Golden Rectangle problemsDate: 9 Dec 1994 11:18:28 WST-8 From: "Helen Mansfield" Organization: Faculty of Education Subject: geometry explorations I am having a visit this morning from some 6th grade students from Swan View Primary School in Perth, Western Australia. They are aged 10/11 and their teacher is Ms Stephanie Winnett. They have been doing geometry explorations and want to share their results and ask for feedback. I have left their mathematics as is. 1. Can you make this shape from the golden rectangle? Use the correct maths language to describe how you did it. Adam Moiler [kite; diagonals 6.3 cm and 6.7 cm; symmetrical about shorter diagonal; diagonals intersect 1.3cm from end of shorter diagonal] 2. I made this shape out of a 3.7 cm by 6 cm rectangle using three shapes. How did I make it? Daniel Lane [kite; same dimensions as #1] 3. This is the golden rectangle. It has the golden ratio of 1.618034 (or pretty close to it). Can you describe in mathematical language how I found this ratio? What would be the length and width of this golden rectangle if you had the exact golden ratio of 1.618034? If my golden rectangle had these symbols on it, what shape would it be? [rectangle; 2 pairs adjacent sides marked equal. One angle marked as a right angle; other 3 angles marked as equal] Orietta Stokes 4. I made this shape out of a 3 by 6 centimetre rectangle. Can you make this shape out of a golden rectangle? Use mathematical language to describe how you made it. Geoffrey Long [Two isos. triangles with one below the other; long axes aligned; orientated the same way] 5. I made these shapes out of a golden rectangle. Can you make them too? Describe how you made it in mathematical terms. David Billing [1. Arrowhead; long sides 6.1 cm; short sides 3.5 cm; angle between long sides 59 deg. 2. Rhombus; diagonals 6.2 cm and 3.6 cm] 6. This parallelogram is made from a 3 cm x 6 cm golden rectangle. How was this shape constructed? Can you describe how by using mathematical vocabulary? Lea Harwood, Melissa Rogers, Jessica Shepley, Cristy Paul [parallelogram. Sides 8.7 cm and 2 cm. Acute angles 45 deg] 7. Can you describe, using mathematical vocabulary, how I made these shapes from a 3 cm x 6 cm rectangle? Ryan Hunter [Arrangement of 8 right triangles around a central point. Arrangement has rot. symmetry. Triangles have perp. sides of 3 cm and 1.5 cm] 8. This shape has been made with a golden rectangle. Can you build this shape? Can you tell how you made it using math. terms? What is the name of the shape? Sam Daly [parallelogram; sides 6.7 cm and 7.2 cm. Acute angles 30 deg.] 9. I constructed this equil. triangle out of one Golden Rectangle. I cut the Golden Rectangle into six triangles (4 right angle and 2 isos) to make this shape. Can you do it? Naomi Scrutton [equil. triangle, sides about 7 cm] 10. This parallelogram has been constructed from a golden rectangle that measured 6 cm by 3.7 cm. It was cut into 3 pieces. Can you construct it? Can you describe how you made it in math. language? Michael Cuccaro [Parallelogram, sides 7 cm and 3.6 cm. Acute angles 60 deg] 11. How many kite shpaes can you make from a goldne rectangle? You have to use paper with a different colour on each side. Once you have constructed the optimum amount of kite shapes, you can record how you constructed the kite shapes. We constructed 16. Karen, Kathy, Susannah [All 16 shapes same kite shape. Each made from 3 pieces. Diagonals 6.4 cm and 5.4 cm. Symmetrical about long diagonals. Diagonals intersect 1.3 cm from end of long diag.] Helen Mansfield Faculty of Education Curtin Univesity of Technology Box U1987 GPO Perth Western Australia 6001 Date: 12 Dec 1994 02:40:45 GMT From: Dr. Math Organization: Swarthmore College Subject: Re: geometry explorations Hello there! We are very impressed with your results! We're also quite happy that you're doing some hands-on work with Geometry. I hope you keep experimenting with geometrical shapes on your own! One thing that is important to remember is that the Golden Rectangle does not refer to one specific size of rectangle. Instead, it refers to a whole family of similar rectangles. For instance, one Golden Rectangle is about 1 inch by 1.62 inches. Another is about 2 inches by 3.24 inches. It looks like the rectangles that you guys cut up were about 3.7 inches by 6 inches. Try drawing these and see how they are all similar. If you need to know more about what the word "similar" means in mathematics, please write back. We liked how you described your constructions. Sometimes, though, it's a little hard to tell what you mean when there's no picture of the shape you describe. I hope that someday soon it will be easy to send pictures over e-mail along with your descriptions! I've gone through and made comments on your constructions. >1. Can you make this shape from the golden rectangle? Use the correct >maths language to describe how you did it. >Adam Moiler >[kite; diagonals 6.3 cm and 6.7 cm; symmetrical about shorter diagonal; >diagonals intersect 1.3cm from end of shorter diagonal] Good job! I like this shape. >2. I made this shape out of a 3.7 cm by 6 cm rectangle using three shapes. How did I make it? >Daniel Lane >[kite; same dimensions as #1] I'm not quite sure what you mean by "three shapes." Do you mean that you cut the Golden Rectangle into three pieces? I assume that you and Adam Moiler did your construction the same way, right? >3. This is the golden rectangle. It has the golden ratio of 1.618034 (or >pretty close to it). Can you describe in mathematical language how I found >this ratio? >What would be the length and width of this golden rectangle if you had the >exact golden ratio of 1.618034? >If my golden rectangle had these symbols on it, what shape would it be? >[rectangle; 2 pairs adjacent sides marked equal. One angle marked as a >right angle; other 3 angles marked as equal] >Orietta Stokes Actually, the Golden Ratio is not exactly 1.618034, but it's pretty close to it. Here are some more decimal places for the Golden Ratio: 1.6180339887498948482... The Golden Ratio has an exact value of 1 + (sqrt) 5 ------------ 2 The way we find the value of the Golden Ratio is kind of complex. First, we define the Golden Ratio as the ratio between x and y if x y ----- = ----- y x+y Let's say y was 1. The we'd have x = 1/(x+1). If we solved this equation to find x, we'd find that it is the value I gave above, which is about 1.62. In the drawing you made, I'm not sure what you mean by your symbols. If two pairs of adjacent (remember that adjacent means "next to each other" or "touchin each other", and opposite means "across from each other") sides were marked equal, then this wouldn't be a Golden Rectangle at all, since a Golden Rectangle must have the Golden Ratio as the ratio between adjacent sides. >4. I made this shape out of a 3 by 6 centimetre rectangle. >Can you make this shape out of a golden rectangle? >Use mathematical language to describe how you made it. >Geoffrey Long >[Two isos. triangles with one below the other; long axes aligned; orientated >the same way] This is a very interesting shape! I think a picture would help a lot when you describe it. I'm not quite sure exactly what the shape is. >5. I made these shapes out of a golden rectangle. Can you make them too? >Describe how you made it in mathematical terms. >David Billing > [1. Arrowhead; long sides 6.1 cm; short sides 3.5 cm; angle between long > sides 59 deg. Is this the same as the kites in the first and second examples above? Good job! I don't think that I've ever seen the word "Arrowhead" in a mathematical context, so remember to describe what you mean. There are lots of different shapes that were used as arrowheads by native hunters. > 2. Rhombus; diagonals 6.2 cm and 3.6 cm] The area of this rhombus is 11.16 (do you see how I found that out?), so it must have come from a Golden Rectangle with area 11.16, right? How would we find out the side lengths of a Golden Rectangle with area 11.16? >6. This parallelogram is made from a 3 cm x 6 cm golden rectangle. >How was this shape constructed? Can you describe how by using >mathematical vocabulary? >Lea Harwood, Melissa Rogers, Jessica Shepley, Cristy Paul >[parallelogram. Sides 8.7 cm and 2 cm. Acute angles 45 deg] If you started out with a 3x6 rectangle, I'm afraid it wasn't a Golden Rectangle. The ratio between the sides of your rectangle is 2, and it would be about 1.62 if it were a Golden Rectangle. >7. Can you describe, using mathematical vocabulary, how I made these >shapes from a 3 cm x 6 cm rectangle? >Ryan Hunter >[Arrangement of 8 right triangles around a central point. Arrangement has >rot. symmetry. Triangles have perp. sides of 3 cm and 1.5 cm] This is a really nice-looking shape. It looks like you took the rectangle, cut it in fourths with cuts parallel to the 3 cm side, and then cut each of the four resulting rectangles along their diagonals. Good job! >8. This shape has been made with a golden rectangle. Can you build this >shape? Can you tell how you made it using math. terms? What is the name of >the shape? >Sam Daly >[parallelogram; sides 6.7 cm and 7.2 cm. Acute angles 30 deg.] It looks like you've taken the Golden Rectangle and cut it along a thirty degree angle with the long side. Then you took the triangle this created and moved it over to the other side of the Golden Rectangle. Is this right? >9. I constructed this equil. triangle out of one Golden Rectangle. I cut the >Golden Rectangle into six triangles (4 right angle and 2 isos) to make this >shape. Can you do it? >Naomi Scrutton >[equil. triangle, sides about 7 cm] This one really interests me. I must admit, I haven't figured out how you did your construction. I got pretty close, but it turned out that what I was working with wasn't able to be cut out of a Golden Rectangle. Great job! >10. This parallelogram has been constructed from a golden rectangle that >measured 6 cm by 3.7 cm. It was cut into 3 pieces. Can you construct it? >Can you describe how you made it in math. language? >Michael Cuccaro >[Parallelogram, sides 7 cm and 3.6 cm. Acute angles 60 deg] I think you've taken the Golden Rectangle, cut it along a diagonal, and then put it back together so that the two long legs are touching. >11. How many kite shpaes can you make from a goldne rectangle? You >have to use paper with a different colour on each side. Once you have >constructed the optimum amount of kite shapes, you can record how you >constructed the kite shapes. We constructed 16. >Karen, Kathy, Susannah >[All 16 shapes same kite shape. Each made from 3 pieces. Diagonals 6.4 cm >and 5.4 cm. Symmetrical about long diagonals. Diagonals intersect 1.3 cm >from end of long diag.] Wow, that's a lot of kites! Are they all different? In the description of your picture, it says that they're all the same. Do you mean that the colors showing are different? Thanks for writing! -Ken "Dr." Math Date: 13 Dec 1994 14:58:10 WST-8 From: "Helen Mansfield" Organization: Faculty of Education Subject: Re: geometry explorations Thanks for your response to the grade 6 children's questions. I was unable to give much explanation when I typed in the message, since all 16 bodies plus Eddy the bus driver were crammed into my office, which is about the size of a broom cupboard. I will pass on your responses to the children, who will be delighted. This trip was the culmination of the mathematical work these children had been doing for the year. It is the end of our school year here in Australia. Of course, both their teacher and I knew the mathematical weaknesses in their work but I decided to leave them. I did correct spelling - easier for me to type the correct spelling than the errors. The children were ecstatic about the idea that their work might be read by people overseas. I told them how long it would take to be transmitted by satellite and they counted this down. They also conducted a terrific conversation in real time with Gary Martin in Hawaii. After they left my office, they went to the university library, used the electronic catalogue to find the shelf numbers for the calculus books and were delighted when they actually touched some calculus books. Finally they had lunch that was NOT healthy (special treat.) BTW, these children will have no idea what inches are. Their teacher will have to translate that part of your response for them. Regards and thanks Helen Mansfield |
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