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Week\Day
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Monday
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Wednesday (Submit no later than by Friday Noon).
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| 1 |
1-20 None
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1-22 Reading |
| 2 |
1-27 Reading
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1-29 Reading
I. Using the seven pieces of the tangram puzzle
create the following figures and indicate with a
sketch your solution.
A. A rectangle and B. A right triangle.
II. Suppose that the square made using the seven
tangram pieces has a side of length 4.
A. Find the area of each
of the seven pieces from the figure without algebra!
Explain your responses.
B. Find the length of the sides
of each of the seven pieces.
Explain your responses.
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| 3 |
2-3 Reading
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2-5 Reading
Ask your adviser or an
instructor you know in your major the following
questions. Submit a summary report of the
interview responses.
A. How do you define
mathematics? (or... What is
mathematics?)
B. Can you give one
example of how mathematics is used in your
discipline? Ask for an explanation of the
response.
C. Has there been a paradigm shift in your
discipline? Ask for an explanation of the
response. |
4
| 2-10 Reading
| 2-12 Reading
Classifications by symmetry:
It is often useful to classify visual
objects by their symmetries. For example, the letter "T"
as it appears on this page has only a reflection
symmetry determined by a vertical line, whereas the
letter "I" has two reflection symmetries and one
rotational symmetry of order 2 (a half turn). Consider
the following list of letters with the specific font style:
A
B C D E F G H I J K L M N
O P Q R S T U V W X Y Z
1. Group the letters together in
different classes determined by the number and types of
symmetries they have as printed on this page. [It is up
to you to determine the appropriate classes.]
2. Explain your classifications briefly by describing the common symmetry of the letters in each class.
Tessellations with triangles:
3. Create a tessellation of the plane using a single isosceles triangle as the template. Is this pattern unique?
If your answer is yes, explain what makes it unique.
If your answer is no, create a distinct tessellation using the same
triangle and explain how you distinguish it from your first
tessellation. |
| 5 |
2-17
Read
Tessellation Day: Wear to class
clothing that has a tiling pattern on it.
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2-19 Read
Submit the following assignment on line or on paper:
- The Sphere has brought the Torus to visit his new friend the
Square in Flatland. Describe two possible successions of different
planar shapes the Torus might appear as while passing through
Flatland. Compare how would these experiences would differ from what the
square saw when the sphere passed through Flatland. You may draw the
sequences as they would be seen in Flatland but you need to describe
these drawings verbally in your response.
- Write a short essay that discusses
one of the following two statements about the relation of perception to
beliefs about truth and reality in the story of Flatland and the Cave.
Use specific information from the stories to illustrate your discussion.
A. If what I believe is true about reality agrees with my perception, then my belief must be correct.
B. If a person says something about reality that does not match what I
believe is true about reality, then that statement must be incorrect.
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6
| 2-24 Reading
Project proposal due by 5 pm. | 2-26
Portfolio Sample(s) due. |
| 7 |
3-3 Reading Continued.
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3-5 Investigating Shadows and Symmetry
1.The sphere is still trying to explain some of the
features of the torus to a Flatlander. This time the
sphere has decided to show the Flatlander different
shadows that are cast by the projection of the torus
onto Flatland.
A.
Draw three different shadows that the torus could cast.
B.
Do you think it is possible to make a torus that would
cast a shadow on Flatland that covers a
circle and its interior exactly? If so, describe some of the
features of such a torus. If not, give some reasons for
your belief. In other words, is it possible that a
Flatlander might mistake a torus for a sphere based on
the shadow it casts?
2.
We are trying to describe the cube to a
Flatlander, this time using the transformation of the
framework made up of the edges and vertices of the cube
onto the plane by central projections from a single light source.
Show
the image of the cube on the plane under the following
projection situations:
A.
One square of the cube is in Flatland and the center of
projection is above the cube directly over the center of
the cube.
B.
Only one edge of the cube is in Flatland and the center
of projection is above the cube directly over the center
of the edge in Flatland.
C.
One square of the cube is in Flatland and the center of
projection is higher than the cube and not directly over
any part of the cube.
3. Write a brief
essay discussing a situation in the contemporary world
where indirect experiences are used to make
observations. How are the observations made? How are
they connected to the actual situation? Do you think
the inferences made from the observations are always
accurate? [3 or 4 paragraphs should be adequate.]
4. Imagine you are a Flatlander talking to a Linelander. Write an explanation of
symmetry to a Linelander from the point of view
of a Flatlander.
In your explanation discuss the kinds of symmetry that are possible
in Lineland.
Discuss which Flatland symmetries (if any)
would you associate with Lineland symmetries? Explain the
association briefly.
Here are some terms you might use in your discussion:
Reflection Rotation Translation
Orientation
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8
| 3-10 Reading TBA
| 3-12
1.
The fourth dimension can be used to visualize and keep
track of many things involving four distinct qualities
that can be measured in some fashion.
A.
For example, a 13 card bridge hand can be thought of as
a point in four dimensions where the coordinates
represent the number of cards of each suit present in
the hand. In this context the point with coordinates (
2, 4, 6, 1) might represent a hand with 2 clubs, 4
diamonds, 6 hearts and 1 spade.
Using
this convention discuss briefly the following
representations of bridge hands: (0, 0, 0, 13),
(0, 0, 6, 7), (3, 3, 3, 4).
Suppose
a bridge hand is represented by the point with
coordinates (x, y, z, w).
Explain
why x + y + z + w = 13.
B.
Describe another context where four dimensions can be
used in representing some features of the context.
2.
Hypercubes in Higher Dimensions.
The
16 vertices of the 4-dimensional hypercube can be
described by the collection of ordered quadruples
(a,b,c,d) where the numbers a, b, c, and d are
either 0 or 1.
Write
a description of the vertices of the 5-dimensional
hypercube. How many vertices does the 5-dimensional
hypercube have? How many vertices does the 6-dimensional
hypercube have? How many vertices does the
10-dimensional hypercube have? What can you say about
the vertices for the hypercube of dimension N.
3. Find and "copy" 3 different world maps.
Describe how each map deals with lines of longitude,
latitude, and the poles.
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9
| 3-17 No Class - Spring
Break
| 3-19 no class |
10
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3-24 Read
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3-26 Read - work on Portfolios
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11
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3-31 No class CC Day
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4-2
1. Find a Sudoku puzzle and show how to add 4 numbers to the puzzle
solution. "Prove" that the 4 numbers you have added must be a part of
the solution to the puzzle. [Note: Since you can find a solution to most
puzzles, the assignment here is to provide the explanation/ proof.
Without the proof you will not be given credit for this work.]
2. a. Following the proof that the square root of 3 is not a rational
number, give a proof that the square root of 5 is not a rational number.
b.Discuss why the outline of the proof to show that the square root 3
and 5 is not a rational number will work for other numbers as long as
they are not perfect squares like 4, 9, 16, 25, 36, ...
3. Make a torus from a single sheet of material using the pattern
 .
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12
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4-7 Reading
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4-9 Work on Portfolio Entries,
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13
| 4-14 Reading. | 4-16
1. Classifications by topological equivalence:
It is often useful to classify visual
objects using topological equivalence. For example, the
letter "T " as it appears on this
page has a single point which separates the letter into
three disconnected pieces, whereas the letter "I "
has only two disconnected pieces for any single point.
Group the following letters as printed
on this page together in different classes
determined by topological equivalence. Indicate what
distinguishes the resulting classes from each other. [It
is up to you to determine the appropriate classes.]
A
B C D E F G H I J K L M N O P Q R S T U V W X Y Z
2. G-Coding the rational numbers.
a.Write an explanation for the following statement: "Every positive natural number is a rational number."
Consider the following G-code for positive rational numbers of the form a/b where a and b are positive counting numbers:
The number a/b is G-coded by the natural number
2a 3b.
b. i.Find the rational numbers G-coded by the number 2, 4, 6, 9, 12, 16.
ii. List the next ten numbers that G-code rational numbers.
iii. Explain why 1000000 is not on the list of numbers G-coding rational numbers.
c. Write an explanation for the following statement:
"There are fewer positive rational numbers than there are natural numbers."
d. Discuss the following statement: "The statements in part a and c contradictory."
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14
| 4-21
| 4-23 Portfolios are due!
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15
| 4-28 Reading
| 4-30 Continue to work on Projects- Due Monday May 5th by noon.
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16
| 5-5 Projects due by Noon.
Reading
| 5-7
Final Assignment.[May be
submitted on paper or on-line until 5-13]
I. Surfaces:
A.Describe 3 distinct physical objects which have topologically
different closed surfaces. For each surface discuss the number of holes
and the Euler characteristic for that surface.
B. Draw two different networks on a map on the mobius band: one that has
five regions that requires five colors. and one with six regions that
requires six colors. Explain your why each of your sketches is
correct. Compute the Euler characteristic for these networks.
II.Over the term we have covered many topics in class
and through the readings. Choose two topics we have
studied for examples in writing a paper (1-3 pages)
discussing one of the following statements:
A. The study of
mathematics is powerful in showing what can be done and also what cannot be done.
B. The amazing thing about
mathematics is how it is able to turn even the
simplest things into abstractions and that very
abstract things can often be coded using just the counting counting
numbers. |
| BELOW
THIS LINE ALL
ASSIGNMENTS ARE TENTATIVE! |
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Unused
old Assignments!
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