New answers tagged geometry
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Here are the set of rules I would adhere to when producing 'not to scale' diagrams:
straight lines on the diagram represent straight lines
acute, right, obtuse, reflex angles on the diagram represent acute, right, obtuse, reflex angles respectively
the relative length order of lines on the diagram is accurate (i.e. if AB is longer than BC on the diagram, ...
3
I noticed in France a trend that did not exist before: there are many exercises (usually at the equivalent of US grade 7-9 = French 5ème-3ème) where the pupils are asked either to work on purposedly incorrect figures (a sketchy version of the left on in your question) or are asked to make quick handwritten sketches (no ruler or compass).
This is to force ...
4
Consider the possibility that the exams will do this
I don't know where you are, or how practices have changed, but when I was sitting my exams at 16 in the UK, the standard practice for the national exams was drawings that were wildly misleading in this manner. I assume the point is to make students concentrate on what is actually known as opposed to ...
19
There is no value in drawing the figure exactly to scale, but the left-hand figure is inaccurate to the point where it is positively misleading.
Since the angle marked 30 degrees is actually drawn greater than 45 degrees, it gives the impression that a is less than 3 (or maybe equal to it, if the student takes the "30 degree" angle as being shown ...
2
This question relates to an interesting feature of ancient geometric proofs: The ancient geometers often drew deliberately distorted figures in order to force the prover to NOT depend on any accidental features of the figures in formulating their proofs. For example, when drawing a chord of a small arc of a circle, which is in fact a straight line segment, ...
7
A mathematics teacher once told me, "at a ratio of 4:5, a rectangle becomes a square."
While this is obviously not true in a strict sense, it holds the deeper truth that a rectangle at 4:5 or worse is not a good example to represent a generic rectangle to young students, unless the goal is to teach or test that squares are also rectangles.
That ...
3
Our SAT problems were frequently not to scale and I think it added to the challenge. Such drawings were labeled clearly "not to scale". If the students will be taking the SAT's then it is good practice.
Whether it is appropriate for this particular quiz/test depends on where the students are in the studies. First they need scale drawings before ...
6
Since your job is helping the teacher with technology and the teacher should be making the pedagogical decisions, ask her her preference and do it that way. Maybe she meant for the pictures not to be to scale, or maybe they came out that way due to her poor technology skills. Just ask.
11
I would split the difference by creating an accurate diagram in Desmos and then giving the students a hand drawing based on that diagram. That way, students could estimate their answer before solving and check their answer against that original estimate. But they couldn't conclude that $b=6$ by measuring the length of the known side.
Pedagogically, the two ...
22
As an answer so that I can paste in a picture: I think that the problem with the one on the left is that it is possibly "good enough" that someone might think that it is to scale and that apparent regularities are real. (Well, maybe not one of those "right angles".) In particular, someone might become under the impression that the length ...
4
The following comes from https://projecteuclid.org/download/pdf_1/euclid.afm/1485893376
Let $(V,|\cdot|)$ be a normed vector space.
Define $v \perp w$ if $|v-w|^2 = |v|^2+|w|^2$.
Define a "normed perpendicularity space" as a normed vector space where the set of vectors orthogonal to a given vector is always a subspace.
Then $(V,|\cdot|)$ arises ...
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In general, manifolds have not a norm, but a metric, which is a function that takes two points as input and gives their distance as output. A space with addition/subtraction and a norm has a metric (the distance between two points is the length of their difference), and metric that respects linearity is a norm (the length of a vector is the distance between ...
2
Here is an approach which works only(?) in two dimensional case and seems reasonably satisfying to me. Hope it helps to clarify the question!
Let $V$ be a two-dimensional real vector space.
Consider a structure on $V$ consisting of:
A linear transformation $r$ on $V$ such that $r^2 = -1$ chosen up to sign.
A nondegenerate skew-symmetric bilinear form $s$ on ...
3
I'm not quite sure if the question is the same as the question in the title.
For the question in the title: Sure, we can certainly develop the concept of length independently from inner product. The result would be a normed vector space. We only need 4 properties for the "norm" function:
Nonnegativity
A vector has zero norm if and only if the ...
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I really like the idea of discovery fictions because they capture an essential component of how mathematics is best understood and communicated. Almost all ideas in mathematics follow simply and inexorably from previous ideas, and understanding any mathematical discipline consists almost entirely of figuring out how you could have developed these ideas on ...
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In axiomatizations of Euclidean plane geometry such as the ones by Hilbert or Tarski, the statement $\overline {AB}\cong\overline {BA}$ is a postulate. In Tarski's system, this congruence axiom is explicitly called "Reflexivity of Congruence."
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For me, this is more of a foundational issue than one of secondary education. I would suggest that $\overline {AB}=\overline {BA}$ is a matter of equality rather than congruence, given the definition of line segment as the locus of points between $A$ and $B$. (Strictly speaking, I suppose, it is the locus of points $C$ such that $AC+BC=AB$.) Based on this, ...
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