Mathematical Literacy: Have we Been Here Before?

Our Math Lit class is nearing the halfway point in the semester; our second test is coming up.  Most of the content now relates to algebraic reasoning; our class work this week dealt with building expressions to model situations and slope.

The ‘modeling’ situation involved variations on the classic problem:

We have tables arranged in a straight line, and chairs are placed (one per open side).  With 4 tables, how many chairs?  With 5 tables, how many chairs?  With n tables, how many chairs?

This problem went quite well; our book does a good job of providing some scaffolding to go from concrete cases to expressions.  The class created at least 3 different descriptions of the pattern, and we showed that all three resulted in the same final expression (2n + 2).

We then looked at different shapes (L-shape, for example … or plus-sign shape).  Because we were dealing with 3 shapes, the process did not work as well; however, the discussion was even more productive.  The book provided a hint: Look at the number of tables with 0 chairs, 1 chair, … 4 chairs (for each shape).  Students did not have much trouble counting.  The challenge came in expressing the unknown for each shape; since the variable category is not the same for each shape, this led to the conversation:

Which type of situation varies for the shape, and which types are always the same?

Several students had the very positive experience of resolving an initial confusion into something that made sense to them.  In the plus-sign shape, for example, the 2-seat table varies, with 1 ‘0 seat’ and 4 ‘1 seat’ tables, and students saw that we needed ‘n – 5’ for the 2 seat count.  We also simplified the expressions for these shapes, and these particular shapes resulted in the same total — 2n + 2.  We talked about whether this is always the case (no), and solicited suggestions for shapes that would work differently; the strangest shape we had was a shape where 2 tables had 1 seat.

The next day focused on slope, and you might think that this would go better; most students had already learned about slope, and many already could say ‘rise over run’.  Two issues got in the way.  First, students wanted to count by coordinates when they started at a point; if the points were (-3, 4) and (2, -2) they would start at (-3, 4) and go right 2 and down 2 (as if they already knew the slope).  Second, the order issue was not obvious for students; they saw every rise as positive and every run as positive.  We had to discuss this for several minutes before it started making sense.

The good thing about this slope work was that we did not start with the classic “m = —–” formula for slope from coordinates.  We have a little better understanding of what slope is measuring, and depend less on memorizing.  Since we had already talked about linear change and exponential change, we could even talk about the slope being a constant for linear situations and slope changing for exponential situations.

Overall, this was a good week in class.  Our assessment (test) will help show how effective the work was.

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