Name ________________ !
Date _______!
Class # ______!
Block ______ Goals
Intro to Calculus
Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
Sangaku 1
No visitor to a foreign country has failed to experience the fascination and unease that accompanies an encounter with unknown traditions and customs. Some visitors attempt to overcome their fears, while the majority quickly retreats to familiar shores, and in this lies a distinction: Those who embrace culture shock are travelers; those who do not are tourists.
The most profound culture shock comes about when one is confronted by a different way of thinking. Most of us can hardly imagine walking into a Western church or cathedral to encounter a stained glass window covered by equations and geometrical figures. Even if we can conceive of it, the thought strikes us as alien, out of place, perhaps sacrilegious. Yet for well over two centuries, Japanese mathematicians— professionals, amateurs, women, children— created what was essentially mathematical stained glass, wooden tablets adorned with beautiful geometric problems that were simultaneously works of art, religious offerings, and a record of what we might call "folk mathematics." The creators of these sangaku— a word that literally means "mathematical tablet"— hung by the thousands in Buddhist temples and Shinto shrines throughout Japan, and for that reason the entire collection of sangaku problems has come to be known as "temple geometry", sacred mathematics. — Tony Rothman, Sacred Mathematics Problem 1— Suanfa Tong Zong (Systematic Tretise on Mathematics) by Chang Da-Wei, 1592 Find the radius of a circle that is inscribed in a right triangle whose short sides are 36 and 27.
Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Name ________________ !
Date _______!
Class # ______!
Intro to Calculus Sangaku 2
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
Problem 1 A square is inscribed in a right triangle whose short sides are in the ratio of 1:2. What is the length of the side of the square in terms of the length of the shortest side of the circumscribed triangle?
Problem 2 A semi-circle is inscribed in a right triangle such that its diameter lies on the shortest side of the triangle. If the short sides of the triangle are 3 and 4, what is the difference between the length of the diameter and the length of the shortest side of the triangle?
Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Name ________________ !
Date _______!
Class # ______!
Block ______ Goals
Intro to Calculus
Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
Sangaku 3
Fujita Sadasuke (1734-1807) Two circles of radii a and b are tangent to each other as well as tangent to line l as shown in the diagram below. Show that DE = 2 ab . Justify with clear and complete work.
a
b l
D
E
Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Name ________________ !
Date _______!
Class # ______!
Intro to Calculus Sangaku 4
Sangaku of the Katayamahiko Shrine The sangaku below is one of 16 puzzles on a dragon-framed tablet which was dedicated by Irie Shinjun in 1873 to the Katayamahiko shrine of Murahisagun Okayama city.
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
Two circles of radius r are tangent to the line l. A square of side t touches both circles. Find t in terms of r.
Scoring Guide Define variables and label drawing. (4 pt.s) Clearly and convincingly guide the reader to the solution. (12 pt.s) Correctly answer the question. (4 pt.s) Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Name ________________ !
Intro to Calculus Sangaku 51
算額
Date _______!
Class # ______!
Block ______ Goals
The following problem was originally proposed in 1800 by Kobata Atsukuni, a student of the Aida school, and presented on a tablet to the Kanzeondo temple of Toba castle town.
Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
A big circle of diameter 2R = 100 inscribes a large and small equilateral triangle. Find the side q in terms of R of the small equilateral triangle ABC if A is the midpoint of one side of the large triangle.
R
A q B
C Scoring Guide Define variables and label drawing. (4 pt.s) Clearly and convincingly guide the reader to the solution. (12 pt.s)
1
This problem is based on material from Sacred Geometry, F. Hidetoshi and T. Rothman, Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Correctly answer the question. (4 pt.s)
Name ________________ !
Intro to Calculus Sangaku 6
算額
Date _______!
Class # ______!
This problem is from the Katayamahiko shrine Sangaku in Murahisagun Okayama city. It was dedicated by Irie Shinjun in 1873.
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
A circle of radius r inscribes three circle of radius t. Find t in terms of r.
Scoring Guide Define variables and label drawing. (4 pt.s) Clearly and convincingly guide the reader to the solution. (12 pt.s) Correctly answer the question. (4 pt.s) Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Name ________________ !
Intro to Calculus Sangaku 71
算額
Date _______!
Class # ______!
Kobayashi Syouta proposed this problem on a table that was hung in the Shimizu shrin, Nagano prefecture, in 1828.
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
A big square of side a, encloses a smaller square of side 2r, and a circle of radius r . The circle is tangent to two sides of the big square and is tangent to the small square at a corner (as shown in the diagram). Find r in terms of a.
Scoring Guide
Extra Challenge Use Geogebra to make this sangaku so that all of the objects move correctly as the result of changing the length of side a. 1
Based on the work of F. Hidetoshi and T. Rothman Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Define variables and label drawing. (4 pt.s) Clearly and convincingly guide the reader to the solution. (12 pt.s) Correctly answer the question. (4 pt.s)
Name ________________ !
Intro to Calculus Sangaku 81
算額
Date _______!
Class # ______!
Watanabe Kiichi proposed this problem, which is on the Sangaku of teh Abe no Monjyuin temple in the Fukushima prefecture. The tablet contains 21 problems. It was hung in 1877 by the students of the Sakuma Y!ken.
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
An equilateral triangle with side t , a square of side s , and a circle touch each other in a right triangle ABC with vertical side a. Find t in terms of a.
C
a
s
t
A
B
Extra Challenge Use Geogebra to make this sangaku so that all of the objects move correctly as the result of changing the length of side a. 1
Based on the work of F. Hidetoshi and T. Rothman Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Scoring Guide Define variables and label drawing. (4 pt.s) Clearly and convincingly guide the reader to the solution. (12 pt.s) Correctly answer the question. (4 pt.s)
Name ________________ !
Intro to Calculus Sangaku 101
算額
Date _______!
Class # ______!
This problem is from the Shimizu shrine from the sangaku presented in 1828 by Kobayashi Nobutomo.
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
As shown in the figure a small circle of radius b sits on the point of contact between two squares of side 2b that in turn sit on a line. A big circle of radius a is tangent to line l, the small circle, and the corner of the nearest square. Find a in terms of b.
Scoring Guide
Extra Challenge Use Geogebra to make this sangaku so that all of the objects move correctly as the result of changing the radius of the small circle.
Define variables and label drawing. (4 pt.s)
1
Correctly answer the question. (4 pt.s)
Based on the work of F. Hidetoshi and T. Rothman Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Clearly and convincingly guide the reader to the solution. (12 pt.s)
Name ________________ !
Intro to Calculus Sangaku 111
算額
Date _______!
Class # ______!
This is another problem from the sangaku of the Katayamahiko shrine dedicated by Irie Shinjun in 1873. Murahisagun Okayama City.
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
On a circular field of diameter 2r = 100 m, we make four lines of length t such that they divide the circle into five equal areas, S, one of which is a square of side d. Find approximate numerical values for t and d using ! = 3.16 (the common value used for ! during this period in Japan.)
Scoring Guide
Extra Challenge Use Geogebra to make this sangaku so that all of the objects move correctly as the result of changing radius of the circle. 1
Adapted from the work of F. Hidetoshi and T. Rothman in Japanese Temple Geometry Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Define variables and label drawing. (4 pt.s) Clearly and convincingly guide through your reasoning (12 pt.s) Correctly answer the question. (4 pt.s)
Sangaku 11
Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Name ________________ !
Intro to Calculus Sangaku 121
算額
Date _______!
Class # ______!
Block ______ Goals
The tablet from which this problem was taken was hung in 1874 in the Akahagi Kannon temple in Ichinoseki city. Its size is 188 cm by 61 cm. The problem itself was proposed by Sat! Naosue, a thirteen-year-old boy.
Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
Two circles of radius r and two of radius t are inscribed in a square, as shown. The square itself is inscribed in a large right triangle and, as illustrated, two circles of radii R and r are inscribed in the small right triangles outside the square. Show that R = 2t .
r
r t R
t r
Extra Challenge Use Geogebra to make this sangaku so that all of the objects move correctly as the result of changing radius of the circle. 1
Adapted from the work of F. Hidetoshi and T. Rothman in Japanese Temple Geometry Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Scoring Guide Define variables and label drawing. (4 pt.s) Clearly and convincingly guide through your reasoning. (12 pt.s) Correctly answer the question. (4 pt.s)
Sangaku 12
r
r t R
t r
Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Name ________________ "
Intro to Calculus Sangaku 131
算額
Date _______"
Class # ______"
The problem below is originally from the Sangaku of the Sugawara shrine of Ueno city, Mie prefecture. It was hung by Hojiroya Sh!emon in 1854.
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
As shown in the figure below, a square of side c is inscribed in an equilateral triangle of side k. Two smaller squares of sides a and b are inscribed between the equilateral triangle and square c. A smaller equilateral triangle of side d is inscribed within square c and a circle of radius r is inscribed within this triangle. Find b, c, d, k, and r in terms of a.
a b
k
r
d c
Extra Challenge Use Geogebra to make this sangaku so that all of the objects move correctly as the result of changing the length of the side of the smallest square.
1
Adapted from the work of F. Hidetoshi and T. Rothman in Japanese Temple Geometry Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Scoring Guide Define variables and label drawing. (4 pt.s) Clearly and convincingly guide through your reasoning. (12 pt.s) Correctly answer the question. (4 pt.s)
Sangaku 13
a b
k
r
d c
Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Name ________________ !
Intro to Calculus Sangaku 141
算額
Date _______!
Class # ______!
Goals
Here we have a rare example of a problem proposed by a woman, Okuda Tsume. Hung in 1865 at the Meiseirinji temple in Ogaki city, Gifu. In a circle of diameter AB = 2R , draw two arcs of radius R with centers A and B, respectively, and ten inscribed circles. Show that the eight R small circles all have equal radii, t, and show that t = . 6
A
Block ______
Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
B
Extra Challenge Use Geogebra to make this sangaku so that all of the objects move correctly as the result of changing AB .
Scoring Guide Define variables and label drawing. (4 pt.s) Clearly and convincingly guide through your reasoning. (12 pt.s)
1
Adapted from the work of F. Hidetoshi and T. Rothman in Japanese Temple Geometry Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
Correctly answer the question. (4 pt.s)
Sangaku 14
A
B
Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
算額
Name ________________ !
Date _______!
Intro to Calculus Sangaku X+1
Class # ______!
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
This problem has survived on an 1824 tablet from the Gumma Prefecture. !A and !C are tangent to each other at one point and are tangent to the same line. !B is tangent to both !A and !C and is also tangent to the same line. How are the radii of the three circles related?
A
C
B
Critical and Informed Thinkers • Effective Communicators • Collaborative Workers
算額
Name ________________ !
Date _______!
Class # ______!
Intro to Calculus Sangaku X
Block ______ Goals Appreciate mathematics as a human activity with a deep and complex history. Improve ability to formulate and solve problems.
From a 1803 Sangaku found in Gumma Prefecture. The base of an isosceles triangle sits on a diameter of the large circle. This diameter also bisects the circle on the left, which is inscribed so that it just touches the inside of the container circle and one vertex of the triangle. The top circle is inscribed so that it touches the outsides of both the left circle and the triangle, as well as the inside of the container circle. A line segment connects the center of the top circle and the intersection point between the left circle and the triangle. Show that this line segment is perpendicular to the drawn diameter of the container circle 1.
1
When solving this problem I found myself being drawn to making assumptions based on the drawing which, on deeper inspection, turned out to be false. Critical and Informed Thinkers • Effective Communicators • Collaborative Workers