Math Applications for Living XX: When a Foot is NOT 12 Inches

Our Math Applications Course is finishing the semester with reviewing and the final exam.   Students have made considerable progress in problem solving and reasoning, although they often can not see their own progress.  One concept from the course continues to create problems, though … so I’ve been thinking about what causes this difficulty.

Here is an example of a problem dealing with the concept:

A poster is 18 inches by 24 inches (rectangular).  Find the area, and covert it to square feet.

Finding the area’s numerical value was easy; knowing that the unit is ‘square inches’ was simple enough.  Converting to square feet?  Not nearly so easy to see.  Most students kept saying “a foot is 12 inches”, so they divided the area by 12.  One student suggested that we convert the original numbers to feet, and find the area in square feet.  This suggestion was seen as being reasonable, so  we did that … and then came back to the original problem.

As we struggled with this problem, we went back to the ‘a foot is 12 inches’ statement.  After a bit, we drew a square on the board — one side labeled ‘1 foot’ and another ’12 inches’.  Yes, we said, those are the same.  We labeled two sides ‘1 foot’ and two sides ’12 inches’.  The area?  1 square foot, or 144 square inches (a few students then understood what to do with our problem).  Some did not see the implication for converting, so I started drawing 1-inch strips in the square.  That might have helped a little; perhaps not.

A foot is not a foot when we are talking about area (or volume). In some ways, this is another example of prior learning being built in an overly simple space … we say ‘1 foot is 12 inches’, instead of saying ‘1 foot long is the same as 12 inches long’.  Conditional statements are critical for accurate learning, and enable problem solving skills to develop; unconditional statements impede future learning as the price for short term results.

Where am I presenting learning without conditional statements, when there should be some?  I fear that my classes routinely omit qualifiers for statements, sometimes due to the focus on the present problem … sometimes out of relative ignorance of where else the concept is used.

Sometimes, we create our own problems by deliberately omitting “if” and “when” statements.  Yes, these statements can impede current results; yes, we can become obsessed with technical accuracy to the point that only mathematicians can understand what we are saying.  However, I suspect that the price for simplicity in the ‘now’ is a set of problems in the future.

 Join Dev Math Revival on Facebook:

Math: Applications for Living XIX: Population Decline with Exponential Models

Percent change is all around us.  Sometimes, the percent change is based on the nature of the quantities involved (such as all those finance formulas).  Other times, the data for a situation assumes an exponential property.  Within our Math: Applications for Living course, we are looking at both half-life models and regular exponential models.  The example today applies these models to data about African elephants.

Although the decline of African elephants is not consistently exponential, as it responds to economic and social forces, the general nature is still exponential.  The current estimates used to predict future populations use differing rates of change, depending on assumptions.  The best case scenario is a 3% reduction in population per year; some recent data shows a spike in poaching activities, which would result in reductions well over 3%.

For class, however, we gave a problem like this.

The population of African elephants is declining at a rate just under 4% per year.  Estimate the percent of the current population which would remain in 100 years.

One way we approached this problem is to use the half-life model.  In order to estimate the half-life, we used the banker’s rule of thumb for doubling-time:  70/P.  In this case, 70/4 is 17.5 years; we will round that to 18 years.  The calculation becomes:

0.5^(100/18)

The result is 0.02 (approximately), or 2%; 2% of the current population would remain in 100 years.

We also used the basic exponential model; with a 4% decrease per year, the multiplier is .96.

(0.96)^100

This result is 0.017 (approximately), also rounded to 2%.

Students were uniformly surprised by this result.  They keep hearing about a half-million elephants declining at 4% per year, and this does not sound serious until we calculate the percent left after a longer period of time.

In our class, we do not explore the sociology of this problem, nor the political components (though it might be fun to combine mathematics and politics into one class).  Our focus is on basic mathematical concepts.  The most common theme in the class is percent change.  This application got their attention!

 Join Dev Math Revival on Facebook:

Math: Applications for Living XVIII “At Least Once”

As you ‘probably’ know, probability reasoning does not come easily for many students.  In our Applications for Living course, we cover some basic probability ideas and situations where they apply.  We have been struggling through a type called ‘at least once’.

Here is an example:

The probability that a particular region in Mexico will be hit by a hurricane in any given year is 0.1.  Find the probability that the region will be hit by a hurricane at least once in a 10 year period.

Students have a tendency to multiply the probability by the value of n, even though we have already discovered that this is not a correct method of a sequence of events.  We do work with a formula that our textbook provides:  P(at least once) = 1 – P(not)^n.  However, progress has only been made by taking this in three steps.

1. If the probability of it happening is 0.1, then the probability of it not happening is 0.9 (1 – 0.1)
2. At least once means all sequences except “not at all” in the 10 years.
3. The probability of “not at all” in 10 years is 0.9^10 (about .3487)
4. 1 – 0.3487 = 0.6513 (the probability of at least once in 10 years)

Students in my class live in Michigan (this is a face-to-face class), so hurricanes are not an event we need to plan for.  However, we use the same approach for floods and tornadoes, which do occur here.

Our course is finishing up our work with statistics and probability.  Students have discovered that these topics are more about reasoning than about calculating, which is not always a pleasant discovery.  Our hope, of course, is that reasoning in one domain (probability) transfers to other domains (in mathematics, and outside as well).

 
Join Dev Math Revival on Facebook:

Math: Applications for Living XVII

One of the big story lines in our ‘Math  — Applications for Living’ course is percent change. The first level is comparing absolute and relative changes in quantities, along with percent increase and decrease expressions.  We began to work on writing a mathematical model for percent increase or decrease as (1+r)^n, and saw compound interest as a variation on this.  The course ends soon after we formalize this work by looking at exponential models for growth and decay.

Recently, I saw an article in our local newspaper that illustrated exponential growth in the world of internet traffic.  With an accompanying story, the following graphic was used:

The original data is part of the Cisco IP modeling report (2011 to 2016); they have a report wizard at http://ciscovni.com/vni_forecast/index.htm

As you can see, the first and third graphs in the article are great examples of showing exponential change; the mobile data chart has the largest rate, but both graphs are delightfully exponential in nature.  The problem is the middle graph — for corporate accounts.  That graph is labeled “21% growth per year”, when the pattern is clearly linear; the data shows a slight decrease in growth rate later in the forecast period.

When we get to the exponential models in class, I plan to show this set of graphs in class and ask … ‘where is the error in this chart’.  I think it is interesting that a journalist writing about internet trends does not understand exponential change enough to clearly communicate about two different patterns.

 Join Dev Math Revival on Facebook: