ALG 03

Week 3: This week is about frieze patterns, ie patterns whose elements repeat and where the focus is on the identity of a particular element in the pattern. This is in contrast to last week's tasks which can be said to involve growth patterns, where the focus is on the number of elements that a particular exemplar contains.
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Monday: We start with a simple frieze pattern with just two repeating elements. Each odd element is a square, each even element is a circle.

The given number, 29, is perhaps just about small enough to tempt some pupils into drawing the first 29 elements. However, a class discussion about the odd-even nature of the pattern should help pupils see that it is not necessary to draw the elements.
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Tuesday: This pattern involves multiples of 3 rather than multiples of 2. It is thus still relatively straightforward. However, some pupils might slip into thinking that the pattern renews on a multiple of 3, ie that the position of each square is a multiple of 3 rather than a multiple of 3, plus 1.

We know that

• the position of each square is 1 more (or 2 less) than a multiple of 3
• the position of each circle is 2 more (or 1 less) than a multiple of 3
• the position of each diamond is a multiple of 3
• 29 is 2 more (or 1 less) than a multiple of 3, so the 29th shape is a circle.

However, because of the possible confusion mentioned above about whether the pattern renews on a multiple of 3 or on 1 more than a multiple of 3, we ask pupils to check their answer by drawing. At the same time this might reinforce the idea that thinking mathematically can be a lot quicker (and more efficient if done carefully) than acting out the task in a concrete way!
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Wednesday: Here we have two patterns in one. The position of each diamond is a multiple of 3; the position of each yellow shape is a multiple of 2 (ie an even number).

Both features of the frieze pattern are familiar so the task allows us to revisit the ideas met in
Monday's and Tuesday's task.
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Thursday: At first sight, this pattern might appear to be of a different type, but in fact it is still a simple repeating pattern, this time involving multiples of 5.

61 is '1 more than a multiple of 5'. This describes the position each time of the first element of the 5-element repeating pattern, and this element is a square.
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Friday: This pattern is genuinely different from the previous frieze patterns. It doesn't involve multiples, but nor are we asked to identify the element occupying a specific position (which would be quite demanding here!).

The key here is that the number of circles that occur immediately after a specific square (and before the next square) is given by the square's position number. So the 100th square is followed by 100 circles and the 101st square is followed by 101 circles.
The task is complicated by asking only about the number of yellow circles between the 100th and 102nd square. These only occur after odd-numbered squares, so there will be 101 yellow circles between the 100th and 102nd square.