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o0XjH2222222DD22222222222222226 ?: Title: Parachute Pattern
Essential questions: How can a 2-dimensional pattern develop into a 3- dimensional surface?
Performance Tasks: 19.04 develop a pattern using surface development techniques.
Sunshine State Standards (LA, MA, and SC): MA.912.G.2.4 Apply transformations (translations, reflections, rotations, dilations, and scale factors) to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons.
Essential Skills (e, m, and s): M34 Understand the properties and classification of polygons
(triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations.
Rigor and Relevance (quadrant): D - Adaptation
Instructions to Teacher:
Teacher Explanation: Explain the underlying mathematics from the Griffiths University Website: http://www.sct.gu.edu.au/~anthony/kites/parafauna/chute_design/
Calculation Method: The design of a gore, even very odd shaped ones, is easy. All you have to remember is a few points...
Width
The radius at a point is proportional to the width of the gore. That is if the width of a gore is half that of the maximum width, then the radius or diameter at that point in the final chute/windsock/whatever, will also be half of the maximum.
Angle
If the end of the chute comes to a point which is `flat' at the end, then all the angles at that end must add up to 360
degrees. As such for a 6 piece gore pattern, the top of the gore must equal a 60 degree angle. EG: 6 gores 6
Similarly if the mouth of the parachute is vertical (or cylindrical) then the edge of the gore must be parallel to the
centerline of the pattern, or 90 degrees to the mouth edge of the gore pattern. Ok them's the rules! So lets study a hemisphere a little more closely! Remember these rules also apply to Windsocks too. First the maximum diameter of the parachute is the mouth opening, the radius of which is r. The circumference of the opening is, from your high school mathematics (it had to be good for something didn't it) is 2 PI r. As we are making 6 gores to create a parachute, the width of the gore at the mouth (which I will call l) is 1/ 6th of this length or PI r/ 3.0 degrees => 360 degree flat at top of chute. Similarly if the mouth of the parachute is vertical (or cylindical) then the edge of the gore must be parallel to the centerline of the pattern, or 90 degrees to the mouth edge of the gore pattern. equal to r. This roughness will make the resulting parachute only 3% smaller in size which is too small to be of real concern. The parachute will still form a proper hemisphere as all gore pattern measurements will be based on the value of l. Also setting l equal to r we simplify all the
calculations, and don't have to deal with PI (apple or otherwise :-). OK the width of the gore is 1/ 6th of the circumference of the parachute radius. The height is also 1/ 4 of the circumference, which you should be able to easily verify. As such the height of the gore will be 1.5 times its maximum width (l). (See the hemisphere gore pattern below) On your gore plan you can now lay out the angles the gore has to have at the mouth (parallel to the center line) and the top (a 60 degree angled end) as per the second design point above. The edge of your gore should curve to match these lines as the edge approaches the top and bottom. Now we come to the tricky part. The radius of the parachute is proportional to the width of the gore. As such, studying the hemisphere, we can look at the radius of the hemisphere half way between the top and
the bottom of the fabric. This is at 45 degree angle to the center point of the hemisphere and thus radius from the centerline at this point is 1/ sqr(2) (you knew math would have to come in somewhere didn't you) or about .71 of the radius of the sphere. This means that the width of the gore half way along must also be .71 of the width of the parachute mouth. (See pattern) The two angles and the center point would probably be enough for you to now sketch out the curved edge of the gore. I have however also worked out the points for the 1/ 3 (.86l) and 2/ 3 positions (.5l) on the gore, to produce a better result. Even more measurements could also be calculated, and for large chutes may be necessary to refine the curve of the gore side. Other angles however will involve trigonometry and its sine and cosine calculations (yuck). I found the above three measurements plenty for the small chute Tuffy required. If you want to make more than 6 gores for your hemispheric chute, the only change is that ALL the widths of the gore pattern will be proportionally smaller. For example an 8 gore parachute will have a smaller length of l approximately equal to 2/ 3r while the gore height will remain as it was (approximately 1.5r). The only other change is that the angle of the tip of the gore will be 45 degrees instead of 60. You may however like to calculate more gore width ratios just to set the curve better.
Demonstrate how to draw the line segments to create a gore. Have the students draw each line segment as you
demonstrate it.
Once the demonstration is finished and the students have completed the pattern, demonstrate scale and explain the scale
factors to be used for the four patterns. The students can do this on their own, but demonstrate the offset command to add
the 6mm tabs to the pattern prior to releasing the students.
Instructions to Students:
1. Draw a horizontal line 10 mm long.
2. Draw a vertical line from the right end of the horizontal line up 30 mm.
3. Draw a horizontal line from the midpoint of the vertical line left 7.07 mm.
4. Draw an arc through 3 points beginning with the top of the vertical line to the left end of the upper horizontal line to the
left end of the bottom horizontal line.
5. Mirror the arc and the bottom horizontal line about the vertical line. The result is a completed gore.
6. Draw a straight line from the top of the vertical line to the left end of the upper horizontal line. Continue with a second
straight line to the left end of the bottom horizontal line. The result is a straight line approximation of the gore curve.
7. Erase the two gore curves and mirror the approximation (2 straight lines) about the vertical line.
8. Erase the vertical line and the upper horizontal line. The result is a completed straight line approximation of a gore.
9. Assign each student a radius of 100 or 200 or 300 or 400 mm. Divide your assigned radius by 20 to derive a scale
factor.
10. Scale the gore using the scale factor.
11. Mirror the gore about one of the upper straight line gore curve approximations.
12. Mirror both of these gores about one of the outside upper straight line gore curve approximations.
13. Mirror 2 gores about one of the outside upper straight line gore curve approximations.
14. Use offset of 6 mm to create tabs for seams.
The pattern should look like the following illustration:
15. Add tabs before printing the pattern onto paper. The tabs should be 6mm wide on both sides of each seam.
16. Start a new Microsoft Word document by listing the step-by-step process for drawing the pattern on the
computer. This will be part of a technical article of 800-1000 words.
Instructions for Learning Styles Modifications:
Circulate computer lab and remediate as required.
Assessment for Activity:
Check completed parachute and log pass/fail. Instruct students to start a
Microsoft Word document by listing the step-by-step process for drawing the pattern. This will become
part of a technical article of 800-1000 words, which will be graded.
Approximate Length of Time for Activity: 90 minutes
Materials Needed:
Paper, scissors, rolls of thin masking tape, hole punches, heavy cotton thread, metric rulers
Resources Needed:
Activity:
1. Students cut out pattern that has been printed on paper.
2. Fold the pattern along the lines.
3. Tape seams along the inside of the parachute.
4. Reinforce the bottom part of each seam with a folded over small piece of masking tape.
4. Using a single hole punch, make a hole near the bottom edge of each seam through the folded over
small piece of tape.
5. Measure 6 strings the length of which should be 4 times the assigned radius.
6. Insert one end of each string in each of the 6 holes created in step 4.
7. Use a very small piece of tape to tape the two ends of the string together.
8. Loop the string through the end of a large paper clip.
9. Once all strings are complete, the parachute is ready for flight testing.
10. Now return to your Microsoft Word document and make a list of all the steps which you used to
make the parachute. This is the second part of your technical article.
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