Questions tagged [geometry]

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

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12 votes
5 answers
3k views

How can I build a protractor without a protractor?

We all know how to use a protractor; it is taught in elementary school. However, I was wondering what type of knowledge is required to build one from scratch. For instance, was the understanding of $\...
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5 votes
2 answers
926 views

Is "Annular Ring" redundant?

I've come across the term annular ring in parentheses following washer in my calculus textbook: "has the shape of a washer (an annular ring)". The definition of the word "annular" ...
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17 votes
4 answers
5k views

Explaining why volume of cone is a third of cylinder

I came across this video explaining to kids why the volume of a cone is a third of the cylinder of same cross-sectional radius and height. Essentially the explainer presents pre-created cylindrical ...
  • 273
10 votes
2 answers
612 views

Real-world applications of taxicab metric

The taxicab metric can be used to measure distances in idealized gridded cities. However, usually this serves only as a fun exercise for students. I'm looking for engaging (as non-technical as ...
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7 votes
1 answer
122 views

Resources for Teaching Parameterization of Curves/Surfaces

In classes like Calc 3 or Computer Graphics, I want my students to be comfortable describing common curves and surfaces parametrically (such as lines, triangles, circles, or surfaces of revolution). ...
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7 votes
7 answers
275 views

Creative problems in 2D vector geometry

What are some "interesting" and creative problems or exercises on specifically 2-dimensional vector geometry that a high school student might find compelling to solve? The class' current ...
0 votes
1 answer
213 views

Why do so many children's book confuse discs with circles? [duplicate]

The difference between a disc (disk) and a circle is crystal clear to me: However, in many children's books, a disc is usually called a circle: Why do many children's book confuse discs with circles?...
  • 4,051
2 votes
2 answers
120 views

Is there a good Animation to explain Rotational Symmetry of Equilateral triangle

I am willing to teach that the Order of rotational Symmetry of Equilateral triangle as $3$ using Animation. Any suggestions of good applet which demonstrates the rotation of equilateral triangle with ...
4 votes
1 answer
167 views

Valid Reasons in Two-Column Geometry Proofs

I'm wondering about the relationship between Eculid's work and modern high school geometry. In "two column proofs," certain reasons are considered acceptible for steps in the proof, such as ...
12 votes
3 answers
3k views

How do I sketch a good gaussian curve freehanded, or by using only common sketching tools?

I'm a lousy artist. If I want my Gaussian curves to be accurately drawn when I use a whiteboard, or work with pen & paper, what are my options? Is there a way to use a straight edge, or compass, ...
3 votes
2 answers
122 views

Line segments on a geoboard

To make a polygon on a traditional geoboard, one usually stretches a rubber band around the vertices. No problem there. When making a simple line segment, however, a rubber band is typically stretched ...
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3 votes
2 answers
478 views

Where can I buy a compass that can hold an "Expo" dry erase whiteboard marker?

Question in title. I'd like to draw a perfect circle on the whiteboard using "expo" dry erase markers. Is there a store I could buy such a compass at? Thank you
0 votes
1 answer
115 views

Names of two circle theorems in English [closed]

There are two theorems: All three angles AC_1B, AC_2B, AC_3B are equal. In general: All angles above a chord are equal. The size of any angle above a chord AB is half the size of angle AOB where O is ...
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3 votes
4 answers
327 views

Explain to 10 year old — Why do 3D mental pictures usually suffice for high-dimensional geometry?

My 10 year old daughter is trying to read this book — please explain in Simple English that she'll grasp. Kindly see the embolded phrases below. The author doesn't expound why the 3D "mental ...
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4 votes
5 answers
366 views

Is it a good idea for elementary school students to observe and discover the "circle perimeter formula" themselves without being dictated to?

Let them discover $\;\ell=2\pi R\;$ by their own, at least the invariance of $\ell/R$ Do you think that the following method works well in a mathematics class of elementary school? What do you think ...
23 votes
9 answers
4k views

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

The assumed age of the students is 10-15 years old. What is the danger in saying that two triangles are "the same" or "equal" instead of saying that they are congruent? It seems to ...
  • 1,691
14 votes
11 answers
6k views

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

The imagined students are in elementary school, say around 9-13 years old. I want to use rather precise terminology when talking to my students. However, it seems like we typically use the same ...
  • 1,691
4 votes
2 answers
215 views

Are there any list of mathematical constructions which can challenge 12-16 year old students?

Mathematical (geometric) constructions are an interesting way to engage students. It also helps in better understanding of different geometrical properties. For example, Sierpinski triangle or square, ...
4 votes
2 answers
235 views

Geometric and Graphical perspective on completing the square

I just read an interesting article that helps to understand completing the square, and prove the quadratic equation from a geomterical perspective. My question is how do I understand the graphical ...
7 votes
7 answers
4k views

What is the preferred way to denote the Pythagorean theorem equation?

I am teaching 12-16 year olds. How should I write down the Pythagorean theorem equation? Some alternatives: $a^2 + b^2 = c^2$ $\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$ $\text{leg}_1^2 + \...
  • 1,691
0 votes
1 answer
180 views

Matriculation exams like in Europe

I just looked at matriculation exams from Finland. They have both basic and advanced level exams. Most US high school seniors could not pass the basic exam. If each US state were to create its own ...
5 votes
7 answers
775 views

What is a good second book in high school geometry?

I have been looking at questions on Math Stack Exchange and I am frequently coming across topics that sound as if they could have been optional chapters in a high school geometry class, but I have ...
27 votes
10 answers
3k views

Should figures be presented to scale?

I've been working with a teacher, helping her with tech. One of the things I help with is to convert PDF formatted quizzes or tests to DeltaMath for the students to take online. The issue that I face ...
4 votes
4 answers
277 views

Can we define length and perpendicularity not via an inner product?

A natural way to reason about Euclidean geometry using modern mathematical language is to define Euclidean space as an affine space $A$ directed by a finite-dimensional real vector space $V$. However, ...
  • 61
7 votes
2 answers
173 views

Is $\overline{AB} \cong \overline{BA}$ usually taught as an instance of the symmetric property of congruence?

I have been tutoring a wide range of math subjects for many years. Recently, I began tutoring a girl in high school geometry (in California, for context). This semester of the course is starting with ...
  • 823
9 votes
3 answers
2k views

Definition of Trapezoid

From one textbook we use in our High School - Transcription: A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases of the trapezoid. And from ...
13 votes
3 answers
531 views

How to teach the Pythagorean theorem in a satisfying way to high school students?

I've been pretty dissatisfied with the way the Pythagorean theorem is usually taught, mainly for two reasons: The chosen proof feels like magic and I don't feel like I have a better understanding of ...
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1 vote
1 answer
269 views

US High School Geometry: What are all the "reasons" allowed in two column proofs?

For a project I am considering on geometry in US high schools, I need a list of all the reasons that are usually allowed in two column geometry proofs there. I have a bachelor's degree in math and ...
3 votes
4 answers
661 views

What is an algebraic explanation of why the product of the slopes of perpendicular lines is $-1$? [duplicate]

Q: What is a succinct, clear and purely algebraic explanation of why the product of the slopes of perpendicular lines is $-1$? Here I am aiming for high-school students (in the U.S.). I have a purely ...
9 votes
5 answers
2k views

Is there a name for paths that follow gridlines?

I'm writing up an activity where students are looking at pathlengths that follow along gridlines. Is there a word or phrase that is commonly used to describe those paths, but doesn't include ...
  • 483
5 votes
3 answers
3k views

Is there online geometric construction software that models physical constructions?

In this weird pandemic school year, I'm doubly interested in technology integration to help my virtual (high school) students as much as my in-person students. I've been particularly eager to get ...
  • 5,539
1 vote
1 answer
214 views

Which Geometry book is more rigorous/harder?

Can anyone please tell me if the AOPS (Art of Problem Solving) Geometry is more rigorous/difficult generally than a book called "Geometry for Enjoyment and Challenge" by authors Rhoad, ...
9 votes
1 answer
228 views

College undergraduate geometry courses

I am interested in learning how a course in geometry is employed today at undergraduate colleges/universities in the U.S. On the one hand, such a course seems to serve as an optional (rarely required) ...
1 vote
0 answers
56 views

Exercises for explaning homothety, homothetic center, similarity on line and plane, free vector and vector space

I need the collection of exercises for such topics as: maps and transformations, composition of maps homothety, rotation homothety, homothetic center similarities of the line and the plane free ...
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7 votes
6 answers
4k views

Why we have to be so precise in Geometry?

Previously I've explained some basic things of graphs to my kid, such as planar, $V-E+F=2$. Now when I introduce geometry, he asked, "Why we have to be so precise in Geometry?" Indeed, in ...
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1 vote
1 answer
158 views

What books are good to study Solid Mensuration?

Preferably I want those that contain the following topics: Solid Figures Polyhedrons Prisms Pyramids Prismatoid Truncated Prisms Cylinders Cones Spheres I've been ...
1 vote
2 answers
238 views

Workbooks for advanced high school math topics

I'm looking for advanced workbooks and exercises for working in class (math high school/undergraduate level) covering the following topics (or some of them): Logic and sets (propositional calculus, ...
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11 votes
11 answers
4k views

Ideas for explaining 4D and higher dimensions

I introduced the hypercube (to undergraduate students in the U.S.) in the context of generalizations of the Platonic solids, explained its structure, showed it rotating. I mentioned Alicia Stott, who ...
4 votes
3 answers
615 views

Should we stop using traditional compass in schools & start/encourage adopting compasses like "Slide N Measure" or "Safe-T" compasses instead?

I think using the traditional compass with those styluses that can literally be used to hurt or accidentally hurt someone are very dangerous. Most people don't use these in day-to-day life anyways, ...
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11 votes
2 answers
211 views

2D drawings of 3D objects in printed school textbooks: orthogonal or perspective?

There is a tradition in the use of orthogonal projections to represent 3D objects in printed school math textbooks. On the other hand, perspective projections represent better the way as we "see" real ...
6 votes
5 answers
2k views

What are strategies for teaching that the altitude of a right triangle creates two similar triangles?

If you draw the altitude to the right triangle as shown, it is easily seen that $$\triangle KLM\sim\triangle KNL\sim\triangle LNM.$$ This in turn leads to several interesting proportional relations ...
  • 5,539
4 votes
2 answers
522 views

Euclid Book 1 Proposition 4 [closed]

In Euclid's The Elements, Book 1, Proposition 4, he makes the assumption that one can create an angle between two lines and then construct the same angle from two different lines. I do not see ...
3 votes
2 answers
194 views

Introducing quadric surfaces in high school

I am presenting an enrichment session on 3D geometry and quadric surfaces to able 15-year-old secondary school students. They know algebra but not calculus. They have learned about equations of ...
  • 1,607
1 vote
1 answer
115 views

How to teach geometric patterns? [closed]

I would like to know how to teach geometric patterns in secondary school. I want to elaborate worksheets, which could include different kinds of strategies related to this topic. Are there resources ...
4 votes
1 answer
389 views

Cinderella vs. GeoGebra

I would be grateful for a comparison between the capabilities of Cinderella and Geogebra, for teaching at all levels, but especially at the college/university-level. I became a reasonably adept user ...
37 votes
12 answers
4k views

Beautiful planar geometry theorems not encountered in high school

I would like to impress college students (undergraduates in the U.S.) that there is more to planar geometry beyond what they learned in high school. I would like to show them beautiful theorems they ...
10 votes
4 answers
1k views

Co-curricular lessons between geometry and chemistry?

My school is hyped about the promise of co-curricular education and they are giving the math and science teachers paid days off to develop lesson plans that synergize our learning goals. I'm on ...
  • 5,539
2 votes
2 answers
1k views

High school maths textbook for talented students

I am looking for a math textbook. I'm 15 and I'd like to complete algebra 2 geometry and perhaps something about probability/ number theory or trigonometry would be nice too. Later I wanna do ...
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28 votes
12 answers
8k views

How to give my students a straightedge instead of a ruler

I'm having a "challenge" in my geometry classes getting students to avoid using rulers as measuring devices in constructions. As natural as that usage is, they're only supposed to use them to connect ...
  • 5,539
8 votes
1 answer
784 views

An alternative to "two column" geometry proofs

I'm a high school teacher in New York State (US), starting in on my first year of teaching Geometry. One of the things that really intrigues me is that the Regents exam (the state-mandated final exam)...
  • 5,539