Class Summary - 11/20/2016

My apologies for this super-late posting. Since you have only today to do the homework, I have made it much shorter than usual.

Instructors:	Mr. Rameel Rizvi
Attendance:	Ayyan, Aiden, Omar, Areej, AbdulRafay
Homework:	Read: Chapter 7.7 (this is a review of all main concepts from the chapter); Problems to solve:  7.39, 7.46, 7.58, 7.66
Challenge Problem:	Recall the definitions of the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) we covered in class. (If you don't remember these and/or didn't write them down, Google is your friend.) Let n be the number of distinct variables in question (i.e. for n = 3 we have x1, x2, and x3). Here's the problem, if you wish to attempt it: Prove that when n = 2, AM >= HM >= GM (so for all x1 and x2, their arithmetic mean is less than or equal to their harmonic mean which is less than or equal to their geometric mean).
Class Activity:	Finished Chapter 7.
Concepts:	Covered ratios, rates, and proportions; for more in-depth review consult Chapter 7.7.
Student Difficulties:	I recall fairly good understanding of ratio/rate problems from class. Remember to read every word in the question carefully!
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.

Class Summary - 11/13/2016

Instructors:	Mr. Ahmed Hefny
Attendance:	Ayyan, Aiden, Omar, Areej, AbdulRafay
Homework:	Read: Chapter 6 Problems to solve:  6.22[(d,e)], 6.29[c], 6.31 [(a,b,c)], 6.30 6.1.10
Class Activity:	Finished Chapter 6.
Concepts:	- Decimal notation. - Rounding. - Using rounding to estimate a range of possible multiplication results.  - Converting a regular fraction to decimal and vice versa. - Repeating decimals.
Student Difficulties:	I haven't noticed a strong difficulty going through the material. I believe students are already familiar with most of the material covered in the chapter. Students, however, had difficulty extending the idea of using rounding for multiplication to division.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.

Class Summary - 11/6/16

Instructors:	Mr. Rameel Rizvi
Attendance:	Ayyan, Aiden, Omar, Areej, AbdulRafay
Homework:	Read: Chapter 5.6 (this is a review of all main concepts from the chapter--as usual, make sure to do this slowly and thoroughly!); Problems to solve:  5.33 (parts (b) and (d)), 5.34 (parts (b), (d), (f), (h), (j)), 5.36, 5.43, 5.44, 5.51 (all parts), 5.52 (all parts), 5.55, 5.60
Challenge Problem:	Before starting to paint, Bill had 130 ounces of blue paint, 164 ounces of red paint, and 188 ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left. (Source: AIME II 2009)
Class Activity:	Finished Chapter 5.
Concepts:	Continued where we left off from Chapter 5 in the previous week, reviewing word problems and solving equations of a single variable.  Covered inequalities, which deal with expressions where the left-hand side (LHS) and right-hand side (RHS) are not equivalent. We use "<" (less than) to express the LHS having a value strictly less than the RHS and we use ">" (greater than) to express the LHS having a value strictly greater than the RHS. Sometimes, we may desire a nonstrict inequality, that is, where it's possible for the LHS to be equivalent to the RHS (in addition to being either less than or greater than the RHS, depending on the situation). We denote such a possibility with a horizontal bar immediately underneath the "<" and ">" symbols. We went through various properties of inequalities and showed that a number line can be used to visually represent them: we shade in black the region on the line corresponding to the values that are permitted by the inequality, perhaps stretching out to positive or negative infinity, in which case we draw a shaded arrow in the appropriate direction. For endpoints, strict inequalities require an open circle whereas nonstrict ones require a shaded circle.  We saw that claims about inequalities could be confirmed using number lines. For example, we showed that transitivity holds: if a > b and b > c, then a > c (this works with other inequality variants as well, of course). In addition, if a > b, then a + k > b + k for any k. We also saw that for k > 0, if a > b then ak > bk, whereas if k < 0 and a > b, then ak < bk.
Student Difficulties:	The biggest difficulty seems to be converting word problems to symbolic equations/inequalities to then be solved (rather than actually solving). The best way to deal with this is to think: what variables do I need (often just one), and what constraints am I being given on these variables? Consider constraints step by step and update equations accordingly for every new constraint you read until there are none left. And remember: Always ask yourself, "Does this make sense?"
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.

Class Summary - 10/30/2016

Instructors:	Mr. Ahmed Hefny
Attendance:	Ayyan, Aiden, Omar, Areej, AbdulRafay,
Homework:	Read: Chapter 5 - Sections 5.1-5.4. Go through examples, especially section 5.4 (word problems). Problems to solve: 5.2.3, 5.2.6, 5.3.6, 5.4.6, 5.4.8, and 5.4.9
Class Activity:	Chapter 5 - Sections 5.1-5.4
Concepts:	- Introduced the notion of expressions, terms, factors, constant and variables. - Differentiated between equations and equivalences.  Equation: An equation is a statement that two expressions are equal. Solving an equation means find the values of the variables that makes an equation true. Equivalence: An equivalence is statement that two expressions are equal for all possible values of the variables. - Solving a linear equation in a single variable. - Translating a word problem into an equation.
Student Difficulties:	Students are getting better at mathematical modeling (translating a problem into an equation) but more practice is needed.  Please try the examples and exercises of section 5.4.  Remember: solving an equation is easy. Coming up with the right equation is the harder part.
Notes

Class Summary - 10/23/16

Instructors:	Mr. Rameel Rizvi
Attendance:	Ayyan, Aiden, Omar, Areej, AbdulRafay,
Homework:	Read: Chapter 4.9 (this is a review of all main concepts from the chapter--like last time, make sure to do this slowly and thoroughly! While memorizing these concepts is important, make sure you also understand each and every one of them so that you can actually apply them in the future (which is, after all, the goal of education)); Problems to solve:  4.55 (parts (b),(d),(f),(h),(j),(l)), 4.56, 4.58, 4.66 (all parts), 4.67, 4.74, 4.82 (all parts), 4.86, 4.87, 4.88 (all parts), 4.89, 4.90 (part (f) only) Note: Again, start early. These shouldn't be too difficult but there is a lot of computational effort involved.
Challenge Problem:	Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is k% acid. From jar C, m/n liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that m and n are relatively prime positive integers, find k + m + n. (Source: AIME I 2011)
Class Activity:	Finished Chapter 4.
Concepts:	Introduced fractions, values of the form "a over b" where "over" indicates a horizontal bar with numerator a immediately above and denominator b immediately below. Said that fractions are the same values we obtain when we divide a by b, and we already covered the concept of division, all of whose rules apply to fractions by definition. For example, reciprocals and exponents behave exactly the same (as they should!) when dealing with fractions as they do when dealing with division.  Showed how the concepts of GCD and LCM we learned previously apply to fractions: GCD negation is used in obtaining simplest form fractions (i.e. coprime, or relatively prime, numerals as the numerator and denominator), and LCM is used in adding/subtracting fractions with unequal denominators (by multiplying each such fraction by some other fraction k/k (unique for each individual fraction) such that the denominators of the resulting fractions are all the LCM of the denominators of the original set of fractions). Also covered mixed numbers, which are, basically, the division-with-remainder representation of a fraction with numerator greater than its denominator. Hence the main left-hand-side numeral of a mixed number corresponds to how many "whole" times the denominator of the fraction in question goes into the numerator, and the fraction on the right of this larger numeral corresponds to the remainder. Recall, for integers m,n with m >= n we can write m = q*n + r for some integers q,r where r < n. In this representation q corresponds to the main numeral of the mixed fraction, and r corresponds to the remainder (in the mixed number this is the right-hand-side fraction with numerator r and denominator n). Arithmetic with mixed vs. non-mixed numbers is, of course, equivalent; it's just a matter of notation.
Student Difficulties:	Glad to see the obvious increase in effort placed in homework! Everyone should keep up this level of effort. In particular, ensure each week that you remember and (more importantly) understand all the concepts previously covered in the course. Everything builds on things we've already covered, and this mirrors the development of mathematics in general. Failure to grasp something now will only begin a buildup of confusion that will surely grow as we progress. Oh, and one more thing: make sure for multi-part homework problems that you do every part that has been assigned. For some reason many of you randomly skipped a part of some multi-part question in the previous assignment--I'm assuming this was by accident, so just remember to double-check that you've done all the assigned parts in the future.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.

Class Summary - 10/16/16

Instructors:	Mr. Rameel Rizvi
Attendance:	Aiden, Omar, Areej, AbdulRafie
Homework:	Read: Chapter 3.8 (this is a review of all main concepts from the chapter--do this slowly and thoroughly); Problems to solve:  3.47, 3.55, 3.56, 3.60, 3.67, 3.71, 3.78, 3.79, 3.82, 3.85, 3.87  Note: start these early--they are not trivial
Challenge Problem:	Find the number of five-digit positive integers, n, that satisfy the following conditions: (a) the number n is divisible by 5; (b) the first and last digits of n are equal; and       (c) the sum of the digits of n is divisible by 5. (Source: AIME I 2013)
Class Activity:	Finished Chapter 3.
Concepts:	Defined divisibility and showed that a divides b (written a|b) iff b is a multiple of a (that is, we can write b = k*a for some integer k). "Iff" is shorthand for "if and only if," which means that either clause being true implies that the other is true. For example, A iff B when, if A is true then B is true and when B is true then A is true. Thus, in this case, if we ever know that m|n, then we also know that n is a multiple of m, and if we ever know that n is a multiple of m, then we know that m|n. Covered the concepts of least common multiple (LCM) and greatest common divisor (GCD), and showed that they are directly related to the prime factorizations of the numbers in question. To find the LCM of a set of numbers, we take all prime factorizations of the numbers in the set and construct the smallest number which is divisible by each such factorization; analogously, for the GCD of a set of numbers, we simply take all prime factorizations of the numbers and "intersect" them such that we obtain the largest number common to all the prime factorizations.
Student Difficulties:	Concepts of number theory, and definitions in general, seem to be hard to reason with. Again, this is to be expected given the stark contrast between the course material and what is traditionally covered in schools. It is important that we continue emphasizing the use of claims we have already proven to generate new ones, and this is done purely through insight. Always think, and if you ever get stuck: keep thinking.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.

Class Summary - 10/09/16

Instructors:	Prof. Isa Hafalir, Mr. Rameel Rizvi
Attendance:	Ayyan, Omar, Areej, AbdulRafie
Homework:	Read: main concepts from Chapter 3.1, 3.2, 3.3, 3.4 (look over the important results); Problems to solve:  3.1.8, 3.2.2, 3.2.7, 3.3.1, 3.3.5 (all parts), 3.4.1 (all parts), 3.4.3, 3.4.8 (all parts)
Challenge Problem:	Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3^k. (Source: AIME II 2006)
Class Activity:	Covered chapters 3.1-3.4
Concepts:	Defined the concept of multiples: an integer a is a multiple of an integer b if and only if a = mb for some integer m. This mirrors the definition of divisibility: a is divisible by b if and only if a = mb for some integer m.  We briefly covered the use of set notation in generalizing numbers with certain properties, such as when we said m is an element of Z, the set of integers (where Z had a second line through it). Read up on this for common set notations: https://www.mathsisfun.com/sets/symbols.html (toward the bottom).  We also briefly delved into proving that the square root of 2 is irrational via the method of contradiction: we assumed that it was rational, so by definition it could be written as a fraction a/b with a,b integers with no common factors besides 1, and then showed that if this assumption were correct, we could deduce that a and b were in fact not in lowest terms, so it followed our initial assumption (the rationality of square root 2) was incorrect. Don't worry if this is hard to grasp: it was meant as a preview for how proving mathematical statements works, which I certainly am not expecting any of you to do. Think of it as supplemental material. Here is a good link to the proof in case you are super interested: https://www.math.utah.edu/~pa/math/q1.html We went on to show some interesting properties of multiples. For example, if a and b are multiples of c, then so are (a+b) and (a-b). We also showed why the sum of digits trick for testing divisibility by 9 works (can be found in the notes). We went on to define the prime and composite numbers: a natural number p is prime if and only if p > 1 and p has no (integer) factors besides 1 and p. A natural number c > 1 is composite if it is not prime, that is, it can be written as ab for two integers a,b where 1 < a,b < c. (1 is neither prime nor composite.) We also showed that any natural number n > 1 can be decomposed into its prime factorization, where all its factors are prime numbers. According to the Fundamental Theorem of Arithmetic, every positive integer can be decomposed into a unique prime factorization. Further, we showed another very nice mathematical proof for the claim that there are infinitely many prime numbers. This was also done by the method of contradiction, where we began by supposing that were were only k primes for some finite number k, then showed that we could construct another prime number such that there were actually k+1 primes, hence our initial assumption (the finitude of primes) was again false. This proof is due to Euclid (c. 300 BC). Here is a good link to check out: http://www.math.utah.edu/~pa/math/q2.html Lastly, I mentioned the Goldbach Conjecture, which states that every even number greater than 2 can be expressed as the sum of two prime numbers (in class I neglected the "even" part of the statement). This is a "conjecture" because nobody has yet to prove the claim, even though a counterexample has not been found. In case you want to learn more about it, here's an informative Wikipedia link: https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
Student Difficulties:	Saw improvement in the homeworks, with more attention paid to detail--good! Continue to do so, and please, no answers without work! Also remember to try everything, especially if you don't know how to do it: that's how you really improve your problem solving skills.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.

Class Summary - 10/2/16

Instructors:	Mr. Rameel Rizvi
Attendance:	Aidan, Ayyan, Omar, AbdulRafie
Homework:	Read: Chapter 2.5 (this is a review of all main concepts covered in Chapter 2); Problems to solve (found at end of Chapter 2, "Review Problems"):  2.37, 2.39, 2.45, 2.47, 2.48, 2.52, 2.61, 2.63, 2.66
Challenge Problem:	2.76 (please attempt this!)
Class Activity:	Covered chapter 2
Concepts:	Introduced the mathematical function of exponentiation, which takes a "base" b and an "exponent" a and performs b^a (pronounced "b to the ath power"), which is defined as a multiplications of b. For example, 2^3 = 2x2x2 = 8, where 2 is the base and 3 is the power (note: when the power is 2 we call that number a square, and when the power is 3 we call it a cube). Briefly covered the sigma and pi notations for expressing patterns of additions and multiplications (here is a very good link that covers these notations in more detail if you are interested: https://mathmaine.wordpress.com/2010/04/01/sigma-and-pi-notation/). Saw that exponentiation is neither commutative nor associative. Showed why some essential properties of exponentiation hold, such as: (1) x^(y+z) = (x^y)(x^z); (2) (x^y)^z = x^(yz); (3) (a^x)(b^x) = (ab)^x; (4) (a/b)^x = a^x / b^x. Further, we showed that, in general, (a+b)^x does NOT equal a^x + b^x. Also covered powers of 0 (ALWAYS 1!) as well as negative exponents (which just mean to take the reciprocal of the base and then apply the power).
Student Difficulties:	Noticed quite a lot of careless mistakes in the last homework. For your benefit, please try not to rush these problems. You will be less likely to make these types of errors and will also gain more insights into problem solving that will stay with you for the rest of your life. If these errors persist I may start addressing homework problems in class.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.

Class Summary - 9/25/16

Instructors:	Mr. Rameel Rizvi
Attendance:	Aidan, Areej, Ayyan, Omar, AbdulRafie
Homework:	Read: Chapter 1.8 (this is a review of all main concepts covered in Chapter 1); Problems to solve (found at end of Chapter 1, "Review Problems"):  1.42, 1.50, 1.54, 1.56, 1.67, 1.68, 1.69, 1.71
Challenge Problem:	1.75 (please attempt this! good practice for AMC 8)
Class Activity:	Covered chapters 1.6 up to 1.8
Concepts:	More review of arithmetic properties. Highlighted how Gauss' formula for sum of first n positive integers is a result of addition being commutative and associative. Introduced idea of division with the definition of the reciprocal, the number r such that for a given number x, xr = 1. Showed that by definition 0 cannot have a reciprocal (as for any number x we have that 0x = 0), and that for all other numbers x, r = 1/x. Also showed that division is neither associative nor commutative (like subtraction), and performed some calculations step-by-step using the definitions we learned (reciprocals, commutativity, associativity, distributivity, prime factorization) rather than through the more quick, direct approach (for example, by changing division by 7 to multiplying by 1/7, the reciprocal of 7, even though we are all capable of directly dividing by 7).
Student Difficulties:	As can be expected, approaching calculations with definitions rather than through the rote methods taught in elementary/middle school is very new to the students. It will take time to process how all of mathematics is essentially built off a set of axioms, which few students this young comprehend.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.

Class Summary - 9/18/16

Instructors:	Mr. Ahmed Hefny
Attendance:	Aidan, Areej, Ayyan, Omar, AbdulRafie
Homework:	"Art of problem solving" Exercises:  1.2.1, 1.2.4, 1.3.1, 1.3.7, 1.4.2, 1.4.3, 1.5.10, 1.5.11,
Class Activity:	Covered chapters 1 up to 1.5
Concepts:	Review of arithmetic properties (commutative, associative, distributive, identity element and inverse).
Student Difficulties:	Students are still facing difficulties in abstraction and reasoning with symbols, but they are getting better at it.
Notes	Please hand-in your homework solutions in the next class.

Class Summary - 8/28/16

Instructors:	Mr. Ahmed Hefny Mr. Rameel Rizvi
Attendance:	Aidan, Areej, Ayyan, Omar
Homework:	None
Class Activity:	Placement test and follow-up discussion.
Concepts:
Student Difficulties:	GCD and LCM
Notes:

- Every whole number (>1) can be written as a product of prime numbers.
- When we divide a whole number X by Y, we essentially remove from X all the prime factors that constitute Y. For example, if X = 12 = 2*2*3 and Y = 4 = 2*2 then X/Y = 3.
- Therefore, for X/Y to be a whole number, X must contain all the prime factors of Y (repeated factors must be repeated with at least the same multiplicity). In this case we say that "Y is a divisor of X" and "X is a multiple of Y". For example, "4 is a divisor of 12" and "12 is a multiple of 4".

- A "common multiple" of two numbers is a number that can be evenly divided by both of them (i.e. is a multiple of both of them). A common multiple of X and Y must contain all the prime factors of X and the prime factors of Y. For example 36=2*2*3*3 is a common multiple of 6=2*3 and 4=2*2*3.

- Given two numbers X and Y, an obvious common multiple is X*Y.

- The "least common multiple (LCM)" of X and Y is the smallest number that can be divided by both of them-- that is, any smaller number will not be a common multiple. To find the LCM, we enumerate the prime factors of X and Y. The LCM must contain each prime factor with the highest multiplicity in X and Y. For example:
If X = 12 = 2*2*3 and Y = 90  = 2*3*3*5
Then the LCM has to contain 2's, 3's and 5's as prime factors.
Since 2 is repeated twice in X but only once in Y, we need to include 2 twice.
Similarly, 3 is repeated twice in Y and once in X, so we need to include 3 twice.
Finally, There is only one 5 in Y and none in X, so we need to include 5 once.
That gives LCM = 2*2*3*3*5 = 180.

If we add more factors to the LCM is won't be "least" anymore. If we remove factors it won't be a common multiple anymore. Hence, it is indeed the least common multiple.

- A "common divisor" of two numbers is a number that evenly divides both of them. A common divisor of X and Y must have its prime factors contained in both X and Y. For example 2 is a common divisor of 6=2*3 and 4=2*2*3.

- The most trivial common divisor between any two numbers is 1.

- The "greatest common divisor (GCD)" of X and Y is the largest number that divides both of them-- that is, any larger number will not be a common divisor. To find the GCD, we enumerate the prime factors of X and Y. The GCD must contain each prime factor with the lowest multiplicity in X and Y. For example:
If X = 12 = 2*2*3 and Y = 90  = 2*3*3*5
Since 2 is repeated twice in X but only once in Y, we need to include 2 once.
Similarly, 3 is repeated twice in Y and once in X, so we need to include 3 once.
Finally, There is only one 5 in Y and none in X, so we need will not include 5.
That gives GCD = 2*3 = 6.

If we add more factors to the GCD is won't be a common divisor anymore. If we remove factors it won't be "greatest" anymore. Hence, it is indeed the greatest common divisor. If there are no common prime factors in X and Y, the GCD is 1.

Note:
We can also talk about GCD and LCM of more than two numbers, the same principle and procedure apply.