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Masterclass Mentorship Team C. 1 dimension and 0 dimensions: A research on the relationship betweeen points and lines. A research and presentation by Cai Yi Zhan, Tan Wei Chuan, Darryll Chong, Lim Jan Jay and Ryan Wee. What we had to do. Our Research Question.

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Masterclass mentorship team c

Masterclass Mentorship Team C

1 dimension and 0 dimensions: A research on the relationship betweeen points and lines

A research and presentation by Cai Yi Zhan, Tan Wei Chuan, Darryll Chong, Lim Jan Jay and Ryan Wee



Our research question
Our Research Question

What is the minimum number of point needed to form n straight lines?

Conditions:

  • Line passes through exactly 2 points

  • Point of intersection not counted as a point,

  • Original point not said to be an intersection point

  • n >/= 1


Illustrations
Illustrations

Counted as a point

4 points are needed for 6 lines

Intersection Point – not a point


Proof
Proof

  • Let us start by showing the results if n= 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.


Proof1
Proof

  • Pattern observed:

    • Periodical increase by 1 of the number of dots needed,

    • Period of 1 for 2 dots, 2 for 3 dots and 3 for 4 dots… … so on and so forth.

  • Prove that this will always happen


Proof2
Proof

  • Each value of no. of points will stay on for (no. of points-1) values of n:

    • Assume no. of points is k

    • When add another point (total now k+1), still k-1 more dots for new dot to join

    • (k+1)-(k-1)=2

    • Therefore, k-2 more lines to be drawn.

    • Add the first line: therefore the no. of dots will always stay on for (no. of dots-1) values of n


Proof3
Proof

  • Will always be consecutive increase

    • Finding minimum number of dots

    • As proven, add 1 dot have (no. of dots – 2) more options

    • So, always increase of 1


Proof formula
Proof - Formula

  • If the following is true, when b is the subtraction that will bring the total to or below zero:

    • [([(n-1)-2]-3)…- (b-1)]

  • Then the number of dots is:

    • ((b-1)+1)+1

    • = b+1


How we found the formula
How we found the formula

  • Already know that no. of dots stays on for (no. of dots-1) values of n

  • Assume:

    • No. of dots 2 – 1st category

    • No. of dots 3 – 2nd category … …

  • The no. of dots is the category+1

  • To find the category:

    • Utilize fact that no. of dots are consecutive

    • Start by subtracting 1, than 2


How we found the formula1
How we found the formula

  • To find the category (continued):

    • Subtraction that brings total to or below 0 then category number, as:

      • 1 number in category 1

      • 2 numbers in category 2 … …

      • In the formula that is b (category number).

    • So, (b-1)+1 is the category number

    • Add another 1 for no. of dots:

      • ((b-1)+1)+1

      • = b+1






Timeline
Timeline

Start

Session 1

Delegated roles and

created a wikispace

Session 2

Posted a few questions

based on a given situation

and presented it

Session 3

Discussed on proving

a question's answer and

learnt how to prove using

mathematical induction and contradiction

Session 4

Posted the problem that we were going to solve and prove and presented it. Listened to other presentations to get ideas in case of rejection.

Session 5

Discussed about our answer and proof and presented 1st draft to Dr Soon

Session 6

End

Session 7

Session 8 (Now)

Present problem,solution, proof......( basically everything)




Acknowledgements
Acknowledgements

  • Dr. Soon, the expert mentor

  • Mr. Goh for helping us

  • Families of members who encouraged them

  • Members of team who did the project

  • Our Math Teachers who guided us


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