V2S2 morning math

Dr. Therese Shelton
Associate Professor
Department of Mathematics and Computer Science
Southwestern University
Georgetown, TX

Summer 2010 high school program, second half, morning math session, 75 minutes a day

a very brief outline of the first portion, by Dr. Alison Marr, in MBH 107:

Daily questions
Euler paths and line puzzles.
Ramsey Theory and party graphs.
Dijkstra's algorithm to find shortest paths
Matrices. products. inverses. powers.
Probability
globe tossing activity.
Randomness and penny flipping.
Birthday Problem. Calculator program.
long run average. independence.
Pig dice game
rock-paper-scissors; win matrix; row sums, column sums.
Matrix Secret Messages
Markov Chains.
Number Bracelets; Modular arithmetic.
Other calculator -- converting decimals to fractions.

Thurs, July 15 in MBH 107: (9am class observed)

"Planets" game/puzzle, from Riddles of the Sphinx.
Homework: look for patterns; try to figure out why it might work.
Calculator trick with 7-11-13.
Led to thinking of the number of triples, each place from 1 to 9. Tree diagram, list some triples.
Homework: think of a system to generate them all.
Mention of combinatorics.
Calculator trick with 1X8+1, 12X8+2, etc. Generated more like these.
Inductive reasoning (and prejudice).
Discussion of how people do math (alone, together, redo, etc.), background needed, what is written (just a number? paragraph explanation?)
In some ways, it is very different from other areas of study; a good reader can catch up in language arts, history, etc. but holes in a math background can make a person feel stupid. Each of us can fill in these holes, and we often need help from a teacher, especially in college.
Brief description of Los Alamos Project, mathematicians performing hand calculations, without technology; working independently and checking with each other; need for error analysis.
Mention that some of the great mathematicians shared mathematics in letters to each other, usually with debate and critique. Sometimes one refused to believe another's approach.
"Faith" in mathematics.
Which soda has the most sugar out of a given selection of products?
Led to what is meant by the question,
list of variables: sugar per serving, number of servings, serving size.
Handout of 2 Sudoku puzzles. Brief discussion of strategies.

Fri, July 16 in MBH 107:

Sudoku puzzles. Answer to one from last time in a handout; answer to the other from student work.
History from the MAA magazine, Focus, Jan 2006. Euler played with Latin squares, but those simpler puzzles are much older.
Number of 9X9 boards of cells with values from 1-9. Number of 2 and 3 cell sequences with different values.
Polygons and polyhedra; regular (equilateral, equiangular) or not; convex or concave.
"octabug" paper model.
In groups, constructed polyhedra using "Roger's Connections" magnetic sticks and balls. Took photos of student constructions.

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Mon, July 19 in FWO 208: (10:30 am class observed)

Discussion: computer lab usage. Students' list of rules. Some specifics of Mac computers: dock, omnidirectional mouse
privacy, secure sites, computer classes in high school, Facebook, etc.
Discussion: Planets game; what makes it work?
parity
Discussion: Polygons and polyhedra.
Paper models. Terms from last time.
http://www.mathsisfun.com/geometry/index.html .
Go to "Plane Geometry". compass and protractor. see "Shape" link. At the bottom of the page, see "Polygons (more detail)" link. Scroll down to see formula for interior angles of a regular polygon.
Return to the geometry index (link above). See "Solid Geometry" link for platonic solids, prisms, pyramids.
Searched the web, using various keywords, for a formula relating vertices, edges, faces. Verified for the platonic solids. The formula is often attributed to Euler but is really much older.
Discussion: vocabulary very important in math, as in other subjects.
Tangrams and Roger's Connection in 10:30 class

Tues, July 20 in FWO 208:

Aided by the "experts," use Excel to generate the list of pairs with digits from 1 to 9.
Then generate triples.
Test the 7-11-13 calculator trick.
More detail: In column B, beginning in row 5, generate arithmetic sequence of 1 through 9; type 1, 2; highlight; drag down. in column A next to these, type a 1, then drag to fill in. Now we have the 9 ordered pairs (1,1), ... (1,9). Copy and paste 1Thru9 once in column B below the current list. Then fill in 2's in column A. Now we have the 18 ordered pairs (1,1), ... (1,9), (2,1) ... (2,9). Copy and paste the set of 2 1Thru9 and paste more until you have as many as we need for all pairs (1,1)...(9,9). Insert a column to the left so we have our values in columns B and C. Type a 1, then drag to fill in. Now we have the 18 ordered triples (1,1,1), ... (1,1,9), (1,2,1) ... (1,2,9). Copy and paste our (1,1)...(9,9); fill in 3's. Continue until we have all ordered triples. Next portion: generate, from these individual sets of 3, a list of all 6-digit values in the pattern abcabc. This is aX10^5 + b*10^4 etc. ÊUse an Excel formula so that we can copy and paste to fill in all others. Next portion: calculate the first value divided by 7, 11, and 13. Use an Excel formula so that we can copy and paste to fill in all others. We have verified the conjecture that we retrieve the ordered triple. We verified it "by exhaustion," which means we checked every value. Think of why our conjecture is true. What is this good for? patterns enumeration; systematic method to generate values so that we get them all, miss none, and do not repeat. Place-value systems. explore Excel: copy, paste, drag to generate a pattern; insert column; window/split; formulas with relative addressing
Tangrams and Roger's Connection in 9am class

Wed, July 21 in FWO 208: (10:30am class observed)

Math raps by Lamar Queen on youtube.com. Listened to slope-intercept and distance-rate-time.
Discussion: what makes 7-11-13 calculator work?
Test the calculator trick with 1X8+1, 12X8+2, etc. Began with a way to generate 1, 12, 123, etc.
Created a Word document with some notes and a copy of the "X8" calculator trick.
Code the instructions and test the calculator trick with the magic number 7. Think of why it works.

Thurs, July 22 in FWO 208:

Numeration systems; handout. Ancient Egyptian, Babylonian, Roman.
Converted several values; addition and subtraction problems that involved exchanging symbols.

Fri, July 23 in FWO 208: (9am class observed; no SU network)

Make paper polyhedra from Math Is Fun -- Platonic Solids printable "net with tabs". Each student had a set of the five solids to color, cut, and glue.

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Mon, July 26 in FWO 208:

Mayan (Olmec) numeration system.
Converted several values; addition and subtraction problems that involved exchanging symbols.
See http://www.ancientscripts.com/ma_ws.html. Especially see the Flash animations of the interlocking calendars
See (with caution) Wikipedia's Timeline of mathematics
Lots of good links including careers, how to succeed, and more. We watched the main video at http://www.whenwilliusemath.com/. Then students browsed careers.

Tues, July 27 in FWO 208:

Meet with Career Services

Wed, July 28 in FWO 208: (9 am class observed)

Unwrapping y on a circle
Materials: pencil, disc (plastic lid), flexible but not-too-stretchy "string" (sewing thread), scissors, graph paper, tape, straight edge
detailed instructions:
1. Draw x and y axes on graph paper. Put the paper on top of the disk with the origin at the middle. Sketch around the lid on the graph paper; cut it out. Attach the circle of graph paper to your disc; double-sided tape works well. Mark a "starting point" on the disc where the positive x-axis points.
2. Tape 2 large pieces of graph paper together to make one long sheet. On the large piece of graph paper, mark x and y axes.
3. Get 2 lengths of differently colored thread; 2 feet each will be plenty.
4. From the starting point, wrap the first thread all around the edge of the circle; cut it. How far is this if we say that our circle is of radius 1 unit? Lay the thread on the x-axis of the graph paper and mark the distance. Label it.
5. Fold the first length of thread in half; how far is that? Fold the thread as many times as needed to measure out &pi/4. Mark distances on your graph paper. Mark the same distances along the outer rim of the disc, measuring counter-clockwise.
6. At each mark on the rim of the disc, use the second thread to measure the vertical distance from that point to the x-axis; notice if the value is positive or negative. Think of how far up or down it is at the starting point and straight across from the starting point.
7. What does it mean to go clockwise?
8. Draw a smooth curve that connects your points.
Sketch the function that I give you on your calculator; some of you know this one. Compare it to yours.
Discuss the shape and properties of the graph.

Thurs, July 29 in FWO 208: (10:30 am class observed)

Towers of Hanoi. See http://www.mazeworks.com/hanoi/index.htm.
Record the number of discs, minimum number of moves, and the number of your moves.
Discussed the formula for the minimum number of moves, history of the problem, strategies.
Survey for choice of presentation topics.

Fri, July 30 in FWO 208:

Prepare for presentations with Dr. Marr and Dr. Shelton.
Surprise activity

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Sat, July 31

Presentations

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