6.17.05 - Correlation - EM 4.2; TM 1.10
Activity: Fraction Darts
Teacher's Page
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
Throws-Team 1
Throws-Team 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. 

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