### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

### Paving the Way

A man paved a square courtyard and then decided that it was too small. He took up the tiles, bought 100 more and used them to pave another square courtyard. How many tiles did he use altogether?

### Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

# Squares in Rectangles

##### Stage: 3 Challenge Level:

A $2$ by $3$ rectangle contains $8$ squares:

six small $1\times 1$ squares and two larger $2 \times 2$ squares

A $3$ by $4$ rectangle contains $20$ squares:

twelve $1\times1$ squares, six $2 \times 2$ squares and two $3 \times 3$ squares

Consider rectangles with a height of $2$ units.
Increase their width by $1$ unit at a time.
What effect does this have on the total number of squares?

What about rectangles with a height of $3, 4, 5, \ldots$?

Make a note of the number of squares in rectangles with a height of $2$ units.

Do you notice anything special?

Use your results to decide whether a rectangle with a height of $2$ units can contain exactly $100$ squares?

What about rectangles with a height of $3, 4, 5, \ldots$?

Draw up a table of results.