• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Maths SL Portfolio - Parallels and Parallelograms

Extracts from this document...

Introduction

IB Standard Level Maths: Portfolio Piece 1

Parallels and Parallelograms

Table of Results for 4 transversals:

Transversals

Number of Parallelograms

Parallelograms

Diagram

4

6

A1, A2, A3

A1 ∪ A2, A2 A3

A1 A2 A3

image05.png

Table of Results for 5, 6 and 7 transversals:

5

10

A1, A2, A3, A4

A1 ∪ A2, A2 A3, A3 ∪ A4

A1 A2 A3 , A2 A3 A4

A1 A2 A3  A4

image06.png

6

15

A1, A2, A3, A4, A5

A1 ∪ A2, A2 A3, A3 ∪ A4, A4 A5

A1 A2 A3 , A2 A3 A4, A3 A4 A5

A1 A2 A3  A4,  A2 A3 A4  A5

A1 A2 A3  A4 A5

image08.png

7

21

A1, A2, A3, A4, A5, A6

A1 ∪ A2, A2 A3, A3 ∪ A4, A4 A5, A5A6

A1 A2 A3 , A2 A3 A4, A3 A4 A5,  A4A5 A6

A1 A2 A3  A4,  A2 A3 A4  A5,  A3 A4 A5  A6

A1 A2 A3  A4 A5,  A2 A3 A4  A5 A6

A1 A2 A3  A4 A5 A6

image09.png

 Let n = number of transversals and letp = number of parallelograms

Transversals (n)

Parallelograms (p)

2

1

3

3 (1 + 2)

4

6 (1 + 2 + 3)

5

10 (1 + 2 + 3 + 4)

6

15 (1 + 2 + 3 + 4 + 5)

7

21 (1 + 2 + 3 + 4 + 5 + 6)

n

1 + 2 + … + (n – 1)

Use of Technology:

Using the TI – 84 Plus, press STAT 1: Edit.

Type in L1, L2:         (2, 1)

                (3, 3)

                (4, 6)

                        …etc.

Using Quadreg, L1, L2,

...read more.

Middle

 A3 A4  A5 A6, A3 A4 A5  A6 A7,A4 A5 A6  A7 A8, A5 A6 A7  A8 A9

        = 5

  • A1 A2 A3  A4 A5 A6,
  • A2 A3 A4  A5 A6 A7,  
  • A3 A4 A5  A6 A7 A8,
  • A4 A5 A6  A7 A8 A9

        = 4

  • A1 A2 A3  A4 A5 A6 A7,
  • A2 A3 A4  A5 A6 A7 A8,
  • A3 A4 A5  A6 A7 A8 A9

        = 3

  • A1 A2 A3  A4 A5 A6 A7A8,  
  • A2 A3 A4  A5 A6 A7 A8A9

        = 2

  •  A2 A3  A4 A5 A6 A7A8 A9

        = 1

 45

p = sum of all integers from 1 to (10 – 1)

= sum of all integers from 1 to 9

= 1 + 2 +3 + 4 + 5 + 6+ 7 + 8 + 9

= 45

p = (n2 – n) ÷ 2

p= 102 – 10 ÷ 2

= 90 ÷ 2

= 45

 Let m = number of horizontal lines

image11.png

If there are three horizontal lines, intersecting two transversals (m = 3, n = 2) then p = 3. Similarly, if there are three transversals, and two horizontal lines, (m= 2, n = 3), then we also obtain p= 3.

Conclusion:

Hence, m horizontal lines and n transversals produce the same amount of parallelograms as n horizontal lines and m transversals.

General Statement:

If there are m horizontal lines, and two transversals, then p = sum of all integers from 1 to (m – 1). Note that this rule is identical to the above investigation of n transversals and two horizontal lines.

Test of Validity for m = 10, n = 2image12.png

We will now prove that if m = 10, where n = 2, we will get the same p value of 45 as example 1 above where m = 2 and n = 10.

e.g. 2) 10 horizontal lines, 2 transversals

Manual method:

  • A1,
  • A2,
  • A3,
  • A4,
  • A5,
  • A6,
  • A7,
  • A8,
  • A9

                = 9

  • A1 ∪ A2,
  • A2 A3,
  • A3 ∪ A4,
  • A4 A5,
  • A5A6,
  • A6A7,
  • A7A8,
  • A8A9

                = 8

  • A1 A2 A3,
  • A2 A3 A4,
  • A3 A4 A5,
  • A4 A5 A6,
  • A5 A6 A7,
  • A6 A7 A8,
  • A7 A8 A9,

                = 7

  • A1 A2 A3  A4,  
  • A2 A3 A4  A5,  
  • A3 A4 A5  A6,
  • A4 A5 A6  A7,
  • A5 A6 A7  A8,
  • A6 A7 A8  A9

                = 6

  • A1 A2 A3  A4 A5,  
  • A2 A3 A4  A5 A6,
  • A3 A4 A5  A6 A7,
  • A4 A5 A6  A7 A8,
  • A5 A6 A7  A8 A9

                = 5

  • A1 A2 A3  A4 A5 A6,  
  • A2 A3 A4  A5 A6 A7,  
  • A3 A4 A5  A6 A7 A8,
  • A4 A5 A6  A7 A8 A9
...read more.

Conclusion

m horizontal lines, and n transversals, the resultant value of p equals the product of p1and p2, where;

p1 = number of parallelograms for m horizontal lines and two transversals

p2 = number of parallelograms for 2 horizontal lines and n transversals.

Hence, for any diagram with m horizontal lines and n transversals,

image04.png

Test of validity for m = 4, n = 3

image13.png

  • A1,
  • A2,
  • A3,
  • A4,
  • A5,
  • A6

        = 6

  • A1 ∪ A2,
  • A2 A3,
  • A4 A5,
  • A5A6,
  • A1 A4,  
  • A2 A5,
  • A3 A6

        = 7

  • A1 A2 A3,
  • A4A5 A6

        = 2

  • A1 A2A4 A5,  
  • A2A3 A5 A6

        = 2

  • A1 A2 A3  A4 A5 A6

= 1

p=6 + 7 + 2 + 2 + 1

        = 18

p = ½ m (m – 1) x ½ n (n – 1)

        = ½ 4 (4 – 1) x ½ 3 (3 – 1)

        = 6 x 3

        = 18

Scope/limitations:

The formula will be valid for m, n ≥ 2. If either value were to be 1 or 0, it would be impossible to create any parallelograms.

image07.png

Explanation of generalisation:

A diagram with m horizontal lines and 2 transversals creates p1parallelograms.

A diagram with 2 horizontal lines and n transversals creates p2parallelograms.

It follows that if a diagram were created, with m horizontal lines, and n transversals, we would be able to fit p1parallelograms vertically and p2 parallelograms horizontally, giving us a total of p1 x p2 parallelograms.

 

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Math IB SL BMI Portfolio

    causing the BMI value of height to remain constant after 20 years of age but the value of weight to keep on increasing until a separate plateau - adult weight, is reached. Therefore, the BMI value would not remain constant after � 20 years of age and it will keep on increasing until both adult height and weight is reached.

  2. Math SL Circle Portfolio. The aim of this task is to investigate positions ...

    O(0, 0), P'(2,0) = = 2 ? = 2, r= 2, = 2 The following graph displayed is when 2, r= 2. Through the same method of calculation as the above, . Using the same method of calculation, the following chart was established r . ' 1 2 2 2 = 2 3 2 4 2 =

  1. Mathematics SL Parellels and Parallelograms. This task will consider the number of parallelograms formed ...

    Second difference between terms 3 2 3 3 x 1 3 3 9 3 x 3 2 6 3 4 18 3 x 6 3 9 3 3 5 30 3 x 10 4 12 3 3 6 45 3 x 15 5 15 3 3 7 63 3 x

  2. Music and Maths Investigation. Sine waves and harmony on the piano.

    Next, the probability that we play the correct middle note for the chord. This time there are 2 valid notes since there are minor and major variations of each chord. Also since we have already played the base note, we now can only choose from 11 notes.

  1. Gold Medal Heights Maths Portfolio.

    To find the points for 1940 and 1944, a graph with a dip is necessary. Since World War 2 occurred throughout 1940 and 1944, the expected values would be lower than 1936 and 1948 due to rationing of food, casualties, loss of practice etc. which would thereby produce a dip.

  2. Math SL Fish Production IA

    Rearranged Years (x) Substituting the x and y values 3 557.3=a34+b33+c32+d3+426.8 4 564.7=a44+b43+c42+d4+426.8 5 575.4=a54+b53+c52+d5+426.8 6 579.8=a64+b63+c62+d6+426.8 Table 4: This shows the rearranged years and the simplified equation of table 3. Rearranged Years (x) Substituting the x and y values 3 81a+27b+9c+3d=130.5 4 256a+64b+16c+4d=137.9 5 625a+125b+25c+5d=148.6 6 1296a+216b+36c+6d=153 These equations

  1. Parallels and Parallelograms Maths Investigation.

    A2 á´ A5 , A2 á´ A6, A3 á´ A4, A3 á´ A5, A3 á´ A6, A4 á´ A5, A4 á´ A6, A5 á´ A6. Adding a seventh transversals gives us a total of twenty-one parallelograms. Transversals Parallelograms 2 1 3 3 4 6 5 10 6 15 7 21

  2. High Jump Gold Medal Heights Type 2 Maths Portfolio

    seem to deviate away from the line of best fit by quite a large amount. By using the coordinates of two points whom this best fit line passes through (E and G), the equation of the line of best fit was found.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work