Name WORDs Next to each step, write down which rule you use.

[Pages:8]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) C _____________ _____________ _____________ _____________ _____________ _____________ ACAC

(2) B _____________ _____________ _____________ _____________ _____________ _____________ ABAB

(3) C _____________ _____________ _____________ _____________ _____________ _____________ B

(4) B _____________ _____________ _____________ _____________ _____________ _____________ BABA

(5) CB _____________ _____________ _____________ _____________ _____________ _____________ AACB

(6) C _____________ _____________ _____________ _____________ _____________ _____________ CBA

(7) CAC _____________ _____________ _____________ _____________ _____________ _____________ CAAAC

(8) CAC _____________ _____________ _____________ _____________ _____________ _____________ CAAC

(9) C _____________ _____________ _____________ _____________ _____________ _____________ BACABC

(10) C _____________ _____________ _____________ _____________ _____________ _____________ AACB

(11) CBC _____________ _____________ _____________ _____________ _____________ _____________ ABAABC

(12) CDCC _____________ _____________ _____________ _____________ _____________ _____________ CDABAB

Compose three problems of your own below.

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

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 _____________ _____________ _____________ _____________ _____________ _____________ CC

(Phineas) CAC _____________ _____________ _____________ _____________ _____________ _____________ CCAAAAAACC

(Max) C _____________ _____________ _____________ _____________ _____________ _____________ BCABBC

(Lily) CCB _____________ _____________ _____________ _____________ _____________ _____________ BBACBBBACB

(Oliver) O _____________ _____________ _____________ _____________ _____________ _____________ AAAA

(Nadia) A _____________ _____________ _____________ _____________ _____________ _____________ CC

(Tucker) C _____________ _____________ _____________ _____________ _____________ _____________ CBACBCA

(Laila) BA _____________ _____________ _____________ _____________ _____________ _____________ BAA

(Nina) C _____________ _____________ _____________ _____________ _____________ _____________ 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)

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

(4)

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

(2)

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

(5)

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

(3)

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

(6)

_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________

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 _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ RAM

(2) BALM _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ LAMB

(3) LUMP _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ PLUM

(4) CAUSE _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ SAUCE

(5) TRACE _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ CATER

(6) SHOTGUN _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ 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) MI _____________ _____________ _____________ _____________ _____________ _____________ MIUIU

(2) MUUI _____________ _____________ _____________ _____________ _____________ _____________ MII

(3) MI _____________ _____________ _____________ _____________ _____________ _____________ MUI

(4) M _____________ _____________ _____________ _____________ _____________ _____________ MIU

(5) MI _____________ _____________ _____________ _____________ _____________ _____________ MIUUI

(6) MUUI _____________ _____________ _____________ _____________ _____________ _____________ MIUIU

(7) MI _____________ _____________ _____________ _____________ _____________ _____________ MU

(8) MUUI _____________ _____________ _____________ _____________ _____________ _____________ MIUU

(9) MUUI _____________ _____________ _____________ _____________ _____________ _____________ 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.

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