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Rhombuses, kites etc

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2

1

$begingroup$

As a high school teacher, I sometimes wonder about the usefulness of certain topics. Some topics seem to be in the textbook because they have always been there, not because they lead anywhere interesting.

For instance, I am fairly sure that rhombuses and kites are pretty useless. In fact, once you get past alt-int angles, parallelograms are not horribly useful later. I do not recall needing any of this in any math afterwards, at least up to and including calculus.

Since I have been told to cut out some geometry to make way for statistics / probability, it seems to me that rhombuses, kites and a good lot of parallelograms are perfect candidates for the chopping block.

Am I right? Or do rhombuses and kites turn out to be really useful in the conceivable future of any random student? I am not denying their beauty etc.

If proofs could be be put back into the state test, of course, rhombuses etc. would just be more practice in proofs.

geometry teaching

share|improve this question

edited 5 hours ago

Jasper

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asked 10 hours ago

Dan MonroeDan Monroe

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New contributor

Dan Monroe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  I see the use in helping students understand definitions and quantifiers; many high school students STILL struggle with the classification of quadrilaterals and recognizing that, for example, every square is a rectangle, but not every rectangle is a square. This facility with language (not just mathematical language) is very important for a logical thinker to have, and I have found that quadrilaterals lead to the best opportunity to develop it in geometry.
  $endgroup$
  – Opal E
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Students who will encounter vectors will also encounter parallelograms in one of the standard definitions of vector addition and in the connection between areas and 2-by-2 determinants.
  $endgroup$
  – Andreas Blass
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The same question on math.se math.stackexchange.com/questions/3290170/…
  $endgroup$
  – Paracosmiste
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Related math.stackexchange.com/questions/650161/…
  $endgroup$
  – Paracosmiste
  7 hours ago
  

  
  
  
  

add a comment | 

2

1

$begingroup$

As a high school teacher, I sometimes wonder about the usefulness of certain topics. Some topics seem to be in the textbook because they have always been there, not because they lead anywhere interesting.

For instance, I am fairly sure that rhombuses and kites are pretty useless. In fact, once you get past alt-int angles, parallelograms are not horribly useful later. I do not recall needing any of this in any math afterwards, at least up to and including calculus.

Since I have been told to cut out some geometry to make way for statistics / probability, it seems to me that rhombuses, kites and a good lot of parallelograms are perfect candidates for the chopping block.

Am I right? Or do rhombuses and kites turn out to be really useful in the conceivable future of any random student? I am not denying their beauty etc.

If proofs could be be put back into the state test, of course, rhombuses etc. would just be more practice in proofs.

geometry teaching

share|improve this question

edited 5 hours ago

Jasper

2,59299 silver badges1919 bronze badges

asked 10 hours ago

Dan MonroeDan Monroe

1411 bronze badge

New contributor

Dan Monroe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  I see the use in helping students understand definitions and quantifiers; many high school students STILL struggle with the classification of quadrilaterals and recognizing that, for example, every square is a rectangle, but not every rectangle is a square. This facility with language (not just mathematical language) is very important for a logical thinker to have, and I have found that quadrilaterals lead to the best opportunity to develop it in geometry.
  $endgroup$
  – Opal E
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Students who will encounter vectors will also encounter parallelograms in one of the standard definitions of vector addition and in the connection between areas and 2-by-2 determinants.
  $endgroup$
  – Andreas Blass
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The same question on math.se math.stackexchange.com/questions/3290170/…
  $endgroup$
  – Paracosmiste
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Related math.stackexchange.com/questions/650161/…
  $endgroup$
  – Paracosmiste
  7 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

As a high school teacher, I sometimes wonder about the usefulness of certain topics. Some topics seem to be in the textbook because they have always been there, not because they lead anywhere interesting.

For instance, I am fairly sure that rhombuses and kites are pretty useless. In fact, once you get past alt-int angles, parallelograms are not horribly useful later. I do not recall needing any of this in any math afterwards, at least up to and including calculus.

Since I have been told to cut out some geometry to make way for statistics / probability, it seems to me that rhombuses, kites and a good lot of parallelograms are perfect candidates for the chopping block.

Am I right? Or do rhombuses and kites turn out to be really useful in the conceivable future of any random student? I am not denying their beauty etc.

If proofs could be be put back into the state test, of course, rhombuses etc. would just be more practice in proofs.

geometry teaching

share|improve this question

edited 5 hours ago

Jasper

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asked 10 hours ago

Dan MonroeDan Monroe

1411 bronze badge

New contributor

Dan Monroe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|improve this question

edited 5 hours ago

Jasper

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asked 10 hours ago

Dan MonroeDan Monroe

1411 bronze badge

New contributor

Dan Monroe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|improve this question

edited 5 hours ago

Jasper

2,59299 silver badges1919 bronze badges

asked 10 hours ago

Dan MonroeDan Monroe

1411 bronze badge

New contributor

Dan Monroe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited 5 hours ago

Jasper

2,59299 silver badges1919 bronze badges

edited 5 hours ago

Jasper

2,59299 silver badges1919 bronze badges

asked 10 hours ago

Dan MonroeDan Monroe

1411 bronze badge

New contributor

Dan Monroe is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 10 hours ago

Dan MonroeDan Monroe

1411 bronze badge

2 Answers
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4

$begingroup$

Parallelograms are useful for understanding:

• Paths taken by light, especially through a layer of a medium with a different refractive coefficient


• Shear, and related deformations


• Area = height * width (but not necessarily the product of the sides' lengths)


• Dot products


• Surface integrals

Paths taken by light are useful for understanding which routes people and other animals will take to minimize their effort.

share|improve this answer

answered 5 hours ago

JasperJasper

2,59299 silver badges1919 bronze badges

$endgroup$

add a comment | 

3

$begingroup$

I am just now close to completing my first time teaching geometry (at a community college, a course that parallels the typical high school course) Using the fact that parallelograms (and hence rhombuses) have bisecting diagonals makes a short proof for why the line segment midpoint construction works. Other than that, I don't see rhombuses as a big part of geometry. I would keep some of parallelograms in. Have you looked at the common core standards to see which topics are meant to be kept?

Henri Picciotto, a former high-school teacher and (current) curriculum developer, has written a well-thought out pair of blog posts, In Defense of Geometry, which you might find useful.

share|improve this answer

answered 9 hours ago

Sue VanHattum♦Sue VanHattum

9,77811 gold badge2121 silver badges6363 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I wish I could upvote, but Henry mentioning NCTM positively and suggesting teaching fractals in public school (a topic that requires university-level education) did not jive with my outlook on school math education.
  $endgroup$
  – Rusty Core
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @RustyCore I'm not sure why you believe that "teaching fractals" (whatever that phrase means) requires a university-level education. Give me any topic in mathematics (with which I, personally, have sufficient knowledge), and I can prepare a lesson that I can teach to middle schoolers, and I can also prepare a lesson which might give some PhDs a hard time (elementary topics can become research level topics very quickly). I have a 50 minute "Introduction to Dimension Theory" talk that I have given successfully to high school freshman and sophomores in the past which touches on fractal geometry.
  $endgroup$
  – Xander Henderson
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @XanderHenderson He says that he does not don’t "the underlying mathematics" of statistics, yet he is going to teach high school students topology? Hausdorff dimension? Extended real numbers? I don't think so. It will be an overview with pretty pictures at best, all at the cost of removing either Algebra II or Geometry topics. This is not what public school students need. They need instead a coherent curriculum without repetitions and omissions, not some fancy stuff that school teachers know nothing about.
  $endgroup$
  – Rusty Core
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can have your opinions, but his work is stellar, and helped me teach a good course.
  $endgroup$
  – Sue VanHattum♦
  2 hours ago
  

  
  
  
  

add a comment | 

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4

$begingroup$

Parallelograms are useful for understanding:

• Paths taken by light, especially through a layer of a medium with a different refractive coefficient


• Shear, and related deformations


• Area = height * width (but not necessarily the product of the sides' lengths)


• Dot products


• Surface integrals

Paths taken by light are useful for understanding which routes people and other animals will take to minimize their effort.

share|improve this answer

answered 5 hours ago

JasperJasper

2,59299 silver badges1919 bronze badges

$endgroup$

add a comment | 

4

$begingroup$

Parallelograms are useful for understanding:

• Paths taken by light, especially through a layer of a medium with a different refractive coefficient


• Shear, and related deformations


• Area = height * width (but not necessarily the product of the sides' lengths)


• Dot products


• Surface integrals

Paths taken by light are useful for understanding which routes people and other animals will take to minimize their effort.

share|improve this answer

answered 5 hours ago

JasperJasper

2,59299 silver badges1919 bronze badges

$endgroup$

add a comment | 

$begingroup$

Parallelograms are useful for understanding:

• Paths taken by light, especially through a layer of a medium with a different refractive coefficient


• Shear, and related deformations


• Area = height * width (but not necessarily the product of the sides' lengths)


• Dot products


• Surface integrals

Paths taken by light are useful for understanding which routes people and other animals will take to minimize their effort.

share|improve this answer

answered 5 hours ago

JasperJasper

2,59299 silver badges1919 bronze badges

$endgroup$

share|improve this answer

answered 5 hours ago

JasperJasper

2,59299 silver badges1919 bronze badges

answered 5 hours ago

JasperJasper

2,59299 silver badges1919 bronze badges

answered 5 hours ago

JasperJasper

2,59299 silver badges1919 bronze badges

3

$begingroup$

I am just now close to completing my first time teaching geometry (at a community college, a course that parallels the typical high school course) Using the fact that parallelograms (and hence rhombuses) have bisecting diagonals makes a short proof for why the line segment midpoint construction works. Other than that, I don't see rhombuses as a big part of geometry. I would keep some of parallelograms in. Have you looked at the common core standards to see which topics are meant to be kept?

Henri Picciotto, a former high-school teacher and (current) curriculum developer, has written a well-thought out pair of blog posts, In Defense of Geometry, which you might find useful.

share|improve this answer

answered 9 hours ago

Sue VanHattum♦Sue VanHattum

9,77811 gold badge2121 silver badges6363 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I wish I could upvote, but Henry mentioning NCTM positively and suggesting teaching fractals in public school (a topic that requires university-level education) did not jive with my outlook on school math education.
  $endgroup$
  – Rusty Core
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @RustyCore I'm not sure why you believe that "teaching fractals" (whatever that phrase means) requires a university-level education. Give me any topic in mathematics (with which I, personally, have sufficient knowledge), and I can prepare a lesson that I can teach to middle schoolers, and I can also prepare a lesson which might give some PhDs a hard time (elementary topics can become research level topics very quickly). I have a 50 minute "Introduction to Dimension Theory" talk that I have given successfully to high school freshman and sophomores in the past which touches on fractal geometry.
  $endgroup$
  – Xander Henderson
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @XanderHenderson He says that he does not don’t "the underlying mathematics" of statistics, yet he is going to teach high school students topology? Hausdorff dimension? Extended real numbers? I don't think so. It will be an overview with pretty pictures at best, all at the cost of removing either Algebra II or Geometry topics. This is not what public school students need. They need instead a coherent curriculum without repetitions and omissions, not some fancy stuff that school teachers know nothing about.
  $endgroup$
  – Rusty Core
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can have your opinions, but his work is stellar, and helped me teach a good course.
  $endgroup$
  – Sue VanHattum♦
  2 hours ago
  

  
  
  
  

add a comment | 

3

$begingroup$

I am just now close to completing my first time teaching geometry (at a community college, a course that parallels the typical high school course) Using the fact that parallelograms (and hence rhombuses) have bisecting diagonals makes a short proof for why the line segment midpoint construction works. Other than that, I don't see rhombuses as a big part of geometry. I would keep some of parallelograms in. Have you looked at the common core standards to see which topics are meant to be kept?

Henri Picciotto, a former high-school teacher and (current) curriculum developer, has written a well-thought out pair of blog posts, In Defense of Geometry, which you might find useful.

share|improve this answer

answered 9 hours ago

Sue VanHattum♦Sue VanHattum

9,77811 gold badge2121 silver badges6363 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I wish I could upvote, but Henry mentioning NCTM positively and suggesting teaching fractals in public school (a topic that requires university-level education) did not jive with my outlook on school math education.
  $endgroup$
  – Rusty Core
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @RustyCore I'm not sure why you believe that "teaching fractals" (whatever that phrase means) requires a university-level education. Give me any topic in mathematics (with which I, personally, have sufficient knowledge), and I can prepare a lesson that I can teach to middle schoolers, and I can also prepare a lesson which might give some PhDs a hard time (elementary topics can become research level topics very quickly). I have a 50 minute "Introduction to Dimension Theory" talk that I have given successfully to high school freshman and sophomores in the past which touches on fractal geometry.
  $endgroup$
  – Xander Henderson
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @XanderHenderson He says that he does not don’t "the underlying mathematics" of statistics, yet he is going to teach high school students topology? Hausdorff dimension? Extended real numbers? I don't think so. It will be an overview with pretty pictures at best, all at the cost of removing either Algebra II or Geometry topics. This is not what public school students need. They need instead a coherent curriculum without repetitions and omissions, not some fancy stuff that school teachers know nothing about.
  $endgroup$
  – Rusty Core
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can have your opinions, but his work is stellar, and helped me teach a good course.
  $endgroup$
  – Sue VanHattum♦
  2 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

I am just now close to completing my first time teaching geometry (at a community college, a course that parallels the typical high school course) Using the fact that parallelograms (and hence rhombuses) have bisecting diagonals makes a short proof for why the line segment midpoint construction works. Other than that, I don't see rhombuses as a big part of geometry. I would keep some of parallelograms in. Have you looked at the common core standards to see which topics are meant to be kept?

Henri Picciotto, a former high-school teacher and (current) curriculum developer, has written a well-thought out pair of blog posts, In Defense of Geometry, which you might find useful.

share|improve this answer

answered 9 hours ago

Sue VanHattum♦Sue VanHattum

9,77811 gold badge2121 silver badges6363 bronze badges

$endgroup$

share|improve this answer

answered 9 hours ago

Sue VanHattum♦Sue VanHattum

9,77811 gold badge2121 silver badges6363 bronze badges

answered 9 hours ago

Sue VanHattum♦Sue VanHattum

9,77811 gold badge2121 silver badges6363 bronze badges

answered 9 hours ago

Sue VanHattum♦Sue VanHattum

9,77811 gold badge2121 silver badges6363 bronze badges