Visualizing three-dimensional geometry

[Belacqua]

Does anybody have suggestions for how a math student can improve at rendering three-dimensional geometry?

 

My student is getting frustrated with his inability to sketch 3D figures when working on a problem. Drawing is really not his strength (as somebody who still makes stick figures, I sympathize and can offer little help), and the geometry problems he's solving often require pretty elaborate diagrams.

 

The problem isn't conceptualizing the figures, exactly, so making actual 3D models hasn't really helped. He just really needs to get better at sketching a diagram (particularly under exam conditions, where he wouldn't be able to build a model, anyway) in order to angle chase, etc.

 

Any ideas gratefully accepted!

[GingerPoppy]

Some people suggest drawing on graph paper so you can be sure all sets of parallel lines are going the exact same way.

 

I prefer to draw on blank paper. Draw one face of the object. For a pyramid, that would be the base. If it is a square base, be sure to draw it more like a parallelogram because that's how we see a square base from our perspective. Put a single dot above the base you drew. Connect the corners of the base to the dot one at a time in perfectly straight lines. Now erase (or turn into dotted lines) all lines that we would NOT see if we were looking straight toward the object.

 

I recommend having a sample of the shape in front of you the first few times. Let's say you're trying to draw a rectangular prism. Set a Kleenex box in front of you at an angle. Try to draw the shape AS YOU SEE IT, not as you "know" in your brain. You'll notice many of the lines become diagonal to show perspective. NOTE: These diagonal lines always are perfectly parallel AND the same length when drawing them.

 

Whew, hard to explain in words. Hope this gave you some ideas. :)

[Tullia]

I was amazed at how doing the lessons in Artistic Pursuits and keeping my own nature journal alongside my son helped me improve my drawing skills. I would expect that there are other art curricula out there which would do the same thing, but I've had a chance to hear the AP author talk about the importance of doing careful observation before drawing. I think there have been some changes in AP since we used it, but Book I of the Jr. High books might be useful. There are observe, think, and draw elements in most of the lessons...that is if I remember correctly.

 

Don't know if you have the time or inclination to do something like that, but art lessons might help. I gave my son a half credit art elective for doing one Artistic Pursuits book. BTW, neither of us was much beyond the stick figure drawing stage when I first looked at AP to improve our nature journal skills. My son still enjoys making hand written graphs, charts, and diagrams for math and science in college. He says it's a skill that helps him on exams.

 

HTH,

Martha

Edited September 30, 2011 by Martha in NM
typos

[Jane in NC]

I once taught a multi-variable Calculus course using a Bradley/Smith text which had 3-D drawing lessons incorporated in the text. My students found these lessons to be quite useful. You really have to know how to sketch some of these 3-D figures to solve problems at this level.

 

What class is your student in?

[Belacqua]

I once taught a multi-variable Calculus course using a Bradley/Smith text which had 3-D drawing lessons incorporated in the text. My students found these lessons to be quite useful. You really have to know how to sketch some of these 3-D figures to solve problems at this level.

 

What class is your student in?

 

Oddly, he didn't grumble too much about having to deal with the figures in MV calculus. Most of his frustration is coming from trying to work with Olympiad-level geometry problems. We got him a white board, thinking a bigger surface might make things better; it has helped somewhat, but there's still an issue with actually rendering the figures in a way that makes solving the problem easier.

 

I'll have a look for that text you mentioned (do you happen to have the title?). While suggesting "art lessons" would undoubtedly result in wholesale rebellion, some instruction in straightforward 3D drawing can only help.

 

Thanks for all the suggestions.

[Jane in NC]

Oddly, he didn't grumble too much about having to deal with the figures in MV calculus. Most of his frustration is coming from trying to work with Olympiad-level geometry problems. We got him a white board, thinking a bigger surface might make things better; it has helped somewhat, but there's still an issue with actually rendering the figures in a way that makes solving the problem easier.

 

I'll have a look for that text you mentioned (do you happen to have the title?). While suggesting "art lessons" would undoubtedly result in wholesale rebellion, some instruction in straightforward 3D drawing can only help.

 

Thanks for all the suggestions.

 

It was simply called Calculus. I taught from the first and second editions which I believe had the drawing lessons. These texts will probably cost more in postage than for the book itself which leads me to wonder about alternative ideas. Do you have a university library at your disposal? Spending some time in the stacks may lead you to find something appropriate.

 

What is he sketching?

[Tullia]

Oddly, he didn't grumble too much about having to deal with the figures in MV calculus. Most of his frustration is coming from trying to work with Olympiad-level geometry problems. We got him a white board, thinking a bigger surface might make things better; it has helped somewhat, but there's still an issue with actually rendering the figures in a way that makes solving the problem easier.

 

I'll have a look for that text you mentioned (do you happen to have the title?). While suggesting "art lessons" would undoubtedly result in wholesale rebellion, some instruction in straightforward 3D drawing can only help.

Thanks for all the suggestions.

 

:lol: I totally understand that! There's a reason why my son had only a 1/2 credit of art on his high school transcript. One semester and he decided he'd BTDT and was ready to move on.

 

ETA: Jane's idea of looking in the library stacks is excellent. Another place to find math resources is library used book sales. Many libraries routinely sell donated books to help raise money, and I once came home with a sack full of wonderful gently used older math texts and books about mathematics for under $5.00. The librarian told me that they'd been donated several years back by a retired math professor and had been sitting in their storage building until I found them. I was browsing for Dorothy Sayers fiction and chatting with the librarian; our library sells mostly fiction so that's what's usually on display. Now I know to ask for the less popular stuff.

Edited September 30, 2011 by Martha in NM

[Belacqua]

It was simply called Calculus. I taught from the first and second editions which I believe had the drawing lessons. These texts will probably cost more in postage than for the book itself which leads me to wonder about alternative ideas. Do you have a university library at your disposal? Spending some time in the stacks may lead you to find something appropriate.

 

What is he sketching?

 

We do live near a few uni libraries, so I'll take a look around. We have several calculus texts lying around, but not, unfortunately, the one you mention.

 

I wonder if wolfram-alpha would be useful, as well.

 

As an example, the most recent problem that comes to mind involved the spherical law of cosines. Just the idea of having to render a sphere and all its attendant details on paper got the poor kid all jittery. He commented that he had trouble representing which "side" of the sphere a point lay on.

[elegantlion]

Draw Squad might be another option. http://www.amazon.com/Mark-Kistlers-Draw-Squad-Kistler/dp/0671656945/ref=sr_1_1?ie=UTF8&qid=1317420773&sr=8-1

 

It's only 30 lessons. My son isn't loving it, but we will start geometry soon and I know the Draw Squad lessons will pay off once he starts.

 

I bought a used copy very inexpensively.