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I am realizing more and more that I just don't fully get geometry. Parts of it are so intuitive that they seem stupidly obvious. Other things are just so much vocabulary, vocabulary and endless formulas. A lot of it is obviously useful, but a surprising amount of it seems really abstract and pointless to me. Other math through basic algebra is all stuff that I think I use pretty often. But things like finding the volume of a cylinder? When does anyone need that? I mean, I know some people need it, but I've made it this far without ever needing to do that. No one I know needs to do that. Everyone I know needs to use simple algebra and lots of arithmetic.

 

Anyway, this year, I tried to open it up a little and do more geometry with the kids. We've, um, not done a great job with it thus far, to be honest. I have been playing around with the Zometools with them. One of the things I've found that is both interesting and confounding is how easy it is to stumble on impossible to name shapes. I mean, who just knows offhand all the various hedrons and dodecasomethingorothers there are? I've looked up a surprising number of shapes they've made only to find they have names and certain properties and I've never heard of them.

 

Other math is not like this. My kids have never accidentally discovered exponents with the C-rods before they'd even learned to add, which is basically how this feels. Like geometry is vastly less linear in progression than other math and the questions that can come up about it are much more complex yet also much more simple. It's just a very weird branch of math for me.

 

I don't have any particular question. I got myself a copy of Jacobs's Geometry to work through and have done a little of it. I like Jacobs's style and I like the book (I should really get back to it and do more). The kids and I also did Dragonbox Elements, though for me that was an example of a duh in geometry. I thought the program was brilliant, but nearly every puzzle was too easy. I didn't feel the learningand connections like I did with the algebra one.

 

I mostly just am musing. Do other people find this? Do other people find geometry is extra hard to teach in elementary and middle school? Do people have deep thoughts about geometry to share?

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Well, the volume of a cylinder...somebody has to do it...otherwise we wouldn't know how much was in our canned goods.  ;)

 

It's true that often we don't use a lot of geometry in daily life, but I still think it's pretty cool.  It also seems rather like logic puzzles.  Nobody I know needs to solve those in daily life either.  But they sure are good for stretching the brain!

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In school, I understood Algebra but found Geometry a big, useless puzzle. I memorized enough to do well in the course but never understood it.

 

I refuse to teach it. (Meaning the high school course.) I have no competence. I will outsource it.

 

I have used Algebra fairly frequently in life but not Geometry.

 

I'm sure there are many folks who find it useful and beautiful, but I'm not one of them.

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This is hard to address without some more specific concerns, geometry is a huge field. Do you know some calculus? That makes understanding parts of it easier (though it is absolutely not necessary for learning basic geometry).

 

Geometry shouldn't feel like a lot of formulas. The formulas show flow from your understanding of the subject. So for your volume of a cylinder example, it's just the area of the circle(base) multiplied by the length. Much like you find the volume of a rectangular prism by multiplying the area of its base by the length. (The why here makes more sense with calculus).

 

Have you looked at Euclid? It's a great place to start.

 

As for usefulness, the vast majority of my social circle are engineers, and they use some level of geometry daily. I use it often, though not daily. Volumes and areas are pretty important whenever constructing something. Knowing how to set a plumb line, or make a door square etc.

 

Edit: fixed a lot of typos. Tying on the iPad is hilarious at times.

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Ashes to ashes,

Dust to dust.

If algebra don't get you

Geometry must.

 

With that, volume of a cylinder is just the area of the circular face times the height, right? Just like the volume of a cube, except that in order to get the area of the circular face you have to remember to use pi.

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Starting this year, I am using it daily to teach my daughter. She might be able to use it when she has her own children. Or she might use some of it in her physics class.

 

Seriously, though, the main lesson I took from my geometry class was the ability to think and present information in an organized way and not try to prove anything based on assumptions.

 

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I thought the same thing exactly, which is why it was much harder for me to teach.

 

The interesting thing is, my husband thought it was about the most interesting class he took in high school.  He just "got it." 

 

To me, algebra is very common sense, and geometry -- except for some of the obvious -- is far more abstract and weird.  I think all my kids felt the same way about it as I did.  Fortunately, most math tests they had to take (ACT, college placement, etc.) seemed to deal more with algebra or the more practical part of geometry.

 

 

 

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We have integrated math here all way through, and we experience geometry as an reinforcement of what we learned otherwise. We used Understanding Geometry during middle school and that is what we liked.

 

The volume of a cylinder is also about gastanks, oxide tanks needed for diving (if a man uses dm3 air per quarter and the size of the tank is x,y,z, how long can he dive), cans (how many beans fit in the can) etc. I don't remember them all, but we got a lot of that type wordproblems in elementary school.

 

We also experience Geometry as a break from other math.

X days geometry y days algebra

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I do not find geometry "endless formulas". Most formulas can be derived from very simple principles and do not need to be memorized. The volume of a cylinder becomes trivial once one realizes that a volume is an area times a height - together with the area of a circle being pi r^2 (which is very useful to know).

 

Aside from geometric proofs as a great tool for developing logical argumentation, there are lots of applications. Any time somebody builds something, geometry is involved. There is geometry in crafting, sewing, baking, cooking.

I am crocheting - lots of geometry: why and how to increase when working in the round? Because the circumference increases by the same amount every round. It becomes very easy to keep track of stitches with some basic geometry understanding.  How much yarn will I need for my triangular shawl if the first x rows took one skein?  Yarn is proportional to the area, area increases quadratic with linear dimension, so I'll need a lot more than it looks like after the first one ;-)

And the volume of the cylinder figures again whenever you try to find out how much stuff fits into a container of cylindrical shape - jar, bucket, bottle... or when judging how much is left in a partially emptied bottle and see whether that would suffice for the recipe: you estimate by height, because the volume is proportional to the height. That's volume of a cylinder right there.

When I estimate how to scale a recipe intended for a 9x13 pan to a square 8x8 pan or a round pie pan I have to calculate areas.

Other applications in daily life I can think of on top of my hat: painting walls, selecting flooring, fencing a garden bed, estimating amount of mulch needed to cover the bed to a certain depth.

My 16 y/o works for a company that makes jewelry supplies. He recently had to use his geometry skills to calculate the amount of wire needed to make  jump rings of a certain shape so he could come up with a price quote.

 

I have some trouble with three dimensional spatial visualization and hit my limits with geometry. But of all the math studied in school, I see geometry most frequently in daily life for normal people (I am phrasing it like this because my job requires daily use of calculus which I realize is a rare exception)

 

ETA:

 

One of the things I've found that is both interesting and confounding is how easy it is to stumble on impossible to name shapes. I mean, who just knows offhand all the various hedrons and dodecasomethingorothers there are?

 

No, you don't need to know all these. But the names really are not complicated once you have a basic knowledge of word roots. A dodecahedron is simply a polyhedron with twelve (dodeca) sides, and octagon is a polygon with eight (oct) sides, a pentagon has five...

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The elementary math program I use for my kids is pretty darn light on geometry -- in terms of shape-knowledge, terminology, and concepts. The K math I've used (or am using) for my last two kids is pretty heavy on shapes & terminology (and pattern-building/recognizing). The difference in how those two (boys vs. girls) see the world during & after that K math is noticeable.

 

I'm doing "geometry" with dd#1 this year. Like I've found in many other areas of our home education, every day there is a tie with other subjects & other parts of life. She is still struggling with wrapping her head around the 'why' of proofs - since we're at the beginning, they seem so 'duh' to her - but she's starting to see the logic of it. I don't know if she'll ever see the beauty, but if she can work through the logic, I'll be happy.

 

At the younger ages, I'd just work with shapes. All of us are fascinated with our set of pentaminoes right now. I can't wait until someone (other than me) can get all 12 of them into a rectangle. I'm ready to celebrate with them.

 

(We also have pattern blocks and tangrams that are used a lot.)

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Ahh, Regentrude, spoken like a true physicist! Actually you make me think of my biostats mentor - she was a great one for first principles and I loved working with her. She made me think.

 

Farrar, how's your spatial perception (interesting that Regentrude brought this up). I ask because DS14 has dyspraxia, so his spatial awareness has been slow to develop. He was at least a year or two behind in geometry until last year (yr8) when his spatial perception improved. I think geometry relies more on visualisation and experiential learning before it clicks, otherwise it is just rote learning of formulas, which is tedious and actually fairly useless long term. Fortunately we have integrated maths in Australia, so geometry just keeps rolling around every year and you can tackle it when the student is ready

D

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My spatial perception is pretty good. I like some spatial puzzles like Tetris and doing jigsaws. I can visualize turned shapes and shadows and things like that when you see those questions.

 

I get that you can re derive the formulas... That's pretty much how I spent my entire math career in school was rediscovering how to do things because I was always forgetting. I just have so rarely needed any of that stuff. And some of it, I'm honestly not sure if I was ever fully taught it. There are definitely some formulas for things that I see in geometry texts that I'm like, oh, I know we never learned that, like volumes of various solids. And in everyday life, nearly every time I need it, I can just estimate it or it's something reasonably simple like the space to put in a table or something in a room. The measurement end of geometry is useful... but in terms of really complex stuff I never need to do more than estimate. I feel like no one ever really taught me a lot of this stuff.

 

I like logic puzzles and logic in general. I really like all the stuff about logic in the Jacobs books and in Jousting Armadillos. I remember thinking it was all pretty easy when I was actually in geometry.

 

I am just finding, especially with the Zomes, that when the kids make unusual solids but which have some element of symmetry to them, that I have no clue what I'm even looking at. And that a lot of the ways that I have guided math discovery for arithmetic and beginning algebra I just can't do... I have no idea even what questions to ask. And we've used a number of different elementary math programs over the years... and they've all had radically different amounts of geometry, approaches to geometry. And, as I posted before, I learned from using MEP that some shapes that are apparently very important to learn about in UK aren't even a thing here. Like, we don't have an equivalent term for those dumb trapeziums. It makes one feel like there's no agreement in what should even constitute basic geometry beyond name the shapes (though beyond certain basics, there seemed to be so little agreement on which ones...) and calculate perimeter or polygons and area of triangles and rectangles. Some programs we've used have had very little on circles, others a lot, some seem to think it's critical you be able to draw and use tools, others never mention it.

 

Like I said, I don't have a specific question... I don't know what I'm looking for... It's just an odd subject to me.

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I am just finding, especially with the Zomes, that when the kids make unusual solids but which have some element of symmetry to them, that I have no clue what I'm even looking at. And that a lot of the ways that I have guided math discovery for arithmetic and beginning algebra I just can't do... I have no idea even what questions to ask. And we've used a number of different elementary math programs over the years... and they've all had radically different amounts of geometry, approaches to geometry. And, as I posted before, I learned from using MEP that some shapes that are apparently very important to learn about in UK aren't even a thing here. Like, we don't have an equivalent term for those dumb trapeziums. It makes one feel like there's no agreement in what should even constitute basic geometry beyond name the shapes (though beyond certain basics, there seemed to be so little agreement on which ones...) and calculate perimeter or polygons and area of triangles and rectangles. Some programs we've used have had very little on circles, others a lot, some seem to think it's critical you be able to draw and use tools, others never mention it.

 

Like I said, I don't have a specific question... I don't know what I'm looking for... It's just an odd subject to me.

 

I'm sure you've had this suggestion before, but have you (re?)considered working through AoPS Geometry on your own? They're not big on memorization, as you know, and go beyond what's typically taught in high schools, so it should be a good fit and not a waste of your time.

 

Regarding the differences in texts: in general, math texts on a subject will all cover certain obviously important topics (like volumes of solids), and then choose which special topics to cover, and there can be a big range of those, but one set of special topics isn't necessarily better than another. It's like taking a cooking class that starts with basics--probably covered in every cooking class--and then you practice and expand those basic skills by actually cooking, and the recipes would vary from to class to class. One class might make typically American food and another typically British, but the point isn't the particular recipes or where they're most popular; the point is to practice making food, and perhaps especially to start to get an intuitive sense of why two recipes with nearly identical ingredients in different ratios produce totally different results. Once you have that intuition, you can think about a similar food that you don't have a recipe for, and figure out what the recipe must be. Similarly, it doesn't matter which shapes or solids your text focuses on, as long as you really get the intuition that volume is area*height, for instance, and then you can apply that concept to whatever solid is in front of you.

 

So you're absolutely correct: beyond the universal basics, there actually isn't agreement on what should be covered. Feel free to choose a text whose special topics particularly appeal to you, like choosing a cooking class with an emphasis on Indian food, if that's your thing (as it is mine), and to pass over the one whose emphasis is on 567 ways to make tofu not really taste like (fill in the blank). 

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I did most of AoPS's algebra on my own, but it was mostly easyish and I didn't love the style. I want to stick with the Jacobs for geometry for me. And maybe for the kids... we'll see... I should keep going and see if it turns on any lights. It's just so odd... in between easy and baffling.

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I'll throw in my two cents.  For elementary, there isn't a whole lot of geometry material to "master" (using that term loosely) and even then, it all comes back around in the high school geometry course anyway.  It's just easier to learn some of these things in elementary so that they're a quick review in the high school course.  Look at the geometry chapters in AoPS Prealgebra (can't remember if you have it) - that about sums up what's necessary for topic coverage for all of elementary, three chapters.

 

For spatial skill development, I'd look at playing around in space as you've been doing already with building tools/toys.  If you're looking for more depth, then there's BA, which you've already done, I think?

 

I've been thinking a lot about approaches to geometry lately, as my ds12's school promises to teach him the course starting next week (he has an Advanced Learning Plan which seems to garner special treatment, though *sigh* the teacher moves too slowly - he's hoping to take two years).  I was trying to get them to use AoPS because he's extremely visual-spatial.  They'll be using Jurgensen instead, which is still rigorous, but I know there is likely to be unpleasantness with regard to the two-column proof.

 

Traditional geometry texts come across to me as awkward because they force this spatial subject into a sequential instructional format.  OTOH, two-column proofs require sequencing and that translation from the spatial to the sequential is a useful mental exercise for someone like my ds, albeit painfully pedantic and jargon-heavy.

 

I hated my dry-as-a-bone geometry course in high school though by then I was already a slacker, so perhaps I'd have enjoyed it more if I had paid attention.  AoPS, on the other hand, was a major revelation for me.  I agree with the suggestion to use AoPS geometry for yourself if you have good spatial skills.  Since you're planning to use Jacobs - probably the next best text for a spatial thinker - I highly, highly recommend adding in Alcumus.

 

From a sequential angle, I see how the algebraic proofs included in older algebra 1 texts are so useful for preparing for the two-column proofs taught in traditional geometry texts.  (AoPS only teaches paragraph style proofs, which I thought I would hate, but I actually find so much easier due to the availability of connecting language to make everything so much easier to communicate and the ability to organize multiple logical sections.)

 

I get that you can re derive the formulas... That's pretty much how I spent my entire math career in school was rediscovering how to do things because I was always forgetting. I just have so rarely needed any of that stuff. And some of it, I'm honestly not sure if I was ever fully taught it.

 

This was my experience too.

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I am realizing more and more that I just don't fully get geometry. Parts of it are so intuitive that they seem stupidly obvious. Other things are just so much vocabulary, vocabulary and endless formulas. A lot of it is obviously useful, but a surprising amount of it seems really abstract and pointless to me. Other math through basic algebra is all stuff that I think I use pretty often. But things like finding the volume of a cylinder? When does anyone need that? I mean, I know some people need it, but I've made it this far without ever needing to do that. No one I know needs to do that. Everyone I know needs to use simple algebra and lots of arithmetic.

 

Anyway, this year, I tried to open it up a little and do more geometry with the kids. We've, um, not done a great job with it thus far, to be honest. I have been playing around with the Zometools with them. One of the things I've found that is both interesting and confounding is how easy it is to stumble on impossible to name shapes. I mean, who just knows offhand all the various hedrons and dodecasomethingorothers there are? I've looked up a surprising number of shapes they've made only to find they have names and certain properties and I've never heard of them.

 

Other math is not like this. My kids have never accidentally discovered exponents with the C-rods before they'd even learned to add, which is basically how this feels. Like geometry is vastly less linear in progression than other math and the questions that can come up about it are much more complex yet also much more simple. It's just a very weird branch of math for me.

 

I don't have any particular question. I got myself a copy of Jacobs's Geometry to work through and have done a little of it. I like Jacobs's style and I like the book (I should really get back to it and do more). The kids and I also did Dragonbox Elements, though for me that was an example of a duh in geometry. I thought the program was brilliant, but nearly every puzzle was too easy. I didn't feel the learningand connections like I did with the algebra one.

 

I mostly just am musing. Do other people find this? Do other people find geometry is extra hard to teach in elementary and middle school? Do people have deep thoughts about geometry to share?

 

No deep thoughts, but I could have written this post.  :D

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(AoPS only teaches paragraph style proofs, which I thought I would hate, but I actually find so much easier due to the availability of connecting language to make everything so much easier to communicate and the ability to organize multiple logical sections.)

 

:smilielol5: I'm going to have to pull this one out (figuratively) if DD complains about two column proofs this year. Writing-phobic girl would really hate proofs if they had to be in paragraph form.  :tongue_smilie:

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I asked this same question to my high school geometry teacher. Her answer: "It teaches you to think logically". The application of logical thinking has been invaluable to me.

 

Also, there are other subjects that have no application to daily life, but are valuable for literate, educated individuals.

 

You also don't want to limit your children's future opportunities by minimizing this subject. Some careers really do need geometry!

 

Best wishes.

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I asked this same question to my high school geometry teacher. Her answer: "It teaches you to think logically". The application of logical thinking has been invaluable to me.

 

Also, there are other subjects that have no application to daily life, but are valuable for literate, educated individuals.

 

You also don't want to limit your children's future opportunities by minimizing this subject. Some careers really do need geometry!

 

Best wishes.

The bolded is why I am outsourcing Geometry at the high school level.  It is too late for my own opportunities, and I am perfectly happy with my career path, which would never have included a math-heavy emphasis.

 

I am teaching literature/writing to high school students this year in a co-op setting, and I give them the same arguments.  (Being a literate, educated individual.)  I find it amusing. :lol:   The world needs all of us, the math-literate and the literature-literate.

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Rose, didn't you guys do an edX geometry course last year? Was that helpful? Or am I misremembering?

 

It was really great.  We both did the class, we both learned a lot, and it definitely demystified Geometry and removed a lot of fear and trepidation about it.  It was a fantastic course. I wish we could just call Geometry done! But I'm afraid there wasn't enough practice to make it stick long-term, and there was no proof writing. They definitely presented proofs, and took you through the logic step by step, it was great. But from my lurking on geometry threads, it doesn't seem like it had everything it would need to call it a full Geometry course.

 

You guys might really enjoy it though, Farar.  I can't speak highly enough of the presentation/teaching style of the Schoolyourself courses. They just don't have enough practice and application/word problems for full mastery/retention or for a full high school credit, IMO.

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It was really great.  We both did the class, we both learned a lot, and it definitely demystified Geometry and removed a lot of fear and trepidation about it.  It was a fantastic course. I wish we could just call Geometry done! But I'm afraid there wasn't enough practice to make it stick long-term, and there was no proof writing. They definitely presented proofs, and took you through the logic step by step, it was great. But from my lurking on geometry threads, it doesn't seem like it had everything it would need to call it a full Geometry course.

 

You guys might really enjoy it though, Farar.  I can't speak highly enough of the presentation/teaching style of the Schoolyourself courses. They just don't have enough practice and application/word problems for full mastery/retention or for a full high school credit, IMO.

 

Got it--but it sounds like a great intro. I'm another algebraic thinker who isn't a fan of geometry. DH is, but he's too lost in his deep naturally mathematically-inclined mind to teach DD.

 

I actually liked proofs quite a bit because they were logical, but I was never ever able to abstract that learning to intuitive thought nor practical purpose. 

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To be clear, I would never limit my kids' education by not teaching geometry. That was never mentioned. I get that some professions use a lot of it.

 

I think it's the lack of linearness to the subject that makes it weird to me. Like I said, these Zome lessons are just hilarious to me, the way they're like, kids will now build 3-dimensional shapes and, of course, they're showing all these super simple ones but my kids build - and I'm not kidding at all about this one - a pentagonal gyrobicupola - and want to know what's *that* shape called? (For anyone else in this situation, Wikipedia has this useful list of shapes you never needed to know exist...) And I'm like... okay. This is what I mean about it going counter to how I've always taught math. The purpose of manipulatives is to give the kids a sandbox to play in. But in the elementary math sandbox, assuming you're not raising super geniuses, there's a limited number of things they'll discover. And whatever they find, like oh, look at this pattern or check out this trick or relationship between numbers, you will pretty much know how that works and what's the use of it - what later math will use it as a base, what applications to the real world it will have. But when I set my kids loose with shapes and angles, half the time they find things or make patterns they like and I have no clue what use or connection it has to anything. So then I want to set aside all those manipulatives and so forth and follow a more linear progression. Except, that none of the textbooks seem to agree on what kids need to know and when.

 

I may end up outsourcing it at the high school level.

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I may end up outsourcing it at the high school level.

I was truly never so relieved as when I outsourced Algebra mid-year for one son.  This year, his Geometry is outsourced.  I pushed that boulder up a mountain for 6 1/2 years, yo.   :lol: The other son is self-teaching AoPS, and I check him using the chapter reviews as tests.  As long as he is doing well, he can rock on with it.  

 

I could do most other subjects if needed.  I could even figure how to teach a good Spanish program respectably (I'm outsourcing that, too!)  But no Geometry.  No.  Just no.   :gnorsi:

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But when I set my kids loose with shapes and angles, half the time they find things or make patterns they like and I have no clue what use or connection it has to anything.

Architecture :)

Fractal geometry as well.

 

ETA:

Plumbing & HVAC

Urban design & traffic engineering

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We just wrapped up HS Geometry. My take on it is it reinforces Algebra 1 concepts, it teaches deconstruction of complex problems into step-by-step processes, and it trains the mind to logically approach problem solving.

 

Most of the pain associated with it stemed from proofs. There is a LOT to memorize in order to really do a proof.  I hated proofs when I took it in HS. My wife (PhD engineer) hated proofs. My son loved the application parts. He hated what little he did of proofs (we steared fairly clear).

 

Obviously, my recommendation is to cover the application stuff, focus on building solutions to problems or conducting an investigation into a subject from the ground up asking, "OK. What do I know and where can I go?" (a sort of mathematical Socratic method), and take a very casual approach to proofs.  But hey, that's just one man's opinion!

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My 12yo is completely asynchronous across the board.  Geometry is his favorite part of math. We just started Patty Paper Geometry.  

 

 

 

If I'm not careful Farrar is going to have me buying Zome manips & lessons along with Bravewriter. (Dangerous woman...I'd put you on ignore for the sake of my bank account, but I love your posts.)

 

 

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Jacob's Geometry is not something you would teach a grade schooler. I would consider it to be quite difficult for a child who is great at math and had completed a very strong algebra 1 program. At your children's ages, maybe Right Start Geometry would be a better way to go. It is supposed to be hands on and fun and set for not high school.

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Jacob's Geometry is not something you would teach a grade schooler. I would consider it to be quite difficult for a child who is great at math and had completed a very strong algebra 1 program. At your children's ages, maybe Right Start Geometry would be a better way to go. It is supposed to be hands on and fun and set for not high school.

The Jacobs was for me, not them. I felt like that was pretty clear.

 

But also, my kids are in middle school. One is on pre-algebra. It's not so far in our future.

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i like the interplay between algebra and geometry.  euclid (especially in Book 2) uses geometry to make algebra visual.  e.g. the algebraic process of squaring is called that literally because you take the length you want to square and you make a square out it, by using the same length for the height as for the width of the square.  And then the algebra formula for (A+B)^2 becomes an expression for a square made out of a length equal to A+B.  In the geometric picture you can see how there are two smaller squares inside the (A+B) square, namely an A square and a B square, and most beautifully of all you can see the two AxB rectangles along the sides that are needed to fill up the (A+B) square.  Without this picture lots of kids are doomed to wondering whether (A+B)^2 equals A^2 + AB + B^2, or A^2 + 2AB + B^2.

 

As regentrude suggested a "cylinder" is actually a general construction, composed of vertical lines erected on a given base.  If the base is a circle it is a (usual) circular cylinder, but the same volume principle applies to all cylinders (sometimes called prisms) - namely the volume is just the area of the base, whatever it is, times the height.

 

A book about a country kentucky teacher I liked, called The thread that runs so true, by Jesse Stuart, tells of a country man not wanting his boy to attend school but to stay home and work, and driving his wagonload of coal to the school to say so.  The teacher takes time to ask the man if he knows how much coal is on his wagon, and he doesn't but he knows how much he is usually paid for the load.  The teacher does the math for him, the volume of the wagon times the price of coal, and he learns he has been cheated every time he sold his coal.  Of course he leaves the child at the school.

 

There are infinite ways algebra and geometry enhance each other, and that geometry is useful in everyday life, designing a house, estimating how much of something to buy for any given job.... I liked regentrudes examples very much.

 

A more subtle volume formula is for a cone, whose volume equals 1/3 of the area of the base times the height, but this formula too holds no matter what shape the base has, as long as the solid is made by joining all point of the base to the same vertex by straight line segments.  If you can visulaize what Archimedes meant when he said a ball is merely a cone whose base is its surface and whose vertex is its center, you can see why the volume of a spherical ball equals 1/3 of the product of the radius and the surface area.  (Well a ball is formed by joining every point of the surface to the center, by a radius.)

 

More basic, is understanding why area formulas are expressed in terms of the square of the length involved and volume formulas are expressed in terms of the cube.  Without this simple insight many college students do not know whether the volume of a ball of radius R is a multiple of R^2 or of R^3.  Euclid knew this much, but I believe it fell to the great Archimedes to actually supply the constant multipliers.

 

Unfortunately geometry is often taught simply as memorizing instead of understanding geometric ideas.  Jacobson is a good source as is the book for elementary school teachers by Sybilla Beckmann.  Of course, as in all cases, we don't use in life anything we don't understand, but we do find ways to use things we do.  When my young son took a math contest after studying algebra with me at home I asked if the test required algebra, and he said "you don't need algebra for this test, but if you know algebra you can see ways to use it."

 

Oddly, considering the reputation of gemetry for proofs among laypersons, many mathematicians regard geometry as the source of intuition compared to algebra as the realm of precise logical argument.  There has a been a powerful movement in the mid 20th century to render geometry more rigorous by translating it entirely into algebra.  I love the famous reaction to this by the great mathematician Herman Weyl (I remind that topology is the most fundamental form of geometry): "In these days, the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics."

 

You may know I am a professional algebraic geometer, and if I knew more literature and philosophy, I could probably make a more compelling argument.

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I took the time to read all of the posts in this thread (there are 40 as I type this) and found it to be EXTREMELY INTERESTING.   Although I frequently think there are too many "Pinned" threads, when I go into a sub forum on WTM, I hope that this thread can be "Pinned" or incorporated into a list of VERY helpful threads about Math.  Just before we ate Lunch, my Wife and DD were in the Kitchen and I told them about this thread. THANK YOU OP for starting this thread!

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While I've been a little grouchy throughout, I got a lot out of the responses too. I'm still not sure if I'm just missing some geometry gene or what, but I'm glad to know some people see the meaning and beauty more clearly.

 

Farrar, how you feel about geometry is how I've always felt about trig. I maintain that most people click better with one or the other. 

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  • 4 weeks later...

Shannon and I were talking about Geometry plans for next year, and I was pushing outsourcing, but she's thinking she wants to work through a text the way she is with Algebra. She said, "After the EdX course, I'm not intimidated by Geometry any more. It's just algebra applied to shapes."

 

I don't know if that's entirely true, but it struck me as a funny addition to the Geometry thinker vs. Algebra thinker discussion!  :lol:

 

I'm feeling a strange compulsion to buy the AoPS Geometry book just to check it out . . . .Must. Resist. The. Urge.  :smilielol5:

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Shannon and I were talking about Geometry plans for next year, and I was pushing outsourcing, but she's thinking she wants to work through a text the way she is with Algebra. She said, "After the EdX course, I'm not intimidated by Geometry any more. It's just algebra applied to shapes."

 

I don't know if that's entirely true, but it struck me as a funny addition to the Geometry thinker vs. Algebra thinker discussion!  :lol:

 

I'm feeling a strange compulsion to buy the AoPS Geometry book just to check it out . . . .Must. Resist. The. Urge.  :smilielol5:

 

 

Rose, I am looking for something D12 can do over the summer. She is currently in 7th grade PS Algebra I. I think a gentle first pass over the summer would be ideal. I am  glad I saw your post EdX course. Can you explain how it works? It there a workbook?

Do they watch videos?

 

I own The great Courses Geometry course. I have not looked at it, but the work book does not have much instruction. Hoping the videos have the instruction. Also was thinking of Keys to Geometry, not sure which way to go.

 

Thank you

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