Why do this problem?
is good for extending pupils' understanding of squares and to challenge their assumptions that a square must be drawn with horizontal and vertical sides. It is a good investigation for those pupils who enjoy practical work.
You could introduce the problem on an interactive whiteboard using this virtual geoboard
. Start by drawing the smallest square from the bottom left corner and ask pupils to find all the other squares that can be drawn from that point. If they are not working at computers using the interactivity, then this sheet of grids might be useful. Discuss the area of each one.
Then indicate a different starting point on the grid (for example the second dot from the left in the bottom row) and ask pupils to work in pairs to find the squares that can be drawn from it. You may find that some children notice that 'tilted' squares can be drawn, or you may have to draw the class' attention to tilted squares by suggesting a second dot to be joined to the first dot. Again,
take some time to share the squares they have found and talk about how you would find the areas of the tilted shapes.
You can then encourage learners to investigate other starting points on the grid, to draw the possible squares and find their areas. Make sure there are pairs of scissors available for pupils to use should they choose. In the plenary you can concentrate on good ways pupils have found to calculate the areas efficiently. This may have involved cutting out and laying pieces on a grid, or it could
have involved annotating their squares in some way.
How do you know that these are squares?
How can you check that these are squares?
Have you found all the squares which have a corner at that point? How do you know?
Invite them to look at squares which are created by overlapping lines, such as:
Learners could also increase the grid size to consider larger sizes of squares.
For more extension work
These pupils could act upon the extension activity that is outlined above but go further and calculate areas. Once they have done that, they could be challenged to find new squares that have an area that is between two values that they already have. They can then work towards making suggestions as to why, in certain circumstances, there are no new squares with 'in between' areas.
There will be a need for some practical resources like a nail board, dotted paper or pegboard.