notebook, scissors, construction papers preprinted with patterns...
Warm Up (5 min.)
“Slide-Together” Geometric Paper Constructions" by George W. Hart
Content Objectives (1 min.)
Students will experience exploring properties and relationships of two-dimensional and three-dimensional geometric figures.
Language Objectives (1 min.)
Activities (35 min.)
Constructing the 30-Squares “Slide-Together”
Copy and Cut. For one model, use five sheets of “card stock” of five different colors.
Using scissors cut on the lines to release thirty squares. Individually cut the four slits in each, i.e., do not stack squares and try to make several slits with one cut as that will be too inaccurate. Neatness counts!
In what follows, keep in mind the following: the squares are planar; you will bend them temporarily during assembly but they should end up flat; when two squares are slid completely into each other, two edges of one square intersect two edges of the other; and each square will join to four squares of the four other colors, e.g., a blue square never touches another blue square.
Notice that there are two long slits and two short slits in each square. You will always slide a long and a short slit into each other. Begin by joining two squares of different colors. Continue the pattern and add a third square, a fourth, and a fifth. Join the fifth to the first to complete a cycle around a five-fold opening. Be sure always to keep the corners of the squares all on the outside of the construction.
To make a three-way corner between squares A and B, you choose a new square C and join it into both A and B. The first issue is to determine what color C should be. The trick is to look directly across the pentagon from where A and B touch and see what color square is there; choose a square C of that same color. The second issue is to make the three-way corner symmetric with a neat little triangle at its center. The trick to this is to first join C into A and B with a kind of rotation of C, and then temporarily bend and unbend the little points of A, B, and C as needed to get around each other and make a sort of spiral.
Completing the Structure. Once this trick is mastered it is straightforward to create another three-way corner, and another, etc, so all five initial joints are locked. In each case, the color of the new square that is added must be determined by looking across the pentagonal opening to match the color of the square opposite. When all five of the original joints are locked in this way, you will have used a total of ten squares, so you are a third done. Completing the structure is just a matter of noticing there are several incomplete pentagonal openings, choosing any one to complete, and locking its joints, etc. until all thirty squares have been used. Double check as you go along that every opening is surrounded by five different colors and each square joins with four other squares of the four other colors. If properly made, the six squares of any color are arranged like an exploded cube.
“Slide-Together” Geometric Paper Constructions" by George W. Hart
Homework (5 min.)
Any missing indicators.
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