Students take out three slips of paper.
Have each student choose any number, write it on a slip of paper, then flip the paper and write the next consecutive number on the back.   On a second slip of paper goes the next consecutive number and so forth until each student has three slips of paper with six numbers on them.   Then they place the slips on their desks with the higher number showing.   They flip any one of the pieces of paper, and add up the three numbers showing.

Here is the "trick":
A student tells me her starting number and I tell her the sum.

I find the sum by multiplying the starting number by 3 and adding 8.

Here is an example:   Let's say the students picked the number 5 to start.
(1) Then the first slip of paper would have 5 and 6 on it.
(2) Then the second slip of paper would have 7 and 8 on it.
(3) Then the third slip of paper would have 9 and 10 on it.
(4) If the student flips the second piece of paper,
      then she would add 7, 6, and 10 to get 23.
(5) I would ask for her starting number, and she would say 5.
(6) Mentally, I would multiply 3 x 5 and add 8 to get 23.

Why does this work?

Use some algebra to prove it.
Let x = the first number.
Here are the three pieces of paper:
(1) x and x + 1
(2) x + 2 and x + 3
(3) x + 4 and x + 5
The student will be finding the sum of one smaller number (the flipped one) and two larger ones.
That sum will always be 3x + 8 (Try all three possibilities).

Here are some ways to extend this problem:
(1) Flip two slips of paper.
(2) Start with the lower numbers facing up.
(3) Use consecutive odd numbers.
(4) Use consecutive even numbers.
(5) Use four slips of paper or five.