High School Algebra & the FOIL method

Factual Questions
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General question for Chronos, Jragon and everyone else in this thread: Just what things, like this, did you learn “back in the day” that they don’t teach any more, or visa versa?

I noted above that my old college-level algebra book (1947) teaches computation by logarithm in extensive detail, but says nothing about set theory. It has a chapter on determinants, but says nothing about matrices other than computing and using determinants. It has an excellent chapter on working with approximate numbers. And it has a chapter on inequalities, which includes instruction on proving unconditional inequalities. (Example: Prove that the arithmetic mean is always greater than the positive geometric mean of two unequal positive numbers, that is:
(a + b) / 2 > √(ab) where a > 0, b > 0, and a ≠ b )

I had Algebra I in 9th grade circa 1963. Set theory was very fashionable then, and the very first chapter was all about sets, terminology and notation, and set operations. Thereafter, solutions to equations always had to be shown in “Solution Set” notation.

When I went back to community college circa 1990, I found out that Linear Interpolation is no longer being taught!

Chronos: Now I’m curious. Why is 𝜋 of 𝜋r[sup]2[/sup] fame the same as 𝜋 of 2𝜋r fame? Is this something that you can expound upon in the space of a reasonable-size post?

Jragon: Okay buster, now you gone done done it! We’re gonna talk about conic sections next, focusing on who learned what when. Okay, I’ll make a separate post about that.

ETA: I privately studied that chapter on Inequalities very thoroughly before I took Calculus I, and I felt, both then and since, that it was a greatly helpful thing to do. They used to teach epsilon-delta in Calculus I, which I think they don’t do any more, and really understanding inequalities REALLY helped make that comprehensible.

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Does this picture help?

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In case it isn’t self-explanatory: It demonstrates the relationship between the circumference and the area of a circle. If you know that the distance around the circle is 2pir, you can “unfold” and rearrange the circle so that it’s in the approximate shape of a rectangle, with length pi*r (halfway around the circle) and height pi.

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Well sorry, dude, you just brought it on yourself! :smiley: Now we gonna talk conics.

In my general consternation over the perceived dumbing-down of math instruction, now I wonder who learned what, when, about that too — and what’s being taught these days. That old 1947 College Algebra book had nothing at all about it.

When I took Albebra III-IV in 11th grade circa 1965 (where I came from the classes Algebra I, II, III, IV were numbered by semester, not full year), we learned conics then. The general form was Ax[sup]2[/sup] + Cy[sup]2[/sup] + Dx + Ey + F = 0 as Jragon says (Note, no Bxy term!). For this too, you had to know how to complete squares.

Was all that being taught when you learned it, Jragon, or anyone else? Is that still being taught? By completing the squares in x and y separately, the general form can be re-written as: A(x-h)[sup]2[/sup] ± B(y-k)[sup]2[/sup] = 0, and from this form, many characteristics of the graph can be read directly by looking at the equation. These always produce conics (unless they are degenerate) with horizontal or vertical axes. We never saw forms with a Bxy term except for a few real simple cases, like xy = k.

Fast forward to college Calculus II or III, also titled “Calculus and Analytic Geometry” which I took circa 1992, with the text by Larson and Hostetler, 5e. Here, we learned about the full form Ax[sup]2[/sup] + Bxy + Cy[sup]2[/sup] + Dx + Ey + F = 0 including the xy term. These simply produce all the same conics as before, but with axes at other angles than horizontal and vertical. By rotating the axes (Ye Gods! Some more formulas full of sines and cosines to memorize!) the xy term can always be eliminated. And there were a few techniques for first determining the angle θ to rotate, to accomplish this. Is that stuff still being taught?

I noticed a few interesting things about those rotation formulas: The coefficients of the quadratic terms are all interrelated, but entirely independent of the coefficients of the linear terms. The coeffiecients of the linear terms are independent of the quadratic coefficients. And the constant term F remains unchanged. Fascinating.

I also discovered that the rotation formulas can be expressed VERY neatly in the form of some matrix products – one 3x3 matrix to rotate the A, B, and C coefficients, and a separate 2x2 to rotate the D and E coefficients. If anyone is interested, I can try to dig those up again (I think I still have all that laying around somewhere) and post them here. OR: Try to develop those matrices yourself!

Oops! Something just came up. Gotta run. No time to proofread the above. Back later today.

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I remember learning some bare rudiments of set theory, but nothing actually relevant to anything useful for mathematics. I get the impression that some educator or educational standard-setter heard that “Set theory is the foundation of modern mathematics; we should teach that”, but didn’t actually understand how it’s the foundation of anything else, and so didn’t know how to teach it properly.

And everything I learned about matrices up through high school was useless. We learned how to use matrices to solve equations using a method that was exactly as much work as how we’d been solving equations all along, but which had the added advantage of being more confusing, and then we learned about determinants, which we used only for Kramer’s rule, which is an even worse way of solving equations. It wasn’t until college that I actually learned how to benefit from using matrices.

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Senegoid: WRT conics… No, as far as I know those things are not being taught at all, really. I study Mathematics (senior year !!!) at an unimpressive state university and I also tutor high school math. What you called Algebra 3-4, we call Algebra 2. When it comes to conic sections the only thing we approach is the circle formula Ax[sup]2[/sup] + By[sup]2[/sup]=0 with variations in A and B to give you the respective parabola, ellipse, hyperbola or circle. I don’t think any of my students would even recognize that what we are manipulating is a conic section.

And to answer your further question, I also have taken calculus 2 and calculus 3 (multivariate calculus). As far as I know, (and looking back at my textbook) that general form has never been given. Perhaps it was phased out of the nationwide curriculum at some point?

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See, this gets at my thesis: One may question the usefulness of learning things like all those conic sections and their equations. But this (if taught right) leads to a more general understanding: For example, that you can translate axes in ANY equation by replacing all x with (x-h) and all y with (y-k); if you have a conic equation given in general form, you can complete squares to put it into (x-h) and (y-k) form.

The calculus book I used for Calc III was Larson & Hostetler 5e (they wrote DAMN GOOD textbooks, IMHO), and had two or three entire chapters devoted to analytic geometry. That’s where I got the conics with xy terms, having axes at angles other than horizontal and vertical. Of course, being at the Calculus level, this also assumed that the student knows his trigonometry, and also included stuff about derivatives (Implicit Differentiation!) and integrals involving those kinds of equations.

Here’s another topic that is never taught in depth, but only touched upon where needed and only as deeply as needed: The algebra of summations using Σ notation. Yes, there are actually a set of algebraic rules useful for manipulating expressions using this notation. These are superficially touched upon, as needed, in Calculus (as an introduction to integration) and in Statistics texts.

But I came across one Stat text that covered Σ rules in some detail, which I found very enlightening. Just like I was very enlightened by that chapter on Inequalities in my old Algebra book, which made epsilon-delta stuff MUCH clearer when I got to that!

My take on all this: Teaching more topics in greater depth, GOOD. Dumbing down (as seems all to prevalent), BAD.

One of my minor hobbies, for many years, was hitting up used book stores and collecting OLD math textbooks. Once upon a time, I have a Trig textbook from 1914. I used it to review trig before I got into Calculus, and learned about things there that I had never seen before. (For example, it gave reduction formulas for sec, csc, and cot functions. I mentioned this to one of my Calculus profs, who said she had never heard of those before either!)

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And from that you can conceptualise. If you’re cutting the circle into sectors and reassembling, then as the number of sectors becomes larger the resemblance to a rectangle improves, and the curved edges of the sectors become less and less convex. They can’t become concave no matter what you do, and so as the number of wedges approaches infinity the shape tends towards a true rectangle.

You can also approach this by considering a regular polygon inscribed inside the circle. An n-gon comprises n identical isosceles triangles, each of area equal to half their base times their height. The sum of these is therefore half the perimeter of the polygon times its apothegm, a term I learned a year ago and meaning the perpendicular height from the middle of any of the sides to the centre (centre in this case being the point where all these heights would intersect). As the number of sides of the polygon approaches infinity, the apothegm approaches the radius of the circle, and of course half the perimeter is just pi times half the diameter, or pi times the radius.

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@Thudlow Boink: Aha. I just noticed your posts about area and circumference of a circle, which I somehow missed before, Malacandra having just called it to my attention in the above post.

Yes, that diagram does it for me. I’ve seen that picture before (as an explanation of how ancient Greek mathematicians developed the formula for area of a circle), but I had never thought about it in the context “why is pi in the perimeter formula the same as the pi in the area formula”.

BTW, something odd: The symbol I used for pi there is 𝜋 but in your post, where you quoted my post, it didn’t render and I just got several black-diamond-question-mark symbols. I wonder WTF?

Everyone else: Is that symbol for pi, 𝜋 rendering legibly for you? How about this π symbol for pi?

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Only the last one (Google Chrome, if that matters). You could always use [noparse]p[/noparse] to give you p instead.

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That’s the only one that works for me. The symbol font doesn’t work for me, either: Most modern browsers don’t recognize that font.

The broad disparity of what works and what doesn’t is why I always just spell out the Greek letters. It’s annoying, but it’s the only thing that works consistently.

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I’m seeing all the correct pi symbols in this thread, including in the place you mention, using Internet Explorer. But I tried looking at the thread using Chrome, and I see the black-diamond-question-marks you describe, and empty squares elsewhere, for the pi symbol.

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Absolutely! I’ve been enjoying the discussion very much.

I posted this question because it seemed that the factoring of polynomials in high school was an objective in and of itself, not that it would lead to anything more useful, or be necessary to comprehend equations applicable to physics or statistics or anything else. I had the feeling that it was essential in more advanced fields of math, but I did not go much further than algebra, geometry, and trigonometry.

This thread has convinced me that it is essential. Thank you, everybody.

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I found it to be an eye-opener when I first began to realize some of these things. Remember those “special products” you had to memorize? Well, you’ll see a lot of them again in Trig. And all those Trig identities (like various formulas for half-angle, double-angle, sum of angles, etc.)? If you get into Calculus, guess what? Those special products and identities and everything all come back to bite you again.

So, the selection of topics to cover in beginning Algebra is not entirely arbitrary. Mathematicians and textbook authors know what topics are important because they will come up again and again. Factoring? Various uses of discriminants? Completing squares? They’re all waiting in the wings to bite you again! :eek:

Here’s another case in point: Statistics. There’s this formula for Standard Deviation that is devised to show clearly the concept of SD, called the “Definitional Formula”. But it’s a PITA to use it if you actually, y’know, want to compute a Standard Deviation. But tweak it with a bunch of fancy-schmancy algebra, and you come up with an even messier grotesque-looking formula, but guess what? Start plugging the data into that and doing the arithmetic, and the same computation actually gets much easier! So they call that the “Computational Formula”.

Now fast-forward half a semester, to Correlation Coefficients and related topics. There too, there is a “Definitional Formula” that actually shows what it means but is a bitch to compute with it, and a MUCH messier-looking “Computational Formula” that is actually easier to compute. But you first need a whole bunch of algebra to get there. Now to be sure, you could just look it up in the book or memorize it – but to best appreciate it, you should really go through the derivation yourself and see it happen!

ETA: And by the way, f.coli, there’s another thread nearby where we are discussing ideas about how math ought to be taught. You might enjoy a peek there too!

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All this stuff has a lot more real-world usefulness than you might think, if you only went through High School math.

It turns out the Real World is mathematically messier than Pythagoras ever imagined! In High School math, you get (relatively) simple word problems, just to demonstrate that there are word problems and give you some practice. Things about when one train overtakes another, or how far a ball travels when you throw it (ignoring air resistance, of course), or how long it takes John and Bill to paint the room.

The farther you go in math, the more you see complicated word problems that are more for real. Mixing to chemicals in a vat and need to compute the percentages? Forget that. In Differential Equations, we got problems that dealt with two (or three) vats of chemicals, with pipes in both directions carrying the mixed contents back and forth at differing rates, and an outflow pipe from the second tank. Now compute how the mixture varies in that! I guess if you’re designing an oil refinery or other major chemical plant, you really have to solve problems like that.

There are problems about rockets flying into space, or spring-loaded weights moving through a viscous chemical fluid where – (hang onto your seats for this!) – you DON’T ignore resistance of the medium. (Did you know that you can propel a rocket infinitely far into space with only a finite amount of fuel? We had one problem that demonstrated that.) There are problems of computing how heat will flow through a three-dimensional block of material. There are problems of computing how vibrations in materials will work, so you can build a bridge that doesn’t do this. (Video of the Tacoma Narrows Bridge, a.k.a. Galloping Gertie, collapsing.) Yes, the same has also happened to commercial airplanes full of passengers too. :eek:

Your basic High School math doesn’t seem too profound because it typically only has (relatively) simple “application problems” that have all been solved since before Isaac Newton. But it all lays the foundation for higher math, where there really are real problems to be solved.

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Don’t you need calc to show that the pi in the circumference and area are the same? The only way I can think of to really “prove” that is to describe the area as a cumulative sum of circumferences of (an infinite amount of) successively smaller concentric circles, which would involve integrals. (And I’m pretty sure is the “standard” way to derive the area of a circle in calc, if I recall correctly).

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It comes down to a matter of definition. Strictly speaking, you’re taking a limit, and so you’re doing calculus… But it’s a simple enough piece of calculus that it can be taught in isolation to grade schoolers, without bringing in the full glory of calculus. Put another way, it was proven long before Newton or Leibnitz.

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But also see my previous post relating the circle area formula to the regular polygon area formula: Area of a polygon is half the perimeter multiplied by the apothegm. This is not dependent on the number of sides of the polygon. For a circle (considered as the limiting case as the number of sides approaches infinity) the semiperimeter is just pi * radius, hence area = pi r[sup]2[/sup] and no need to understand a thing about integral calculus.

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It’s easy to describe with words, but working it out formally/symbolically is kind of gnarly here’s the best version of it which looks similar to how I did the same proof in class. The sum-of-circumferences proof is way more transparent about understanding why the pi’s are the same, even if you have to do some handwaving about integrals. I also think it’s easier to conceptualize adding a bunch of progressively tinier circles than it is to really grasp what it truly means to have a bunch of polygons split into triangles trending to infinite vertices.

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Well, the relationship between the volume and the surface area of a sphere (the former being the latter times the radius over three, because the volume of a pyramid is its height times its base over three [picture fitting three pyramids in a cube]]) is just as easy to see as the relationship between the area and the perimeter of a circle (the former being the latter times the radius over two, because the area of a triangle is its height times its base over two [picture fitting two triangles in a square]).

So I suppose the concern is how to demonstrate that the surface area of a sphere is its circumference times its diameter; i.e., equal to that of a cylindrical tube just touching the sphere at its “equator” and extending as far out as its “poles”. But this can be understood easily enough as well:

Imagine projecting the sphere out horizontally onto that enclosing cylinder. At lattitude L from the equator, the projection stretches lattitudinal-into-horizontal distances and squeezes longitudinal-into-vertical distances by the same factor, 1/cos(L). Thus, as the stretch and squeeze cancel each other out, this projection is area-preserving.