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What is the minimum number of squares a 13 by 13 square can be dissected into? ### 2001 Spatial Oddity

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

# Square Coordinates

### Why do this problem?

This problem emphasises to students that squares don't just exist in their usual orientation. It goes well with the game Square It

The context offers an ideal opportunity to challenge students to visualise relationships between coordinates.
The interactivity could also be useful when introducing Pythagoras' Theorem and when working on the gradients of perpendicular lines.

### Possible approach

Display the interactivity. Ask for volunteers to move the corners to make a different square.

Fix a couple of corners and challenge students to complete the square.

Offer them a chance to see the coordinates.
Choose two points where all the coordinates are either all even or all odd. Challenge students to complete the square with these as opposite vertices.

Set students to work in pairs (ideally at computers) practising making squares until they can answer the key questions below. Suggest they make a variety of squares of different sizes and note down the sets of coordinates of their completed squares.

This could lead to a plenary discussion or, when appropriate, challenge students to work away from the computer on the final questions in the problem. This sheet provides further practice with tilted squares, but without reference to their co-ordinates.

### Key questions

How can we construct a square when we are given two adjacent corners?
How can we construct a square when we are given two opposite corners?
How can we construct a square when we are given the centre and one corner?
If we are given four points, how can we tell if they will make a square or not?
Can we do all this without plotting the points?

### Possible extension

How does this extend to rectangles?
If you are given three coordinates, work out how to determine if they will define a right angle.
Draw squares with as many different areas (under 50) as is possible. Which areas are possible and which aren't?

### Possible support

Provide students with a handout of some tilted squares drawn on squared paper and ask them to box each one in with a non-tilted square. Students can look at the four right angled triangles which result around the edge and will see that these triangles are congruent.

Students can answer the last four questions by plotting the points provided and boxing them in to decide whether they make a square.