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## 'A Cartesian Puzzle' printed from http://nrich.maths.org/

## A Cartesian Puzzle

Here are the coordinates of some quadrilaterals but in each case one coordinate is missing!

- $(2,11), \; (0,9),\; (2,7),\; (?,?)$
- $(3,7),\; (3,4),\; (8,4),\; (?,?)$
- $(18,3),\; (16,5), \;(12,5),\; (?,?)$
- $(13,12),\; (15,14),\; (12,17),\; (?,?)$
- $(7,14),\; (6,11),\; (7,8),\; (?,?)$
- $(15,9),\; (19,9),\; (16,11),\; (?,?)$
- $(11,3),\; (15,2),\; (16,6),\; (?,?)$
- $(9,16),\; (2,9),\; (9,2),\; (?,?)$

The quadrilaterals are all symmetrical. This may be rotational or line symmetry or both. Can you work out what the missing coordinates are if you know they are all positive? Is there more than one way to find out?

Now plot those eight missing coordinates on a graph like this. What shape do they make and what sort of symmetry does it have?

### Why do this problem?

This problem is one that requires some understanding of coordinates in the first quadrant. It will also call on knowledge of both rotational and line symmetry, and the properties of various quadrilaterals.

Possible approach

You could play a game of 'twenty questions' to begin with so that pupils get a chance to familiarise themselves with properties of shapes. Choose a quadrilateral and write the name of it on a piece of paper. Invite the class to ask questions to guess what your quadrilateral is, but you can only answer yes or no. Keep a tally of the number of questions asked - if they get it in less than
twenty, they win, otherwise you win. You could repeat this a few times with pupils choosing shapes.

You could start on the problem itself by showing it to the group on an interactive whiteboard or data projector. Alternatively, if they are already very familiar with coordinates in the first quadrant, you could get them to work in pairs from a printed sheet of the problem from the beginning. It is important that they are able to talk through their ideas with a partner while doing the
problem.

This sheet of the first quadrant could be used for both rough working and the final results. Otherwise supply plenty of squared paper! It might help learners to know that the coordinates of each quadrilateral are given going round in an anti-clockwise
direction. You could choose not to give this information to some pupils and then ask them to select the coordinates which give the most symmetry for the plotted shape.
One of the nice things about this problem is that learners will know that they have solved it correctly. In the plenary, therefore, you can concentrate on asking some pairs to explain the way they tackled the problem, rather than focusing on the answer. Were some ways more efficient than others?

### Key questions

What kind of quadrilateral do you think this one is?

Where is its fourth vertex?

What kind of symmetry do you think this quadrilateral has?

Possible extension

Learners could plot their own quadrilaterals with one vertex of each forming a hexagon and so make a similar problem for a friend to try.

Possible support

You might want to tell some children that the shapes include one parallelogram, one trapezium and one rhombus, and are otherwise squares and rectangles.