Math Exploration: Clock Hands Problem

The Problem: At 12:00:00, the hour and the minute hands of a clock coincide. How much time passes before the next instant that the hour and minute hands coincide?

My solution:
The minute and hour hand will collide again after one but before two complete revolutions of the minute hand.

Let x be the number of minutes the hour hand travels and t be the amount of time until they next coincide.

Then 60+x is the number of minutes the minute hand travels.

The minute hand travels at a rate of 1 (in terms of minutes passed).

The hour hand travels at a rate of 5/60 = 1/12 (since it is on the 5 minute mark after 60 minutes).

Since distance = speed*time
x = (1/12)t
60 + x = t
Therefore,

60 + (1/12)t = t
60 = (11/12)t
60*(12/11) = t

Since this is in minutes, we have to divide by 60 to get hours.

t=12/11

It takes 12/11 hours for the clock hands to next coincide. In other words, it takes 1 hour, 5 minutes and 27+27/99 seconds.

Topic Consolidation: Geometrical Properties of Circles

Chord- lines by joining two points on a circle
Tangent- line touching the circle at its circumference at only 1 point
Secant- An extended chord on one side

Chord Properties
1. If the radius cuts the chord into half, then the radius is perpendicular to the chord. (and vice versa)

Tangent Properties
1. The radius is perpendicular to the tangent at the point of tangent.

Angle properties of circle
1. Angle at center = 2x angle at circumference
2. Angles in the same segment are equal
3. Angle in a semi-circle is 90 degrees
4. Opposite angles of a cyclic quadrilateral add up to 180 degrees.
5. Exterior angles of a cyclic quadrilateral is equal to its interior opposite angle

Topic Consolidation: Plane Geometry

Midpoint Theorem

The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side.

Intercept Theorem

If a line is parallel to one side of a triangle, it divides the other two sides proportionally.

Alternate Segment Theorem

An angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

The alternate segment theorem states that the red angles are equal to each other and the green angles are equal to each other.

Intersecting Chords Theorem

The Intersecting Chords theorem states that when two chords intersect, no matter where, A*B = C*D.

Tangent-Secant Theorem

DC*CA = DB*DB

If a secant segment and tangent segment are drawn to a circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment.

Topic Consolidation: Congruency and Similarity

CONGRUENCY

A pair of triangles are considered to be congruent if they fulfil any of the following criteria:

SSS (Side-Side-Side)

Each side of the first triangle must be the same length as the corresponding side of the second triangle.

SAS (Side-Angle-Side)

Two pairs of sides of the first triangle must be equal to another pair of sides on the second, the angle that they enclose must also be the same.

AAS (Angle-Angle-Side)

Two angles and one side must be the same.

RHS (Right Angle-Hypotenuse-Side)

The triangles must have a right angle (angle that is 90 degrees) and their hypotenuse and a side must be equal.

SIMLARITY

Triangles are said to be similar if they fulfil any of the following criteria:
SSS (Side-Side-Side)

Each side of the first triangle must be the same length as the corresponding side of the second triangle.

SAS (Side-Angle-Side)

Two pairs of sides of the first triangle must be equal to another pair of sides on the second, the angle that they enclose must also be the same.

AA (Angle-Angle)

Two angles in both triangles must be the same.

APPLICATION

If two triangles are congruent, their sides will equal each other.

If two triangles are similar, SIDE A of triangle 1 over SIDE A of triangle 2 will be equal to SIDE B of triangle 1 over SIDE B of triangle 2.

Side A and Side B being the corresponding sides.

Math Exploration: Birthday Problem

The birthday problem is as such: In a class of n people, what are the chances that one of them share the same birthday as any other?

At first, this seems like a tough question. However, using a formula the answer is actually extremely easy to find out.

First, to have a 100% chance that two of them share the same birthday, one can use the Pidgeon Hole principle. You will find that if you have 366 people, at least 2 of them will share the same birthday (excluding February 29).

However, it is surprising because to achieve 99% that two of them share the same birthday, only 57 people are needed. It is even more surprising that to achieve 50%, only 23 people are needed!

To calculate the formula, we must first calculate the chances of each person not sharing a birthday. The first person will definitely not share his birthday with any one as his birthday is the only birthday known. The next person will have a 364/365 chance of not sharing his birthday, because he there is a chance he might share it with the first person. The next person would have a 363/365 chance of not sharing it with the first two and it goes on and on...

Therefore, in a group of 23 people, the chances of one of them sharing a birthday with another would be :

1-[(365/365)*(364/365)*(363/365)...(343/365)]

=1-[(1/365)23 * (365 * 364 * 363 * ... * 343)]

=1-0.49270276

=0.507297%

Maths for Fun Video 3

Title: 13x7=28
Resource type: Video
URL: http://www.youtube.com/watch?v=rLprXHbn19I&feature=related
Reasons for recommendation: This is a very hilarious video of a man proving how 13 times 7 equals to 28. I found this interesting because it is basic math and if one does not learn his foundations well, he/she may end up like the guy in the video. That is why I feel that it is important to learn math now so in the future we can understand harder concepts.

Maths for Fun Video 2

Title: Math Math Baby
Resource type: video
URL: http://www.youtube.com/watch?v=zNLRfOF_Ygk&feature=related
Reasons for recommendation: This is a amazing song parody of "Ice Ice Baby" that talks about math. I found it extremely interesting and cool! It would be nice if students could be able to parody songs for ACE points... *hint*

Maths for Fun Video 1

Title: 5 Ways to use Trigonometry in Everyday Life
Resource type: video
URL: http://www.youtube.com/watch?v=T_19ZxaCP3g&feature=related
Reasons for recommendation: I found this interesting and funny video that shows how trigonometry can be used in one's everyday life! It is a humorous video that I enjoyed. With nice background music and pictures of Lady Gaga, this video will be sure to leave an impact on you.

Math Exploration: Monty Hall Problem

The Monty Hall Problem is a probability puzzle. It goes as such: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

The way to determine whether to switch or not can be derived from listing out all the possibilities.

If you picked Goat #1: The host opens Goat #2 and switching would give you the Car

If you picked Goat #2: The host opens Goat #1 and switching would give you the Car
If you picked the Car: The host opens one goat and switching would give you the other goat.

It is easy to see now how switching gives you a higher chance of getting the Car. However, this problem is interesting because many people disagree with this solution and still think that there is no difference. Many people think that the chance of getting a car is still 1/3 as there are 3 doors or that after opening a Goat, there are only 2 doors therefore the chances of having a car is 1/2.

Maths for Fun 3

Title: Digits of Pi
Resource type: Website
URL: http://www.eveandersson.com/pi/digits/
Reasons for recommendation: As I have always been interested in Pi, I tried to find a website that could show as many digits of Pi as possible. This website shows up to a million digits! So far, I can only memorise up to 27 digits.

Maths for Fun 2

Title: Pythagorean Triplets
Resource type: tool
URL: http://www.mathsisfun.com/pythagorean_triples.html
Reasons for recommendation: As we have to learn about Pythagorean triplets, I found this website extremely useful for learning. In addition to using easy to understand terms, they also provide pictures for one to learn. Learning in advance is never bad.

Maths for Fun 1

Title: Wolfram Alpha
Resource type: Tool
URL: http://www.wolframalpha.com/

Reasons for recommendation: This is an excellent tool that everyone should know about. It allows you to solve almost any math question! It even allows you to use unknowns and will still get the proper answer for you! I feel that this is useful as if you are stuck on a math question, you can enter it here to get the answer and work backwards. It is a very easy way to learn.

Lesson Reflection on Plane Geometry

How did you feel about this lesson?
I found this lesson extremely interesting! However, at first I did not understand why it was so but after I did some research on my own and asked around a bit, I finally understood and could see why certain properties were so.
What have you learnt in this lesson?
I have learnt several important theorems such as the Alternate Segment Theorem, Intersecting Chords Theorem and Tangent-secant Theorem and I have also learnt how to use and apply them.
What are your thoughts about this lesson?

Originally, I had problems with the Alternate Segment Theorem as I just could not understand how it worked. However, after doing more and more work I understood it as I gained more experience by doing questions.

Lesson Reflection on Geometrical Properties of Circles

How did you feel about this lesson?
I feel that I throughly enjoyed this lesson. This is one of the more interesting math topics we have. Also, this is building on the foundations of angles we learned last year. I believe that we should try to master this topic as in the future there will be future lessons that build on these properties.

What have you learnt in this lesson?
I have learnt many different circle properties, from the angle at the centre equals to twice the angle at the circumference to angles in the same segment to the tangent being perpendicular to the radius.
What are your thoughts about this lesson?
I think that the lesson is interesting and I hope to see more such topics in the future. I would not mind learning more about this topic as it seems very interesting. However, I can also see that this topic may not be as easy as I think and can be quite tricky, therefore I will start revising early for it.

Lesson Reflection on Congruency and Similarity

How did you feel about this lesson?
Although this is a new topic, I actually felt that it is quite easy. I found it easy on my part to grasp the concepts involved as I have always been a visual person. This topic seems easy and I think that I will just have to focus on lessening my careless mistakes.

What have you learnt in this lesson?

I have learnt how to find out the sides of a triangle by using another triangle which is similar to it! I think this will be extremely useful in the future if I ever have to find out something using a small/bigger object.

What are your thoughts about this lesson?

I found this lesson kind of interesting as I liked last term's lessons on Trigonometry. I think that this lesson is an expansion on last term. I can already foresee questions that require you to combine both Trigonometry and Congruency or Similarity in the end of year paper, therefore I shall start early by finding some questions on that and start revising.

Target Setting

Previous terms
Term 2: A1
Current term
Target Grade: A1
Results obtained:
Test 1: 31/40

Test 2: 34/40
Reflection for class test
Test 1
This test made me realise how much more I needed to study in order to get a clear A1. This time round, I was lucky and thanks to the moderation I moved up from A2 to A1. It showed that I lacked practice in similar and congruent triangles which I will definitely do my best to improve for test 2. I hope to see a score of at least 36 for test 2.

Test 2

I was extremely relieved when I got back my marks for this test - my revision paid off. Although I did quite well, I feel that I could have done better, losing a total from 4 marks due to careless or stupid mistakes. One reason why I lose marks was because I failed to write the degree signs many times while another was that I wrote the wrong symbol for similar triangles. Also, I shaded the symmetry question wrongly which was foolish on my part. As this is the last test, I am ready to forge ahead full steam and do well for the end of year by revising now.