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Questions tagged [geometry]

For questions related to geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

37
votes
30answers
14k views

Real-world examples of more “obscure” geometric figures

As part of my secondary geometry class I like to hook students by presenting real-world examples (usually images I find online or have taken myself) of different geometric shapes from real life. For ...
32
votes
16answers
10k views

Why are triangles so prevalent in high school geometry?

A colleague and I recently discussed what we call the "Triangle Trap." High school geometry covers a very large unit reflecting the common core: Classifying Triangles Triangle Angle Properties ...
29
votes
10answers
9k views

Is this homework problem on counting triangles within a 4x4 grid too vague?

My six-year old daughter was given this maths problem for her homework: Given a regular square grid of 4 × 4 dots, how many different triangles with one dot in the middle can you draw? We were ...
28
votes
11answers
21k views

Why should kids learn how to use a compass and straightedge, and not rely on a drawing program?

I am curious why it is necessary for people to learn how to use compasses and straightedges in geometry, and not just rely on a drawing program. I have a couple ideas, but I might be missing ...
28
votes
10answers
4k views

Is Euclid dead? or Should Euclidean geometry be taught to high school students?

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (...
26
votes
17answers
13k views

Given a 3 4 5 triangle, how do you know that it is a right triangle?

Without knowing the Pythagorean theorem, and in presenting reasons why the theorem might be true (without giving a full proof), is there any way to give examples of triangles that are intuitively ...
24
votes
7answers
3k views

Why do we care about multiple proofs of the same theorem?

I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in ...
20
votes
10answers
4k views

Pi or Tau? How should the circle constant be taught?

Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent ...
20
votes
3answers
418 views

At what point is it a disservice to pass someone on to the next math class?

Background information I'm currently teaching common core geometry, which assumes that a student has algebraic knowledge coming in. Clearly, we shouldn't expect students to retain everything from ...
19
votes
8answers
2k views

Are there disadvantages to teaching complex numbers as purely geometrical objects?

Complex numbers are, or at least were to me, generally introduced like this: There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! Set $i^2=-1$ and declare ...
19
votes
4answers
968 views

Teaching manifolds to high schoolers

I would like to introduce the concept of a manifold to a high school student. This person isn't familiar with the axioms of set theory nor topology. However, despite these deficits, I would like to ...
18
votes
10answers
4k views

A parabolic arc is not semicircular. But students think so

I'm teaching a Calc 2 class now (integration and applications) and I'm surprised that more than a handful of students seem to think the graph of $y=x^2$ on $-1\le x\le 1$ is part of a circle! Here is ...
18
votes
12answers
8k views

How to explain that we live in a three-dimensional world?

How does one explain, clearly and simply, that we live in a three-dimensional world? The explanation has to be understandable for a twelve year old child.
17
votes
11answers
1k views

Should we teach abstract affine spaces?

In France at least, there is quite an ancient tradition of teaching abstract affine spaces (e.g. as a triple $(\mathcal{E}, E, -)$ where $\mathcal{E}$ is a set, $E$ is a vector space and $-:\mathcal{E}...
17
votes
2answers
519 views

Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?

When I taught courses on algebra giving a first exposition to Galois theory I usually included some discussion of classical results showing the impossibility of constructing certain points with ruler ...
16
votes
5answers
1k views

Should my 8th graders see a proof of the Pythagorean Theorem?

I've been teaching the Pythagorean Theorem in my 8th grade class, and I noticed something odd. In the book I'm using, the sequence goes something like this: Motivate the idea of distances on a grid ...
16
votes
6answers
1k views

A text-based program to draw geometric figures

I found this question on another forum. Are there any hints how one could make pictures from geometric problems by coding? I mean, if one has a handicap in his hands and he can't use mouse very well, ...
16
votes
3answers
448 views

Good lessons/activities for one-day subs

In my school district, and I'm sure most others, every teacher needs to have a set of "emergency lesson plans", in case they are sick or need to be out for a day, so that the substitute can have ...
16
votes
3answers
393 views

Evidence for or against the claim that some students are “algebra people” and others are “geometry people”

Where I live and work, there is a widely-accepted and often-repeated claim that there are two kinds of students: "algebra people" and "geometry people". This claim sometimes gets expressed in ...
15
votes
6answers
1k views

Are precise drawings important in geometry?

In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also ...
15
votes
4answers
371 views

How to teach affine geometry to future high-school teachers?

This question is a follow-up to that one, where I expressed doubt about the use of abstract affine geometry in undergraduate education. However, future high-school teachers need to be able to relate ...
13
votes
6answers
3k views

Visual Pythagorean demonstration

I know there is some visual demonstration of a² + b² = c² with smalĺ piece of paper, but there is a lot of variations. Which visual or drawing demonstration of the ...
13
votes
4answers
362 views

Simpler explanation for finding the vertex of a parabola

I'm tutoring a Grade 11 Math student in BC, Canada, and we're going over parabolas. He's having difficulty with finding the vertex of a parabola - not how to find it, but WHY it works. And I'm having ...
13
votes
4answers
714 views

Wiggins' question #12

There's an interesting read: Conceptual Understanding in Mathematics by Grant Wiggins. In that text the author proposes "a test for conceptual understanding" which should be given "to 10th, 11th, and ...
12
votes
4answers
399 views

Is there an elementary way to explain that a map of the earth cannot preserve distances?

I am teaching a short "topics in geometry" course to future high school math teacher in France. I plan to cover some spherical geometry. I will be treating the following topics: volume of the ball ...
12
votes
4answers
504 views

What is currently called geometry in high school?

I've encountered classes of 25 or so undergraduates in which more than half -- maybe two-thirds -- of the students claim to have had a high-school geometry course but none has seen a proof of the ...
12
votes
6answers
542 views

Can we explain to undergraduates how points make a line?

Many of my students arrive in college believing that lines are (in some way) made out of points. They also believe that points have no length. They want to know how a bunch of zero length points ...
12
votes
5answers
477 views

Geometry with a view towards differential geometry textbook

I am scheduled to teach an upper-division undergraduate class on "Geometry" and I get to choose more or less what that means. Common choices seem to be non-Euclidean, hyperbolic, projective, or ...
12
votes
2answers
307 views

What prerequisites would college students need for a course based primarily on Euclid's elements?

I love Euclid's elements, and would like to base a course around them. Before I can pitch it to my supervisors, I need to know where it would fit in the curriculum. While it begins from elementary ...
11
votes
8answers
724 views

Symmetry - practical usages

My girlfriend is studying to become a math teacher, and asked me what I have ever used symmetries for. I'm a web developer, and do some designing from time to time, so I answered that symmetry have ...
11
votes
5answers
1k views

Rigorously defining the concept of an angle for high school students

Arriving at a rigorous definition of the concept of angle for high school students is not as easy as expected. Google search provided me with many definition that are too technical or too vague IMO. ...
11
votes
4answers
980 views

Phrasing the Van Hiele levels in student-friendly language

I teach high school geometry and see many of my students fall in to the trap of "it looks like it, so it's true" -- a Van Hiele Level 0 to 1 thought process. For instance, when talking about parallel ...
11
votes
3answers
257 views

When Euclid was used as a textbook, what exercises did students do?

Until fairly recently, it was common for students in school to learn Euclidean geometry from a translation of Euclid. I get the impression that ca. 1700 this would have been in college and only for a ...
11
votes
4answers
257 views

Helping students use constructions in geometry

I work as a teaching assistant in a high school and geometry gives most students headaches. I emphasize understanding the problem by constructing lines and using elementary properties to arrive at ...
11
votes
4answers
842 views

Reasons to teach Thales' theorem

In a classical course on Euclidean, compass-and-ruler geometry, Thales' theorem has always had a prominent place. However, as the Wikipedia article says, It is equivalent to the theorem about ...
11
votes
2answers
628 views

Physical vs. Virtual manipulatives in the high school classroom

I notice that geometry students frequently have difficulty with representations of 3-dimensional objects in 2 dimensions. Today, we worked with physical manipulatives in order to help visualize where ...
11
votes
1answer
359 views

Using number theory instead geometry to introduce proof in Basic School?

It seems there is an overall agreement that Geometry is the right place to introduce proof in Basic School. However, number theory (arithmetic) looks like to be a more simple environment (consider, ...
10
votes
5answers
213 views

Lesson-planning: Teaching probability concepts via geometry

I am intending to teach a lesson covering some topic related to "Probability via Geometry" and, if possible, I would appreciate references or materials (or good ideas) that can help me. The target ...
10
votes
4answers
632 views

Examples of Mathematical Beauty in School Mathematics

Various branches of mathematics have mathematical beauty. Some of this are visual, such as the mandelbrot set, while others are logically sublime, such as the recursive simplicities of peano ...
10
votes
3answers
249 views

Appropriate education level for this geometry problem

What's the appropriate education level for the following concise but non-trivial geometry problem? Points $A$, $B$, $C$ are collinear; $\|AB\|=\|BD\|=\|CD\|=1$; $\|AC\|=\|AD\|$. What is the set ...
10
votes
2answers
1k views

Project/Assessment based on Flatland for Secondary Geometry Class

One of my favorite books is Flatland by Edwin Abbott, which is incredibly rich in geometric concepts especially relating to dimensionality. I would really like to assign it to my secondary geometry ...
10
votes
1answer
523 views

How to denote angle?

I'm teaching mathematics on my free time for young pupils. Once I have seen that they denote angles like $\angle ABC$. But sometimes I have difficulties to understand whether they mean an angle or its ...
10
votes
0answers
184 views

Use of Lockhart's *Measurement* in a course?

I greatly admire Paul Lockhart's Measurement (Harvard Press). Many of you know him through A Mathematician's Lament. One review of Measurement said, “Here Lockhart offers the positive side of the ...
9
votes
5answers
192 views

Extensions beyond Euclidian Geometry for Secondary students

My secondary geometry class is really amazing and is likely going to finish everything I have for the curriculum with a few weeks left in the school year. I was thinking to use these last few weeks ...
9
votes
4answers
1k views

Secondary Geometry Curriculum Sequencing?

I am currently student teaching, and the main class that I am focusing on is a secondary geometry class. I am currently following my classroom mentors curriculum sequence which looks something like: ...
9
votes
2answers
211 views

Teaching geometric transformations through fractals?

In roughly two weeks, my secondary geometry class will have reached the part of the year where we discuss geometric translations: translation, rotation, and reflection. Dilation i am going to use as ...
8
votes
5answers
229 views

How do you explain straightness to a 5-year-old?

How do you explain straightness to a 5-year-old? I am having a hard time trying to explain straightness to little kids. I have been asked, "How do you know that the ruler is straight?" Are there any ...
8
votes
3answers
741 views

Gifs of finding the volume of 3d shapes?

I'm looking for some animations (videos or gifs) of finding the volume of different 3d shapes. It would be super helpful if I could find something which stacks unit cubes into a rectangular prism to ...
8
votes
2answers
263 views

Learning Math like Euclid

I'm posting this here as it's off-topic for the Math stack exchange, and I'm hoping there will be some educators here that can point me in the right direction. I enjoyed watching the YouTube series ...
8
votes
2answers
194 views

Coverage of Fundamentals of Lines and Planes

I was helping a high school student with some fundamental concepts of planes and lines, when I realized I am rusty on some definitions myself. I found some minimal coverage in his high school math and ...