This category is for questions. The question here is: how many US or other students are taught, as described?
How many readers here have been in school recently, and been taught as described?
Honestly? A number approaching zero.
We have very few active posters under the age of 30, and even fewer who have taken U.S. high school geometry in, say, the past decade.
But several with children/grandchildren and others who do teach.
We can say with something approaching certainty that the situation the OP describes is not universally true even within the US.
For how many students it is true, I doubt anybody can actually provide a number. It would require polling every school district in the country for their practices to know for certain. School districts are incredibly local in the US. Sure, there are some common federal level and state level standards, but textbook choice (within a limited set of state approved texts) is local.
ETA: the OP boils down to a false premise. It maybe be true for some, but certainly not all. One generalization that can be made is that there is almost nothing related to primary school education that can be generalized to the entire country from any particular local example.
We do, however, have high school teachers.
Really, though, usually when someone complains about how math is taught “these days”, and “why can’t they just do it the same way we did when I was in school?”, the way we’re doing it now is the same way it was done when they were in school.
How so? It is simply a set of standards. And I have seen a lot worse in my career.
I disagree. In math at least, there is a lot more experiential learning. In the old days it was, “Here is the formula. Sit down, STFU and do the worksheet. Homework is pp 134-5: 1-53 odd.”
At the risk of highjacking, I love this and wish I was back teaching geometry.
Deriving the Pythagorean Theorem with similar triangles.
That’s the exact proof that was used in my high school geometry book, back in the early 90s. And we had experiential learning, too.
It seems to me that the thread boils down to “what is the definition of similar triangles?”
AFAIK it was that it contained the same set of angles, implying that ABC was similar to DEF and DFE - that is, a triangle was similar even if it was reflected. Clarify this first.
Similarly SSA defines a triangle. If you cannot construct such a triangle, it’s no different than defining AAA that do not add up to 180, it’s not valid. The only gotcha is that a valid SSA can define two different triangles, whether the one side intercepts the unknown side at an an acute or obtuse angle. (No ambiguity if it defines a right triangle.)
How it’s taught today? What’s defined as similar today? I have no ##### clue, my experience was almost 60 years ago, not in the USA.
Two consecutive sides and then an angle does not always define a triangle up to congruence, as you remark. But, if you cannot uniquely determine the angles (or a triangle at all) then there will not be a unique solution up to similarity anyway.
I mentioned reflections, rotations, translations, and homotheties. I do not imagine the definition is going to change much in 60 years, or 600, or 6000. However it is absolutely worth clarifying that an actual correspondence is required, as in @Chronos 's Post #5 — triangles are not a very good illustration of this since you do not have room for different ways to match up a set of angles. (Cf. Proposition VI.4)
Figures other than triangles can be similar, but having “the same set of angles” is not a sufficient condition for, for example, two rectangles to be similar to each other.
I agree. I have two kids in school and there is way more experiential learning and the methods are different. There is more emphasis on presenting multiple ways to solve a problem, and the way they mechanically do arithmetic is a bit different than the way my wife and I were taught. This is not a criticism. I actually like the more holistic philosophy, but the methods of arithmetic are taught differently, so much so that my wife (who is a data scientist and very strong in math) and I (math was my strongest subject) have to review the textbook to see how a concept is being taught, because sometimes we can’t make heads or tails of a problem otherwise. Unlike other parents, we don’t bitch about it though, we realize there’s more than one way to skin a cat. And this is two kids in two different school systems where I grew up.
So, no, it’s being taught differently (though this depends on how you define “taught differently”) than it was in the 80s. Better for the most part , in my opinion, but people tend not to like what they’re not familiar with which is why you hear so much bitching.
Yet it is absurd to bitch about different ways of doing and understanding mathematics.
Nowadays, some students are taught that if scalene triangle ABC is similar to DEF, then it’s not similar to DFE. My question is, how many students are so taught. Any, outside the US?
You’re unlikely to get a solid answer to this, here on a general-interest message board, for the reasons already discussed. What you’re going to wind up with – and already have – are anecdotes.
If any are not taught that, then what can they do with similar triangles? If I just show you a picture of two triangles, and tell you that “these two triangles are similar”, and maybe even give you all of the sides of one of them, and one side of the other, what then?
From IXL, a popular online platform used in schools for learning and practicing math (and other sibjects):
So this is one example of what they are taught about the meaning and use of the “is similar to” symbol.
But that’s not the same thing as saying that they’re not similar if you list the vertices in a different order. If similarity is a relation between triangles themselves, and if DEF is the same triangle as DFE, then it follows that any triangle that is similar to DEF must be similar to DFE.
Triangles are rather unique in that you cannot arrange them any other way - DEF and DFE are simply reflections. (Unlike rectangles etc. where ABCD can look nothing like ACBD) A triangle rotated, reflected, and scaled is still “similar” by the definition I learned. (Indeed, the whole point of having the category “similar” is the scale is different but the angles are the same.)
If there is a new improved definition that is being taught, please enlighten me.
(In the above, I assume a consistent angle naming, either clockwise or counterclockwise, does not matter)
Geometry with triangles relies on similarity and the “sum of angles is 180°” whereas other constructs tend to rely on the properties of things like parallel lines and right angles.
As to the question of “how many students are taught this way”, as above, there is no practical way of knowing.
There is no national level or even state level syllabus that is applicable to all schools, even limiting only to publicly funded schools, much less private schools. Much of it is handled locally, subject only to following general standards.
There is almost no single specific thing you can confidently say “US public school students are always taught to do X in manner Y”. Maybe the alphabet and counting to 10? I wouldn’t even have 100% confidence in those. At best, you can say “I know for schools in this particular district, students are (or are not) taught this way”. To know how many across the country would require polling all school districts nationwide.
In this case they’re relabelings, not reflections. 3-4-5 right triangle ABC is similar to 6-8-10 triangle DEF only if angle A equals angle D and B equals E, according to the textbook; otherwise, 3-4-5 triangle ABC is not similar to 6-8-10 triangle DEF.
You need to show us a photo of the textbook page in question, along with its full title and edition.