First of all, congratulations to the following students in our class who have won in the recent school wide Math contest:

Karen Li is the sole 1st place winner in the  6th grade category with $20 prize.

Eugene Jiang, Robert Sheng, Alexander Deng, Jiming Xu are among other prize winners in the 6th grade category and will each get $10 prize.

Congratulations again!

Since we have finished covering the regular content in the textbook, I am adding a little more today. For the last two lessons, we will have review and final exam.

We study combinations and permutations, with the introduction of factorial. A combination is a group of outcomes where the order does not matter, while permutation is a group of outcomes where the order does matter. Factorial is a way of computation. The factorial of a number is the product of the number and all the natural numbers less than the number.

n! = n x (n-1) x (n-3) x … x 3 x 2 x 1

The number of permutations of choosing r things out of n things at a time is denoted as nPr and

nPr = n! / (n-r)!

The number of combinations of choosing r things out of n things at a time is denoted as nCr and

nCr = n! / [r! x (n-r)!]

Today’s material can be found here for Section 10.8.

Homework: download this page

Page 765, #12 – #23.

We study probabilities for two events that are dependent to each other and we have the following:

If A and B are dependent events, then P(A and B) = P(A) * P(B after A).

We go over quite some examples in the classroom.

Homework:

HomeWorkL13

We continue to study probabilities with more than one events.

Two events, A and B, are said to be mutually exclusive if they cannot occur at the same time. In genenral, if A and B are two mutually exclusive events, then the probability of A or B occurring is: P(A or B) = P(A) + P(B).

Two events are said to be independent events if the occurrence or non-occurrence of one event does not affect the probability of the occurrence of the other event. In general, if A and B are independent events, the probability of both events A and B occurring is the product of their individual probabilities: P(A and B) = P(A) x P(B).

Homework:

Page 66, #9 – #18.

We can use set notation to describe probabilities.

The sample space, usually denoted by S, of a random experiment is the set of all possible outcomes. Thus, each outcome is an element of the sample space,and each event, being a collection of some outcomes, is a subset of the sample space.

Thus P(E) = n(E)/n(S) where n(E) is the number of outcomes in the event E, and n(S) is the number of outcomes in the sample space S.

And we study the basic properties of probabilities: 0 <= P(E) <= 1, and P(E’) = 1 – P(E).

We further study how to use a possibility diagram or a tree diagram to represent a sample space to help us figure out the probability problem.

Homework:

P61, #24 – #29; P65, #3 – #8.

We study the meaning of probability.

Probability is a branch of mathematics that studies the likelihood, or chance, of a phenomenon happening.

A random experiment is a process in which the result cannot be predicated with certainty. The result is called an outcome of the experiment. An experiment is likely to have more than one possible outcome.

A collection of outcomes is called an event. For example, rolling a dice, you get an odd number.

A measure of how likely an event E will take place is called the probability of that event, and it is denoted by P(E). Mathematically, it is defined as:

The probability of an event E, P(E), in a random experiment with equally likely outcomes is:

P(E) = (number of outcomes favorable to event E) / (total number of possible outcomes)

We also distinguish experimental probability from theoretical probability. Experimental probably refers to the probability of an event occurring when an experiment was conducted, as P(E) = (number of times E occurs) / (number of trials).

Theoretical probability, or simply  probability, P(E), is determined by noting all possible outcomes theoretically, and determine how likely the given outcome is for the event E. P(E) = (# of outcomes for event E) / (total # of possible outcomes).

Homework:

P58, #7, 8, 9, 10, 1, 14, 15, 16, 21, 23.

We start the new chapter on Probability of Simple Events. Today we study the notation of set.

We use the term set to mean a collection of well-defined distinct objects. The objects in a set are called elements or members. Note, a set only contains distinct elements, no duplicates, and the order in which the element appear is insignificant.

Two sets A and A are equal, written as A = B, if they have exactly the same elements.

If every element of a set A is also an element of a set B, then A is said to be a subset of B. And if A is a subset of B, but A is not equal to B, then A is said to be a proper subset of B.

We define a universal set as an overall set of a particular problem when all sets for the problem are subsets of this set. Note, different problems may have different universal sets, and a problem can have more than one universal set.

We also define the empty set which contains no elements and is a subset of any set.

Last we define  the compliment of a set A as the set that contains elements that do not belong to the set A but belong to a universal set of A.

Note, pretty everything we study is by definition, so it is important that students understand the definitions.

Homework:

Page 57, #1 – #6, #17 – #20.

We study Mode as another way to represent the center in a data set. The mode of a data set is the value that occurs most often. In a data set, there can be more than one mode, or there can be no mode (when no number occurs more often than others).

We further comparebetween Mean, Median, and Mode.

Homework:

Page 50, #8, #10, #15, #21 – 25.

We study some basic data analysis methods.

There are two common days to measure the center of a data set: the mean, and the median.

The Mean of a data set is the sum of all data values divided by the total number of data.

While the mean tells the average, we need more to know how the data are distributed, or data deviation.

The mean absolute deviation (MAD) of a data set is the total distance of all data values from the mean value divided by the number of values. The bigger this values is, the more scattered the data; the smaller this value is, the more clustered the data.

Mean does not always tell a good story of the average, as some smaller set of data can affect the outcome. Another way to calculate the center is the median, which is the middle value when the data are are arranged in order from the smallest to the largest.

The mean, the median, and the MAD, together they give us better understanding of the data.

Homework:

P50, #4(i)(iii)(v)(vii), #5, #6(b, d, f, h), #7, #16 – #20.

We will spend the next few weeks study basic statistics.

Statistics is a branch of mathematics that involves collecting, organizing, interpreting, presenting, and analyzing data. Based on studies of data obtained, people are able to draw conclusions, make decisions, and plan wisely.

We talk about different ways of collecting data, including taking measurement in experiments, observing outcomes of events, conducting surveys, and reading statistical publications.

We further look at dot plots to represent data collected.

Homework: Page 49. #1, #2, #3, #12, #13, #14.

We study direct proportion and inverse proportion.

When two quantities are related to each other, their changes may follow certain patterns. We study two kinds.

When two quantities, x and y, vary in such a way that y/x is a constant, they are said to be in direct proportion, or y is said to be directly proportional to x.

We have y/x = k, where k is a constant. Or y = kx. On a drawing, it is a straight line passing through the origin.

For example, for a car traveling at a constant speed, the distance traveled  is directly proportional to the time taken. The constant here is known as the speed.

When two quantities, x and y, vary in such a way that xy is a constant, they are said to be in inverse proportion, or y is said to be inversely proportional to x.

We have xy = k, where k is a constant. Or y = k * 1/x. On a drawing, it is a hyperbola curve.

An example can be the speed and the time for a car travelling uniformly from A to B.

Homework:

Page 45: #15 – #20, #23 – #25.