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I consider pupils 12-14 years old, who are new to translation. On the other hand, they have been accustomed to placing points in a coordinate system, especially when studying relative numbers. In addition, they know the addition and subtraction of relatives like $$ (-2.5)+7$$

They can be offered the following activity and many others of the same type: enter image description here $$\color{Blue} {\begin{pmatrix} 3\\5 \end{pmatrix}}+\begin{pmatrix}\color{Green} 6 \\\color{Red} 7\end{pmatrix}:=\begin{pmatrix}3+6 \\5+7\end{pmatrix}$$

There are real difficulties with the many mathematical notations that should not be overlooked but with the awareness of these difficulties and patience, nothing insurmountable. All the more so if this type of activity in reference points was done in primary school.

The other classic way to introduce translation is to use pupils' knowledge of parallelograms:$$M'=t_{A\to B}(M)\iff AMM'B \text{ is a parallelogram}\iff A'=s_D(A)$$where $D:=m(M,B)$ and $s_D $ is symmetry.

What do you think? Should one be preferred over the other? Should you practice both? An argument for the former is that it effectively prepares for higher education and here for the $\mathbb R^2$ affine space(here $A+u$ is illustrated). But I am often told that secondary education is not intended to train future mathematicians, neither engineers nor ... One argument for the second is that it leaves students in a familiar environment but, in my experience, I wonder: really that familiar?

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  • $\begingroup$ 1. What are relative numbers/relatives in this context? 2.I think I can decode what the chain of equivalencies means, but explaining it a bit more would be nice. 3. How old are the pupils or where are they in their education? $\endgroup$
    – Tommi
    Commented Feb 24 at 13:11
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    $\begingroup$ 1) For example, $(-2.5)+7$; 2)$M'$ is the image of $M$ by translation that transforms $A$ in $B\iff A'$ is the symetric of $A$ around $D$; 3)12-14 years old $\endgroup$ Commented Feb 24 at 13:15
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    $\begingroup$ If you start with #2, will it inhibit generalization to 3D or higher? Also it seems trickier to establish that the composition of two transitions is a transition. $\endgroup$
    – user52817
    Commented Feb 24 at 14:10
  • $\begingroup$ I might start by saying that translating means moving something from one place to another without rotating it. $\endgroup$ Commented Feb 25 at 21:00

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My recommendation is to show both concepts, but with the pictorial representation of vector addition more as "motivation". Of course, even before doing that, you ought to start even simpler with discussion of vectors, themselves.

"There's these things called vectors. Vector means magnitude and direction. Think of a car moving (how fast it goes, but also what direction it goes). Other vectors can be forces (how hard you push and what direction you push)."

IOW, don't jump to addition as the first concept...it needs to come after grokking vectors themselves.

In my experience, in the US, (and looking at a couple texts), both ways of thinking about it are brought up, pretty close to the same time. The parallelogram is more of a rationale to help understand a bit of an abstract topic, rather than the retained algorithm.

In math class, the student will pretty quickly turn to using the algebraic reasoning for most problems. But in physics classes, the geometric view of vectors can still be helpful. At least draw a picture, label it, before starting to crank algebra. And in many cases, you really do care about the difference between axes (height versus distance over ground, having very different physical implications).

I wouldn't worry about "inhibit generalization to 3D or higher". [comment to head post] There is an expert bias tendency of many math grad students to think that lower classes should be taught in generalized form. That it will somehow either save time or eliminate confusion. But the opposite is true (for mortal non geniuses). I mean, would you teach multivariable calculus first? How would that go over? And while div/grad/curl/and all that are sorta intrinsically hard...do we really think they are easier for kids unexposed to any (single-variable) calculus, versus having had a basic single variable class first?

FWIW, in the US, your timing for introducing this topic seems early. Traditionally, it was not even an "algebra 2" (11th grade, 16-17 year olds) topic, but was part of analytic geometry (or "pre-calculus"). Even if they have moved vectors earlier into the curriculum, than my dated experience/texts, it's not a 13 year old topic. Not generally.

Also, I would be a little wary of how you are approaching this question. It sounds very theoretical, versus observational. "I consider pupils 12-14 years old" and "Should one be preferred" and "An argument for". Instead of trying to come up with de novo insights from thinking about pedagogy, go look at examples and ask for experiences. I'm not saying to NEVER try to be deductive in pedagogy, but it probably should NOT be your first or primary frame of reference. You will miss a lot of insights that you might have gotten by "looking at the animals" instead of "I consider" or just brainstormed possible rationales ("An argument for"). And you will develop a lot of crazy Ivory Tower fiascos if you try to design things de novo, versus looking at practice/experience.

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  • $\begingroup$ Good answer. Do note the asker has located themselves in Paris and the name is also French, so I would not read too much into a detailed analysis of writing style and choices there. It reads to my non-native eyes as not written by a native. $\endgroup$
    – Tommi
    Commented Feb 24 at 20:00
  • $\begingroup$ D'accord. ;) ;) $\endgroup$ Commented Feb 24 at 20:53
  • $\begingroup$ Thank you very much for your answer; I am very interested in the lines of thought you suggest. (yes, I teach in Paris) $\endgroup$ Commented Feb 26 at 14:47
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TTo enhance the understanding and teaching of symmetry and invariants, employing transparency sheets can be a highly effective and interactive method. This technique not only visually demonstrates these concepts but also engages learners in hands-on exploration.

enter image description here

TThis method not only visualizes the concept but also allows learners to physically engage with the material, facilitating a deeper understanding through embodied learning.

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