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Parallels and Parallelograms. Aim: To find the effects of increasing the number of intersecting transversal and horizontal parallel lines on the number of parallelograms formed.

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Introduction IB Standard Mathematics

Internal Assessment

Candidate Name: Alejandro Ortigas Vásquez

Candidate Number: -

Title: Parallels and Parallelograms

 D DCP CE

Aim:

To find the effects of increasing the number of intersecting transversal and horizontal parallel lines on the number of parallelograms formed.

Procedure:

There are two types of lines to consider in this investigation:

When the lines intersect, they are able to form a parallelogram. The most basic way to do this is without changing the orientation of the lines is shown below.

2 lines

A

Let us call the parallelogram enclosed by the lines “A”.

Now let’s see what happens when we add one transversal line.

3 lines

A                     B

We now have 3 parallelograms: A, B, and A U B.

Though we can continue to add lines indefinitely, for the purpose of this investigation, we will stop at 7 transversal lines.

4 lines

A             B                           C

This would give us 6 parallelograms: A, B, C, A U B, B U C, and A U B U C.

5 lines

A        B                      C                     D

This would give us 10 parallelograms: A, B, C, D, A U B, B U C, C U D, A U B U C, B U C U D, and A U B U C U D.

6 lines

A        B                    C                      D                         E

Middle

We can repeat this process like last time.

3 lines

A        B

C          D

This would give us 9 parallelograms: A, B, C, D, A U B, C U D, A U C, B U D, and A U B U C U D.

4 lines

A                      B                      C

D                        E                      F

This would give us 18 parallelograms: A, B, C, D, E, F, A U B, B U C, D U E, E U F, A U D, B U E, C U F, A U B U C, D U E U F, A U B U D U E, B U C U E U F, A U B U C U D U E U F.

After doing, this we can say that:

 N0 of transversal lines N0 of parallelograms 1 0 2 3 3 9 4 18

We can also develop a general statement from these values:

1. We analyze the first three values.
2. We notice that the # of parallelograms is three times as many for each value as # of parallelograms from having only 2 intersecting horizontal lines.
3. Thus, the general expression is simply:

p =   3 (n (n-1))

2

After all this we see another pattern unfolding, this being the relation between the # of parallelograms as we increase the # of horizontal lines.

Conclusion

The program Autograph was used to illustrate the relation between two or more variables through graphs. It also provided a means of easily obtaining the equation of a trend in the form of the Display Information tool. This was crucial when validating the results.

Conclusion:

The general statement can be used in almost any situation since ‘n’ is a value decided by the individual, in the sense that when asking “how many parallelograms are there when there are m horizontal parallel lines and n transversal parallel lines” the ‘m’ and ‘n’ values act as independent variables while the # of parallelograms is dependant.

However, it cannot be reversed (such as if given only the # of parallelograms and told to find the values of m and n) since there are two variables and none of them cancels out.

This investigation was carried out by analyzing three different variables and trying to understand the mathematical relationship between them. The statement was found after first observing the effect of an increase in the # of parallel transversal lines on the number of enclosed parallelograms and then doing the same for the # of horizontal lines. Finally, an analysis on the effects of the # of one type of line on the other helped to develop an equation that included them both.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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