ALG 12

Week 12: Here we explore a form of diagram called a lattice that allows us to generate all the factors of a number, given its prime factors, and to represent the structure of the set of factors.
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Monday: How are the factors 'connected' in this diagram? How can we use the line segments joining the various dots to find the two missing factors?

Each dot represents a factor. In general, a factor is directly connected to two factors below it in the diagram, and its value is the product of these two factors.
One missing factor is connected to the two factors 2 and 7 directly below it, so its value is 14. The other missing factor is connected to the two factors 3 and 7 directly below it, so its value is 21.
Three of the factors, 2, 3 and 7, are only connected to one factor directly below them, namely to the factor 1. What is special about the numbers 2, 3 and 7?
Another way of looking at the diagram is to group it into sets of parallel line segments; for each set, moving upwards along a segment is an instruction to multiply by a given number (eg ×2, if the direction is up-left); moving downwards along a segment is an instrcution to divide (eg ÷3 if the direction is vertically down).
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Tuesday: Which missing factor should we look for first?

We can choose various paths to find the missing factors. One approach is to find the missing 'red' number, ie the missing prime factor first. Given that we are looking for the factors of 70, this must be 7, which is represented by the lowest far-right dot. We can then work up the page: directly up give us 7×5=35; up-left gives us 7×2=15.
Or we could have started at 70 and worked down the page. Directly down one step gives us 70÷5= 15 (why?); down-right one step gives us 70÷2=35 (why?).
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Wednesday: These lattices have a different shape from the lattices for 42 and 70 (why?). But the basic rules are the same for finding (or generating) factors.

The missing factor in the first diagram is 14 (2×7 or 28÷2). In the second diagram, the missing factors are 5 and 15 (why?).
You might want to ask pupils what is 'special' about numbers with lattices of this shape. Expressed formally, we can say they are of the form a×a×b, where a and b are prime factors (though you might want to use more natural language with pupils). The smallest of this family of numbers is of course 12, which features in Thursday's task.
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Thursday: The lattice diagrams for 24 and 36 are different from this diagram, though they could both be said to contain it.

It is helpful to think of the numbers whose lattice diagram (and factors) we are trying to find, in terms of their prime factors. 24 = 2×2×2×3, which means its lattice diagram has a line segment representing ×2, starting from the dot representing 1 and occuring three times, and another line segment representing ×3, starting from the dot representing 1 and occuring just once, and drawn in another direction.
In the case of 36 = 2×2×3×3 we get line segments from 1 representing ×2 twice in one direction and representing ×3 twice in another direction.
The complete lattices look like this:

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Friday: This lattice has the same shape as the lattices considered on Monday and Tuesday, but extended in one direction, so instead of the form a×b×c, it is of the form a×a×b×c.

The lattice is for the number 2×2×5×3 = 60. The missing factors are 15, 30 and 60 along one branch and 10 and 20 along another.
Other numbers that have this lattice diagram include 84, 90, 132 and 150...
You might want to extend this work by looking at numbers that share some other lattice shape.