2. The King’s Slice

This is the story of The King’s Slice. To honor the King’s birthday, the royal head chef makes him a huge cake in the shape of his kingdom, a triangle! (Big cardboard triangle.) But, as the aides to the royal head chef are en route to deliver the cake, the Queen stops them and says, “The King doesn’t need that much cake! Make it less, by half. That is the order of your queen!”

Fearing for their lives, the aides appeal, “But, your Majesty, the shape is that of the Kingdom. The royal chef thought the King would be so delighted. We mustn’t destroy the shape, … the royal chef would have our heads!”  To which the Queen responds, “Nonsense. With a single straight cut, you can preserve the shape of our kingdom, the desires of your Queen, and the attachments of your heads!” (She must have been schooled in the appropriate mathematics.)

Thus, each team of students is handed a cardboard triangle and asked to perform a single straight cut in such a way that, of the two resulting pieces, one is similar to the original whole, and is exactly half the size (by area) of the original.

Of course, any straight cut parallel to a side of the triangle will preserve the angles of the original and the resulting triangles, thus establishing that they will be similar. So, the question reduces to that of determining how far from a vertex, along a perpendicular to the opposite side (the base), should such a parallel cut be made. For students without the prerequisite algebraic skills, a good approximation can be made using trial and error measurements. Thus, such matters as: the notions of area, of half (as a variation of the MathLab, the queen might insist on only a third, or on three-eighths), of triangle similarity; the importance of measurement; the formula A = (1/2)bh; etc., are brought to light for the student participants. For students aware of the algebra, a very important number, the square root of 2, naturally emerges (if H is the distance from vertex to base—along a perpendicular—then, the cut should be made at H/sqrt(2) of this distance from the vertex).

Again, we observe that the Q, R, V requirements are met:

Q. The students are to determine how to make a straight cut that results in a triangle that is half the size of, and similar to, the triangle from which it was cut.

R. The mathematics required to obtain a solution can range from simple trial-and-error measurements (using the formula for the area of a triangle) to resolving the equation: (1/2)(kB)(kH) = (1/2)(1/2)BH (here B is the base and H the corresponding height of the original triangle; k is the proportionality constant), for which k results in the reciprocal of the square root of 2.

V. The result is probably best verified by comparing the weights of the cut segments (an electronic scale works well; a balance scale is okay)—they should be close to equal. Alternatively, one piece can be cut up and used to tile the other piece. The pieces should cover it nicely.

Collaborative School Activities for the Teaching and Learning of Mathematics