Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University

Geometry and Spatial Reasoning

Activity: Envelope Tetrahedrons

TEKS: (7.6) Geometry and spatial reasoning. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. The student is expected to: (A) use angle measurements to classify pairs of angles as complementary or supplementary; (B) use properties to classify triangles and quadrilaterals; (C) use properties to classify three-dimensional figures, including pyramids, cones, prisms, and cylinders; and

(7.8) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world. The student is expected to: (B) make a net (two-dimensional model) of the surface area of a three-dimensional figure; and

(7.9) Measurement. The student solves application problems involving estimation and measurement. The student is expected to: (A) estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes;

(7.15) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to: (B) validate his/her conclusions using mathematical properties and relationships.

Overview: Students will use ordinary business envelopes and paper folding to investigate attributes of two- and three-dimensional figures. Properties of two- and three-dimensional figures will be reviewed in the context of paper folding.

Materials: An envelope (or two) for each participant Rulers Scissors Set of geometric solids

Grouping: Large group activity with small group break outs

Time: 30-45 minute class

Geometry and Spatial Reasoning Grade 7 Envelope Tetrahedrons Page 1 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University

Lesson: Procedures Notes 1. Have each student seal his/her envelope. When one diagonal is drawn, ask Using the ruler, draw one diagonal of the what shapes are now created. rectangle. triangles

Do the two triangles have any special characteristics? They are congruent.

After asking questions about the two When the second diagonal is triangles formed, have students draw the drawn, ask what is formed. two second diagonal. pairs of congruent isosceles triangles

Ask if students can justify why the triangles are congruent. There are actually 8 triangles, 4 overlapping large and 4 small.

What is the area of each triangle compared to the area of the original envelope? Each triangle has the same area, thus each triangle is one-fourth the area of the envelope.

2. The students should now fold the envelope Using the ruler as a creasing along the diagonals. guide helps. Scoring along the diagonal with a ball point pen helps also, but you have to be very careful not to cut the paper.

3. Cut out one of the triangles that is formed by Ask students to identify the type of two half diagonals and the longer edge of the triangle just cut from the envelope envelope. (isosceles obtuse triangle) and verify the type. Fold to show that two sides of the triangle are actually equal in length.

Ask students to identify the fold line. line of symmetry

4. Now have students open the triangle. What geometric shape do they have? Most will say parallelogram, but it is actually a

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Procedures Notes special parallelogram, a rhombus.

If needed, clarify the more specific name. Why is it a rhombus? Opposite angles congruent makes it a parallelogram; adjacent sides congruent makes it a rhombus.

What are other attributes of a rhombus? The diagonals of a rhombus are perpendicular. The diagonals of a rhombus bisect the angles of the rhombus.

If students identify attributes that are common to all parallelograms, be sure to point out that those characteristics do not apply only to the rhombus.

What is the area of the rhombus compared to the original envelope? The rhombus has area equal to one-half the area of the original envelope.

5. Refold the rhombus back to the obtuse What is the area of the triangle isosceles triangle. compared to the area of the original envelope? The triangle is Encourage students to justify how they one-fourth the area of the determined the area. envelope.

How does the area of the obtuse isosceles How does the area of the obtuse triangle compare to the area of the acute isosceles triangle compare to the isosceles triangle? area of the acute isosceles triangle? Areas are equal! The area of each triangle is one-fourth the area of the original envelope.

6. Using the envelope, ask what geometric shape it is. Concave pentagon

Fold the envelope in half by matching the Ask what geometric shape the two shorter edges. Open, turn the envelope folded envelope is. trapezoid over, and make the same fold again. This

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Procedures Notes fold should be a very crisp crease. How does the area of the trapezoid compare to the area of the original envelope? The area of the trapezoid is three-eighths the area of the original envelope.

7. Now open the envelope until the new fold is This three-dimensional figure is a straight line. At this point, the right half will also called a skewed pyramid. fit inside the left half and form a tetrahedron. Identify the vertices, edges and faces of the tetrahedron as a review (or introduction) of three- dimensional geometry.

8. How does the surface are of the tetrahedron The surface area of the (pyramid) compare to the area of the original tetrahedron is equal to the area of envelope? the original envelope.

Have students “open” the figure and cut it This works really well if students apart to make a net of the skewed pyramid. start with two envelopes. They can keep one intact, and cut one to make a net and verify that they really have a net.

9. Put students in groups and distribute Have students sketch each solid, geometric solids to each group. and make a table that identifies how many vertices, edges, and faces each geometric solid figure has.

Homework: Have students find three or four three-dimensional shapes in their homes. They should sketch the shape; identify number of vertices, edges and faces; and draw a net of the shape.

Resources: 1. Any size envelope will work except square envelopes. The average "letter" size envelope will produce an almost regular tetrahedron. 2. Instead of using new envelopes, collect the return envelopes often included in promotions. 3. This basic idea for this activity came from "Tetrahedral Models from Envelopes" by Charles W. Trigg in Mathematics Magazine, Vol.51, No. 1, January 1978.

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