When we talk about number patterns in Geometry I think most of us miss an opportunity to tie it in with the Algebraic concept of systems of equations. In my opinion, for most students, linking the two concepts will make it easier to understand and remember.
Nowadays most high school Geometry students have not finished an Algebra II class and so should not be able to understand any system of equations problems that have exponents in them. So don't talk about those. If we limit ourselves to linear patterns the students will do fine. The discussion should go something like this:
'Now the first thing to do with these patterns is to just look at them and try to imagine what is being done to the first one to make it equal the second one, and what is being done to the second one to make it equal the third one, and so on. If the number is getting bigger then something is being added to it, or it is being multiplied by some number, or it is being raised to a power (probably squared, maybe cubed...) or some combination of these things. If the numbers are getting really big in a hurry then they are probably being squared or cubed. If they are getting big at a slower rate then they are probably being added to or multiplied by something.
Ex: 2 4 8 16 32 64
The numbers are getting bigger. What rule would account for it? Each time we are multiplying the number by two to get the next number. The next number in the pattern would be 128.
Ex: 2 4 16 256 65536
The numbers are getting really big really fast. What rule would account for it? Each time we are squaring the number to get the next number. The next number in the pattern is too big for my calculator to figure.
And oftentimes you will have a pattern that combines multiplication and addition.
Ex. 1 3 7 15 31
The numbers are getting bigger. You might be able to look at the pattern and just 'see' that the rule is two times the number plus one to equal the next number. But if you don't we can form a system of equations and solve it that way.
We let any of the numbers except for the first number in our pattern (NOT EVER the first number in our pattern) be our constant and we will multiply it by a variable A and add to it a variable B and this will equal the next number in the pattern.
So we have: 3A+B=7
And solving for A we get A=2 So we've discovered that we're multiplying our number by two each time to get the next number. But wait--there's more!!
And solving for B we get B=1. So we're also adding one each time to get the next number.
So our rule is 2x+1=the next number. So the next number in the pattern would be 63.
This method works if we are multiplying/dividing and/or adding/subtracting something to our numbers. If we happen to be dividing our number A would equal a fraction and if we happen to be subtracting from our number B would be negative.
The rules for solving a system of equations where the number is being raised to a power is a task for students who have finished an Algebra II class. So if you are faced with one of those just look at it and try to figure it out.
Be prepared to answer the questions: 'Why can't we use the first number in the pattern as part of a system of equations?' Poetic Metaphysical Answer: 'The second number in the pattern and all of the others are like a road with the rule telling it where to go. The first number is just an access ramp.'
'What happens if we use the form xA+B=next number in the pattern and we really aren't adding a number to it? Answer B would equal Zero. 'What happens if we use the form xA+B=next number in the pattern and we really aren't multiplying the number by anything, we're just adding a number to it?' Answer: A would equal 1.
And finally give the kids a difficult pattern and see if they can solve it with a system of equations.
*Don't forget to laugh diabolically...*