I have implemented the conceptual progression, from easy proportional situations to harder ones, here: https://ijkijkevin.wordpress.com/2016/11/03/teaching-linear-functions-for-meaning/

In harder situations, the y-intercept and the slope are not objects of different types. For example, suppose a puppy weighs 3 lbs when it is born and then gains 1 lb per week. The 3 lbs of birth-weight and the pound-per-week of gained weight are not visually distinguishable. They mix together. You can’t look at a picture of a 5-week old puppy and point to the 3 lbs that were there at birth. Similarly, when you turn on an oven and let it heat up to 400 degrees, you can’t point to the 71 degrees that were there from ambient temperature before you turned the oven on.

Harder non-proportional situations can’t really be understood by students until they really get that the y-intercept represents the amount when time = 0. Or, more simply, until they get that in Algebra we always decompose a single quantity into its original value plus its added value, even though that may feel like an unnatural thing to do.

As a result, you can teach identifying simpler non-proportional situations in your first introduction to linear functions (e.g., Stacking Cups or Domino Effect). But you can’t graduate to identifying harder ones until you’ve talked about graphing stories and really nailed the idea that the y-intercept is the (temporal) starting value.

Watched your milk video. Nice. What if you shot it from the top (so that you saw the weight but only a 2D circle of milk) so that students had to predict number of cups based on weight? Seems like we usually change x and predict y but rarely vice-versa. If this doesn’t muddle too much the independent/dependent relationship, maybe this could be a cool alternate angle…pardon the pun.

Disagree, though, about changing it to be continuous. If the activity serves as the turn from proportional relationships to linear functions, it seems reasonable to keep things easily countable. I wasn’t able to access Kate’s video, but I assume cumin is measured in something like tablespoons. Students may find a slope of “y grams per x tablespoons” a bit confusing. Not only is the weight-weight thing a bit tough, but I’d worry that — especially if the video shows someone scooping cumin into a glass — they might just treat the spoon as a discrete object…which would be problematic unless the scooping was extraordinarily precise. Marbles are easy to count, and lend themselves to a pretty intuitive “grams per marble” discussion.

Oh, and I agree with Hunter. Of course.