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OPPOSITE CORNERS (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation is about the difference between the products of the numbers in the opposite corners of a square window on a grid.
To calculate the opposite difference for any window:
• multiply the numbers in the opposite corners
• subtract the smaller answer from the larger answer.
[Table_1]
Consecutive even numbers fill a grid of width 10 as shown. The grid continues downwards.
A 2 by 2 window moves on the grid.
Example
This is the first window.
$22 \times 4 = 88$
$2 \times 24 = 48$
$88 - 48 = 40$
The opposite difference is 40.
Use the grid to complete each window and find the opposite difference.
$34 \times \text{.........} = \text{.........}$
$14 \times 36 = \text{.........}$
$\text{.........} - \text{.........} = \text{.........}$
Opposite difference = ........................................................
...........................................................
........................................................... [3]
558
546
504
A 3 by 3 window moves on the same grid.
(a) Complete the corner squares in the first window.
(b) Complete the opposite difference calculations for this window.
......... \times 6 = .........
2 \times ......... = ......... ......... - ......... = .........
(c) Complete the corner squares for each window and find the opposite difference.
548
502
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(a) Copy the opposite differences that you have found and complete the table.
[Table_1]
| Size of window | Opposite difference |
|----------------|---------------------|
| 2 by 2 | $(2-1)^2=1$ |
| 3 by 3 | $(3-1)^2=4$ |
| 4 by 4 | $(4-1)^2=9$ |
| 5 by 5 | |
| $w$ by $w$ | $40($ $)$ |
You may use this grid, which continues downwards, to help you.
[Table_2]
|
| 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
|:----|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 | 40 |
| 42 | 44 | 46 | 48 | 50 | 52 | 54 | 56 | 58 | 60 |
| 62 | 64 | 66 | 68 | 70 | 72 | 74 | 76 | 78 | 80 |
| 82 | 84 | 86 | 88 | 90 | 92 | 94 | 96 | 98 |100 |
| 102 |104 |106 |108 |110 |112 |114 |116 |118 |120 |
[4]
(b) Find the greatest possible opposite difference for a window on this grid.
............................................. [2]
523
586
525
Another grid of consecutive even numbers has width 7 units. The diagram shows the start of the grid.
The diagram shows a 2 by 2 window on the grid. $n$ is the first number in the window.
(a) Complete the window using expressions in terms of $n$.
(b) Use part (a) to show that the opposite difference for a 2 by 2 window is 28.
560
516
539
A 2 by 2 window moves on a grid of width $g$, with squares numbered 2, 4, 6, \ldots .
Use algebra to find an expression for the opposite difference on this grid. Give your answer in its simplest form.
You may use this diagram to help you.
588
595
543
(a) A square window of side $x + 1$ moves on a grid of width $g$, with squares numbered 2, 4, 6, \ldots\, $n$ is the first number in the window.
Show that an expression for the opposite difference using this window is $4gx^2$, where $x$ can be any positive integer.
(b) A square window moves on a grid numbered 2, 4, 6, \ldots\ The opposite difference is 144.
Use your answer to part (a) to find all the ways this is possible.
519
575
596
(a) Calculate the mean temperature and the mean number of chirps.
Temperature .........................................
Number of chirps ..................................... [2]
(b) Plot the data on the grid on page 11.
The first five points have been plotted for you. [2]
(c) A model for the number of chirps, $N$, is the regression line for $N$ in terms of $T$.
Find this model.
.................................................... [2]
(d) Draw the graph of the model on the axes. [2]
(e) A Snowy Tree Cricket chirps 170 times in 60 seconds.
Use your model to find the temperature.
.................................................... [2]
(f) Amos Dolbear originally counted the number of chirps in 13 seconds and measured the temperature in degrees Fahrenheit, $F$.
To change the temperature from $T^\circ C$ to $F$, use $F = 1.8T + 32$.
Complete the table to change the data to the form Amos Dolbear used.
All the data in the table is correct to the nearest integer.
[Table_1]
(ii) Amos Dolbear suggested a simple model to find the temperature, $F$, from the number of chirps.
Add 40 to the number of chirps in 13 seconds to find the temperature, $F$.
Is this a suitable model for the data?
Give a reason for your answer.
....................................................................................................
.................................................................................................... [1]
596
529
545
This is another model for the data on page 10.
$$N = 0.5T^2 + aT + b$$ where $a$ and $b$ are constants
(a) Use the information in the table on page 10 for temperatures of 16°C and 27°C to find the value of $a$ and the value of $b$, each correct to the nearest integer. Write down the model.
$a = \text{...............................................}$
$b = \text{...............................................}$
$$N = 0.5T^2 \text{...............................................}$$ [5]
(b) Sketch the graph of the model on the axes on page 11. [2]
(c) Explain how suitable the model is for the data on page 10.
..............................................................................................................................
.............................................................................................................................. [1]
571
510
586
The number of chirps per second, $A$, made by a type of African cricket is counted.
These crickets do not chirp at higher or lower temperatures.
The graph shows the results and a linear model.
(a) (i) Use the points (17.5, 2.35) and (21, 3.5) to find a linear model for $A$ in terms of $T$.
.................................................. [4]
(ii) $A$ is the number of chirps per second.
$N$ is the number of chirps per minute.
Use your model for $A$ to write a linear model for $N$.
................................................. [1]
(b) Write two statements comparing the chirping of the Snowy Tree Cricket and this African cricket.
Statement 1 .........................................................
........................................................................................
........................................................................................
Statement 2 .........................................................
........................................................................................
........................................................................................ [2]
509
530
517
