IMP Day 43 - Making "formal" connections

I have thoroughly enjoyed the Moving Along activity and decided to "sit down" here and make some formal connections to the slope formula and slope-intercept form. However, we first did the activity without using either formulas and it was AMAZING! We actually have been using slope-intercept form but they just didn't know it as y=mx+b. The IMP book uses y=ax+b and calls b the "starting point" and a the "rate of change."

Today I gave the class a "warm-up" where they were asked to find the slope of a line and write the equation of the line given 2 points. There was no context given and they did great! There were no formulas on the board but most of the students proceeded to put the 2 points in a table and then fill in the x-values from 0 to the highest x-value given. They go back to 0 for x because we have drilled the fact that x-coordinate of the starting point is always 0. We have also tried to drill that the starting point is always on the y-axis but they seem to forget that sometimes...

After the warm-up, which most of them got with a table, I had them add the slope formula and slope-intercept formula to their notes. I told them that they would receive a reference page on their end-of-course (EOC) algebra exam and that I wanted them to be familiar with the formulas. We went over how we could have used the formulas for the warm-up and then I gave them another problem.

I used this weird effect on my picture to make it a little more readable. 

He writes so light I know it is a little difficult to read. However, Raul (whose paper is above) was the first student finished finding the slope and equation of the line! Then I gave the class a problem where the slope was a fraction thinking that they would resort to using the formula (and most of them did). However, Raul and one other student STILL used the table to get the equation. Their "number sense" is very good and thinking in fractions (or decimals) did not bother those 2 a bit! We have only been writing equations of lines given 2 points for 2 days and there are many more students who do it correctly than I have seen in the past.

It is exciting to realize that there are students who really benefit from their exposure to the different methods that can be used to write equations. I guess that since I am so accustomed to using the formulas I thought they would automatically start using them. However, the majority of my students are still using tables! I don't think I have ever used a table to find the slope or y-intercept so I am getting an education too!