That’s a great challenge!
The simplest practical way to have mobility on 2D plane with one motor is what @Deicer proposed - ratcheting one side in one direction.
However, what your teacher wants is, probably, some sort of gear shifting transmission. When you run motor in one direction it shifts among 4 transmission states:
- both sides are powered in forward direction
- sides are powered for turning left
- both sides are powered in reverse direction
- sides are powered for turning right
And then when you run the motor in other direction it actually powers them
And I just realized @Deicer beat me to this answer again 
Long time ago when I was in middle school I was given as a present remote control with a single button and receiver board that could turn on and off one motor in only one direction. I tried to implement this scheme, when it would switch modes on click-back when power was cut off - it was clearly too complex to me to finish.
Also, if your teacher didn’t say if it is legal to use pneumatics, then you can have shifting implemented with pneumatics. Then everything will be much simpler: you need to have both sides powered in ether same or opposite direction switched with pneumatic gear shift.
Yes you will. unless you have a two-wheel drive. Two for the middle one on one side, this is because you will need to run two chains to each of the big sprockets. Same for the other side. That means that each side has four sprockets. 4 + 4 = 8.
when your motor spins say, clockwise, the motor will move along a rail. Along this rail the counter-clockwise spinning gear falls into other ratcheting mechanisms that power your drive in one direction, left, right, forward, or backward. Have one for each. The clockwise rotation pushes your motor into the desired gearbox like a selector and the counter-clockwise actually powers it
since you can’t turn you can just connect both sides with your axle. Additionally, if your motor powers one of the wheels directly you could use just two total sprockets to connect the the other axle to it.
Oh yeah, oof. How did I not think of that before? 
Like a single half of the drive base with one motor? Yeah it’s very possible and even a terrible builder such as myself have done it two different ways, but if you mean the entire drive base with 1 motor then still yeah, just the amount of power output by the single motor will be less for each wheel and you’ll have to do some fancy gear ratios so everything moves in the right direction.
i know this is not for a comp but i was just currious
For IQ, there is this design:
If I’m not mistaken that’s built like a car - one driving motor plus front wheel steering actuation (aka +1 motor)
Pneumatic drive. It’d only last about 30 seconds, but it’d be a cool train-style motion. Then you have your one motor left for a spinning arm of destruction.
That’s 2 motors tho, gyro walker, if this was in a shooting game you can rotate your flywheel and use the torque steer too move forward and turn
Double flywheel. Attach flaps to the flywheels and vary their speed to swim through the air and drag the robot forward.
You could have a three wheel tricycle kind of robot with the front swerve wheel powered by the motor. You can then add pneumatics to change the direction of the front swerve wheel for turning.
Well if you have 3 motors, you’d just be able to build a kiwi so no pneumatics necessary for turning
I meant three wheel not three motor. Sorry.
Did you have the Cubli in mind?
It needs, at least, 3 motors, however.
More like the robot in blue here
“had no wheels, instead operating on the principle of gyroscopic procession. Its weapon was attached to an apparatus which tilted it back and forth. This rocking motion caused a large “foot” mounted at the back of the robot to move, causing the robot to move forward.”
The problem with these gyroscopes is that they seem to take a long time to complete an action, and appear to have to “charge up”
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