Sometimes the best way to do an problem is the easist. The easiest way is simply to guess and check, adjust and repeat until you have the right answer.
There are three integer values of x that make the equation x3 + 6x2 + 11x + 6 = 0.
This is a difficult problem for any child in grade school. It's nearly impossible for a 10 year old. This class of problems shows up on all tests, including the SAT. Yet my kids have figured out a way to get these problems right and frustrate my ability to teach them math concepts. In this article, I'm going to describe the direct way of doing this problem (their way), the problem solving way (the right way), and the math way which I teach after they get the problem right quickly with no discernible effort (the penalty box way).
If this problem shows up on the SAT, my lazy child will just try all 4 answer choices and circle the one that works. This is a real skill with a name and this is how it works.
- I recognize the math concepts in the problem and go about solving the equation.
- My son, meanwhile, notices that he doesn't recognize the math concepts and goes about plugging in random numbers until he gets one that works.
- In the process of plugging in random numbers, sometimes the result is too high, sometimes too low, so he'll adjust accordingly. Sometimes he sees a pattern between 'too high' and 'too low' and adjusts with intent.
- Just as I'm beginning my work, he announces the correct answer and makes me look stupid and slow.
The problem solving skill is called estimation, and the sub-skill is to recognize a problem that makes this technique the most expedient. Advanced concepts usually fit this category, and younger children have an advantage recognizing things they haven't learned versus an adult who has already learned the concept.
The second sub-skill is iterate and adjust. This takes some practice. A child will adjust the wrong way and make things worse and start to learn what adjusting is all about. In this case, starting with -1, 0, and 1 should point to an 'ah ha' moment because -1 works but the other's don't. A survey of the equation again (aka reread the question) shows why this is the case.
The acceptable approach is to spend some time understanding the equation under the heading 'read the question'. To understand how this equation works, one plugs in a few numbers to build a mental graph. Since this is a timed test, there are 3 numbers that need to be plugged in; any other number is usually wrong. These numbers are -1, 0, and 1. The correct numbers are the easiest to work with - thus small or trivial ones.
I'm OK with 1,2 and 3, provide it's followed by -1, 0, and 1, but 4 is definitely the wrong approach and results in a lecture. The little one always says 4 to buy some thinking time while I lecture him. I always fall for it.
In every problem we've seen so far, the numbers we plug in are very close to the answers. I think the test makers intentionally put a time waster like this on the test to give the advantage to kids who stop, think, and go through the Big Five problem solving techniques. 'Estimation' and 'Try An Easier Problem' are somewhat related on these problems.In the event that a child guesses correctly within seconds and ruins my plans for a good 20 minute math problem, I'll follow up the problem with a lecture in question form. They should know all of the answers, because they've heard them before, but they don't regurgitate terminology like '3rd order polynomial with complex roots' quickly no matter how many times they've heard it. Nonetheless, sitting through a lecture - in question form - is good practice for getting thrown into a math class beyond their comfort level.
I'll spare you the questions. Here are the answers.
- This is a 3rd degree polynomial and therefore has 3 roots according to the fundamental theorem of algebra.
- I'll draw a few examples. If it only crosses the x axis 2 times, this means that one root is complex
- New York gives a test in high school with a famous name I forget that expects you to decompose the equation into it's roots, but we are not going to do that because it's tedious and boring.
- In a 3rd degree polynomial, like a 2nd degree polynomial, once you get past the outside roots, y continues to grow in magnitude.
- If I'm in a bad mood, we'll delve into the slope of the curve, a 2nd degree polynomial on it's side is not a function, or anything else I can think of until we get to the 20 minute mark.
There's a big difference between a kid sitting in class hearing '3rd degree polynomial with complex roots' for the first time and one who's heard it all before and finally gets to learn what it really means.
The answer to this question is the roots -1, -2, and -3. The purpose of this question is to test the child's ability to apply basic arithmetic to a complicated, unfamiliar topic and move on quickly to the next question.
I've outlined 3 approaches above that you can use as you're child's academic coach to impart learning skills. If you take the time to not teach math, your child will pick up a variety of really create cognitive and learning skills. The tree examples above involve cognitive sub-skills, resilience in the face of complexity, and becoming inured to tops that are age-inappropriate.
My todo list includes preparation for a future math competition. Based on past experience, we'll never get that far down on the priority list.
Final note - I 'borrowed' the picture above from https://theemotionallearner.com. I like this website. It's thought provoking.
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