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This investigation looks at the number of dots inside rectangles drawn on square dotty paper.
Rectangles with a height of 1 unit have no dots inside.
(a) (i) These rectangles all have a height of 2 units.
Complete the table.
[Table_1]
(ii) Find an expression, in terms of $w$, for the number of dots, $d$, inside rectangles of height 2.
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(b) Complete the tables below. You may use the square dotty paper on this page to help you.
[Table_2]
[Table_3]
(c) (i) Find a formula, in terms of $h$ and $w$, for the number of dots, $d$, inside a rectangle of height $h$ and width $w$.
$d = ext{.....................................................}$
(ii) Use your formula in part (i) to show that there are 30 dots inside a rectangle of height 6 and width 7.
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(iii) There are 33 dots inside a rectangle. Find the height and the width of this rectangle.
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580
509
539
(a) Squares are formed when $h = w$. Change your formula in question 1(c)(i) to find the number of dots, $d$, inside a square of side $s$.
(b) Find the number of dots inside a square of side 10.
568
538
543
(a) Complete the tables below.
[Table_1]
Height of diagonal rectangle (h) | Width of diagonal rectangle (w) | Number of dots inside diagonal rectangle (d)
1 | 1 | 1
1 | 2 | 2
1 | 3 | 3
1 | 4 |
[Table_2]
Height of diagonal rectangle (h) | Width of diagonal rectangle (w) | Number of dots inside diagonal rectangle (d)
2 | 1 | 2
2 | 2 |
2 | 3 |
2 | 4 |
2 | w | 3w - 1
[Table_3]
Height of diagonal rectangle (h) | Width of diagonal rectangle (w) | Number of dots inside diagonal rectangle (d)
3 | 1 |
3 | 2 |
3 | 3 |
3 | 4 |
(b) Use your results from part (a) and any patterns you notice to complete the following table with expressions, in terms of the width w, for the number of dots inside diagonal rectangles.
[Table_4]
Height of diagonal rectangle (h) | Number of dots inside diagonal rectangle (d)
1 |
2 | 3w - 1
3 |
4 |
5 | 9w - 4
(c) (i) Find a formula, in terms of h and w, for the number of dots, d, inside a diagonal rectangle.
..................................................................................................
(ii) Use your formula in part (i) to find the number of dots inside the diagonal rectangle of height 10 and width 3.
..................................................................................................
517
600
595
(a) \textit{Diagonal} squares are formed when $h = w$.
Change your formula in \textbf{question 3(c)(i)} to find the number of dots, $d$, inside a \textit{diagonal} square of side $s$.
.........................................................
(b) There are 181 dots in a \textit{diagonal} square.
Find the length of one side.
.........................................................
544
559
547
