Coordinate Tan

This problem combines a skill - reading coordinates in all four quadrants - with mathematical thinking in the form of pattern spotting, i.e. generalising.

The problem is probably best introduced to the group of children in your class who are reasonably confident with coordinates in all four quadrants, as children who are struggling with the idea of negative coordinates may become confused. It is more engaging if done using the interactivity but it is possible to do the same activity just with pencil and squared paper.

Check that all the children understand what is meant by the origin and ask them to work in pairs to answer the first part of the problem. They then check their answers with another pair and resolve any differences. Still working in pairs, allow the children some time to investigate the effect of changing the origin. Bring the group together and ask the children to share their findings. Highlight examples of systematic working, for example drawing up a table, or listing the pairs of coordinates in some way which makes spotting a pattern easier.

Can they make any general statements about the changes? Set up a shared place for them to record these, for example on an A3 piece of paper or the whiteboard. The rest of the children then check and record whether the statement is always true, sometimes true or never true. Facilitate the conversations so that the children come to an agreed conclusion about each.

What ways of recording do we know that will help us to spot patterns easily?

Does moving the origin systematically help?

Can you predict what changes will happen to the coordinates if you know what has happened to the origin?

Some children will have noticed a pattern between old and new coordinates but will not have linked this to the shift of the origin. Others will have realised that changes in the coordinates will 'look different' according to whether they are above or below the origin, or left or right. For example moving the origin to the right one unit will change $(5,1)$ to $(4,1)$ and $(-1,1)$ to $(-2,1)$. Both of course are the same change to the first coordinate (one less) but most children will think that changing $-1$ to $-2$ is an increase. A discussion using the number line can help them to realise that bigger numbers are to the right, smaller to the left.

Some children will need support in recording in an appropriate way. The accompanying solution illustrates that some children will find patterns which are specific. Sharing findings can help them to understand the difference between the specific and the general.