Math – Applications for Living II

Another day in “Math119-Land” (applications for living), though students were not enthusiastic about today’s work.

Percent are standardized rates; percents are ‘evil’ (in the way that people describe things that do not make sense).  We were doing some puzzles today, to prepare us for the good stuff.  For example, today we had “after an 8% discount, the price was $75.60”.  Today, this was a puzzle … meaning the answer was known by the person who wrote it, and our job is to find it.  Most students wanted to multiply 8% by 75.60 and then add … sadly, close in numeric value but awfully wrong in terms of relationships.

The whole point of these puzzles, in this class, is to connect a percent increase to a growth model and a percent decrease to a decay model.  Eventually, we will write the exponential models in this class … calculate various outputs, graph a bit, and even find the original value or the multiplier in limited types.  We went through the usual ‘8% less means take off .08 times the original, which gives us 0.92 times the original’.  A large portion of the class did not think this made any sense at all.

On days like this, I wonder if we should delay all percent work until we are in a setting where we can use algebra.  Students have learned one method (operations on the numbers given), and resist a transition to a formalized method.  This resistance handicaps their problem solving skills, which would show in other classes besides math (science in particular but including ‘social sciences’).

Earlier, I had a post on ‘stealth percents’, and today’s post is related — our students really struggle with percents.  Another example from today —
    “A survey last year gave the mayor an 84% approval rating, and the recent survey showed a 33% approval rating.  What is the relative change?”

Students easily subtracted the percent values, though some thought that the percents HAD to be converted first.  However, few of them saw how to make this a ‘relative change’ — which is often the only measure that makes a significant difference. 

Unfortunately, percents are commonly used in a variety of problems and situations; today’s troubles with percents will show again when we talk about finance formulas in a couple of weeks.  Later in the semester, we will bring up the exponential growth and decay, where we connect a percent change to an algebraic model.    Each time, I can hope to make this percent mess clearer to the students.  Wish me luck!

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Math Applications for Living I

I’ll be sharing some applications that you might find interesting from the Math119 class that I am teaching this semester.  The title of this class is “Math – Applications for Living”, and that will tend to be the title of the post; this is number I in this ‘series’.

Here is the problem:  You are figuring out if it makes more sense to take a bath or take a shower, in terms of conserving money or energy (or both!).  The approximate dimensions of the bath tub are 6 feet by 3 feet by 2.5 feet, and you guess that it is filled halfway.  Your shower head, according to the manufacturer, has a flow rate of 1.75 gallons per minute; you guess that you take 10 minutes with the shower with the water running. 

A key step in solving the problem is being able to convert from cubic feet to gallons (or the reverse).  In our class, we focus on flexible use of fractions — proportional reasoning.  For this problem, we have the fact listed that there are 7.5 gallons in one cubic foot.  If you want the values, this bath scenario uses about 170 gallons of water while the shower uses about 17.5 gallons.  [We have not covered precision and significant digits yet, so I did not worry about whether they rounded the bath volume correctly.]

   0.5 * (6ft * 3ft * 2.5ft) = 22.5 ft³;     22.5 ft³ * (7.5 gal/1 ft³) = 168.75 gal  [Bath]           10 min * (1.75 gal/1 min) = 17.5 gal

The students were a bit surprised by the magnitude of the difference, and they seem to be getting more skilled at using these fractions.  We will see how they do next week when I give them a more challenging follow-up problem.

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