ALG 17

Week 17: This week we look more closely at the Cartesian graph. We plot points that fit a story about numbers of 2p and 5p coins. It turns out that the points lie on a straight line, and we use, and try to make sense of, this fact.
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Monday: Here we introduce the story that will feature throughout the week, and find some values (of numbers of 2p and 5p coins) that we will subsequently plot on a graph.

This task is straightforward [apart from the fact, perhaps, that in part c) the given number refers to Ella's coins rather than Dan's]. We end up with the ordered pairs 50,60 and 100,40 and 125,30 for the numbers of Dan's and Ella's coins respectively.
It is possible that some pupils will start looking for patterns, for example that if we increase the number of Dan's coins by 5, we decrease the number of Ella's coins by 2. Is this always true? Why?! The existence of this pattern could be said to be the reason why we get a straight line when we represent these values as points on a graph (as we do in Tuesday's task). This is a nice insight - and useful for us as teachers! However, we don't take things to this depth in the current set of tasks.
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Tuesday: Here we provide a reminder of how we can represent pairs of values as points on a Cartesian graph. Pupils might notice that the points seem to form a pattern - namely that they lie on a straight line...

We have deliberately asked pupils merely to sketch the graph and we suggest that it is worth getting them to do this on plain paper (you could follow this up by asking pupils to use graph paper if you wish!). It is then interesting to see

• whether pupils place the points in a straight line in their sketches, or 
• whether, if they don't place them in a straight line, they nonetheless expect that they would lie on a straight line on a 'proper' graph, or 
• whether they have no expectations about this at all.

Wednesday: It seems fairly clear that the line cuts the 2p axis at the 200 mark, but the cut on the other axis is less clear. How do we find its value?

A primary aim of this task is to help pupils experience the fact that graphs usually make sense! So, here we have a set of points that fit a particular story. However, for this to happen, pupils will need to realise (or be made aware) that the 200 mark actually represents the pair of numbers 200,0 (ie 200 2p coins, 0 5p coins). Similarly, the mark on the 5p axis also represents a pair of numbers. But this mark represents more of a challenge. Do pupils estimate its value by eye and then check it fits the story (assuming they appreciate that it should fit the story!)? Or do they derive its value from the story, namely that it is the number of 5p coins that make 400p, ie 400÷5 = 80.
[Note: We have used informal notation for coordinate pairs, to convey the fact that the notation is arbitrary. However, you might prefer to use the conventional notation, eg (200, 0) rather than 200,0, especially if pupils are already familiar with this convention.]
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Thursday: It is likely that pupils will focus on the number of 2p coins first, when looking for the 4th point - so it will represent 75 coins, as 75 is midway between 50 and 100. How will they go about finding the second number (of 5p coins)? Use the story or use the numbers (60 and 40) shown on the graph?

Again, the task can help pupils appreciate that graphs can make sense! Here the 4th point fits the pattern shown on the graph (in that the numbers 75,50 lie midway between 50,60 and 100,40) and it fits the story: 75×2p + 50×5p = 150p + 250p = 400p.
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Friday: Would a 45˚ line through the origin help?!

There is no exact solution to this task, since the number that fits the desired condition (namely, 400÷7 = 571⁄7) is not a whole number.
The coordinates 55,55 provide quite a good visual estimate for a point on the graph's line. However 55×2 + 55×5 = 385, so we have not quite reached 400 and so 55,55 is not quite on the line. We can get the extra 15 needed to reach 400 by having three more 5p coins, giving us the ordered pair 55,58. From here we can get other pairs by exchanging two 5p coins for five 2p coins. Notice that this is equivalent to hopping in small steps along the line! So two other solutions would be 60,56 and 50,60. However, neither has a gap between the numbers as small as the gap for 55,58. It turns out that this is infact the closest pair.