Folding Polygons From a Circle A circle cut from a regular sheet of typing paper is a marvelous manipulative for the mathematics classroom. Instead of

Folding Polygons Folding Polygons From a CircleFrom a Circle

A circle cut from a regular sheet of typing paper is A circle cut from a regular sheet of typing paper is a marvelous manipulative for the mathematics a marvelous manipulative for the mathematics classroom. Instead of placing an emphasis on classroom. Instead of placing an emphasis on

manipulating expressions and practicing manipulating expressions and practicing algorithms, it provides a hands-on approach fro algorithms, it provides a hands-on approach fro

the visual and tactile learner. the visual and tactile learner.

1. Mark the center of your circular disk with a pencil. Fold 1. Mark the center of your circular disk with a pencil. Fold the circle in half. What is the creased line across the the circle in half. What is the creased line across the disk called? Fold in half again to determine the true disk called? Fold in half again to determine the true center. What are these two new segments called? center. What are these two new segments called? What angle have you formed? Unfold the circle. How What angle have you formed? Unfold the circle. How many degrees are there in a circle? Was your many degrees are there in a circle? Was your estimate of the center of the circle a good one? estimate of the center of the circle a good one? Compare your result with that of your neighbor.Compare your result with that of your neighbor.

VocabularyVocabularyplaneplanecirclecirclearc of a circlearc of a circledegrees in a circledegrees in a circlesemicirclesemicircledegrees in a semicircledegrees in a semicirclecenter of circlecenter of circlediameterdiameterendpointendpointline segmentline segmentmidpoint of a line segmentmidpoint of a line segmentradius radius

2. Place a point on the circumference of the 2. Place a point on the circumference of the circle. Fold the point to the center. What is circle. Fold the point to the center. What is this new segment called? this new segment called?

VocabularyVocabulary

circumference of a circlecircumference of a circle

area of a circlearea of a circle

chordchord

3. Fold again to the center, using one 3. Fold again to the center, using one endpoint of the chord as an endpoint for endpoint of the chord as an endpoint for your new chord.your new chord.


sector of a circle sector of a circle

4. Fold the remaining arc to the center. 4. Fold the remaining arc to the center. What have you formed? Compare your What have you formed? Compare your equilateral triangle with that of your equilateral triangle with that of your neighbor. Throughout of the rest of this neighbor. Throughout of the rest of this activity suppose that the area of your activity suppose that the area of your triangle is one unit.triangle is one unit.

VocabularyVocabularyarea of a triangle = 1/2 base x heightarea of a triangle = 1/2 base x heighttriangletriangleequilateral triangleequilateral triangleisosceles triangleisosceles triangleequiangular triangleequiangular trianglesum of the measures of the angles in a triangle = 180 sum of the measures of the angles in a triangle = 180 degreesdegreesbasebasevertexvertexpointpointaltitudealtitudemediancircumcentermediancircumcenterincenterincenter


orthocenterorthocentercentroid centroid angle bisectorangle bisectorperpendicular bisectorperpendicular bisectorperimeter of a triangleperimeter of a trianglescalene trianglescalene triangleright triangleright trianglehypotenusehypotenuselegs of a right trianglelegs of a right trianglespecial 30-60-90 degree trianglespecial 30-60-90 degree trianglePythagorean theoremPythagorean theoremtriangle inscribed in a circletriangle inscribed in a circle

5. Find the midpoint of one of the sides of 5. Find the midpoint of one of the sides of your triangle. Fold the opposite vertex to your triangle. Fold the opposite vertex to the midpoint. What have you formed? the midpoint. What have you formed? What is the area of the isosceles What is the area of the isosceles trapezoid if the area of the original trapezoid if the area of the original triangle is one unit?triangle is one unit?


trapezoidtrapezoidparallel vs not parallel sidesparallel vs not parallel sidesisosceles trapezoidisosceles trapezoidarea of a trapezoid = 1/2 height (top base + bottom area of a trapezoid = 1/2 height (top base + bottom base)base)quadrilateralquadrilateralfractionsfractionsrectanglerectangleright angleright anglearea of a rectangle = length x widtharea of a rectangle = length x widthperimeter of a rectangleperimeter of a rectangle

6. Notice that the trapezoid consists of three 6. Notice that the trapezoid consists of three congruent triangles. Fold one of these congruent triangles. Fold one of these triangles over the top of the middle triangles over the top of the middle triangle. What have you formed? What triangle. What have you formed? What is its area?is its area?


parallelogramparallelogramparallel linesparallel linesarea of a parallelogramarea of a parallelogrampolygonpolygonregular polygonregular polygonperimeter of any polygonperimeter of any polygonrhombusrhombusarea of a rhombusarea of a rhombuslengthlength

7. Fold the remaining triangle over the top 7. Fold the remaining triangle over the top of the other two. What shape do you of the other two. What shape do you now have? What is its area? The now have? What is its area? The triangle is similar to the unit triangle we triangle is similar to the unit triangle we started with.started with.


similarsimilar

congruentcongruent

8. Place the three folded over triangles in 8. Place the three folded over triangles in the palm of your hand and open it up to the palm of your hand and open it up to form a three dimensional figure. What form a three dimensional figure. What new shape have you made? What is its new shape have you made? What is its surface area?surface area?


pyramidpyramid

surface areasurface area

facesfaces

basebase

edgeedge

9. Open it back up to the large equilateral 9. Open it back up to the large equilateral triangle you first made. Fold each of the triangle you first made. Fold each of the vertices to the center of the circle. What vertices to the center of the circle. What have you formed? What is its area?have you formed? What is its area?


hexagonhexagon

pentagonpentagon

central angles of polygonscentral angles of polygons

sum of the measures of the interior angles sum of the measures of the interior angles of a polygonof a polygon

10. Turn the hexagon over and with a 10. Turn the hexagon over and with a crayon, pen, or pencil shade the crayon, pen, or pencil shade the hexagon. Remember what the area of hexagon. Remember what the area of this hexagon is when compared to the this hexagon is when compared to the original equilateral triangle. Turn the original equilateral triangle. Turn the figure over again. Push gently toward figure over again. Push gently toward the center so that the hexagon folds up the center so that the hexagon folds up to form a truncated tetrahedron. What is to form a truncated tetrahedron. What is its surface area?its surface area?


tetrahedrontetrahedron

platonic solidplatonic solid

truncated tetrahedrontruncated tetrahedron

11. Using only the fold lines already 11. Using only the fold lines already determined, create different polygonal determined, create different polygonal figures and determine their area. Using figures and determine their area. Using only the existing fold lines, can you only the existing fold lines, can you construct figures with the following areas? construct figures with the following areas? Are there any others? If so, sketch them.Are there any others? If so, sketch them.

1 , 1 , 19 , 2 , 3 , 7 , 8 , 7 , 231 , 1 , 19 , 2 , 3 , 7 , 8 , 7 , 23

4 2 36 3 4 9 9 18 364 2 36 3 4 9 9 18 36


fractions fractions

12. You can tape twenty truncated tetrahedra 12. You can tape twenty truncated tetrahedra together to make an icosahedron. There will together to make an icosahedron. There will be five on the top, five on the bottom, and be five on the top, five on the bottom, and ten around the middle.ten around the middle.

VocabularyVocabularyicosahedronicosahedron

Other vocabulary that might be used:Other vocabulary that might be used:

common denominatorcommon denominatorarithmetic of fractions arithmetic of fractions closed setclosed setbounded setbounded setcompact setcompact setinterior of a setinterior of a setquadrantsquadrantssecant linesecant lineEuler LineEuler Line

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Folding Polygons From a Circle A circle cut from a regular sheet of typing paper is a marvelous manipulative for the mathematics classroom. Instead of