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.

We study scale drawings and scale maps today.

A scale drawing is a reduced or enlarged drawing of an actual object. It is used in science, engineering, construction, and architecture to show how various parts of an object should be made and assembled.

The Scale Factor of a scale drawing is the ratio of a length in the drawing to the corresponding length of the actual object. When a scale factor is bigger than 1, you have an enlarged dawing; when the scale factor is less than 1, you have a reduced drawing.

The scale factor is also known as its scale. A Scale us usually expressed as a ratio of 1:n or a fraction 1/n where i unit length on the drawing represents an actual length of n units.

Note, units must be the same when finding the ratio. Ratios have no unit. And, a scale factor is the same as the scale of a drawing.

Give the scale factor and some measurement for some sides in the drawing for an object, we can calculate the corresponding sides for the actual object. And vice versa. We can also use them to estimate.

Scale map is a special type of scale drawing. A scale map usually comes with a bar scale or a linear scale when it shows you a unit length on the map is equal to how much length in the actual land.

With the scale from a map, we can calculate the actual length or estimate some length.

If a map’s scale is 1:r, then the ratio of an area of an object on the map to the area of the real object on land is 1 : r^2.

Homework: P43. #1 (a, c, e, g), #2 (b, d, f), #3, (a, c, e), $4 (b, d, f), #5 (a, c, e). #11, #12, #13, #14.

We went over such typical mistakes in the final exam last week. It appears that solving inequalities with fractions are a bis issues for quite some students.

I still need some time to finalize the exam score and will give them back next week.

For the homework today, I am posting the exam again below. You can have your kids do the inequality question again and watch how they do. I am also posting the answers below in case YOU want to look at them:

http://www.newtonchineseschool.org/teachers/wangweidong/Math7FinalExamFall2014.pdf