Many math or science units come with simple balance scales. Remove one balloon from the scale, carefully noting the exact position of the tape or pushpin if one was used. Inflate the balloon. Put it back on the balance scale, exactly where it was when it was deflated. When the volume of a given mass of gas increases, its pressure decreases.
Heat causes air particles to expand, lowering how densely it is packed, thus lowering the pressure.
Since this activity involves the use of fire, it is suggested that it be done as a demonstration. Use the vegetable oil to lightly lubricate the hard boiled egg. Place the egg on the rim of the flask or bottle to demonstrate that it is definitely too large to fall into the bottle. Light a wood splint or piece of paper, lift the egg, drop in the burning paper or wood and replace the egg quickly back onto the container's mouth. As the air inside the bottle warms, it will expand and exit the bottle, causing the egg to vibrate as the expanding air escapes.