6th Grade Math Summer Reading - Vignone
The Adventures of Penrose - The Mathematical Cat
By Theoni Pappas
Penrose, a cat with a knack for math, takes children on an adventurous tour of
mathematical concepts from fractals to infinity. When the fractal dragon jumps off the computer
screen and threatens to grow larger than the room itself, Penrose must find out if fractal patterns
can work in reverse, getting smaller instead of larger. Penrose the cat explores and experiences
a variety of mathematical concepts, including infinity, the golden rectangle, and impossible
figures.
Your summer reading assignment is due the first day of school to Mrs. Vignone. Your
summer reading assignment will count as 10% of your grade for the first trimester. For every
day the assignment has not be turned in on the appropriate due date, a penalty of five points
will be dropped per day. Please do your summer reading assignment and have your assignment
prepared to turn in when you return on the first day. Manage your time and do not wait until
the last week of August to do all your reading assignments. If you have any questions, feel free
to email me at jvignone@mdacademy.org.
You may use a composition notebook to complete both assignments for this reading, as
this will be easier as you are reading…
Pre-reading Exercises:
1. Look up the following terms:
a. Fractal:
b. Infinity:
c.
Binary System
d. Rectangle:
i. The Golden Rectangle:
e. Impossible figures:
f.
g.
Fibonacci:
Polyhedral:
h. Abacus:
6th Grade Math Summer Reading - Vignone
The following exercises can be done after your completion of the book:
For each of the following chapters in the book, which are listed below, you either have a
choice of (1) explaining one mathematical concept you read about in the chapter, or (2)
creating a word problem related to the chapter read with the solution. You must include the
mathematical operation read throughout the chapter, as well as utilizing the characters from
that chapter.
Relax! You can mix and match – picking to explain the concept of one chapter, and
creating a word problem for another. This assignment is definitely challenging, but I encourage
you to read this amazing book with interest and challenge yourself while reading it and have
fun with it! I assure you; you will learn exciting new mathematical concepts and enjoy the
illusions and tricks within the book, as well as looking at mathematics in a new light! Try to do
the problems given with each chapter and check your answers at the back of the book.
After the listing of chapters, I have provided an example for each.
Table of Contents:
2 Penrose meets the O’s and 1’s
4 Penrose is captured by the numbers
8 Penrose discoveries mathematical stars
10 Penrose discovers pancake world
12 Penrose meets the fractal dragon
16 Penrose discovers the mathematics of soap bubbles
20 Penrose learns the truth about infinity
22 Penrose meets Fibonacci Rabbit
26 Penrose watches the puzzling egg hatch
28 The polyhedral connects with Penrose
32 The Golden Rectangle dazzles Penrose
36 A square becomes a bird right before Penrose’s eyes
40 Penrose meets Mr. Abacus
44 Penrose discoveries the mystery of the triangle
48 Penrose meets the Tangramians
52 Penrose solves the case of the missing square
55 Penrose sees into the invisible nano-world
58 Penrose loves the games numbers play
62 Penrose flips over the Mobius strip
6th Grade Math Summer Reading - Vignone
66 Penrose discovers mathematics in the forest
70 Penrose meets Lo-Shu
74 Mauritus teaches Penrose a tessellation trick
78 Penrose discovers Penrose
82 Penrose tangles with the impossible figures
86 Penrose learns more number antics
Examples:
2 Penrose meets the O’s and 1’s: The Concept
The Binary System uses zero and ones, or bits to represent numbers. We use the
Decimal System, which is based on place value of 1, 10, 100, 1000, 100000, etc. The
Metric System is based on a value of 10, and place value of 1000 (kilo) , 100 (hector), 10
(Deka), 1, .1 (deci), .01 (centi), .001 (milli). The binary system was used during ancient
times, and it represented numbers with 0’s and 1’s, with place value also, example
below.
Decimal
Binary
0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
10
1010
4 Penrose is captured by the numbers: Word Problem
(Square root four) √4 walked up to Penrose and said, “The square root of me is
three”. Penrose stretched and raised his tail in the air and said, “ I do not think
that is correct.” Who is correct? What is the square root of four?
Solution: Penrose is correct. Two is solution to √4 (2 * 2 = 4).
6th Grade Math Summer Reading - Vignone
Mrs. Vignone Summer Reading:
Pre-reading Exercises:
2. Look up the following terms:
a. Fractal: a curve or geometric figure, each part of which has the same statistical
character as the whole. Fractals are useful in modeling structures in which
similar patterns recur at progressively smaller sales, and in describing partly
random or chaotic phenomena such as crystal growth.
b. Infinity: a number greater than any assignable quantity or countable number.
c.
Binary System:
a system in which information can be expressed by
combinations of the digits zero and one. (10101010).
d. Rectangle: a plane figure with four straight sides and four straight angles.
i. The Golden Rectangle: The Golden rectangle has been known since
antiquity as one having a pleasing shape, and is frequently found in art
and architecture as a rectangular shape that seems 'right' to the eye. It is
mentioned in Euclid's Elements and was known to artists and philosophers
such as Leonardo da Vinci.
e. Impossible figures:
the figure uses pictorial rules to create the illusion
of three dimensions, but then breaks some of these rules to make the object impossible
to construct.
f.
Fibonacci: In mathematics, the Fibonacci numbers are the numbers in the
following sequence:
6th Grade Math Summer Reading - Vignone
By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent
number is the sum of the previous two. Some sources omit the initial 0, instead
beginning the sequence with two 1s.
The bee ancestry code
Fibonacci numbers also appear in the description of the reproduction of a
population of idealized bees, according to the following rules:

If an egg is laid by an unmated female, it hatches a male.

If, however, an egg was fertilized by a male, it hatches a female.
Thus, a male bee will always have one parent, and a female bee will have two.
If one traces the ancestry of any male bee (1 bee), he has 1 female parent (1
bee). This female had 2 parents, a male and a female (2 bees). The female had
two parents, a male and a female, and the male had one female (3 bees). Those
two females each had two parents, and the male had one (5 bees). This sequence
of numbers of parents is the Fibonacci sequence.[50]
g.
Polyhedral:
A polyhedron (plural polyhedral or polyhedrons) is
a geometric solid in three dimensions with flat faces and straight edges. The
word polyhedron comes from the Classical Greekπολύεδρον, as poly- (stem of
πολύς, "many") + -edron (form of έδρα, "base", "seat", or "face").
h. Abacus: The abacus, also called a counting frame, is a calculating tool used
primarily in parts of Asia for performing arithmetic processes. Today, abacuses
are often constructed as a bamboo frame with beads sliding on wires, but
originally they were beans or stones moved in grooves in sand or on tablets of
wood, stone, or metal. The abacus was in use centuries before the adoption of the
written modern numeral system and is still widely used by merchants, traders and
clerks in Asia, Africa, and elsewhere. The user of an abacus is called an
abacist.[2]
6th Grade Math Summer Reading - Vignone
The following exercises can be done after your completion of the book:
For each of the following chapters in the book, which are listed below, you either have a
choice of (1) explaining one mathematical concept you read about in the chapter, or (2)
creating a word problem related to the chapter read with the solution. You must include the
mathematical operation read throughout the chapter, as well as utilizing the characters from
that chapter.
Relax! You can mix and match – picking to explain the concept of one chapter, and
creating a word problem for another. This assignment is definitely challenging, but I encourage
you to read this amazing book with interest and challenge yourself while reading it and have
fun with it! I assure you; you will learn exciting new mathematical concepts and enjoy the
illusions and tricks within the book, as well as looking at mathematics in a new light! Try to do
the problems given with each chapter and check your answers at the back of the book.
After the listing of chapters, I have provided an example for each.
Table of Contents:
2 Penrose meets the O’s and 1’s
4 Penrose is captured by the numbers
8 Penrose discoveries mathematical stars
10 Penrose discovers pancake world
12 Penrose meets the fractal dragon
16 Penrose discovers the mathematics of soap bubbles
20 Penrose learns the truth about infinity
22 Penrose meets Fibonacci Rabbit
26 Penrose watches the puzzling egg hatch
28 The polyhedral connects with Penrose
32 The Golden Rectangle dazzles Penrose
36 A square becomes a bird right before Penrose’s eyes
40 Penrose meets Mr. Abacus
44 Penrose discoveries the mystery of the triangle
48 Penrose meets the Tangramians
52 Penrose solves the case of the missing square
55 Penrose sees into the invisible nano-world
58 Penrose loves the games numbers play
62 Penrose flips over the Mobius strip
6th Grade Math Summer Reading - Vignone
66 Penrose discovers mathematics in the forest
70 Penrose meets Lo-Shu
74 Mauritus teaches Penrose a tessellation trick
78 Penrose discovers Penrose
82 Penrose tangles with the impossible figures
86 Penrose learns more number antics
Examples:
2 Penrose meets the O’s and 1’s: The Concept
The Binary System uses zero and ones, or bits to represent numbers. We use the
Decimal System, which is based on place value of 1, 10, 100, 1000, 100000, etc. The
Metric System is based on a value of 10, and place value of 1000 (kilo) , 100 (hector), 10
(Deka), 1, .1 (deci), .01 (centi), .001 (milli). The binary system was used during ancient
times, and it represented numbers with 0’s and 1’s, with place value also, example
below.
Decimal
Binary
0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
10
1010
4 Penrose is captured by the numbers: Word Problem
(Square root four) √4 walked up to Penrose and said, “The square root of me is
three”. Penrose stretched and raised his tail in the air and said, “ I do not think
that is correct.” Who is correct? What is the square root of four?
Solution: Penrose is correct. Two is solution to √4 (2 * 2 = 4).