## Thursday, June 17, 2010

### Math problem of the week: Core-Plus Math vs. traditional math on parallelograms

I. From Core Plus Mathematics, Contemporary Math in Context (a unified high school "core curriculum appropriate for all students") Course 2 (year 2), Part B, p. 378:

A common method of designing a pop-up page or greeting card is based on a parallelogram. Tabs attached to facing pages form two consecutive faces of a prism with a parallelogram base. The other two faces are attached to the tabs and serve as props for the picture. As a page or card is turned the prism unfolds and the picture pops up.

a. Use this method to construct a paper model of a pop-up picture. Use piece of paper folded in the middle as consecutive pages of a book. A second piece of paper is needed to carry the picture. Make your model carefully. Test it. Modify it until it works well.

b. Why does the picture in your model lie flat when the book is closed?

c. Why does the picture in your model lie flat when the book is wide open?

d. When will the picture be positioned perpendicular to the right-hand page?

e. When will the picture mke a 120o angle with the right-hand page?

f. Could the parallelogram base of the prism be relaced by any other quadrilteral? If so, which ones? If not, explain your reasoning.

II. From Weeks & Atkins A Course in Geometry (a high school geometry text first published in 1961), p. 153:

1. The diagonals of a parallelogram ABCD meet at O. Any line through O meets AB and DC at X and Y respectively. Prove that XO = OY.

2. Prove that the perpendiculars from the vertices A nd C of the parallelogram ABCD to the diagonl BD are equal.

3. The side AB of the parallelogram ABCD is extended to E so that BE = AB. Prove that BC and DE bisect each other.

III. Extra Credit

From proofs to popups: discuss how far geometry instruction has come in 20th-21st century America.