Many of us probably memorized the slope intercept formula (y=mx+b) in middle school without fully understanding how the formula represents linear growth. I was not able to visualize how the formula translated into lines until 11th or 12th grade.
It turns out that, in Japan, some teachers give students an open ended (but supported) problem called the “growing dot” problem to try and build conceptual knowledge that relates to the slope intercept formula. Check out the slideshow below (adapted from a presentation for a Math and Technology class at Boston College) to see how one teacher applied this method to his middle-school students.
Source: Goldsmith, L. T., and Seago, N. M. (2013). Examining mathematics practice through classroom artifacts. 88-96, Boston. MA: Pearson