Mathematical Investigation: Paper Size

Mathematical Investigation: The Perfect Rectangle

Topic: Angles, Polygons and Geometric Constructions

The purpose of this worksheet is to investigate what is so nice about perfect rectangles.

Section A: The Perfect Rectangle

1. Which one of the following rectangles do you like best? It is o.k. if your opinion is different from your classmates or friends. So no mark for this question.

Your choice: ______

2. The diagram below shows a series of squares. The two smallest squares S1 and S2 have lengths of 1 unit. Complete the middle column of the table below. [1]

Square Sn / Length of Square Ln / Ln / Ln–1 (to 8 s.f.)
S1 / 1 / –
S2 / 1 / 1 / 1 = 1
S3
S4
S5
S6
S7
S8

3. What do you notice about the numbers in the middle column of the above table? [1]

4. Complete the last column of the above table, leaving the answer to 8 significant figures where applicable. [1]

5. What is the significance of the ratio L8 / L7? [1]

Hint: Ratio of something for some rectangle. Use a pencil to highlight the rectangle in Q2. [1]

6. Is the rectangle in Q5 about the same shape as Rectangle C in Q1? [1]

7. Is your favourite rectangle Rectangle C in Q1? It is o.k. if your choice is different. [no mark]

8. Rectangle C is called the Perfect Rectangle because it is supposed to be the most pleasing to the eye for most people. Measure the length and the width of Rectangle C and calculate the ratio of its length to its width. [1]

9. The ratio Ln/Ln–1 in the last column of the above table converges to the value 1.6180339887 49894848204586834365… This value is called the Golden Ratio F (pronounced as ‘phi’). Is this value about the same as your answer in Q8? What does this mean? [1]

10. Use a calculator to evaluate and write down its value. What do you notice? [1]

Section B: Some Interesting Properties of the Golden Ratio

11. Use a calculator to evaluate F2 and write down its value, correct to 5 decimal places. What do you notice? How is F2 related to F (other than squaring F)? [1]

12. Use a calculator to evaluate 1/F and write down its value, correct to 5 decimal places. What do you notice? How is 1/F related to F (other than taking the reciprocal of F)? [1]

Section C: Perfect Rectangle and Golden Ratio in Real Life

13. Many buildings and structures are shaped like the Perfect Rectangle or linked to the Golden Ratio. For example, the diagrams below show a photo of the Parthenon at Athens. Measure the length and width of the rectangle in the first diagram below (must take into account the height of the original roof) and calculate the ratio of its length to its width. What type of rectangle do you get? [2]

14. Another example is the Great Pyramid built by the Egyptians in Giza (see below). Calculate the ratio of the “distance up the middle of one side” to the “distance from the edge to the centre”. What do you get? [2]

15. Nature also has some interesting shapes. A nautilus is a creature that lives in the sea and it grows to about 25 cm (see photos below). As its shell develops, it creates one of the most fascinating shapes in mathematics. To see how the shell looks like, an arc of a quadrant is drawn with its centre at the bottom left hand corner of S8 (see diagram below).

(a) Draw a similar arc for S7 but with the centre at the bottom right hand corner of S7.

(b) Draw a similar arc for S6 but with the centre at the top right hand corner of S6.

(d) Continue the process until you reach S1. [2]

Further Investigation

16. One of the most important questions in any mathematical investigation that you should ask yourself is: “What else is there for me to investigate?” List one thing related to this topic that you would like to investigate further and if time permits, investigate it. [2]

Question: ______

Answer:

Conclusion

17. Write down one main lesson that you have learnt from this worksheet. [1]

Final Score:

/ 20

Teacher’s Comments (if any):

Mathematical Investigation: The Perfect Rectangle

Topic: Angles, Polygons and Geometric Constructions

Answers & Scoring Rubric

(Marks allocated in square brackets; total marks = 20)

Section A: The Perfect Rectangle

1. No mark for this question because it is a matter of personal preference. The teacher is to collate the results for the class and observe whether most students choose Rectangle C. But don’t tell them the significance of C (Perfect Rectangle – see later) but can tell them the significance of B (shape of A4 paper) and F (square – a special rectangle). It does not matter if the majority does not prefer Rectangle C because they may be so used to A4 paper nowadays that they may prefer Rectangle B. But one of the authors has surveyed some students recently and most of them still prefer Rectangle C.

Q2. [1 mark for middle column] Q4. [1 mark for last column]

Square Sn / Length of Square Ln / Ln / Ln–1 (to 8 s.f.)
S1 / 1 / –
S2 / 1 / 1 / 1 = 1
S3 / 2 / 2 / 1 = 2
S4 / 3 / 3 / 2 = 1.5
S5 / 5 / 5 / 3 = 1.6666667
S6 / 8 / 8 / 5 = 1.6
S7 / 13 / 13 / 8 = 1.625
S8 / 21 / 21 / 13 = 1.6153846

3. Fibonacci numbers or sequence [1]

5. The rectangle is highlighted below: its length is L8 and its width is L7. [1]

Therefore the ratio L8 / L7 is the ratio of the length to the width of this rectangle. [1]

6. Yes [1]

7. See Comments for Q1 above. Teacher can now tell the students the significance of Rectangle C and compare it with their preference. [No mark for this question]

8. 2.8 / 1.75 = 1.6 [1]

9. Yes. Ratio of length of Perfect Rectangle to its width is the Golden Ratio. [1]

10. is the exact value of the Golden Ratio. [1]

Section B: Some Interesting Properties of the Golden Ratio

11. F2 = 2.61803 (to 5 decimal places) = F + 1 [1]

[For your info: ]

12. 1/F = 0.61803 (to 5 decimal places) = F – 1 [1]

[For your info: ]

Section C: Perfect Rectangle and Golden Ratio in Real Life

13. 5.6 / 3.5 = 1.6 » F [1]

Perfect Rectangle [1]

14. 186.4 / 115.2 = » F [1]

Golden Ratio [1]

15. See diagram below. OHT template is on page 9.

[2 marks for correct construction; 1 mark for incomplete construction or minor mistake]

Further Investigation

16. Some suggestions for further investigations: [2] {Full marks for creative or original idea}

ï How do you find the limit of convergence of the ratio Ln/Ln–1?

Ans: One way is to use an Excel spreadsheet. The gist of it is: B4=1, B5=1, B6=B4+B5, and then auto-complete for B7 onwards. For more info, download Fibonacci.xls from http://math.nie.edu.sg/bwjyeo/aa.

ï How do you construct a perfect rectangle with straightedge and compasses?

Ans: See below or Rectangle.gsp (download from http://math.nie.edu.sg/bwjyeo/aa). The gist of it is: ABCD is a square with AB = 1 unit; E is the midpoint of AD so that EC = and AF = AE + EF = = .

Conclusion

17. Some suggestions for main lesson learnt from this worksheet: [1]

ï Perfect Rectangle is the most pleasing to the eye.

ï Perfect Rectangle and Golden Ratio occur in man-made structures and in nature.

ï Mathematics occurs in nature.

Shell of Nautilus