Lesson: Tricky Tables Part 2
If you haven’t yet read the first post in this series, please do before continuing here.
We left off with the students/interview panel having completed an activity involving “filling in” the missing values in tables that increase or decrease at a constant rate. I neglected to tell you that,on the day prior to this activity in class, students were assigned the following homework:
I had also handed it out to the interview panel to work on briefly before the rest of the “lesson.” So, coming in, most of the students had “tried out” these kinds of problems in advance.
What were some common strategies (both on the homework and the slide problems?) Guess and check was popular with the eighth graders – especially since most of the rates were whole numbers. The further apart the “known values” drifted, the more challenging it became for them to apply this method. However, some continued to stick with that strategy, even after others shared more efficient methods. Just “hearing about” another way to solve a problem doesn’t automatically change your mindset if you are not ready.
Many focused on how far apart the two y values were, and then students (and interview panel) started counting the “spaces” between the values. If there were three spaces between the y values, they divided the difference in the y values by three. Then they realized that didn’t work and adjusted up to dividing by four (often without really thinking about why.) One member of the interview panel came up with a formula for find the amount of increase or decrease: d/(g+1). We then had to define the variables in the discussion portion: d = difference in the y values and g = “gap” (the number of spaces in the gap.)
In class (and during the interview) we pulled back and added some terminology.
I chose to use “rate of change” at this point – reinforcing the idea that we are interested in how “fast” the values are changing. The “non-example” was an eye-opener for some, as they were entirely focused on the y-values and not really looking at x at all.
The next section deals with activities we did in the classroom. The interview “mini-lesson” wrap-up will resume further down.
Expanding the Concept
Before moving on to finding more challenging rates of change, we moved to how this relates to graphs and story situations. In each of these activities I use a “Think-Pair-Share” routine.
Most (but not all) students were able to “see” the rate of change by looking at y-coordinates for consecutive x-coordinates. (My words, not theirs!) They often (but not always) started at the y-intercept (not a “known vocabulary term yet”.) Those who are not as comfortable with reading coordinates off of a graph struggled more. Marking in the points on the graph and identifying coordinates definitely helped. (I use Absolute Board to draw on screen shots of the slides, or shine the projector on the whiteboard instead of the screen and “write” on the image.) We segued from there to defining another term:
I am sure this is not the “classic” definition of slope people are looking for, but it follows from how we had been approaching the problems.
Next up was connecting to story situations:
Students whipped through these like lightning. For future planning I want to stretch them a bit by addressing situations like the ones given below, but maybe not until we reach the next level.
(1) Stu had $100 in savings. Four weeks later he had $200. On average, how much had he saved each week?
(2) On March 3, Sherry put an iTunes gift card on her account that was worth $100. By March 8 her balance was down to $70. On average, how much did she spend each day?
Disclaimer: These are all very much “pseudo context” but at this point we are developing concepts and not “problem-solving.”
At this point I want them to pause and reflect on what they understand and know how to do. Most are feeling quite confident regarding this concept. The biggest challenges were working though finding a process to “calculate” the rate of change, and some are still not there (yet!) Keeping the values fairly simple has allowed them to determine this process on their own rather than being “told what to do.”
Student writing prompt:
Interview panel writing prompt: (no discussion of slope, graphs, or stories)
Writing about math is always a challenge. Even if the students feel comfortable “knowing what to do,” it is still hard to put it into words.
Students have taken the first steps towards developing Enduring Mathematical Understanding regarding the concept of “rate of change” or “slope.” Even if it is just the idea of “how much it goes up to down by,” the concept will be expanded and refined in the days to come.