Heart of Mathematics, New Edition Proposed Interactive Explorations (Mathlets)

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Oct 28, 2013 (3 years and 5 months ago)

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Heart of Mathematics, New Edition

Proposed Interactive Explorations (Mathlets)


Layout and Navigation:


The layout and navigational scheme will be similar to the design of the National Library of Virtual
Manipulatives (
http://matti.usu.edu/nlvm/nav/index.html
) where the instructions, comments, and activities
appear on the right side of the mathlet. The navigation buttons (Home, Print, Save, Instructions, Activities)
will appear along the top.


For

the Heart of Math CD, we will provide a new Save option for many of the explorations. When the
Save button is clicked, the student is asked to provide a name for the current activity and then the exact
state of the applet is saved. When the student re
turns to the program, she can retrieve her work by clicking
on the Activities button. A list of the saved activities will appear at the right and by selecting her activity
the mathlet will be launched and will appear exactly as she left it.


Listed belo
w are the proposed explorations for each chapter. This list reflects our review of the entire text,
including the revised Chapter 7, and the new Chapter 8. We have also carefully considered the list of
suggested changes that was provided by Key College (
11/7/03) and have included those recommendations
when feasible.


Those applets that will include the Save feature are marked by
(S)
.


Chapter 1: Fun and Games


Counterfeit Coin
(revised)

Link:
http://matti.usu.edu/tadriola/server/nav/toc.jsp?sid=key&cid=heart&lid=11



Let’s Make a Deal

(formerly Stick or Switch)

A substantial revision has been developed. Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=12



Fill and Pour
(new)

This applet will support the Fountain of Knowledge story.

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=13




Dodge Ball
(new)

Player one will be the computer and will begin the game by randomly placing X’s and O’s in the first row
.
As they take turns, player two (student) will try to “dodge” the computer’s sequences. We will provide
game boards for column sizes 6
-
10, with the default set to 6, the size in the book.



Suggestion:
The CD could have a video of Ed dropping trou!



Chapter 2: Number Contemplation


Golden Ratios and Rectangles
(revised)

We will modify this mathlet so that the student will have more control over the animation. Instead of just
watching the rectangles and spiral being drawn, the student will step thro
ugh the animation and there will
be short explanations at each step.

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=21



Fibo
nacci Numbers and the Golden Ratio

(new)

The Fibonacci sequence, the fractions of adjacent numbers, and the decimal equivalents will be displayed
in three rows, 10 at a time. The student will be able to adjust the display with arrow buttons to show large
r
and larger Fibonacci numbers along with the fractions and decimals. Not only will the students be able to
see more Fibonacci numbers but they will also be able to see the decimal equivalents get ever closer to



.



Cyber Sue:

Carson Kid’s Little Sister
(revised)

We should modify this one so that we have just 6 colored cards and allow the student to enter a message on
the cards (6 letters), and then step through each perfect shuffle until the original message appears.

Link:



RSA Coding Scheme (S)
(new)

This exploration will demonstrate the coding scheme described in the text (page 101). The trick here will
be to allow the student to use the web to announce the public numbers and to allow friends to encode
simple messages. An

encoded message can be displayed at the same website (available to anyone) and
then the student can use the method to decode the message. Of course the two prime numbers selected will
not be all that large and so it will be possible for others to break t
he code.



Chapter 3: Infinity


Rolling Up the Real Line

We should revise this so that the coordinates for both points are displayed. Link:
http://matti.us
u.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=31









Connect the Dots (S)
(new)

Using the mouse, the student will be asked to “draw” a one
-
to
-
one correspondence between the natural
numbers and the set of points in the plane with positi
ve integer coordinates. The challenge will be to draw
a path that demonstrates a pattern that will establish the one
-
to
-
one correspondence. See the diagram
below. A second problem will be to “draw” a one
-
to
-
one correspondence between the natural number
s
and the set of all points in the plane with integer coefficients.







Dodge Ball
( new)

See the description in Chapter 1.



Triangle of Positive Rational Numbers
(new)

This exploration could provide several nice activities:

Guess the left
-
hand and
right
-
hand rules.

Show that each fraction listed is in reduced form.

Show that the product of the entries in any given row is 1.

Show that all positive rational numbers are listed exactly once.

You can count the set of positive rational numbers by st
arting at the top and going along the rows. One
concern is that this array is not in the book. It will still be 100 times better than Rolling up the Real Line.



Chapter 4: Geometric Gems


Pythagorean Puzzle
(new)

The existing puzzle at
http://matti.usu.edu/nlvm/nav/frames_asid_164_g_3_t_3.html?open=instructions

will be improved and we will provide the puzzle proof found in the text.


Art Gallery

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=42



Platonic Solids and Duals
(revised)

The student will construct the duals for the
cube, tetrahedron, and octahedron (while in wire frame mode)
by clicking on a face to identify the center point. As center points are constructed, the line segments
between them will also be drawn. Finally, the dual can be filled in.

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=43


Slicing Solids with a Plane
(revised)

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=44



Pattern Blocks (S)
(new)

A variety of tilings of the plane can be constructed using the pattern block mathlet.

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=45




Tiling the Plane with the Pinwheel Pattern (S)
(new)

This exploration will be an adapta
tion of our Pattern Blocks manipulative in which the student can zoom in
and zoom out, group pieces together, and clone additional pieces. Starting with a single Pinwheel Triangle,
four others are cloned and pieced together to create a 5
-
unit Pinwheel Tri
angle. The process is repeated
until a nice rendition of the Pinwheel Pattern is achieved.



Chapter 5: Contortions of Space


Solids and the Euler Characteristic

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=51



Feeling Edgy?
(revised)

Link:

http://matti.usu.edu/tadriola/serv
er/nav/activity.jsp?sid=key&cid=heart&lid=52


In addition to the current exploration that allows the student to create their own connected graph, we will
build in several graphs that will ask the student to identify (by clicking with the mouse) the vertic
es, edges,
and faces of the graph. As each piece is identified,

V


E + F will be updated.





Chapter 6: Chaos and Fractals


Julia and Mandelbrot Sets

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=61



Conway’ Game of Life (S)
(revised)

Link:
http://matti.usu.edu/tadriola/s
erver/nav/activity.jsp?sid=key&cid=heart&lid=62


Given a particular initialization, the student will be asked to mark in green those squares that represent
births in the next generation, and mark in black those squares that represent death. When the Next

Generation key is pressed, the student will learn if the rules were applied correctly.


Cobweb Plots

Link:
http://matti.usu.edu/tadriola/server/nav/activity
.jsp?sid=key&cid=heart&lid=63



Koch Curve and Siepinski Carpet

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=64



Barnsl
ey Fern and Fractal Garden

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=65



Polygonal Probability Fractals

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=66



Chapter 7: Risky Business


Dueling Calculators

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=71



Law of Large Numbers
(revised)

The student will be able to place numbers in the box from a larger list. W
ith this option, the mathlet will
have the capability to simulate the sum of the tosses of two fair dice and other simple games.

Link:
http://matti.usu.edu/tadr
iola/server/nav/activity.jsp?sid=key&cid=heart&lid=72



Unusual Coincidences

This activity will be replaced by the Hamlet Happens mathlet described below.


Hamlet Happens
(new)

The student will be asked to input their favorite number in base 2. For e
xample, 1943 in base 2 is
11110001101. The monkey will random draw from the box [ 0, 1 ] until the base 2 number occurs. For
this to happen, the student (not the monkey) may have to go to lunch.



Heads Have the Runs (new)

With this exploration, th
e student can set the probability of heads, and toss the coin until a desired number
of heads has occurred. There will be a nice connection to free throw shooting in the NBA.

Link: http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&
lid=75


Let’s Make a Deal

This also is found in Chapter 1.

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=76



Chapter 8:

Deciding Wisely


Scatterplot (S)
(revised)

The new version has a better layout and will provide several interesting examples and activities that can be
immediately loaded.

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=81



Bar Chart (S)
(new)

Link:
http://matti.usu.edu/tadriola/
server/nav/activity.jsp?sid=key&cid=heart&lid=82



Histogram (S)
(new)

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=83



B
ox Plot (S)
(new)

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=84



Pie Chart (S)
(new)

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=85



Box Model
(new)

Mendel’s experiments can be simulated using this mathlet.

Link:
http://matti.usu.edu/tadriola/server/nav/activity.jsp?sid=key&cid=heart&lid=86



Voting Strategies: The Whammy Awards (S)
(new)

There are 3 choices: Classical, Country, and Pop. Nine v
oters are each asked to give a preference ranking.
(The student fills in the spreadsheet by typing in 1, 2, or 3.) Once the spreadsheet is completed, the
winners are announced (displayed) according to the various voting strategies (Plurality, Borda Count
,
Runoff, Least Favorite, Vote for Two). Any of the preferences can be tweaked. Another feature is to
increase the number of choices (include Jazz and or Rock) and to conduct the election with different
numbers of voters (3
-
15).



Heart Savings and Loan
(new)

A dynamic calculation tool to explore savings growth and monthly payments.