Table Of Content6”x9” b3505 Solving Problems in Our Spatial World
b131520055__F97M8.9in8d1d3 2 178035_tp.indd 1 27-Ma2y4-/159/1 9 1 : 487:2:231 A PMM
b3505 Solving Problems in Our Spatial World 6”x9”
Problem Solving in Mathematics and Beyond
Print ISSN: 2591-7234
Online ISSN: 2591-7242
Series Editor: Dr. Alfred S. Posamentier
Distinguished Lecturer
New York City College of Technology - City University of New York
There are countless applications that would be considered problem solving in
mathematics and beyond. One could even argue that most of mathematics in one
way or another involves solving problems. However, this series is intended to be of
interest to the general audience with the sole purpose of demonstrating the power
and beauty of mathematics through clever problem-solving experiences.
Each of the books will be aimed at the general audience, which implies that the
writing level will be such that it will not engulfed in technical language — rather
the language will be simple everyday language so that the focus can remain on the
content and not be distracted by unnecessarily sophiscated language. Again, the
primary purpose of this series is to approach the topic of mathematics problem-
solving in a most appealing and attractive way in order to win more of the
general public to appreciate his most important subject rather than to fear it. At
the same time we expect that professionals in the scientific community will also
find these books attractive, as they will provide many entertaining surprises for the
unsuspecting reader.
Published
Vol. 11 Solving Problems in Our Spatial World
by Guenter Maresch and Alfred S. Posamentier
Vol. 10 A Problem-Solving Approach to Supporting Mathematics Instruction
in Elementary School: A Guide for Parents, Teachers, and Students
by Sheldon Rothman
Vol. 9 The Mathematics Coach Handbook
by Alfred S. Posamentier and Stephen Krulik
Vol. 8 Tools to Help Your Children Learn Math: Strategies, Curiosities, and
Stories to Make Math Fun for Parents and Children
by Alfred S. Posamentier, Gavrielle Levine, Aaron Lieberman and
Danielle Sauro Virgadamo
For the complete list of volumes in this series, please visit www.worldscientific.com/series/psmb
bR3o5k0 T5i_nFgM - .1in1d2d0 5 2- Solving Problems in Our Spatial World.indd 1 2274-M-0a5y--1199 41::5477::5281 PPMM 11205_9789813278035_tp.indd 2 24/5/19 8:23 AM
6”x9” b3505 Solving Problems in Our Spatial World
Problem Solving in Mathematics and Beyond
Print ISSN: 2591-7234
Online ISSN: 2591-7242
Series Editor: Dr. Alfred S. Posamentier
Distinguished Lecturer
New York City College of Technology - City University of New York
There are countless applications that would be considered problem solving in
mathematics and beyond. One could even argue that most of mathematics in one
way or another involves solving problems. However, this series is intended to be of
interest to the general audience with the sole purpose of demonstrating the power
and beauty of mathematics through clever problem-solving experiences.
Each of the books will be aimed at the general audience, which implies that the
writing level will be such that it will not engulfed in technical language — rather
the language will be simple everyday language so that the focus can remain on the
content and not be distracted by unnecessarily sophiscated language. Again, the
primary purpose of this series is to approach the topic of mathematics problem-
solving in a most appealing and attractive way in order to win more of the
general public to appreciate his most important subject rather than to fear it. At
the same time we expect that professionals in the scientific community will also
find these books attractive, as they will provide many entertaining surprises for the
unsuspecting reader.
Published
Vol. 11 Solving Problems in Our Spatial World
by Guenter Maresch and Alfred S. Posamentier
Vol. 10 A Problem-Solving Approach to Supporting Mathematics Instruction
in Elementary School: A Guide for Parents, Teachers, and Students
by Sheldon Rothman
Vol. 9 The Mathematics Coach Handbook
by Alfred S. Posamentier and Stephen Krulik
Vol. 8 Tools to Help Your Children Learn Math: Strategies, Curiosities, and
Stories to Make Math Fun for Parents and Children
by Alfred S. Posamentier, Gavrielle Levine, Aaron Lieberman and
Danielle Sauro Virgadamo
For the complete list of volumes in this series, please visit www.worldscientific.com/series/psmb
Rok Ting - 11205 - Solving Problems in Our Spatial World.indd 1 24-05-19 4:57:58 PM b131520055__F97M8.9in8d1d3 2 378035_tp.indd 2 27-Ma2y4-/159/1 9 1 : 487:2:231 A PMM
b3505 Solving Problems in Our Spatial World 6”x9”
“7x10” b3505 Solving Problems in Our Spatial World FA
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Preface
Library of Congress Cataloging-in-Publication Data
Names: Maresch, Guenter, author.
Title: Solving problems in our spatial world / by Guenter Maresch (University of Salzburg, Austria)
and Alfred S Posamentier (City University of New York, USA).
Description: New Jersey : World Scientific, [2019] | Series: Problem Solving in Mathematics
and Beyond ; volume 11 | Includes bibliographical references.
Most people, when they hear the word geometry, tend to recall their study of the subject
Identifiers: LCCN 2019009733 | ISBN 9789813278035 (hc : alk. paper)
during school years. Of course, it is expected that when you think back to the study of
LC record available at https://lccn.loc.gov/2019009733
geometry, you will recall some of the relationships that dominated this study. Probably,
the most often recalled relationship is the Pythagorean Theorem. Ambitious teachers
may even have gone through the enriching experience of exposing students to several
British Library Cataloguing-in-Publication Data
proofs of this famous theorem — of which today the number of these exceeds 400.
A catalogue record for this book is available from the British Library.
Although the study of geometry in high school has focused mostly on proving geo-
metric relationships, this curriculum was limited largely to plane geometry. However,
we live in a three-dimensional (3D) world and so it is appropriate that we investigate
how we see things presented in two dimensions when they are actually representing
Copyright © 2019 by World Scientific Publishing Co. Pte. Ltd. three dimensions. This is the focus of this book. Our goal is to train the reader in an
entertaining fashion to be able to visualize 3D representations in two dimensions, that
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
is, on a planar surface. Since education should not be a “spectator sport”, you will be
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher. asked to become an active participant in this extended training experience. We present
this activity in a problem-solving mode so that the reader remains motivated and active.
You are about to embark on an unusual approach to solving problems — one which
will enhance your geometric spatial ability. The 45 problems — including more than
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
120 tasks — are arranged in a very special and well-balanced order, which have been
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher. tested with a variety of populations. Both students and adults can enhance their spatial
intelligence by working with the problems presented in this book. More importantly,
they have been compiled according to specific scientific criteria.
We invite you to immerse yourself in the fascinating world of geometry and spatial
For any available supplementary material, please visit
ability — either individually or in small groups, either as challenges or play problems.
https://www.worldscientific.com/worldscibooks/10.1142/11205#t=suppl
These are in the form of geometric problems that will leave you with a more powerful
command to tackle geometric tasks.
Desk Editors: V. Vishnu Mohan/Rok Ting Enjoy your journey through the book.
Typeset by Stallion Press
Email: enquiries@stallionpress.com
Printed in Singapore
v
b3505_FM.indd 5 29-03-2019 14:45:54
bR3o5k0 T5i_nFgM - .1in1d2d0 5 4- Solving Problems in Our Spatial World.indd 2 2274-M-0a5y--1199 41::5477::5282 PPMM
6”x9” b3505 Solving Problems in Our Spatial World
“7x10” b3505 Solving Problems in Our Spatial World FA
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Preface
Library of Congress Cataloging-in-Publication Data
Names: Maresch, Guenter, author.
Title: Solving problems in our spatial world / by Guenter Maresch (University of Salzburg, Austria)
and Alfred S Posamentier (City University of New York, USA).
Description: New Jersey : World Scientific, [2019] | Series: Problem Solving in Mathematics
and Beyond ; volume 11 | Includes bibliographical references.
Most people, when they hear the word geometry, tend to recall their study of the subject
Identifiers: LCCN 2019009733 | ISBN 9789813278035 (hc : alk. paper)
during school years. Of course, it is expected that when you think back to the study of
LC record available at https://lccn.loc.gov/2019009733
geometry, you will recall some of the relationships that dominated this study. Probably,
the most often recalled relationship is the Pythagorean Theorem. Ambitious teachers
may even have gone through the enriching experience of exposing students to several
British Library Cataloguing-in-Publication Data
proofs of this famous theorem — of which today the number of these exceeds 400.
A catalogue record for this book is available from the British Library.
Although the study of geometry in high school has focused mostly on proving geo-
metric relationships, this curriculum was limited largely to plane geometry. However,
we live in a three-dimensional (3D) world and so it is appropriate that we investigate
how we see things presented in two dimensions when they are actually representing
Copyright © 2019 by World Scientific Publishing Co. Pte. Ltd. three dimensions. This is the focus of this book. Our goal is to train the reader in an
entertaining fashion to be able to visualize 3D representations in two dimensions, that
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
is, on a planar surface. Since education should not be a “spectator sport”, you will be
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher. asked to become an active participant in this extended training experience. We present
this activity in a problem-solving mode so that the reader remains motivated and active.
You are about to embark on an unusual approach to solving problems — one which
will enhance your geometric spatial ability. The 45 problems — including more than
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
120 tasks — are arranged in a very special and well-balanced order, which have been
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher. tested with a variety of populations. Both students and adults can enhance their spatial
intelligence by working with the problems presented in this book. More importantly,
they have been compiled according to specific scientific criteria.
We invite you to immerse yourself in the fascinating world of geometry and spatial
For any available supplementary material, please visit
ability — either individually or in small groups, either as challenges or play problems.
https://www.worldscientific.com/worldscibooks/10.1142/11205#t=suppl
These are in the form of geometric problems that will leave you with a more powerful
command to tackle geometric tasks.
Desk Editors: V. Vishnu Mohan/Rok Ting Enjoy your journey through the book.
Typeset by Stallion Press
Email: enquiries@stallionpress.com
Printed in Singapore
v
b3505_FM.indd 5 29-03-2019 14:45:54
Rok Ting - 11205 - Solving Problems in Our Spatial World.indd 2 24-05-19 4:57:58 PM b3505_FM.indd 5 27-May-19 1:47:22 PM
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk
b2530_FM.indd 6 01-Sep-16 11:03:06 AM
6”x9” b3505 Solving Problems in Our Spatial World
“7x10” b3505 Solving Problems in Our Spatial World FA
About the Authors
Guenter J. Maresch is currently Associated Professor of
Mathematics Education at the University of Salzburg (Austria).
His research topics are spatial ability, descriptive geometry,
computer-aided design (CAD), new digital media, didactical
principles, and curriculum development. Dr. Maresch studied
descriptive geometry and mathematics at the Vienna University
of Technology (Austria) and after completing his Master’s degree
(1994), he took a position as a teacher of mathematics, descrip-
tive geometry, and information and communication technol-
ogy (ICT) at secondary schools in the province of Salzburg (Austria). Since that time
Dr. Maresch has provided lecturers at more than 250 conferences and teacher in-service
training programs in the form of keynotes, lectures, workshops, and seminars, which
were focused on geometry (spatial ability, curriculum, new media, competence mod-
el(s), CAD trainings, etc.), mathematics (educational standards, competence model,
didactical principles, etc.), ICT (new digital media, learning platforms, computer-aided
design, etc.), principal conferences and school inspector conferences (sustainability of
teacher in-service trainings, etc.), and supervising teacher internship (centralized final
exams, prescientific research paper, pedagogical competences, etc.).
Also, after receiving his Master’s degree, Dr. Maresch completed studies to earn
a teaching certificate for ICT at the local school board of Salzburg (1998). He further
qualified for the Intel Master Teacher Certificate (2001), and completed the Leadership
Academy Program (2008–2009) offered by the Austrian Ministry of Education.
From 1998 to 2007, Dr. Maresch was the Teachers’ Mentor for teachers of descrip-
tive geometry, mathematics, and ICT in the province of Salzburg. He also organized
more than 50 conferences and teacher in-service training programs during this time.
Dr. Maresch completed his Ph.D. studies at the University of Salzburg (Austria) in
2004. His Ph.D. thesis focused on e-learning and computer-aided design as well as the
development of a didactical concept for increasing the quality of geometry instruction.
He continues to provide professional programs in many countries, including: Austria,
vii
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viii About the Authors About the Authors ix
Germany, Finland, Netherlands, Denmark, Spain, Hungary, and South Africa. In 2012, Immediately upon joining the faculty of the City College in 1970, he began to
Dr. Maresch was named Educator of the Year at the worldwide competition of Bentley develop in-service courses for secondary school mathematics teachers, including
Systems. such special areas as recreational mathematics and problem-solving in mathematics.
From 2007 to 2012, Dr. Maresch was a member of the local education management As Dean of the City College School of Education for 10 years, his scope of interest
board and the head of the institute for lifelong learning for upper secondary schools at in educational issues covered the full gamut. During his tenure as dean, he took the
the Salzburg University College of Teacher Education and during this time he orga- School of Education from the bottom of the New York State rankings to the top with
nized more than 700 teacher in-service training programs for teachers and headmasters a perfect NCATE accreditation assessment in 2009. He achieved the same success in
of Austrian lower and upper secondary schools per year. Also during that time he was 2014 at Mercy College, which received both NCATE and CAEP accreditation during
chair of the national final exams at upper secondary schools. Since 2002, he has been his leadership as Dean of the School of Education.
a member of the curriculum board of the Austrian Ministry of Education—responsible In 1973, Dr. Posamentier received his Ph.D. from Fordham University (New York)
for the development of the curricula for the secondary schools. in mathematics education and has since extended his reputation in mathematics education
Dr. Maresch was a lecturer at the University of Salzburg from 1998 to 2014, to Europe. He has been visiting professor at several European universities in Austria,
where he taught courses on spatial ability, descriptive geometry, computer-aided design England, Germany, Czech Republic, and Poland, while at the University of Vienna he
(CAD), new digital media, and curriculum development. From 2014 to 2016, he was was Fulbright Professor (1990). In 1989 he was awarded an Honorary Fellow at the
an Assistant Professor for Mathematics Education and Geometry at the University South Bank University (London, England). In recognition of his outstanding teaching,
of Salzburg. In 2016, Dr. Maresch was awarded the rank of Associate professor for the City College Alumni Association named him Educator of the Year in 1994, and in
Mathematics Education at the University of Salzburg. In summary, he has published 58 2009. New York City had the day, May 1, 1994, named in his honor by the President of
research papers and 12 books for students, researchers and teachers of mathematics and the New York City Council. In 1994, he was also awarded the Grand Medal of Honor
books on geometry for the general readership. from the Republic of Austria, and in 1999, upon approval of Parliament, the President of
the Republic of Austria awarded him the title of University Professor of Austria. In 2003
he was awarded the title of Ehrenbürger (Honorary Fellow) of the Vienna University of
Alfred S. Posamentier is currently Distinguished Lecturer at Technology, and in 2004 was awarded the Austrian Cross of Honor for Arts and Science,
New York City College of Technology of the City University First Class from the President of the Republic of Austria. In 2005 he was inducted into
of New York. Previously, he was the Executive Director for the Hunter College Alumni Hall of Fame, and in 2006 he was awarded the prestigious
Internationalization and Sponsored Programs at Long Island Townsend Harris Medal by the City College Alumni Association. He was inducted into the
University, New York. This was preceded by a five-year period, New York State Mathematics Educator’s Hall of Fame in 2009, in 2010 he was awarded
where he was Dean of the School of Education and tenured the coveted Christian-Peter-Beuth Prize in Berlin, and in 2017 he received the Summa
Professor of Mathematics Education at Mercy College, New York. Cum Laude nemine discrepante Award from Fundacion Sebastian, A.C. in Mexico City.
He is now also Professor Emeritus of Mathematics Education He has taken on numerous important leadership positions in mathematics
at The City College of the City University of New York, and education locally. He was a member of the New York State Education Commissioner’s
former Dean of the School of Education, where he was tenured for 40 years. He is Blue-Ribbon Panel on the Math-A Regents Exams in 2003, and in 2014–15 he was
the author and co-author of more than 60 mathematics books for teachers, secondary on the Commissioner’s Mathematics Standards Committee, which redefined the
and elementary school students, and the general readership. Dr. Posamentier is also a Standards for New York State, and he also served on the New York City schools’
frequent commentator in newspapers and journals on topics relating to education. After Chancellor’s Math Advisory Panel. Dr. Posamentier is a leading commentator on
completing his B.A. degree in mathematics at Hunter College of the City University of educational issues and continues his long-time passion of seeking ways to make
New York in 1964, he took a position as a teacher of mathematics at Theodore Roosevelt mathematics interesting to both teachers, students and the general public — as can be
High School (Bronx, NY), where he focused his attention on improving the students’ seen from some of his more recent books:
problem-solving skills and at the same time enriching their instruction far beyond what
the traditional textbooks offered. During his six-year tenure there, he also developed the 1. Math Makers: The Lives and Works of 50 Famous Mathematicians (Prometheus,
school’s first mathematics teams (both at the junior and senior level). He is still involved 2019)
in working with mathematics teachers and supervisors, nationally and internationally, 2. Tools to Help Your Children Learn Math (World Scientific, 2019)
to help them maximize their effectiveness. During this time, he earned an M.A. degree 3. The Mathematics Coach Handbook (World Scientific, 2019)
at the City College of the City University of New York in 1966. 4. The Mathematics of Everyday Life (Prometheus, 2018)
b3505_FM.indd 8 29-03-2019 14:45:56 b3505_FM.indd 9 29-03-2019 14:45:56
b3505_FM.indd 8 27-May-19 1:47:25 PM
6”x9” b3505 Solving Problems in Our Spatial World
FA b3505 Solving Problems in Our Spatial World “7x10” “7x10” b3505 Solving Problems in Our Spatial World FA
viii About the Authors About the Authors ix
Germany, Finland, Netherlands, Denmark, Spain, Hungary, and South Africa. In 2012, Immediately upon joining the faculty of the City College in 1970, he began to
Dr. Maresch was named Educator of the Year at the worldwide competition of Bentley develop in-service courses for secondary school mathematics teachers, including
Systems. such special areas as recreational mathematics and problem-solving in mathematics.
From 2007 to 2012, Dr. Maresch was a member of the local education management As Dean of the City College School of Education for 10 years, his scope of interest
board and the head of the institute for lifelong learning for upper secondary schools at in educational issues covered the full gamut. During his tenure as dean, he took the
the Salzburg University College of Teacher Education and during this time he orga- School of Education from the bottom of the New York State rankings to the top with
nized more than 700 teacher in-service training programs for teachers and headmasters a perfect NCATE accreditation assessment in 2009. He achieved the same success in
of Austrian lower and upper secondary schools per year. Also during that time he was 2014 at Mercy College, which received both NCATE and CAEP accreditation during
chair of the national final exams at upper secondary schools. Since 2002, he has been his leadership as Dean of the School of Education.
a member of the curriculum board of the Austrian Ministry of Education—responsible In 1973, Dr. Posamentier received his Ph.D. from Fordham University (New York)
for the development of the curricula for the secondary schools. in mathematics education and has since extended his reputation in mathematics education
Dr. Maresch was a lecturer at the University of Salzburg from 1998 to 2014, to Europe. He has been visiting professor at several European universities in Austria,
where he taught courses on spatial ability, descriptive geometry, computer-aided design England, Germany, Czech Republic, and Poland, while at the University of Vienna he
(CAD), new digital media, and curriculum development. From 2014 to 2016, he was was Fulbright Professor (1990). In 1989 he was awarded an Honorary Fellow at the
an Assistant Professor for Mathematics Education and Geometry at the University South Bank University (London, England). In recognition of his outstanding teaching,
of Salzburg. In 2016, Dr. Maresch was awarded the rank of Associate professor for the City College Alumni Association named him Educator of the Year in 1994, and in
Mathematics Education at the University of Salzburg. In summary, he has published 58 2009. New York City had the day, May 1, 1994, named in his honor by the President of
research papers and 12 books for students, researchers and teachers of mathematics and the New York City Council. In 1994, he was also awarded the Grand Medal of Honor
books on geometry for the general readership. from the Republic of Austria, and in 1999, upon approval of Parliament, the President of
the Republic of Austria awarded him the title of University Professor of Austria. In 2003
he was awarded the title of Ehrenbürger (Honorary Fellow) of the Vienna University of
Alfred S. Posamentier is currently Distinguished Lecturer at Technology, and in 2004 was awarded the Austrian Cross of Honor for Arts and Science,
New York City College of Technology of the City University First Class from the President of the Republic of Austria. In 2005 he was inducted into
of New York. Previously, he was the Executive Director for the Hunter College Alumni Hall of Fame, and in 2006 he was awarded the prestigious
Internationalization and Sponsored Programs at Long Island Townsend Harris Medal by the City College Alumni Association. He was inducted into the
University, New York. This was preceded by a five-year period, New York State Mathematics Educator’s Hall of Fame in 2009, in 2010 he was awarded
where he was Dean of the School of Education and tenured the coveted Christian-Peter-Beuth Prize in Berlin, and in 2017 he received the Summa
Professor of Mathematics Education at Mercy College, New York. Cum Laude nemine discrepante Award from Fundacion Sebastian, A.C. in Mexico City.
He is now also Professor Emeritus of Mathematics Education He has taken on numerous important leadership positions in mathematics
at The City College of the City University of New York, and education locally. He was a member of the New York State Education Commissioner’s
former Dean of the School of Education, where he was tenured for 40 years. He is Blue-Ribbon Panel on the Math-A Regents Exams in 2003, and in 2014–15 he was
the author and co-author of more than 60 mathematics books for teachers, secondary on the Commissioner’s Mathematics Standards Committee, which redefined the
and elementary school students, and the general readership. Dr. Posamentier is also a Standards for New York State, and he also served on the New York City schools’
frequent commentator in newspapers and journals on topics relating to education. After Chancellor’s Math Advisory Panel. Dr. Posamentier is a leading commentator on
completing his B.A. degree in mathematics at Hunter College of the City University of educational issues and continues his long-time passion of seeking ways to make
New York in 1964, he took a position as a teacher of mathematics at Theodore Roosevelt mathematics interesting to both teachers, students and the general public — as can be
High School (Bronx, NY), where he focused his attention on improving the students’ seen from some of his more recent books:
problem-solving skills and at the same time enriching their instruction far beyond what
the traditional textbooks offered. During his six-year tenure there, he also developed the 1. Math Makers: The Lives and Works of 50 Famous Mathematicians (Prometheus,
school’s first mathematics teams (both at the junior and senior level). He is still involved 2019)
in working with mathematics teachers and supervisors, nationally and internationally, 2. Tools to Help Your Children Learn Math (World Scientific, 2019)
to help them maximize their effectiveness. During this time, he earned an M.A. degree 3. The Mathematics Coach Handbook (World Scientific, 2019)
at the City College of the City University of New York in 1966. 4. The Mathematics of Everyday Life (Prometheus, 2018)
b3505_FM.indd 8 29-03-2019 14:45:56 b3505_FM.indd 9 29-03-2019 14:45:56
b3505_FM.indd 9 27-May-19 1:47:25 PM
