Golden Pentagon

AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC) AIMING HIGH

GOLDEN PENTAGON In each of these diagrams of a regular pentagon find the ratio of the length shown in � (�√�) red to the length in blue in terms of the Golden Ratio � = � Find all the angles in the diagram. Let AE = 1 unit and BE = x units. Which triangles are isosceles? Which triangles are similar? Use similar triangles to give an equation for x and solve the equation.

Prove = � Prove = = �

Prove �� = � ��

Prove �� = ∅ ��

HELP You may find it helpful to mark all the angles of 36o in one colour, angles of 54o in another colour and angles of 72o in a third colour. This will help you to see which triangles are isosceles, and also to see which pairs of triangles are similar. Write the ratios in terms of x and 1 + x and simplify the expressions to give quadratic equations. For some of these proofs you will need to manipulate surds.

NEXT Using the same diagram write down the value of cos 36o in terms of √5.

NOTES FOR TEACHERS SOLUTION The angles of a regular pentagon are 108° so all the angles can be found. The edges of the regular pentagon are

36o 36o 36o 1 unit and BE = x units.

1 1 Triangle ABR is isosceles so BA = BR = 1 x -1 x -1 and RE = x – 1 units. o o o o B 36 108 72 72 x -1 E Triangles ABP and ARE are congruent 36o 72o 36o isosceles triangles. AR = RE = AP = x – 1.

U 1. Triangles ABE, ABP and ARE are similar (108o, 36o, 36o) hence

BE/AE = AB/AP � 1 36o 36o = 1 � − 1 36o

Hence x2 - x - 1 = 0 and this quadratic equation has solutions

� = (±√)

As the lengths must be positive � = (√) .

1. BE/AE = � = √ the Golden Ratio.

2. BE/BR = BE/BS because triangle BRS is isosceles. = � = the Golden Ratio ∅. � Alternatively, using the fact that triangles AEB and SBE are congruent: BE/BR = BE/BS = BE/AE = ∅.

3. AE/EU= � which we have proved equal to � the �� Golden Ratio ∅.

4. AP/PR = �� (��) � (√��) (√��)(�√�) (��√�) = � = = = 1 + √5 = � � (�√�)(�√�) � � (�√�) By similar triangles: AP/PR = AE/EU = ∅ the Golden Ratio because triangles AEU and APR are similar (36o,72o,72o) NEXT SOLUTION

o o cos 36 = sin54 = x = (1 + √5) so 2cos36o= ∅ the Golden Ratio

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DIAGNOSTIC ASSESSMENT This should take about 5–10 minutes. Write the question on the board, say to the class: “Put up 1 finger if you think the answer is A, 2 fingers for B, 3 fingers for C and 4 fingers for D”. 1. Notice how the learners respond. Ask a learner who gave answer A to explain why he or she gave that answer. DO NOT say whether it is right or wrong but simply thank the learner for giving the answer. 2. It is important for learners to explain the reasons for their answers. Putting thoughts into words may help them to gain better understanding and improve their communication skills. 3. Then do the same for answers B, C and D. Try to make sure that learners listen to these reasons and try to decide if their own answer was right or wrong. 4. Ask the class to vote for the right answer by putting up 1, 2, 3 or 4 fingers. Notice if there is a change and who gave right and wrong answers. The correct answer is: B and G – all squares are similar to each other.

Possible misconceptions: Some learners will just guess. Encourage them to look at the square grid and

to decide if the proportions are the same. This activity is about the Golden Ratio. All Golden Rectangles https://diagnosticquestions.com have edges in the ratio Ø: 1 where Ø = (√) https://diagnosticquestions.com

Why do this activity? This activity offers a blend of algebra and geometry suitable for 14 year olds and older students. There is scaffolding to guide students to solve the problem using the solutions to a quadratic equation.

Learning objectives In doing this activity students will have an opportunity to: • review and deepen their knowledge and understanding of similar triangles; • review and deepen their knowledge and understanding of ratios; • review and deepen their knowledge and understanding of how to solve a quadratic equation; • review and deepen their knowledge and understanding of surds.

Generic competences In doing this activity students will have an opportunity to: • develop problem solving skills; • develop visualization skills; • make connections between different topics and use algebra to solve a problem in geometry.

Suggestions for teaching Start with the diagnostic quiz and discuss the different ratios between the lengths of the long and short edges of the rectangles. Then show this painting of the Last Supper, painted by the Spanish artist Salvador Dali in 1955. It is designed to show the Golden Rectangle and the Golden Ratio. Notice also the pentagonal windows that Dali used because of their golden proportions.

The yellow frame is a golden rectangle split into a square with Christ at the lower left hand corner, and another rectangle that is also a golden rectangle. If you measure the lengths of the edges you will find that both rectangles have the proportions 1.62 : 1 ( The golden ratio is 1.618 to 3 decimal places). Use the 1 – 2 – 4 – more teaching strategy, first asking students to work individually, then in pairs, then to compare their findings in groups of four students and finally have a plenary where students give presentations on the work of their group of four, to the whole class. The easiest way to solve the problem is to use similar triangles and the ratios of lengths which lead to the quadratic equation x2 - x - 1 = 0

Key questions • have you found all the angles? • which triangles are isosceles? • have you listed all the pairs of similar triangles? • how would you describe the symmetries of the pentagon/pentagram diagram? • what happens if you draw a pentagram inside the small inner pentagon? Can you repeat this process again and again on a smaller and smaller scale? Try it.

Follow up Elephant Dreaming https://aiminghigh.aimssec.ac.za/years-8-12-elephant-dreaming/ One Step Two Steps https://aiminghigh.aimssec.ac.za/years-7-10-one-step-two-steps/ Fibonacci’s Rabbits https://aiminghigh.aimssec.ac.za/years-9-to-13-fibonaccis-rabbits/ Sheep Talk https://aiminghigh.aimssec.ac.za/years-7-12-sheep-talk/ Great Pyramid https://aiminghigh.aimssec.ac.za/years-11-12-great-pyramid/

Go to the AIMSSEC AIMING HIGH website for lesson ideas, solutions and curriculum links: http://aiminghigh.aimssec.ac.za Subscribe to the MATHS TOYS YouTube Channel https://www.youtube.com/c/mathstoys Download the whole AIMSSEC collection of resources to use offline with the AIMSSEC App see https://aimssec.app or find it on Google Play.

Note: The Grades or School Years specified on the AIMING HIGH Website correspond to Grades 4 to 12 in South Africa and the USA, to Years 4 to 12 in the UK and school years up to Secondary 5 in East Africa. New material will be added for Secondary 6. For resources for teaching A level mathematics (Years 12 and 13) see https://nrich.maths.org/12339 Mathematics taught in Year 13 (UK) & Secondary 6 (East Africa) is beyond the SA CAPS curriculum for Grade 12 Lower Primary Upper Primary Lower Secondary Upper Secondary Approx. Age 5 to 8 Age 8 to 11 Age 11 to 15 Age 15+ South Africa Grades R and 1 to 3 Grades 4 to 6 Grades 7 to 9 Grades 10 to 12 East Africa Nursery and Primary 1 to 3 Primary 4 to 6 Secondary 1 to 3 Secondary 4 to 6 USA Kindergarten and G1 to 3 Grades 4 to 6 Grades 7 to 9 Grades 10 to 12 UK Reception and Years 1 to 3 Years 4 to 6 Years 7 to 9 Years 10 to 13