Today we are onto geometric series, which talks about the sum of the first n terms of a geometric sequence. Given a geometric seuqnece with a as its first term, and r as its common ratio, the sum of its first n terms can be written as:

Sn = a(r^n – 1)/(r-1)

We talk about whether a series is convergent (which means the sum is bounded by a constant), or divergent (the series always go bigger or smaller), or it is un-determinant.

When r > 1, the series is divergent;

When r = 1, it is a constant sequence, Sn = a*n, so it is divergent;

When -1 < r < 1, or |r| < 1, the series converges to a/(1-r);

When r = -1, the series jumps between 0 and a, so it is undeterminant;

When r < -1, the series jumps between positive and negative, so it is also undeterminant.

The homework is as follows:

http://www.newtonchineseschool.org/teachers/wangweidong/HomeworkGeometricSeries.pdf

We started goemetric sequence today. A geometric sequence is defined by its first term and a common ratio. For a geometric sequence, any two consecutive numbers have the same ratio, the common ratio.

With the first term as a and the common ratio as r, the nth term is: an = ar^(n-1).

With this, we can proof the following properties for a geometric sequence:

1.  the square of nth term equals the product of (n-k)th term and (n+k)th term;

2. if {an} and {bn} are two geometri sequences, then {anbn} and {an/bn} are geometric sequences as well.

We can use the given nth term formulae to solve a variety of problems about geometric sequences.

Note, when working on sequences, it is important to make it right with the subscripts.

Homework is as follows:

http://www.newtonchineseschool.org/teachers/wangweidong/HomeworkGeometricSequence.pdf

We covered Arithmetic Series today.

When we add a group of consecutive terms of an arithmetic sequence, we form an arithmetic series. For an arithmetic sequence with the first term as a and the common difference d, the sum of the first n terms of the sequenece is:

Sn = n(a1 + an)/2

That is, the sum is the average of the first and last terms, times the number of terms. We want the students to understand how we get to that formulae, not just memorize it (which is the least they should do).

We did quite some in-class exercises mixing the arithmetic sequence formulae for the nth term and the sum of the first n terms. Usually the steps go like this: first figure of how many terms we are talking about, then see if we have the first term and the last term, if so, we can already find the sum; if not, look at the givens and figure out the first term and the common difference.

Another set of problems usually give two conditions about some terms in an arithmetic sequence. Because we want to find out a and d, two conditions usually is all we need to find the unknown of a and d. With a and d, we can find any term in an arithmetic sequnce.

Homework is the following:

http://www.newtonchineseschool.org/teachers/wangweidong/HomeworkArithmeticSeries.pdf

Today we started the 2nd half of the Spring semester where we will go over new material that is not in the “Algebra I” text book. The material can be found under:

http://www.newtonchineseschool.org/teachers/wangweidong/SequenceSeries_Part1.pdf

http://www.newtonchineseschool.org/teachers/wangweidong/SequenceSeries_Part2.pdf

http://www.newtonchineseschool.org/teachers/wangweidong/SequenceSeries_Part3.pdf

We talked about arithmetic sequence where any two consecutive numbers in the sequence have the common difference. For an arithmetic sequence with the first term as a and the common difference as d, its nth term is:

      an = a + (n-1)d

As I was telling the students, it takes two parameters, the first term and the common difference, to uniquely identify an arithmetic sequence.

We did some in-class exercise in determining such parameters.

For homework, click the link below:

http://www.newtonchineseschool.org/teachers/wangweidong/HomeworkArithmeticSequence.pdf

Today is our review day as we will have the midterm exam next week.

First, we went over the making square part (Ch 12.4) that the substitute teacher did not get to finish last week. In particular, we talked about how to derive the general formulae for solving a generic quadratic equation ax^2 + bx +c. I hope every student really understands this.

Some students missed last week’s homework because they didn’t see it in my last post (I posted two articles last week, the first one with homework, the second one didn’t). If so, please get it this week and do it. It will help you prepare for the midterm exam.

We then went over what we learned so far this semester, mainly Chapter 10 and 11, plus 12.4 (making squares).

For homework, please do the Chapter 11 standarized test in the book. This is to help you prepare. You should also look back to Chapter 10’s test to help you review as well.

Sorry that I miss the school today due to Lexington Town business.

The material covered today are the following:

Ch 11.7 on dividing polynomials; Ch 11.8 on rational equations.

It is important to remember that when solving rational equation, always check your answer and throw away any that makes any denominator zero.

We also cover Ch 12.4 on conpleting squares. This is to let students learn how to derive the general solution for a quadratic equation of ax^2 + bx + c = 0

For homework, visit the link below:

http://blog.newtonchineseschool.org/wangweidong/files/2012/03/Math7_Spring_HW_L6.pdf

We started the topic of rational expression operation today. A rational expression is a fraction with either numerator or denominator or both being a polynomial. As such, all rules for dealing with factions apply.

A rational expression is simplified if there is no common factor between the numerator and denominator (other than 1 or -1).

We covered rational expression multiplication and division, followed by addition and substraction. The goal is to reach a rational expression in its simplified form after the operation.

Homework link can be found with the link below:

http://blog.newtonchineseschool.org/wangweidong/files/2012/03/Math7_Spring_HW_L5.pdf

Sorry I could not go to school to teach today, as I had two important campaign events to go to.

As some of you know, I am running for Lexington Housing Authority and Town Meeting Member this time and the election day is coming Tuesday 3/6. This weekend is the last weenend before the election.

So, if you are a Lexington resident, I hope you will go out to vote on Tuesday and vote for me.

That said, here is the material covered today by a substitute teacher:

Section 11.1 on Ratio and Propertion, two important properties to remember:

1. Reciprocal Property: if a/b = c/d, then b/a = d/c if they exist

2. Cross Product Property: if a/b = c/d, then ad = bc

Section 11.2 on Percents: number being compared to base = percent of base number

Section 11.3 on Direct and Inverse Variation:

The variables x and y vary directly if for a constant k, y/x = k or y = kx, k not equal to 0.

The variables x and y vary inversely if for a constant k, xy = k, or y = k/x, k not euqal to 0.

Here is the homework:

http://blog.newtonchineseschool.org/wangweidong/files/2012/03/Math7_Spring_HW_L4.pdf

We spent more time on factorization today, as we really want the students to get good at. We did more classroom exercises.

For homework, it is to take the Chapter 10 Stanaridized Test:

http://blog.newtonchineseschool.org/wangweidong/files/2012/02/Math7_Spring_HW_L3.pdf

Note:

I won’t be able to come in to teach next Sunday. There will be a substitute teacher. We will start Chapter 11.

Today we continued to cover the content of factorization, with focus on factoring ax^2 + bx + c.

If it can be factored, then we can write the following:

ax^2 + bx +c = (mx + p) (nx + q) = mnx^2 + (mq+np)x + pq

Then we have:

a = mn

b = mq + np

c = pq

That is, we want to find two factors m and n from a, so that a = mn, we want to find two factors p and q from c, so that c = pq, and such that the mq + np = b.

If you list m, n, p, q the following way:

m        p

X

n        q

It is the sum of the cross-product, m * q + n * p that must equal to b.

We also talked abut factoring by group. For example:

x^3 +2x^2 +3x +6 = x^2(x+2) + 3(x+2) = (x+2)(x^2 + 2)

Factorization takes time to get good at and only practice can get you there. So here is the homework:

http://blog.newtonchineseschool.org/wangweidong/files/2012/02/Math7_Spring_HW_L2.pdf