Week 14: We have used an ordered table and a Cartesian graph to represent the changing score in a football match. This week we look at yet more representations. Most of these are not commonly used (and inded some could be said to have been invented for this blog) and pupils certainly don't have to become 'fluent' in their use. The set of tasks might help pupils to see
- that mathematical representations are not handed down from on high but are invented by us;
- that we might like some representions more than others (and different representations have different affordances), and
- that we can learn to read and make sense of representations.
We use information from some of the matches that we met in Week 13, so pupils can refer back to these,
at appropriate times, to check their work.
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Monday: Here we use arrows to show the football scores. How is the arrow diagram related to the graph?
Instead of drawing dots, we have drawn the lines that join successive dots. The arrow-heads help to emphase the direction in which the scores build up.
If we rotate the new diagram through 45˚ anticlockwise, it maps neatly onto the graph.
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Tuesday: Here we consolidate the ideas from Monday's task by asking pupils to interpret an arrow diagram.
Pupils can check their reading of the arrow diagram by looking back at the Monday and Tuesday tasks of Week 13. We also show the complete table in the next task.
You might want to ask pupils to describe how the arrow diagram conveys the fluctuating fortunes of the two teams.
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Wednesday: This new representation might look very strange! But the Cartesian graph probably did too, when pupils first met it! What exactly is going on?!
Here we represent a score not by a single dot, but by a line segment, or rather by its two end points. This is rather like a mapping diagram (see Friday's task) excpet that the axes are at right angles ('orthogonal') rather than parallel. We could convert our new representation back into a Cartesian graph by replacing each slanting line segment with a a vertical and horizontal line segment connecting the two endpoints, and by putting a dot where the new segments join.
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Thursday: This shows the completed mapping diagram from Wednesday's task. How can we tell which lines represent a score where Rangers are in the lead?
It is interesting to compare this mapping diagram to the kind of graph-with-added-red-line that we had in last Thursday's task. The diagrams below carry the same information about which team is in the lead or whether the scores are level, but how effectively does each one convey the information to us?
In the second slide we have changed the colour of some of the mapping lines.
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Friday: Here is another mapping diagram, but this time with parallel axes. Is this form easier to read?
The final score was Valencia 2:4 Arsenal (see Friday's task of Week 13). The vertical lines show scores that are level: 0,0 and 1,1 and 2,2. Down-Left lines show Valencia in the lead, of which there is only one, when the score is 1,0. Down-Right lines show Arsenal in the lead: 2,1 and 3,2 and 4,3.
What are the strengths and weaknesses of the different representations, ie the score card, ordered table, Cartesian graph, mapping diagram with orthogonal axes, mapping diagram wih parallel axes?
