Free Ride Game Meeting
(Creating Ratios/Adding Fractions)
Provided by NCTM/Illuminations
http://illuminations.nctm.org
Topic
Using an NCTM Illuminations applet, students will calculate the correct gear sizes that will enable a bike to
complete a trip while hitting specific target points along the way.
Materials Needed
♦♦ Equipment for accessing the Illuminations Free Ride applet (http://illuminations.nctm.org/mathcounts)
Meeting Plan
For this activity, it will be helpful for students to have an idea of how gears of different sizes work together.
For instance, in this activity, the gear on the back wheel of a bike is connected to the gear on the pedals.
Before setting your students loose to play the game, you may wish to review the following ideas with them.
When the gears are the same size (as shown to the right), one complete revolution
of the pedals will result in one complete rotation of the back wheel. Assuming the
circumference of the back wheel is 1 yard, the bike would travel 1 yard. However, if
the gears are different sizes, this 1:1 ratio does not hold. Ask your students, “What
happens if the radius of the gear attached to the pedals is twice the radius of the
gear attached to the back wheel?” This scenario is shown to
the left. [One complete revolution of the pedals will result in two complete rotations of
the back wheel; thus the bike will travel twice as far, or 2 yards.] If students are okay
with this, the next question to ask might be, “What if the gear attached to the pedals
has 40 teeth and the gear attached to the back wheel has 25 teeth?” [The ratio of the
two gears is 40/25 = 8/5. Therefore, the bike will travel 8/5 of the circumference of
the back wheel (or 8/5 of a yard) for each full revolution of the pedals.] This scenario is shown below as it
appears in the Free Ride game.
Because the game works with half-revolutions of the pedals, it may be helpful to ask a couple of questions
about half-revolutions, too. For instance, “If the ratio of the number of teeth on the pedals’ gear to the
number of teeth on the back wheel’s gear is 4:1, how many yards will the bike travel with half a revolution
of the pedals? [With the 4:1 ratio, the back wheel goes around 4 times for every full revolution of the
pedals. Therefore, a half-revolution of the pedals will result in the back wheel completing 2 full rotations,
making the bike travel 2 yards.]
2010–2011 MATHCOUNTS Club Resource Guide
37
Your students now are ready for the interactive Free Ride game! Instruct them to navigate to http://illuminations.nctm.org/mathcounts, and select the link to the Free Ride applet.
Selecting the Free Ride option allows students to get a feel for how the game works. The Choose Route
and Random Route options are challenging, interactive games. Note that students can move the bike
backward by clicking on the bottom pedal if they are not satisfied with their moves. However, a backward
move still counts in the pedal count!
Here is a screen shot of a game in progress. The player chose Random Route, so the flags she must
land on are spaced randomly along the path. Notice that the player has just captured the first flag (at 1 4/5 yards) by
landing her back tire
exactly at that point.
The Pedal Count is
1, so two half-turns of
the pedals have been
done. (The first turn
used the Gear Ratio
40/25, or 8/5, and the
second Gear Ratio,
still showing, is 40/20,
or 2/1.) The next flag
is at 3 3/5 yards. The
Course Par value of
4 indicates that 4 full
pedal-turns is a good
goal for completing
the entire course.
(Note: The Course
Par is not always the
least number of pedal
revolutions possible.)
Possible Next Steps
At the top of the Free Ride page, there is an option for Exploration that gives a series of questions for
students to consider (shown below). Answering these questions reinforces the concepts of reducing
fractions, counting possible outcomes and adding fractions.
►►How many different gear ratios are possible?
►►If you use two different gear ratios on consecutive pedals, what possible distances can you travel? How
could this information help you in completing a route?
Resources for Teaching Math
38 This meeting plan is based on an activity
from the NCTM Illuminations project.
http://illuminations.nctm.org
2010–2011 MATHCOUNTS Club Resource Guide