# Teaching linear functions for meaning

For years I’ve been rearranging the pieces of my linear functions unit like a jigsaw puzzle, trying to optimize comprehension for weaker students.  Weaker students see math as a giant bag of disconnected steps to memorize, right?  Changing that can require a cultural shift in the classroom that I’m not usually able to pull off.  It’s not that student engagement is so hard — there are lots of tasks that kids get excited about.  But while those tasks might motivate kids to learn something like slope, they don’t always help kids internalize what slope really means.

And even if you can give them an aha! moment today, it may be lost by tomorrow.  In fact, it probably will be.  Once you introduce the slope formula, slope becomes that formula.  It barely even matters if today’s lesson created a nice footpath in students’ brains between “slope” and the change in one quantity per unit of change in another.  Once that formula comes out, your measly footpath is no competition for the 8-lane highway that’s opened up between “slope” and (y2–y1)/(x-x1).

If the intuitive meaning is going to compete at all, you’re going to have to write a lot of drill and assessment questions that force students to traverse that footpath over and over until students notice what nice scenery it has.

Here’s a screenshot of the key question type I’ve developed to make that happen:

• You can visually interpret why m = y/x doesn’t work when the y-intercept isn’t zero.  In the top picture, y/x would be 105/2=52.5 grams.  Does each M&M weigh 52.5 g?  No, because 52.5 g would represent 1 M&M plus half the fro-yo.  On the other hand, if you just had a bunch of candies on the scale without any fro-yo, y/x would make perfect sense.
• This question imposes a time cost to missing the conceptual point, but still allows students to get the question right if you don’t get the concept.  If you really can’t tell whether something’s direct variation, you can always divide y/x for both examples and see if you get the same answer both times.  (Sadly, that’s mainly what my state wants kids to learn about direct variation: that given a table of values, you should divide y/x and see if you get the same answer for all the ordered pairs).  There will always be a few students who need to do this.  But it’s much faster to notice visually that this example is not direct variation because there’s a non-zero y-intercept: the fro-yo.  So I’m teaching what the state wants me to teach, but allowing students to use comprehension as a shortcut if they can see it.
• You can even visually interpret why non-direct variation scenarios give you different answers to  y/x.  In the top picture, 105/2 represents 1 M&M and half the fro-yo.  In the bottom picture, 108/6 represents 1 M&M and just one-sixth of the fro-yo.  Kids can see that it should be a smaller answer.
• Repeated practice with this question type causes students to associate m=(y2-y1)/(x-x1) and y=mx+b with each other, and m=y/x and y=mx with each other.  I want that association.

Soooooo, drumroll please, I now present my new linear functions unit outline:

• Linear dot patterns.
• The focus is on noticing which part of the pattern is repeating, and which part is staying the same, and what to do with that information.  We don’t use the words “slope” and “y-intercept” yet.
• Note: some dot patterns should be direct variation. Direct variation patterns can even be tricky by having 2 parts to the pattern that both repeat:
• Linear story problems. These are your usual algebra story problems: a tree was 4 ft tall when it was planted, and it grows at a rate of 1.5 ft per year.  Students have to interpret key words that indicate initial value or rate of change.  Here we use y=mx+b, but not the words “slope” and “y-intercept”.  Instead, students use their own words to describe what the m and b mean.
• Linear graphing stories.  I lead off with some of Dan Meyer’s graphing stories (Kenneth Lawler’s bench-press and Adam Poetzel’s Height of Waist off Ground), focusing on how the starting value shows up on the graph as the y-intercept.  Then we do my own chubby bunny lesson (video below) which is more geared toward slope-intercept form.  Now that rate is showing up as steepness on a graph, and the starting amount is showing up on the y-axis, it’s okay to start calling them the “slope” and b the “y-intercept”.
• Slope from 2 points, conceptually, which explores the concepts shown in the fro-yo question above.  Here is my slope-from-2-points lesson.  I do it as a Pear Deck lesson now, but at some point will probably convert it over to Desmos now that Desmos has the classroom conversation toolkit.   After this I start giving questions like the fro-yo assessment question.
• Identifying proportional scenarios.  Given a scenario, can students identify it as a proportional or non-proportional situation?
• Here’s a screenshot of this question type:

The idea here is to combine “Graphing stories” with “Slope from 2 points, conceptually.”   The multiple choice graphing question scaffolds kids’ thinking, but it’s not just a crutch: it also improves learning by signaling to kids that the most important thing is to use common sense to tell whether the y-intercept would be zero or not.   So in the question above, would a pizza with zero toppings cost \$0.00 ?  Compare this to the current Khan Academy exercise on identifying proportional situations:

• Slope-intercept form formalism, including the all the goodies we need kids to know: graphing lines given in slope-intercept form, applying the formula for slope to random pairs of points, etc.  My kids need lots of focus on distinguishing between y=2x and y=2+x, and also y=2 or x=2.  Yours, too, right?  The formalism can also include some more advanced work on slope and direct variation.

I’m very hopeful that at each phase of this unit outline,  I’ll be able to ask quiz questions that check real comprehension of the meaning.  And for a certain type of kid, if it ain’t on the quiz (and the quiz after that, and…), then you never really taught it.

**Yes, that quiz question is part of my new adaptive paper-based quiz generator.