Name _____________________
WORDs
Use the following rules to change each “word” (each string of letters) into the other.
Next to each step, write down which rule you use.
1) Any letter can change into an A.
2) Two Cs in a row can turn into a B.
3) Any word can be doubled.
(1)
(2)
(3)
C
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ACAC
B
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ABAB
C
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B
(4)
(5)
(6)
B
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BABA
CB
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AACB
C
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CBA
(7)
CAC
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CAAAC
(8)
CAC
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CAAC
(9)
C
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BACABC
(10)
C
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AACB
(11)
CBC
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ABAABC
(12)
CDCC
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CDABAB
Compose three problems of your own below.
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What do you find appealing or unappealing about this mathematical structure?
Think up three rules that we could add as Rule #4 and write these below. Pick one and
give an argument as to why this would be a good addition to the system.
Name _____________________
WORDs
Use the following rules to change each “word” (each string of letters) into the other.
Write the number of the rule you use next to each step.
1) Any letter can change into an A.
2) Two Cs in a row can turn into a B.
3) Any word can be doubled.
(Louis)
B
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CC
(Phineas)
CAC
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CCAAAAAACC
(Max)
C
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BCABBC
(Lily)
CCB
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BBACBBBACB
(Oliver)
(Nadia)
O
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AAAA
A
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CC
(Tucker)
(Laila)
(Nina)
C
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CBACBCA
BA
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BAA
C
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BXAC
For Nina’s:
4) Four Cs can turn into an X.
Created by ______________!
Name _____________________
Use the following rules to change each “word” (each string of letters) into the other.
1) Any letter can change into an A.
2) Two Cs in a row can turn into a B.
3) Any word can be doubled.
4)
(1)
(2)
(3)
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(4)
(5)
(6)
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Name _____________________
WORDs Switcheroo!
Use the following rules to change each “word” (each string of letters) into the other.
1) Neighboring letters can switch places. MATH  MTAH
2) The first and last letters of a word can switch places. GLAD  DLAG
(1)
ARM
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RAM
(2)
BALM
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LAMB
(3)
LUMP
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PLUM
(4)
CAUSE
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SAUCE
(5)
TRACE
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CATER
(6)
SHOTGUN
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GUNSHOT
Can you come up with a problem of you own on the back?
Name _____________________
WORDs #4
A NEW SYSYEM
Use the following rules to change each “word” (each string of letters) into the other.
1) If the string ends in I, you can add a U onto the end. MUUI—›MUUIU
2) The string Msomething can become Msomethingsomething. MIIU—›MIIUIIU
3) III can change into U. MUIII—›MUU
4) Any UU can be eliminated. MIIUUI—›MIII
(1)
(2)
(3)
MI
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MIUIU
MUUI
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MII
MI
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MUI
(4)
(5)
(6)
M
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MIU
MI
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MIUUI
MUUI
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MIUIU
(7)
(8)
(9)
MI
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MU
MUUI
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MIUU
MUUI
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MIIIUII
On the back, describe three ways that this system is different from our original ABC system.
Name _____________________
Consider the rules that you and your classmates have added as Rule #4 to the ABC
system. Which new rule is…
 the most powerful (makes solving problems much easier)?
 the least powerful (doesn’t add many new possibilities to the system)?
 the most useful (can be used in many circumstances)?
 the least useful (hardly ever gets used)?
 the trickiest?
 the most creative?
 the somethingelsest?
Be sure to give reasons for your choices and examples!!!
Name __________________
Isometries #1
Describe what isometry or combination of isometries takes the first triangle in each pair to the
second.
c to h __________________________
b to d __________________________
a to b __________________________
f to b __________________________
g to f __________________________
c to a __________________________
a to g __________________________
e to d __________________________
Make up a problem of your own inspired by the one above.
Investigation #4
Transformations
1) Complete a transformation challenge:

Isometries: Decide under what conditions a planar shape can be mapped to a congruent
copy of itself lying in the same plane through a series of reflections. When this is
possible, describe how the process of reflecting transpires and give bounds for how many
reflections it takes. Some cases to consider are. Geogebra and patty paper with a stencil
could be good tools to use for this challenge.

Cut and paste: Decide what triangles can be cut and pasted into what quadrilaterals.
Describe some classes of these and show how the transformation can be accomplished.
For example, A special, interesting, and challenging example is transforming and
equilateral triangle into a square. Graph paper and scissors could be good tools for this
challenge.

Projection: Describe what kinds of projections can be produced by a robust subset of the
following objects, plus any others you would like to try out: square, equilateral triangle,
arbitrary triangle, tetrahedron, cube. The projector (or the sun) and some cutouts and 3D
objects could be good tools to use for this challenge. You may want to take some
photographs, as well.