**Mathematics Lesson Notes SS2 Third Term**

**SCHEME OF WORK**

**WEEK 1: SS2 MATHEMATICS THIRD TERM: REVISION: TRIGONOMETRICAL RATIOS**

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**WEEK 2: SS2 MATHEMATICS THIRD TERM: CIRCLE THEOREM: TANGENT PROPERTIES OF CIRCLE**

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**WEEK 3: SS2 MATHEMATICS THIRD TERM: TRIGONOMETRY: DERIVATION OF SINE RULE AND APPLICATION; DERIVATION OF COSINE RULE AND APPLICATION**

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**WEEK 4: SS2 MATHEMATICS THIRD TERM: BEARINGS: ANGLES OF ELEVATION AND DEPRESSION**

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**WEEK 5: SS2 MATHEMATICS THIRD TERM: STATISTICS- CALCULATION OF CLASS BOUNDARIES AND INTERVALS**

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**WEEK 6: SS2 MATHEMATICS THIRD TERM: CUMULATIVE FREQUENCY GRAPH**

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**WEEK 7: SS2 MATHEMATICS THIRD TERM: DETERMINATION OF THE MEAN, MEDIAN AND MODE OF GROUPED FREQUENCY DATA (REVISION)**

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**WEEK 8: SS 2 MATHEMATICS THIRD TERM: PROBLEM-SOLVING ON NUMBER BASES EXPANSION, CONVERSION AND RELATIONSHIP**

**WEEK 9: SS 2 MATHEMATICS THIRD TERM INDICIAL EQUATION**

# Mathematics Lesson Note For SS2 (ThirdTerm)

Below are the 2022 complete SS2 Third Term Mathematics Lesson Note

Week 1

# Introduction to Trigonometric Ratios

Trigonometry is the study of triangles in relation to their sides and angles and many other areas which find applications in many disciplines. In particular, trigonometry functions have come to play great roles in science. For example, in physics, it is used when we want to analyse different kinds of waves, like sound waves, radio waves, light waves, etc. Also trigonometric ideas are of great importance to surveying, navigation and engineering.

We shall, however at this stage be concerned with the elementary ideas of trigonometry and their application.

Trigonometrical Ratios (Sine, Cosine and Tangent)

The basic trigonometric ratios are defined in terms of the sides of a right-angled triangle. It is necessary to recall that in a right-angled ∆ABC, with *L*C = 90^{0}, the side AB opposite the 90^{0} is called the hypotenuse.

The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express. To learn more, click **here **

**WEEK 2**

Introduction

A Tangent of a Circle has two defining properties:

- A tang
- ent intersects a circle in exactly one place
- The tangent intersects the circle’s radius at a 90° angle

Since a tangent only touches the circle at exactly one and only one point, that point must be perpendicular to a radius.

To test out the interconnected relationship of these two defining traits of a tangent, try the interactive applet.

The point where the tangent and the circle intersect is called the point of tangency. To learn more, click **here**

**WEEK 3**

# Trigonometry: Derivation of Sine Rule and Application; Derivation of Cosins Rule and Application

# Introduction

The sine rule formula states that the ratio of a side to the sine function applied to the corresponding angle is same for all sides of the triangle. For a triangle ABC, sine rule can be stated as given below:

The sine rule formula can be used to find the measure of unknown angle or side of a triangle. It can be used to predict unknown values for two congruent triangles.

If for a given triangle, a, b, and c are the lengths of sides , and A, B, and C are the opposite angles then the sine rule formula is also stated as the reciprocal of this equation:

Derivation of Sine Rule

To derive the formula, erect an altitude through B and label it h_{B} as shown below. Expressing h_{B} in terms of the side and the sine of the angle will lead to the formula of the sine law. To learn more, click **here**

**WEEK 4 **

# Bearings: Angles of Elevation and Depression

# Introduction to Angles of Elevation and Depression

Any surface which is parallel to the surface of the earth is said to be horizontal. For example, the surface of liquid in a container is always horizontal, even if the container is held at an angle.

The floor of your classroom is horizontal. Any line drawn on horizontal surface is will also be horizontal. Any line or surface which is perpendicular to a surface is said to be vertical. The walls of your classroom are vertical. A plum-line is a mass which hangs freely on a thread.

1. Say whether the following are horizontal or vertical, or neither:

a. the table top

b. the door

c. the pictures

d. the floor boards

e. the back of the chair

f. the table legs

g. the ruler (on the table)

h. the line where the walls meet

i. the brush handle

j. the top edge of the small To learn more, click **here**

**WEEK 5**

# calculation of Class Boundaries and Intervals

# Calculation of Class Boundaries and Intervals

Class Boundaries: Class Boundaries are the midpoints between the upper class limit of a class and the lower class limit of the next class in the sequence. Therefore, each class has an upper and lower class boundary.

Class | Frequency |

200 – 299 | 12 |

300 – 399 | 19 |

400 – 499 | 6 |

500 – 599 | 2 |

600 – 699 | 11 |

700 – 799 | 7 |

800 – 899 | 3 |

Total Frequency | 60 |

Using the frequency table above, determine the class boundaries of the first three classes.

The lower class boundary is the midpoint between 199 and 200, that is *199.5*

The upper class boundary is the midpoint between 299 and 300, that is *299.5*

The lower class boundary is the midpoint between 299 and 300, that is *299.5*

The upper class boundary is the midpoint between 399 and 400, that is *399.5*

The lower class boundary is the midpoint between 399 and 400, that is *399.5*

The upper class boundary is the midpoint between 499 and 500, that is *499.5 *To learn more, click **here**

*WEEK 6*

# Cumulative Frequency Graph

# Cumulative Frequency Graphs: What is it?

Cumulative frequency is the running total of the frequencies. On a graph, it can be represented by a cumulative frequency polygon, where straight lines join up the points, or a cumulative frequency curve.

Example

Frequency: | Cumulative Frequency: | |

4 | 4 | |

6 | 10 | (4 + 6) |

3 | 13 | (4 + 6 + 3) |

2 | 15 | (4 + 6 + 3 + 2) |

6 | 21 | (4 + 6 + 3 + 2 + 6) |

4 | 25 | (4 + 6 + 3 + 2 + 6 + 4) |

This short video shows you how to plotting a cumulative frequency curve from the frequency distribution. How to find the median and inter-quartile range.

**The Median Value**

**The Median Value**

The median of a group of numbers is the number in the middle, when the numbers are in order of magnitude. For example, if the set of numbers is 4, 1, 6, 2, 6, 7, 8, the median is 6:

1, 2, 4, **6**, 6, 7, 8 (6 is the middle value when the numbers are in order)

If you have n numbers in a group, the median is the (n + 1)/2 th value. For example, there are 7 numbers in the example above, so replace n by 7 and the median is the (7 + 1)/2 th value = 4th value. The 4th value is 6.

When dealing with a cumulative frequency curve, “n” is the cumulative frequency (25 in the above example). Therefore the median would be the 13th value. To find this, on the cumulative frequency curve, find 13 on the y-axis (which should be labelled cumulative frequency). The corresponding ‘x’ value is an estimation of the median. To learn more, click **here**

**WEEK 7**

# Determination of the Mean, Median and Mode of Grouped Frequency Data (Revision)

# How to Determine the Mean, Median and Mode from Grouped Frequencies

To better explain how to determine the mean, median and mode of grouped frequency data, we will work with common, relatable examples as you can see below-

### Ade timed 21 people in the sprint race, to the nearest second:

59, 65, 61, 62, 53, 55, 60, 70, 64, 56, 58, 58, 62, 62, 68, 65, 56, 59, 68, 61, 67

To find the Mean, Ade adds up all the numbers, then divides by how many numbers:

Mean = |
59+65+61+62+53+55+60+70+64+56+58+58+62+62+68+65+56+59+68+61+67 |

21 | |

= | 61.38095… |

To find the Median Ade places the numbers in value order and finds the middle number.

In this case the median is the 11^{th} number:

53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70

**Median = 61 **To learn more, click **here**

Week 8

Topic: PROBLEM-SOLVING ON NUMBER BASES EXPANSION, CONVERSION AND RELATIONSHIP

Converting from base b to base 10

The next natural question is: how do we convert a number from another base into base 10? For example, what does 4201_{5} mean? Just like base 10, the first digit to the left of the decimal place tells us how many 5^{0}’s we have, the second tells us how many 5^{1}’s we have, and so forth. Therefore:

4201_{5 }= (4.5^{3 }+ 2.5^{2 }+ 0.5^{1 }+ 1.5^{0})_{10}

= 4.125 + 2.25 + 1

=551_{10}

From here, we can generalize. Let x =(a_{n}a_{n-1} … a_{1}a_{0})_{b }be an n + 1 -digit number in base b. In our example (2746_{10}) a_{3 }= 2, a_{2 }= 7, a_{1 }= 4 and a_{0 }= 6. We convert this to base 10 as follows:

x = (a_{n}a_{n-1} … a_{1}a_{0})_{b}

= (b^{n}.a_{n }+ b^{n-1 }. a_{n-1 }+ … + b.a_{1 }+ a_{0})_{10}

Converting from base 10 to base b

It turns out that converting from base 10 to other bases is far harder for us than converting from other bases to base 10. This shouldn’t be a suprise, though. We work in base 10 *all the time* so we are naturally less comfortable with other bases. Nonetheless, it is important to understand how to convert from base 10 into other bases. To learn more, click **here**

Week 9

Topic: INDICIAL EQUATION

Concept and Relationship with Quadratic Equation

Exponential or Indicial Equation is a combination of indices and all other forms of equations, it is very easy to solve provided you have excellent knowledge of the laws of Indices.

Rules for Solving Exponential (Indicial) Equations

1. The two sides i.e LHS and RHS of the equation must be expressed in index form.

2. The two sides of the equation must also have the same values for you to cancel them out.

3. You’ll always solve for an unknown value which can be represented by any letter of the alphabet.

Note

You will need to master all the laws of indices, if you must properly understand exponential equations.

Examples

1. If 3^{2} = 3^{2}, find x. To learn more, click **here**