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INVESTIGATION SQUARES IN RECTANGLES
This investigation looks at finding the total number of squares inside a rectangle drawn on a grid.
In this investigation:
• the sides of the rectangles are on the grid lines
• the length of a rectangle is never less than its width.
Squares in Rectangles
Rectangles of width 1
Complete the statements.
The number of squares in a rectangle of width 1 and length 4 is ................................................
The number of squares in a rectangle of width 1 and length $L$ is ................................................ [2]
560
520
588
(a) Draw lines on these rectangles and write the number of squares under each one to show there is a total of 11 squares.
Length 4
[Image_1: Diagram of rectangles with labels for squares of side 1 and side 2]
.................... squares of side 1 + .................... squares of side 2 + .................... square of side 2
Total = 11 squares [3]
(b) (i) Complete the table.
You may use the grid below the table to help you.
[Table_1: Rectangles of width 2]
| Length of rectangle | Total number of squares |
|----------------------|-------------------------|
| 2 | 5 |
| 3 | 8 |
| 4 | 11 |
| 5 | |
[2]
(ii) Find the total number of squares inside a rectangle of width 2 and length 8.
........................................................ [2]
(c) Can a rectangle of width 2 have a total of exactly 30 squares inside?
Show how you decide.
........................................................................................................ [1]
(d) (i) Find an expression, in terms of $L$, for the total number of squares in a rectangle of width 2 and length $L$.
........................................................ [2]
(ii) Calculate the total number of squares in a rectangle of width 2 and length 170.
........................................................ [2]
535
598
552
(a) Draw lines on these rectangles and write the number of squares under each one to find the total number of squares in a rectangle of width 3 and length 4. You may not need to use all the rectangles.
(b) (i) Complete the table. You may use the grid to help you.
[Table_1: Rectangles of width 3
Length of rectangle | Total number of squares
3 | 14
4 |
5 | 26
6 | 32
7 | ]
(ii) Find an expression, in terms of $L$, for the number of squares in a rectangle of width 3 and length $L$.
[Grid Image]
513
521
569
(a) Use this method to show that the total number of squares in a rectangle of width 4 and length 4 is 30. [2]
(b) (i) Complete the table.
[Table_1]
| Rectangles of width 4 |
| Length of rectangle | Total number of squares |
| 4 | 30 |
| 5 | |
| 6 | |
| 7 | |
| 8 | 70 |
[3]
(ii) Find an expression, in terms of $L$, for the number of squares in a rectangle of width 4 and length $L$.
...................................................... [2]
563
544
543
(a) Complete the table.
Use your answers to Question 1, Question 2(d)(i), Question 3(b)(ii) and Question 4(b)(ii) to help you.
| Width of rectangle | Expression for total number of squares in terms of $L$ |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | $21L - 35$ |
(b) Find all the rectangles that have a total of 50 squares.
Give the length and width of each rectangle.
522
590
569
