AOPS Algebra 1 Online Course, Lesson23 , 04/05/2020

Today, students studied the graph of quadratic equations y=ax^2+bx+c and understand the feature of the graphs in the shape of “parabola”, such as, leading coefficient “a” determines the graph open direction, upward or downward, vertex point represents the minimum point or maximum points, symmetric graph with the axis of symmetry passing through the vertex point. In order to figure out the feature easily, students learned to convert standard quadratic form ax^2+bx+c to vertex form a(x-h)^2+k, where (h,k) is vertex point, x=h is the axis of symmetry. To solve quadratic equation ax^2+bx+c=0 in graph method, plot y=ax^2+bx+c, then check if there is x-intercepts of the graph with x-axis that are the solution of the quadratic equation.
Here is the homework for today’s class:

P.397, 13.4.2

P.414, 14.1.1, additionally find the solutions of quadratic equation when y=0

14.1.2, additionally find the solutions of quadratic equation when y=0


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Singapore Math 8th Online Class, Lesson 23, 04/05/2020

Today, students worked on the graphing method to solve the quadratic equations, understanding the relation of quadratic function y=ax^2+bx+c and quadratic equation, ax^2+bx+c=0. First of all, students learned to plot the quadratic functions, mostly using traditional way, connecting some ordered pairs. Then, find the graph x-intercepts which are real solutions of the equation of ax^2+bx+c=0. If there is no intersections of graph and x-axis, it indicates that there are no real solutions in that quadratic equation. Besides the graphing method in solving the quadratic equations, students also reviewed the factorization method to solve the equations. It is clear that factorization method is limited to those that are able to be written in the factor form. The graph method extends to all that have real number solutions, although the results are not accurate due to the error in reading graph.

Here is the homework to cover today’s contents:
workbook 8B
Ch. 14
4. (a) (ii) (iii) (v) (vi); (b) (ii) (iii) (v) (vi), if the equation can be written in factor form, get the solutions to verify the results from the graphs.

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Singapore Math 8th Online Class, Lesson 22, 03/29/2020

Today, students studied the graph application in practical situation, distance vs. time, related with linear graph. The slope in the linear relation is speed of the movement of distance with time. Because of the straight line of the relationship, the speed is constant. On the other hand, when the relation of distance vs time is not linear, the slope is varying, indicating that the speed of movement is changing. The change is identified as decrease and increase of speed, depending upon the slope changing trend, increasing or decreasing. The graph of distance vis time reveals the information of movement status and the speed. Also, the multiple movement graph can be used to determine the relations among the different cases, such as when/where they meet.

Here is the homework assignment for today’s contents:
Workbook 8B
Ch. 9
1, 2, 4, 5, 6, 7, 8, 9, 14, 15, 17, 18, 19, 20, 21, 23.

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AOPS Algebra 1 Online Course, Lesson22 , 03/29/2020

Today, students continued to work on “completing the square” and solving the quadratic equations by using “Completing the square”. Also, quadratic formula was also introduced in the class. As I mentioned, the formula was derived from solving the quadratic equation ax^2+bx+c=0, by using the “completing the square” method. In other words, the formula is the final solutions applicable for all  quadratic equations. Except for request, people directly use the formula, instead of using completing the square. In addition, using formula to solve equation to find the roots could provide another way to factorize a quadratic equation. In the formula approach, the new term “discriminant”=b^2-4ac is used to evaluate any quadratic equation with two real roots, double root, or two non-real roots, as it is positive, zero, or negative, respectively.
This is an topic with so much fun during the studying. They are two more methods to solve quadratic equations along with the first method students learned before, factorization. There is one more method to be introduced in next class time.
Here is the homework covering today’s contents.
P.384, 13.2.1, 13.2.2(a)

P.391, 13.3.1, 13.3.2, 13.3.4, 13.3.5

P.397, 13.4.1, 13.4.4

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AOPS Algebra 1 Online Course, Lesson21 , 03/22/2020

Today is the 3rd formal online class on AOPS Algebra 1 course. In today’s class, students studied the complex numbers and the operation of complex numbers, such as addition/subtraction, multiplication and division. In addition and subtraction, we only need to add/subtract real parts together and imaginary parts together. In division, it is very similar to the rationalization of denominators discussed in previous class, that is, multiplying the conjugate complex number to generate the real numbers. The concept of complex number will be used in solving quadratic equations by the completing the square. Regarding the completing the square, students need to understand how to create the perfect square by using the formula a^2+2ab+b^2=(a+b)^2. Based on this foundation, we are able to extend the knowledge into the method of solving quadratic equation, to be taught in next class.
Here is the homework assignment:

P.362, 12.2.1, 12.2.2, 12.2.3, 12.2.5

P.368, 12.3.1, 12.3.2, 12.3.3, 12.3.4

P.378, 13.1.1, 13.1.2, 13.1.3

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Singapore Math 8th Online Class, Lesson 21, 03/22/2020

Today, students started to study the function, linear function, and quadratic function, primarily focusing on the features of graphs of each function. It is the new chapter in book 8B and workbook 8B. In linear function, slope, and intercepts are main keys to represent the feature of a straight line. In quadratic function, students need to understand the cores of the function, such as the shape of the graph, vertex point, symmetric-line of symmetry, intercepts and the relation of function and equation. Linear function and quadratic function knowledge is applied to the questions in real work, such as driving distance, speed, time, suspension in bridge, throwing ball in the air, etc. In other word, graphs are useful tool in solving some engineering type of questions without requirement of accuracy.

Here is the homework assignment for today:
Workbook 8B

1, 2, 3, 5, 6, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 24.

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AOPS Algebra 1 Online Course, Lesson20 , 03/15/2020

In today’s class, students continued to work on the special factorization, such as the sum and difference of cubes and their applications in factoring, simplifications, number computations, and importantly the rationalization of denominators. I strongly ask students to remember those formulas. With the information in their mind, people would pick up the knowledge quickly and form an effective method to solve problems, besides, it may cause less mistakes.  
Again, solving problem is not the only goal, grasping the core in any question and applying the ideas to other questions are what students should keep pursuing in their studying.
Here is the homework to cover today’s contents:
P.343, 11.3.1, 11.3.2, 11.3.3, 11.3.4(a)

P.348, 11.4.1, 11.4.2, 11.4.3, 11.4.4

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Singapore Math 8th Online Class, Lesson 20, 03/15/2020

Today, students continued to study the applications of angle systems in parallel lines, triangle, quadrilateral to the different types of the multiple sides figures. In addition, polygons were introduced, primarily the convex polygons, including the sum of interior angles of n-sided polygons, and the sum of exterior angles of polygons.

Students need to focus on the analysis of the questions and find the approaches to reach the solutions. Especially in geometry study, statements/reasons are critical for students to practice their logical thinking and presentations of their ideas.

Here is today’s homework assignment:

Workbook 8A
Ch. 7:
5.(b), 8.(a), 9.(a)(b)(e), 10, 11, 16, 18, 19, 20, 21.

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AOPS Algebra 1 Online Course, Lesson 19, 03/8/2020

In today’s class, students continued to work on solving quadratic equations by factoring. In addition, understanding the sum of solutions and product of the solutions is helpful to understand the relation of solutions and equations. In solving problems, pursuing more approaches for one question can train students to open their minds and look for the efficient way. Besides the regular factorization in solving quadratic equation, students learned two special expressions for factorization, a^2+2ab+b^2=(a+b)^2, and a^2-b^2=(a+b)(a-b). These special factorization formulas are also useful in some number computing, solving algebraic questions, such as doing big number square, changing the expressions from addition/subtraction form to multiplication form, etc.
Here is the homework to cover today’s contents:
P.307, 10.3.4, 10.3.5

P.311, 10.4.3, 10.4.4, 10/4.5

P.331, 11.1.1, 11.1.2(a)(b)(c)(d), 11.1.5(a)

P.338, 11.2.1, 11.2.2, 11.2.3, 11.2.4

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Singapore Math 8th Online Class, Lesson 19, 03/8/2020

Today, for the first time, we had a successful online class and most of students were able to join the class.

In the class, I continued to show to solve angles problems in parallel line system, most importantly, the procedure of presenting the prove/calculation work. The interior/exterior angles in triangle were introduced and applied to calculate the angles in different types of figures. Some examples regarding the quadrilateral figures were combined with the principles in triangles and parallel line theorems in order to compute the size of length and angles in the figures.

Here is the homework to cover today’s contents:

workbook 8A. Ch. 7

  1. (a)(c)(f); 4. (a)(c)(d); 5. (a)(c); 6; 7.(a)(c); 8.(b)(c)(d); 14.(a)(b)(d)(e); 15.(b)(d).
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