MGTutoring

Solving Systems of Linear Equations: Graphing

Solving Systems of Linear Equations: Substitution and Elimination

Multiplying Polynomials of Several Variables

Here is a note I sent to a student after a tutoring session (getting a public high school student ready for a standardized math test):

Remember to look for first principles in math -- as in history, science, biology and life.

We frequently use facts like a number times one equals the number; any number times zero is zero; zero in the denominator of a fraction is not a defined fraction; and zero in the numerator of a fraction is zero.

1. So when we need to add fractions, we must multiply by, e.g., 3/3, because 3/3 = 1 and so does not change the number. For example, 7/3 x 3/3 = 21/9, whereas 7/3 x 3 = 7, which is three times greater than 7/3!!!

2. And if we have to solve an equation like (x - 4)(x + 5) = 0, we know that either x + 4 must equal 0 or x - 5 must equal 0, so either x must equal -4 or it must equal 5. (Because a number plus its opposite is zero (-4 + 4 = 0) and a number minus itself is zero (5 - 5 = 0).)

3. And if we have to work with a function like f(x) = (x - 4)/(x + 5), then we know x cannot equal -5, because we would then have a zero in the denominator -- an impossibility, since we cannot divide anything into zero parts.

4. And if we have to work with an equation like (x - 4)/(x + 5) = 0, then we know the only way to make the equation equal to zero is by making the numerator equal to zero. So (x - 4)/(x + 5) is zero only when x - 4 is zero, which means that x = 4.

The fundamentals of math come up in algebra, precal -- and calculus -- just as the principle of individual rights comes up in having friends over to watch a movie, in making a movie, and in signing a billion dollar business contract.

Fundamentals and first principles make things more intelligible and easier.

(c) 2009 Michael Gold

Here is a note I sent to some parents after a geometry tutoring session (I am editing out the students' names, and calling them A and B; as the students were homeschooled, I was their primary and only geometry teacher):

A and B did good in tutoring today. A and I covered six proofs -- that's a first. We have not done that many proofs in one sitting before. He's doing good! We also discussed the nature of definitions (as having a genus and differentia, which we had discussed weeks/months ago) and their importance; we discussed and analyzed the definitions of polygon and triangle, and from there discussed quadrilateral and pentagon, midpoint and bisector, and, to discuss the concept "genus", capitalism, democracy, communism (genus political system), and then dog and cat (genus mammal). And we discussed how, in learning a subject, it is important to review material that came before what one is doing now; A had forgotten what a polygon was, what a midpoint was, what a definition was, and what some of the axioms were, and so could not use them when he needed to. This was a great discussion for learning how to learn and how reason works. And we discussed Mill's Methods of induction: the method of agreement, the method of difference, the joint method of agreement and difference, and the method of concomitant variation. These are very important for science -- and every day life.

B and I analyzed four theorems in the book, and went over one proof (two proofs?). The proof was several steps more complex than others he had done, so it was one I needed to help him on. We had to do a bit more reasoning and add more to the diagram than what he's used to. Most diagrams so far had given him all the information he needed. One theorem we covered proved that the hypotenuse of a right triangle (with a 30 degree angle) is twice the length of the leg opposite to the 30 degree angle. This triangle is one that we build on and use in trigonometry -- which fact I pointed out to B. We find 30 degree angles and 60 degree angles used around the unit circle; now B will know why we use those angles in particular.

Very good sessions today. A and B are both making clear and definite progress; they have come a long way.

(c) 2009 Michael Gold

Elluminate's vRoom allows us to have video contact, audio contact, a place to chat, and a white board to do work on and take notes on. I tried to copy the white board notes into a post, but the copies were not in a form where I could do that.

I will not write up the whole session, but only some highlights from my notes, with some commentary. We covered parallel lines this session. We had already done an introductory chapter (basic terms of geometry, axioms, postulates, etc.) and a chapter on congruent triangles. So the student had a few months of context and of working together that won't see here.

We started the session by reviewing the idea of parallel, of transversal, and of angles around a transversal, then started chapter three, which started off with two proofs:

1. If alternate interior angles are congruent, then the two lines are parallel.

2. If two lines are parallel, then alternate interior angles are congruent.

We discussed the proofs as laid out in the textbook (we both -- the student in San Antonio and I here in Houston -- could look at the same material this way), going over what they said, how the proofs were laid out, and why the proofs were laid out the way they were. The student got a good, solid presentation of the proofs -- not the kind of thing that is done very much in modern education. First off, the very important proofs of geometry are usually neglected or devalued/deemphasized today, which results in students who do not appreciate -- and who do not know how to engage in the process of -- reasoning. And second, the very important "hows" and "whys" of proofs are even more infrequently taught.

I focus on the "how" and the "why" of proofs, as well as the "what." Geometry should be used to help raise a student's thinking from the oral and semi-literate to the literate; it should be used to raise a student's thinking from the subconscious and perceptual to the volitional and conceptual.

I milk theorems and proofs for their methods and logic, as well as their content. Cannot have the former without the latter.

But I pointed out how the first proof depended on the basic idea (which we learned prior to this) that an exterior angle of a triangle is greater than either remote interior angle, while the second proof depended on the idea (which we learned prior to this) that you can have only one parallel to a given line through a point not on the line. As this illustrates, my students learn hierarchy (the structure of knowledge), an important principle of logic.

A fascinating thing about these two theorems is a new method used: reductio ad absurdum. My students learn that reductio is a form of argument very important in philosophy, as well as law and mathematics. They learn that it has played an important role in physics, too -- which role my students learn about in detail. I concretize reductio; I do not leave students floating or frustrated intellectually with empty promises or empty leads. As this illustrates, my students learn integration (the interrelationship of all knowledge), an important principle of logic. And they learn to concretize their abstractions, to not make statements not backed up by facts.

I take logic seriously, so I take hierarchy and integration seriously -- I walk the walk and talk the talk.

It takes someone well-versed in geometry -- and well-versed beyond -- to make these points. There is no taking the place of a really good teacher. The reductio arguments in these two proofs would go by the student; very rare would be the student who grasped the importance of reductio -- students (and most adults) just don't have the context. Invaluable is the teacher who stops the student, points out the nature of the argument, names it reductio ad absurdum, identifies the importance of the argument, and gives examples of where the argument has been used in history.

It is critical to show the student how knowledge of reductio helps him/her (1) in reasoning and (2) in life. And this is the role only the master teacher can fill.

After going over those two theorems, we discussed some corollaries of them -- while discussing what a corollary was.

And at this point we were able to prove -- finally -- that the angles of a triangle add to 180! I like this proof. Yeah, it's not as interesting or complex as another favorite proof of mine: the geometric proof that y = x^2 in a parabola, but it's a nice, solid, fundamental proof. The notes from the whiteboard in Elluminate's vRoom show a diagram of this theorem, as well as the exterior angles theorem.

And as a corollary, we were able to prove that the exterior angle of a triangle is equal to the sum of the remote interior angles. And here's a picture for the exterior angles theorem.

At this point, we were out of time, so I took a few questions (he wanted to review a proof of the Pythagorean Theorem, of which he has learned two so far) and gave the student his assignments for the week. All in all, an excellent class.

It is a joy -- and quite a sense of accomplishment -- to see students grow cognitively/intellectually, and to see them enthusiastic about it.

Today we started class by looking at one reason why we need to learn to graph linear inequalities (which topic we covered last week): so we can graph, evaluate, and criticize the graphs we use and find in statistics. Graphs of inequalities play an important role in statistics. Then we worked an inequality together: p. 434 #22.

Then instead of doing more inequalities, we started working on graphing linear systems, to make sure we'd have time to cover that topic first. We started out with some motivation: knowing how to find points of intersection is a critical part of understanding how some people navigate, or locate a position on the earth's surface, using LORAN. LORAN works by finding the intersection of two hyperbolas, as we saw in class. We are not yet ready for working with hyperbolas or systems of hyperbolas, of course; we need to work with lines first. We will build up to hyperbolas one step at a time. (In looking at the LORAN example, we were able to introduce some classic properties of hyperbolas, ellipses, and circles, so we had an idea how a hyperbola was generated and how it was different from other conic sections. And we were able to see how LORAN depends on the basic idea D = RT.)

We read some of the introductory material on p. 453 of our book, then worked three examples of solving linear systems by graphing:

1.  the system 3x - y = 5 and y = x + 1, which has a solution of (3, 4);
2.  the system 3x + 4y = 12 and 6x = 18 - 8y, which has no solution;
3.  the system y = (1/3)x - 1 and -9x + 3y = -3, which has a solution of (0, -1) -- no work involved since that point is immediately seen to be the y-intercept of both lines once you have both equations in slope-intercept form. (THINK! We don't have to waste time actually graphing this one if we take into account our background knowledge of algebra and graphing!)

I pointed out how solving linear systems builds on graphing single lines and solving algebraic equations -- mathematical skills build on old knowledge and skills, just as in martial arts and dancing. We can do things like this -- math, dancing, martial arts -- because we are different from the rest of the animals: we are conceptual beings.

Then, to wrap up, we read and discussed p. 456, covering the three comprehensive and mutually exclusive situations we could have with two linear algebraic equations: one point of intersection; no points of intersection; overlapping lines.

We were then able to do more review work: to take time to work another inequality (p. 434 #23), and to work some direct and inverse variations (p. 439 #10 and p. 441 #32, 34, 28). The variations were good, real-life, practical exercises: wage, pressure-volume, pump speed-time, etc.

Everyone understood and was satisfied, so we stopped there.

Assignments for the Week

Day 1:
Read, study and take notes on "Systems of Equations in Two Variables," pp. 453-456.  Do p. 457 #1-17 odd.
Do "Cumulative Review: Chapters 1-6," pp. 449 #1-10 all.

Day 2:
Do p. 457 #2-18 even.
Do "Cumulative Review: Chapters 1-6," pp. 449 #11-20 all.

Day 3:
Reread and study "Systems of Equations in Two Variables," pp. 453-456.   Do p. 458 #19-33 odd.
Do "Cumulative Review: Chapters 1-6," pp. 449 #21-30 all.

Day 4:
Reread and study "Systems of Equations in Two Variables," pp. 453-456. Do p. 458 #20-34 even.
Do "Cumulative Review: Chapters 1-6," pp. 450 #31-40.
ead, study, and take notes on Section 7.2, "The Substitution Method."

Day 5:
Do "Cumulative Review: Chapters 1-6," pp. 450 #41-62.
Read, study, and take notes on Section 7.3, "The Elimination Method."

Remember: THINK!! REASON!! MAKE CONNECTIONS!!

See you all next Tuesday at 11:30 at the Library!! :)