Confusion involving less than or equal

Every year a number of my first-year students are confused by the “less than or equals” relation.

Some students are confused when I claim that 2 \leq 3 or when I claim that 1 \leq 1 (or both).

You can see what happened this year when I mentioned that 2 \leq 3:

https://www.youtube.com/watch?v=EgC-08j-6As&index=14&list=PLMvnJkyFoLy1H52LFymg2IqOHNQ2FE8vN&t=4m34s

As I said, I get questions about this every year.

I think that the confusion goes via a subtle conversion of “or” to “and”, as in  “it might be less than and it might be equal”, and this jars when they see examples where it is blatantly one and not the other.

Some students like diagrams a lot, and so number line sketches could help illustrate the concept more clearly.

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