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This investigation looks at the number of ways of connecting dots using straight lines.
This diagram shows 1 dot.
There is 1 row and 1 column.
This is a 1 by 1 diagram.
There are no connections to other dots.
This diagram shows 4 dots.
There are 2 rows and 2 columns.
This is a 2 by 2 diagram.
There are 6 ways to join 2 dots.
These are:
• 2 vertical connectors (solid lines)
• 2 horizontal connectors (solid lines)
• 1 up diagonal connector (dashed line)
• 1 down diagonal connector (dashed line).
(a) This is a 3 by 3 diagram.
The diagram shows:
• 6 horizontal connectors
• 4 up diagonal connectors.
Each connector joins 2 dots.
Write down the number of vertical connectors and the number of down diagonal connectors that join 2 dots.
Vertical ................................................
Down diagonal ............................................. [2]
(b) Complete the table for the numbers of connectors that join 2 dots.
Use part (a) and any patterns you notice.
You may use the square dotty paper on page 12 for diagrams.
[Table_1]
Size of diagram (n by n) | Horizontal | Vertical | Up diagonal | Down diagonal | Total
1 by 1 | 0 | 0 | 0 | 0 | 0
2 by 2 | 2 | 2 | 1 | 1 | 6
3 by 3 | 6 | | 4 | |
4 by 4 | | | | 9 |
5 by 5 | 20 | | | |
6 by 6 | | | | | 110
[3]
(c) In an n by n diagram there are n rows and n columns.
(i) Find an expression, in terms of n, for the number of up diagonal connectors that join 2 dots on an n by n diagram.
......................................................... [2]
(ii) Find an expression, in terms of n, for the number of horizontal connectors that join 2 dots on an n by n diagram.
......................................................... [3]
(iii) Use your answers in part (i) and part (ii) to find an expression for the total number of connectors that join 2 dots.
Do not simplify your expression.
......................................................... [1]
513
541
558
This is a 3 by 3 diagram.
There are 8 ways to join 3 dots together.
These are:
• 3 vertical connectors
• 3 horizontal connectors
• 1 up diagonal connector
• 1 down diagonal connector.
(a) This is a 4 by 4 diagram.
Find the number of horizontal, vertical, up diagonal and down diagonal connectors that join 3 dots.
Two horizontal connectors have been drawn for you.
Horizontal ................................................
Vertical ...................................................
Up diagonal ..............................................
Down diagonal ........................................ [2]
(b) Complete the table for the numbers of connectors that join 3 dots.
Use your answers to part (a) and any patterns you notice.
You may use the square dotty paper on page 12 for diagrams.
[Table_1]
[2]
(c) (i) This is an expression for the number of up diagonal connectors that join 3 dots on an $n \times n$ diagram.
$$(n - 2)^2$$
Work out the number of up diagonal connectors that join 3 dots on a 20 by 20 diagram.
............................................................. [1]
(ii) This is an expression for the number of horizontal connectors that join 3 dots on an $n \times n$ diagram.
$$n^2 + an$$
Find the value of $a$ and write down the expression.
............................................................. [3]
(d) Find an expression, in terms of $n$, for the total number of connectors that join 3 dots on an $n \times n$ diagram.
Do not simplify your expression.
............................................................. [1]
572
596
524
(a) Complete the table for the numbers of connectors that join 4 dots.
[Table: Numbers of connectors that join 4 dots]
| Size of diagram (n by n) | Horizontal | Vertical | Up diagonal | Down diagonal | Total |
|---|---|---|---|---|---|
| 3 by 3 | 0 | 0 | 0 | 0 | 0 |
| 4 by 4 | 10 | ||||
| 5 by 5 | 10 | ||||
| 6 by 6 | 54 | ||||
| n by n |
(b) Find an expression, in terms of $n$, for the total number of connectors that join 4 dots on an $n$ by $n$ diagram.
Do not simplify your answer. [1]
592
521
512
(a) This is an expression for the total number of connectors that join $m$ dots on an $n$ by $n$ diagram.
$$2n(n-k) + 2(n-k)^2$$
What is the relationship between $k$ and $m$?
(b) (i) Use part (a) to show that when $n = 5$ and $m = 2$ there is a total of 72 connectors.
(ii) Find all the possible values for $n$ and $m$ that give a total of 72 connectors.
560
559
548
(a) (i) At the end of the first year the farmer sends 20\% of the 20 deer to other farmers. He sends an equal number of male and female deer.
Show that at the start of the second year he has 36 deer in his herd.
(a) (ii) $P_n$ = the number of deer in the herd at the start of year $n$.
$P_{n+1}$ = the number of deer in the herd at the start of year $n+1$.
At the end of each year the farmer sends 20\% of $P_n$ to other farmers. He always sends an equal number of male and female deer.
Show that $P_{n+1} = 1.8 P_n$.
(b) (i) When the farmer finds $P_{n+1}$ he rounds the value to the nearest even integer. The table shows the number of deer in the herd, $P_n$, at the start of year $n$. Use part (a)(ii) to complete the table.
[Table_1]
| Year (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---------|----|----|----|----|----|----|----|----|
| Number in herd ($P_n$) | 20 | 36 | 116 | 374 | | | | |
(b) (ii) Use your answers to part (b)(i) to plot the four missing points.
[Graph_1]
(c) The farmer models the data using $P = ab^{n-1}$.
Use the first two years in the table in part (b)(i) to find the value of $a$ and the value of $b$ and write down the model.
553
513
557
The farmer now models the data in the table on page 8 using $P = a(n-1)^2 + b(n-1) + c$.
(a) (i) Use the first year in the table in Question 5(b)(i) to show that $c = 20$. [1]
(ii) Use year 2 and year 4 in the table in Question 5(b)(i) to write down a pair of simultaneous equations in terms of $a$ and $b$.
.....................................................
..................................................... [2]
(b) (i) Solve your simultaneous equations from part (a)(ii) and write down the model.
.....................................................
..................................................... [4]
(ii) Sketch your model on the grid in Question 5(b)(ii) for $1 \le n \le 8$. [2]
(c) Is this a suitable model for the number of deer in the herd?
Give one reason for your answer.
........................................................................
........................................................................ [1]
512
567
535
The farmer now models the data in the table on page 8 using $P = a(n-1)^b$.
(a) Use the data for year 2 and year 4 in the table in Question 5(b)(i) to
(i) find $a$, ............................................................... [2]
(ii) find $b$ and write down the model. ............................................................... [3]
(b) Give two reasons why the model in part (a) is not suitable for the number of deer in the herd.
1 ............................................................................................................................
2 ............................................................................................................................ [2]
513
519
593
