GROUP WORK

Spring 2013

**Thirteenth Group Work (April 15, 2013):**
Use geometric arguments to deduce trigonometric formulas. See Section 4.7
(Please I could not remember what I had assigned to tables 2, 3 and 4... I
know the topics, can't remember the order... please confirm.)

**Twelveth (spelling?) Group Work (April 10, 2013):**
Use similarity of triangles to show the following:

Assume a and b are positive numbers: The square root of ab is called the "geometric mean" of a and b. The "arithmetic mean" of a and b is (a+b)/2 The "harmonic mean" of a and b is the reciprocal of the arithmetic mean of the reciprocals of a and b.

**Eleventh Group Work (April 1, 2013):**
We took an axiomatic approach to area (see pages 118-119):

With those 5 axioms each table was tasked to show:

Tables 5 and 6 discovered that it was simpler to derive formula for the rectangle from the axioms (Theorem 3.1 in the book, see picture 3.7). Then use that to get the area of a right triangle (half a rectangle), and use those to calculate the area of a general triangle (Theorem 3.3).

One could also calculate the area of a parallelogram (Theorem 3.2) and then use that to find the area of a general triangle as half of the area of an appropriate parallelogram.

For the Trapezoid we dropped some heigth and broke into a rectangle and two right triangles, and carefully calculated the total area in terms of the bases an the height (Theorem 3.4). We also broke into a parallelogram and a triangle and calculated those areas in terms of bases and height.

For the regular hexagon, we observed its inscribed in a circle, and the 6 triangles created by joining the center to the vertices are all equilateral with side a. The area then is 6 times the area of the equilateral triangle. But we wanted the area of the triangle only in terms of the sidelength. It was necessary to calculate the height. For that Pythagorean theorem came handy... We observed that with the area axioms the proof that Kendall presented at the beginning of the semester worked (Proof 1 in the book page 131).

Finally for the kite we know the diagonals are perpendicular to each other, can calculate the area of the two isosceles triangles in terms of those diagonal lengths, after a little algebra conclude that the area of a kite is the product of the diagonal's lengths divided by 2.

**Tenth Group Work (March 6, 2013):**
We worked on problems 25, 31 and 32 in Problem set 1.3 (p. 61-62).
I want each one of you to write carefully the solutions of the three
problems discussed today.

**Ninth Group Work (March 4, 2013):**
The last 17 minutes we worked in groups on Now Solve this 1.13 (A proof
that your conjecture on the Treasure Island Problem was correct). You are
asked to use Figure 1.61 in page 56 to get the proof. In that picture M is
the midpoint of the segment S_2S_1. I think all the tables understood that
triangles T_1AS_1 and GT_1C are congruent and so are triangles S_2BT_2
and T_2CG... with that info we could show that N bisects T_2T_1. It
remains to be shown that MN is congruent to T_2N (or to NT_1). I want this
problem carefully written by Monday after Spring Break, I said one paper
per table, but I actually want each one you to write down the whole
argument.

**Eigth Group Work (Feb 25, 2013):**
From the book: Now Do This 1.7(page 36)
Tables were tasked to write up for Wednesday the results of their work
(please include the names of all the team-mates).

**Seventh Group Work (Feb 18, 2013):**
Discuss your proofs for the Converse to Pons Asinorum
(this was assigned as a homework, collected after the discussion).
All together we had 5 different arguments:

**Sixth Group Work (Feb 13, 2013):**
We play a little bit with propositional calculus. Each table has to
verify whether two given statements are equivalent or not,
by calculating their truth tables and comparing them.

We used this as a springboard to talk about converse of an implication, contrapositive, argument by contradiction, and de Morgan's Laws in logic.

**Fifth Group Work (Feb 11, 2013):**
From Problem Set 1 in the book:

**Fourth Group Work (Jan 30, 2013):**
Think about how to define:

Here we realized the need for the parallelogram postulate to construct a parallelogram.

**Third Group Work (Jan 28, 2013):**
Define using the undefined objects (points. lines) and the incidence axioms, and
what tables before yours have defined. For example, to define a triangle,
Table 3 can use the definitions of segment and ray provided by Tables 1 and 2.

We realized the need to introduce other postulates such as Ruler Postulate to define segment, and the Plane-separation axiom to define the interior of an angle.

**Second Group Work (Jan 23, 2013):**
Each table (numbered from 1 to 6) will create an
ornament S with N wires (where N is the Table number) and beads
constructed using the following rules or axioms:

I asked you to focus on two questions: How many beads are there in S? How many beads on each wire?

**First Group Work (week of Jan 15):**
Each group (defined by the people sitting in one round table)
will read and discuss one of the following problems described in
Chapter Zero
then a spokesperson will describe the problem to the class and any additional
insight the group gained on the problem (a solution or proof in general or for
some particular configuration)

Return to: Department of Mathematics and Statistics, University of New Mexico

Last updated: April 24th, 2013