Today some homeschoolers had their weekly algebra class. They are coming along well, and getting a good grasp of the subject. After class and after I got home, I sent them their assignments for the week. In today’s email, however, I decided to add some detail as to what we covered in class — sometimes I’ll write up a summary like this, sometimes I won’t. So sometimes parents get a good, detailed, written report of what a class or a student in private tutoring covered.
But this email sketches out what typically happens in my classes and private tutoring sessions, so I decided to post it. (But in this post I added a few things to the email to illustrate more of what we had actually done in class, instead of reporting only some of the highlights.)
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
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.
Do p. 457 #2-18 even.
Do “Cumulative Review: Chapters 1-6,” pp. 449 #11-20 all.
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.
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.
Read, study, and take notes on Section 7.2, “The Substitution Method.”
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!! 🙂
A parent responded to this email a few hours after I sent it: “I just have to say it again – Ryan REALLY enjoyed this class today. :)”
Ryan had said out loud after class (to paraphrase): “I understand all this much better than I had before class.” (He had completed some pre-algebra and algebra before I started working with him.)