hsbeth Posted February 14, 2013 Share Posted February 14, 2013 As my oldest went through the sequence we covered all books cover to cover.We use LOF for high school. She is now successfully tackling math at the local uni. and planning on a STEM major. With my second child (who isn't particularly mathy) I'm wondering if it's okay to skip over a topic from time to time. She is very set on pursuing a nursing major and the programs we've looked at seriously all require only College Algebra and Stats.We are at the tail end of Algebra 2 and LOF includes two topics which I remember struggling through with our older dd (her only trouble spot in that course). The topics are resolving polynomials into partial fractions and doing some inductive math proofs. After speaking with my oldest who at the tail end of Calc 1 it seems that she hasn't needed these skills (at least up until this point). She also says that she has no real memory of them and would have to re-study them at this point if she did encounter them again. I studied Calc 1 and 2 for non-majors in college and took college stats and don't recall needing any of these skills either. Would it be okay to skip these or is it a bad idea? It certainly doesn't seem like something that comes up on SAT or ACT either. Thoughts? Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted February 14, 2013 Share Posted February 14, 2013 Partial fractions will be used in Calc 2 classes when techniques of integration are studied. If there's any chance at all that she might need to take two semesters of college calculus, then it might be a good idea to take a stab at them now. Induction proofs probably won't come up again for kids who aren't math majors, but the reasoning behind them is good for the brain in the same way that geometry proofs were good for her. :001_smile: That said, she's only in algebra 2. I've taught math long enough to know that most alg 2 students are at most going to get a flavor of these topics and not master them completely yet. They'll be exposed more and gain that mastery in future courses if they continue in math. I wouldn't sweat it too much now with your daughter. Quote Link to comment Share on other sites More sharing options...
Teachin'Mine Posted February 14, 2013 Share Posted February 14, 2013 Maybe you could take the pressure off by considering Algebra 2 done without those two sections, and then just have her work through them without counting the grade. Personally, I think it's great to stretch a student, especially since there's no risk. You never know if she'd suddenly decide to pursue more math in college, and if she ends up in a Discrete course, she'll be happy to have had the exposure to induction proofs. Any familiarity with a topic, no matter how superficial, is helpful when attacking it a second time. These "side topics" also show the breadth of math and can be fascinating. Quote Link to comment Share on other sites More sharing options...
kiana Posted February 14, 2013 Share Posted February 14, 2013 Partial fractions are needed in calc 2 and diffeq but I think it'd be fine to skip them for her. I think when your older dd sees them in those classes she'll be like 'oohhhh, yeah, THOSE' after a brief reminder. Personally I would prefer to cover the proof by induction in an open-book manner but omit it from any exams. IOW, stress for an 'appreciation' and not complete understanding. It is occasionally (VERY occasionally) taught in college algebra or precalculus, but most usually in an intro to proofs class for math majors. However, I think it's one of the few chances for a high school student to see what mathematicians actually do. Quote Link to comment Share on other sites More sharing options...
hsbeth Posted February 14, 2013 Author Share Posted February 14, 2013 Thank you all for chiming in, it's much appreciated. I may introduce these, but will penalize on the test for any of those she gets wrong or include them on the final I make for her. Quote Link to comment Share on other sites More sharing options...
regentrude Posted February 14, 2013 Share Posted February 14, 2013 Induction proofs probably won't come up again for kids who aren't math majors Kathy, question for you: when we studied sequences and series in high school, we had to use proof by induction: you guess the sum, but need to prove it by induction. What other technique would one use to do that? I can't think of any way. (I still vividly remember our 11th grade math teacher demonstrating induction by a chain of falling dominoes) Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted February 14, 2013 Share Posted February 14, 2013 Kathy, question for you: when we studied sequences and series in high school, we had to use proof by induction: you guess the sum, but need to prove it by induction. What other technique would one use to do that? I can't think of any way. (I still vividly remember our 11th grade math teacher demonstrating induction by a chain of falling dominoes) Hi regentrude! Basic summation formulas for arithmetic and geometric series (finite or infinite) can be proved without mathematical induction. Are you thinking of series of sums of square numbers and the like (neither arithmetic nor geometric)? For sure, they'd require induction proofs. Just checked my old Dolciani intermediate algebra text on my shelf, an excellent standard of what could be taught at the 'honors' level today. It covers arithmetic and geometric sequences & series, but does mathematical induction as an "extra for experts" section. AoPS intermediate algebra also covers its sequences and series chapter before math induction proofs. As a mathematician, of course I'm always eager to interest others in what I think are the more beautiful & advanced topics, but that's not going to happen for everyone. Just trying to answer the OP's concerns in the case of her particular daughter's needs. In the course of tutoring math with lots of kids over that years, especially from the nearby county-wide math & science gifted magnet school and also from various colleges (most recently kids in STEM tracks at top 30 universities like W&M and Vanderbilt), I can guarantee that these kids have no clue about induction proofs unless they did math competitions as an extracurricular. Now when I was in high school (not even a college prep kind of school; most kids went to the work force after graduating), I did have a unit on mathematical induction in my honors section of elementary functions class in my 11th grade year (what would now be the precalculus year). My teacher, Mr. Grimm, also used the domino analogy. :001_smile: One day he made me go to the board and prove the binomial theorem by mathematical induction on the fly. Haha, I can still relive that day (fear does that to you! he was a scary teacher...), but I learned a lot that year. I don't think that there are many Mr. Grimms left teaching, and those that are have to deal with kids who are less prepared due to the watering down of the elementary and middle school math curricula. FWIW. Quote Link to comment Share on other sites More sharing options...
kiana Posted February 14, 2013 Share Posted February 14, 2013 I did not learn as a chain of falling dominoes, but my instructor used the analogy of teaching a robot to climb a ladder. You teach it to put its foot on the first rung. Then you teach it to move from one rung to the next. Then, no matter how high the ladder is, the robot can climb it. :) Quote Link to comment Share on other sites More sharing options...
regentrude Posted February 15, 2013 Share Posted February 15, 2013 Hi regentrude! Basic summation formulas for arithmetic and geometric series (finite or infinite) can be proved without mathematical induction. Are you thinking of series of sums of square numbers and the like (neither arithmetic nor geometric)? For sure, they'd require induction proofs. Yes, I am thinking of all kinds of "strange" series, not basic arithmetic/geometric ones. We covered such series summation and proof by induction in our German college prep high school in the mandatory math course for all college bound students (not a STEM magnet school and not a course specifically for STEM interested students), shortly before calculus. As a mathematician, of course I'm always eager to interest others in what I think are the more beautiful & advanced topics, but that's not going to happen for everyone. Just trying to answer the OP's concerns in the case of her particular daughter's needs. I completely understand, and I agree that the OP's DD probably won't need them - I was merely puzzled by your remark that only a math major would need proof by induction. Just though I'd ask because I was curious whether there is another clever way to prove series sums. Quote Link to comment Share on other sites More sharing options...
regentrude Posted February 15, 2013 Share Posted February 15, 2013 I did not learn as a chain of falling dominoes, but my instructor used the analogy of teaching a robot to climb a ladder. You teach it to put its foot on the first rung. Then you teach it to move from one rung to the next. Then, no matter how high the ladder is, the robot can climb it. :) That's a neat analogy; I had not heard this one. Our teacher always said: we have to show that the first domino tips, and that, if the k-th domino falls, the k+1st domino falls as a consequence. And I still remember how he sounded because he had a funny pronounciation of "k+1st" that we made fun of. Quote Link to comment Share on other sites More sharing options...
jhschool Posted February 18, 2013 Share Posted February 18, 2013 PP said (Kathy in Richmond): "As a mathematician, of course I'm always eager to interest others in what I think are the more beautiful & advanced topics..." What are the more beautiful & advanced topics, and where do I find a book or books that teach them? Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted February 18, 2013 Share Posted February 18, 2013 A terrific starting point is the book What Is Mathematics? by Courant, Robbins, and Stewart. It's accessible to anyone who's interested in what math really is all about. I'd write more but I'm in the middle of packing for a week-long vacation trip, and it's hectic around here (& I need to get off the 'puter!). Search for my past posts, and also for quark's past posts for more ideas, or nudge me again at the end of the month. Quote Link to comment Share on other sites More sharing options...
jhschool Posted February 18, 2013 Share Posted February 18, 2013 Thanks Kathy in Richmond! I just ordered it--I'll bug you again when DC has finished it. Have a good vacation! Quote Link to comment Share on other sites More sharing options...
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