Math 9, Lesson 10, Fall 2019, 11/10/2019

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We study multiplication of a matrix by a scalar and multiplication of two matrices.

A matrix can be multiplied by a real number (usually called a scalar). If k is a scalar, then the scalar multiplication of a matrix A by k, denoted by kA, is obtained by multiplying every element of A by k.

Matrices multiplication is defined as follows:

If A is a matrix of order m x n and B a matrix of order n x p, then the product AB is a matrix of order m x p whose element at the ith row and jth column is the sum of the products of the corresponding elements in the ith row of A and jth column of B.

If the column number of A is not equal to the row number of B, then AB is undefined.

We introduce Identity Matrix of order n, which has 1 on its major diagonal line and 0 anywhere else..

Homework:

Workbook Page 21, #7, #10 – #12, #18, #19. Please use last lesson’s homework pages for the questions.

Answers to last week’s homework:

Answers page1, answers page2

Math 9, Lesson 9, Fall 2019, 11/03/2019

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We start a new topic today. We study matrices.

In real life, we often use tables to help organize data. If we abstract the concept by extracting the data from a table and arrange them in a rows and columns with brackets, we call this rectangular array of numbers a matrix. The numbers in a matrix are called entries or elements. An element is identified by its row and column positions in the matrix. If a matrix has m rows and n columns, we say that the order of this matrix is m x n. A matrix having the same number of rows and columns is called a square matrix. For a square matrix, we can simply say its order with the number of rows.

We usually use capital letters to represent matrices.

Two matrices A and B are equal, written as A = B, if they have the same order and their corresponding elements are equal.

We then discuss the addition and subtraction of two same-order matrices.

If A and B are two matrices of the same order, then sum A+B is the matrix obtained by adding the corresponding elements in A and B.

Similarly, we define subtraction of two same-order matrices.

We talk about zero matrix where all elements are zero. It is often represented  as O.

Homework:

Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8 (please keep these pages, we will need them for the next two lessons)

Workbook Page 19, #1 – #6, #12, #14, #16

Math9, Lesson 8, Fall 2019, 10/27/2019

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We have our midterm exam today.

Math 9, Lesson 7, Fall 2019, 10/19/2019

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We have a review lesson today, in preparation for the midterm exam the week after.

Students should review their homework.

For answers to the last week’s homework, please see the answers sheet on the last week’s post.

 

Math 9, Lesson 6, Fall 2019, 10/13/2019

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We continue study of standard deviation. We first review the formula for ungrouped data and grouped data.

We apply the standard deviation to help analyze two data sets. While means can give us the picture of average, standard deviation helps us to determine how consistent the data are. For example, if we look at two sets of data representing two basketball players’ scores, the lower the standard deviation, the more consistent the player is.

Homework:

Print these pages (they are links, click to download): Page 1Page 2Page 3Page 4

Workbook Page 2, #7, 8, 9, 10, 12, 14, 16.

Answers to the last week’s homework:

Answers

 

Math 9, Lesson 5, Fall 2019, 10/6/2019

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We study standard deviation. In statistics, we have learned using mean to find the average, and we have learned to use range and interquartile range to see the spread of the data. But both range and interquartile range are not very good as the tool to represent the spread of the data.

A better way is the measure called standard deviation:

Standard Deviation SD = sqrt(sum of (Xi – AvgX)^2/N)

where N is the total number of data, AvgX is the mean of the data. Xi – AvgX is called the deviation of Xi from the mean AvgX for each i = 1, 2, …, N

For a set of grouped data in the form of a frequency table, we have

Mean AvgX = sum of fx / sum of f

where x is the class mark of each class and f is the frequency of the corresponding class.

Standard Deviation SD = sqrt(sum of f(Xi – AvgX)^2 / sum of f)

[Sorry, I can’t type using the summation notation sigma, which is what we learn in the class. This will simplify the writing.]

Here is a written proof of going from the definition to a formulae often used:

sqrt(∑(Xi – AvgX)^2 / N) = ∑Xi^2 / N – (∑Xi / N)^2

I showed the proof in the classroom, but students may not get it.

Homework (pages are links, click to download):

Page 1  and Page 2: #1, #2, #3, #4, #5, #6.

Math 9, Lesson 4, Fall 2019, 9/29/2019

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We continue our study of complex number. Today we talk about representing complex numbers in Complex Plane and how to calculate the magnitude of a complex number. We also look at various properties of a complex number with its conjugate.

Homework (pages are links. Click to download): Page 1Page 2Page 3,

3.2.1 – 3.2.7, 3,3.1 – 3.3.1 – 3.3.7

AoPS Algebra 2, Lesson 4, Fall 2019, 9/29/2019

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We study circles and ellipses. We will study hyperbolas next time.

More info is communicated via email.

AoPS Algebra 2, Lesson 3, Fall 2019, 9/22/2019

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We study parabola, with its relationship to quadratic functions.

We will study circles and ellipses next time.

More info is communicated via email.

AoPS Algebra 2, Lesson 2, Fall 2019, 9/15/2019

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We finish the chapter on complex number.

More info is communicated via email.