Just a Simple Protractor

Sometimes the simple things misunderstood create academic mountains where there should not even be molehills. Take proper use of a protractor.
A simple exercise using a good display can most probably stop these problems before they begin.

I suspect that is some teacher out there who could profit by using something as simple as this.
Find the original here.

 

Some thoughts on "progress"

So now the Massachusetts Turnpike is joining the movement to "cashless" tolls, which I see as a euphemism for "no more salaries and sick days and health insurance and no more PEOPLE!" Such is progress.  This progress will have 3 levels of charges from least expensive to most expensive: Massachusetts EZ-Pass, out-of-state EZ-Pass, and the rest. They justify higher tolls by claiming additional costs, but..Free Ride: New Mass. Pike tolling system will allow you to bypass tolls in Springfield, Worcester points out that in many instances short runs on the Mass Pike will be toll-free. Seems to me that charging "others" more will help subsidize this. 

Speaking of the "others", another article, from WVBC5, states "For drivers who lack any type of electronic tolling transponder, the cost of driving from New York to Logan Airport will nearly double from its current $7.10 to $13.40 plus a 60-cent billing fee." Doesn't that sound like a strong push to sell EZPass? Be advised that to get an EZPass, automatic deduction must be started from an account (could be a credit card), and the tolls are actually prepaid with the requirement of an initial payment of at least $20.  They actually take your money up front. 

FYI: With the advent of  "cashless", the Mass Pike toll for the entire eastbound run, end-to-end, for EZPassMA holders will go from $6.60 to $7.65. Talk about the economies of progress. People will pay more even though people will be losing jobs.

I have a NY EZPass, and I got it originally out of a wish to get in shorter lines. 

It won't be long before before your driverless Uber takes you via cashless tolls to visit your college student son or daughter who is just months away from huge debt and no job.  Such is progress.

It has been a while since I began to see "self checkouts" at grocery stores and Walmart and Target, etc., giving you the opportunity to perform the cashiers job at no pay and with no discount. Take a moment to read Self-checkouts: Who really benefits from the technology? courtesy of CBC. 

Amazon's attempts on drone delivery are disguised as "faster delivery" but are actually just an attempt to cut back on the need to pay people such as deliverymen. They are already making their "human" delivery less efficient by frequently bypassing UPS and FedEx and opting for USPS, whereby your items could be ordered Thursday afternoon, arrive at your local post office distribution center Friday, then sit until Monday until your local post office delivery time. Drone technology just might be accepted if the customers see it as a 'speed-up" in delivery.  Delivery times to my home have gotten appreciably longer despite using Amazon Prime.

I see more and more instances of people losing jobs not because of jobs becoming obsolete (such as farriers with the advent of the automobile) but rather because technology has found a way to keep the job but make it robotic.  The only people who can consider some of these changes as progress are the investors, as they see payroll and benefits decrease, increasing their own bottom line.

To read some opposing viewpoints on this issue, check out these two articles:

How Technology is Destroying Jobs from MIT Technology Review

Automation and Technology Increase Living Standards from www.heritage.org.

Get the Cart Behind the Horse

Many times during the flash of time when I was in the classroom (even though sometimes it felt like an eternity), I questioned what I was doing as I was doing it, and in the process it seemed like I was living the distinction between school and education.

One thing that always bugged me was that I had to repeatedly refer to the commutative property of addition before students had adequately internalized the facts that in addition the order of the two numbers is irrelevant, but with subtraction order matters. Similarly with multiplication and division. In my school, students would become versed in arithmetic well before getting bogged down in the legalese vocabulary of the laws of arithmetic. (Just as we expect people to be able to drive legally and safely without needing to quote government statutes on transportation)

As another example, I refer to the distance between two points on the coordinate plane. For those non-mathematically inclined, that amounts to placing two dots on a grid such as shown here, and determining the distance between them using the grid's scale.

Quite frequently you would see a textbook begin by giving the formula, then include a short dissertation on where it came from (normally making reference to the Pythagorean theorem), and then jump into an application or two.

Just like the real world! First the formula is found, then it is verified to be true, then it is used! That is exactly how life goes, isn't it?  Of course not!

In my personal school I would never make mention of the distance formula until the time came to compare the distance method to that for slope. As a matter of fact, I would introduce the formulas together, AFTER students already had a solid grasp of calculating distances and slopes on the coordinate grid.  Prior to that I would make copious use of the phrases "change in y", and "change in x" interlaced with references to "rise" and "run" until the students picked out their seeming interchangeability, at which point we would shorten them to Δx and  Δy, at which point there would be discussion of the Greek letter delta (any connection to a river's delta?).

Somewhere buried withing the classroom conversation would be the discovery that in the question of distance the signs of Δx and  Δy were irrelevant, but in the case of slope they were very relevant. To help us with this issue, we would establish a tradition of referring to one of the points as (x_1,y₁) and (x_2, y₂). Only then, when appropriate, would we actually convert our methods for distance and slope into actual formulas. The formulas would appear as short hand for what they were already doing, not as a prescriptive rule to be mindlessly followed. (Could you imagine if a student driver's first lesson was all about the cruise control buttons?) 

Welcome to a new school year

I have a creation here that I made today with my usual goal in mind: hoping to help teachers teach and students learn. My hope is that somewhere sometime this school year, some teacher or some student might find this helpful.  That is the "some" of my intentions.

The original file can be found here: https://ggbm.at/p48kEWCp.

Part of Math is Just Being Amazed.

This I put together this morning just as a thought provoker. It shows the essence of a relationship between circles and ellipses that I have not seen in any book, be it textbook or not.

The basic relationship is accessible to anyone who knows what midpoints of segments and rotation around a point mean.  The file shown here can be downloaded at https://ggbm.at/nmNGpVs6