• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
• Level: GCSE
• Subject: Maths
• Word count: 2509

# Investigate calendars, and look for any patterns.

Extracts from this document...

Introduction

Maths Coursework

Introduction: I was given a task to investigate calendars, and look for any patterns. I noticed several patterns, the first of which was the relationship between the starting days of different months, also I noticed the relationship between numbers in columns of the calendar, the relationship between numbers in the rows, Studying diagonal relationships, and Studying relationships between adjacent numbers.

1. Days on which months start

First, I explore the days on which different months start.

Which months are the same/ have a pattern?

Ex 1.1 Study Sample Calendar:

 Month Starting day 1 Friday 2 Monday 3 Monday 4 Thursday 5 Saturday 6 Tuesday 7 Thursday 8 Sunday 9 Wednesday 10 Friday 11 Monday 12 Wednesday

From the above table, I can see that some of the months start on the same day, which means there may be a pattern when compared with other years. If so, then that means there is a pattern of which months start on the same day each year. The results of this first test are as follows:

1, 10 = same

2, 3, 11 = same

4, 7 = same

5

6

8

9, 12 = same

Now I must investigate to find out if the pattern is the same in other years. In order to do this, I check a calendar of the year 2004.

## Ex 1.2: Results for 2004

 Months ( n ) Start day 1 Thursday 2 Sunday 3 Monday 4 Thursday 5 Saturday 6 Tuesday 7 Thursday 8 Sunday 9 Wednesday 10 Friday 11 Monday 12 Wednesday

Therefore, we can see that the dates are thus:

1, 4, 7 = same

2, 8 = same

3, 11 = same

5

6

9, 12 = same

10

Middle

3. Exploring the relationship between numbers in the rows

Now I explore the horizontal numbers…

Ex 3.1 The first row in June:

 n 1 1 2 2 3 3 4 4 5 5

The formula is obviously n = n here, but will it change in the next few rows?

Ex 3.2 The second row in June:

 n 1 6 3 7 4 8 5 9 6 11 7 12

As you can see from the table above, the formula is quite simple:

n = n + 5

Ex 3.3 The third row in June:

 n 1 13 2 14 3 15 4 16 5 17 6 18 7 19

The above table gives another formula: n = n + 12

Ex 3.4, Finally the fourth row in June:

 n 1 20 2 21 3 22 4 23 5 24 6 25 7 26

The final formula is n = n + 19

Ex 3.5:These are the formulae for the rows in the month of June:

First row: n = n

Second row: n = n+5

Third row: n = n+12

## Fourth row: n = n+19 I didn’t draw the table for 5th, but predict it as:

Fifth row: n = n+26

The general expression is n = n + x depending on the start date.

4. Studying diagonal relationships

Ex 4.1 Diagonally descending to the left, starting with 2nd January.

 n date 1 2 2 8 3 14 4 20 5 26

The difference is always 6. n x 6 gives the sequence:

6     12     18     24     30, but to get the correct sequence subtract four:

n = 6n – 4

I will test the formula on 2 other diagonals descending to the left: one more January, one July.

Ex 4.2: Diagonally descending to the left, starting with 9th January.

 n date 1 9 2 15 3 21 4 27

## The above table gives the formula n = 6n + 3

Ex 4.3: Diagonally descending to the left, starting with 5th June.

 n date 1 5 2 11 3 17 4 23 5 29

Conclusion

st February
 1 2 3 1 x 24 = 243 x 22 = 66Difference = 42 8 9 10 15 16 17 22 23 24

Finally, I decided to test a box of 14 numbers.

Ex.5.15 Starting 1st August

 1 2  3   4  5   6   7 1 x 14 = 14 8 9 10 11 12 13 14 7 x   8 = 56 Difference = 42

Ex.5.16 Testing a box of 16 numbers, starting 1st June.

 1 2 3 4 1 x 25 = 25 8 9 10 11 4 x 22 = 88 15 16 17 18 Difference = 63 22 23 24 25

Ex.5.17 A box of 18 numbers.

 1 2 3 4 5 6 1 x 20 = 20 8 9 10 11 12 13 6 x 15 = 90 15 16 17 18 19 20 Difference = 70

Ex.5.18 A box of 20 numbers

 1 2 3 4 5 1 x 26 = 26 8 9 10 11 12 5 x 22 = 110 15 16 17 18 19 Difference = 84 22 23 24 25 26

I decided to plot a table summarising the above results:

## Ex. 5.19

### Number of figures in box

Difference between multiplied diagonals

4

7

6

14

8

21

9

28

10

28

12

42

14

42

16

63

18

70

20

84

Note that the above table includes the 7 x table, which is once again significant as there are seven days a week. Therefore, there is a relationship between the figures involved in calendar calculations.

However, I must comment on this last table, Ex 5.19– it’s odd that certain numbers you would logically expect to be in the sequence are not present, as you never get the numbers 35, 49, 56 or 77. All these numbers are multiples of 7, and yet they do not follow an orderly sequence as you increase after 9 figures in a box: the pattern is not regular. I am unable to work out why this is so.

Areas of possible further work

I could have explored the following:

• Division of diagonals, but this would be difficult because they would not be whole numbers.
• Addition or subtraction of diagonals in boxes, e.g.
1 2                1 + 9 = 10
8 9                2 + 8 = 10
1 + 8 + 15 + 22
• A write up of whether the last day of each month follows the same pattern as the first month, in each year except leap years, i.e. an investigation similar to that of Ex 1.

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers 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

# Related GCSE Consecutive Numbers essays

1. ## GCSE Maths Coursework - Maxi Product

6 2/7 and 5 5/7 6 2/7+5 5/7 6 2/7x5 5/7 35.92 (2dp) 12 6 2/9 and 5 7/9 6 2/9+5 7/9 6 2/9x5 7/9 35.95 (2dp) Number Two numbers used Two numbers added Two numbers multiplied Maxi Product 13 1 and 12 1+12 1x12 12 13 2 and 11

2. ## Investigate the Maxi Product of numbers

I will now try in decimal numbers if I can get a number higher than 56. (7.1,7.9)= 15 à 7.1+7.9 à 7.1x7.9=56.09 (7.2,7.8)= 15 à 7.2+7.8 à 7.2x7.8=56.16 (7.3,7.7)= 15 à 7.3+7.7 à 7.3x7.7=56.21 (7.4,7.6)= 15 à 7.4+7.6 à 7.4x7.6=56.24 (7.5,7.5)= 15 à 7.5+7.5 à 7.5x7.5= 56.25 I have found a number higher than 56 in decimal numbers.

1. ## Nth Term Investigation

50 x 50 x 50 8 392 2401 100 x 100 x 100 8 792 9801 Cuboids Another example of 3-d shapes are the cuboids. Here is a table showing my results: n x n x n 1 x 2 x 2 8 8 2 0 2 x 2 x

2. ## Investigating a Sequence of Numbers.

and Tn, when I add (n + 1)! to (n + 2)!, there is a constant difference of 3 between the sum and Tn. According to what I have found out, Tn can be express mathematically like this: Tn = (1 + 2)! - 1! + (2 + 2)!

1. ## In this investigation I will explore the relationship between a series of straight, non-parallel, ...

= 4 CORRECT! OR (3) using the formula 2(n) let n = 3 2(3) = 6 CORRECT! OR (5) using the formula 2(n) let n = 5 2(5) = 10 CORRECT! I predict: OR (7) using the formula 2(n) let n = 7 2(7) = 14 To test the prediction I made I draw a diagram with 7 lines.

2. ## I'm going to investigate the difference between products on a number grid first I'm ...

44 45 46 52 53 54 55 56 62 63 64 65 66 72 73 74 75 76 82 83 84 85 86 The difference between 3772 and 3612 is 160 because 3772 - 3612 = 160. 46 x 82 = 3772 42 x 86 = 3612 I predicted that

1. ## I am to conduct an investigation involving a number grid.

the two numbers is 28 � 4 x 4 Boxes Box 1 7 8 9 10 X X+1 X+2 X+3 14 15 16 17 X+7 X+8 X+9 X+10 21 22 23 24 X+14 X+15 X+16 X+17 28 29 30 31 X+21 X+22 X+23 X+24 [image001.gif] [image041.gif] 7 x 31 = 217 x (x + 24)

2. ## Investigate the sequence of squares in a pattern.

I will now use the formula to find the number of squares in a higher sequence. Sequence 6 N=6 N={2+10+26+50+82}+61 N=231 Successful Instead of illustrating the pattern I am going to use the method I used at the start of this piece of this part of the coursework.

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