This lesson is one of the activities that has been used in Julia Robinson Mathematics Festivals. At most of these festivals there are many pages listing interesting problems, and students come in and choose to work on the problems. The hard part for the teacher is to say more by saying less. It is best to ask a few questions and just encourage the students to think and explore. The list of questions does most of the work.
This lesson is based on a paper by E. Demaine, M. Demaine, and V. Hart.
Even young students can figure out how to design a cube made from balloons. This is a great way to start the activity. It is worth putting two uninflated balloons together in an X and three together in an *. After students discover that it takes four balloons to make a cube, they should still try to draw the plan. There are at least two different ways to make a cube - one is a cycle of four U's, the other has each balloon in a non planar configuration.
Students also like to design their own balloon figures - a pentagonal pyramid, a house, or...
A follow-up with young students would be to have them find a walk of the puppy
that uses each sidewalk exactly once. See the handout at https://www.mathcircles.org/files/puppy-walk.pdf This can be done on a later day if they have a short attention span. The right hint is to ask which locations require a balloon end. Older students may enjoy making up their own maze problems.
One can increase the level by talking about Kuratowski's theorem. There is even an open problem here - give a similar characterization of genus one graphs. A handout leading in this direction may is available. See: https://www.mathcircles.org/files/deep-graphs.pdf
The following two videos show Dave Auckly leading this activity with a group of teachers.