1/2 + 1/4 + 1/6 + 1/12
1/2 + 5/12 + 1/12
1/3 + 5/12 + 1/6 + 1/12
1/3 + 1/4 + 5/12
I found out all the points by multiplying
all the numbers by 12 and then adding them
up to see which combination add up to 12.
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Reflections:
The biggest problem was with question 3
was finding the term of least common denominator,
or at least explaining it correctly. Many
students stated that they were all factors
of 12 or were divisible by the primes of
2 and 3. Fractions aren't really divisible
by whole numbers, and the prime factors
of 2 and 3 only apply if the denominator
is changed to 12. Many students simply stated
that they all had a common denominator and
were divisible by the primes, without indicating...
A) the LCD found;
B) that the primes of 2 and 3 only apply
after the fractions have been converted.
Some students didn't connect the LCD principle
at all or used 60 instead of 12.
Questions 1 and 2 generally went well,
although some students were misled by question
2 and simply assumed that it must be possible
without including why they 'knew' so. The
bonus was fairly unsuccessful, but mostly
because of question 3. If the earlier connection
had been made, the bonus would have been
easier.
I tried to explain my difficulty with the
wording used in the explanations for question
3, but did not deny credit if this was the
only thing wrong with the solution. One
of the schools, Caroline Davis, must have
done the problem as a class since all of
the submissions contained the 2,3 prime
factor explanation. I understand their idea
but it was not explained or presented well.
The students did not seem to understand
that the numerator is divisible by a whole
number, but the denominator can be made
into a "common denominator" in
order to add or subtract various fractions.
It's a subtle point, but the group effort
on the problem did not foster the correct
thought.
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