ALG 20

Week 20: Here we look at some figurative patterns where there is (or could be) a rule between different elements of the pattern (eg between the number of blue tiles and yellow tiles).
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Monday: Can we invent a simple (linear) rule that allows us to determine the number of yellow tiles when we know the number of blue tiles?

From the limited information that we are given, there is no unique rule connecting the numbers of blue and yellow tiles. However we can invent rules that fit our two examples (Terry and Pete) and assume they apply to Nona.
The simplest rule can be expressed like this:
every Dogtile has 4 yellow tiles for its feet, plus the same number of yellow tiles for its neck as it has blue tiles, and one fewer yellow tile for its tail.
For Terry and Pete, this would give 4 + 4 + 3 and 4 + 2 + 1 yellow tiles respectively.
For Nona, it would give 4 + 9 + 8 = 21 yellow tiles.
We can express this rule in slightly different but equivalent ways, and also in ways that show the structure more explicity - as we do in Tuesday's task.
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Tuesday: Can pupils see what the various numbers in Ella's expressions refer to? Nona has 9 blue tiles. Where would 9 appear in Ella's expression?

The two 4s in Ella's expression for Terry, and the two 2s in her expression for Pete, play very different roles. In each case one of the 4s and one of the 2s is a 'quasi variable' and would be replaced by 9 in the expression for Nona, as here: 2×9 – 1 + 4.
We can show this more clearly by putting the quasi variable in blue, say:
Terry: 2×4 – 1 + 4.  Pete: 2×2 – 1 + 4.  Nona: 2×9 – 1 + 4.
Notice that we are quite close here to having a general formula for the number of yellow tiles when there are, say, B blue tiles: 2B – 1 + 4.
Can pupils explain Ella's rule? You might want to consider equivalent rules, such as this:
Terry: 4 + (4 – 1) + 4.  Pete: 2 + (2 – 1) + 4.  Nona: 9 + (9 – 1) + 4.
Here the first term refers to the neck, the second to the tail and the third to the feet.
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Wednesday: Meet the Blocks... Can pupils find a simple rule?

As before, there is no unique rule connecting the numbers of blue and yellow tiles. However we can invent rules that fit our two examples (Tertius and Juno) and assume they apply to Octavia.
The simplest rule can be expressed in this (perhaps rather long winded) way:
Each person has 2 yellow tiles for the ears and then the same number of yellow tiles for each hand and each foot as there are blue tiles for the body.
For Octavia, who is made of 8 blue tiles, we could express the rule in ways such as these:
2 + (8–1) + (8–1) + (8–1) + (8–1), or
2+ 4×(8–1).
We could write these expressions more simply as 4×8–2. Does this still make sense in terms of our tile pattern? 
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Thursday: The stick pattern here is reminiscent of the matchstick pattern in tasks 10C and 10D.

Here it is reasonable to assume that the sticks needed for the head, neck, arms and legs don't change but that adding an extra blue body tile requires 3 extra sticks (as in task 10D). So Octavia, who has 5 more body tiles than Tertious, will need 5 lots of 3 extra sticks, making 31 sticks in all.
This is quite challenging. It would be more challenging still to show the full structure of the relationship between the number of blue tiles and sticks, but we could do it with expressions like this for Octavia's sticks:
4 + 1 + 2 + 2 + 4 + (8–2)×3; or
9 + 4 + (8–2)×3, where the head, neck, arms and legs require 9 sticks, one of the body tiles requires 4 sticks and the remaining (8–2) body tiles require 3 sticks; or
9 + (8–1)×3 + 1, where we are now saying the body tiles each require 3 sticks, plus 1 extra stick.
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Friday: "Quincy, how many yellow parts do you have?"
"Excuse me, I will ask Alexa."

Here, the default rule would seem to be that each robot has 2 yellow parts for each foot, twice the number of arms as blue parts, and this number of yellow parts for each hand.
So Quincy, who has 5 blue parts, would have this number of yellow parts:
2 + 2 + 5×2×5, or
4 + 2×5²,
which makes a total of 54 yellow parts.
If we accept this rule, then Quincy will look something like this:

Bye, bye, Quincy. Goodbye dear reader.