15

Speaking from an American perspective, your son's approach strikes me as much more natural. For instance, to solve your problem We have three points $M(7;-2)$, $N(0;t)$, $P(3;1)$. Find $t$ so that they are aligned. my students would start by observing that the slope (gradient) of $\overline{MN}$ would have to be the same as that of $\overline{MP}$ and ...


9

Calculate when you want an answer. Solve algebraically when you seek patterns. If the problem was "show that for any two points, you can find a third point along the y-axis that is collinear with them," then symbolic logic is the right way to go. But if you have the points, just plugging them in simplifies the problem dramatically and makes it easier to ...


5

I don't think this is a nationalistic difference (I'm in the US), but I also don't think your example is optimal. As an example where the correct technique is more well defined, let's say we have a physics problem like this: A bug starts from rest and accelerates with constant acceleration for 0.53 s, traveling 1.37 m. Find the bug's acceleration. I would ...


5

If these are exercises from a published textbook, then it's probably self-deluding to imagine that the students don't have access to them already. Chegg et al. probably already have the solutions available to anyone willing to pay the monthly membership fee. Your lead instructor has already chosen a philosophy and a set of rules. They've made the homework ...


4

WebWork maintains an Open Problem Library. It is not necessarily "easy" to construct your own problems on WebWork, but the system is constructed with purposes like these in mind. In particular, if you have set up a WebWork server, you can maintain a local problem database, with a file structure, course structure, tags, difficulty levels, and so on. In ...


3

It costs $5/month (for educators) to use Wolfram Alpha in its practice worksheets model. It will generate a lot of problems for you, but I'm not 100% sure it gives you the granularity you want. I really like it. https://www.wolframalpha.com/pro-for-educators/ Also, Math.com has a worksheet generator, which allows some specification of fraction use and ...


2

You can't generalize a single, isolated question, which is what your high school question about collinear points is. Of course you can invent a set of similar problems, create a general solution for the whole set, and then solve the particular problem given - but why do all that unnecessary work? If your child's homework set contains several similar ...


2

Typical constrained optimization (useful to "get it") is asking for the rectangle of largest area that can be enclosed in a fence of given length. Sure, it can be reduced to one-dimensional, but leave that option out. Or ask for the largest volume box with given surface area.


2

I think the difference between the two approaches is the goals. Your solution is exactly what I would do, but I'm an engineer. In high school, the goal of learning is usually more about understanding the general concept and gaining some practice with the mechanics. I would think that plugging in the values early on is "easier" to grasp the ideas than ...


1

My own experience in math classes (as a student in Germany) and tutoring my peers would lead me to the following conclusion regarding your question: Many students have problems with the generalized formulas, because they find it rather unintuitive to calculate with "letters" rather than numbers and they would frequently ask for real world examples and ...


1

Something like this? $$a. \left({a+b \over a-b} + {a-b \over a+b}\right) \div \left({a^2 \over a^2-b^2} + {1 \over {a^2 \over b^2}-1}\right)$$ $$b. \left({x^2y - xy^2 \over x-y} + xy\right) \times \left({y \over x} + {x \over y}\right)$$ $$c. \left({n \over m-n} + {m \over {m + n}} \right) \times \left({m^2 \over n^2} + {n^2 \over m^2} - 2\right)$$ $$d. \...


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