The
object of the Fraction
Darts challenge is to "pop" balloons located on a number line
between 0 and 1. The darts are "thrown" by entering a number in
fractional form. Here is a glimpse of a game in progress. Two
darts (5/8 and 3/4) have been thrown so far. Notice that 3/4 is too big
and 5/8 is too small.
What
would be your next throw? Explain your strategy.

Resource:
Microworlds Applet.
Requires webplayer download. See http://mathforum.org/mathtools/activity/15465/
Environment:
One computer or classroom lab
Strategy:
In one computer enviroments Students work in groups taking turns
throwing darts.
Standards
Connections: Understanding fractions

Description
Darts is a Microworlds activity that requires the player to pop a
balloon
that appears on a number line between 0 and 1 by throwing fractional
darts.
Setting the Stage
Before doing the computer game, play this guessing game with the
students. I'm thinking of a fraction
between 0 and 1. Can you guess it? It must be stated in
fractional
form with a whole numbers for the numerator and denominator. (Choose
3/4
as the fraction to be guessed.) A student may guess 1/2. Your
response would be
that the number is too small. Another student may try 7/8 to
which
you respond with: "That's too large". At this point the students may be
stuck
because they need to find a fraction that is between 1/2 and 7/8. Some
students may realize that 3/4 is in between and would guess correctly.
At this point show them the challenge problem and explain how Fraction
Darts
works.
Doing the Activity
Hand out the student page and ask the students working in groups to
explain
mathematically
how they found a number between 3/4 and 5/8. Some possible anwers
from three students might be:
 7 / 10. Student may explain: "I changed the numbers to
decimals .75 and .625 and
saw that .7 is
in between. I would enter 7/10."
 4 / 6. Student may explain: "4 is between 3 and 6 is
between 4 and 8. So I think
4/6 or 2/3
would
work."
 5.5 / 8. Student may explain: "Make the two fractions have
a common denominator
like 6/8
and 5/8. So 5.5 over 8 should be in between." (How would you
respond to that student?)
Debriefing
Finding a number between 3/4 and 5/8 is challenging for students who
have a fragile understanding of fractions. For teachers the obvious
strategy would be to find a common denominator. If you use 8 then you
run into the problem that that the third you need a number between 5/8
and 6/8. With fractional notation
this
is a difficult problem to "intuit" since it is not obvious what is in
between
5/8 & 6/8. Of course if one uses 16 as the common denominator you
get 10/16 and
12/16
then the number in between is easy to determine 11/16. What about
an answer like 5.5 / 8? One way to find out would be in the
fraction darts microworld.
Fraction Darts microworld*
*If the
activity does not work, you may need to download the
Microworld EX web player.
Extensions & Additional
Activities
Play a round of 5.
Balloon

ThrowsTeam 1

ThrowsTeam 2

1



2



3



4



5



TOTAL= 


An Interesting Extension
An interesting (though not necessarily
pedagogically sound) way to find
a number between 3/4 and 5/8 is to add the two numerators (3+5=8) and
the
two denominators (4+8=12) and form the fraction 8/12 (or 2/3) which
just happens
to fall between the two original fractions. 2/3’s middle stature can be
easily confirmed by looking at the three fractions with the common
denominator
24. (18/24; 16/24; 15/24.). Was that just a coincidence or does that
always
work? It turns out that it always works. Here’s a proof.
Back
to Student Page
Copyright ©
19992004 Stevens
Institute of Technology Center for
Innovation in
Engineering and Science Education, All Rights Reserved. 