What does Balance of Equations mean, and what does it have to do with marshmallows?
Balance of Equations is exactly what it sounds like: the value on both sides of an equal sign must be equal to each other, like a scale balancing.
In school, we grow to think of the equal sign to mean "gives us an answer of".
For example, what's 2+3 = ? Did you say "5"?
That's true, adding 2 and 3 does give us an answer of 5, but is 5 the only solution that balances that equation?
Of course not! What about 4+1? 7-2? 20/4? -50+55? All of these answers are creative ways of balancing the equation. The equal sign is not meant to say "just write the answer here", but we often forget that.
Students exploring Marshmallow Geometry can discover for themselves how to balance equations by exploring the amazing formula discovered by mathematician Leonhard Euler ("oil-er").
If we define the following:
V = the number of Vertexes: the points or corners of an object,
F = the number of Faces: the flat surfaces of an object,
E = the number of Edges: the straight lines that connect the points or corners of an object,
then, for any non-self-intersecting polyhedron (a closed, 3-dimensional shape made up of flat edges [no curves], points, and faces), all polyhedra (plural of polyhedron) will follow this rule:
V + F = E + 2
For example, a cube has 8 corners, 6 faces, and 12 edges. 8 + 6 = 12 + 2 (14 = 14).
In the world of marshmallows and toothpicks, we re-define the variables to be:
M = the number of Marshmallows, or vertexes
F = the number of Faces
T = the number of Toothpicks, or edges
And so we have:
M + F = T + 2
Do you believe Euler? Put the formula to the test! Start grabbing some toothpicks and marshmallows, and create all kinds of 3-D structures! When you're done, count the the number of marshmallows (vertexes), the number of faces, and the number of toothpicks (edges), and see if your equation is balanced!