Math – Applications for Living XII

In our ‘math – applications for living’ class, we are reviewing what we have learned this semester.  Some parts (like probability) are still tough for students, partially because there is some memorizing to do with new material.  Truth is … I like to cover probability mostly because the process encourages reasoning about quantities.  [For example, we had a problem to solve about the probability of having 5 children — 2 girls followed by 3 boys; some of us wanted to look at this as dependent probability: 2/5 for the first girl, 1/4 for the second, and then confusion about what to do with the boys.  Clearly, knowing that events are independent is critical.]

The best problem we worked on today was one with almost no practical value: 

We had to really work on this problem.  The intent is to have students focus on the units (we need ‘square feet’ for area; we have cubic feet and feet … how can we do this?).  When students asked how to do this problem, I would ask them “How do you measure area?” (to get them thinking about units).  Every student (individually) said “length times width”; clearly, we are still too focused on one formula, and not thinking about what we are measuring. 

Of course, we could follow up on the “length times width” idea with something more reasonable. 

S: Area is length times width.
I: Okay, for a rectangle we calculate area that way.  How do we calculate the volume of a box?
S: Multiply (writes V = LWH)
I: So, the volume is L*W times H; right?
S: Yes
I: We know that L*W is the area of a rectangle.  Think of that volume formula as “V = area * Height”.  How would we solve this for the height, which is like the depth of the lake?
S: Hmmm (thinking) … we would divide
I: Yep — divide both sides by area.  Does that give you an idea how to solve the lake problem?

Most students originally decided that they had better multiply the numbers in the problem; of course, they only dealt with the value not the units.  They did not think about getting “feet to the fourth power”, and what this might mean.  A couple of students thought that the ‘cubic’ in ‘cubic feet’ meant that that value needed to be cubed.  [More evidence of a ‘messy landscape’ of math knowledge.]

The good news from today’s class was that students actually did a reasonably good job figuring out a complicated ‘unit conversion problem’ (given dimensions of a box, the flow in gallons per minute, and rate of gallons per cubic feet … how long would it take to fill the box).  Prolonged effort on related problems with diverse settings has paid off.  We are having more difficulties with geometry (concepts) than we are with proportional reasoning.

 
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