Name : ________________________________ Class : _____________ Date:______________ The Golden Ratio Investigation GOLDEN RATIO IN RECTANGLES The golden ratio is the ratio of length to width of a rectangle that is most pleasing to the eye. A rectangle that has sides in the golden ratio is called a golden rectangle. The ratio of length to width in a golden rectangle is about 8: 5 or 1.6: 1. Examples: A rectangle that has a length of 16 cm and a width of 10 cm can be categorized as a golden rectangle. (*Not drawn to scale) Many ancient buildings are said to have floors or sides which were built in the golden ratio. 1 of 5 GOLDEN RATIO IN THE HUMAN BODY Leonardo’s Conjecture: Leonardo Da Vinci suggested the ratio of certain body measurements is close to the golden ratio. Keeping in mind that we are all different shapes and sizes, let us see if we can find the golden ratio using our body measurements. What to do: 1. Take your body measurement (in mm) as indicated in the following diagrams. Record these measurements and calculate the required ratios as shown by the example below. Your teacher’s data: A. Height Height of navel So, Height: Navel height (1670 ÷ 990) B. Length of hand (fingertip to armpit) Length of forearm (fingertip to elbow) So, arm : forearm (670 ÷ 410) = = = = 1670 mm 990 mm 1670 : 990 1.7 : 1 = 670 mm = 410 mm = 670 : 410 = 1.7: 1 My own data: C. Height Height of navel So, Height: Navel height = = = = 1430 mm 850 mm 1430 mm : 850 mm ____: 1 D. Length of arm = ________ mm (Fingertip to armpit) Length of forearm = ________ mm (Fingertip to elbow) So, arm: forearm = _____ : _____ (Arm divided by forearm) = _____: 1 1430mm 850mm 600mm 380mm 2 of 5 2. Get other 9 measurements from your classmates and fill in the table below. 6A https://binusianorgmy.sharepoint.com/personal/jhobal_sebarrotin_binus_edu/_layouts/15/guestaccess.as px?docid=1991d06d10a3b4e0791ef1f5870ec77c4&authkey=AQgzSz3o0RUHw_baFxmgh Mg&e=wGwRru Height (in mm) Height of navel (in mm) Length of arm (in mm) Length of forearm (in mm) My data Classmate 1: Classmate 2: Classmate 3: Classmate 4: Classmate 5: Classmate 6: Classmate 7: Classmate 8: Classmate 9: *Average *To compute for the average, add aal numbers in each column and then divide by 10. 3. From the averages of the height and the height of navel, Height : Height of navel = _______ : _______ Height divided by Height of navel= _______ : 1 4. From the averages of the length of arm and the length of forearm, Length of arm : Length of forearm Length of arm divided Length of forearm = _______ : _______ = _______ : 1 5. What general rule can you generate from your investigation regarding the golden ratio and Leonardo’s conjecture? (Write in words.) From my investigation regarding the golden ratio and Leonardo’s conjecture, I can say that... 3 of 5 ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 6. Justify the general rule that you have mentioned above by using three more data from other classmates and comment on what you find out. Height of navel (in mm) Height (in mm) Length of arm (in mm) Length of forearm (in mm) Classmate 1: Classmate 2: Classmate 3: Average From the averages of the height and the height of navel, Height : Height of navel = _______ : _______ = _______ : 1 From the averages of the length of arm and the length of forearm, Length of arm : Length of forearm = _______ : _______ = _______ : 1 Compare and comment on the ratio of the additional three sets of data to the general rule you suggested: ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 4 of 5 6A Day/Date Activity/ Expected Output Feb 14 Task Introduction/ Part 1 (Taking measurements)- page 1-3 Feb 16 Questions 3-5 (answering questions on Page 3) Feb 21 Question 6 (answering questions on Page 4) Feb 23 Completion (individually)and uploading in ManageBac Criterion B: Investigating Patterns Level Indicators Level of Achievement (LoA):_______ Task-Specific Clarification The student does not reach a standard described by any of the descriptors below The student does not reach a standard described by any of the descriptors below The student applies, with some guidance, mathematical problemsolving techniques to recognize simple patterns similar to previously seen examples The student measures his/her own body and records the results The student applies mathematical problem-solving techniques to recognize simple patterns similar to previously seen examples, and suggests relationships or general rules The student calculates the ratios of his own data LoA 0 1–2 The student collects data of his/her friends and completes the table The student calculates the ratios of the averages from the tables The student generates the general rule The student applies mathematical problem-solving techniques to recognize based on the data table simple patterns similar to previously seen examples, describes them as relationships or general rules using words The student justifies the general rule The student applies mathematical problem-solving techniques to recognize that he/she generates by using 3 other examples simple patterns similar to previously 3–4 5–6 7-8 5 of 5 seen examples, describes them as relationships or general rules using words, and justify using other examples with guidance from the teacher The comment that the student writes on the ratio when justifying corresponds to the general rule. 6 of 5