Welcome back! Hope everyone had a wonderful Christmas and New Year!

We talk about inequalities today, how to represent an inequality on the number line, how to solve an inequality by using the Addition and Multiplication Properties of Order.

The most important thing to remember is, when you multiply or divide a negative number on both sides, you need to remember to change the inequality sign, from ‘<‘ to ‘>’, or from ‘>’ to ‘<‘, or from “<=” to “>=”, or from “>=” to “<=”.

Here is this week’s homework from Page 64, 65 and 66:

3a, 3c 3e, 4 (a – g), 5a, 5c, 5e, 5g, 5i, 6 (a – j), 7 (a – d)

We continue to study system of linear equations, with the focus on the word problems involving systems of linear equations.

Such work problem usually can be modeled with two variables x and y, and the given conditions should allow you to come up with two equations using x and y. Then you solve the equations to come up with the solution. Be careful to check if the solution makes sense for the real life.

We look at several challenge problems where by transformation, we can solve them by systems of linear equations.

Homework:

Page 55: #34 – #48

We study system of linear equations. Here we talk about linear equations with two unknown. So a system of linear equations is with two linear equations. The solution of x and y that satisfies both equation is called the solution to the system.

To solve it in algebraic way, there are basically two methods:

With “Elimination Method”, we try to multiply some numbers to one or both equations, so that by adding or subtracting the two equations, one unknown is gone. Once we solve the first unknown, we put in back into one of the original equation to solve the other.

In the simple elimination case, you can just add or subtract the two given equations. Often you need to pick an unknown first, look at the coefficients of that unknown in both equations and come up with the LCM. Then multiply to each equation some number so that the new equations have that unknown with the same coefficient. Now you can do add or subtract.

With “Substitution Method”, you start with one given equation, rewrite the equation using one unknown to represent the other. Now substitute this unknown in the second equation with what you have, so now you have an equation with only one unknown.

We look at the special case where there is no solution to a system of linear equations, which shows two parallel lines in you graph them, and the special case where there can be infinite number of solutions, basically the two equations are identical. In the graph, you have the same line.

Here is this week’s homework:

Page 53, #1 – #12; Page 54, #21 – #25, #31

More graphing and graphs.

We first examined travel graphs. A travel graph is a graph showing the relationship between the distance traveled and the time taken. With a travel graph, we can calculate the average speed (total distance by total time), a speed for a particular period, measuring the total distance traveled, etc. When a travel graph has information about two objects (like two cars), we can further see when a faster catches a slower one, or when two cars meet, etc.

We then looked at solving system of linear equations by graphs. Here we talk about two linear equations with x and y variables and find a solution which satisfies both equations. On the graph, it should be the intersection point of the two lines representing the two linear equations. Of course, there are some special cases where there is no solution due to the two lines being parallel to each other, or there are infinite number of solutions when two lines are the same.

Lastly we examined the graphs for a quadratic equation, taking the form of y = ax^2 + bx + c. Here we just need to student to have a basic knowledge of the shape of a quadratic function, its symmetry axis, the existence of a maximum or minimum point.

Here is this week’s homework:

Page 40: #6, #7, #8, #9, #21, #22, #23, #24, #34, #35, #36

Problems involving financial transactions is the first topic we went over today.

Here we talk about profit (selling price – cost) and profit percentage (profit / cost), discounted price (usual price – discount percentage * usual price), interests. This is mainly the continuation of what we learned last week, about rate, ratio and percentage.

We move on to coordinate plane and representing ordered pairs in the coordinate plane, drawing graphs for linear equation with two variables, and how to use linear graph to solve simple problems.

Homework:

Page 26, #22 – #28

Page 38, #3, #4, #5, #16

I have communicate with every parent about students’ test score and ran an extra session on Saturday 11/10/2012 to go over the tests.

Today we first cover the section for word problems leading to quadratic equations, question like find the number such that the sum of the number and its square is 156. We go from the given conditions to come up with a quadratic equation, solve it. With real world problem, we need to be care to pick the right answer as a quadratic equation can have two solutions, one is possible, the other can be impossible in the real world.

We move on to talk about rate, ratio and percentage, and the corresponding word problems.

Rate is used to describe how a quantity change with regard to another quantity. For example, a worker is paid $84 for 6 hours of work, we say that he is paid $14 for one hour, or at the rate of $14/h.

Ratio is used to compare the magnitude of two similar quantities. It indicates what fraction one quantity is of the other, or how many times one quantity is as much as the other. Here is an example: a sum of money is divided between X and Y in the ratio of 3:5. if X gets $x and Y gets $y, then we can write: x:y = 3:5, or x/y = 3/5, etc.

Percentage is a special fraction with one hundred as its denominator. For example, 15% = 15/100 = 0.15

There are quite some word problems involving rate, ratio, and percentage, and some of them can be challenging,. We talk about several examples in the classroom.

Homework:

Page 21: #27 – #32

Page 24: #2, #4, #5, #9, #12, #14, #18, #19, #21, #25

Today is the review day and we went over what we have learned this semester in the first 6 lessons: Indices (Exponents); Algebraic manipulations; Leanr and quadratic equations.

We will have our midterm exam next week on 11/4. Please go over your homework. To help you prepare, you can also do the following on your workbook:

Page 11, Test Paper 1, question 1 through 8, and Page 32, Test Paper 2, question 1.

Today we mainly focused on quadratic expression factorization (因式分解). This is a skill that is very useful in quickly solving quadratic equations. While it is not hard, it does take practice to get good at. So we spent most of the time letting the students going exercises.

Basically, for the simple form of x^2 + bx + c (that is, the coefficient in front of x^2 is 1) to be able to be factored as (x + p) (x + q), then p + q must be equal to b and pq (p times q) must be equal to c. So the basic steps are, first look at the factors for c and find two whose product is c, then check if their sum equals to b, if so, you have it. If not, try different pairs.

Then for the general form of ax^2 + bx + c (a <> 0), it is the same idea but a little more complicated. For it to be factored into (px + m)(qx + n), then pq (p times q) must be equal to a, mn( m times n) must be equal to c and pn + qm must be equal to b. The basic steps are, first look at the factors for a and pick two so that its product is a, then pick two factors of c so that its product is c, then check if the cross multiplication of those two pairs is b.

A cross multiplication chart will help

px         +m        | +pnx

X             |

qx          +n        | + qmx

—————————————-

pqx^2 + mn   | (pn + qm)x

With this skill, we looked at solving quadratic equations where we factor first.

Note, we have not got into the general solution of a quadratic equation yet.

Homework:

Page 19 18 a c e g i k m  o, 19 b d f h j l n, 20 a ii, f ii, g ii, i ii, j ii, k ii

We talked fractional equations and word problems with fractional equations.

Any equation involving any fraction, either x (unknown) either in enumerator or denominator, is called a fractional equation.

We studied fractional equations with x in enumerators, and we studied equations with x in denominators. When x is in denominators, one important step is to check the answer to make sure that it does not make any denominator zero and throw away any answer that does.

For word problems, it is important to come up with the right equation with the given information.

Homework is as follows:

Page 9: 49 a c d e g i k m n o, 52 a, 53, 54, 55, 56

Note, someone left an umbrella in the classroom. I have it.

We covered the following material today:

• Scientific Notation or Standard Form
• Special Algebraic Rules for Calculations
• Expanding Products

To represent a number, scientists use the form of A * 10^n where 1 <= A < 10, n is an integer. This is called scientific notation or standard form. The key point is, A must be smalled than 10.

We talked about how to convert from a decimal form to the standard form, and vice versa. We talked about how to perform addition, subtraction, multiplication, and division with two numbers in standard form.

For special algebraic rules, we established the following three rules:

• (a + b)^2 = a^2 + 2ab + b^2
• (a – b)^2 = a^2 -2ab + b^2
• (a^2 – b^2) = (a + b)*(a – b)

For expanding products, the key point we went over is the distributive law, using which one should be able to expanding products in any form.

For homework, it is on the Workbook:

• Page 4: 11b, 11d, 11f, 12d, 12f, 12h, 15a, 15b, 15c, 15d
• Page 5: 18a, 18b
• Page 6: 1b, 1d, 2c, 2d, 3d, 3f
• Page 7: 4a, 4c, 4e, 5c, 5e, 5f

http://blog.newtonchineseschool.org/wangweidong/files/2012/09/HW_L2_P4.pdf

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