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The battery is what actually produces the electrical current and so the alligator clips provide the pathway for transporting that current to the water. You need to attach one clip to the positive terminal and one to the negative terminal.  Use a 6-volt battery but if you cannot find one, use a 9-volt battery. You can find this size of battery at pretty much any convenience store or supermarket. Make sure that the metallic part of the alligator clip is connected to the graphite of the pencil. You may have to shave down a little bit more of the wood from the pencil just to make sure that the alligator clips are completely connected to the graphite. Doing this completes the connection to the battery and allows the current to be transferred from the battery all the way through into the water. The way you cut the cardboard earlier means that it should sit nicely on top of the glass. Try to do this gently so that the pencils that are stuck through the cardboard don’t get disturbed from their position. For this experiment to work, the graphite of the pencils needs to not be touching the side of the glass so just double check that here and adjust the pencils if need be. At this point, bubbles start to rise from the submerged points of graphite. This is the hydrogen and oxygen gas being split. Hydrogen gas will be bubbling from the pencil connected to the negative terminal and oxygen will be bubbling from the pencil connected to the positive terminal.  Once you connect the alligator clips to the battery and graphite, the current begins to flow immediately. There will be more bubbles coming from the hydrogen pencil because there is twice as much hydrogen as oxygen in each water molecule.
Connect one end of each alligator clip to the terminals of the battery. Connect the other end of each alligator clip to each pencil. Place the cardboard on top of the glass so the pencils are submerged. Watch the separation of hydrogen and oxygen occur.