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This investigation looks at the number of squares drawn on square grids. All squares are drawn using gridlines.
Here is a 2 by 2 grid.
Explain why there are 5 squares on a 2 by 2 grid.
532
501
519
Here is a 3 by 3 grid.
Complete these statements about the numbers of different sized squares on a 3 by 3 grid.
The number of 1 by 1 squares on a 3 by 3 grid is .................
The number of 2 by 2 squares on a 3 by 3 grid is 4
The number of 3 by 3 squares on a 3 by 3 grid is .................
So the total number of squares on a 3 by 3 grid is ................
594
539
562
Complete these statements about the numbers of different sized squares on a 4 by 4 grid. You may use the grids below to help you.
The number of 1 by 1 squares on a 4 by 4 grid is ..................
The number of 2 by 2 squares on a 4 by 4 grid is ..................
The number of 3 by 3 squares on a 4 by 4 grid is ..................
The number of 4 by 4 squares on a 4 by 4 grid is ..................
So the total number of squares on a 4 by 4 grid is 30
525
556
561
(a) Use your results from questions 1, 2 and 3 to help you complete this table.
[Table_1: Size of grid and Number of squares by size, 1 by 1 grid has 1 total square, 2 by 2 grid has 5 total squares, 3 by 3 grid has 30 total squares]
(b) What is the mathematical name for the numbers in the 1 by 1 squares column?
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(c) Work out the total number of squares on an 8 by 8 grid.
(d) Here is part of a table for an $n$ by $n$ grid. It only has columns for 1 by 1 squares up to 6 by 6 squares.
Complete the table using expressions in terms of $n$.
[Table_2: Size of grid $n$ by $n$, Number of 1 by 1 to 6 by 6 squares, last column for (5 by 5 squares) is $(n-4)^2$]
(e) Write an expression, in terms of $n$, for the number of 12 by 12 squares on an $n$ by $n$ grid.
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(f) (i) Find the number of 5 by 5 squares on a 20 by 20 grid.
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(ii) The number of 5 by 5 squares on an $n$ by $n$ grid is 36. Find the value of $n$.
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571
524
536
Here is a formula for the total number of squares, $T$, on an $n$ by $n$ grid.
$$T = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6} + d$$
(a) The total number of squares on a 1 by 1 grid is 1.
Show that $d = 0$.
(b) Show that the formula gives the correct total number of squares on a 4 by 4 grid.
(c) Find the total number of squares on a 10 by 10 grid.
507
556
522
(a) There are nine 7 by 7 squares on a 9 by 9 grid.
The diagrams show a 7 by 7 square drawn in two positions on a 9 by 9 grid.
In each diagram the same 2 by 2 square is shaded.
Consider the possible positions of the 7 by 7 square.
Explain how the shaded 2 by 2 square can be used to calculate the number of 7 by 7 squares on a 9 by 9 grid.
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(b) A square which is bigger than 9 by 9 is drawn on a square grid.
It is only possible to draw 25 of these squares on the square grid.
Find two possible sizes for the square and the grid it is drawn on.
You may use the grid below to help you.
Square size ........... by ........... on a ........... by ........... grid.
Square size ........... by ........... on a ........... by ........... grid.
586
573
512
