Recap/HW for 02/12/17 Class

Instructor:	Mr. Rameel Rizvi
Attendance:	Aiden, Omar, Ayyan
Material Covered:	Special topic: introduction to basic set theory
Problems Solved in Class:	Examples of set unions, intersections, differences, and symmetric differences. Also briefly talked about Euler's number, e, which is approximately 2.71828, and we saw that the function f(x) = e^x has some very interesting properties.
Concepts:	a set is a collection of objects; there is no restriction whatsoever on these objects elements of a set do not repeat we use brackets to define a set, separating the listed elements with commas (for example, A = {1,2,3} is a set containing exactly the numbers 1, 2, and 3) a null set, or empty set (i.e. a set containing no elements), is denoted by { } or ∅ the union of two sets A and B, denoted A ∪ B, is a set containing all elements from A and all elements from B  the intersection of two sets A and B, denoted A ∩ B, is a set containing those elements that belong to both A and B, and no other elements the difference of two sets A and B, denoted A \ B, is a set containing those elements that belong to A but not to B, and no other elements the symmetric difference of two sets A and B, often denoted A ⊕ B, is a set containing those elements that belong to either A or B, but not to A ∩ B, and no other elements the cardinality of a set A, denoted |A|, is the number of elements that are members of A a proper subset or strict subset S of a set A, denoted S ⊂ A, is a set that is strictly smaller (i.e. has a smaller cardinality) than A and all of whose elements are also members of A we may allow a subset S to have the same cardinality as A, and denote this possibility by S ⊆ A (so S could be the same size as or smaller than A, and we simply call it a subset rather than a strict subset or proper subset) the power set of a set A, denoted P(A), is the set of all subsets of A if |A| = k, then |P(A)| = 2^k (we saw this in class by considering binary strings of length k and observing that we have 2 choices for each digit of the string, either 0 or 1)
Student Difficulties:	Students were slightly familiar with sets and operations union and intersection, but the rest was new and was understandably not the easiest to grasp, but I was quite happy with what was learned.
Homework:	Let U be the set of all lower-case letters of the English alphabet. Let A = {a,b,f,h,w,y} and B = {w,c,h,q,e,m}. Answer all of the following: 1) What is |U|? 2) Write the smallest subset of U. 3) How many proper subsets of U are there? 4) How many proper subsets of U are there that have maximal cardinality? 5) What is U \ {a,e,i,o,u}? What is this set commonly called? 6) Write A ∪ B. 7) Write A ∩ B. 8) Write A \ B. 9) Write B \ A. 10) Write A ⊕ B. 11) Write P(U ∩ {x,y,z}).
Notes:

Instructor:	Mr. Rameel Rizvi
Attendance:	Aiden, Omar
Material Covered:	Pre-Algebra Chapter 11, 12 (up til and including 12.2)
Problems Solved in Class:	Chosen exercises from chapters 11 and 12
Concepts:	* Covered perimeter of bounded figures.  For a rectangle, the total perimeter length is given by 2(l + w) where l is the length, w the width For a circle, this is the circumference, whose length is 2πr with r the radius * Covered area of bounded figures For a rectangle, the total area is given by l*w square units, where l is the length, w the width For a circle, this is πr^2 with r the radius For a triangle, this is (1/2)bh where b is the base length and h is the height * Triangle inequality: For any 3 points A,B,C on a plane, AB + BC >=  AC, and this is only equal for a line * Pythagorean Theorem: For a right triangle with sides a,b,c we have a^2 + b^2 = c^2    - Proved this using triangles enclosed in a square * Special triangles    - 45-45-90 (side lengths a and a*sqrt(2))    - 30-60-90 (side lengths a, a*sqrt(3), 2a) * Briefly covered a quadrilateral known as a rhombus, whose defining property is all sides having equal length. The diagonals of a rhombus bisect the vertex angles (split them in half), and they intersect  perpendicularly (forming right angles). The area of a rhombus can be found by multiplying the lengths of the diagonals and halving the result.
Student Difficulties:	Students seemed familiar to some extent with the shapes covered, but there may have been trouble understanding the proofs that were presented (such as for Pythagoras' Theorem or for the area of a rhombus).
Homework:	Chapter 11: Read chapter summary. Problems: 11.40, 11.45, 11.47, 11.50 Chapter 12 Problems: 12.1.2, 12.1.5, 12.2.6
Notes:

Class on 1/15/2017

Instructor:	Mr. Ahmed Hefny
Attendance:	Ayaan, Aiden, Omar
Material Covered:	Pre-Algebra Chapter 10
Problems Solved in Class:	Chosen exercises from chapters 9 and 10
Concepts:	* Review on square roots. * Angles:      - Plane Geometry as an example of axiomatic method.      - Difference between a definition, an axiom and a theorem.      - Points, line segments, rays, lines, angles.      - Highly composite numbers and why we use 60 minutes and 360 degrees.      - Complementary angles, supplementary angles and vertical angles.      - Angles and parallel lines. Remember these 3 patterns:      - Interior angles of convex polygons.          -- Sum of interior angles of a polygon with n vertices.
Student Difficulties:
Homework:	Chapter 10: Please watch the videos and make sure you can solve example problems. Chapter 9: . Solve exercises: 9.2.8,9.3.9. . Solve review exercises: 9.43,9.63 Chapter 10: . Solve review exercises: 10.24, 10.28, 10.37, 10.38, 10.39
Notes:	If you are curious, this website provides a nice exposition of Euclid's Elements. As you can see, everything is based on 10 axioms (called postulates or common notions) from which theorems (called propositions) are derived. The parallel line axiom we discussed today is postulate 5 but is stated a bit differently. Can you see how postulate 5 implies what we said in class ?

Tentative Syllabus

Overview:
This course develops creative problem solving skills for middle school students. For instance, consider the following math problems: 
(i) I have 120 blocks. Each block is one of 2 different materials, 3 different colors, 4 different sizes, and 5 different shapes. No two blocks have exactly the same of all four properties. I take two blocks at random. What is the probability the two blocks have exactly two of these four properties the same? (Source: AIME)
(ii) Prove that √2 (square root of 2) is irrational.

In this course, students will practice solving problems such as those listed above. By the end of this course, students should be able to approach challenging problems (using elementary techniques) and write simple proofs.

Instructors:
Professor Isa Hafalir (http://www.andrew.cmu.edu/user/isaemin/)
Mr. Ahmed Hefny (http://www.cs.cmu.edu/~ahefny/)
Mr. Rameel Rizvi (http://ri.cmu.edu/person.html?person_id=4563)


Location, Dates, Time:
MCCGP 233 Seaman Ln, Monroeville, PA 15146
Jan 8, 2017 to May 14, 2017
Sunday 2:30 pm to 4:30 pm

Text & Topics (tentative)
Prealgebra by Richard Rusczyk, David Patrick, Ravi Boppana

Review & Problem-Solving Strategies
Percents
Square Roots
Angles
Perimeter and Area
Right Triangles and Quadrilaterals
Data and Statistics
Counting

Competition Preparation (time permitting):
AMC 8
MATHCOUNTS
Purple Comet