# The sequence machine

The fun we have had with a Lakeshore Learning Multiplication Machine in our house is well documented. Not once has that fun been based in the machine’s original purpose, and I am here once again to report to you on a new off-label use for these things.

You should know that Lakeshore Learning makes a machine for each of the four basic operations: addition, subtraction, multiplication, and division. You should further know that they got the structure of the subtraction and division machines wrong (more on that at end of post). So today, I’ll focus on the addition and multiplication machines.

These have the same format as each other: 9 rows of 9 buttons. Each button labeled on top with m+n or m×n  accordingly, in the mth row and nth column. The button pops up when you press it; down when you press it again, like a ball-point clicky pen. On the front of the button, visible only when popped up, is the corresponding sum or product. Lakeshore Learning views them as mechanical flashcards and nothing more.

I convert them to Pattern Machines by covering all the numbers with colored vinyl. These are a ton of fun at home and in classrooms—great for patterning, counting, making pixel pictures, etc.

Now I am playing with a Sequence Machine. I have covered all of the tops of the buttons on a Multiplication Machine.

Now you can generate number sequences, without being distracted by the multiplication facts. Above, you see a familiar sequence—the squares.

Things quickly get more complicated, though. If you read each of the sequences below from left to right, ask yourself What comes next?What would be the 10th or 23rd term in the sequence? and What is the general relationship between the term number and its value (i.e. What is the nth term?)

There are many more sequences to be made on these; many more questions to explore.

I’m working on a follow up question about the relationship between the sequence generated by the same button patterns on Sequence Machines built from an Addition Machine, and from properly redesigned Subtraction and Division Machines.

What is a properly designed Subtraction Machine? you ask? I’m glad you did! Such a thing would have mn or m÷n on the button in row m, column n. This is not how Lakeshore Learning makes them. They make them like this: