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*