Seeing patterns in data

Suppose a sample of student data looked like the data in the table below. After circling the 2s and the 4s and the 6s, what patterns do students see?

 Distance input moves

1”

2”

3”

 Distance output moves

2”, 2 ½ ”, 2”

4”, 5”, 4”

5”, 6”, 6”

Typically, students see patterns such as these:

The output moves more than the input.
The input goes 1, 2, 3, and the output goes 2, 4, 6.
Each time the input gets bigger by 1 and the output gets bigger by 2 Students see relations of bigger than / less than, they see counting relationships, and they see additive relationships. In general, they have a more difficult time seeing multiplicative relations, which is what ratios and proportions are.

Questions such as these can help students see that they can multiply to establish a relation between the numbers.

What can I do to the top number to get the bottom number?

Force and motion