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what you know about the angles of a triangle<br />
About the angles of a quadrilateral<br />
The final point is probably the key.<br />
One technique that facilitates exploration is to<br />
take a sheet of paper, fold it in half three times,<br />
then draw a triangle or quadrilateral on it. If you<br />
cut it out, you get eight identical copies, and<br />
that’s usually enough to conduct experiments.<br />
Be careful, though, that you don’t turn over any<br />
of the triangles or quadrilaterals.To avoid that<br />
problem, make sure all the copies are placed on<br />
the table with the same face up, and mark or<br />
color that face.<br />
The explanations in Problems 2 and 4 are not<br />
easy to write and may take you beyond one<br />
class period.You may ask students to write<br />
only one of the two. In any case, encourage<br />
your students to discuss their approaches with<br />
each other and perhaps with you and to write<br />
drafts before embarking on the final version.<br />
Hopefully, the points listed in the hints above<br />
will appear in the students’ write-ups. Remind<br />
them that well-labeled illustrations are a must<br />
in a report of this sort.<br />
Answers<br />
1. Answers will vary.<br />
2. No. Possible explanation:Any two copies<br />
of a triangle can be arranged to make a<br />
parallelogram.The parallelograms can be<br />
arranged to make strips.<br />
3. Answers will vary.<br />
4. Copies of the quadrilaterals must be arranged<br />
so that matching sides are adjacent and all<br />
four angles are represented at each vertex.<br />
An example is shown here.<br />
1<br />
2<br />
4<br />
3 4<br />
2 1<br />
Discussion Answers<br />
A. Answers will vary.<br />
B. Answers will vary.<br />
C. Answers will vary.<br />
Lab 7.4: Tiling with Regular Polygons<br />
Problem 1 revisits work students may have<br />
already done in Lab 7.2 (Tiling with Pattern<br />
Blocks). In fact, for some of the regular polygons,<br />
it is convenient to use the pattern blocks. For<br />
others, see the Note in Lab 7.3 (Tiling with<br />
Triangles and Quadrilaterals) about cutting out<br />
eight copies of the tiles in order to experiment.<br />
If students are stuck on Problem 2, you may<br />
suggest as a hint that they read the text<br />
following Problem 3.<br />
Problem 4 is yet another take on the “angles<br />
around a point” concept. See Labs 1.1<br />
(Angles Around a Point), 7.2, and 7.3.<br />
Note that an arrangement of regular<br />
polygons around a point is a necessary<br />
but not sufficient condition for a tiling.<br />
In Question C, eight of the eleven tilings can<br />
be made with pattern blocks, and all can be<br />
made with the template.<br />
If some students are particularly interested<br />
in these questions, you could ask about the<br />
possibility of tiling with regular polygons that<br />
are not on the template. It is not too difficult to<br />
dismiss 9- and 11-gons by numerical arguments<br />
based on their angles.<br />
Answers<br />
1. Equilateral triangle, square, regular hexagon<br />
2. Their angles are not factors of 360°.<br />
3. Answers will vary.<br />
4. There are twelve ways to do this, including<br />
three one-polygon solutions, six two-polygon<br />
solutions, and three three-polygon solutions:<br />
6 triangles<br />
4 squares<br />
3 hexagons<br />
206 Notes and Answers Geometry Labs<br />
© 1999 Henri Picciotto, www.MathEducationPage.org
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