High School Math - How is this used in the real world?

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RLovell

Not your usual poster here. I am a high school teacher. I teach kids how to factor x^2 + 5x +6 or how to graph rational functions (polynomial over a polynomial) and inevitably someone asks how this abstract stuff is actually used by people out there.

The problem is that math teachers (myself included) usually have no clue! They are not specialists or experts that have an job that applies the math. But I bet people on this forum do all the time (even if they don't realize they learned the foundation of their math back in high school).

So if you have any examples, please let me know!! Please start high level and drill down, like:

A plane has a nose-wheel steering system. The system is controlled by a box with a computer in it. The computer has something called a PI controller which decides how much current to send to steer the wheel. The engineers need to figure out how to set certain parameters, P and I. To do this they work with second-order polynomials, specifically a rational function called "transfer function". One of the thing the I-value does is eliminate steady-state error, so the nose wheel is centred when the pilot has hit foot off the steering pedal. To set the I-value, engineers use a 2-dimensional graph called Pole-Zero. By finding the right zeroes/roots, they can find the best I-value.

Something like that...

I want to research this area. There is plenty of examples on how jobs use math but often that math is not in the high school curriculum. There is also the "mental gymnastics" / "learn to think systematically" argument. But often kids want to know examples of how a specific topic they are learning (factoring, quadratic equation, polynomials, radical functions, rational functions, domain/range) is used.

(And not how it could hypothetically be used in some dumb artificial word problem, nor how it was used thousands of years ago..)

Thanks!!
RL

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CuriousOne

Thermocouples are widely used in industry to measure temperature. In order to control temperature, one must be able accurately measure a temperature from the non-linear feedback of the thermocouple.

Although most modern controllers have the polynomial built in, making it easier for the technician, we still know it exists.

Example: http://www.mosaic-industries.com/em...re-measurement:thermocouple:calibration-table

<b>Moderator's Note:</b> When copying a pasting a long URL, please copy the entire URL and delete any spaces placed in the URL by the forum software.

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CuriousOne

Anything that is non-linear must have a equation that represents it or we cannot find the limits to control the process.

Electrical protective devices must use current transformer and voltage transformers to measure the process and alarm or trip when needed.

Without protective relaying the infinite electrical grid would not be very stable.

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Peter Nachtwey

> I bet people on this forum do all the time (even if they don't realize they
> learned the foundation of their math back in high school).

I bet not. One of my big complaints is that mechanical engineers rarely, in my experience never, provide a transfer function for a piece of machinery. I must do system identification to determine a transfer function. The it is easy to compute controller gains.

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CuriousOne

Thank you for making me aware of tinyurl.

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Bruce Durdle

The behaviour of a car's suspension system - in terms of bounce period and damping - are given by the roots of a second-order polynomial found from its differential equation. Useful information for any motorheads in the class.

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Curt Wuollet

Thermocouple linearization is a good example. They are cheap, they are sensitive, they require a system of 3 equations in 3 variables to linearize. Of course there are published tables, but it you need to interpolate or discern small changes such as for a mass flow sensor, you do the math. It was after I wrote code to work with undocumented thermistors that I found a 3VLE function on the Sharp calculator I had left over from teaching electronics.

Regards
cww

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pvbrowser

Back at university i learned control engineering.
You need to be familiar with the frequency domain (Fourier, Laplace, Z transformation).

In the real world in most cases you use PID controllers.
In our OSS project we have implemented this class
http://pvbrowser.org/pvbrowser/sf/manual/rlli/html/classrlController.html

or as tiny url
http://tinyurl.com/aue4f2w

But i remember a case where i had to use a Smith Predictor
http://en.wikipedia.org/wiki/Smith_predictor

The application was the flatness controller within a hot strip mill which was implemented in a process computer. The flatness of the hot strip was measured by a laser based system running on a QNX computer. Our OpenVMS based process computer got the measurements via network and send the setpoint for the CVC rolls (See google: "cvc rolls") to the basic automation system.

The big time delay within the closed loop here results from the distance of the last stand with the CVC rolls and the position of the laser measurement device which is more than 5 meters divided by the speed of the strip.

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William Sturm

> Not your usual poster here. I am a high school teacher. I teach kids how to factor x^2 + 5x +6 or how to graph
> rational functions (polynomial over a polynomial) and inevitably someone asks how this abstract stuff is actually used
> by people out there.

Much of the math that I use involves your basic linear equations with a slope and intercept. Those are used almost daily. They get multiplied and divided and translated. An example would be reading an analog input that has a range of 0-4095 and converting that to a user unit, such as position in inches. I might divide the value and then translate it so that it passes through the origin. A value may come to me with a range of 300 to 2500 and I have to scale it into percent, for instance.

I have also done a fair amount of differentiation and integration in computer programs. I typically use a secant line and your classic rise/run to calculate a slope. This is not a true derivative because computers are working with data that come in discrete steps. You cannot make the dt smaller than the sample time of the data, the dt cannot go to zero. To increase precision, one must increase the sampling rate of the computer. At some point, you have enough precision and increasing the sampling rate no longer improves the process.

I also use integrals occasionally. Much like the derivative, I have time sampled data to work with. The dt cannot go to zero. I basically use repeated addition, better known as a Riemann sum. Again, a higher sampling rate results in a smaller dt, although we cannot sample infinitely fast and we cannot have a dt truly approach zero. It is all about "good enough" in the computer world.

Another key issue in computers is the limited size of numbers. You have to deal with overflow and undeflow. I refer to computer numbers as a number circle instead of a number line. If you keep increasing the size of a 16 or 32 bit number, it will "roll over" and become very small. We have to remember to restrict the domains and ranges of our calculations.

Another key issue in computers is varying number bases. This concept goes all the way back to
basic addition with "carry and borrow". It would be helpful to teach students how to add numbers in different bases other than ten.

In my humble opinion, if you students understand these concepts, they will be off to a good start.

Bill Sturm

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William Sturm

Rotary encoders make a nice example for a Calculus lesson. You could think of the encoder as a light beam that counts each spoke on a bicycle as the wheel turns. Or distance that a robot travels. Or a car... Each pulse represents a specific distance (like a constant function). You add (integrate?) the pulses to get total distance traveled. To calculate velocity, you find the derivative of position, in this case you count pulses over a fixed time period (delta p / delta t). There is always some error when using a digital signal like an encoder to calculate velocity or position. This error is because the encoder is like a Greatest Integer Function, we cannot count partial pulses and lose some precision to rounding.

We used to use Tachogenerators or tachometers to measure velocity directly, the tach outputs a voltage that is directly proportional to velocity. This can easily be scaled into an actual unit of velocity. One could attempt to integrate the voltage over time to calculate position. This is not usually very accurate however, because the measurement errors are also integrated and result in a significant error in most cases. When integrating pulses from an encoder, our error is typically one encoder pulse.

Maybe this will give you some more ideas of real world applications.

Bill Sturm

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Curt Wuollet

Too long a day. I meant to say Thermistor.

Regards
cww

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Gerald Beaudoin

We recently had an application where a linear signal (imagine a sight glass on the side of a tank) was used to represent the contents of the tank. However the tanks were not entirely cylindrical, and so not directly proportional to the value seen in the sight glass. The bottoms of the tanks were conical and others were hemispherical. We therefore needed an expression to describe the how much liquid (volume) was in the tank, based on the linear value as seen in the sight glass (in practice it was a 4-20 signal, but you get the idea). I found all those simple volumetric calculations, specific gravity concepts, algebraic equations, and trigonometry, all left over from way back.....suddenly had a verrry practical use. It was a great example to demonstrate that learning this stuff, does eventually pay off, much more than just getting a course passing grade.

CSA

Hi,

This is such a great question!!! I think all of us when we were learning the basics in primary and secondary schools had this very same thought--how is this stuff actually used in the "real world?" The fact that you, RLovell, actually listen to the students and are trying to find examples for them is very admirable, indeed.

The best example I can provide was that in my final algebra course in secondary school one of the student's fathers was a roofing contractor and was trying to estimate how much roofing material was required to re-roof a local church building. The church's roof was curved from the eaves to the gutter, in an exponential curve, and the roof was u-shaped (it wrapped around the building on three sides and did not have right angled corners). The question posed to the class was how to find the equation of the angle of the roof, and then the area of the roof, and then to estimate how much roofing material would be required. He wanted to minimize unused material and provide the best estimate to the church.

The student's father was simply presuming the roof was flat and "square" and was making some assumptions about the angular area where the roof curved around the building and was "ball-parking" the amount of roofing material required (which also affected his total labor costs!). So, he had a figure, and was interested in how much different his estimate would be from a more formal, formulaic calculation of the roof area.

It turns out his estimate was almost 40% higher than the actual area we calculated after taking some measurements of the roof used to derive the formula. He used our calculated roof area to bid the job--and won it. And, even though he added 10% extra to the material he ordered it turned out that he had only about 8% extra material left over when the job was done. He was happy; the parishioners were happy; and the students and teacher were happy. Everyone got something very useful out of that experience.

That's kind of an unusual application (the roof was unique and older, and the construction plans were long since lost--hence the need to make our own measurements and calculations), but it was certainly educational and informative.

In my field most of the applications are simple straight-line, linear applications. Sometimes, there are multiple straight lines used to approximate a curve, and it's necessary to be able to solve for intersections and check for continuity of two or more simple y-mx+b equations. As the control systems become more capable and have more horsepower I'm sure the equations will become more indicative of real factors and use exponential functions for reference calculations, but not too much at this time.

But, I applaud RLovell for asking this question and for looking for real-world applications. I only wish I had more applicable examples and values and data that I could provide. That's what this teacher and the students need--real world examples with actual data and parameters that can be used to help understand why this stuff isn't irrelevant. I know lots of computer programmers who have had to go back to university to get their maths skills up to speed for a particular project or application to be able to model and/or simulate something using mathematical formulae.

I can also think there would be some examples that might be applicable from the actuarial (insurance) world. And, many of the "rocket scientists" hired by financial institutions probably use some formulae which might be applicable in trying to "beat the market."

I hope you can find some good, concrete examples, RLovell--and I sincerely wish you had been one of my maths instructors!

Oh--how about the volume of a tank with curved ends, which is horizontal?

fluidflow

Helo There,

1. Y=aX^2 + bX + C this algebraic equation can be used to find the excess oxygen present in the the combustion product (flue gases) of a Boiler. here Y represents The load in (Te/Hr) and X represents the percentage Excess air. As the load on the Boiler increases the percentage excess air decreases because more oxygen is utilized for combustion. Basically the stated equation represents a Parabola and thus solution for different loads provides a great insight in how we produce steam.

2. Linear equations (at least at introducary level) are used a lot to optimize many systems like share holding and risk assessment. You can ask your students to minimize there expenses using these linear equations and this is basically a real world problem, where a given amount of money is utilized for a many aspirations, each with different degree of satisfaction. Tell them to find it. Construct these aspiration lines (let us say 2 aspirations for keeping things simple) on a graph paper and from feasible solutions find which solution provides maximum satisfaction. This very subject in higher studies develops in full scale Operations Research. This very procedure is applied to find how the workforce should be optimized. i.e. how many people should be placed for a job at a given order of time so that payment is minimized for the day.

3.Y=aX^2+bX + C this very equation basically represents a projectile motion and and it used almost everywhere whether we fire a gun or jump over a bridge. The same equation is at work when we use a Anti-Missile system.

Every equation has some physical significance, patiently waiting for our wits to grow sharper.

Lorentz Equations were there since yore, but it took genius of Einstien to give them a meaning and change the way we look at Universe.

"Mathematics is the language in which God has written this Universe"
Today is 5 sep. its Teachers day here in India and on this occasion I would like to thanks all the teachers (including CSA, Phil Corso and all other prolific contributors on this forum) all over the world for making a world a better place to live.

Regards
fluidflow

PhilCorso

Thanks you FluidFlow,

I finally reached 80! Clearly it feels like two 40's.

I'm still active enough to help others, so if anyone wants to avail themselves of 60+ years of Electrical Power experience, contact me at:

[email protected]

Best Regards to all, Phil

Ps: If anyone wants advice on "How to reach your 1st 80, I can still provide a few tips!