We move on to the chapter about mathematics in practical situations, and today we study profit, loss, simple interest, and compound interest. These topics are not new for most students.

Profit is achieved when the selling price is bigger than the cost price. The difference is the profit. Profit over the cost defines the profit percentage.

When the selling price is lower than the cost, you have a loss, which is defined as the difference between the cost and the selling price. The loss over the cost defines the cost percentage.

Banking usually deals with interest. The original amount you deposit into a bank account is called the principal. The interest is calculated based on an interest rate, which usually is given as an annual percentage.

In simple interest calculation, the interest calculation is based solely on the principal. So interest $I earned on the principal $P at the interest rate of i% annually for a period of n years is given by:

I = P x i% x n

In compound interest calculation, interest is calculated on prior interest accumulated over a period of time. Suppose an initial principal $P is invested at i% per period with compound interest for n periods, let An be the amount after n periods, then:

An = P(1 + i%)^n

Homework:

Workbook Page 41, #1 – #4, #13 – #16.

We review measuration, with the focus on:

• Areas of parallelogram and trapezium;
• Volume and surface area of a prism;
• Volume and surface area of a cylinder;
• Volume and surface area of a pyramid;
• Volume and surface area of a cone;
• Volume and surface area of a sphere;
• Length of an arc, area of a sector in a circle

Homework: from Page1, Page2, Page3:

#5- #18

We review Pythagoras’ Theorem and basic trigonometry. We talk about the Sine Rule and Cosine Rule and how to use them to help solve some geometry problems.

We talk about terminologies like Angel of Elevation, Angle of Depression, the Bearing.

For homework, download: Page 1, Page 2, Page 3

#4 – #17.

We continue our study of vectors.

We can simplify the position of a point P on a plane by the vector OP where O is a reference point on the plane. The vector OP is called the position vector of P with respect to the reference point O. That is, every point on the plane can be represented by its position vector wrt the reference point O.

Then vector PQ = vector OQ – vector OP.

Furthermore, when we look at the coordinate plane for a point P (x, y), we can use its coordinates x and y to represent its position vector with respect to the origin O. We introduce column vector notation. A column vector is like a special form of a matrix (2 rows, 1 column). All matrix operations apply to column vectors.

We can apply what we have learned about vectors to help us solve geometry problems.

Notice when vector AB = vector DC, it means line AB is parallel to line DC and AB = DC.

Homework:

Workbook Page 33: # 8, #9, #13, #19 – #26

For a given vector a, a vector which has the same magnitude but in the opposite direction of a is called the negative of vector a , and is denoted by –a.

The subtraction of a vector v from a vector u can be denoted as adding the vector –v to u: u – v = u + (-v).

A vector with the same initial point and the terminal point has zero magnitude. It is called a zero vector, or a null vector, and is denoted by 0. a + (-a) = a  – a = 0.

When a vector a is multiplied by a constant k, the product ka is called a scalar multiplication of a and is defined as follows:

* If k > 0, ka is a vector with magnitude k|a| and in the same direction as a;

* If k < 0, ka is a vector with magnitude -k|a| and in the opposite direction of a;

* If k = 0, ka is a zero vector 0.

Homework:

Workbook Page 33, #6, #7, #10, #11, #14 – #18, #20, #21.

We start the new chapter on vectors.

While a scalor is a quantity that has only magnitude, a vector is a quantity that has both magnitude and direction. We usually use directed line segment to represent a vector. The direction of the line segment, indicated by an arrow, represents the direction of the vector, and the length of the line segment represents the magnitude of the vector.

Two vectors are equal if they have equal magnitude and are in the same direction.

For vector additions, we study the triangle law and parallelogram law of vector additions.

Homework:

Workbook P 31, #1, #2, #3, #4, #5, #12, #15.

Students will bring home their final exam papers. I have uploaded answers for the exam here.

Apart from going over some exam questions, we review the properties of circles.

In the symmetry properties category, we have the following:

• Equal chords are  equidistant from the center;
• The perpendicular bisector of a chord passes through the center;
• Tangents from an external point are equal in length;
• The line joining an external point to the center bisects the angle between the tangents

In the category of angle properties, we have:

• Angle in a semicircle is a right angle;
• Angle between tangents and radius is a right angle;
• Angle at the center is twice the angle at the circumference;
• Angle in the same segment are equal;
• Angle in opposite segments are supplementary.

Homework:

Page 1, Page 2, Page 3

#2, #4, #6, #8, #10, #12, #14, #16, #18